Subgroups — C₃₁×#61215 #21517974

Order 7936.

2200 conjugacy classes of subgroups.

The subgroup lattice has 932 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3511	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3333	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3332	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3325	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3324	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3305	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3304	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3295	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3294	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3096	C₂	1	2	yes	maximal, complemented
13	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3062	C₂	1	2	yes	maximal, complemented
15	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#3012	C₂	1	2	yes	maximal, complemented
17	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
18	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
19	C₃₁×#2897	C₂	1	2	yes	maximal, complemented
20	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	—
21	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
22	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#131	C₂²	1	2	yes	complemented
24	C₃₁×#556	C₂²	1	2	yes	complemented
25	C₃₁×#555	C₂²	1	2	yes	complemented
26	C₃₁×#554	C₂²	1	2	yes	complemented
27	C₃₁×#553	—	2	2	no	—
28	C₃₁×#550	C₂²	1	2	yes	—
29	C₃₁×#550	C₂²	1	2	yes	—
30	C₃₁×#548	—	2	2	no	—
31	C₃₁×#547	C₂²	1	2	yes	complemented
32	D₄²×C₃₁	—	2	2	no	—
33	D₄²×C₃₁	—	2	2	no	—
34	D₄²×C₃₁	C₂²	1	2	yes	—
35	D₄²×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#542	—	2	2	no	—
37	C₃₁×#537	C₂²	1	2	yes	complemented
38	C₃₁×#535	—	2	2	no	—
39	C₃₁×#535	C₂²	1	2	yes	—
40	C₃₁×#533	—	2	1	no	—
41	C₃₁×#532	—	2	1	no	—
42	C₃₁×#532	—	2	1	no	—
43	C₃₁×#532	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₃₁×#127	—	2	1	no	—
46	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
47	C₃₁×#520	C₂²	1	2	yes	complemented
48	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
49	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic, complemented
50	C₃₁×#486	C₂²	1	2	yes	complemented
51	C₃₁×#486	C₂²	1	2	yes	complemented
52	C₃₁×#486	C₂²	1	2	yes	complemented
53	C₃₁×#486	C₂²	1	2	yes	complemented
54	C₃₁×#482	C₂²	1	4	yes	complemented
55	C₃₁×#482	C₂²	1	4	yes	complemented
56	C₃₁×#481	C₂²	1	4	yes	complemented
57	C₃₁×#480	C₂²	1	4	yes	complemented
58	C₃₁×#477	C₂²	1	2	yes	complemented
59	C₃₁×#476	C₂²	1	2	yes	complemented
60	C₃₁×#475	C₂²	1	2	yes	complemented
61	C₃₁×#474	C₂²	1	2	yes	complemented
62	C₃₁×#469	C₂²	1	4	yes	complemented
63	C₃₁×#468	C₂²	1	4	yes	complemented
64	C₃₁×#466	C₂²	1	4	yes	complemented
65	C₃₁×#465	C₂²	1	4	yes	complemented
66	C₃₁×#460	C₂²	1	2	yes	complemented
67	C₃₁×#460	C₂²	1	2	yes	complemented
68	C₃₁×#459	C₂²	1	2	yes	complemented
69	C₃₁×#459	C₂²	1	2	yes	complemented
70	C₃₁×#452	C₂²	1	4	yes	complemented
71	C₃₁×#450	C₂²	1	4	yes	complemented
72	C₃₁×#449	C₂²	1	4	yes	complemented
73	C₃₁×#447	C₂²	1	4	yes	complemented
74	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
75	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
76	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
77	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
78	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
79	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
80	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
81	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
82	C₃₁×#436	C₂²	1	2	yes	complemented
83	C₃₁×#436	C₂²	1	2	yes	complemented
84	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
85	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
86	C₃₁×#428	C₂²	1	2	yes	complemented
87	C₃₁×#427	C₂²	1	1	yes	characteristic, complemented
88	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
89	C₃₁×#418	C₂²	1	2	yes	complemented
90	C₃₁×#418	C₂²	1	4	yes	complemented
91	C₃₁×#417	C₂²	1	2	yes	—
92	C₃₁×#417	C₂²	1	4	yes	complemented
93	C₃₁×#416	C₂²	1	1	yes	characteristic
94	C₃₁×#416	C₂²	1	4	yes	complemented
95	C₃₁×#416	C₂²	1	1	yes	characteristic, complemented
96	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
97	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
98	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
99	C₃₁×#408	C₂²	1	2	yes	complemented
100	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
101	C₃₁×H(5,𝔽₂)	—	2	8	no	—
102	C₃₁×H(5,𝔽₂)	—	4	4	no	—
103	C₂×#41×C₃₁	—	2	2	no	—
104	C₂×#41×C₃₁	—	2	2	no	—
105	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×#41×C₃₁	C₂³	1	4	yes	—
107	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
108	C₂×#41×C₃₁	C₂³	1	2	yes	—
109	C₂²×Q₈×C₃₁	—	2	1	no	—
110	C₂²×D₄×C₃₁	—	2	2	no	—
111	C₂²×D₄×C₃₁	—	2	4	no	—
112	C₂²×D₄×C₃₁	—	2	2	no	—
113	C₂²×D₄×C₃₁	—	2	2	no	—
114	C₂²×D₄×C₃₁	—	2	1	no	—
115	C₂²×D₄×C₃₁	—	2	1	no	—
116	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
117	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
118	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
119	C₃₁×#135	—	2	4	no	—
120	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
124	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×D₈×C₃₁	C₂³	1	4	yes	—
126	C₃₁×#131	—	4	4	no	—
127	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
128	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
129	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
130	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
131	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
132	C₃₁×C₄⋊Q₈	—	2	1	no	—
133	C₃₁×C₄⋊Q₈	—	2	1	no	—
134	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
135	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
136	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
137	C₃₁×#127	—	2	2	no	—
138	C₃₁×#127	—	2	2	no	—
139	C₃₁×#127	—	2	2	no	—
140	C₃₁×#127	—	2	1	no	—
141	C₃₁×#127	—	2	1	no	—
142	C₃₁×#127	D₄	1	1	yes	characteristic
143	C₃₁×#127	C₂³	1	2	yes	—
144	C₃₁×#127	C₂³	1	2	yes	—
145	C₃₁×#126	—	2	4	no	—
146	C₃₁×#125	—	2	2	no	—
147	C₃₁×#124	—	2	2	no	—
148	C₃₁×#124	—	2	2	no	—
149	C₃₁×#124	—	2	4	no	—
150	C₃₁×#124	D₄	1	2	yes	—
151	C₃₁×#123	—	2	4	no	—
152	C₃₁×C₂²⋊Q₈	—	2	2	no	—
153	C₃₁×C₂²⋊Q₈	—	2	2	no	—
154	C₃₁×C₂²⋊Q₈	—	2	4	no	—
155	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
156	C₃₁×#121	—	2	2	no	—
157	C₃₁×#121	—	2	2	no	—
158	C₃₁×#121	—	2	4	no	—
159	C₃₁×#121	—	2	2	no	—
160	C₃₁×#121	—	2	2	no	—
161	C₃₁×#121	—	2	4	no	—
162	C₃₁×#121	—	2	4	no	—
163	C₃₁×#121	D₄	1	2	yes	—
164	C₃₁×#121	D₄	1	2	yes	—
165	C₃₁×#121	C₂³	1	4	yes	—
166	C₃₁×#121	C₂³	1	4	yes	—
167	C₃₁×#121	C₂³	1	4	yes	—
168	C₃₁×#120	—	2	4	no	—
169	C₃₁×#120	—	2	4	no	—
170	C₃₁×#120	—	2	4	no	—
171	C₄×Q₈×C₃₁	—	2	2	no	—
172	C₄×Q₈×C₃₁	—	2	2	no	—
173	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
174	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
175	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
176	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
177	C₄×D₄×C₃₁	—	2	2	no	—
178	C₄×D₄×C₃₁	—	2	4	no	—
179	C₄×D₄×C₃₁	—	2	2	no	—
180	C₄×D₄×C₃₁	—	2	1	no	—
181	C₄×D₄×C₃₁	—	2	1	no	—
182	C₄×D₄×C₃₁	—	2	1	no	—
183	C₄×D₄×C₃₁	—	2	1	no	—
184	C₄×D₄×C₃₁	—	2	2	no	—
185	C₄×D₄×C₃₁	—	2	2	no	—
186	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
187	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
188	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
189	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
190	C₄×D₄×C₃₁	C₂³	1	2	yes	—
191	C₄×D₄×C₃₁	C₂³	1	2	yes	—
192	C₃₁×#117	—	2	2	no	—
193	C₃₁×#117	—	2	1	no	—
194	C₃₁×#117	—	2	1	no	—
195	C₃₁×#117	C₂³	1	1	yes	characteristic
196	C₃₁×#117	C₂³	1	4	yes	—
197	C₃₁×#117	C₂³	1	1	yes	characteristic
198	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
199	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
200	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
201	C₂×C₄²×C₃₁	—	2	1	no	—
202	C₂×C₄²×C₃₁	—	2	1	no	—
203	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
204	C₃₁×#107	C₂³	1	1	yes	characteristic
205	C₃₁×#107	C₂³	1	1	yes	characteristic
206	C₃₁×#107	C₂³	1	4	yes	—
207	C₃₁×#106	C₂³	1	1	yes	characteristic
208	C₃₁×#106	C₂³	1	1	yes	characteristic
209	C₃₁×#106	C₂³	1	4	yes	—
210	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
211	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
212	C₃₁×#103	C₂³	1	1	yes	characteristic
213	C₃₁×#103	C₂³	1	1	yes	characteristic
214	C₃₁×#103	C₂³	1	4	yes	—
215	C₃₁×#103	C₂³	1	1	yes	characteristic
216	C₃₁×#103	C₂³	1	1	yes	characteristic
217	C₃₁×#102	C₂³	1	1	yes	characteristic
218	C₃₁×#102	C₂³	1	4	yes	—
219	C₃₁×#102	C₂³	1	4	yes	—
220	C₃₁×#102	C₂³	1	4	yes	—
221	C₃₁×#102	C₂³	1	4	yes	—
222	C₃₁×#102	C₂³	1	4	yes	—
223	C₃₁×#102	C₂³	1	4	yes	—
224	C₃₁×#102	C₂³	1	1	yes	characteristic
225	C₃₁×#102	C₂³	1	4	yes	—
226	C₃₁×#102	C₂³	1	1	yes	characteristic
227	C₃₁×#102	C₂³	1	1	yes	characteristic
228	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
229	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
230	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
231	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
232	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
233	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
234	C₂⁴×C₃₁	—	2	4	no	—
235	C₂⁴×C₃₁	—	2	2	no	—
236	#41×C₃₁	—	2	4	no	—
237	#41×C₃₁	—	2	4	no	—
238	#41×C₃₁	—	2	2	no	—
239	#41×C₃₁	—	2	8	no	—
240	#41×C₃₁	—	2	2	no	—
241	#41×C₃₁	—	2	2	no	—
242	#41×C₃₁	—	4	4	no	—
243	#41×C₃₁	—	4	8	no	—
244	#41×C₃₁	—	8	4	no	—
245	#41×C₃₁	#41	1	4	yes	complemented
246	C₂×Q₈×C₃₁	—	2	2	no	—
247	C₂×Q₈×C₃₁	—	2	2	no	—
248	C₂×Q₈×C₃₁	—	8	2	no	—
249	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
250	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
251	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
252	C₂×D₄×C₃₁	—	2	1	no	—
253	C₂×D₄×C₃₁	—	2	8	no	—
254	C₂×D₄×C₃₁	—	2	1	no	—
255	C₂×D₄×C₃₁	—	2	4	no	—
256	C₂×D₄×C₃₁	—	2	4	no	—
257	C₂×D₄×C₃₁	—	2	4	no	—
258	C₂×D₄×C₃₁	—	2	4	no	—
259	C₂×D₄×C₃₁	—	2	2	no	—
260	C₂×D₄×C₃₁	—	2	2	no	—
261	C₂×D₄×C₃₁	—	2	4	no	—
262	C₂×D₄×C₃₁	—	2	2	no	—
263	C₂×D₄×C₃₁	—	2	2	no	—
264	C₂×D₄×C₃₁	—	2	2	no	—
265	C₂×D₄×C₃₁	—	2	4	no	—
266	C₂×D₄×C₃₁	—	2	4	no	—
267	C₂×D₄×C₃₁	—	4	4	no	—
268	C₂×D₄×C₃₁	—	4	8	no	—
269	C₂×D₄×C₃₁	—	4	16	no	—
270	C₂×D₄×C₃₁	—	8	2	no	—
271	C₂×D₄×C₃₁	—	8	2	no	—
272	C₂×D₄×C₃₁	—	8	4	no	—
273	C₂×D₄×C₃₁	—	8	4	no	—
274	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
275	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
276	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
277	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
278	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
279	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
280	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
281	C₂²×C₄×C₃₁	—	2	1	no	—
282	C₂²×C₄×C₃₁	—	2	2	no	—
283	C₂²×C₄×C₃₁	—	2	4	no	—
284	C₂²×C₄×C₃₁	—	2	1	no	—
285	C₂²×C₄×C₃₁	—	2	1	no	—
286	C₂²×C₄×C₃₁	—	2	1	no	—
287	C₂²×C₄×C₃₁	—	2	1	no	—
288	C₂²×C₄×C₃₁	—	2	1	no	—
289	C₂²×C₄×C₃₁	—	2	4	no	—
290	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
291	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
292	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
293	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
294	Q₁₆×C₃₁	—	2	2	no	—
295	SD₁₆×C₃₁	—	2	2	no	—
296	SD₁₆×C₃₁	—	2	2	no	—
297	SD₁₆×C₃₁	—	4	4	no	—
298	D₈×C₃₁	—	2	2	no	—
299	D₈×C₃₁	—	4	4	no	—
300	M₁₆×C₃₁	—	2	4	no	—
301	M₁₆×C₃₁	—	4	4	no	—
302	C₂×C₈×C₃₁	—	2	2	no	—
303	C₂×C₈×C₃₁	—	2	2	no	—
304	C₂×C₈×C₃₁	—	4	4	no	—
305	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
306	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
307	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
308	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
309	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
310	C₄⋊C₄×C₃₁	—	2	2	no	—
311	C₄⋊C₄×C₃₁	—	2	2	no	—
312	C₄⋊C₄×C₃₁	—	2	2	no	—
313	C₄⋊C₄×C₃₁	—	2	4	no	—
314	C₄⋊C₄×C₃₁	—	2	1	no	—
315	C₄⋊C₄×C₃₁	—	2	1	no	—
316	C₄⋊C₄×C₃₁	—	2	1	no	—
317	C₄⋊C₄×C₃₁	—	2	1	no	—
318	C₄⋊C₄×C₃₁	—	2	4	no	—
319	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
320	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
321	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
322	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
323	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
324	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
325	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
326	C₂²⋊C₄×C₃₁	—	2	1	no	—
327	C₂²⋊C₄×C₃₁	—	2	1	no	—
328	C₂²⋊C₄×C₃₁	—	2	4	no	—
329	C₂²⋊C₄×C₃₁	—	2	4	no	—
330	C₂²⋊C₄×C₃₁	—	2	1	no	—
331	C₂²⋊C₄×C₃₁	—	2	1	no	—
332	C₂²⋊C₄×C₃₁	—	2	1	no	—
333	C₂²⋊C₄×C₃₁	—	2	1	no	—
334	C₂²⋊C₄×C₃₁	—	2	4	no	—
335	C₂²⋊C₄×C₃₁	—	2	4	no	—
336	C₂²⋊C₄×C₃₁	—	2	4	no	—
337	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
338	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
339	C₄²×C₃₁	—	2	1	no	—
340	C₄²×C₃₁	—	2	1	no	—
341	C₄²×C₃₁	—	2	1	no	—
342	C₄²×C₃₁	—	2	1	no	—
343	C₄²×C₃₁	—	2	1	no	—
344	C₄²×C₃₁	—	2	1	no	—
345	C₄²×C₃₁	—	2	4	no	—
346	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
347	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
348	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
349	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
350	#61215	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
351	C₂³×C₃₁	—	2	1	no	—
352	C₂³×C₃₁	—	2	1	no	—
353	C₂³×C₃₁	—	2	4	no	—
354	C₂³×C₃₁	—	4	8	no	—
355	C₂³×C₃₁	—	4	4	no	—
356	C₂³×C₃₁	—	8	2	no	—
357	C₂³×C₃₁	—	8	2	no	—
358	C₂³×C₃₁	—	8	4	no	—
359	C₂³×C₃₁	—	8	4	no	—
360	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
361	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
362	Q₈×C₃₁	—	2	4	no	—
363	Q₈×C₃₁	—	2	1	no	—
364	Q₈×C₃₁	—	2	1	no	—
365	Q₈×C₃₁	—	2	1	no	—
366	Q₈×C₃₁	—	4	4	no	—
367	Q₈×C₃₁	—	8	2	no	—
368	Q₈×C₃₁	C₂×#41	1	2	yes	—
369	D₄×C₃₁	—	2	4	no	—
370	D₄×C₃₁	—	2	8	no	—
371	D₄×C₃₁	—	2	4	no	—
372	D₄×C₃₁	—	2	4	no	—
373	D₄×C₃₁	—	2	4	no	—
374	D₄×C₃₁	—	2	1	no	—
375	D₄×C₃₁	—	2	4	no	—
376	D₄×C₃₁	—	2	1	no	—
377	D₄×C₃₁	—	2	1	no	—
378	D₄×C₃₁	—	2	4	no	—
379	D₄×C₃₁	—	4	8	no	—
380	D₄×C₃₁	—	4	8	no	—
381	D₄×C₃₁	—	4	2	no	—
382	D₄×C₃₁	—	4	4	no	—
383	D₄×C₃₁	—	4	4	no	—
384	D₄×C₃₁	—	4	4	no	—
385	D₄×C₃₁	—	8	2	no	—
386	D₄×C₃₁	—	8	2	no	—
387	D₄×C₃₁	—	8	4	no	—
388	D₄×C₃₁	—	8	4	no	—
389	D₄×C₃₁	—	8	4	no	—
390	D₄×C₃₁	C₂×#41	1	2	yes	—
391	C₂×C₄×C₃₁	—	2	1	no	—
392	C₂×C₄×C₃₁	—	2	4	no	—
393	C₂×C₄×C₃₁	—	2	2	no	—
394	C₂×C₄×C₃₁	—	2	2	no	—
395	C₂×C₄×C₃₁	—	2	4	no	—
396	C₂×C₄×C₃₁	—	2	4	no	—
397	C₂×C₄×C₃₁	—	2	1	no	—
398	C₂×C₄×C₃₁	—	2	1	no	—
399	C₂×C₄×C₃₁	—	2	1	no	—
400	C₂×C₄×C₃₁	—	2	1	no	—
401	C₂×C₄×C₃₁	—	2	1	no	—
402	C₂×C₄×C₃₁	—	2	4	no	—
403	C₂×C₄×C₃₁	—	4	8	no	—
404	C₂×C₄×C₃₁	—	4	2	no	—
405	C₂×C₄×C₃₁	—	4	2	no	—
406	C₂×C₄×C₃₁	—	4	2	no	—
407	C₂×C₄×C₃₁	—	4	2	no	—
408	C₂×C₄×C₃₁	—	4	2	no	—
409	C₂×C₄×C₃₁	—	8	2	no	—
410	C₂×C₄×C₃₁	—	8	4	no	—
411	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
412	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
413	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
414	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
415	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
416	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
417	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
418	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
419	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
420	C₈×C₃₁	—	2	1	no	—
421	C₈×C₃₁	—	2	1	no	—
422	C₈×C₃₁	—	2	1	no	—
423	C₈×C₃₁	—	2	1	no	—
424	C₈×C₃₁	—	2	4	no	—
425	#3511	C₂×C₃₁	1	1	yes	characteristic, complemented
426	#3498	C₂×C₃₁	1	1	yes	characteristic, complemented
427	#3333	C₂×C₃₁	1	2	yes	complemented
428	#3332	C₂×C₃₁	1	2	yes	complemented
429	#3325	C₂×C₃₁	1	2	yes	complemented
430	#3324	C₂×C₃₁	1	2	yes	complemented
431	#3305	C₂×C₃₁	1	2	yes	complemented
432	#3304	C₂×C₃₁	1	2	yes	complemented
433	#3295	C₂×C₃₁	1	2	yes	complemented
434	#3294	C₂×C₃₁	1	2	yes	complemented
435	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
436	#3096	C₂×C₃₁	1	2	yes	complemented
437	#3079	C₂×C₃₁	1	1	yes	characteristic, complemented
438	#3062	C₂×C₃₁	1	2	yes	complemented
439	#3038	C₂×C₃₁	1	1	yes	characteristic, complemented
440	#3012	C₂×C₃₁	1	2	yes	complemented
441	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
442	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
443	#2897	C₂×C₃₁	1	2	yes	complemented
444	C₂²×C₃₁	—	2	1	no	—
445	C₂²×C₃₁	—	2	1	no	—
446	C₂²×C₃₁	—	2	4	no	—
447	C₂²×C₃₁	—	2	2	no	—
448	C₂²×C₃₁	—	2	4	no	—
449	C₂²×C₃₁	—	2	4	no	—
450	C₂²×C₃₁	—	4	1	no	—
451	C₂²×C₃₁	—	4	1	no	—
452	C₂²×C₃₁	—	4	4	no	—
453	C₂²×C₃₁	—	4	1	no	—
454	C₂²×C₃₁	—	4	1	no	—
455	C₂²×C₃₁	—	4	4	no	—
456	C₂²×C₃₁	—	4	1	no	—
457	C₂²×C₃₁	—	4	4	no	—
458	C₂²×C₃₁	—	16	4	no	—
459	C₂²×C₃₁	—	16	8	no	—
460	C₂²×C₃₁	#546	1	2	yes	—
461	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
462	C₄×C₃₁	—	2	2	no	—
463	C₄×C₃₁	—	2	4	no	—
464	C₄×C₃₁	—	2	1	no	—
465	C₄×C₃₁	—	2	1	no	—
466	C₄×C₃₁	—	4	1	no	—
467	C₄×C₃₁	—	4	1	no	—
468	C₄×C₃₁	—	4	1	no	—
469	C₄×C₃₁	—	4	4	no	—
470	C₄×C₃₁	—	4	1	no	—
471	C₄×C₃₁	—	4	1	no	—
472	C₄×C₃₁	#546	1	2	yes	—
473	C₄×C₃₁	#576	1	2	yes	—
474	C₄×C₃₁	#577	1	2	yes	—
475	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
476	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
477	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	—
478	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	complemented
479	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
480	C₂×#131	C₂²×C₃₁	1	2	yes	complemented
481	#556	C₂²×C₃₁	1	2	yes	complemented
482	#555	C₂²×C₃₁	1	2	yes	complemented
483	#554	C₂²×C₃₁	1	2	yes	complemented
484	#553	—	2	2	no	—
485	#550	C₂²×C₃₁	1	2	yes	—
486	#550	C₂²×C₃₁	1	2	yes	—
487	#548	—	2	2	no	—
488	#547	C₂²×C₃₁	1	2	yes	complemented
489	D₄²	—	2	2	no	—
490	D₄²	—	2	2	no	—
491	D₄²	C₂²×C₃₁	1	2	yes	—
492	D₄²	C₂²×C₃₁	1	2	yes	—
493	#542	—	2	2	no	—
494	#537	C₂²×C₃₁	1	2	yes	complemented
495	#535	—	2	2	no	—
496	#535	C₂²×C₃₁	1	2	yes	—
497	#533	—	2	1	no	—
498	#532	—	2	1	no	—
499	#532	—	2	1	no	—
500	#532	C₂²×C₃₁	1	1	yes	characteristic
501	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
502	C₂×#127	—	2	1	no	—
503	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
504	#520	C₂²×C₃₁	1	2	yes	complemented
505	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
506	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic, complemented
507	#486	C₂²×C₃₁	1	2	yes	complemented
508	#486	C₂²×C₃₁	1	2	yes	complemented
509	#486	C₂²×C₃₁	1	2	yes	complemented
510	#486	C₂²×C₃₁	1	2	yes	complemented
511	#482	C₂²×C₃₁	1	4	yes	complemented
512	#482	C₂²×C₃₁	1	4	yes	complemented
513	#481	C₂²×C₃₁	1	4	yes	complemented
514	#480	C₂²×C₃₁	1	4	yes	complemented
515	#477	C₂²×C₃₁	1	2	yes	complemented
516	#476	C₂²×C₃₁	1	2	yes	complemented
517	#475	C₂²×C₃₁	1	2	yes	complemented
518	#474	C₂²×C₃₁	1	2	yes	complemented
519	#469	C₂²×C₃₁	1	4	yes	complemented
520	#468	C₂²×C₃₁	1	4	yes	complemented
521	#466	C₂²×C₃₁	1	4	yes	complemented
522	#465	C₂²×C₃₁	1	4	yes	complemented
523	#460	C₂²×C₃₁	1	2	yes	complemented
524	#460	C₂²×C₃₁	1	2	yes	complemented
525	#459	C₂²×C₃₁	1	2	yes	complemented
526	#459	C₂²×C₃₁	1	2	yes	complemented
527	#452	C₂²×C₃₁	1	4	yes	complemented
528	#450	C₂²×C₃₁	1	4	yes	complemented
529	#449	C₂²×C₃₁	1	4	yes	complemented
530	#447	C₂²×C₃₁	1	4	yes	complemented
531	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
532	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
533	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
534	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
535	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
536	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
537	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
538	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
539	#436	C₂²×C₃₁	1	2	yes	complemented
540	#436	C₂²×C₃₁	1	2	yes	complemented
541	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
542	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
543	#428	C₂²×C₃₁	1	2	yes	complemented
544	#427	C₂²×C₃₁	1	1	yes	characteristic, complemented
545	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
546	#418	C₂²×C₃₁	1	2	yes	complemented
547	#418	C₂²×C₃₁	1	4	yes	complemented
548	#417	C₂²×C₃₁	1	2	yes	—
549	#417	C₂²×C₃₁	1	4	yes	complemented
550	#416	C₂²×C₃₁	1	1	yes	characteristic
551	#416	C₂²×C₃₁	1	4	yes	complemented
552	#416	C₂²×C₃₁	1	1	yes	characteristic, complemented
553	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
554	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
555	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
556	#408	C₂²×C₃₁	1	2	yes	complemented
557	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
558	C₂×C₃₁	—	2	2	no	—
559	C₂×C₃₁	—	4	4	no	—
560	C₂×C₃₁	—	4	2	no	—
561	C₂×C₃₁	—	8	1	no	—
562	C₂×C₃₁	—	8	4	no	—
563	C₂×C₃₁	C₂×#546	1	1	yes	characteristic
564	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
565	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
566	H(5,𝔽₂)	—	2	8	no	—
567	H(5,𝔽₂)	—	4	4	no	—
568	C₂×#41	—	2	2	no	—
569	C₂×#41	—	2	2	no	—
570	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
571	C₂×#41	C₂³×C₃₁	1	4	yes	—
572	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
573	C₂×#41	C₂³×C₃₁	1	2	yes	—
574	C₂²×Q₈	—	2	1	no	—
575	C₂²×D₄	—	2	2	no	—
576	C₂²×D₄	—	2	4	no	—
577	C₂²×D₄	—	2	2	no	—
578	C₂²×D₄	—	2	2	no	—
579	C₂²×D₄	—	2	1	no	—
580	C₂²×D₄	—	2	1	no	—
581	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
582	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
583	AGL(1,ℤ/8)	—	4	4	no	—
584	#135	—	2	4	no	—
585	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
586	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
587	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
588	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
589	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
590	C₂×D₈	C₂³×C₃₁	1	4	yes	—
591	#131	—	4	4	no	—
592	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
593	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
594	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
595	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
596	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
597	C₄⋊Q₈	—	2	1	no	—
598	C₄⋊Q₈	—	2	1	no	—
599	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
600	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
601	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
602	#127	—	2	2	no	—
603	#127	—	2	2	no	—
604	#127	—	2	2	no	—
605	#127	—	2	1	no	—
606	#127	—	2	1	no	—
607	#127	D₄×C₃₁	1	1	yes	characteristic
608	#127	C₂³×C₃₁	1	2	yes	—
609	#127	C₂³×C₃₁	1	2	yes	—
610	#126	—	2	4	no	—
611	#125	—	2	2	no	—
612	#124	—	2	2	no	—
613	#124	—	2	2	no	—
614	#124	—	2	4	no	—
615	#124	D₄×C₃₁	1	2	yes	—
616	#123	—	2	4	no	—
617	C₂²⋊Q₈	—	2	2	no	—
618	C₂²⋊Q₈	—	2	2	no	—
619	C₂²⋊Q₈	—	2	4	no	—
620	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
621	#121	—	2	2	no	—
622	#121	—	2	2	no	—
623	#121	—	2	4	no	—
624	#121	—	2	2	no	—
625	#121	—	2	2	no	—
626	#121	—	2	4	no	—
627	#121	—	2	4	no	—
628	#121	D₄×C₃₁	1	2	yes	—
629	#121	D₄×C₃₁	1	2	yes	—
630	#121	C₂³×C₃₁	1	4	yes	—
631	#121	C₂³×C₃₁	1	4	yes	—
632	#121	C₂³×C₃₁	1	4	yes	—
633	#120	—	2	4	no	—
634	#120	—	2	4	no	—
635	#120	—	2	4	no	—
636	C₄×Q₈	—	2	2	no	—
637	C₄×Q₈	—	2	2	no	—
638	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
639	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
640	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
641	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
642	C₄×D₄	—	2	2	no	—
643	C₄×D₄	—	2	4	no	—
644	C₄×D₄	—	2	2	no	—
645	C₄×D₄	—	2	1	no	—
646	C₄×D₄	—	2	1	no	—
647	C₄×D₄	—	2	1	no	—
648	C₄×D₄	—	2	1	no	—
649	C₄×D₄	—	2	2	no	—
650	C₄×D₄	—	2	2	no	—
651	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
652	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
653	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
654	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
655	C₄×D₄	C₂³×C₃₁	1	2	yes	—
656	C₄×D₄	C₂³×C₃₁	1	2	yes	—
657	#117	—	2	2	no	—
658	#117	—	2	1	no	—
659	#117	—	2	1	no	—
660	#117	C₂³×C₃₁	1	1	yes	characteristic
661	#117	C₂³×C₃₁	1	1	yes	characteristic
662	#117	C₂³×C₃₁	1	4	yes	—
663	C₂×C₄⋊C₄	—	2	1	no	—
664	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
665	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
666	C₂×C₄²	—	2	1	no	—
667	C₂×C₄²	—	2	1	no	—
668	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
669	#107	C₂³×C₃₁	1	1	yes	characteristic
670	#107	C₂³×C₃₁	1	1	yes	characteristic
671	#107	C₂³×C₃₁	1	4	yes	—
672	#106	C₂³×C₃₁	1	1	yes	characteristic
673	#106	C₂³×C₃₁	1	1	yes	characteristic
674	#106	C₂³×C₃₁	1	4	yes	—
675	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
676	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
677	#103	C₂³×C₃₁	1	1	yes	characteristic
678	#103	C₂³×C₃₁	1	1	yes	characteristic
679	#103	C₂³×C₃₁	1	4	yes	—
680	#103	C₂³×C₃₁	1	1	yes	characteristic
681	#103	C₂³×C₃₁	1	1	yes	characteristic
682	#102	C₂³×C₃₁	1	1	yes	characteristic
683	#102	C₂³×C₃₁	1	4	yes	—
684	#102	C₂³×C₃₁	1	4	yes	—
685	#102	C₂³×C₃₁	1	4	yes	—
686	#102	C₂³×C₃₁	1	4	yes	—
687	#102	C₂³×C₃₁	1	4	yes	—
688	#102	C₂³×C₃₁	1	4	yes	—
689	#102	C₂³×C₃₁	1	1	yes	characteristic
690	#102	C₂³×C₃₁	1	4	yes	—
691	#102	C₂³×C₃₁	1	1	yes	characteristic
692	#102	C₂³×C₃₁	1	1	yes	characteristic
693	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
694	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
695	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
696	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
697	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
698	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
699	C₃₁	#61215	1	1	yes	characteristic, direct product factor, Sylow
700	C₂⁴	—	2	4	no	—
701	C₂⁴	—	2	2	no	—
702	#41	—	2	4	no	—
703	#41	—	2	4	no	—
704	#41	—	2	2	no	—
705	#41	—	2	8	no	—
706	#41	—	2	2	no	—
707	#41	—	2	2	no	—
708	#41	—	4	4	no	—
709	#41	—	4	8	no	—
710	#41	—	8	4	no	—
711	#41	#41×C₃₁	1	4	yes	complemented
712	C₂×Q₈	—	2	2	no	—
713	C₂×Q₈	—	2	2	no	—
714	C₂×Q₈	—	8	2	no	—
715	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
716	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
717	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
718	C₂×D₄	—	2	1	no	—
719	C₂×D₄	—	2	8	no	—
720	C₂×D₄	—	2	1	no	—
721	C₂×D₄	—	2	4	no	—
722	C₂×D₄	—	2	4	no	—
723	C₂×D₄	—	2	4	no	—
724	C₂×D₄	—	2	4	no	—
725	C₂×D₄	—	2	2	no	—
726	C₂×D₄	—	2	2	no	—
727	C₂×D₄	—	2	4	no	—
728	C₂×D₄	—	2	2	no	—
729	C₂×D₄	—	2	2	no	—
730	C₂×D₄	—	2	2	no	—
731	C₂×D₄	—	2	4	no	—
732	C₂×D₄	—	2	4	no	—
733	C₂×D₄	—	4	4	no	—
734	C₂×D₄	—	4	8	no	—
735	C₂×D₄	—	4	16	no	—
736	C₂×D₄	—	8	2	no	—
737	C₂×D₄	—	8	2	no	—
738	C₂×D₄	—	8	4	no	—
739	C₂×D₄	—	8	4	no	—
740	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
741	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
742	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
743	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
744	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
745	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
746	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
747	C₂²×C₄	—	2	1	no	—
748	C₂²×C₄	—	2	2	no	—
749	C₂²×C₄	—	2	4	no	—
750	C₂²×C₄	—	2	1	no	—
751	C₂²×C₄	—	2	1	no	—
752	C₂²×C₄	—	2	1	no	—
753	C₂²×C₄	—	2	1	no	—
754	C₂²×C₄	—	2	1	no	—
755	C₂²×C₄	—	2	4	no	—
756	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
757	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
758	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
759	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
760	Q₁₆	—	2	2	no	—
761	SD₁₆	—	2	2	no	—
762	SD₁₆	—	2	2	no	—
763	SD₁₆	—	4	4	no	—
764	D₈	—	2	2	no	—
765	D₈	—	4	4	no	—
766	M₁₆	—	2	4	no	—
767	M₁₆	—	4	4	no	—
768	C₂×C₈	—	2	2	no	—
769	C₂×C₈	—	2	2	no	—
770	C₂×C₈	—	4	4	no	—
771	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
772	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
773	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
774	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
775	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
776	C₄⋊C₄	—	2	2	no	—
777	C₄⋊C₄	—	2	2	no	—
778	C₄⋊C₄	—	2	2	no	—
779	C₄⋊C₄	—	2	4	no	—
780	C₄⋊C₄	—	2	1	no	—
781	C₄⋊C₄	—	2	1	no	—
782	C₄⋊C₄	—	2	1	no	—
783	C₄⋊C₄	—	2	1	no	—
784	C₄⋊C₄	—	2	4	no	—
785	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
786	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
787	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
788	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
789	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
790	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
791	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
792	C₂²⋊C₄	—	2	1	no	—
793	C₂²⋊C₄	—	2	1	no	—
794	C₂²⋊C₄	—	2	4	no	—
795	C₂²⋊C₄	—	2	4	no	—
796	C₂²⋊C₄	—	2	1	no	—
797	C₂²⋊C₄	—	2	1	no	—
798	C₂²⋊C₄	—	2	1	no	—
799	C₂²⋊C₄	—	2	1	no	—
800	C₂²⋊C₄	—	2	4	no	—
801	C₂²⋊C₄	—	2	4	no	—
802	C₂²⋊C₄	—	2	4	no	—
803	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
804	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
805	C₄²	—	2	1	no	—
806	C₄²	—	2	1	no	—
807	C₄²	—	2	1	no	—
808	C₄²	—	2	1	no	—
809	C₄²	—	2	1	no	—
810	C₄²	—	2	1	no	—
811	C₄²	—	2	4	no	—
812	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
813	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
814	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
815	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
816	C₂³	—	2	1	no	—
817	C₂³	—	2	1	no	—
818	C₂³	—	2	4	no	—
819	C₂³	—	4	8	no	—
820	C₂³	—	4	4	no	—
821	C₂³	—	8	2	no	—
822	C₂³	—	8	2	no	—
823	C₂³	—	8	4	no	—
824	C₂³	—	8	4	no	—
825	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
826	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
827	Q₈	—	2	4	no	—
828	Q₈	—	2	1	no	—
829	Q₈	—	2	1	no	—
830	Q₈	—	2	1	no	—
831	Q₈	—	4	4	no	—
832	Q₈	—	8	2	no	—
833	Q₈	C₂×#41×C₃₁	1	2	yes	—
834	D₄	—	2	4	no	—
835	D₄	—	2	8	no	—
836	D₄	—	2	4	no	—
837	D₄	—	2	4	no	—
838	D₄	—	2	4	no	—
839	D₄	—	2	1	no	—
840	D₄	—	2	4	no	—
841	D₄	—	2	1	no	—
842	D₄	—	2	1	no	—
843	D₄	—	2	4	no	—
844	D₄	—	4	8	no	—
845	D₄	—	4	8	no	—
846	D₄	—	4	2	no	—
847	D₄	—	4	4	no	—
848	D₄	—	4	4	no	—
849	D₄	—	4	4	no	—
850	D₄	—	8	2	no	—
851	D₄	—	8	2	no	—
852	D₄	—	8	4	no	—
853	D₄	—	8	4	no	—
854	D₄	—	8	4	no	—
855	D₄	C₂×#41×C₃₁	1	2	yes	—
856	C₂×C₄	—	2	1	no	—
857	C₂×C₄	—	2	4	no	—
858	C₂×C₄	—	2	2	no	—
859	C₂×C₄	—	2	2	no	—
860	C₂×C₄	—	2	4	no	—
861	C₂×C₄	—	2	4	no	—
862	C₂×C₄	—	2	1	no	—
863	C₂×C₄	—	2	1	no	—
864	C₂×C₄	—	2	1	no	—
865	C₂×C₄	—	2	1	no	—
866	C₂×C₄	—	2	1	no	—
867	C₂×C₄	—	2	4	no	—
868	C₂×C₄	—	4	8	no	—
869	C₂×C₄	—	4	2	no	—
870	C₂×C₄	—	4	2	no	—
871	C₂×C₄	—	4	2	no	—
872	C₂×C₄	—	4	2	no	—
873	C₂×C₄	—	4	2	no	—
874	C₂×C₄	—	8	2	no	—
875	C₂×C₄	—	8	4	no	—
876	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
877	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
878	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
879	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
880	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
881	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
882	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
883	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
884	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
885	C₈	—	2	1	no	—
886	C₈	—	2	1	no	—
887	C₈	—	2	1	no	—
888	C₈	—	2	1	no	—
889	C₈	—	2	4	no	—
890	C₂²	—	2	1	no	—
891	C₂²	—	2	1	no	—
892	C₂²	—	2	4	no	—
893	C₂²	—	2	2	no	—
894	C₂²	—	2	4	no	—
895	C₂²	—	2	4	no	—
896	C₂²	—	4	1	no	—
897	C₂²	—	4	1	no	—
898	C₂²	—	4	4	no	—
899	C₂²	—	4	1	no	—
900	C₂²	—	4	1	no	—
901	C₂²	—	4	4	no	—
902	C₂²	—	4	1	no	—
903	C₂²	—	4	4	no	—
904	C₂²	—	16	4	no	—
905	C₂²	—	16	8	no	—
906	C₂²	C₃₁×#546	1	2	yes	—
907	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
908	C₄	—	2	2	no	—
909	C₄	—	2	4	no	—
910	C₄	—	2	1	no	—
911	C₄	—	2	1	no	—
912	C₄	—	4	1	no	—
913	C₄	—	4	1	no	—
914	C₄	—	4	1	no	—
915	C₄	—	4	4	no	—
916	C₄	—	4	1	no	—
917	C₄	—	4	1	no	—
918	C₄	C₃₁×#546	1	2	yes	—
919	C₄	C₃₁×#576	1	2	yes	—
920	C₄	C₃₁×#577	1	2	yes	—
921	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
922	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
923	C₂	—	2	2	no	—
924	C₂	—	4	4	no	—
925	C₂	—	4	2	no	—
926	C₂	—	8	4	no	—
927	C₂	—	8	1	no	—
928	C₂	C₂×C₃₁×#546	1	1	yes	characteristic
929	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
930	C₂	C₂×C₃₁×#577	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×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3511	C₂	semidirect
2. C₃₁×#3498	C₂	semidirect
3. C₃₁×#3333	C₂	semidirect
4. C₃₁×#3332	C₂	semidirect
5. C₃₁×#3325	C₂	semidirect
6. C₃₁×#3324	C₂	semidirect
7. C₃₁×#3305	C₂	semidirect
8. C₃₁×#3304	C₂	semidirect
9. C₃₁×#3295	C₂	semidirect
10. C₃₁×#3294	C₂	semidirect
11. C₂×C₃₁×#486	C₂	semidirect
12. C₃₁×#3096	C₂	semidirect
13. C₃₁×#3079	C₂	semidirect
14. C₃₁×#3062	C₂	semidirect
15. C₃₁×#3038	C₂	semidirect
16. C₃₁×#3012	C₂	semidirect
17. C₄×C₃₁×#135	C₂	semidirect
18. C₃₁×#2931	C₂	semidirect
19. C₃₁×#2897	C₂	semidirect
21. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
23. C₂×C₃₁×#131	C₂²	semidirect
24. C₃₁×#556	C₂²	semidirect
25. C₃₁×#555	C₂²	semidirect
26. C₃₁×#554	C₂²	semidirect
31. C₃₁×#547	C₂²	semidirect
37. C₃₁×#537	C₂²	semidirect
44. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
47. C₃₁×#520	C₂²	semidirect
49. C₄×#41×C₃₁	C₂²	semidirect
50. C₃₁×#486	C₂²	semidirect
51. C₃₁×#486	C₂²	semidirect
52. C₃₁×#486	C₂²	semidirect
53. C₃₁×#486	C₂²	semidirect
54. C₃₁×#482	C₂²	semidirect
55. C₃₁×#482	C₂²	semidirect
56. C₃₁×#481	C₂²	semidirect
57. C₃₁×#480	C₂²	semidirect
58. C₃₁×#477	C₂²	semidirect
59. C₃₁×#476	C₂²	semidirect
60. C₃₁×#475	C₂²	semidirect
61. C₃₁×#474	C₂²	semidirect
62. C₃₁×#469	C₂²	semidirect
63. C₃₁×#468	C₂²	semidirect
64. C₃₁×#466	C₂²	semidirect
65. C₃₁×#465	C₂²	semidirect
66. C₃₁×#460	C₂²	semidirect
67. C₃₁×#460	C₂²	semidirect
68. C₃₁×#459	C₂²	semidirect
69. C₃₁×#459	C₂²	semidirect
70. C₃₁×#452	C₂²	semidirect
71. C₃₁×#450	C₂²	semidirect
72. C₃₁×#449	C₂²	semidirect
73. C₃₁×#447	C₂²	semidirect
74. C₄×Q₁₆×C₃₁	C₂²	semidirect
75. C₄×Q₁₆×C₃₁	C₂²	semidirect
76. C₄×SD₁₆×C₃₁	C₂²	semidirect
77. C₄×SD₁₆×C₃₁	C₂²	semidirect
78. C₄×SD₁₆×C₃₁	C₂²	semidirect
79. C₄×SD₁₆×C₃₁	C₂²	semidirect
80. C₄×D₈×C₃₁	C₂²	semidirect
81. C₄×D₈×C₃₁	C₂²	semidirect
82. C₃₁×#436	C₂²	semidirect
83. C₃₁×#436	C₂²	semidirect
84. C₈×D₄×C₃₁	C₂²	semidirect
85. C₈×D₄×C₃₁	C₂²	semidirect
86. C₃₁×#428	C₂²	semidirect
87. C₃₁×#427	C₂²	semidirect
88. C₃₁×#423	C₂²	semidirect
89. C₃₁×#418	C₂²	semidirect
90. C₃₁×#418	C₂²	semidirect
92. C₃₁×#417	C₂²	semidirect
94. C₃₁×#416	C₂²	semidirect
95. C₃₁×#416	C₂²	semidirect
96. C₂×C₃₁×#102	C₂²	semidirect
97. C₂×C₃₁×#102	C₂²	semidirect
98. C₂×C₃₁×#102	C₂²	semidirect
99. C₃₁×#408	C₂²	semidirect
100. C₂×C₄×C₈×C₃₁	C₂²	semidirect
245. #41×C₃₁	#41	semidirect
350. #61215	C₃₁	direct (N × H)
425. #3511	C₂×C₃₁	semidirect
426. #3498	C₂×C₃₁	semidirect
427. #3333	C₂×C₃₁	semidirect
428. #3332	C₂×C₃₁	semidirect
429. #3325	C₂×C₃₁	semidirect
430. #3324	C₂×C₃₁	semidirect
431. #3305	C₂×C₃₁	semidirect
432. #3304	C₂×C₃₁	semidirect
433. #3295	C₂×C₃₁	semidirect
434. #3294	C₂×C₃₁	semidirect
435. C₂×#486	C₂×C₃₁	semidirect
436. #3096	C₂×C₃₁	semidirect
437. #3079	C₂×C₃₁	semidirect
438. #3062	C₂×C₃₁	semidirect
439. #3038	C₂×C₃₁	semidirect
440. #3012	C₂×C₃₁	semidirect
441. C₄×#135	C₂×C₃₁	semidirect
442. #2931	C₂×C₃₁	semidirect
443. #2897	C₂×C₃₁	semidirect
478. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
480. C₂×#131	C₂²×C₃₁	semidirect
481. #556	C₂²×C₃₁	semidirect
482. #555	C₂²×C₃₁	semidirect
483. #554	C₂²×C₃₁	semidirect
488. #547	C₂²×C₃₁	semidirect
494. #537	C₂²×C₃₁	semidirect
501. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
504. #520	C₂²×C₃₁	semidirect
506. C₄×#41	C₂²×C₃₁	semidirect
507. #486	C₂²×C₃₁	semidirect
508. #486	C₂²×C₃₁	semidirect
509. #486	C₂²×C₃₁	semidirect
510. #486	C₂²×C₃₁	semidirect
511. #482	C₂²×C₃₁	semidirect
512. #482	C₂²×C₃₁	semidirect
513. #481	C₂²×C₃₁	semidirect
514. #480	C₂²×C₃₁	semidirect
515. #477	C₂²×C₃₁	semidirect
516. #476	C₂²×C₃₁	semidirect
517. #475	C₂²×C₃₁	semidirect
518. #474	C₂²×C₃₁	semidirect
519. #469	C₂²×C₃₁	semidirect
520. #468	C₂²×C₃₁	semidirect
521. #466	C₂²×C₃₁	semidirect
522. #465	C₂²×C₃₁	semidirect
523. #460	C₂²×C₃₁	semidirect
524. #460	C₂²×C₃₁	semidirect
525. #459	C₂²×C₃₁	semidirect
526. #459	C₂²×C₃₁	semidirect
527. #452	C₂²×C₃₁	semidirect
528. #450	C₂²×C₃₁	semidirect
529. #449	C₂²×C₃₁	semidirect
530. #447	C₂²×C₃₁	semidirect
531. C₄×Q₁₆	C₂²×C₃₁	semidirect
532. C₄×Q₁₆	C₂²×C₃₁	semidirect
533. C₄×SD₁₆	C₂²×C₃₁	semidirect
534. C₄×SD₁₆	C₂²×C₃₁	semidirect
535. C₄×SD₁₆	C₂²×C₃₁	semidirect
536. C₄×SD₁₆	C₂²×C₃₁	semidirect
537. C₄×D₈	C₂²×C₃₁	semidirect
538. C₄×D₈	C₂²×C₃₁	semidirect
539. #436	C₂²×C₃₁	semidirect
540. #436	C₂²×C₃₁	semidirect
541. C₈×D₄	C₂²×C₃₁	semidirect
542. C₈×D₄	C₂²×C₃₁	semidirect
543. #428	C₂²×C₃₁	semidirect
544. #427	C₂²×C₃₁	semidirect
545. #423	C₂²×C₃₁	semidirect
546. #418	C₂²×C₃₁	semidirect
547. #418	C₂²×C₃₁	semidirect
549. #417	C₂²×C₃₁	semidirect
551. #416	C₂²×C₃₁	semidirect
552. #416	C₂²×C₃₁	semidirect
553. C₂×#102	C₂²×C₃₁	semidirect
554. C₂×#102	C₂²×C₃₁	semidirect
555. C₂×#102	C₂²×C₃₁	semidirect
556. #408	C₂²×C₃₁	semidirect
557. C₂×C₄×C₈	C₂²×C₃₁	semidirect
699. C₃₁	#61215	direct (N × H)
711. #41	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3511	2
C₃₁×#3498	2
C₃₁×#3333	2
C₃₁×#3332	2
C₃₁×#3325	2
C₃₁×#3324	2
C₃₁×#3305	2
C₃₁×#3304	2
C₃₁×#3295	2
C₃₁×#3294	2
C₂×C₃₁×#486	2
C₃₁×#3096	2
C₃₁×#3079	2
C₃₁×#3062	2
C₃₁×#3038	2
C₃₁×#3012	2
C₄×C₃₁×#135	2
C₃₁×#2931	2
C₃₁×#2897	2
#61215	31

Subgroups — C31×#61215 #21517974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61215 #21517974