Subgroups — C₃₁×#61166 #21517925

Order 7936.

2182 conjugacy classes of subgroups.

The subgroup lattice has 978 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3329	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3325	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3321	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3319	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3305	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3303	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3093	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3055	C₂	1	2	yes	maximal, complemented
13	C₃₁×#3036	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#3006	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
16	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#2896	C₂	1	2	yes	maximal, complemented
18	C₂²×#41×C₃₁	C₂²	1	2	yes	—
19	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
21	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	complemented
22	C₃₁×#555	—	2	1	no	—
23	C₃₁×#555	—	2	1	no	—
24	C₃₁×#555	—	2	2	no	—
25	C₃₁×#555	C₂²	1	2	yes	—
26	C₃₁×#555	C₂²	1	2	yes	complemented
27	C₃₁×#555	C₂²	1	4	yes	complemented
28	C₃₁×#550	C₂²	1	1	yes	characteristic, complemented
29	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#548	C₂²	1	2	yes	complemented
31	C₃₁×#547	—	2	1	no	—
32	C₃₁×#547	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#546	—	2	2	no	—
34	C₃₁×#546	C₂²	1	2	yes	complemented
35	D₄²×C₃₁	—	2	1	no	—
36	D₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#532	—	2	1	no	—
38	C₃₁×#532	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₃₁×#124	—	2	2	no	—
41	C₃₁×#525	—	2	2	no	—
42	C₃₁×#525	C₂²	1	2	yes	—
43	C₂×C₃₁×#121	—	2	1	no	—
44	C₂×C₃₁×#121	C₂²	1	2	yes	complemented
45	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
46	C₂×C₄×D₄×C₃₁	—	2	1	no	—
47	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
48	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₃₁×#117	—	2	1	no	—
50	C₂×C₃₁×#117	C₂²	1	2	yes	complemented
51	C₃₁×#489	C₂²	1	4	yes	complemented
52	C₃₁×#487	C₂²	1	2	yes	complemented
53	C₃₁×#486	C₂²	1	2	yes	complemented
54	C₃₁×#485	C₂²	1	4	yes	complemented
55	C₃₁×#485	C₂²	1	4	yes	complemented
56	C₃₁×#482	C₂²	1	4	yes	complemented
57	C₃₁×#482	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×#463	C₂²	1	4	yes	complemented
67	C₃₁×#461	C₂²	1	2	yes	complemented
68	C₃₁×#459	C₂²	1	2	yes	complemented
69	C₃₁×#449	C₂²	1	4	yes	complemented
70	C₃₁×#449	C₂²	1	4	yes	complemented
71	C₃₁×#448	C₂²	1	4	yes	complemented
72	C₃₁×#447	C₂²	1	4	yes	complemented
73	C₃₁×#442	C₂²	1	2	yes	complemented
74	C₃₁×#440	C₂²	1	2	yes	complemented
75	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
76	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
77	C₃₁×#436	C₂²	1	2	yes	complemented
78	C₃₁×#435	C₂²	1	4	yes	complemented
79	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
80	C₃₁×#428	C₂²	1	2	yes	complemented
81	C₃₁×#428	C₂²	1	1	yes	characteristic, complemented
82	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
83	C₃₁×#419	C₂²	1	4	yes	complemented
84	C₃₁×#418	C₂²	1	4	yes	complemented
85	C₃₁×#418	C₂²	1	1	yes	characteristic, complemented
86	C₃₁×#417	C₂²	1	2	yes	—
87	C₃₁×#417	C₂²	1	1	yes	characteristic
88	C₃₁×#417	C₂²	1	2	yes	complemented
89	C₃₁×#417	C₂²	1	2	yes	complemented
90	C₃₁×#417	C₂²	1	2	yes	complemented
91	C₃₁×#416	C₂²	1	4	yes	complemented
92	C₃₁×#416	C₂²	1	4	yes	complemented
93	C₃₁×#407	C₂²	1	2	yes	complemented
94	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
95	C₂×#41×C₃₁	—	2	2	no	—
96	C₂×#41×C₃₁	—	2	2	no	—
97	C₂×#41×C₃₁	—	2	8	no	—
98	C₂×#41×C₃₁	—	4	4	no	—
99	C₂×#41×C₃₁	C₂³	1	2	yes	—
100	C₂×#41×C₃₁	C₂³	1	2	yes	—
101	C₂×#41×C₃₁	C₂³	1	2	yes	—
102	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
103	C₂²×Q₈×C₃₁	—	2	1	no	—
104	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
105	C₂²×D₄×C₃₁	—	2	2	no	—
106	C₂²×D₄×C₃₁	—	2	2	no	—
107	C₂²×D₄×C₃₁	—	2	1	no	—
108	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
109	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
111	C₂³×C₄×C₃₁	—	2	1	no	—
112	C₂³×C₄×C₃₁	—	2	2	no	—
113	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
114	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
115	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
116	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
117	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
118	C₂×D₈×C₃₁	C₂³	1	2	yes	—
119	C₂×D₈×C₃₁	C₂³	1	4	yes	—
120	C₂×M₁₆×C₃₁	—	4	4	no	—
121	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
124	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
125	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
126	C₃₁×C₄⋊Q₈	—	2	1	no	—
127	C₃₁×C₄⋊Q₈	—	2	1	no	—
128	C₃₁×C₄⋊Q₈	—	2	1	no	—
129	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
130	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
131	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
132	C₃₁×#127	—	2	1	no	—
133	C₃₁×#127	C₂³	1	1	yes	characteristic
134	C₃₁×#124	—	2	4	no	—
135	C₃₁×#124	—	2	4	no	—
136	C₃₁×#124	—	2	4	no	—
137	C₃₁×#124	—	2	4	no	—
138	C₃₁×#124	—	2	2	no	—
139	C₃₁×#124	C₂³	1	2	yes	—
140	C₃₁×#123	—	2	2	no	—
141	C₃₁×#123	—	2	2	no	—
142	C₃₁×#123	—	2	2	no	—
143	C₃₁×C₂²⋊Q₈	—	2	2	no	—
144	C₃₁×C₂²⋊Q₈	—	2	2	no	—
145	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
146	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
147	C₃₁×#121	—	2	2	no	—
148	C₃₁×#121	—	2	2	no	—
149	C₃₁×#121	—	2	4	no	—
150	C₃₁×#121	—	2	2	no	—
151	C₃₁×#121	—	2	1	no	—
152	C₃₁×#121	—	2	2	no	—
153	C₃₁×#121	—	2	2	no	—
154	C₃₁×#121	—	2	1	no	—
155	C₃₁×#121	—	2	1	no	—
156	C₃₁×#121	—	2	1	no	—
157	C₃₁×#121	—	2	2	no	—
158	C₃₁×#121	—	2	2	no	—
159	C₃₁×#121	C₂³	1	4	yes	—
160	C₃₁×#121	C₂³	1	4	yes	—
161	C₃₁×#121	C₂³	1	2	yes	—
162	C₃₁×#121	C₂³	1	2	yes	—
163	C₃₁×#120	—	2	2	no	—
164	C₃₁×#120	—	2	2	no	—
165	C₃₁×#120	—	2	2	no	—
166	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
167	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
168	C₄×D₄×C₃₁	—	2	1	no	—
169	C₄×D₄×C₃₁	—	2	2	no	—
170	C₄×D₄×C₃₁	—	2	1	no	—
171	C₄×D₄×C₃₁	—	2	1	no	—
172	C₄×D₄×C₃₁	—	2	2	no	—
173	C₄×D₄×C₃₁	—	2	1	no	—
174	C₄×D₄×C₃₁	—	2	2	no	—
175	C₄×D₄×C₃₁	—	2	1	no	—
176	C₄×D₄×C₃₁	—	2	1	no	—
177	C₄×D₄×C₃₁	—	2	1	no	—
178	C₄×D₄×C₃₁	—	2	1	no	—
179	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
180	C₄×D₄×C₃₁	D₄	1	2	yes	—
181	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
182	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
183	C₄×D₄×C₃₁	C₂³	1	2	yes	—
184	C₄×D₄×C₃₁	C₂³	1	2	yes	—
185	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
186	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
187	C₄×D₄×C₃₁	C₂³	1	2	yes	—
188	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
189	C₃₁×#117	—	2	2	no	—
190	C₃₁×#117	—	2	2	no	—
191	C₃₁×#117	—	2	2	no	—
192	C₃₁×#117	—	2	2	no	—
193	C₃₁×#117	—	2	4	no	—
194	C₃₁×#117	—	2	1	no	—
195	C₃₁×#117	C₂³	1	4	yes	—
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	—	2	1	no	—
200	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
201	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
202	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
203	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
206	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
208	C₂×C₄²×C₃₁	—	2	2	no	—
209	C₂×C₄²×C₃₁	—	2	1	no	—
210	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
211	C₃₁×#107	C₂³	1	2	yes	—
212	C₃₁×#107	C₂³	1	4	yes	—
213	C₃₁×#106	C₂³	1	2	yes	—
214	C₃₁×#106	C₂³	1	4	yes	—
215	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
216	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
217	C₃₁×#103	C₂³	1	2	yes	—
218	C₃₁×#103	C₂³	1	4	yes	—
219	C₃₁×#103	C₂³	1	4	yes	—
220	C₃₁×#103	C₂³	1	4	yes	—
221	C₃₁×#103	C₂³	1	4	yes	—
222	C₃₁×#102	C₂³	1	2	yes	—
223	C₃₁×#102	C₂³	1	4	yes	—
224	C₃₁×#102	C₂³	1	4	yes	—
225	C₃₁×#102	C₂³	1	4	yes	—
226	C₃₁×#102	C₂³	1	4	yes	—
227	C₃₁×#102	C₂³	1	2	yes	—
228	C₃₁×#102	C₂³	1	2	yes	—
229	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
230	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
231	C₃₁×#97	C₂³	1	2	yes	—
232	C₄×C₈×C₃₁	C₂³	1	2	yes	—
233	C₂⁴×C₃₁	—	2	1	no	—
234	C₂⁴×C₃₁	—	2	2	no	—
235	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
236	#41×C₃₁	—	2	2	no	—
237	#41×C₃₁	—	2	4	no	—
238	#41×C₃₁	—	2	4	no	—
239	#41×C₃₁	—	2	4	no	—
240	#41×C₃₁	—	2	4	no	—
241	#41×C₃₁	—	4	8	no	—
242	#41×C₃₁	—	4	2	no	—
243	#41×C₃₁	—	4	2	no	—
244	#41×C₃₁	—	4	8	no	—
245	#41×C₃₁	#41	1	4	yes	—
246	C₂×Q₈×C₃₁	—	2	2	no	—
247	C₂×Q₈×C₃₁	—	2	2	no	—
248	C₂×Q₈×C₃₁	—	2	2	no	—
249	C₂×Q₈×C₃₁	—	2	4	no	—
250	C₂×Q₈×C₃₁	—	4	2	no	—
251	C₂×Q₈×C₃₁	—	8	2	no	—
252	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
253	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
254	C₂×D₄×C₃₁	—	2	1	no	—
255	C₂×D₄×C₃₁	—	2	4	no	—
256	C₂×D₄×C₃₁	—	2	2	no	—
257	C₂×D₄×C₃₁	—	2	2	no	—
258	C₂×D₄×C₃₁	—	2	2	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	4	no	—
263	C₂×D₄×C₃₁	—	2	1	no	—
264	C₂×D₄×C₃₁	—	2	1	no	—
265	C₂×D₄×C₃₁	—	2	2	no	—
266	C₂×D₄×C₃₁	—	2	1	no	—
267	C₂×D₄×C₃₁	—	2	2	no	—
268	C₂×D₄×C₃₁	—	2	2	no	—
269	C₂×D₄×C₃₁	—	4	2	no	—
270	C₂×D₄×C₃₁	—	4	2	no	—
271	C₂×D₄×C₃₁	—	4	8	no	—
272	C₂×D₄×C₃₁	—	4	4	no	—
273	C₂×D₄×C₃₁	—	8	4	no	—
274	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
275	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
276	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
277	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
278	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
279	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
280	C₂²×C₄×C₃₁	—	2	1	no	—
281	C₂²×C₄×C₃₁	—	2	4	no	—
282	C₂²×C₄×C₃₁	—	2	4	no	—
283	C₂²×C₄×C₃₁	—	2	1	no	—
284	C₂²×C₄×C₃₁	—	2	2	no	—
285	C₂²×C₄×C₃₁	—	2	2	no	—
286	C₂²×C₄×C₃₁	—	2	1	no	—
287	C₂²×C₄×C₃₁	—	2	2	no	—
288	C₂²×C₄×C₃₁	—	2	2	no	—
289	C₂²×C₄×C₃₁	—	2	2	no	—
290	C₂²×C₄×C₃₁	—	2	2	no	—
291	C₂²×C₄×C₃₁	—	2	2	no	—
292	C₂²×C₄×C₃₁	—	2	1	no	—
293	C₂²×C₄×C₃₁	—	4	8	no	—
294	C₂²×C₄×C₃₁	—	4	4	no	—
295	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
296	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
297	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
298	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
299	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
300	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
301	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
302	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
303	SD₁₆×C₃₁	—	2	4	no	—
304	SD₁₆×C₃₁	—	4	4	no	—
305	D₈×C₃₁	—	2	4	no	—
306	D₈×C₃₁	—	4	4	no	—
307	M₁₆×C₃₁	—	2	2	no	—
308	M₁₆×C₃₁	—	2	2	no	—
309	M₁₆×C₃₁	—	2	4	no	—
310	M₁₆×C₃₁	—	4	4	no	—
311	C₂×C₈×C₃₁	—	4	4	no	—
312	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
313	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
314	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
315	C₄⋊C₄×C₃₁	—	2	2	no	—
316	C₄⋊C₄×C₃₁	—	2	2	no	—
317	C₄⋊C₄×C₃₁	—	2	1	no	—
318	C₄⋊C₄×C₃₁	—	2	2	no	—
319	C₄⋊C₄×C₃₁	—	2	1	no	—
320	C₄⋊C₄×C₃₁	—	2	2	no	—
321	C₄⋊C₄×C₃₁	—	2	1	no	—
322	C₄⋊C₄×C₃₁	—	2	1	no	—
323	C₄⋊C₄×C₃₁	—	2	1	no	—
324	C₄⋊C₄×C₃₁	—	2	2	no	—
325	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
326	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
327	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
328	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
329	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
330	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
331	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
332	C₂²⋊C₄×C₃₁	—	2	4	no	—
333	C₂²⋊C₄×C₃₁	—	2	4	no	—
334	C₂²⋊C₄×C₃₁	—	2	2	no	—
335	C₂²⋊C₄×C₃₁	—	2	1	no	—
336	C₂²⋊C₄×C₃₁	—	2	2	no	—
337	C₂²⋊C₄×C₃₁	—	2	4	no	—
338	C₂²⋊C₄×C₃₁	—	2	4	no	—
339	C₂²⋊C₄×C₃₁	—	2	2	no	—
340	C₂²⋊C₄×C₃₁	—	2	2	no	—
341	C₂²⋊C₄×C₃₁	—	2	2	no	—
342	C₂²⋊C₄×C₃₁	—	2	2	no	—
343	C₂²⋊C₄×C₃₁	—	2	1	no	—
344	C₂²⋊C₄×C₃₁	—	2	1	no	—
345	C₂²⋊C₄×C₃₁	—	2	2	no	—
346	C₂²⋊C₄×C₃₁	—	2	2	no	—
347	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
348	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
349	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
350	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
351	C₄²×C₃₁	—	2	4	no	—
352	C₄²×C₃₁	—	2	4	no	—
353	C₄²×C₃₁	—	2	1	no	—
354	C₄²×C₃₁	—	2	1	no	—
355	C₄²×C₃₁	—	2	1	no	—
356	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
357	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
358	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
359	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
360	#61166	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
361	C₂³×C₃₁	—	2	1	no	—
362	C₂³×C₃₁	—	2	4	no	—
363	C₂³×C₃₁	—	2	2	no	—
364	C₂³×C₃₁	—	2	2	no	—
365	C₂³×C₃₁	—	4	2	no	—
366	C₂³×C₃₁	—	4	2	no	—
367	C₂³×C₃₁	—	4	4	no	—
368	C₂³×C₃₁	—	4	2	no	—
369	C₂³×C₃₁	—	4	4	no	—
370	C₂³×C₃₁	—	4	2	no	—
371	C₂³×C₃₁	—	4	2	no	—
372	C₂³×C₃₁	—	8	2	no	—
373	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
374	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
375	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
376	Q₈×C₃₁	—	2	1	no	—
377	Q₈×C₃₁	—	2	2	no	—
378	Q₈×C₃₁	—	2	2	no	—
379	Q₈×C₃₁	—	4	4	no	—
380	Q₈×C₃₁	—	8	2	no	—
381	Q₈×C₃₁	—	8	2	no	—
382	Q₈×C₃₁	C₂×#41	1	2	yes	—
383	D₄×C₃₁	—	2	4	no	—
384	D₄×C₃₁	—	2	2	no	—
385	D₄×C₃₁	—	2	2	no	—
386	D₄×C₃₁	—	2	2	no	—
387	D₄×C₃₁	—	2	2	no	—
388	D₄×C₃₁	—	2	1	no	—
389	D₄×C₃₁	—	2	2	no	—
390	D₄×C₃₁	—	2	2	no	—
391	D₄×C₃₁	—	4	2	no	—
392	D₄×C₃₁	—	4	4	no	—
393	D₄×C₃₁	—	4	4	no	—
394	D₄×C₃₁	—	4	4	no	—
395	D₄×C₃₁	—	4	4	no	—
396	D₄×C₃₁	—	4	4	no	—
397	D₄×C₃₁	—	4	2	no	—
398	D₄×C₃₁	—	4	2	no	—
399	D₄×C₃₁	—	8	1	no	—
400	D₄×C₃₁	—	8	2	no	—
401	D₄×C₃₁	—	8	4	no	—
402	D₄×C₃₁	—	8	4	no	—
403	D₄×C₃₁	C₂×#41	1	2	yes	—
404	C₂×C₄×C₃₁	—	2	4	no	—
405	C₂×C₄×C₃₁	—	2	1	no	—
406	C₂×C₄×C₃₁	—	2	4	no	—
407	C₂×C₄×C₃₁	—	2	4	no	—
408	C₂×C₄×C₃₁	—	2	4	no	—
409	C₂×C₄×C₃₁	—	2	2	no	—
410	C₂×C₄×C₃₁	—	2	2	no	—
411	C₂×C₄×C₃₁	—	2	2	no	—
412	C₂×C₄×C₃₁	—	2	2	no	—
413	C₂×C₄×C₃₁	—	2	2	no	—
414	C₂×C₄×C₃₁	—	2	2	no	—
415	C₂×C₄×C₃₁	—	2	1	no	—
416	C₂×C₄×C₃₁	—	2	1	no	—
417	C₂×C₄×C₃₁	—	2	4	no	—
418	C₂×C₄×C₃₁	—	2	2	no	—
419	C₂×C₄×C₃₁	—	4	1	no	—
420	C₂×C₄×C₃₁	—	4	4	no	—
421	C₂×C₄×C₃₁	—	4	4	no	—
422	C₂×C₄×C₃₁	—	4	4	no	—
423	C₂×C₄×C₃₁	—	4	4	no	—
424	C₂×C₄×C₃₁	—	4	2	no	—
425	C₂×C₄×C₃₁	—	4	4	no	—
426	C₂×C₄×C₃₁	—	4	4	no	—
427	C₂×C₄×C₃₁	—	4	4	no	—
428	C₂×C₄×C₃₁	—	4	2	no	—
429	C₂×C₄×C₃₁	—	8	1	no	—
430	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
431	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
432	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
433	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
434	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
435	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
436	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
437	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
438	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
439	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
440	C₈×C₃₁	—	2	2	no	—
441	C₈×C₃₁	—	2	2	no	—
442	C₈×C₃₁	—	2	4	no	—
443	C₂×#555	C₂×C₃₁	1	1	yes	characteristic, complemented
444	D₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
445	#3329	C₂×C₃₁	1	2	yes	complemented
446	#3325	C₂×C₃₁	1	2	yes	complemented
447	#3321	C₂×C₃₁	1	4	yes	complemented
448	#3319	C₂×C₃₁	1	4	yes	complemented
449	#3305	C₂×C₃₁	1	2	yes	complemented
450	#3303	C₂×C₃₁	1	2	yes	complemented
451	#3141	C₂×C₃₁	1	1	yes	characteristic, complemented
452	#3093	C₂×C₃₁	1	2	yes	complemented
453	#3079	C₂×C₃₁	1	1	yes	characteristic, complemented
454	#3055	C₂×C₃₁	1	2	yes	complemented
455	#3036	C₂×C₃₁	1	1	yes	characteristic, complemented
456	#3006	C₂×C₃₁	1	2	yes	complemented
457	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic, complemented
458	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
459	#2896	C₂×C₃₁	1	2	yes	complemented
460	C₂²×C₃₁	—	2	1	no	—
461	C₂²×C₃₁	—	2	1	no	—
462	C₂²×C₃₁	—	2	2	no	—
463	C₂²×C₃₁	—	2	2	no	—
464	C₂²×C₃₁	—	2	2	no	—
465	C₂²×C₃₁	—	2	2	no	—
466	C₂²×C₃₁	—	2	2	no	—
467	C₂²×C₃₁	—	2	2	no	—
468	C₂²×C₃₁	—	2	2	no	—
469	C₂²×C₃₁	—	4	1	no	—
470	C₂²×C₃₁	—	4	1	no	—
471	C₂²×C₃₁	—	4	2	no	—
472	C₂²×C₃₁	—	4	2	no	—
473	C₂²×C₃₁	—	4	2	no	—
474	C₂²×C₃₁	—	4	2	no	—
475	C₂²×C₃₁	—	4	2	no	—
476	C₂²×C₃₁	—	4	2	no	—
477	C₂²×C₃₁	—	8	4	no	—
478	C₂²×C₃₁	—	8	4	no	—
479	C₂²×C₃₁	—	8	8	no	—
480	C₂²×C₃₁	#546	1	2	yes	—
481	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
482	C₄×C₃₁	—	2	1	no	—
483	C₄×C₃₁	—	2	1	no	—
484	C₄×C₃₁	—	2	2	no	—
485	C₄×C₃₁	—	2	2	no	—
486	C₄×C₃₁	—	2	2	no	—
487	C₄×C₃₁	—	2	1	no	—
488	C₄×C₃₁	—	4	4	no	—
489	C₄×C₃₁	—	4	1	no	—
490	C₄×C₃₁	—	4	2	no	—
491	C₄×C₃₁	—	4	2	no	—
492	C₄×C₃₁	—	4	1	no	—
493	C₄×C₃₁	#546	1	2	yes	—
494	C₄×C₃₁	#575	1	2	yes	—
495	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
496	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
497	C₂²×#41	C₂²×C₃₁	1	2	yes	—
498	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
499	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	complemented
500	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	complemented
501	#555	—	2	1	no	—
502	#555	—	2	1	no	—
503	#555	—	2	2	no	—
504	#555	C₂²×C₃₁	1	2	yes	—
505	#555	C₂²×C₃₁	1	2	yes	complemented
506	#555	C₂²×C₃₁	1	4	yes	complemented
507	#550	C₂²×C₃₁	1	1	yes	characteristic, complemented
508	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
509	#548	C₂²×C₃₁	1	2	yes	complemented
510	#547	—	2	1	no	—
511	#547	C₂²×C₃₁	1	1	yes	characteristic, complemented
512	#546	—	2	2	no	—
513	#546	C₂²×C₃₁	1	2	yes	complemented
514	D₄²	—	2	1	no	—
515	D₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
516	#532	—	2	1	no	—
517	#532	C₂²×C₃₁	1	1	yes	characteristic
518	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
519	C₂×#124	—	2	2	no	—
520	#525	—	2	2	no	—
521	#525	C₂²×C₃₁	1	2	yes	—
522	C₂×#121	—	2	1	no	—
523	C₂×#121	C₂²×C₃₁	1	2	yes	complemented
524	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
525	C₂×C₄×D₄	—	2	1	no	—
526	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
527	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
528	C₂×#117	—	2	1	no	—
529	C₂×#117	C₂²×C₃₁	1	2	yes	complemented
530	#489	C₂²×C₃₁	1	4	yes	complemented
531	#487	C₂²×C₃₁	1	2	yes	complemented
532	#486	C₂²×C₃₁	1	2	yes	complemented
533	#485	C₂²×C₃₁	1	4	yes	complemented
534	#485	C₂²×C₃₁	1	4	yes	complemented
535	#482	C₂²×C₃₁	1	4	yes	complemented
536	#482	C₂²×C₃₁	1	4	yes	complemented
537	#477	C₂²×C₃₁	1	2	yes	complemented
538	#476	C₂²×C₃₁	1	2	yes	complemented
539	#475	C₂²×C₃₁	1	2	yes	complemented
540	#474	C₂²×C₃₁	1	2	yes	complemented
541	#469	C₂²×C₃₁	1	4	yes	complemented
542	#468	C₂²×C₃₁	1	4	yes	complemented
543	#466	C₂²×C₃₁	1	4	yes	complemented
544	#465	C₂²×C₃₁	1	4	yes	complemented
545	#463	C₂²×C₃₁	1	4	yes	complemented
546	#461	C₂²×C₃₁	1	2	yes	complemented
547	#459	C₂²×C₃₁	1	2	yes	complemented
548	#449	C₂²×C₃₁	1	4	yes	complemented
549	#449	C₂²×C₃₁	1	4	yes	complemented
550	#448	C₂²×C₃₁	1	4	yes	complemented
551	#447	C₂²×C₃₁	1	4	yes	complemented
552	#442	C₂²×C₃₁	1	2	yes	complemented
553	#440	C₂²×C₃₁	1	2	yes	complemented
554	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
555	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
556	#436	C₂²×C₃₁	1	2	yes	complemented
557	#435	C₂²×C₃₁	1	4	yes	complemented
558	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
559	#428	C₂²×C₃₁	1	2	yes	complemented
560	#428	C₂²×C₃₁	1	1	yes	characteristic, complemented
561	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
562	#419	C₂²×C₃₁	1	4	yes	complemented
563	#418	C₂²×C₃₁	1	4	yes	complemented
564	#418	C₂²×C₃₁	1	1	yes	characteristic, complemented
565	#417	C₂²×C₃₁	1	2	yes	—
566	#417	C₂²×C₃₁	1	1	yes	characteristic
567	#417	C₂²×C₃₁	1	2	yes	complemented
568	#417	C₂²×C₃₁	1	2	yes	complemented
569	#417	C₂²×C₃₁	1	2	yes	complemented
570	#416	C₂²×C₃₁	1	4	yes	complemented
571	#416	C₂²×C₃₁	1	4	yes	complemented
572	#407	C₂²×C₃₁	1	2	yes	complemented
573	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
574	C₂×C₃₁	—	2	2	no	—
575	C₂×C₃₁	—	4	2	no	—
576	C₂×C₃₁	—	4	2	no	—
577	C₂×C₃₁	—	4	2	no	—
578	C₂×C₃₁	—	8	2	no	—
579	C₂×C₃₁	—	8	2	no	—
580	C₂×C₃₁	C₂×#546	1	1	yes	characteristic
581	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
582	C₂×C₃₁	#3585	1	1	yes	characteristic
583	C₂×#41	—	2	2	no	—
584	C₂×#41	—	2	2	no	—
585	C₂×#41	—	2	8	no	—
586	C₂×#41	—	4	4	no	—
587	C₂×#41	C₂³×C₃₁	1	2	yes	—
588	C₂×#41	C₂³×C₃₁	1	2	yes	—
589	C₂×#41	C₂³×C₃₁	1	2	yes	—
590	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
591	C₂²×Q₈	—	2	1	no	—
592	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
593	C₂²×D₄	—	2	2	no	—
594	C₂²×D₄	—	2	2	no	—
595	C₂²×D₄	—	2	1	no	—
596	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
597	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
598	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
599	C₂³×C₄	—	2	1	no	—
600	C₂³×C₄	—	2	2	no	—
601	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
602	AGL(1,ℤ/8)	—	2	4	no	—
603	AGL(1,ℤ/8)	—	4	4	no	—
604	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
605	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
606	C₂×D₈	C₂³×C₃₁	1	2	yes	—
607	C₂×D₈	C₂³×C₃₁	1	4	yes	—
608	C₂×M₁₆	—	4	4	no	—
609	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
610	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
611	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
612	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
613	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
614	C₄⋊Q₈	—	2	1	no	—
615	C₄⋊Q₈	—	2	1	no	—
616	C₄⋊Q₈	—	2	1	no	—
617	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
618	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
619	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
620	#127	—	2	1	no	—
621	#127	C₂³×C₃₁	1	1	yes	characteristic
622	#124	—	2	4	no	—
623	#124	—	2	4	no	—
624	#124	—	2	4	no	—
625	#124	—	2	4	no	—
626	#124	—	2	2	no	—
627	#124	C₂³×C₃₁	1	2	yes	—
628	#123	—	2	2	no	—
629	#123	—	2	2	no	—
630	#123	—	2	2	no	—
631	C₂²⋊Q₈	—	2	2	no	—
632	C₂²⋊Q₈	—	2	2	no	—
633	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
634	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
635	#121	—	2	2	no	—
636	#121	—	2	2	no	—
637	#121	—	2	4	no	—
638	#121	—	2	2	no	—
639	#121	—	2	1	no	—
640	#121	—	2	2	no	—
641	#121	—	2	2	no	—
642	#121	—	2	1	no	—
643	#121	—	2	1	no	—
644	#121	—	2	1	no	—
645	#121	—	2	2	no	—
646	#121	—	2	2	no	—
647	#121	C₂³×C₃₁	1	4	yes	—
648	#121	C₂³×C₃₁	1	4	yes	—
649	#121	C₂³×C₃₁	1	2	yes	—
650	#121	C₂³×C₃₁	1	2	yes	—
651	#120	—	2	2	no	—
652	#120	—	2	2	no	—
653	#120	—	2	2	no	—
654	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
655	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
656	C₄×D₄	—	2	1	no	—
657	C₄×D₄	—	2	2	no	—
658	C₄×D₄	—	2	1	no	—
659	C₄×D₄	—	2	1	no	—
660	C₄×D₄	—	2	2	no	—
661	C₄×D₄	—	2	1	no	—
662	C₄×D₄	—	2	2	no	—
663	C₄×D₄	—	2	1	no	—
664	C₄×D₄	—	2	1	no	—
665	C₄×D₄	—	2	1	no	—
666	C₄×D₄	—	2	1	no	—
667	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
668	C₄×D₄	D₄×C₃₁	1	2	yes	—
669	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
670	C₄×D₄	C₂³×C₃₁	1	2	yes	—
671	C₄×D₄	C₂³×C₃₁	1	2	yes	—
672	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
673	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
674	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
675	C₄×D₄	C₂³×C₃₁	1	2	yes	—
676	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
677	#117	—	2	2	no	—
678	#117	—	2	2	no	—
679	#117	—	2	2	no	—
680	#117	—	2	2	no	—
681	#117	—	2	4	no	—
682	#117	—	2	1	no	—
683	#117	C₂³×C₃₁	1	4	yes	—
684	#117	C₂³×C₃₁	1	4	yes	—
685	#117	C₂³×C₃₁	1	1	yes	characteristic
686	C₂×C₄⋊C₄	—	2	1	no	—
687	C₂×C₄⋊C₄	—	2	1	no	—
688	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
689	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
690	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
691	C₂×C₂²⋊C₄	—	2	2	no	—
692	C₂×C₂²⋊C₄	—	2	2	no	—
693	C₂×C₂²⋊C₄	—	2	1	no	—
694	C₂×C₂²⋊C₄	—	2	2	no	—
695	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
696	C₂×C₄²	—	2	2	no	—
697	C₂×C₄²	—	2	1	no	—
698	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
699	#107	C₂³×C₃₁	1	2	yes	—
700	#107	C₂³×C₃₁	1	4	yes	—
701	#106	C₂³×C₃₁	1	2	yes	—
702	#106	C₂³×C₃₁	1	4	yes	—
703	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
704	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
705	#103	C₂³×C₃₁	1	2	yes	—
706	#103	C₂³×C₃₁	1	4	yes	—
707	#103	C₂³×C₃₁	1	4	yes	—
708	#103	C₂³×C₃₁	1	4	yes	—
709	#103	C₂³×C₃₁	1	4	yes	—
710	#102	C₂³×C₃₁	1	2	yes	—
711	#102	C₂³×C₃₁	1	4	yes	—
712	#102	C₂³×C₃₁	1	4	yes	—
713	#102	C₂³×C₃₁	1	4	yes	—
714	#102	C₂³×C₃₁	1	4	yes	—
715	#102	C₂³×C₃₁	1	2	yes	—
716	#102	C₂³×C₃₁	1	2	yes	—
717	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
718	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
719	#97	C₂³×C₃₁	1	2	yes	—
720	C₄×C₈	C₂³×C₃₁	1	2	yes	—
721	C₃₁	#61166	1	1	yes	characteristic, direct product factor, Sylow
722	C₂⁴	—	2	1	no	—
723	C₂⁴	—	2	2	no	—
724	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
725	#41	—	2	2	no	—
726	#41	—	2	4	no	—
727	#41	—	2	4	no	—
728	#41	—	2	4	no	—
729	#41	—	2	4	no	—
730	#41	—	4	8	no	—
731	#41	—	4	2	no	—
732	#41	—	4	2	no	—
733	#41	—	4	8	no	—
734	#41	#41×C₃₁	1	4	yes	—
735	C₂×Q₈	—	2	2	no	—
736	C₂×Q₈	—	2	2	no	—
737	C₂×Q₈	—	2	2	no	—
738	C₂×Q₈	—	2	4	no	—
739	C₂×Q₈	—	4	2	no	—
740	C₂×Q₈	—	8	2	no	—
741	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
742	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
743	C₂×D₄	—	2	1	no	—
744	C₂×D₄	—	2	4	no	—
745	C₂×D₄	—	2	2	no	—
746	C₂×D₄	—	2	2	no	—
747	C₂×D₄	—	2	2	no	—
748	C₂×D₄	—	2	2	no	—
749	C₂×D₄	—	2	2	no	—
750	C₂×D₄	—	2	4	no	—
751	C₂×D₄	—	2	4	no	—
752	C₂×D₄	—	2	1	no	—
753	C₂×D₄	—	2	1	no	—
754	C₂×D₄	—	2	2	no	—
755	C₂×D₄	—	2	1	no	—
756	C₂×D₄	—	2	2	no	—
757	C₂×D₄	—	2	2	no	—
758	C₂×D₄	—	4	2	no	—
759	C₂×D₄	—	4	2	no	—
760	C₂×D₄	—	4	8	no	—
761	C₂×D₄	—	4	4	no	—
762	C₂×D₄	—	8	4	no	—
763	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
764	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
765	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
766	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
767	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
768	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
769	C₂²×C₄	—	2	1	no	—
770	C₂²×C₄	—	2	4	no	—
771	C₂²×C₄	—	2	4	no	—
772	C₂²×C₄	—	2	1	no	—
773	C₂²×C₄	—	2	2	no	—
774	C₂²×C₄	—	2	2	no	—
775	C₂²×C₄	—	2	1	no	—
776	C₂²×C₄	—	2	2	no	—
777	C₂²×C₄	—	2	2	no	—
778	C₂²×C₄	—	2	2	no	—
779	C₂²×C₄	—	2	2	no	—
780	C₂²×C₄	—	2	2	no	—
781	C₂²×C₄	—	2	1	no	—
782	C₂²×C₄	—	4	8	no	—
783	C₂²×C₄	—	4	4	no	—
784	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
785	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
786	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
787	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
788	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
789	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
790	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
791	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
792	SD₁₆	—	2	4	no	—
793	SD₁₆	—	4	4	no	—
794	D₈	—	2	4	no	—
795	D₈	—	4	4	no	—
796	M₁₆	—	2	2	no	—
797	M₁₆	—	2	2	no	—
798	M₁₆	—	2	4	no	—
799	M₁₆	—	4	4	no	—
800	C₂×C₈	—	4	4	no	—
801	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
802	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
803	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
804	C₄⋊C₄	—	2	2	no	—
805	C₄⋊C₄	—	2	2	no	—
806	C₄⋊C₄	—	2	1	no	—
807	C₄⋊C₄	—	2	2	no	—
808	C₄⋊C₄	—	2	1	no	—
809	C₄⋊C₄	—	2	2	no	—
810	C₄⋊C₄	—	2	1	no	—
811	C₄⋊C₄	—	2	1	no	—
812	C₄⋊C₄	—	2	1	no	—
813	C₄⋊C₄	—	2	2	no	—
814	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
815	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
816	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
817	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
818	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
819	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
820	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
821	C₂²⋊C₄	—	2	4	no	—
822	C₂²⋊C₄	—	2	4	no	—
823	C₂²⋊C₄	—	2	2	no	—
824	C₂²⋊C₄	—	2	1	no	—
825	C₂²⋊C₄	—	2	2	no	—
826	C₂²⋊C₄	—	2	4	no	—
827	C₂²⋊C₄	—	2	4	no	—
828	C₂²⋊C₄	—	2	2	no	—
829	C₂²⋊C₄	—	2	2	no	—
830	C₂²⋊C₄	—	2	2	no	—
831	C₂²⋊C₄	—	2	2	no	—
832	C₂²⋊C₄	—	2	1	no	—
833	C₂²⋊C₄	—	2	1	no	—
834	C₂²⋊C₄	—	2	2	no	—
835	C₂²⋊C₄	—	2	2	no	—
836	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
837	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
838	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
839	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
840	C₄²	—	2	4	no	—
841	C₄²	—	2	4	no	—
842	C₄²	—	2	1	no	—
843	C₄²	—	2	1	no	—
844	C₄²	—	2	1	no	—
845	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
846	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
847	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
848	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
849	C₂³	—	2	1	no	—
850	C₂³	—	2	4	no	—
851	C₂³	—	2	2	no	—
852	C₂³	—	2	2	no	—
853	C₂³	—	4	2	no	—
854	C₂³	—	4	2	no	—
855	C₂³	—	4	4	no	—
856	C₂³	—	4	2	no	—
857	C₂³	—	4	4	no	—
858	C₂³	—	4	2	no	—
859	C₂³	—	4	2	no	—
860	C₂³	—	8	2	no	—
861	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
862	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
863	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
864	Q₈	—	2	1	no	—
865	Q₈	—	2	2	no	—
866	Q₈	—	2	2	no	—
867	Q₈	—	4	4	no	—
868	Q₈	—	8	2	no	—
869	Q₈	—	8	2	no	—
870	Q₈	C₂×#41×C₃₁	1	2	yes	—
871	D₄	—	2	4	no	—
872	D₄	—	2	2	no	—
873	D₄	—	2	2	no	—
874	D₄	—	2	2	no	—
875	D₄	—	2	2	no	—
876	D₄	—	2	1	no	—
877	D₄	—	2	2	no	—
878	D₄	—	2	2	no	—
879	D₄	—	4	2	no	—
880	D₄	—	4	4	no	—
881	D₄	—	4	4	no	—
882	D₄	—	4	4	no	—
883	D₄	—	4	4	no	—
884	D₄	—	4	4	no	—
885	D₄	—	4	2	no	—
886	D₄	—	4	2	no	—
887	D₄	—	8	1	no	—
888	D₄	—	8	2	no	—
889	D₄	—	8	4	no	—
890	D₄	—	8	4	no	—
891	D₄	C₂×#41×C₃₁	1	2	yes	—
892	C₂×C₄	—	2	4	no	—
893	C₂×C₄	—	2	1	no	—
894	C₂×C₄	—	2	4	no	—
895	C₂×C₄	—	2	4	no	—
896	C₂×C₄	—	2	4	no	—
897	C₂×C₄	—	2	2	no	—
898	C₂×C₄	—	2	2	no	—
899	C₂×C₄	—	2	2	no	—
900	C₂×C₄	—	2	2	no	—
901	C₂×C₄	—	2	2	no	—
902	C₂×C₄	—	2	2	no	—
903	C₂×C₄	—	2	1	no	—
904	C₂×C₄	—	2	1	no	—
905	C₂×C₄	—	2	4	no	—
906	C₂×C₄	—	2	2	no	—
907	C₂×C₄	—	4	1	no	—
908	C₂×C₄	—	4	4	no	—
909	C₂×C₄	—	4	4	no	—
910	C₂×C₄	—	4	4	no	—
911	C₂×C₄	—	4	4	no	—
912	C₂×C₄	—	4	2	no	—
913	C₂×C₄	—	4	4	no	—
914	C₂×C₄	—	4	4	no	—
915	C₂×C₄	—	4	4	no	—
916	C₂×C₄	—	4	2	no	—
917	C₂×C₄	—	8	1	no	—
918	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
919	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
920	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
921	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
922	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
923	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
924	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
925	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
926	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
927	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
928	C₈	—	2	2	no	—
929	C₈	—	2	2	no	—
930	C₈	—	2	4	no	—
931	C₂²	—	2	1	no	—
932	C₂²	—	2	1	no	—
933	C₂²	—	2	2	no	—
934	C₂²	—	2	2	no	—
935	C₂²	—	2	2	no	—
936	C₂²	—	2	2	no	—
937	C₂²	—	2	2	no	—
938	C₂²	—	2	2	no	—
939	C₂²	—	2	2	no	—
940	C₂²	—	4	1	no	—
941	C₂²	—	4	1	no	—
942	C₂²	—	4	2	no	—
943	C₂²	—	4	2	no	—
944	C₂²	—	4	2	no	—
945	C₂²	—	4	2	no	—
946	C₂²	—	4	2	no	—
947	C₂²	—	4	2	no	—
948	C₂²	—	8	4	no	—
949	C₂²	—	8	4	no	—
950	C₂²	—	8	8	no	—
951	C₂²	C₃₁×#546	1	2	yes	—
952	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
953	C₄	—	2	2	no	—
954	C₄	—	2	2	no	—
955	C₄	—	2	2	no	—
956	C₄	—	2	1	no	—
957	C₄	—	2	1	no	—
958	C₄	—	2	1	no	—
959	C₄	—	4	2	no	—
960	C₄	—	4	2	no	—
961	C₄	—	4	1	no	—
962	C₄	—	4	4	no	—
963	C₄	—	4	1	no	—
964	C₄	C₃₁×#546	1	2	yes	—
965	C₄	C₃₁×#575	1	2	yes	—
966	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
967	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
968	C₂	—	2	2	no	—
969	C₂	—	4	2	no	—
970	C₂	—	4	2	no	—
971	C₂	—	4	2	no	—
972	C₂	—	8	2	no	—
973	C₂	—	8	2	no	—
974	C₂	C₂×C₃₁×#546	1	1	yes	characteristic
975	C₂	C₂×C₃₁×#575	1	1	yes	characteristic
976	C₂	C₃₁×#3585	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 ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#555	C₂	semidirect
2. D₄×#41×C₃₁	C₂	semidirect
3. C₃₁×#3329	C₂	semidirect
4. C₃₁×#3325	C₂	semidirect
5. C₃₁×#3321	C₂	semidirect
6. C₃₁×#3319	C₂	semidirect
7. C₃₁×#3305	C₂	semidirect
8. C₃₁×#3303	C₂	semidirect
9. C₃₁×#3141	C₂	semidirect
10. C₃₁×#3093	C₂	semidirect
11. C₃₁×#3079	C₂	semidirect
12. C₃₁×#3055	C₂	semidirect
13. C₃₁×#3036	C₂	semidirect
14. C₃₁×#3006	C₂	semidirect
15. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
16. D₄×M₁₆×C₃₁	C₂	semidirect
17. C₃₁×#2896	C₂	semidirect
20. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
21. C₂²×M₁₆×C₃₁	C₂²	semidirect
26. C₃₁×#555	C₂²	semidirect
27. C₃₁×#555	C₂²	semidirect
28. C₃₁×#550	C₂²	semidirect
29. D₄×Q₈×C₃₁	C₂²	semidirect
30. C₃₁×#548	C₂²	semidirect
32. C₃₁×#547	C₂²	semidirect
34. C₃₁×#546	C₂²	semidirect
36. D₄²×C₃₁	C₂²	semidirect
39. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
44. C₂×C₃₁×#121	C₂²	semidirect
48. C₂×C₄×D₄×C₃₁	C₂²	semidirect
50. C₂×C₃₁×#117	C₂²	semidirect
51. C₃₁×#489	C₂²	semidirect
52. C₃₁×#487	C₂²	semidirect
53. C₃₁×#486	C₂²	semidirect
54. C₃₁×#485	C₂²	semidirect
55. C₃₁×#485	C₂²	semidirect
56. C₃₁×#482	C₂²	semidirect
57. C₃₁×#482	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×#463	C₂²	semidirect
67. C₃₁×#461	C₂²	semidirect
68. C₃₁×#459	C₂²	semidirect
69. C₃₁×#449	C₂²	semidirect
70. C₃₁×#449	C₂²	semidirect
71. C₃₁×#448	C₂²	semidirect
72. C₃₁×#447	C₂²	semidirect
73. C₃₁×#442	C₂²	semidirect
74. C₃₁×#440	C₂²	semidirect
75. C₄×SD₁₆×C₃₁	C₂²	semidirect
76. C₄×D₈×C₃₁	C₂²	semidirect
77. C₃₁×#436	C₂²	semidirect
78. C₃₁×#435	C₂²	semidirect
79. C₈×D₄×C₃₁	C₂²	semidirect
80. C₃₁×#428	C₂²	semidirect
81. C₃₁×#428	C₂²	semidirect
82. C₃₁×#423	C₂²	semidirect
83. C₃₁×#419	C₂²	semidirect
84. C₃₁×#418	C₂²	semidirect
85. C₃₁×#418	C₂²	semidirect
88. C₃₁×#417	C₂²	semidirect
89. C₃₁×#417	C₂²	semidirect
90. C₃₁×#417	C₂²	semidirect
91. C₃₁×#416	C₂²	semidirect
92. C₃₁×#416	C₂²	semidirect
93. C₃₁×#407	C₂²	semidirect
94. C₄×M₁₆×C₃₁	C₂²	semidirect
360. #61166	C₃₁	direct (N × H)
443. C₂×#555	C₂×C₃₁	semidirect
444. D₄×#41	C₂×C₃₁	semidirect
445. #3329	C₂×C₃₁	semidirect
446. #3325	C₂×C₃₁	semidirect
447. #3321	C₂×C₃₁	semidirect
448. #3319	C₂×C₃₁	semidirect
449. #3305	C₂×C₃₁	semidirect
450. #3303	C₂×C₃₁	semidirect
451. #3141	C₂×C₃₁	semidirect
452. #3093	C₂×C₃₁	semidirect
453. #3079	C₂×C₃₁	semidirect
454. #3055	C₂×C₃₁	semidirect
455. #3036	C₂×C₃₁	semidirect
456. #3006	C₂×C₃₁	semidirect
457. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
458. D₄×M₁₆	C₂×C₃₁	semidirect
459. #2896	C₂×C₃₁	semidirect
499. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
500. C₂²×M₁₆	C₂²×C₃₁	semidirect
505. #555	C₂²×C₃₁	semidirect
506. #555	C₂²×C₃₁	semidirect
507. #550	C₂²×C₃₁	semidirect
508. D₄×Q₈	C₂²×C₃₁	semidirect
509. #548	C₂²×C₃₁	semidirect
511. #547	C₂²×C₃₁	semidirect
513. #546	C₂²×C₃₁	semidirect
515. D₄²	C₂²×C₃₁	semidirect
518. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
523. C₂×#121	C₂²×C₃₁	semidirect
527. C₂×C₄×D₄	C₂²×C₃₁	semidirect
529. C₂×#117	C₂²×C₃₁	semidirect
530. #489	C₂²×C₃₁	semidirect
531. #487	C₂²×C₃₁	semidirect
532. #486	C₂²×C₃₁	semidirect
533. #485	C₂²×C₃₁	semidirect
534. #485	C₂²×C₃₁	semidirect
535. #482	C₂²×C₃₁	semidirect
536. #482	C₂²×C₃₁	semidirect
537. #477	C₂²×C₃₁	semidirect
538. #476	C₂²×C₃₁	semidirect
539. #475	C₂²×C₃₁	semidirect
540. #474	C₂²×C₃₁	semidirect
541. #469	C₂²×C₃₁	semidirect
542. #468	C₂²×C₃₁	semidirect
543. #466	C₂²×C₃₁	semidirect
544. #465	C₂²×C₃₁	semidirect
545. #463	C₂²×C₃₁	semidirect
546. #461	C₂²×C₃₁	semidirect
547. #459	C₂²×C₃₁	semidirect
548. #449	C₂²×C₃₁	semidirect
549. #449	C₂²×C₃₁	semidirect
550. #448	C₂²×C₃₁	semidirect
551. #447	C₂²×C₃₁	semidirect
552. #442	C₂²×C₃₁	semidirect
553. #440	C₂²×C₃₁	semidirect
554. C₄×SD₁₆	C₂²×C₃₁	semidirect
555. C₄×D₈	C₂²×C₃₁	semidirect
556. #436	C₂²×C₃₁	semidirect
557. #435	C₂²×C₃₁	semidirect
558. C₈×D₄	C₂²×C₃₁	semidirect
559. #428	C₂²×C₃₁	semidirect
560. #428	C₂²×C₃₁	semidirect
561. #423	C₂²×C₃₁	semidirect
562. #419	C₂²×C₃₁	semidirect
563. #418	C₂²×C₃₁	semidirect
564. #418	C₂²×C₃₁	semidirect
567. #417	C₂²×C₃₁	semidirect
568. #417	C₂²×C₃₁	semidirect
569. #417	C₂²×C₃₁	semidirect
570. #416	C₂²×C₃₁	semidirect
571. #416	C₂²×C₃₁	semidirect
572. #407	C₂²×C₃₁	semidirect
573. C₄×M₁₆	C₂²×C₃₁	semidirect
721. C₃₁	#61166	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#555	2
D₄×#41×C₃₁	2
C₃₁×#3329	2
C₃₁×#3325	2
C₃₁×#3321	2
C₃₁×#3319	2
C₃₁×#3305	2
C₃₁×#3303	2
C₃₁×#3141	2
C₃₁×#3093	2
C₃₁×#3079	2
C₃₁×#3055	2
C₃₁×#3036	2
C₃₁×#3006	2
C₄×C₃₁×AGL(1,ℤ/8)	2
D₄×M₁₆×C₃₁	2
C₃₁×#2896	2
#61166	31

Subgroups — C31×#61166 #21517925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61166 #21517925