Subgroups — C₃₁×#61942 #21518701

Order 7936.

1762 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3548	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3399	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3397	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3376	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3370	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3221	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3220	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3213	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3126	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3122	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3115	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2987	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#562	—	2	2	no	—
15	C₃₁×#559	C₂²	1	4	yes	—
16	C₃₁×#556	C₂²	1	4	yes	—
17	C₃₁×#554	C₂²	1	4	yes	complemented
18	C₃₁×#551	—	2	4	no	—
19	C₃₁×#550	C₂²	1	4	yes	complemented
20	C₃₁×#547	—	2	2	no	—
21	C₃₁×#542	—	2	2	no	—
22	C₃₁×#540	C₂²	1	2	yes	—
23	C₃₁×#540	C₂²	1	2	yes	—
24	C₃₁×#537	—	2	4	no	—
25	C₃₁×#536	C₂²	1	1	yes	characteristic
26	C₃₁×#535	C₂²	1	1	yes	characteristic
27	C₃₁×#533	C₂²	1	1	yes	characteristic
28	C₃₁×#532	C₂²	1	2	yes	—
29	C₃₁×#529	—	2	2	no	—
30	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
31	C₃₁×#519	C₂²	1	1	yes	characteristic
32	C₃₁×#518	C₂²	1	1	yes	characteristic
33	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#496	C₂²	1	2	yes	—
35	C₃₁×#496	C₂²	1	4	yes	—
36	C₃₁×#496	C₂²	1	2	yes	complemented
37	C₃₁×#493	C₂²	1	2	yes	—
38	C₃₁×#492	C₂²	1	2	yes	complemented
39	C₃₁×#491	C₂²	1	2	yes	complemented
40	C₃₁×#490	C₂²	1	2	yes	—
41	C₃₁×#489	C₂²	1	4	yes	—
42	C₃₁×#488	C₂²	1	2	yes	—
43	C₃₁×#488	C₂²	1	2	yes	complemented
44	C₃₁×#482	C₂²	1	4	yes	—
45	C₃₁×#479	C₂²	1	4	yes	complemented
46	C₃₁×#478	C₂²	1	4	yes	—
47	C₃₁×#475	C₂²	1	2	yes	complemented
48	C₃₁×#474	C₂²	1	2	yes	complemented
49	C₃₁×#463	C₂²	1	4	yes	—
50	C₃₁×#463	C₂²	1	4	yes	—
51	C₃₁×#463	C₂²	1	4	yes	—
52	C₃₁×#463	C₂²	1	4	yes	complemented
53	C₃₁×#461	C₂²	1	4	yes	complemented
54	C₃₁×#460	C₂²	1	2	yes	complemented
55	C₃₁×#460	C₂²	1	4	yes	complemented
56	C₃₁×#459	C₂²	1	4	yes	—
57	C₃₁×#459	C₂²	1	4	yes	—
58	C₃₁×#459	C₂²	1	2	yes	complemented
59	C₃₁×#449	C₂²	1	4	yes	—
60	C₃₁×#448	C₂²	1	4	yes	—
61	C₃₁×#447	C₂²	1	4	yes	—
62	C₃₁×#446	C₂²	1	4	yes	complemented
63	C₃₁×#446	C₂²	1	4	yes	complemented
64	C₃₁×#442	C₂²	1	4	yes	—
65	C₃₁×#442	C₂²	1	4	yes	complemented
66	C₃₁×#440	C₂²	1	4	yes	complemented
67	C₃₁×#440	C₂²	1	4	yes	complemented
68	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
69	C₄×D₈×C₃₁	C₂²	1	4	yes	—
70	C₃₁×#433	C₂²	1	2	yes	—
71	C₃₁×#433	C₂²	1	2	yes	—
72	C₃₁×#432	C₂²	1	2	yes	—
73	C₂×#41×C₃₁	—	2	4	no	—
74	C₂×#41×C₃₁	—	2	2	no	—
75	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
79	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
80	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
81	C₂×D₈×C₃₁	C₂³	1	4	yes	—
82	C₂×D₈×C₃₁	C₂³	1	4	yes	—
83	C₂×D₈×C₃₁	C₂³	1	4	yes	—
84	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
85	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
87	C₃₁×#127	C₂³	1	2	yes	—
88	C₃₁×#127	C₂³	1	2	yes	—
89	C₃₁×#127	C₂³	1	2	yes	—
90	C₃₁×#127	C₂³	1	2	yes	—
91	C₃₁×#126	—	2	4	no	—
92	C₃₁×#126	—	2	4	no	—
93	C₃₁×#125	D₄	1	2	yes	—
94	C₃₁×#125	C₂³	1	2	yes	—
95	C₃₁×#125	C₂³	1	2	yes	—
96	C₃₁×#124	—	2	4	no	—
97	C₃₁×#124	—	2	1	no	—
98	C₃₁×#124	—	2	1	no	—
99	C₃₁×#124	—	2	2	no	—
100	C₃₁×#124	C₂³	1	2	yes	—
101	C₃₁×#124	C₂³	1	4	yes	—
102	C₃₁×#124	C₂³	1	2	yes	—
103	C₃₁×#123	—	2	4	no	—
104	C₃₁×#123	—	2	4	no	—
105	C₃₁×#123	—	2	4	no	—
106	C₃₁×#123	—	2	4	no	—
107	C₃₁×#123	—	2	2	no	—
108	C₃₁×C₂²⋊Q₈	—	2	4	no	—
109	C₃₁×C₂²⋊Q₈	—	2	4	no	—
110	C₃₁×C₂²⋊Q₈	—	2	2	no	—
111	C₃₁×C₂²⋊Q₈	—	2	4	no	—
112	C₃₁×C₂²⋊Q₈	—	2	4	no	—
113	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
114	C₃₁×#121	—	2	2	no	—
115	C₃₁×#121	—	2	2	no	—
116	C₃₁×#121	—	2	4	no	—
117	C₃₁×#121	—	2	4	no	—
118	C₃₁×#121	—	2	4	no	—
119	C₃₁×#121	—	2	4	no	—
120	C₃₁×#121	—	2	4	no	—
121	C₃₁×#121	C₂³	1	4	yes	—
122	C₃₁×#120	—	2	2	no	—
123	C₄×Q₈×C₃₁	—	2	2	no	—
124	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
125	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
126	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
127	C₄×D₄×C₃₁	—	2	2	no	—
128	C₄×D₄×C₃₁	—	2	2	no	—
129	C₄×D₄×C₃₁	—	2	4	no	—
130	C₄×D₄×C₃₁	—	2	2	no	—
131	C₄×D₄×C₃₁	C₂³	1	4	yes	—
132	C₄×D₄×C₃₁	C₂³	1	4	yes	—
133	C₄×D₄×C₃₁	C₂³	1	4	yes	—
134	C₄×D₄×C₃₁	C₂³	1	4	yes	—
135	C₄×D₄×C₃₁	C₂³	1	2	yes	—
136	C₃₁×#117	—	2	4	no	—
137	C₃₁×#117	—	2	2	no	—
138	C₃₁×#117	—	2	1	no	—
139	C₃₁×#117	C₂³	1	1	yes	characteristic
140	C₃₁×#117	C₂³	1	1	yes	characteristic
141	C₃₁×#117	C₂³	1	1	yes	characteristic
142	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
143	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
144	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
145	C₂×C₄²×C₃₁	—	2	1	no	—
146	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
147	C₃₁×#107	C₂³	1	4	yes	—
148	C₃₁×#106	C₂³	1	4	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
150	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
151	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
152	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
153	C₃₁×#103	C₂³	1	4	yes	—
154	C₃₁×#103	C₂³	1	4	yes	—
155	C₃₁×#103	C₂³	1	4	yes	—
156	C₃₁×#102	C₂³	1	4	yes	—
157	C₃₁×#102	C₂³	1	4	yes	—
158	C₃₁×#102	C₂³	1	4	yes	—
159	C₃₁×#102	C₂³	1	4	yes	—
160	C₃₁×#102	C₂³	1	4	yes	—
161	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
162	C₃₁×#97	C₂³	1	4	yes	—
163	C₃₁×#97	C₂³	1	2	yes	—
164	C₃₁×#97	C₂³	1	2	yes	—
165	C₄×C₈×C₃₁	C₂³	1	2	yes	—
166	C₄×C₈×C₃₁	C₂³	1	2	yes	—
167	C₂⁴×C₃₁	—	2	1	no	—
168	#41×C₃₁	—	4	2	no	—
169	#41×C₃₁	—	8	4	no	—
170	#41×C₃₁	—	8	8	no	—
171	C₂×Q₈×C₃₁	—	2	1	no	—
172	C₂×Q₈×C₃₁	—	2	2	no	—
173	C₂×Q₈×C₃₁	—	4	2	no	—
174	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
175	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
176	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂×D₄×C₃₁	—	2	1	no	—
178	C₂×D₄×C₃₁	—	2	4	no	—
179	C₂×D₄×C₃₁	—	2	1	no	—
180	C₂×D₄×C₃₁	—	2	2	no	—
181	C₂×D₄×C₃₁	—	2	4	no	—
182	C₂×D₄×C₃₁	—	2	4	no	—
183	C₂×D₄×C₃₁	—	2	4	no	—
184	C₂×D₄×C₃₁	—	2	2	no	—
185	C₂×D₄×C₃₁	—	4	2	no	—
186	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
187	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
188	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₃₁	—	2	1	no	—
190	C₂²×C₄×C₃₁	—	2	1	no	—
191	C₂²×C₄×C₃₁	—	2	4	no	—
192	C₂²×C₄×C₃₁	—	2	2	no	—
193	C₂²×C₄×C₃₁	—	2	4	no	—
194	C₂²×C₄×C₃₁	—	2	4	no	—
195	C₂²×C₄×C₃₁	—	2	4	no	—
196	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
198	SD₁₆×C₃₁	—	4	4	no	—
199	SD₁₆×C₃₁	—	4	4	no	—
200	SD₁₆×C₃₁	—	4	4	no	—
201	D₈×C₃₁	—	4	4	no	—
202	D₈×C₃₁	—	4	4	no	—
203	D₈×C₃₁	—	4	4	no	—
204	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
205	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
206	C₄⋊C₄×C₃₁	—	2	2	no	—
207	C₄⋊C₄×C₃₁	—	2	4	no	—
208	C₄⋊C₄×C₃₁	—	2	4	no	—
209	C₄⋊C₄×C₃₁	—	2	2	no	—
210	C₄⋊C₄×C₃₁	—	2	4	no	—
211	C₄⋊C₄×C₃₁	—	2	4	no	—
212	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
213	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
214	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
215	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
217	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
218	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
219	C₂²⋊C₄×C₃₁	—	2	1	no	—
220	C₂²⋊C₄×C₃₁	—	2	4	no	—
221	C₂²⋊C₄×C₃₁	—	2	4	no	—
222	C₂²⋊C₄×C₃₁	—	2	4	no	—
223	C₂²⋊C₄×C₃₁	—	2	2	no	—
224	C₂²⋊C₄×C₃₁	—	2	4	no	—
225	C₂²⋊C₄×C₃₁	—	2	4	no	—
226	C₂²⋊C₄×C₃₁	—	2	4	no	—
227	C₂²⋊C₄×C₃₁	—	2	4	no	—
228	C₂²⋊C₄×C₃₁	—	2	4	no	—
229	C₂²⋊C₄×C₃₁	—	2	4	no	—
230	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
231	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
232	C₄²×C₃₁	—	2	2	no	—
233	C₄²×C₃₁	—	2	1	no	—
234	C₄²×C₃₁	—	2	4	no	—
235	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
237	C₄²×C₃₁	C₂⁴	1	2	yes	—
238	C₄²×C₃₁	C₂⁴	1	2	yes	—
239	C₄²×C₃₁	C₂⁴	1	2	yes	—
240	#61942	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂³×C₃₁	—	2	2	no	—
242	C₂³×C₃₁	—	2	1	no	—
243	C₂³×C₃₁	—	2	4	no	—
244	C₂³×C₃₁	—	4	2	no	—
245	C₂³×C₃₁	—	8	2	no	—
246	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
247	Q₈×C₃₁	—	2	4	no	—
248	Q₈×C₃₁	—	2	2	no	—
249	Q₈×C₃₁	—	4	4	no	—
250	Q₈×C₃₁	—	4	2	no	—
251	Q₈×C₃₁	—	8	2	no	—
252	D₄×C₃₁	—	2	4	no	—
253	D₄×C₃₁	—	2	4	no	—
254	D₄×C₃₁	—	2	2	no	—
255	D₄×C₃₁	—	2	4	no	—
256	D₄×C₃₁	—	4	2	no	—
257	D₄×C₃₁	—	4	4	no	—
258	D₄×C₃₁	—	4	2	no	—
259	D₄×C₃₁	—	8	4	no	—
260	D₄×C₃₁	—	8	2	no	—
261	D₄×C₃₁	—	8	4	no	—
262	D₄×C₃₁	—	8	4	no	—
263	D₄×C₃₁	—	8	4	no	—
264	C₂×C₄×C₃₁	—	2	2	no	—
265	C₂×C₄×C₃₁	—	2	2	no	—
266	C₂×C₄×C₃₁	—	2	2	no	—
267	C₂×C₄×C₃₁	—	2	1	no	—
268	C₂×C₄×C₃₁	—	2	4	no	—
269	C₂×C₄×C₃₁	—	4	2	no	—
270	C₂×C₄×C₃₁	—	4	2	no	—
271	C₂×C₄×C₃₁	—	4	8	no	—
272	C₂×C₄×C₃₁	—	4	2	no	—
273	C₂×C₄×C₃₁	—	8	2	no	—
274	C₂×C₄×C₃₁	—	8	4	no	—
275	C₂×C₄×C₃₁	—	8	4	no	—
276	C₂×C₄×C₃₁	—	8	4	no	—
277	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
280	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
282	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
283	C₈×C₃₁	—	2	4	no	—
284	C₈×C₃₁	—	2	4	no	—
285	#3548	C₂×C₃₁	1	1	yes	characteristic, complemented
286	#3510	C₂×C₃₁	1	1	yes	characteristic, complemented
287	#3399	C₂×C₃₁	1	4	yes	complemented
288	#3397	C₂×C₃₁	1	4	yes	complemented
289	#3376	C₂×C₃₁	1	4	yes	complemented
290	#3370	C₂×C₃₁	1	4	yes	complemented
291	#3221	C₂×C₃₁	1	2	yes	complemented
292	#3220	C₂×C₃₁	1	2	yes	complemented
293	#3213	C₂×C₃₁	1	2	yes	complemented
294	#3126	C₂×C₃₁	1	2	yes	complemented
295	#3122	C₂×C₃₁	1	2	yes	complemented
296	#3115	C₂×C₃₁	1	2	yes	complemented
297	#2987	C₂×C₃₁	1	1	yes	characteristic, complemented
298	C₂²×C₃₁	—	2	2	no	—
299	C₂²×C₃₁	—	2	1	no	—
300	C₂²×C₃₁	—	4	4	no	—
301	C₂²×C₃₁	—	4	4	no	—
302	C₂²×C₃₁	—	4	2	no	—
303	C₂²×C₃₁	—	4	4	no	—
304	C₂²×C₃₁	—	4	1	no	—
305	C₂²×C₃₁	—	4	2	no	—
306	C₂²×C₃₁	—	4	4	no	—
307	C₂²×C₃₁	—	16	2	no	—
308	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
309	C₄×C₃₁	—	2	2	no	—
310	C₄×C₃₁	—	2	2	no	—
311	C₄×C₃₁	—	2	4	no	—
312	C₄×C₃₁	—	2	4	no	—
313	C₄×C₃₁	—	2	1	no	—
314	C₄×C₃₁	—	4	4	no	—
315	C₄×C₃₁	—	4	2	no	—
316	C₄×C₃₁	—	4	1	no	—
317	C₄×C₃₁	—	4	2	no	—
318	C₄×C₃₁	#585	1	2	yes	—
319	#562	—	2	2	no	—
320	#559	C₂²×C₃₁	1	4	yes	—
321	#556	C₂²×C₃₁	1	4	yes	—
322	#554	C₂²×C₃₁	1	4	yes	complemented
323	#551	—	2	4	no	—
324	#550	C₂²×C₃₁	1	4	yes	complemented
325	#547	—	2	2	no	—
326	#542	—	2	2	no	—
327	#540	C₂²×C₃₁	1	2	yes	—
328	#540	C₂²×C₃₁	1	2	yes	—
329	#537	—	2	4	no	—
330	#536	C₂²×C₃₁	1	1	yes	characteristic
331	#535	C₂²×C₃₁	1	1	yes	characteristic
332	#533	C₂²×C₃₁	1	1	yes	characteristic
333	#532	C₂²×C₃₁	1	2	yes	—
334	#529	—	2	2	no	—
335	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
336	#519	C₂²×C₃₁	1	1	yes	characteristic
337	#518	C₂²×C₃₁	1	1	yes	characteristic
338	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
339	#496	C₂²×C₃₁	1	2	yes	—
340	#496	C₂²×C₃₁	1	4	yes	—
341	#496	C₂²×C₃₁	1	2	yes	complemented
342	#493	C₂²×C₃₁	1	2	yes	—
343	#492	C₂²×C₃₁	1	2	yes	complemented
344	#491	C₂²×C₃₁	1	2	yes	complemented
345	#490	C₂²×C₃₁	1	2	yes	—
346	#489	C₂²×C₃₁	1	4	yes	—
347	#488	C₂²×C₃₁	1	2	yes	—
348	#488	C₂²×C₃₁	1	2	yes	complemented
349	#482	C₂²×C₃₁	1	4	yes	—
350	#479	C₂²×C₃₁	1	4	yes	complemented
351	#478	C₂²×C₃₁	1	4	yes	—
352	#475	C₂²×C₃₁	1	2	yes	complemented
353	#474	C₂²×C₃₁	1	2	yes	complemented
354	#463	C₂²×C₃₁	1	4	yes	—
355	#463	C₂²×C₃₁	1	4	yes	—
356	#463	C₂²×C₃₁	1	4	yes	—
357	#463	C₂²×C₃₁	1	4	yes	complemented
358	#461	C₂²×C₃₁	1	4	yes	complemented
359	#460	C₂²×C₃₁	1	2	yes	complemented
360	#460	C₂²×C₃₁	1	4	yes	complemented
361	#459	C₂²×C₃₁	1	4	yes	—
362	#459	C₂²×C₃₁	1	4	yes	—
363	#459	C₂²×C₃₁	1	2	yes	complemented
364	#449	C₂²×C₃₁	1	4	yes	—
365	#448	C₂²×C₃₁	1	4	yes	—
366	#447	C₂²×C₃₁	1	4	yes	—
367	#446	C₂²×C₃₁	1	4	yes	complemented
368	#446	C₂²×C₃₁	1	4	yes	complemented
369	#442	C₂²×C₃₁	1	4	yes	—
370	#442	C₂²×C₃₁	1	4	yes	complemented
371	#440	C₂²×C₃₁	1	4	yes	complemented
372	#440	C₂²×C₃₁	1	4	yes	complemented
373	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
374	C₄×D₈	C₂²×C₃₁	1	4	yes	—
375	#433	C₂²×C₃₁	1	2	yes	—
376	#433	C₂²×C₃₁	1	2	yes	—
377	#432	C₂²×C₃₁	1	2	yes	—
378	C₂×C₃₁	—	4	1	no	—
379	C₂×C₃₁	—	8	4	no	—
380	C₂×C₃₁	—	8	1	no	—
381	C₂×C₃₁	—	8	2	no	—
382	C₂×C₃₁	#3487	1	1	yes	characteristic
383	C₂×C₃₁	#3585	1	2	yes	—
384	C₂×#41	—	2	4	no	—
385	C₂×#41	—	2	2	no	—
386	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
387	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
388	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
389	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
390	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
391	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
392	C₂×D₈	C₂³×C₃₁	1	4	yes	—
393	C₂×D₈	C₂³×C₃₁	1	4	yes	—
394	C₂×D₈	C₂³×C₃₁	1	4	yes	—
395	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
396	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
397	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
398	#127	C₂³×C₃₁	1	2	yes	—
399	#127	C₂³×C₃₁	1	2	yes	—
400	#127	C₂³×C₃₁	1	2	yes	—
401	#127	C₂³×C₃₁	1	2	yes	—
402	#126	—	2	4	no	—
403	#126	—	2	4	no	—
404	#125	D₄×C₃₁	1	2	yes	—
405	#125	C₂³×C₃₁	1	2	yes	—
406	#125	C₂³×C₃₁	1	2	yes	—
407	#124	—	2	4	no	—
408	#124	—	2	1	no	—
409	#124	—	2	1	no	—
410	#124	—	2	2	no	—
411	#124	C₂³×C₃₁	1	4	yes	—
412	#124	C₂³×C₃₁	1	2	yes	—
413	#124	C₂³×C₃₁	1	2	yes	—
414	#123	—	2	4	no	—
415	#123	—	2	4	no	—
416	#123	—	2	4	no	—
417	#123	—	2	4	no	—
418	#123	—	2	2	no	—
419	C₂²⋊Q₈	—	2	4	no	—
420	C₂²⋊Q₈	—	2	4	no	—
421	C₂²⋊Q₈	—	2	2	no	—
422	C₂²⋊Q₈	—	2	4	no	—
423	C₂²⋊Q₈	—	2	4	no	—
424	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
425	#121	—	2	2	no	—
426	#121	—	2	2	no	—
427	#121	—	2	4	no	—
428	#121	—	2	4	no	—
429	#121	—	2	4	no	—
430	#121	—	2	4	no	—
431	#121	—	2	4	no	—
432	#121	C₂³×C₃₁	1	4	yes	—
433	#120	—	2	2	no	—
434	C₄×Q₈	—	2	2	no	—
435	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
436	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
437	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
438	C₄×D₄	—	2	2	no	—
439	C₄×D₄	—	2	2	no	—
440	C₄×D₄	—	2	4	no	—
441	C₄×D₄	—	2	2	no	—
442	C₄×D₄	C₂³×C₃₁	1	2	yes	—
443	C₄×D₄	C₂³×C₃₁	1	4	yes	—
444	C₄×D₄	C₂³×C₃₁	1	4	yes	—
445	C₄×D₄	C₂³×C₃₁	1	4	yes	—
446	C₄×D₄	C₂³×C₃₁	1	4	yes	—
447	#117	—	2	4	no	—
448	#117	—	2	2	no	—
449	#117	—	2	1	no	—
450	#117	C₂³×C₃₁	1	1	yes	characteristic
451	#117	C₂³×C₃₁	1	1	yes	characteristic
452	#117	C₂³×C₃₁	1	1	yes	characteristic
453	C₂×C₄⋊C₄	—	2	4	no	—
454	C₂×C₂²⋊C₄	—	2	2	no	—
455	C₂×C₂²⋊C₄	—	2	2	no	—
456	C₂×C₄²	—	2	1	no	—
457	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
458	#107	C₂³×C₃₁	1	4	yes	—
459	#106	C₂³×C₃₁	1	4	yes	—
460	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
461	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
462	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
463	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
464	#103	C₂³×C₃₁	1	4	yes	—
465	#103	C₂³×C₃₁	1	4	yes	—
466	#103	C₂³×C₃₁	1	4	yes	—
467	#102	C₂³×C₃₁	1	4	yes	—
468	#102	C₂³×C₃₁	1	4	yes	—
469	#102	C₂³×C₃₁	1	4	yes	—
470	#102	C₂³×C₃₁	1	4	yes	—
471	#102	C₂³×C₃₁	1	4	yes	—
472	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
473	#97	C₂³×C₃₁	1	4	yes	—
474	#97	C₂³×C₃₁	1	2	yes	—
475	#97	C₂³×C₃₁	1	2	yes	—
476	C₄×C₈	C₂³×C₃₁	1	2	yes	—
477	C₄×C₈	C₂³×C₃₁	1	2	yes	—
478	C₃₁	#61942	1	1	yes	characteristic, direct product factor, Sylow
479	C₂⁴	—	2	1	no	—
480	#41	—	4	2	no	—
481	#41	—	8	4	no	—
482	#41	—	8	8	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	2	no	—
485	C₂×Q₈	—	4	2	no	—
486	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
487	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
488	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	2	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	4	2	no	—
498	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
499	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
500	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	2	1	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	4	no	—
507	C₂²×C₄	—	2	4	no	—
508	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
509	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
510	SD₁₆	—	4	4	no	—
511	SD₁₆	—	4	4	no	—
512	SD₁₆	—	4	4	no	—
513	D₈	—	4	4	no	—
514	D₈	—	4	4	no	—
515	D₈	—	4	4	no	—
516	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
517	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
525	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
526	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
527	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
528	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
529	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
530	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
531	C₂²⋊C₄	—	2	1	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	2	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	—	2	4	no	—
542	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
543	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
544	C₄²	—	2	2	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	4	no	—
547	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
548	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
549	C₄²	C₂⁴×C₃₁	1	2	yes	—
550	C₄²	C₂⁴×C₃₁	1	2	yes	—
551	C₄²	C₂⁴×C₃₁	1	2	yes	—
552	C₂³	—	2	2	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	4	no	—
555	C₂³	—	4	2	no	—
556	C₂³	—	8	2	no	—
557	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
558	Q₈	—	2	4	no	—
559	Q₈	—	2	2	no	—
560	Q₈	—	4	4	no	—
561	Q₈	—	4	2	no	—
562	Q₈	—	8	2	no	—
563	D₄	—	2	4	no	—
564	D₄	—	2	4	no	—
565	D₄	—	2	2	no	—
566	D₄	—	2	4	no	—
567	D₄	—	4	2	no	—
568	D₄	—	4	4	no	—
569	D₄	—	4	2	no	—
570	D₄	—	8	4	no	—
571	D₄	—	8	2	no	—
572	D₄	—	8	4	no	—
573	D₄	—	8	4	no	—
574	D₄	—	8	4	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	2	no	—
582	C₂×C₄	—	4	8	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	8	2	no	—
585	C₂×C₄	—	8	4	no	—
586	C₂×C₄	—	8	4	no	—
587	C₂×C₄	—	8	4	no	—
588	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
589	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
590	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
591	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
592	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
593	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
594	C₈	—	2	4	no	—
595	C₈	—	2	4	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	2	no	—
598	C₂²	—	4	1	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	4	no	—
601	C₂²	—	4	4	no	—
602	C₂²	—	4	4	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	4	no	—
605	C₂²	—	16	2	no	—
606	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
607	C₄	—	2	1	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	2	4	no	—
611	C₄	—	2	4	no	—
612	C₄	—	4	1	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	4	no	—
615	C₄	—	4	2	no	—
616	C₄	C₃₁×#585	1	2	yes	—
617	C₂	—	4	1	no	—
618	C₂	—	8	2	no	—
619	C₂	—	8	1	no	—
620	C₂	—	8	4	no	—
621	C₂	C₃₁×#3487	1	1	yes	characteristic
622	C₂	C₃₁×#3585	1	2	yes	—

	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×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3548	C₂	semidirect
2. C₃₁×#3510	C₂	semidirect
3. C₃₁×#3399	C₂	semidirect
4. C₃₁×#3397	C₂	semidirect
5. C₃₁×#3376	C₂	semidirect
6. C₃₁×#3370	C₂	semidirect
7. C₃₁×#3221	C₂	semidirect
8. C₃₁×#3220	C₂	semidirect
9. C₃₁×#3213	C₂	semidirect
10. C₃₁×#3126	C₂	semidirect
11. C₃₁×#3122	C₂	semidirect
12. C₃₁×#3115	C₂	semidirect
13. C₃₁×#2987	C₂	semidirect
17. C₃₁×#554	C₂²	semidirect
19. C₃₁×#550	C₂²	semidirect
36. C₃₁×#496	C₂²	semidirect
38. C₃₁×#492	C₂²	semidirect
39. C₃₁×#491	C₂²	semidirect
43. C₃₁×#488	C₂²	semidirect
45. C₃₁×#479	C₂²	semidirect
47. C₃₁×#475	C₂²	semidirect
48. C₃₁×#474	C₂²	semidirect
52. C₃₁×#463	C₂²	semidirect
53. C₃₁×#461	C₂²	semidirect
54. C₃₁×#460	C₂²	semidirect
55. C₃₁×#460	C₂²	semidirect
58. C₃₁×#459	C₂²	semidirect
62. C₃₁×#446	C₂²	semidirect
63. C₃₁×#446	C₂²	semidirect
65. C₃₁×#442	C₂²	semidirect
66. C₃₁×#440	C₂²	semidirect
67. C₃₁×#440	C₂²	semidirect
68. C₄×SD₁₆×C₃₁	C₂²	semidirect
240. #61942	C₃₁	direct (N × H)
285. #3548	C₂×C₃₁	semidirect
286. #3510	C₂×C₃₁	semidirect
287. #3399	C₂×C₃₁	semidirect
288. #3397	C₂×C₃₁	semidirect
289. #3376	C₂×C₃₁	semidirect
290. #3370	C₂×C₃₁	semidirect
291. #3221	C₂×C₃₁	semidirect
292. #3220	C₂×C₃₁	semidirect
293. #3213	C₂×C₃₁	semidirect
294. #3126	C₂×C₃₁	semidirect
295. #3122	C₂×C₃₁	semidirect
296. #3115	C₂×C₃₁	semidirect
297. #2987	C₂×C₃₁	semidirect
322. #554	C₂²×C₃₁	semidirect
324. #550	C₂²×C₃₁	semidirect
341. #496	C₂²×C₃₁	semidirect
343. #492	C₂²×C₃₁	semidirect
344. #491	C₂²×C₃₁	semidirect
348. #488	C₂²×C₃₁	semidirect
350. #479	C₂²×C₃₁	semidirect
352. #475	C₂²×C₃₁	semidirect
353. #474	C₂²×C₃₁	semidirect
357. #463	C₂²×C₃₁	semidirect
358. #461	C₂²×C₃₁	semidirect
359. #460	C₂²×C₃₁	semidirect
360. #460	C₂²×C₃₁	semidirect
363. #459	C₂²×C₃₁	semidirect
367. #446	C₂²×C₃₁	semidirect
368. #446	C₂²×C₃₁	semidirect
370. #442	C₂²×C₃₁	semidirect
371. #440	C₂²×C₃₁	semidirect
372. #440	C₂²×C₃₁	semidirect
373. C₄×SD₁₆	C₂²×C₃₁	semidirect
478. C₃₁	#61942	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×#3548	2
C₃₁×#3510	2
C₃₁×#3399	2
C₃₁×#3397	2
C₃₁×#3376	2
C₃₁×#3370	2
C₃₁×#3221	2
C₃₁×#3220	2
C₃₁×#3213	2
C₃₁×#3126	2
C₃₁×#3122	2
C₃₁×#3115	2
C₃₁×#2987	2
#61942	31

Subgroups — C31×#61942 #21518701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61942 #21518701