Subgroups — C₃₁×#61012 #21517771

Order 7936.

1776 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3260	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3231	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3228	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3208	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3206	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3200	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3182	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3167	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3126	C₂	1	4	yes	maximal, complemented
12	C₃₁×#3123	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#559	—	2	2	no	—
15	C₃₁×#557	—	2	2	no	—
16	C₃₁×#556	—	2	2	no	—
17	C₃₁×#552	—	2	2	no	—
18	C₃₁×#552	—	2	2	no	—
19	C₃₁×#550	—	2	2	no	—
20	C₃₁×#546	—	2	2	no	—
21	C₃₁×#546	—	2	2	no	—
22	C₃₁×#541	C₂²	1	2	yes	—
23	C₃₁×#541	C₂²	1	2	yes	complemented
24	C₃₁×#540	C₂²	1	2	yes	—
25	C₃₁×#540	C₂²	1	2	yes	—
26	C₃₁×#536	C₂²	1	2	yes	—
27	C₃₁×#535	C₂²	1	2	yes	—
28	C₃₁×#533	C₂²	1	2	yes	—
29	C₃₁×#533	C₂²	1	1	yes	characteristic
30	C₃₁×#533	C₂²	1	2	yes	complemented
31	C₃₁×#533	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#532	C₂²	1	2	yes	—
33	C₃₁×#532	C₂²	1	2	yes	—
34	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
36	C₃₁×#519	C₂²	1	2	yes	complemented
37	C₃₁×#518	C₂²	1	2	yes	—
38	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#500	C₂²	1	2	yes	complemented
40	C₃₁×#498	C₂²	1	2	yes	complemented
41	C₃₁×#492	C₂²	1	2	yes	—
42	C₃₁×#492	C₂²	1	2	yes	—
43	C₃₁×#487	C₂²	1	2	yes	complemented
44	C₃₁×#487	C₂²	1	2	yes	complemented
45	C₃₁×#486	C₂²	1	2	yes	—
46	C₃₁×#486	C₂²	1	2	yes	—
47	C₃₁×#485	C₂²	1	8	yes	—
48	C₃₁×#482	C₂²	1	8	yes	complemented
49	C₃₁×#479	C₂²	1	8	yes	complemented
50	C₃₁×#478	C₂²	1	8	yes	complemented
51	C₃₁×#468	C₂²	1	8	yes	complemented
52	C₃₁×#465	C₂²	1	8	yes	complemented
53	C₃₁×#461	C₂²	1	8	yes	complemented
54	C₃₁×#460	C₂²	1	8	yes	—
55	C₃₁×#450	C₂²	1	8	yes	—
56	C₃₁×#448	C₂²	1	8	yes	complemented
57	C₃₁×#440	C₂²	1	8	yes	complemented
58	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
59	C₃₁×#433	C₂²	1	2	yes	—
60	C₃₁×#433	C₂²	1	2	yes	complemented
61	C₃₁×#431	C₂²	1	2	yes	—
62	C₃₁×#431	C₂²	1	2	yes	complemented
63	C₃₁×#424	C₂²	1	2	yes	complemented
64	C₃₁×#408	C₂²	1	2	yes	—
65	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
66	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
67	C₂×#41×C₃₁	—	2	2	no	—
68	C₂×#41×C₃₁	—	2	2	no	—
69	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
71	C₂²×D₄×C₃₁	—	2	2	no	—
72	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
73	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
74	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
75	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
76	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
77	C₃₁×C₄⋊Q₈	—	2	2	no	—
78	C₃₁×C₄⋊Q₈	—	2	2	no	—
79	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
80	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₃₁×#127	—	2	2	no	—
84	C₃₁×#127	—	2	2	no	—
85	C₃₁×#127	C₂³	1	2	yes	—
86	C₃₁×#127	C₂³	1	2	yes	—
87	C₃₁×#126	—	2	4	no	—
88	C₃₁×#126	—	2	4	no	—
89	C₃₁×#125	—	2	4	no	—
90	C₃₁×#125	—	2	4	no	—
91	C₃₁×#125	—	2	2	no	—
92	C₃₁×#125	—	2	2	no	—
93	C₃₁×#125	C₂³	1	2	yes	—
94	C₃₁×#125	C₂³	1	2	yes	—
95	C₃₁×#124	—	2	2	no	—
96	C₃₁×#124	—	2	2	no	—
97	C₃₁×#124	—	2	2	no	—
98	C₃₁×#124	—	2	2	no	—
99	C₃₁×#123	—	2	2	no	—
100	C₃₁×#123	—	2	2	no	—
101	C₃₁×#123	—	2	4	no	—
102	C₃₁×#123	—	2	4	no	—
103	C₃₁×C₂²⋊Q₈	—	2	2	no	—
104	C₃₁×C₂²⋊Q₈	—	2	2	no	—
105	C₃₁×C₂²⋊Q₈	—	2	2	no	—
106	C₃₁×C₂²⋊Q₈	—	2	2	no	—
107	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
108	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
109	C₃₁×#121	—	2	2	no	—
110	C₃₁×#121	—	2	2	no	—
111	C₃₁×#121	—	2	4	no	—
112	C₃₁×#121	—	2	4	no	—
113	C₃₁×#121	C₂³	1	4	yes	—
114	C₃₁×#121	C₂³	1	4	yes	—
115	C₃₁×#120	—	2	2	no	—
116	C₃₁×#120	—	2	2	no	—
117	C₄×Q₈×C₃₁	—	2	4	no	—
118	C₄×Q₈×C₃₁	—	2	2	no	—
119	C₄×Q₈×C₃₁	—	2	2	no	—
120	C₄×Q₈×C₃₁	—	2	2	no	—
121	C₄×Q₈×C₃₁	—	2	2	no	—
122	C₄×Q₈×C₃₁	D₄	1	2	yes	—
123	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
124	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
125	C₄×D₄×C₃₁	—	2	4	no	—
126	C₄×D₄×C₃₁	—	2	2	no	—
127	C₄×D₄×C₃₁	—	2	2	no	—
128	C₄×D₄×C₃₁	D₄	1	2	yes	—
129	C₄×D₄×C₃₁	D₄	1	2	yes	—
130	C₄×D₄×C₃₁	D₄	1	2	yes	—
131	C₄×D₄×C₃₁	C₂³	1	4	yes	—
132	C₄×D₄×C₃₁	C₂³	1	4	yes	—
133	C₃₁×#117	—	2	2	no	—
134	C₃₁×#117	—	2	2	no	—
135	C₃₁×#117	—	2	4	no	—
136	C₃₁×#117	C₂³	1	2	yes	—
137	C₃₁×#117	C₂³	1	2	yes	—
138	C₃₁×#117	C₂³	1	1	yes	characteristic
139	C₃₁×#117	C₂³	1	1	yes	characteristic
140	C₃₁×#117	C₂³	1	1	yes	characteristic
141	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
143	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
145	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
146	C₃₁×#107	C₂³	1	8	yes	—
147	C₃₁×#106	C₂³	1	8	yes	—
148	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
150	C₃₁×#103	C₂³	1	8	yes	—
151	C₃₁×#103	C₂³	1	8	yes	—
152	C₃₁×#102	C₂³	1	8	yes	—
153	C₃₁×#102	C₂³	1	8	yes	—
154	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
155	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
156	C₃₁×#97	C₂³	1	2	yes	—
157	C₄×C₈×C₃₁	C₂³	1	2	yes	—
158	C₄×C₈×C₃₁	C₂³	1	2	yes	—
159	C₄×C₈×C₃₁	C₂³	1	2	yes	—
160	C₂⁴×C₃₁	—	2	2	no	—
161	#41×C₃₁	—	4	2	no	—
162	#41×C₃₁	—	8	4	no	—
163	#41×C₃₁	—	8	4	no	—
164	C₂×Q₈×C₃₁	—	2	2	no	—
165	C₂×Q₈×C₃₁	—	2	2	no	—
166	C₂×Q₈×C₃₁	—	2	2	no	—
167	C₂×Q₈×C₃₁	—	4	4	no	—
168	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
170	C₂×D₄×C₃₁	—	2	4	no	—
171	C₂×D₄×C₃₁	—	2	4	no	—
172	C₂×D₄×C₃₁	—	2	2	no	—
173	C₂×D₄×C₃₁	—	2	2	no	—
174	C₂×D₄×C₃₁	—	2	2	no	—
175	C₂×D₄×C₃₁	—	2	4	no	—
176	C₂×D₄×C₃₁	—	4	4	no	—
177	C₂×D₄×C₃₁	—	8	4	no	—
178	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₃₁	—	2	2	no	—
183	C₂²×C₄×C₃₁	—	2	4	no	—
184	C₂²×C₄×C₃₁	—	2	4	no	—
185	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
191	SD₁₆×C₃₁	—	4	8	no	—
192	SD₁₆×C₃₁	—	4	8	no	—
193	M₁₆×C₃₁	—	4	2	no	—
194	C₂×C₈×C₃₁	—	4	2	no	—
195	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
196	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
197	C₄⋊C₄×C₃₁	—	2	2	no	—
198	C₄⋊C₄×C₃₁	—	2	2	no	—
199	C₄⋊C₄×C₃₁	—	2	4	no	—
200	C₄⋊C₄×C₃₁	—	2	2	no	—
201	C₄⋊C₄×C₃₁	—	2	4	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	2	no	—
204	C₄⋊C₄×C₃₁	—	2	4	no	—
205	C₄⋊C₄×C₃₁	—	2	4	no	—
206	C₄⋊C₄×C₃₁	—	2	4	no	—
207	C₄⋊C₄×C₃₁	—	2	4	no	—
208	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
209	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
210	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
211	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
212	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
213	C₂²⋊C₄×C₃₁	—	2	2	no	—
214	C₂²⋊C₄×C₃₁	—	2	2	no	—
215	C₂²⋊C₄×C₃₁	—	2	4	no	—
216	C₂²⋊C₄×C₃₁	—	2	4	no	—
217	C₂²⋊C₄×C₃₁	—	2	4	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	—	2	4	no	—
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	2	yes	—
224	C₄²×C₃₁	—	2	4	no	—
225	C₄²×C₃₁	—	2	4	no	—
226	C₄²×C₃₁	—	2	2	no	—
227	C₄²×C₃₁	—	2	2	no	—
228	C₄²×C₃₁	C₂×D₄	1	2	yes	—
229	C₄²×C₃₁	C₂×D₄	1	2	yes	—
230	C₄²×C₃₁	C₂×D₄	1	2	yes	—
231	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
232	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
233	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
234	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
235	#61012	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂³×C₃₁	—	2	4	no	—
237	C₂³×C₃₁	—	4	4	no	—
238	C₂³×C₃₁	—	8	4	no	—
239	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
241	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	Q₈×C₃₁	—	2	8	no	—
243	Q₈×C₃₁	—	4	1	no	—
244	Q₈×C₃₁	—	4	4	no	—
245	Q₈×C₃₁	—	4	4	no	—
246	Q₈×C₃₁	—	4	4	no	—
247	D₄×C₃₁	—	2	8	no	—
248	D₄×C₃₁	—	4	1	no	—
249	D₄×C₃₁	—	4	1	no	—
250	D₄×C₃₁	—	4	1	no	—
251	D₄×C₃₁	—	4	4	no	—
252	D₄×C₃₁	—	4	4	no	—
253	D₄×C₃₁	—	4	4	no	—
254	D₄×C₃₁	—	8	4	no	—
255	D₄×C₃₁	—	8	4	no	—
256	D₄×C₃₁	—	8	4	no	—
257	C₂×C₄×C₃₁	—	2	4	no	—
258	C₂×C₄×C₃₁	—	2	4	no	—
259	C₂×C₄×C₃₁	—	2	4	no	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	4	no	—
264	C₂×C₄×C₃₁	—	2	4	no	—
265	C₂×C₄×C₃₁	—	2	4	no	—
266	C₂×C₄×C₃₁	—	4	4	no	—
267	C₂×C₄×C₃₁	—	8	4	no	—
268	C₂×C₄×C₃₁	—	8	4	no	—
269	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
276	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
277	C₈×C₃₁	—	2	4	no	—
278	C₈×C₃₁	—	2	4	no	—
279	#3509	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#3508	C₂×C₃₁	1	1	yes	characteristic
281	#3260	C₂×C₃₁	1	4	yes	complemented
282	#3231	C₂×C₃₁	1	2	yes	complemented
283	#3228	C₂×C₃₁	1	2	yes	complemented
284	#3208	C₂×C₃₁	1	2	yes	complemented
285	#3206	C₂×C₃₁	1	2	yes	complemented
286	#3200	C₂×C₃₁	1	4	yes	complemented
287	#3182	C₂×C₃₁	1	2	yes	complemented
288	#3167	C₂×C₃₁	1	2	yes	complemented
289	#3126	C₂×C₃₁	1	4	yes	complemented
290	#3123	C₂×C₃₁	1	4	yes	complemented
291	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
292	C₂²×C₃₁	—	2	1	no	—
293	C₂²×C₃₁	—	2	1	no	—
294	C₂²×C₃₁	—	2	1	no	—
295	C₂²×C₃₁	—	2	2	no	—
296	C₂²×C₃₁	—	2	2	no	—
297	C₂²×C₃₁	—	2	2	no	—
298	C₂²×C₃₁	—	4	4	no	—
299	C₂²×C₃₁	—	4	4	no	—
300	C₂²×C₃₁	—	4	4	no	—
301	C₂²×C₃₁	—	16	4	no	—
302	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
303	C₄×C₃₁	—	2	1	no	—
304	C₄×C₃₁	—	2	1	no	—
305	C₄×C₃₁	—	2	1	no	—
306	C₄×C₃₁	—	2	2	no	—
307	C₄×C₃₁	—	2	2	no	—
308	C₄×C₃₁	—	2	2	no	—
309	C₄×C₃₁	—	4	4	no	—
310	C₄×C₃₁	—	4	4	no	—
311	C₄×C₃₁	—	4	4	no	—
312	C₄×C₃₁	#577	1	4	yes	—
313	C₄×C₃₁	#585	1	2	yes	—
314	#559	—	2	2	no	—
315	#557	—	2	2	no	—
316	#556	—	2	2	no	—
317	#552	—	2	2	no	—
318	#552	—	2	2	no	—
319	#550	—	2	2	no	—
320	#546	—	2	2	no	—
321	#546	—	2	2	no	—
322	#541	C₂²×C₃₁	1	2	yes	—
323	#541	C₂²×C₃₁	1	2	yes	complemented
324	#540	C₂²×C₃₁	1	2	yes	—
325	#540	C₂²×C₃₁	1	2	yes	—
326	#536	C₂²×C₃₁	1	2	yes	—
327	#535	C₂²×C₃₁	1	2	yes	—
328	#533	C₂²×C₃₁	1	2	yes	—
329	#533	C₂²×C₃₁	1	1	yes	characteristic
330	#533	C₂²×C₃₁	1	2	yes	complemented
331	#533	C₂²×C₃₁	1	1	yes	characteristic, complemented
332	#532	C₂²×C₃₁	1	2	yes	—
333	#532	C₂²×C₃₁	1	2	yes	—
334	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
335	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
336	#519	C₂²×C₃₁	1	2	yes	complemented
337	#518	C₂²×C₃₁	1	2	yes	—
338	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
339	#500	C₂²×C₃₁	1	2	yes	complemented
340	#498	C₂²×C₃₁	1	2	yes	complemented
341	#492	C₂²×C₃₁	1	2	yes	—
342	#492	C₂²×C₃₁	1	2	yes	—
343	#487	C₂²×C₃₁	1	2	yes	complemented
344	#487	C₂²×C₃₁	1	2	yes	complemented
345	#486	C₂²×C₃₁	1	2	yes	—
346	#486	C₂²×C₃₁	1	2	yes	—
347	#485	C₂²×C₃₁	1	8	yes	—
348	#482	C₂²×C₃₁	1	8	yes	complemented
349	#479	C₂²×C₃₁	1	8	yes	complemented
350	#478	C₂²×C₃₁	1	8	yes	complemented
351	#468	C₂²×C₃₁	1	8	yes	complemented
352	#465	C₂²×C₃₁	1	8	yes	complemented
353	#461	C₂²×C₃₁	1	8	yes	complemented
354	#460	C₂²×C₃₁	1	8	yes	—
355	#450	C₂²×C₃₁	1	8	yes	—
356	#448	C₂²×C₃₁	1	8	yes	complemented
357	#440	C₂²×C₃₁	1	8	yes	complemented
358	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
359	#433	C₂²×C₃₁	1	2	yes	—
360	#433	C₂²×C₃₁	1	2	yes	complemented
361	#431	C₂²×C₃₁	1	2	yes	—
362	#431	C₂²×C₃₁	1	2	yes	complemented
363	#424	C₂²×C₃₁	1	2	yes	complemented
364	#408	C₂²×C₃₁	1	2	yes	—
365	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
366	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
367	C₂×C₃₁	—	4	1	no	—
368	C₂×C₃₁	—	4	1	no	—
369	C₂×C₃₁	—	4	1	no	—
370	C₂×C₃₁	—	8	4	no	—
371	C₂×C₃₁	#3449	1	1	yes	characteristic
372	C₂×C₃₁	C₂×#577	1	2	yes	—
373	C₂×#41	—	2	2	no	—
374	C₂×#41	—	2	2	no	—
375	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
376	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
377	C₂²×D₄	—	2	2	no	—
378	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
379	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
380	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
381	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
382	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
383	C₄⋊Q₈	—	2	2	no	—
384	C₄⋊Q₈	—	2	2	no	—
385	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
386	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
387	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
388	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
389	#127	—	2	2	no	—
390	#127	—	2	2	no	—
391	#127	C₂³×C₃₁	1	2	yes	—
392	#127	C₂³×C₃₁	1	2	yes	—
393	#126	—	2	4	no	—
394	#126	—	2	4	no	—
395	#125	—	2	4	no	—
396	#125	—	2	4	no	—
397	#125	—	2	2	no	—
398	#125	—	2	2	no	—
399	#125	C₂³×C₃₁	1	2	yes	—
400	#125	C₂³×C₃₁	1	2	yes	—
401	#124	—	2	2	no	—
402	#124	—	2	2	no	—
403	#124	—	2	2	no	—
404	#124	—	2	2	no	—
405	#123	—	2	2	no	—
406	#123	—	2	2	no	—
407	#123	—	2	4	no	—
408	#123	—	2	4	no	—
409	C₂²⋊Q₈	—	2	2	no	—
410	C₂²⋊Q₈	—	2	2	no	—
411	C₂²⋊Q₈	—	2	2	no	—
412	C₂²⋊Q₈	—	2	2	no	—
413	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
414	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
415	#121	—	2	2	no	—
416	#121	—	2	2	no	—
417	#121	—	2	4	no	—
418	#121	—	2	4	no	—
419	#121	C₂³×C₃₁	1	4	yes	—
420	#121	C₂³×C₃₁	1	4	yes	—
421	#120	—	2	2	no	—
422	#120	—	2	2	no	—
423	C₄×Q₈	—	2	4	no	—
424	C₄×Q₈	—	2	2	no	—
425	C₄×Q₈	—	2	2	no	—
426	C₄×Q₈	—	2	2	no	—
427	C₄×Q₈	—	2	2	no	—
428	C₄×Q₈	D₄×C₃₁	1	2	yes	—
429	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
430	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
431	C₄×D₄	—	2	4	no	—
432	C₄×D₄	—	2	2	no	—
433	C₄×D₄	—	2	2	no	—
434	C₄×D₄	D₄×C₃₁	1	2	yes	—
435	C₄×D₄	D₄×C₃₁	1	2	yes	—
436	C₄×D₄	D₄×C₃₁	1	2	yes	—
437	C₄×D₄	C₂³×C₃₁	1	4	yes	—
438	C₄×D₄	C₂³×C₃₁	1	4	yes	—
439	#117	—	2	2	no	—
440	#117	—	2	2	no	—
441	#117	—	2	4	no	—
442	#117	C₂³×C₃₁	1	2	yes	—
443	#117	C₂³×C₃₁	1	2	yes	—
444	#117	C₂³×C₃₁	1	1	yes	characteristic
445	#117	C₂³×C₃₁	1	1	yes	characteristic
446	#117	C₂³×C₃₁	1	1	yes	characteristic
447	C₂×C₄⋊C₄	—	2	2	no	—
448	C₂×C₂²⋊C₄	—	2	4	no	—
449	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
452	#107	C₂³×C₃₁	1	8	yes	—
453	#106	C₂³×C₃₁	1	8	yes	—
454	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
455	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
456	#103	C₂³×C₃₁	1	8	yes	—
457	#103	C₂³×C₃₁	1	8	yes	—
458	#102	C₂³×C₃₁	1	8	yes	—
459	#102	C₂³×C₃₁	1	8	yes	—
460	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
461	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
462	#97	C₂³×C₃₁	1	2	yes	—
463	C₄×C₈	C₂³×C₃₁	1	2	yes	—
464	C₄×C₈	C₂³×C₃₁	1	2	yes	—
465	C₄×C₈	C₂³×C₃₁	1	2	yes	—
466	C₃₁	#61012	1	1	yes	characteristic, direct product factor, Sylow
467	C₂⁴	—	2	2	no	—
468	#41	—	4	2	no	—
469	#41	—	8	4	no	—
470	#41	—	8	4	no	—
471	C₂×Q₈	—	2	2	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	2	2	no	—
474	C₂×Q₈	—	4	4	no	—
475	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	4	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	4	4	no	—
484	C₂×D₄	—	8	4	no	—
485	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
493	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
494	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
495	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
498	SD₁₆	—	4	8	no	—
499	SD₁₆	—	4	8	no	—
500	M₁₆	—	4	2	no	—
501	C₂×C₈	—	4	2	no	—
502	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
503	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	4	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
516	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
517	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
518	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
519	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	4	no	—
523	C₂²⋊C₄	—	2	4	no	—
524	C₂²⋊C₄	—	2	4	no	—
525	C₂²⋊C₄	—	2	4	no	—
526	C₂²⋊C₄	—	2	4	no	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
529	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
530	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
531	C₄²	—	2	4	no	—
532	C₄²	—	2	4	no	—
533	C₄²	—	2	2	no	—
534	C₄²	—	2	2	no	—
535	C₄²	C₂×D₄×C₃₁	1	2	yes	—
536	C₄²	C₂×D₄×C₃₁	1	2	yes	—
537	C₄²	C₂×D₄×C₃₁	1	2	yes	—
538	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
539	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
540	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
541	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
542	C₂³	—	2	4	no	—
543	C₂³	—	4	4	no	—
544	C₂³	—	8	4	no	—
545	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
546	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
547	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	Q₈	—	2	8	no	—
549	Q₈	—	4	1	no	—
550	Q₈	—	4	4	no	—
551	Q₈	—	4	4	no	—
552	Q₈	—	4	4	no	—
553	D₄	—	2	8	no	—
554	D₄	—	4	1	no	—
555	D₄	—	4	1	no	—
556	D₄	—	4	1	no	—
557	D₄	—	4	4	no	—
558	D₄	—	4	4	no	—
559	D₄	—	4	4	no	—
560	D₄	—	8	4	no	—
561	D₄	—	8	4	no	—
562	D₄	—	8	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	8	4	no	—
574	C₂×C₄	—	8	4	no	—
575	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
576	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
577	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
580	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
582	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
583	C₈	—	2	4	no	—
584	C₈	—	2	4	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	2	1	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	4	4	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	16	4	no	—
595	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
596	C₄	—	2	1	no	—
597	C₄	—	2	1	no	—
598	C₄	—	2	1	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	2	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	4	no	—
604	C₄	—	4	4	no	—
605	C₄	C₃₁×#577	1	4	yes	—
606	C₄	C₃₁×#585	1	2	yes	—
607	C₂	—	4	1	no	—
608	C₂	—	4	1	no	—
609	C₂	—	4	1	no	—
610	C₂	—	8	4	no	—
611	C₂	C₃₁×#3449	1	1	yes	characteristic
612	C₂	C₂×C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3508	C₂
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×#3509	C₂	semidirect
3. C₃₁×#3260	C₂	semidirect
4. C₃₁×#3231	C₂	semidirect
5. C₃₁×#3228	C₂	semidirect
6. C₃₁×#3208	C₂	semidirect
7. C₃₁×#3206	C₂	semidirect
8. C₃₁×#3200	C₂	semidirect
9. C₃₁×#3182	C₂	semidirect
10. C₃₁×#3167	C₂	semidirect
11. C₃₁×#3126	C₂	semidirect
12. C₃₁×#3123	C₂	semidirect
13. C₃₁×#2922	C₂	semidirect
23. C₃₁×#541	C₂²	semidirect
30. C₃₁×#533	C₂²	semidirect
31. C₃₁×#533	C₂²	semidirect
34. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
36. C₃₁×#519	C₂²	semidirect
39. C₃₁×#500	C₂²	semidirect
40. C₃₁×#498	C₂²	semidirect
43. C₃₁×#487	C₂²	semidirect
44. C₃₁×#487	C₂²	semidirect
48. C₃₁×#482	C₂²	semidirect
49. C₃₁×#479	C₂²	semidirect
50. C₃₁×#478	C₂²	semidirect
51. C₃₁×#468	C₂²	semidirect
52. C₃₁×#465	C₂²	semidirect
53. C₃₁×#461	C₂²	semidirect
56. C₃₁×#448	C₂²	semidirect
57. C₃₁×#440	C₂²	semidirect
58. C₄×SD₁₆×C₃₁	C₂²	semidirect
60. C₃₁×#433	C₂²	semidirect
62. C₃₁×#431	C₂²	semidirect
63. C₃₁×#424	C₂²	semidirect
65. C₄×M₁₆×C₃₁	C₂²	semidirect
66. C₂×C₄×C₈×C₃₁	C₂²	semidirect
235. #61012	C₃₁	direct (N × H)
279. #3509	C₂×C₃₁	semidirect
281. #3260	C₂×C₃₁	semidirect
282. #3231	C₂×C₃₁	semidirect
283. #3228	C₂×C₃₁	semidirect
284. #3208	C₂×C₃₁	semidirect
285. #3206	C₂×C₃₁	semidirect
286. #3200	C₂×C₃₁	semidirect
287. #3182	C₂×C₃₁	semidirect
288. #3167	C₂×C₃₁	semidirect
289. #3126	C₂×C₃₁	semidirect
290. #3123	C₂×C₃₁	semidirect
291. #2922	C₂×C₃₁	semidirect
323. #541	C₂²×C₃₁	semidirect
330. #533	C₂²×C₃₁	semidirect
331. #533	C₂²×C₃₁	semidirect
334. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
336. #519	C₂²×C₃₁	semidirect
339. #500	C₂²×C₃₁	semidirect
340. #498	C₂²×C₃₁	semidirect
343. #487	C₂²×C₃₁	semidirect
344. #487	C₂²×C₃₁	semidirect
348. #482	C₂²×C₃₁	semidirect
349. #479	C₂²×C₃₁	semidirect
350. #478	C₂²×C₃₁	semidirect
351. #468	C₂²×C₃₁	semidirect
352. #465	C₂²×C₃₁	semidirect
353. #461	C₂²×C₃₁	semidirect
356. #448	C₂²×C₃₁	semidirect
357. #440	C₂²×C₃₁	semidirect
358. C₄×SD₁₆	C₂²×C₃₁	semidirect
360. #433	C₂²×C₃₁	semidirect
362. #431	C₂²×C₃₁	semidirect
363. #424	C₂²×C₃₁	semidirect
365. C₄×M₁₆	C₂²×C₃₁	semidirect
366. C₂×C₄×C₈	C₂²×C₃₁	semidirect
466. C₃₁	#61012	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×#3509	2
C₃₁×#3508	2
C₃₁×#3260	2
C₃₁×#3231	2
C₃₁×#3228	2
C₃₁×#3208	2
C₃₁×#3206	2
C₃₁×#3200	2
C₃₁×#3182	2
C₃₁×#3167	2
C₃₁×#3126	2
C₃₁×#3123	2
C₃₁×#2922	2
#61012	31

Subgroups — C31×#61012 #21517771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61012 #21517771