Subgroups — C₃₁×#62021 #21518780

Order 7936.

1828 conjugacy classes of subgroups.

The subgroup lattice has 1064 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3561	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3522	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3401	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3369	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3316	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3306	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3298	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3282	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3271	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3270	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3263	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3246	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#3225	C₂	1	2	yes	maximal, complemented
14	C₃₁×#3213	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#3125	C₂	1	2	yes	maximal, complemented
16	C₃₁×#3113	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#3046	C₂	1	2	yes	maximal, complemented
18	C₂×C₃₁×#464	C₂	1	1	yes	maximal, characteristic, complemented
19	C₃₁×#2992	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×C₃₁×#137	C₂²	1	2	yes	complemented
21	C₂×C₃₁×#135	C₂²	1	2	yes	—
22	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#563	—	2	1	no	—
25	C₃₁×#562	—	2	1	no	—
26	C₃₁×#559	—	2	1	no	—
27	C₃₁×#559	C₂²	1	1	yes	characteristic, complemented
28	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#556	C₂²	1	2	yes	—
30	C₃₁×#554	—	2	1	no	—
31	C₃₁×#554	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#551	—	2	2	no	—
33	C₃₁×#550	C₂²	1	1	yes	characteristic, complemented
34	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
35	C₃₁×#548	C₂²	1	2	yes	—
36	C₃₁×#548	C₂²	1	2	yes	—
37	C₃₁×#547	—	2	2	no	—
38	C₃₁×#547	C₂²	1	2	yes	—
39	C₃₁×#546	C₂²	1	2	yes	—
40	C₃₁×#543	—	2	1	no	—
41	C₃₁×#542	—	2	2	no	—
42	C₃₁×#542	—	2	2	no	—
43	C₃₁×#542	C₂²	1	2	yes	—
44	C₃₁×#542	C₂²	1	2	yes	complemented
45	C₃₁×#538	—	2	1	no	—
46	C₃₁×#536	C₂²	1	2	yes	—
47	C₃₁×#533	C₂²	1	1	yes	characteristic
48	C₃₁×#532	—	2	1	no	—
49	C₃₁×#532	C₂²	1	1	yes	characteristic
50	C₃₁×#529	—	2	1	no	—
51	C₃₁×#529	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
53	C₃₁×#520	C₂²	1	2	yes	complemented
54	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₃₁×#497	C₂²	1	2	yes	complemented
56	C₃₁×#496	C₂²	1	2	yes	—
57	C₃₁×#495	C₂²	1	2	yes	—
58	C₃₁×#489	C₂²	1	2	yes	—
59	C₃₁×#489	C₂²	1	2	yes	—
60	C₃₁×#487	C₂²	1	2	yes	complemented
61	C₃₁×#483	C₂²	1	2	yes	—
62	C₃₁×#483	C₂²	1	2	yes	complemented
63	C₃₁×#481	C₂²	1	2	yes	—
64	C₃₁×#476	C₂²	1	2	yes	—
65	C₃₁×#476	C₂²	1	2	yes	complemented
66	C₃₁×#475	C₂²	1	1	yes	characteristic, complemented
67	C₃₁×#475	C₂²	1	1	yes	characteristic, complemented
68	C₃₁×#475	C₂²	1	2	yes	complemented
69	C₃₁×#475	C₂²	1	2	yes	complemented
70	C₃₁×#470	C₂²	1	2	yes	complemented
71	C₃₁×#469	C₂²	1	2	yes	—
72	C₃₁×#465	C₂²	1	2	yes	—
73	C₃₁×#465	C₂²	1	2	yes	—
74	C₃₁×#465	C₂²	1	2	yes	—
75	C₃₁×#465	C₂²	1	2	yes	complemented
76	C₃₁×#464	C₂²	1	2	yes	—
77	C₃₁×#464	C₂²	1	2	yes	—
78	C₃₁×#464	C₂²	1	2	yes	—
79	C₃₁×#464	C₂²	1	2	yes	—
80	C₃₁×#464	C₂²	1	2	yes	—
81	C₃₁×#464	C₂²	1	2	yes	complemented
82	C₃₁×#463	C₂²	1	2	yes	—
83	C₃₁×#463	C₂²	1	2	yes	—
84	C₃₁×#463	C₂²	1	2	yes	—
85	C₃₁×#463	C₂²	1	2	yes	—
86	C₃₁×#462	C₂²	1	1	yes	characteristic, complemented
87	C₃₁×#462	C₂²	1	1	yes	characteristic, complemented
88	C₃₁×#462	C₂²	1	2	yes	complemented
89	C₃₁×#461	C₂²	1	2	yes	complemented
90	C₃₁×#461	C₂²	1	2	yes	complemented
91	C₃₁×#460	C₂²	1	2	yes	complemented
92	C₃₁×#452	C₂²	1	2	yes	—
93	C₃₁×#452	C₂²	1	2	yes	complemented
94	C₃₁×#451	C₂²	1	2	yes	—
95	C₃₁×#451	C₂²	1	2	yes	complemented
96	C₃₁×#449	C₂²	1	2	yes	—
97	C₃₁×#449	C₂²	1	2	yes	—
98	C₃₁×#448	C₂²	1	2	yes	—
99	C₃₁×#448	C₂²	1	2	yes	complemented
100	C₃₁×#447	C₂²	1	2	yes	—
101	C₃₁×#446	C₂²	1	2	yes	complemented
102	C₃₁×#442	C₂²	1	2	yes	—
103	C₃₁×#441	C₂²	1	2	yes	complemented
104	C₃₁×#441	C₂²	1	2	yes	complemented
105	C₃₁×#441	C₂²	1	2	yes	complemented
106	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
107	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
108	C₃₁×#435	C₂²	1	2	yes	—
109	C₃₁×#435	C₂²	1	2	yes	complemented
110	C₈×D₄×C₃₁	C₂²	1	2	yes	—
111	C₃₁×#433	C₂²	1	2	yes	complemented
112	C₃₁×#432	C₂²	1	1	yes	characteristic
113	C₃₁×#424	C₂²	1	2	yes	complemented
114	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
115	C₃₁×#417	C₂²	1	2	yes	—
116	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
117	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
118	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
119	C₂×#41×C₃₁	—	2	2	no	—
120	C₂×#41×C₃₁	—	2	1	no	—
121	C₂×#41×C₃₁	C₂³	1	2	yes	—
122	C₂×#41×C₃₁	C₂³	1	2	yes	—
123	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
124	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
126	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₃₁×#137	—	4	4	no	—
128	C₃₁×#135	—	4	4	no	—
129	C₂×Q₁₆×C₃₁	—	4	2	no	—
130	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
131	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
132	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
133	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
134	C₂×SD₁₆×C₃₁	—	4	2	no	—
135	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
136	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
137	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
138	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
139	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
140	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
141	C₂×D₈×C₃₁	C₂³	1	2	yes	—
142	C₂×D₈×C₃₁	C₂³	1	2	yes	—
143	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
144	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
145	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
146	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
147	C₃₁×C₄⋊Q₈	—	2	1	no	—
148	C₃₁×C₄⋊Q₈	—	2	2	no	—
149	C₃₁×C₄⋊Q₈	—	2	2	no	—
150	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
151	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
152	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
153	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
154	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
155	C₃₁×#127	—	2	1	no	—
156	C₃₁×#127	C₂³	1	1	yes	characteristic
157	C₃₁×#126	—	2	2	no	—
158	C₃₁×#126	—	2	2	no	—
159	C₃₁×#126	—	2	2	no	—
160	C₃₁×#126	—	2	2	no	—
161	C₃₁×#126	D₄	1	2	yes	—
162	C₃₁×#125	—	2	1	no	—
163	C₃₁×#125	—	2	1	no	—
164	C₃₁×#125	D₄	1	1	yes	characteristic
165	C₃₁×#124	—	2	1	no	—
166	C₃₁×#124	—	2	2	no	—
167	C₃₁×#124	—	2	2	no	—
168	C₃₁×#124	—	2	2	no	—
169	C₃₁×#124	—	2	2	no	—
170	C₃₁×#124	—	2	1	no	—
171	C₃₁×#124	—	2	1	no	—
172	C₃₁×#124	—	2	1	no	—
173	C₃₁×#124	D₄	1	1	yes	characteristic
174	C₃₁×#124	C₂³	1	2	yes	—
175	C₃₁×#124	C₂³	1	2	yes	—
176	C₃₁×#124	C₂³	1	2	yes	—
177	C₃₁×#124	C₂³	1	2	yes	—
178	C₃₁×#123	—	2	2	no	—
179	C₃₁×#123	—	2	2	no	—
180	C₃₁×#123	—	2	2	no	—
181	C₃₁×#123	—	2	2	no	—
182	C₃₁×#123	—	2	2	no	—
183	C₃₁×#123	—	2	2	no	—
184	C₃₁×#123	—	2	2	no	—
185	C₃₁×#123	—	2	2	no	—
186	C₃₁×#123	D₄	1	2	yes	—
187	C₃₁×C₂²⋊Q₈	—	2	2	no	—
188	C₃₁×C₂²⋊Q₈	—	2	1	no	—
189	C₃₁×C₂²⋊Q₈	—	2	2	no	—
190	C₃₁×C₂²⋊Q₈	—	2	2	no	—
191	C₃₁×C₂²⋊Q₈	—	2	2	no	—
192	C₃₁×C₂²⋊Q₈	—	2	2	no	—
193	C₃₁×C₂²⋊Q₈	—	2	2	no	—
194	C₃₁×C₂²⋊Q₈	—	2	2	no	—
195	C₃₁×C₂²⋊Q₈	—	2	2	no	—
196	C₃₁×C₂²⋊Q₈	—	2	2	no	—
197	C₃₁×C₂²⋊Q₈	—	2	2	no	—
198	C₃₁×C₂²⋊Q₈	—	2	2	no	—
199	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
200	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
201	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
202	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
203	C₃₁×#121	—	2	2	no	—
204	C₃₁×#121	—	2	2	no	—
205	C₃₁×#121	—	2	1	no	—
206	C₃₁×#121	—	2	2	no	—
207	C₃₁×#121	—	2	2	no	—
208	C₃₁×#121	D₄	1	1	yes	characteristic
209	C₃₁×#121	C₂³	1	2	yes	—
210	C₃₁×#121	C₂³	1	2	yes	—
211	C₃₁×#121	C₂³	1	2	yes	—
212	C₃₁×#120	—	2	2	no	—
213	C₄×Q₈×C₃₁	—	2	1	no	—
214	C₄×Q₈×C₃₁	—	2	1	no	—
215	C₄×Q₈×C₃₁	—	2	1	no	—
216	C₄×Q₈×C₃₁	—	2	1	no	—
217	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
218	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
219	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
220	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
221	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
222	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
223	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
224	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
225	C₄×D₄×C₃₁	—	2	1	no	—
226	C₄×D₄×C₃₁	—	2	2	no	—
227	C₄×D₄×C₃₁	—	2	1	no	—
228	C₄×D₄×C₃₁	—	2	1	no	—
229	C₄×D₄×C₃₁	—	2	2	no	—
230	C₄×D₄×C₃₁	—	2	2	no	—
231	C₄×D₄×C₃₁	—	2	2	no	—
232	C₄×D₄×C₃₁	—	2	2	no	—
233	C₄×D₄×C₃₁	—	2	1	no	—
234	C₄×D₄×C₃₁	C₂³	1	2	yes	—
235	C₄×D₄×C₃₁	C₂³	1	2	yes	—
236	C₄×D₄×C₃₁	C₂³	1	2	yes	—
237	C₄×D₄×C₃₁	C₂³	1	2	yes	—
238	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
239	C₃₁×#117	—	2	1	no	—
240	C₃₁×#117	—	2	2	no	—
241	C₃₁×#117	—	2	2	no	—
242	C₃₁×#117	C₂³	1	2	yes	—
243	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
244	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
245	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
246	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
247	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
248	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
249	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
250	C₂×C₄²×C₃₁	—	2	1	no	—
251	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
252	C₃₁×#106	C₂³	1	2	yes	—
253	C₃₁×#106	C₂³	1	2	yes	—
254	C₃₁×#106	C₂³	1	2	yes	—
255	C₃₁×#106	C₂³	1	2	yes	—
256	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
257	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
258	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
259	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
260	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
261	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
262	C₃₁×#103	C₂³	1	2	yes	—
263	C₃₁×#103	C₂³	1	2	yes	—
264	C₃₁×#103	C₂³	1	2	yes	—
265	C₃₁×#103	C₂³	1	2	yes	—
266	C₃₁×#103	C₂³	1	2	yes	—
267	C₃₁×#103	C₂³	1	2	yes	—
268	C₃₁×#103	C₂³	1	2	yes	—
269	C₃₁×#103	C₂³	1	2	yes	—
270	C₃₁×#103	C₂³	1	2	yes	—
271	C₃₁×#103	C₂³	1	2	yes	—
272	C₃₁×#102	C₂³	1	2	yes	—
273	C₃₁×#102	C₂³	1	2	yes	—
274	C₃₁×#102	C₂³	1	2	yes	—
275	C₃₁×#102	C₂³	1	2	yes	—
276	C₃₁×#102	C₂³	1	2	yes	—
277	C₃₁×#102	C₂³	1	2	yes	—
278	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
279	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
280	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
281	C₃₁×#97	C₂³	1	2	yes	—
282	C₃₁×#97	C₂³	1	2	yes	—
283	C₄×C₈×C₃₁	C₂³	1	2	yes	—
284	C₂⁴×C₃₁	—	2	1	no	—
285	#41×C₃₁	—	2	4	no	—
286	#41×C₃₁	—	4	2	no	—
287	#41×C₃₁	—	4	2	no	—
288	#41×C₃₁	—	4	2	no	—
289	#41×C₃₁	—	8	2	no	—
290	#41×C₃₁	—	8	4	no	—
291	C₂×Q₈×C₃₁	—	2	1	no	—
292	C₂×Q₈×C₃₁	—	2	1	no	—
293	C₂×Q₈×C₃₁	—	2	2	no	—
294	C₂×Q₈×C₃₁	—	2	1	no	—
295	C₂×Q₈×C₃₁	—	2	2	no	—
296	C₂×Q₈×C₃₁	—	2	1	no	—
297	C₂×Q₈×C₃₁	—	2	2	no	—
298	C₂×Q₈×C₃₁	—	4	1	no	—
299	C₂×Q₈×C₃₁	—	4	1	no	—
300	C₂×Q₈×C₃₁	—	4	2	no	—
301	C₂×Q₈×C₃₁	—	4	1	no	—
302	C₂×Q₈×C₃₁	—	4	2	no	—
303	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
304	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
305	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
306	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
307	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
308	C₂×D₄×C₃₁	—	2	2	no	—
309	C₂×D₄×C₃₁	—	2	2	no	—
310	C₂×D₄×C₃₁	—	2	2	no	—
311	C₂×D₄×C₃₁	—	2	2	no	—
312	C₂×D₄×C₃₁	—	2	2	no	—
313	C₂×D₄×C₃₁	—	2	1	no	—
314	C₂×D₄×C₃₁	—	2	1	no	—
315	C₂×D₄×C₃₁	—	2	1	no	—
316	C₂×D₄×C₃₁	—	4	1	no	—
317	C₂×D₄×C₃₁	—	4	1	no	—
318	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
319	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
320	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
321	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
322	C₂²×C₄×C₃₁	—	2	1	no	—
323	C₂²×C₄×C₃₁	—	2	1	no	—
324	C₂²×C₄×C₃₁	—	2	1	no	—
325	C₂²×C₄×C₃₁	—	2	2	no	—
326	C₂²×C₄×C₃₁	—	2	2	no	—
327	C₂²×C₄×C₃₁	—	2	2	no	—
328	C₂²×C₄×C₃₁	—	2	2	no	—
329	C₂²×C₄×C₃₁	—	2	2	no	—
330	C₂²×C₄×C₃₁	—	2	2	no	—
331	C₂²×C₄×C₃₁	—	2	2	no	—
332	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
333	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
334	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
335	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
336	Q₁₆×C₃₁	—	2	4	no	—
337	Q₁₆×C₃₁	—	2	4	no	—
338	Q₁₆×C₃₁	—	4	2	no	—
339	Q₁₆×C₃₁	—	4	2	no	—
340	SD₁₆×C₃₁	—	2	4	no	—
341	SD₁₆×C₃₁	—	4	2	no	—
342	SD₁₆×C₃₁	—	4	2	no	—
343	SD₁₆×C₃₁	—	4	2	no	—
344	SD₁₆×C₃₁	—	4	2	no	—
345	SD₁₆×C₃₁	—	4	2	no	—
346	D₈×C₃₁	—	4	2	no	—
347	D₈×C₃₁	—	4	2	no	—
348	M₁₆×C₃₁	—	4	2	no	—
349	C₂×C₈×C₃₁	—	4	1	no	—
350	C₂×C₈×C₃₁	—	4	1	no	—
351	C₂×C₈×C₃₁	—	4	2	no	—
352	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
353	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
354	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
355	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
356	C₄⋊C₄×C₃₁	—	2	1	no	—
357	C₄⋊C₄×C₃₁	—	2	2	no	—
358	C₄⋊C₄×C₃₁	—	2	2	no	—
359	C₄⋊C₄×C₃₁	—	2	1	no	—
360	C₄⋊C₄×C₃₁	—	2	1	no	—
361	C₄⋊C₄×C₃₁	—	2	1	no	—
362	C₄⋊C₄×C₃₁	—	2	1	no	—
363	C₄⋊C₄×C₃₁	—	2	2	no	—
364	C₄⋊C₄×C₃₁	—	2	1	no	—
365	C₄⋊C₄×C₃₁	—	2	1	no	—
366	C₄⋊C₄×C₃₁	—	2	1	no	—
367	C₄⋊C₄×C₃₁	—	2	2	no	—
368	C₄⋊C₄×C₃₁	—	2	2	no	—
369	C₄⋊C₄×C₃₁	—	2	2	no	—
370	C₄⋊C₄×C₃₁	—	2	2	no	—
371	C₄⋊C₄×C₃₁	—	2	1	no	—
372	C₄⋊C₄×C₃₁	—	2	2	no	—
373	C₄⋊C₄×C₃₁	—	2	2	no	—
374	C₄⋊C₄×C₃₁	—	2	2	no	—
375	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
376	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
377	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
378	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
379	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
380	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
381	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
382	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
383	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
384	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
385	C₂²⋊C₄×C₃₁	—	2	2	no	—
386	C₂²⋊C₄×C₃₁	—	2	2	no	—
387	C₂²⋊C₄×C₃₁	—	2	2	no	—
388	C₂²⋊C₄×C₃₁	—	2	2	no	—
389	C₂²⋊C₄×C₃₁	—	2	2	no	—
390	C₂²⋊C₄×C₃₁	—	2	2	no	—
391	C₂²⋊C₄×C₃₁	—	2	2	no	—
392	C₂²⋊C₄×C₃₁	—	2	2	no	—
393	C₂²⋊C₄×C₃₁	—	2	2	no	—
394	C₂²⋊C₄×C₃₁	—	2	2	no	—
395	C₂²⋊C₄×C₃₁	—	2	2	no	—
396	C₂²⋊C₄×C₃₁	—	2	2	no	—
397	C₂²⋊C₄×C₃₁	—	2	2	no	—
398	C₂²⋊C₄×C₃₁	—	2	2	no	—
399	C₂²⋊C₄×C₃₁	—	2	2	no	—
400	C₂²⋊C₄×C₃₁	—	2	2	no	—
401	C₂²⋊C₄×C₃₁	—	2	2	no	—
402	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
403	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
404	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
405	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
406	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
407	C₄²×C₃₁	—	2	1	no	—
408	C₄²×C₃₁	—	2	1	no	—
409	C₄²×C₃₁	—	2	1	no	—
410	C₄²×C₃₁	—	2	1	no	—
411	C₄²×C₃₁	—	2	2	no	—
412	C₄²×C₃₁	—	2	2	no	—
413	C₄²×C₃₁	—	2	2	no	—
414	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
415	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
416	C₄²×C₃₁	C₂⁴	1	2	yes	—
417	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
418	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
419	#62021	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
420	C₂³×C₃₁	—	2	2	no	—
421	C₂³×C₃₁	—	2	2	no	—
422	C₂³×C₃₁	—	4	2	no	—
423	C₂³×C₃₁	—	8	1	no	—
424	C₂³×C₃₁	—	8	1	no	—
425	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
426	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
427	Q₈×C₃₁	—	2	1	no	—
428	Q₈×C₃₁	—	2	2	no	—
429	Q₈×C₃₁	—	2	1	no	—
430	Q₈×C₃₁	—	2	2	no	—
431	Q₈×C₃₁	—	2	2	no	—
432	Q₈×C₃₁	—	2	1	no	—
433	Q₈×C₃₁	—	2	2	no	—
434	Q₈×C₃₁	—	4	1	no	—
435	Q₈×C₃₁	—	4	2	no	—
436	Q₈×C₃₁	—	8	1	no	—
437	Q₈×C₃₁	—	8	2	no	—
438	Q₈×C₃₁	—	8	1	no	—
439	Q₈×C₃₁	—	8	1	no	—
440	Q₈×C₃₁	—	8	2	no	—
441	Q₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
442	D₄×C₃₁	—	2	4	no	—
443	D₄×C₃₁	—	2	2	no	—
444	D₄×C₃₁	—	2	2	no	—
445	D₄×C₃₁	—	2	2	no	—
446	D₄×C₃₁	—	2	2	no	—
447	D₄×C₃₁	—	4	1	no	—
448	D₄×C₃₁	—	4	4	no	—
449	D₄×C₃₁	—	4	2	no	—
450	D₄×C₃₁	—	4	4	no	—
451	D₄×C₃₁	—	8	1	no	—
452	D₄×C₃₁	—	8	1	no	—
453	D₄×C₃₁	—	8	2	no	—
454	D₄×C₃₁	—	8	2	no	—
455	D₄×C₃₁	—	8	2	no	—
456	C₂×C₄×C₃₁	—	2	2	no	—
457	C₂×C₄×C₃₁	—	2	1	no	—
458	C₂×C₄×C₃₁	—	2	1	no	—
459	C₂×C₄×C₃₁	—	2	2	no	—
460	C₂×C₄×C₃₁	—	2	1	no	—
461	C₂×C₄×C₃₁	—	2	2	no	—
462	C₂×C₄×C₃₁	—	2	1	no	—
463	C₂×C₄×C₃₁	—	2	2	no	—
464	C₂×C₄×C₃₁	—	2	1	no	—
465	C₂×C₄×C₃₁	—	2	1	no	—
466	C₂×C₄×C₃₁	—	2	2	no	—
467	C₂×C₄×C₃₁	—	2	2	no	—
468	C₂×C₄×C₃₁	—	2	2	no	—
469	C₂×C₄×C₃₁	—	4	1	no	—
470	C₂×C₄×C₃₁	—	4	4	no	—
471	C₂×C₄×C₃₁	—	4	2	no	—
472	C₂×C₄×C₃₁	—	4	2	no	—
473	C₂×C₄×C₃₁	—	8	2	no	—
474	C₂×C₄×C₃₁	—	8	2	no	—
475	C₂×C₄×C₃₁	—	8	2	no	—
476	C₂×C₄×C₃₁	—	8	2	no	—
477	C₂×C₄×C₃₁	—	8	2	no	—
478	C₂×C₄×C₃₁	—	8	1	no	—
479	C₂×C₄×C₃₁	—	8	2	no	—
480	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
481	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
482	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
483	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
484	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
485	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
486	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
487	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
488	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
489	C₈×C₃₁	—	2	2	no	—
490	C₈×C₃₁	—	2	2	no	—
491	C₈×C₃₁	—	2	2	no	—
492	C₈×C₃₁	—	2	2	no	—
493	#3561	C₂×C₃₁	1	1	yes	characteristic, complemented
494	#3522	C₂×C₃₁	1	1	yes	characteristic, complemented
495	#3401	C₂×C₃₁	1	2	yes	complemented
496	#3369	C₂×C₃₁	1	2	yes	complemented
497	#3316	C₂×C₃₁	1	2	yes	complemented
498	#3306	C₂×C₃₁	1	2	yes	complemented
499	#3298	C₂×C₃₁	1	2	yes	complemented
500	#3282	C₂×C₃₁	1	2	yes	complemented
501	#3271	C₂×C₃₁	1	2	yes	complemented
502	#3270	C₂×C₃₁	1	2	yes	complemented
503	#3263	C₂×C₃₁	1	2	yes	complemented
504	#3246	C₂×C₃₁	1	1	yes	characteristic, complemented
505	#3225	C₂×C₃₁	1	2	yes	complemented
506	#3213	C₂×C₃₁	1	1	yes	characteristic, complemented
507	#3125	C₂×C₃₁	1	2	yes	complemented
508	#3113	C₂×C₃₁	1	1	yes	characteristic, complemented
509	#3046	C₂×C₃₁	1	2	yes	complemented
510	C₂×#464	C₂×C₃₁	1	1	yes	characteristic, complemented
511	#2992	C₂×C₃₁	1	1	yes	characteristic, complemented
512	C₂²×C₃₁	—	2	1	no	—
513	C₂²×C₃₁	—	2	1	no	—
514	C₂²×C₃₁	—	2	2	no	—
515	C₂²×C₃₁	—	2	2	no	—
516	C₂²×C₃₁	—	2	2	no	—
517	C₂²×C₃₁	—	4	2	no	—
518	C₂²×C₃₁	—	4	2	no	—
519	C₂²×C₃₁	—	4	2	no	—
520	C₂²×C₃₁	—	4	2	no	—
521	C₂²×C₃₁	—	4	2	no	—
522	C₂²×C₃₁	—	4	2	no	—
523	C₂²×C₃₁	—	16	2	no	—
524	C₂²×C₃₁	#577	1	2	yes	—
525	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
526	C₄×C₃₁	—	2	1	no	—
527	C₄×C₃₁	—	2	1	no	—
528	C₄×C₃₁	—	2	2	no	—
529	C₄×C₃₁	—	2	2	no	—
530	C₄×C₃₁	—	2	1	no	—
531	C₄×C₃₁	—	2	2	no	—
532	C₄×C₃₁	—	2	1	no	—
533	C₄×C₃₁	—	2	1	no	—
534	C₄×C₃₁	—	2	2	no	—
535	C₄×C₃₁	—	4	2	no	—
536	C₄×C₃₁	—	4	1	no	—
537	C₄×C₃₁	—	4	2	no	—
538	C₄×C₃₁	—	4	2	no	—
539	C₄×C₃₁	—	4	1	no	—
540	C₄×C₃₁	—	4	1	no	—
541	C₄×C₃₁	—	4	2	no	—
542	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
543	C₄×C₃₁	#585	1	1	yes	characteristic
544	C₂×#137	C₂²×C₃₁	1	2	yes	complemented
545	C₂×#135	C₂²×C₃₁	1	2	yes	—
546	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
547	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
548	#563	—	2	1	no	—
549	#562	—	2	1	no	—
550	#559	—	2	1	no	—
551	#559	C₂²×C₃₁	1	1	yes	characteristic, complemented
552	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
553	#556	C₂²×C₃₁	1	2	yes	—
554	#554	—	2	1	no	—
555	#554	C₂²×C₃₁	1	1	yes	characteristic, complemented
556	#551	—	2	2	no	—
557	#550	C₂²×C₃₁	1	1	yes	characteristic, complemented
558	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
559	#548	C₂²×C₃₁	1	2	yes	—
560	#548	C₂²×C₃₁	1	2	yes	—
561	#547	—	2	2	no	—
562	#547	C₂²×C₃₁	1	2	yes	—
563	#546	C₂²×C₃₁	1	2	yes	—
564	#543	—	2	1	no	—
565	#542	—	2	2	no	—
566	#542	—	2	2	no	—
567	#542	C₂²×C₃₁	1	2	yes	—
568	#542	C₂²×C₃₁	1	2	yes	complemented
569	#538	—	2	1	no	—
570	#536	C₂²×C₃₁	1	2	yes	—
571	#533	C₂²×C₃₁	1	1	yes	characteristic
572	#532	—	2	1	no	—
573	#532	C₂²×C₃₁	1	1	yes	characteristic
574	#529	—	2	1	no	—
575	#529	C₂²×C₃₁	1	1	yes	characteristic
576	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
577	#520	C₂²×C₃₁	1	2	yes	complemented
578	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
579	#497	C₂²×C₃₁	1	2	yes	complemented
580	#496	C₂²×C₃₁	1	2	yes	—
581	#495	C₂²×C₃₁	1	2	yes	—
582	#489	C₂²×C₃₁	1	2	yes	—
583	#489	C₂²×C₃₁	1	2	yes	—
584	#487	C₂²×C₃₁	1	2	yes	complemented
585	#483	C₂²×C₃₁	1	2	yes	—
586	#483	C₂²×C₃₁	1	2	yes	complemented
587	#481	C₂²×C₃₁	1	2	yes	—
588	#476	C₂²×C₃₁	1	2	yes	—
589	#476	C₂²×C₃₁	1	2	yes	complemented
590	#475	C₂²×C₃₁	1	1	yes	characteristic, complemented
591	#475	C₂²×C₃₁	1	1	yes	characteristic, complemented
592	#475	C₂²×C₃₁	1	2	yes	complemented
593	#475	C₂²×C₃₁	1	2	yes	complemented
594	#470	C₂²×C₃₁	1	2	yes	complemented
595	#469	C₂²×C₃₁	1	2	yes	—
596	#465	C₂²×C₃₁	1	2	yes	—
597	#465	C₂²×C₃₁	1	2	yes	—
598	#465	C₂²×C₃₁	1	2	yes	—
599	#465	C₂²×C₃₁	1	2	yes	complemented
600	#464	C₂²×C₃₁	1	2	yes	—
601	#464	C₂²×C₃₁	1	2	yes	—
602	#464	C₂²×C₃₁	1	2	yes	—
603	#464	C₂²×C₃₁	1	2	yes	—
604	#464	C₂²×C₃₁	1	2	yes	—
605	#464	C₂²×C₃₁	1	2	yes	complemented
606	#463	C₂²×C₃₁	1	2	yes	—
607	#463	C₂²×C₃₁	1	2	yes	—
608	#463	C₂²×C₃₁	1	2	yes	—
609	#463	C₂²×C₃₁	1	2	yes	—
610	#462	C₂²×C₃₁	1	1	yes	characteristic, complemented
611	#462	C₂²×C₃₁	1	1	yes	characteristic, complemented
612	#462	C₂²×C₃₁	1	2	yes	complemented
613	#461	C₂²×C₃₁	1	2	yes	complemented
614	#461	C₂²×C₃₁	1	2	yes	complemented
615	#460	C₂²×C₃₁	1	2	yes	complemented
616	#452	C₂²×C₃₁	1	2	yes	—
617	#452	C₂²×C₃₁	1	2	yes	complemented
618	#451	C₂²×C₃₁	1	2	yes	—
619	#451	C₂²×C₃₁	1	2	yes	complemented
620	#449	C₂²×C₃₁	1	2	yes	—
621	#449	C₂²×C₃₁	1	2	yes	—
622	#448	C₂²×C₃₁	1	2	yes	—
623	#448	C₂²×C₃₁	1	2	yes	complemented
624	#447	C₂²×C₃₁	1	2	yes	—
625	#446	C₂²×C₃₁	1	2	yes	complemented
626	#442	C₂²×C₃₁	1	2	yes	—
627	#441	C₂²×C₃₁	1	2	yes	complemented
628	#441	C₂²×C₃₁	1	2	yes	complemented
629	#441	C₂²×C₃₁	1	2	yes	complemented
630	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
631	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
632	#435	C₂²×C₃₁	1	2	yes	—
633	#435	C₂²×C₃₁	1	2	yes	complemented
634	C₈×D₄	C₂²×C₃₁	1	2	yes	—
635	#433	C₂²×C₃₁	1	2	yes	complemented
636	#432	C₂²×C₃₁	1	1	yes	characteristic
637	#424	C₂²×C₃₁	1	2	yes	complemented
638	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
639	#417	C₂²×C₃₁	1	2	yes	—
640	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
641	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
642	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
643	C₂×C₃₁	—	2	2	no	—
644	C₂×C₃₁	—	4	2	no	—
645	C₂×C₃₁	—	8	2	no	—
646	C₂×C₃₁	—	8	2	no	—
647	C₂×C₃₁	#3488	1	1	yes	characteristic
648	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
649	C₂×C₃₁	#3586	1	1	yes	characteristic
650	C₂×#41	—	2	2	no	—
651	C₂×#41	—	2	1	no	—
652	C₂×#41	C₂³×C₃₁	1	2	yes	—
653	C₂×#41	C₂³×C₃₁	1	2	yes	—
654	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
655	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
656	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
657	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
658	#137	—	4	4	no	—
659	#135	—	4	4	no	—
660	C₂×Q₁₆	—	4	2	no	—
661	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
662	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
663	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
664	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
665	C₂×SD₁₆	—	4	2	no	—
666	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
667	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
668	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
669	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
670	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
671	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
672	C₂×D₈	C₂³×C₃₁	1	2	yes	—
673	C₂×D₈	C₂³×C₃₁	1	2	yes	—
674	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
675	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
676	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
677	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
678	C₄⋊Q₈	—	2	1	no	—
679	C₄⋊Q₈	—	2	2	no	—
680	C₄⋊Q₈	—	2	2	no	—
681	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
682	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
683	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
684	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
685	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
686	#127	—	2	1	no	—
687	#127	C₂³×C₃₁	1	1	yes	characteristic
688	#126	—	2	2	no	—
689	#126	—	2	2	no	—
690	#126	—	2	2	no	—
691	#126	—	2	2	no	—
692	#126	D₄×C₃₁	1	2	yes	—
693	#125	—	2	1	no	—
694	#125	—	2	1	no	—
695	#125	D₄×C₃₁	1	1	yes	characteristic
696	#124	—	2	1	no	—
697	#124	—	2	2	no	—
698	#124	—	2	2	no	—
699	#124	—	2	2	no	—
700	#124	—	2	2	no	—
701	#124	—	2	1	no	—
702	#124	—	2	1	no	—
703	#124	—	2	1	no	—
704	#124	D₄×C₃₁	1	1	yes	characteristic
705	#124	C₂³×C₃₁	1	2	yes	—
706	#124	C₂³×C₃₁	1	2	yes	—
707	#124	C₂³×C₃₁	1	2	yes	—
708	#124	C₂³×C₃₁	1	2	yes	—
709	#123	—	2	2	no	—
710	#123	—	2	2	no	—
711	#123	—	2	2	no	—
712	#123	—	2	2	no	—
713	#123	—	2	2	no	—
714	#123	—	2	2	no	—
715	#123	—	2	2	no	—
716	#123	—	2	2	no	—
717	#123	D₄×C₃₁	1	2	yes	—
718	C₂²⋊Q₈	—	2	2	no	—
719	C₂²⋊Q₈	—	2	1	no	—
720	C₂²⋊Q₈	—	2	2	no	—
721	C₂²⋊Q₈	—	2	2	no	—
722	C₂²⋊Q₈	—	2	2	no	—
723	C₂²⋊Q₈	—	2	2	no	—
724	C₂²⋊Q₈	—	2	2	no	—
725	C₂²⋊Q₈	—	2	2	no	—
726	C₂²⋊Q₈	—	2	2	no	—
727	C₂²⋊Q₈	—	2	2	no	—
728	C₂²⋊Q₈	—	2	2	no	—
729	C₂²⋊Q₈	—	2	2	no	—
730	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
731	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
732	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
733	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
734	#121	—	2	2	no	—
735	#121	—	2	2	no	—
736	#121	—	2	1	no	—
737	#121	—	2	2	no	—
738	#121	—	2	2	no	—
739	#121	D₄×C₃₁	1	1	yes	characteristic
740	#121	C₂³×C₃₁	1	2	yes	—
741	#121	C₂³×C₃₁	1	2	yes	—
742	#121	C₂³×C₃₁	1	2	yes	—
743	#120	—	2	2	no	—
744	C₄×Q₈	—	2	1	no	—
745	C₄×Q₈	—	2	1	no	—
746	C₄×Q₈	—	2	1	no	—
747	C₄×Q₈	—	2	1	no	—
748	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
749	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
750	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
751	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
752	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
753	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
754	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
755	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
756	C₄×D₄	—	2	1	no	—
757	C₄×D₄	—	2	2	no	—
758	C₄×D₄	—	2	1	no	—
759	C₄×D₄	—	2	1	no	—
760	C₄×D₄	—	2	2	no	—
761	C₄×D₄	—	2	2	no	—
762	C₄×D₄	—	2	2	no	—
763	C₄×D₄	—	2	2	no	—
764	C₄×D₄	—	2	1	no	—
765	C₄×D₄	C₂³×C₃₁	1	2	yes	—
766	C₄×D₄	C₂³×C₃₁	1	2	yes	—
767	C₄×D₄	C₂³×C₃₁	1	2	yes	—
768	C₄×D₄	C₂³×C₃₁	1	2	yes	—
769	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
770	#117	—	2	1	no	—
771	#117	—	2	2	no	—
772	#117	—	2	2	no	—
773	#117	C₂³×C₃₁	1	2	yes	—
774	C₂×C₄⋊C₄	—	2	1	no	—
775	C₂×C₄⋊C₄	—	2	2	no	—
776	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
777	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
778	C₂×C₂²⋊C₄	—	2	1	no	—
779	C₂×C₂²⋊C₄	—	2	1	no	—
780	C₂×C₂²⋊C₄	—	2	2	no	—
781	C₂×C₄²	—	2	1	no	—
782	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
783	#106	C₂³×C₃₁	1	2	yes	—
784	#106	C₂³×C₃₁	1	2	yes	—
785	#106	C₂³×C₃₁	1	2	yes	—
786	#106	C₂³×C₃₁	1	2	yes	—
787	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
788	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
789	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
790	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
791	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
792	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
793	#103	C₂³×C₃₁	1	2	yes	—
794	#103	C₂³×C₃₁	1	2	yes	—
795	#103	C₂³×C₃₁	1	2	yes	—
796	#103	C₂³×C₃₁	1	2	yes	—
797	#103	C₂³×C₃₁	1	2	yes	—
798	#103	C₂³×C₃₁	1	2	yes	—
799	#103	C₂³×C₃₁	1	2	yes	—
800	#103	C₂³×C₃₁	1	2	yes	—
801	#103	C₂³×C₃₁	1	2	yes	—
802	#103	C₂³×C₃₁	1	2	yes	—
803	#102	C₂³×C₃₁	1	2	yes	—
804	#102	C₂³×C₃₁	1	2	yes	—
805	#102	C₂³×C₃₁	1	2	yes	—
806	#102	C₂³×C₃₁	1	2	yes	—
807	#102	C₂³×C₃₁	1	2	yes	—
808	#102	C₂³×C₃₁	1	2	yes	—
809	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
810	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
811	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
812	#97	C₂³×C₃₁	1	2	yes	—
813	#97	C₂³×C₃₁	1	2	yes	—
814	C₄×C₈	C₂³×C₃₁	1	2	yes	—
815	C₃₁	#62021	1	1	yes	characteristic, direct product factor, Sylow
816	C₂⁴	—	2	1	no	—
817	#41	—	2	4	no	—
818	#41	—	4	2	no	—
819	#41	—	4	2	no	—
820	#41	—	4	2	no	—
821	#41	—	8	2	no	—
822	#41	—	8	4	no	—
823	C₂×Q₈	—	2	1	no	—
824	C₂×Q₈	—	2	1	no	—
825	C₂×Q₈	—	2	2	no	—
826	C₂×Q₈	—	2	1	no	—
827	C₂×Q₈	—	2	2	no	—
828	C₂×Q₈	—	2	1	no	—
829	C₂×Q₈	—	2	2	no	—
830	C₂×Q₈	—	4	1	no	—
831	C₂×Q₈	—	4	1	no	—
832	C₂×Q₈	—	4	2	no	—
833	C₂×Q₈	—	4	1	no	—
834	C₂×Q₈	—	4	2	no	—
835	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
836	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
837	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
838	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
839	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
840	C₂×D₄	—	2	2	no	—
841	C₂×D₄	—	2	2	no	—
842	C₂×D₄	—	2	2	no	—
843	C₂×D₄	—	2	2	no	—
844	C₂×D₄	—	2	2	no	—
845	C₂×D₄	—	2	1	no	—
846	C₂×D₄	—	2	1	no	—
847	C₂×D₄	—	2	1	no	—
848	C₂×D₄	—	4	1	no	—
849	C₂×D₄	—	4	1	no	—
850	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
851	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
852	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
853	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
854	C₂²×C₄	—	2	1	no	—
855	C₂²×C₄	—	2	1	no	—
856	C₂²×C₄	—	2	1	no	—
857	C₂²×C₄	—	2	2	no	—
858	C₂²×C₄	—	2	2	no	—
859	C₂²×C₄	—	2	2	no	—
860	C₂²×C₄	—	2	2	no	—
861	C₂²×C₄	—	2	2	no	—
862	C₂²×C₄	—	2	2	no	—
863	C₂²×C₄	—	2	2	no	—
864	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
865	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
866	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
867	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
868	Q₁₆	—	2	4	no	—
869	Q₁₆	—	2	4	no	—
870	Q₁₆	—	4	2	no	—
871	Q₁₆	—	4	2	no	—
872	SD₁₆	—	2	4	no	—
873	SD₁₆	—	4	2	no	—
874	SD₁₆	—	4	2	no	—
875	SD₁₆	—	4	2	no	—
876	SD₁₆	—	4	2	no	—
877	SD₁₆	—	4	2	no	—
878	D₈	—	4	2	no	—
879	D₈	—	4	2	no	—
880	M₁₆	—	4	2	no	—
881	C₂×C₈	—	4	1	no	—
882	C₂×C₈	—	4	1	no	—
883	C₂×C₈	—	4	2	no	—
884	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
885	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
886	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
887	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
888	C₄⋊C₄	—	2	1	no	—
889	C₄⋊C₄	—	2	2	no	—
890	C₄⋊C₄	—	2	2	no	—
891	C₄⋊C₄	—	2	1	no	—
892	C₄⋊C₄	—	2	1	no	—
893	C₄⋊C₄	—	2	1	no	—
894	C₄⋊C₄	—	2	1	no	—
895	C₄⋊C₄	—	2	2	no	—
896	C₄⋊C₄	—	2	1	no	—
897	C₄⋊C₄	—	2	1	no	—
898	C₄⋊C₄	—	2	1	no	—
899	C₄⋊C₄	—	2	2	no	—
900	C₄⋊C₄	—	2	2	no	—
901	C₄⋊C₄	—	2	2	no	—
902	C₄⋊C₄	—	2	2	no	—
903	C₄⋊C₄	—	2	1	no	—
904	C₄⋊C₄	—	2	2	no	—
905	C₄⋊C₄	—	2	2	no	—
906	C₄⋊C₄	—	2	2	no	—
907	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
908	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
909	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
910	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
911	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
912	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
913	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
914	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
915	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
916	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
917	C₂²⋊C₄	—	2	2	no	—
918	C₂²⋊C₄	—	2	2	no	—
919	C₂²⋊C₄	—	2	2	no	—
920	C₂²⋊C₄	—	2	2	no	—
921	C₂²⋊C₄	—	2	2	no	—
922	C₂²⋊C₄	—	2	2	no	—
923	C₂²⋊C₄	—	2	2	no	—
924	C₂²⋊C₄	—	2	2	no	—
925	C₂²⋊C₄	—	2	2	no	—
926	C₂²⋊C₄	—	2	2	no	—
927	C₂²⋊C₄	—	2	2	no	—
928	C₂²⋊C₄	—	2	2	no	—
929	C₂²⋊C₄	—	2	2	no	—
930	C₂²⋊C₄	—	2	2	no	—
931	C₂²⋊C₄	—	2	2	no	—
932	C₂²⋊C₄	—	2	2	no	—
933	C₂²⋊C₄	—	2	2	no	—
934	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
935	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
936	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
937	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
938	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
939	C₄²	—	2	1	no	—
940	C₄²	—	2	1	no	—
941	C₄²	—	2	1	no	—
942	C₄²	—	2	1	no	—
943	C₄²	—	2	2	no	—
944	C₄²	—	2	2	no	—
945	C₄²	—	2	2	no	—
946	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
947	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
948	C₄²	C₂⁴×C₃₁	1	2	yes	—
949	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
950	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
951	C₂³	—	2	2	no	—
952	C₂³	—	2	2	no	—
953	C₂³	—	4	2	no	—
954	C₂³	—	8	1	no	—
955	C₂³	—	8	1	no	—
956	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
957	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
958	Q₈	—	2	1	no	—
959	Q₈	—	2	2	no	—
960	Q₈	—	2	1	no	—
961	Q₈	—	2	2	no	—
962	Q₈	—	2	2	no	—
963	Q₈	—	2	1	no	—
964	Q₈	—	2	2	no	—
965	Q₈	—	4	1	no	—
966	Q₈	—	4	2	no	—
967	Q₈	—	8	1	no	—
968	Q₈	—	8	2	no	—
969	Q₈	—	8	1	no	—
970	Q₈	—	8	1	no	—
971	Q₈	—	8	2	no	—
972	Q₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
973	D₄	—	2	4	no	—
974	D₄	—	2	2	no	—
975	D₄	—	2	2	no	—
976	D₄	—	2	2	no	—
977	D₄	—	2	2	no	—
978	D₄	—	4	1	no	—
979	D₄	—	4	4	no	—
980	D₄	—	4	2	no	—
981	D₄	—	4	4	no	—
982	D₄	—	8	1	no	—
983	D₄	—	8	1	no	—
984	D₄	—	8	2	no	—
985	D₄	—	8	2	no	—
986	D₄	—	8	2	no	—
987	C₂×C₄	—	2	2	no	—
988	C₂×C₄	—	2	1	no	—
989	C₂×C₄	—	2	1	no	—
990	C₂×C₄	—	2	2	no	—
991	C₂×C₄	—	2	1	no	—
992	C₂×C₄	—	2	2	no	—
993	C₂×C₄	—	2	1	no	—
994	C₂×C₄	—	2	2	no	—
995	C₂×C₄	—	2	1	no	—
996	C₂×C₄	—	2	1	no	—
997	C₂×C₄	—	2	2	no	—
998	C₂×C₄	—	2	2	no	—
999	C₂×C₄	—	2	2	no	—
1000	C₂×C₄	—	4	1	no	—
1001	C₂×C₄	—	4	4	no	—
1002	C₂×C₄	—	4	2	no	—
1003	C₂×C₄	—	4	2	no	—
1004	C₂×C₄	—	8	2	no	—
1005	C₂×C₄	—	8	2	no	—
1006	C₂×C₄	—	8	2	no	—
1007	C₂×C₄	—	8	2	no	—
1008	C₂×C₄	—	8	2	no	—
1009	C₂×C₄	—	8	1	no	—
1010	C₂×C₄	—	8	2	no	—
1011	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1012	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1013	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1014	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1015	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1016	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1017	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
1018	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
1019	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
1020	C₈	—	2	2	no	—
1021	C₈	—	2	2	no	—
1022	C₈	—	2	2	no	—
1023	C₈	—	2	2	no	—
1024	C₂²	—	2	1	no	—
1025	C₂²	—	2	1	no	—
1026	C₂²	—	2	2	no	—
1027	C₂²	—	2	2	no	—
1028	C₂²	—	2	2	no	—
1029	C₂²	—	4	2	no	—
1030	C₂²	—	4	2	no	—
1031	C₂²	—	4	2	no	—
1032	C₂²	—	4	2	no	—
1033	C₂²	—	4	2	no	—
1034	C₂²	—	4	2	no	—
1035	C₂²	—	16	2	no	—
1036	C₂²	C₃₁×#577	1	2	yes	—
1037	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
1038	C₄	—	2	2	no	—
1039	C₄	—	2	1	no	—
1040	C₄	—	2	2	no	—
1041	C₄	—	2	1	no	—
1042	C₄	—	2	1	no	—
1043	C₄	—	2	2	no	—
1044	C₄	—	2	1	no	—
1045	C₄	—	2	1	no	—
1046	C₄	—	2	2	no	—
1047	C₄	—	4	1	no	—
1048	C₄	—	4	2	no	—
1049	C₄	—	4	2	no	—
1050	C₄	—	4	1	no	—
1051	C₄	—	4	1	no	—
1052	C₄	—	4	2	no	—
1053	C₄	—	4	2	no	—
1054	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
1055	C₄	C₃₁×#585	1	1	yes	characteristic
1056	C₂	—	2	2	no	—
1057	C₂	—	4	2	no	—
1058	C₂	—	8	2	no	—
1059	C₂	—	8	2	no	—
1060	C₂	C₃₁×#3488	1	1	yes	characteristic
1061	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
1062	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#529 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#3561	C₂	semidirect
2. C₃₁×#3522	C₂	semidirect
3. C₃₁×#3401	C₂	semidirect
4. C₃₁×#3369	C₂	semidirect
5. C₃₁×#3316	C₂	semidirect
6. C₃₁×#3306	C₂	semidirect
7. C₃₁×#3298	C₂	semidirect
8. C₃₁×#3282	C₂	semidirect
9. C₃₁×#3271	C₂	semidirect
10. C₃₁×#3270	C₂	semidirect
11. C₃₁×#3263	C₂	semidirect
12. C₃₁×#3246	C₂	semidirect
13. C₃₁×#3225	C₂	semidirect
14. C₃₁×#3213	C₂	semidirect
15. C₃₁×#3125	C₂	semidirect
16. C₃₁×#3113	C₂	semidirect
17. C₃₁×#3046	C₂	semidirect
18. C₂×C₃₁×#464	C₂	semidirect
19. C₃₁×#2992	C₂	semidirect
20. C₂×C₃₁×#137	C₂²	semidirect
27. C₃₁×#559	C₂²	semidirect
28. Q₈²×C₃₁	C₂²	semidirect
31. C₃₁×#554	C₂²	semidirect
33. C₃₁×#550	C₂²	semidirect
44. C₃₁×#542	C₂²	semidirect
53. C₃₁×#520	C₂²	semidirect
55. C₃₁×#497	C₂²	semidirect
60. C₃₁×#487	C₂²	semidirect
62. C₃₁×#483	C₂²	semidirect
65. C₃₁×#476	C₂²	semidirect
66. C₃₁×#475	C₂²	semidirect
67. C₃₁×#475	C₂²	semidirect
68. C₃₁×#475	C₂²	semidirect
69. C₃₁×#475	C₂²	semidirect
70. C₃₁×#470	C₂²	semidirect
75. C₃₁×#465	C₂²	semidirect
81. C₃₁×#464	C₂²	semidirect
86. C₃₁×#462	C₂²	semidirect
87. C₃₁×#462	C₂²	semidirect
88. C₃₁×#462	C₂²	semidirect
89. C₃₁×#461	C₂²	semidirect
90. C₃₁×#461	C₂²	semidirect
91. C₃₁×#460	C₂²	semidirect
93. C₃₁×#452	C₂²	semidirect
95. C₃₁×#451	C₂²	semidirect
99. C₃₁×#448	C₂²	semidirect
101. C₃₁×#446	C₂²	semidirect
103. C₃₁×#441	C₂²	semidirect
104. C₃₁×#441	C₂²	semidirect
105. C₃₁×#441	C₂²	semidirect
107. C₄×SD₁₆×C₃₁	C₂²	semidirect
109. C₃₁×#435	C₂²	semidirect
111. C₃₁×#433	C₂²	semidirect
113. C₃₁×#424	C₂²	semidirect
117. C₂×C₃₁×#103	C₂²	semidirect
419. #62021	C₃₁	direct (N × H)
493. #3561	C₂×C₃₁	semidirect
494. #3522	C₂×C₃₁	semidirect
495. #3401	C₂×C₃₁	semidirect
496. #3369	C₂×C₃₁	semidirect
497. #3316	C₂×C₃₁	semidirect
498. #3306	C₂×C₃₁	semidirect
499. #3298	C₂×C₃₁	semidirect
500. #3282	C₂×C₃₁	semidirect
501. #3271	C₂×C₃₁	semidirect
502. #3270	C₂×C₃₁	semidirect
503. #3263	C₂×C₃₁	semidirect
504. #3246	C₂×C₃₁	semidirect
505. #3225	C₂×C₃₁	semidirect
506. #3213	C₂×C₃₁	semidirect
507. #3125	C₂×C₃₁	semidirect
508. #3113	C₂×C₃₁	semidirect
509. #3046	C₂×C₃₁	semidirect
510. C₂×#464	C₂×C₃₁	semidirect
511. #2992	C₂×C₃₁	semidirect
544. C₂×#137	C₂²×C₃₁	semidirect
551. #559	C₂²×C₃₁	semidirect
552. Q₈²	C₂²×C₃₁	semidirect
555. #554	C₂²×C₃₁	semidirect
557. #550	C₂²×C₃₁	semidirect
568. #542	C₂²×C₃₁	semidirect
577. #520	C₂²×C₃₁	semidirect
579. #497	C₂²×C₃₁	semidirect
584. #487	C₂²×C₃₁	semidirect
586. #483	C₂²×C₃₁	semidirect
589. #476	C₂²×C₃₁	semidirect
590. #475	C₂²×C₃₁	semidirect
591. #475	C₂²×C₃₁	semidirect
592. #475	C₂²×C₃₁	semidirect
593. #475	C₂²×C₃₁	semidirect
594. #470	C₂²×C₃₁	semidirect
599. #465	C₂²×C₃₁	semidirect
605. #464	C₂²×C₃₁	semidirect
610. #462	C₂²×C₃₁	semidirect
611. #462	C₂²×C₃₁	semidirect
612. #462	C₂²×C₃₁	semidirect
613. #461	C₂²×C₃₁	semidirect
614. #461	C₂²×C₃₁	semidirect
615. #460	C₂²×C₃₁	semidirect
617. #452	C₂²×C₃₁	semidirect
619. #451	C₂²×C₃₁	semidirect
623. #448	C₂²×C₃₁	semidirect
625. #446	C₂²×C₃₁	semidirect
627. #441	C₂²×C₃₁	semidirect
628. #441	C₂²×C₃₁	semidirect
629. #441	C₂²×C₃₁	semidirect
631. C₄×SD₁₆	C₂²×C₃₁	semidirect
633. #435	C₂²×C₃₁	semidirect
635. #433	C₂²×C₃₁	semidirect
637. #424	C₂²×C₃₁	semidirect
641. C₂×#103	C₂²×C₃₁	semidirect
815. C₃₁	#62021	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3561	2
C₃₁×#3522	2
C₃₁×#3401	2
C₃₁×#3369	2
C₃₁×#3316	2
C₃₁×#3306	2
C₃₁×#3298	2
C₃₁×#3282	2
C₃₁×#3271	2
C₃₁×#3270	2
C₃₁×#3263	2
C₃₁×#3246	2
C₃₁×#3225	2
C₃₁×#3213	2
C₃₁×#3125	2
C₃₁×#3113	2
C₃₁×#3046	2
C₂×C₃₁×#464	2
C₃₁×#2992	2
#62021	31

Subgroups — C31×#62021 #21518780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62021 #21518780