Subgroups — C₃₁×#62538 #21519297

Order 7936.

1962 conjugacy classes of subgroups.

The subgroup lattice has 720 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×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3367	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3330	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3308	C₂	1	8	yes	maximal, complemented
6	C₃₁×#3230	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3226	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3207	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3182	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3141	C₂	1	2	yes	maximal, complemented
13	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
16	C₃₁×#556	C₂²	1	2	yes	—
17	C₃₁×#555	—	2	1	no	—
18	C₃₁×#554	—	2	1	no	—
19	C₃₁×#553	—	2	2	no	—
20	D₄×Q₈×C₃₁	—	2	1	no	—
21	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#548	C₂²	1	4	yes	—
23	C₃₁×#547	—	2	2	no	—
24	C₃₁×#547	—	2	1	no	—
25	C₃₁×#547	C₂²	1	2	yes	—
26	C₃₁×#547	C₂²	1	4	yes	—
27	C₃₁×#547	C₂²	1	2	yes	—
28	C₃₁×#542	—	2	2	no	—
29	C₃₁×#537	—	2	1	no	—
30	C₃₁×#536	—	2	1	no	—
31	C₃₁×#536	—	2	2	no	—
32	C₃₁×#533	C₂²	1	2	yes	—
33	C₃₁×#532	C₂²	1	2	yes	—
34	C₃₁×#532	C₂²	1	1	yes	characteristic
35	C₃₁×#532	C₂²	1	1	yes	characteristic
36	C₃₁×#532	C₂²	1	2	yes	—
37	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
40	C₃₁×#520	—	2	1	no	—
41	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#500	C₂²	1	1	yes	characteristic
43	C₃₁×#500	C₂²	1	2	yes	—
44	C₃₁×#493	C₂²	1	1	yes	characteristic
45	C₃₁×#492	C₂²	1	2	yes	—
46	C₃₁×#489	C₂²	1	8	yes	—
47	C₃₁×#487	C₂²	1	4	yes	—
48	C₃₁×#487	C₂²	1	2	yes	—
49	C₃₁×#487	C₂²	1	2	yes	—
50	C₃₁×#487	C₂²	1	2	yes	—
51	C₃₁×#486	C₂²	1	1	yes	characteristic
52	C₃₁×#486	C₂²	1	2	yes	—
53	C₃₁×#486	C₂²	1	1	yes	characteristic
54	C₃₁×#484	C₂²	1	8	yes	—
55	C₃₁×#480	C₂²	1	4	yes	—
56	C₃₁×#477	C₂²	1	8	yes	—
57	C₃₁×#474	C₂²	1	4	yes	—
58	C₃₁×#468	C₂²	1	8	yes	—
59	C₃₁×#466	C₂²	1	4	yes	—
60	C₃₁×#463	C₂²	1	8	yes	—
61	C₃₁×#461	C₂²	1	4	yes	—
62	C₃₁×#449	C₂²	1	8	yes	—
63	C₃₁×#448	C₂²	1	4	yes	—
64	C₃₁×#446	C₂²	1	2	yes	—
65	C₃₁×#440	C₂²	1	8	yes	—
66	C₄×D₈×C₃₁	C₂²	1	4	yes	—
67	C₃₁×#436	C₂²	1	2	yes	—
68	C₃₁×#435	C₂²	1	4	yes	—
69	C₃₁×#431	C₂²	1	1	yes	characteristic
70	C₃₁×#431	C₂²	1	2	yes	—
71	C₃₁×#417	C₂²	1	8	yes	—
72	C₂×C₃₁×#103	C₂²	1	4	yes	—
73	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
74	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
75	C₃₁×2₋¹⁺⁴	—	4	8	no	—
76	C₂×#41×C₃₁	—	2	4	no	—
77	C₂×#41×C₃₁	—	2	2	no	—
78	C₂×#41×C₃₁	—	2	1	no	—
79	C₂×#41×C₃₁	—	2	2	no	—
80	C₂×#41×C₃₁	—	2	1	no	—
81	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
82	C₂×#41×C₃₁	C₂³	1	4	yes	—
83	C₂²×Q₈×C₃₁	—	2	2	no	—
84	C₂²×Q₈×C₃₁	—	2	2	no	—
85	C₂²×Q₈×C₃₁	—	2	1	no	—
86	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
87	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
88	C₂×D₈×C₃₁	C₂³	1	4	yes	—
89	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
90	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
91	C₃₁×C₄⋊Q₈	—	2	2	no	—
92	C₃₁×C₄⋊Q₈	—	2	1	no	—
93	C₃₁×C₄⋊Q₈	—	2	1	no	—
94	C₃₁×C₄⋊Q₈	—	2	2	no	—
95	C₃₁×C₄⋊Q₈	—	2	1	no	—
96	C₃₁×C₄⋊Q₈	—	2	1	no	—
97	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
98	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
99	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
100	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
101	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
102	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
103	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
104	C₃₁×#127	C₂³	1	1	yes	characteristic
105	C₃₁×#127	C₂³	1	1	yes	characteristic
106	C₃₁×#126	—	2	4	no	—
107	C₃₁×#125	—	2	2	no	—
108	C₃₁×#124	—	2	4	no	—
109	C₃₁×#124	—	2	2	no	—
110	C₃₁×#124	—	2	1	no	—
111	C₃₁×#124	—	2	1	no	—
112	C₃₁×#124	C₂³	1	4	yes	—
113	C₃₁×#123	—	2	4	no	—
114	C₃₁×#123	—	2	4	no	—
115	C₃₁×#123	—	2	4	no	—
116	C₃₁×#123	—	2	4	no	—
117	C₃₁×C₂²⋊Q₈	—	2	2	no	—
118	C₃₁×C₂²⋊Q₈	—	2	2	no	—
119	C₃₁×C₂²⋊Q₈	—	2	2	no	—
120	C₃₁×C₂²⋊Q₈	—	2	4	no	—
121	C₃₁×C₂²⋊Q₈	—	2	2	no	—
122	C₃₁×C₂²⋊Q₈	—	2	4	no	—
123	C₃₁×C₂²⋊Q₈	—	2	4	no	—
124	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
125	C₃₁×#121	—	2	2	no	—
126	C₃₁×#121	—	2	4	no	—
127	C₃₁×#121	—	2	2	no	—
128	C₃₁×#121	—	2	1	no	—
129	C₃₁×#121	—	2	2	no	—
130	C₃₁×#121	—	2	1	no	—
131	C₃₁×#121	—	2	2	no	—
132	C₃₁×#121	C₂³	1	2	yes	—
133	C₃₁×#121	C₂³	1	4	yes	—
134	C₃₁×#121	C₂³	1	2	yes	—
135	C₄×Q₈×C₃₁	—	2	2	no	—
136	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
137	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
138	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₄×D₄×C₃₁	—	2	4	no	—
140	C₄×D₄×C₃₁	—	2	2	no	—
141	C₄×D₄×C₃₁	—	2	2	no	—
142	C₄×D₄×C₃₁	—	2	1	no	—
143	C₄×D₄×C₃₁	—	2	2	no	—
144	C₄×D₄×C₃₁	—	2	1	no	—
145	C₄×D₄×C₃₁	D₄	1	2	yes	—
146	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
147	C₄×D₄×C₃₁	C₂³	1	2	yes	—
148	C₄×D₄×C₃₁	C₂³	1	2	yes	—
149	C₄×D₄×C₃₁	C₂³	1	4	yes	—
150	C₄×D₄×C₃₁	C₂³	1	2	yes	—
151	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
152	C₃₁×#117	—	2	2	no	—
153	C₃₁×#117	—	2	2	no	—
154	C₃₁×#117	D₄	1	1	yes	characteristic
155	C₃₁×#117	D₄	1	2	yes	—
156	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
162	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
163	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
164	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
165	C₃₁×#107	C₂³	1	4	yes	—
166	C₃₁×#107	C₂³	1	8	yes	—
167	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
168	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
169	C₃₁×#103	C₂³	1	8	yes	—
170	C₃₁×#103	C₂³	1	8	yes	—
171	C₃₁×#103	C₂³	1	8	yes	—
172	C₃₁×#102	C₂³	1	4	yes	—
173	C₃₁×#102	C₂³	1	8	yes	—
174	C₃₁×#102	C₂³	1	4	yes	—
175	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
176	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
177	C₃₁×#97	C₂³	1	4	yes	—
178	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
180	C₄×C₈×C₃₁	C₂³	1	2	yes	—
181	C₄×C₈×C₃₁	C₂³	1	2	yes	—
182	#41×C₃₁	—	2	4	no	—
183	#41×C₃₁	—	4	8	no	—
184	#41×C₃₁	—	4	8	no	—
185	#41×C₃₁	—	4	16	no	—
186	#41×C₃₁	—	4	8	no	—
187	#41×C₃₁	—	4	4	no	—
188	#41×C₃₁	—	4	4	no	—
189	C₂×Q₈×C₃₁	—	2	1	no	—
190	C₂×Q₈×C₃₁	—	2	4	no	—
191	C₂×Q₈×C₃₁	—	2	2	no	—
192	C₂×Q₈×C₃₁	—	2	1	no	—
193	C₂×Q₈×C₃₁	—	2	2	no	—
194	C₂×Q₈×C₃₁	—	2	2	no	—
195	C₂×Q₈×C₃₁	—	4	8	no	—
196	C₂×Q₈×C₃₁	—	4	4	no	—
197	C₂×Q₈×C₃₁	—	4	8	no	—
198	C₂×Q₈×C₃₁	—	4	4	no	—
199	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
201	C₂×D₄×C₃₁	—	2	2	no	—
202	C₂×D₄×C₃₁	—	2	4	no	—
203	C₂×D₄×C₃₁	—	2	2	no	—
204	C₂×D₄×C₃₁	—	2	4	no	—
205	C₂×D₄×C₃₁	—	2	1	no	—
206	C₂×D₄×C₃₁	—	2	1	no	—
207	C₂×D₄×C₃₁	—	2	1	no	—
208	C₂×D₄×C₃₁	—	2	1	no	—
209	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
210	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
212	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₃₁	—	2	4	no	—
214	C₂²×C₄×C₃₁	—	2	2	no	—
215	C₂²×C₄×C₃₁	—	2	2	no	—
216	C₂²×C₄×C₃₁	—	2	2	no	—
217	C₂²×C₄×C₃₁	—	2	4	no	—
218	C₂²×C₄×C₃₁	—	2	2	no	—
219	C₂²×C₄×C₃₁	—	2	4	no	—
220	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
221	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
222	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
223	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
224	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
225	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
226	SD₁₆×C₃₁	—	4	8	no	—
227	D₈×C₃₁	—	4	4	no	—
228	M₁₆×C₃₁	—	4	4	no	—
229	C₂×C₈×C₃₁	—	4	2	no	—
230	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
231	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
232	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
233	C₄⋊C₄×C₃₁	—	2	2	no	—
234	C₄⋊C₄×C₃₁	—	2	2	no	—
235	C₄⋊C₄×C₃₁	—	2	1	no	—
236	C₄⋊C₄×C₃₁	—	2	2	no	—
237	C₄⋊C₄×C₃₁	—	2	2	no	—
238	C₄⋊C₄×C₃₁	—	2	4	no	—
239	C₄⋊C₄×C₃₁	—	2	4	no	—
240	C₄⋊C₄×C₃₁	—	2	2	no	—
241	C₄⋊C₄×C₃₁	—	2	2	no	—
242	C₄⋊C₄×C₃₁	—	2	2	no	—
243	C₄⋊C₄×C₃₁	—	2	4	no	—
244	C₄⋊C₄×C₃₁	—	2	1	no	—
245	C₄⋊C₄×C₃₁	—	2	4	no	—
246	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
247	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
248	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
249	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
250	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
251	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
252	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
253	C₂²⋊C₄×C₃₁	—	2	4	no	—
254	C₂²⋊C₄×C₃₁	—	2	4	no	—
255	C₂²⋊C₄×C₃₁	—	2	4	no	—
256	C₂²⋊C₄×C₃₁	—	2	2	no	—
257	C₂²⋊C₄×C₃₁	—	2	4	no	—
258	C₂²⋊C₄×C₃₁	—	2	2	no	—
259	C₂²⋊C₄×C₃₁	—	2	4	no	—
260	C₂²⋊C₄×C₃₁	—	2	2	no	—
261	C₂²⋊C₄×C₃₁	—	2	2	no	—
262	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
263	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
264	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
265	C₄²×C₃₁	—	2	4	no	—
266	C₄²×C₃₁	C₂×D₄	1	2	yes	—
267	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
268	C₄²×C₃₁	C₂×D₄	1	2	yes	—
269	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
270	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
271	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
272	C₄²×C₃₁	C₂⁴	1	2	yes	—
273	#62538	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂³×C₃₁	—	2	2	no	—
275	C₂³×C₃₁	—	2	2	no	—
276	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
277	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
278	Q₈×C₃₁	—	2	2	no	—
279	Q₈×C₃₁	—	2	8	no	—
280	Q₈×C₃₁	—	4	8	no	—
281	Q₈×C₃₁	—	4	2	no	—
282	Q₈×C₃₁	—	4	4	no	—
283	Q₈×C₃₁	—	4	4	no	—
284	Q₈×C₃₁	—	4	2	no	—
285	Q₈×C₃₁	—	8	2	no	—
286	D₄×C₃₁	—	2	4	no	—
287	D₄×C₃₁	—	2	2	no	—
288	D₄×C₃₁	—	2	4	no	—
289	D₄×C₃₁	—	2	4	no	—
290	D₄×C₃₁	—	4	8	no	—
291	D₄×C₃₁	—	4	4	no	—
292	D₄×C₃₁	—	4	4	no	—
293	D₄×C₃₁	—	4	2	no	—
294	D₄×C₃₁	—	4	2	no	—
295	D₄×C₃₁	—	4	2	no	—
296	D₄×C₃₁	—	4	2	no	—
297	D₄×C₃₁	—	8	4	no	—
298	C₂×C₄×C₃₁	—	2	4	no	—
299	C₂×C₄×C₃₁	—	2	2	no	—
300	C₂×C₄×C₃₁	—	2	4	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	2	4	no	—
303	C₂×C₄×C₃₁	—	2	2	no	—
304	C₂×C₄×C₃₁	—	2	4	no	—
305	C₂×C₄×C₃₁	—	2	4	no	—
306	C₂×C₄×C₃₁	—	2	2	no	—
307	C₂×C₄×C₃₁	—	4	8	no	—
308	C₂×C₄×C₃₁	—	4	8	no	—
309	C₂×C₄×C₃₁	—	4	4	no	—
310	C₂×C₄×C₃₁	—	4	4	no	—
311	C₂×C₄×C₃₁	—	4	4	no	—
312	C₂×C₄×C₃₁	—	8	4	no	—
313	C₂×C₄×C₃₁	—	8	2	no	—
314	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
315	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
316	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
317	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
318	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
320	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
322	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
324	C₈×C₃₁	—	2	2	no	—
325	C₈×C₃₁	—	2	2	no	—
326	C₈×C₃₁	—	2	4	no	—
327	#3511	C₂×C₃₁	1	1	yes	characteristic, complemented
328	#3499	C₂×C₃₁	1	1	yes	characteristic, complemented
329	#3367	C₂×C₃₁	1	4	yes	complemented
330	#3330	C₂×C₃₁	1	4	yes	complemented
331	#3308	C₂×C₃₁	1	8	yes	complemented
332	#3230	C₂×C₃₁	1	2	yes	complemented
333	#3226	C₂×C₃₁	1	1	yes	characteristic, complemented
334	#3207	C₂×C₃₁	1	2	yes	complemented
335	#3206	C₂×C₃₁	1	1	yes	characteristic, complemented
336	#3182	C₂×C₃₁	1	2	yes	complemented
337	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
338	#3141	C₂×C₃₁	1	2	yes	complemented
339	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
340	#2965	C₂×C₃₁	1	1	yes	characteristic, complemented
341	C₂²×C₃₁	—	2	1	no	—
342	C₂²×C₃₁	—	2	1	no	—
343	C₂²×C₃₁	—	2	2	no	—
344	C₂²×C₃₁	—	2	2	no	—
345	C₂²×C₃₁	—	2	2	no	—
346	C₂²×C₃₁	—	4	2	no	—
347	C₂²×C₃₁	—	4	2	no	—
348	C₂²×C₃₁	—	4	2	no	—
349	C₂²×C₃₁	—	4	2	no	—
350	C₂²×C₃₁	—	4	2	no	—
351	C₂²×C₃₁	—	4	2	no	—
352	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
353	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
354	C₄×C₃₁	—	2	2	no	—
355	C₄×C₃₁	—	2	1	no	—
356	C₄×C₃₁	—	2	2	no	—
357	C₄×C₃₁	—	2	2	no	—
358	C₄×C₃₁	—	2	1	no	—
359	C₄×C₃₁	—	4	4	no	—
360	C₄×C₃₁	—	4	4	no	—
361	C₄×C₃₁	—	4	2	no	—
362	C₄×C₃₁	—	4	2	no	—
363	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
364	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
365	C₄×C₃₁	#578	1	2	yes	—
366	C₄×C₃₁	#585	1	2	yes	—
367	C₂×2₋¹⁺⁴	—	2	1	no	—
368	#556	C₂²×C₃₁	1	2	yes	—
369	#555	—	2	1	no	—
370	#554	—	2	1	no	—
371	#553	—	2	2	no	—
372	D₄×Q₈	—	2	1	no	—
373	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
374	#548	C₂²×C₃₁	1	4	yes	—
375	#547	—	2	2	no	—
376	#547	—	2	1	no	—
377	#547	C₂²×C₃₁	1	2	yes	—
378	#547	C₂²×C₃₁	1	2	yes	—
379	#547	C₂²×C₃₁	1	4	yes	—
380	#542	—	2	2	no	—
381	#537	—	2	1	no	—
382	#536	—	2	1	no	—
383	#536	—	2	2	no	—
384	#533	C₂²×C₃₁	1	2	yes	—
385	#532	C₂²×C₃₁	1	2	yes	—
386	#532	C₂²×C₃₁	1	1	yes	characteristic
387	#532	C₂²×C₃₁	1	1	yes	characteristic
388	#532	C₂²×C₃₁	1	2	yes	—
389	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
390	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
391	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
392	#520	—	2	1	no	—
393	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
394	#500	C₂²×C₃₁	1	1	yes	characteristic
395	#500	C₂²×C₃₁	1	2	yes	—
396	#493	C₂²×C₃₁	1	1	yes	characteristic
397	#492	C₂²×C₃₁	1	2	yes	—
398	#489	C₂²×C₃₁	1	8	yes	—
399	#487	C₂²×C₃₁	1	4	yes	—
400	#487	C₂²×C₃₁	1	2	yes	—
401	#487	C₂²×C₃₁	1	2	yes	—
402	#487	C₂²×C₃₁	1	2	yes	—
403	#486	C₂²×C₃₁	1	1	yes	characteristic
404	#486	C₂²×C₃₁	1	2	yes	—
405	#486	C₂²×C₃₁	1	1	yes	characteristic
406	#484	C₂²×C₃₁	1	8	yes	—
407	#480	C₂²×C₃₁	1	4	yes	—
408	#477	C₂²×C₃₁	1	8	yes	—
409	#474	C₂²×C₃₁	1	4	yes	—
410	#468	C₂²×C₃₁	1	8	yes	—
411	#466	C₂²×C₃₁	1	4	yes	—
412	#463	C₂²×C₃₁	1	8	yes	—
413	#461	C₂²×C₃₁	1	4	yes	—
414	#449	C₂²×C₃₁	1	8	yes	—
415	#448	C₂²×C₃₁	1	4	yes	—
416	#446	C₂²×C₃₁	1	2	yes	—
417	#440	C₂²×C₃₁	1	8	yes	—
418	C₄×D₈	C₂²×C₃₁	1	4	yes	—
419	#436	C₂²×C₃₁	1	2	yes	—
420	#435	C₂²×C₃₁	1	4	yes	—
421	#431	C₂²×C₃₁	1	1	yes	characteristic
422	#431	C₂²×C₃₁	1	2	yes	—
423	#417	C₂²×C₃₁	1	8	yes	—
424	C₂×#103	C₂²×C₃₁	1	4	yes	—
425	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
426	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
427	C₂×C₃₁	—	2	2	no	—
428	C₂×C₃₁	—	4	2	no	—
429	C₂×C₃₁	—	8	2	no	—
430	C₂×C₃₁	—	8	2	no	—
431	C₂×C₃₁	#3500	1	1	yes	characteristic
432	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
433	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
434	2₋¹⁺⁴	—	4	8	no	—
435	C₂×#41	—	2	2	no	—
436	C₂×#41	—	2	4	no	—
437	C₂×#41	—	2	1	no	—
438	C₂×#41	—	2	2	no	—
439	C₂×#41	—	2	1	no	—
440	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
441	C₂×#41	C₂³×C₃₁	1	4	yes	—
442	C₂²×Q₈	—	2	2	no	—
443	C₂²×Q₈	—	2	2	no	—
444	C₂²×Q₈	—	2	1	no	—
445	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
446	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
447	C₂×D₈	C₂³×C₃₁	1	4	yes	—
448	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
449	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
450	C₄⋊Q₈	—	2	2	no	—
451	C₄⋊Q₈	—	2	1	no	—
452	C₄⋊Q₈	—	2	1	no	—
453	C₄⋊Q₈	—	2	2	no	—
454	C₄⋊Q₈	—	2	1	no	—
455	C₄⋊Q₈	—	2	1	no	—
456	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
457	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
458	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
459	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
460	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
461	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
462	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
463	#127	C₂³×C₃₁	1	1	yes	characteristic
464	#127	C₂³×C₃₁	1	1	yes	characteristic
465	#126	—	2	4	no	—
466	#125	—	2	2	no	—
467	#124	—	2	4	no	—
468	#124	—	2	2	no	—
469	#124	—	2	1	no	—
470	#124	—	2	1	no	—
471	#124	C₂³×C₃₁	1	4	yes	—
472	#123	—	2	4	no	—
473	#123	—	2	4	no	—
474	#123	—	2	4	no	—
475	#123	—	2	4	no	—
476	C₂²⋊Q₈	—	2	2	no	—
477	C₂²⋊Q₈	—	2	2	no	—
478	C₂²⋊Q₈	—	2	2	no	—
479	C₂²⋊Q₈	—	2	4	no	—
480	C₂²⋊Q₈	—	2	2	no	—
481	C₂²⋊Q₈	—	2	4	no	—
482	C₂²⋊Q₈	—	2	4	no	—
483	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
484	#121	—	2	2	no	—
485	#121	—	2	4	no	—
486	#121	—	2	2	no	—
487	#121	—	2	1	no	—
488	#121	—	2	2	no	—
489	#121	—	2	1	no	—
490	#121	—	2	2	no	—
491	#121	C₂³×C₃₁	1	2	yes	—
492	#121	C₂³×C₃₁	1	4	yes	—
493	#121	C₂³×C₃₁	1	2	yes	—
494	C₄×Q₈	—	2	2	no	—
495	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
496	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
497	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
498	C₄×D₄	—	2	4	no	—
499	C₄×D₄	—	2	2	no	—
500	C₄×D₄	—	2	2	no	—
501	C₄×D₄	—	2	1	no	—
502	C₄×D₄	—	2	2	no	—
503	C₄×D₄	—	2	1	no	—
504	C₄×D₄	D₄×C₃₁	1	2	yes	—
505	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
506	C₄×D₄	C₂³×C₃₁	1	2	yes	—
507	C₄×D₄	C₂³×C₃₁	1	4	yes	—
508	C₄×D₄	C₂³×C₃₁	1	2	yes	—
509	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
510	C₄×D₄	C₂³×C₃₁	1	2	yes	—
511	#117	—	2	2	no	—
512	#117	—	2	2	no	—
513	#117	D₄×C₃₁	1	1	yes	characteristic
514	#117	D₄×C₃₁	1	2	yes	—
515	C₂×C₄⋊C₄	—	2	1	no	—
516	C₂×C₄⋊C₄	—	2	1	no	—
517	C₂×C₄⋊C₄	—	2	2	no	—
518	C₂×C₄⋊C₄	—	2	1	no	—
519	C₂×C₄⋊C₄	—	2	2	no	—
520	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
521	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
522	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
523	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
524	#107	C₂³×C₃₁	1	4	yes	—
525	#107	C₂³×C₃₁	1	8	yes	—
526	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
527	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
528	#103	C₂³×C₃₁	1	8	yes	—
529	#103	C₂³×C₃₁	1	8	yes	—
530	#103	C₂³×C₃₁	1	8	yes	—
531	#102	C₂³×C₃₁	1	4	yes	—
532	#102	C₂³×C₃₁	1	8	yes	—
533	#102	C₂³×C₃₁	1	4	yes	—
534	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
535	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
536	#97	C₂³×C₃₁	1	4	yes	—
537	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
538	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
539	C₄×C₈	C₂³×C₃₁	1	2	yes	—
540	C₄×C₈	C₂³×C₃₁	1	2	yes	—
541	C₃₁	#62538	1	1	yes	characteristic, direct product factor, Sylow
542	#41	—	2	4	no	—
543	#41	—	4	8	no	—
544	#41	—	4	8	no	—
545	#41	—	4	16	no	—
546	#41	—	4	8	no	—
547	#41	—	4	4	no	—
548	#41	—	4	4	no	—
549	C₂×Q₈	—	2	1	no	—
550	C₂×Q₈	—	2	4	no	—
551	C₂×Q₈	—	2	2	no	—
552	C₂×Q₈	—	2	1	no	—
553	C₂×Q₈	—	2	2	no	—
554	C₂×Q₈	—	2	2	no	—
555	C₂×Q₈	—	4	8	no	—
556	C₂×Q₈	—	4	4	no	—
557	C₂×Q₈	—	4	8	no	—
558	C₂×Q₈	—	4	4	no	—
559	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
560	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
561	C₂×D₄	—	2	2	no	—
562	C₂×D₄	—	2	4	no	—
563	C₂×D₄	—	2	2	no	—
564	C₂×D₄	—	2	4	no	—
565	C₂×D₄	—	2	1	no	—
566	C₂×D₄	—	2	1	no	—
567	C₂×D₄	—	2	1	no	—
568	C₂×D₄	—	2	1	no	—
569	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
570	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
571	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
572	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
573	C₂²×C₄	—	2	4	no	—
574	C₂²×C₄	—	2	2	no	—
575	C₂²×C₄	—	2	2	no	—
576	C₂²×C₄	—	2	2	no	—
577	C₂²×C₄	—	2	4	no	—
578	C₂²×C₄	—	2	2	no	—
579	C₂²×C₄	—	2	4	no	—
580	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
581	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
582	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
583	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
584	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
585	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
586	SD₁₆	—	4	8	no	—
587	D₈	—	4	4	no	—
588	M₁₆	—	4	4	no	—
589	C₂×C₈	—	4	2	no	—
590	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
591	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
592	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
593	C₄⋊C₄	—	2	2	no	—
594	C₄⋊C₄	—	2	2	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	2	2	no	—
597	C₄⋊C₄	—	2	2	no	—
598	C₄⋊C₄	—	2	4	no	—
599	C₄⋊C₄	—	2	4	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	2	no	—
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	1	no	—
605	C₄⋊C₄	—	2	4	no	—
606	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
607	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
608	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
609	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
610	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
611	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
612	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
613	C₂²⋊C₄	—	2	4	no	—
614	C₂²⋊C₄	—	2	4	no	—
615	C₂²⋊C₄	—	2	4	no	—
616	C₂²⋊C₄	—	2	2	no	—
617	C₂²⋊C₄	—	2	4	no	—
618	C₂²⋊C₄	—	2	2	no	—
619	C₂²⋊C₄	—	2	4	no	—
620	C₂²⋊C₄	—	2	2	no	—
621	C₂²⋊C₄	—	2	2	no	—
622	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
623	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
624	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
625	C₄²	—	2	4	no	—
626	C₄²	C₂×D₄×C₃₁	1	2	yes	—
627	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
628	C₄²	C₂×D₄×C₃₁	1	2	yes	—
629	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
630	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
631	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
632	C₄²	C₂⁴×C₃₁	1	2	yes	—
633	C₂³	—	2	2	no	—
634	C₂³	—	2	2	no	—
635	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
636	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
637	Q₈	—	2	2	no	—
638	Q₈	—	2	8	no	—
639	Q₈	—	4	8	no	—
640	Q₈	—	4	2	no	—
641	Q₈	—	4	4	no	—
642	Q₈	—	4	4	no	—
643	Q₈	—	4	2	no	—
644	Q₈	—	8	2	no	—
645	D₄	—	2	4	no	—
646	D₄	—	2	2	no	—
647	D₄	—	2	4	no	—
648	D₄	—	2	4	no	—
649	D₄	—	4	8	no	—
650	D₄	—	4	4	no	—
651	D₄	—	4	4	no	—
652	D₄	—	4	2	no	—
653	D₄	—	4	2	no	—
654	D₄	—	4	2	no	—
655	D₄	—	4	2	no	—
656	D₄	—	8	4	no	—
657	C₂×C₄	—	2	4	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	4	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	4	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	4	no	—
664	C₂×C₄	—	2	4	no	—
665	C₂×C₄	—	2	2	no	—
666	C₂×C₄	—	4	8	no	—
667	C₂×C₄	—	4	8	no	—
668	C₂×C₄	—	4	4	no	—
669	C₂×C₄	—	4	4	no	—
670	C₂×C₄	—	4	4	no	—
671	C₂×C₄	—	8	4	no	—
672	C₂×C₄	—	8	2	no	—
673	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
674	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
675	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
676	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
677	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
678	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
679	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
680	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
681	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
682	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
683	C₈	—	2	2	no	—
684	C₈	—	2	2	no	—
685	C₈	—	2	4	no	—
686	C₂²	—	2	1	no	—
687	C₂²	—	2	1	no	—
688	C₂²	—	2	2	no	—
689	C₂²	—	2	2	no	—
690	C₂²	—	2	2	no	—
691	C₂²	—	4	2	no	—
692	C₂²	—	4	2	no	—
693	C₂²	—	4	2	no	—
694	C₂²	—	4	2	no	—
695	C₂²	—	4	2	no	—
696	C₂²	—	4	2	no	—
697	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
698	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
699	C₄	—	2	2	no	—
700	C₄	—	2	1	no	—
701	C₄	—	2	2	no	—
702	C₄	—	2	2	no	—
703	C₄	—	2	1	no	—
704	C₄	—	4	4	no	—
705	C₄	—	4	4	no	—
706	C₄	—	4	2	no	—
707	C₄	—	4	2	no	—
708	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
709	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
710	C₄	C₃₁×#578	1	2	yes	—
711	C₄	C₃₁×#585	1	2	yes	—
712	C₂	—	2	2	no	—
713	C₂	—	4	2	no	—
714	C₂	—	8	2	no	—
715	C₂	—	8	2	no	—
716	C₂	C₃₁×#3500	1	1	yes	characteristic
717	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
718	C₂	C₂×C₃₁×#578	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 ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3511	C₂	semidirect
2. C₃₁×#3499	C₂	semidirect
3. C₃₁×#3367	C₂	semidirect
4. C₃₁×#3330	C₂	semidirect
5. C₃₁×#3308	C₂	semidirect
6. C₃₁×#3230	C₂	semidirect
7. C₃₁×#3226	C₂	semidirect
8. C₃₁×#3207	C₂	semidirect
9. C₃₁×#3206	C₂	semidirect
10. C₃₁×#3182	C₂	semidirect
11. C₃₁×#3169	C₂	semidirect
12. C₃₁×#3141	C₂	semidirect
13. C₂×C₃₁×#487	C₂	semidirect
14. C₃₁×#2965	C₂	semidirect
273. #62538	C₃₁	direct (N × H)
327. #3511	C₂×C₃₁	semidirect
328. #3499	C₂×C₃₁	semidirect
329. #3367	C₂×C₃₁	semidirect
330. #3330	C₂×C₃₁	semidirect
331. #3308	C₂×C₃₁	semidirect
332. #3230	C₂×C₃₁	semidirect
333. #3226	C₂×C₃₁	semidirect
334. #3207	C₂×C₃₁	semidirect
335. #3206	C₂×C₃₁	semidirect
336. #3182	C₂×C₃₁	semidirect
337. #3169	C₂×C₃₁	semidirect
338. #3141	C₂×C₃₁	semidirect
339. C₂×#487	C₂×C₃₁	semidirect
340. #2965	C₂×C₃₁	semidirect
541. C₃₁	#62538	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×#3511	2
C₃₁×#3499	2
C₃₁×#3367	2
C₃₁×#3330	2
C₃₁×#3308	2
C₃₁×#3230	2
C₃₁×#3226	2
C₃₁×#3207	2
C₃₁×#3206	2
C₃₁×#3182	2
C₃₁×#3169	2
C₃₁×#3141	2
C₂×C₃₁×#487	2
C₃₁×#2965	2
#62538	31

Subgroups — C31×#62538 #21519297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62538 #21519297