Subgroups — C₃₁×#19780 #21476539

Order 7936.

622 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3329	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3303	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3298	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1718	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1684	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1683	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1654	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1651	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1646	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1637	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1581	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#555	C₂²	1	1	yes	characteristic
17	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#547	C₂²	1	1	yes	characteristic
20	C₃₁×#500	—	2	1	no	—
21	C₃₁×#500	C₂²	1	1	yes	characteristic
22	C₃₁×#498	C₂²	1	1	yes	characteristic
23	C₃₁×#487	C₂²	1	1	yes	characteristic
24	C₃₁×#487	C₂²	1	1	yes	characteristic
25	C₃₁×#487	C₂²	1	1	yes	characteristic
26	C₃₁×#487	C₂²	1	1	yes	characteristic
27	C₃₁×#485	—	2	1	no	—
28	C₃₁×#485	—	2	1	no	—
29	C₃₁×#483	—	2	1	no	—
30	C₃₁×#477	C₂²	1	1	yes	characteristic
31	C₃₁×#476	C₂²	1	1	yes	characteristic
32	C₃₁×#475	C₂²	1	1	yes	characteristic
33	C₃₁×#474	—	2	1	no	—
34	C₃₁×#474	—	2	1	no	—
35	C₃₁×#474	C₂²	1	1	yes	characteristic
36	C₃₁×#474	C₂²	1	1	yes	characteristic
37	C₃₁×#468	—	2	1	no	—
38	C₃₁×#465	—	2	1	no	—
39	C₃₁×#465	—	2	1	no	—
40	C₃₁×#461	C₂²	1	1	yes	characteristic
41	C₃₁×#461	C₂²	1	1	yes	characteristic
42	C₃₁×#461	C₂²	1	1	yes	characteristic
43	C₃₁×#449	—	2	1	no	—
44	C₃₁×#448	—	2	1	no	—
45	C₃₁×#448	—	2	1	no	—
46	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
47	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
48	C₄×D₈×C₃₁	—	2	1	no	—
49	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
50	C₃₁×#436	C₂²	1	1	yes	characteristic
51	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
52	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
53	C₃₁×#419	—	2	1	no	—
54	C₃₁×#416	—	2	1	no	—
55	C₂×C₃₁×#103	—	2	1	no	—
56	C₃₁×#335	C₂²	1	1	yes	characteristic
57	C₃₁×#335	C₂²	1	1	yes	characteristic
58	C₃₁×#333	C₂²	1	1	yes	characteristic
59	C₃₁×#332	C₂²	1	1	yes	characteristic
60	C₃₁×#332	C₂²	1	1	yes	characteristic
61	C₃₁×#332	C₂²	1	1	yes	characteristic
62	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
64	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₃₁×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	—	2	1	no	—
68	C₂²×Q₈×C₃₁	—	2	1	no	—
69	C₂²×Q₈×C₃₁	—	2	1	no	—
70	C₂×SD₁₆×C₃₁	—	2	1	no	—
71	C₂×SD₁₆×C₃₁	—	2	1	no	—
72	C₂×SD₁₆×C₃₁	—	2	1	no	—
73	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
74	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
75	C₂×M₁₆×C₃₁	—	2	1	no	—
76	C₂²×C₈×C₃₁	—	2	1	no	—
77	C₂²×C₈×C₃₁	—	2	1	no	—
78	C₃₁×C₄⋊Q₈	—	2	1	no	—
79	C₃₁×C₄⋊Q₈	—	2	1	no	—
80	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₃₁×#124	—	4	1	no	—
85	C₃₁×#123	—	4	1	no	—
86	C₃₁×C₂²⋊Q₈	—	2	1	no	—
87	C₃₁×C₂²⋊Q₈	—	2	1	no	—
88	C₃₁×C₂²⋊Q₈	—	2	1	no	—
89	C₃₁×C₂²⋊Q₈	—	4	1	no	—
90	C₃₁×C₂²⋊Q₈	—	4	1	no	—
91	C₃₁×#121	—	2	1	no	—
92	C₃₁×#121	—	2	1	no	—
93	C₃₁×#121	—	2	1	no	—
94	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₃₁×#107	—	2	1	no	—
104	C₃₁×#107	—	2	1	no	—
105	C₃₁×#107	—	2	1	no	—
106	C₃₁×#107	—	2	1	no	—
107	C₃₁×#107	—	2	1	no	—
108	C₃₁×#107	D₄	1	1	yes	characteristic
109	C₃₁×#106	—	2	1	no	—
110	C₃₁×#106	—	2	1	no	—
111	C₃₁×#106	—	4	1	no	—
112	C₃₁×#106	D₄	1	1	yes	characteristic
113	C₃₁×C₄⋊C₈	—	2	1	no	—
114	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₃₁×#103	—	2	1	no	—
119	C₃₁×#103	—	2	1	no	—
120	C₃₁×#103	—	2	1	no	—
121	C₃₁×#103	—	2	1	no	—
122	C₃₁×#103	—	2	1	no	—
123	C₃₁×#103	—	2	1	no	—
124	C₃₁×#103	—	2	1	no	—
125	C₃₁×#103	—	2	1	no	—
126	C₃₁×#103	—	2	1	no	—
127	C₃₁×#103	—	2	1	no	—
128	C₃₁×#103	—	2	1	no	—
129	C₃₁×#103	—	4	1	no	—
130	C₃₁×#103	D₄	1	1	yes	characteristic
131	C₃₁×#102	—	2	1	no	—
132	C₃₁×#102	—	2	1	no	—
133	C₃₁×#102	—	2	1	no	—
134	C₃₁×#102	D₄	1	1	yes	characteristic
135	C₃₁×#102	D₄	1	1	yes	characteristic
136	C₃₁×#102	D₄	1	1	yes	characteristic
137	C₃₁×C₂²⋊C₈	—	2	1	no	—
138	C₃₁×C₂²⋊C₈	—	2	1	no	—
139	C₃₁×C₂²⋊C₈	—	2	1	no	—
140	C₄×C₈×C₃₁	—	2	1	no	—
141	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	#41×C₃₁	—	4	2	no	—
145	#41×C₃₁	—	8	1	no	—
146	C₂×Q₈×C₃₁	—	2	1	no	—
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	—	2	1	no	—
149	C₂×Q₈×C₃₁	—	2	1	no	—
150	C₂×Q₈×C₃₁	—	4	1	no	—
151	C₂×Q₈×C₃₁	—	4	1	no	—
152	C₂×Q₈×C₃₁	—	4	2	no	—
153	C₂×Q₈×C₃₁	—	4	2	no	—
154	C₂×Q₈×C₃₁	—	8	1	no	—
155	C₂×Q₈×C₃₁	—	8	1	no	—
156	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂×D₄×C₃₁	—	4	1	no	—
158	C₂×D₄×C₃₁	—	4	1	no	—
159	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	4	1	no	—
166	C₂²×C₄×C₃₁	—	4	1	no	—
167	C₂²×C₄×C₃₁	—	4	1	no	—
168	SD₁₆×C₃₁	—	2	2	no	—
169	SD₁₆×C₃₁	—	8	1	no	—
170	SD₁₆×C₃₁	—	8	1	no	—
171	SD₁₆×C₃₁	—	8	1	no	—
172	D₈×C₃₁	—	2	2	no	—
173	M₁₆×C₃₁	—	8	1	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	1	no	—
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	8	1	no	—
180	C₂×C₈×C₃₁	—	8	1	no	—
181	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	4	1	no	—
195	C₄⋊C₄×C₃₁	—	4	1	no	—
196	C₄⋊C₄×C₃₁	—	4	1	no	—
197	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₂²⋊C₄×C₃₁	—	2	1	no	—
202	C₂²⋊C₄×C₃₁	—	2	1	no	—
203	C₂²⋊C₄×C₃₁	—	2	1	no	—
204	C₂²⋊C₄×C₃₁	—	4	1	no	—
205	C₂²⋊C₄×C₃₁	—	4	1	no	—
206	C₂²⋊C₄×C₃₁	—	4	1	no	—
207	C₂²⋊C₄×C₃₁	—	4	1	no	—
208	C₂²⋊C₄×C₃₁	—	4	1	no	—
209	C₄²×C₃₁	—	2	1	no	—
210	C₄²×C₃₁	—	4	1	no	—
211	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
212	#19780	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₃₁	—	2	1	no	—
214	C₂³×C₃₁	—	2	1	no	—
215	C₂³×C₃₁	—	2	1	no	—
216	Q₈×C₃₁	—	2	1	no	—
217	Q₈×C₃₁	—	4	1	no	—
218	Q₈×C₃₁	—	4	1	no	—
219	Q₈×C₃₁	—	4	1	no	—
220	Q₈×C₃₁	—	4	1	no	—
221	Q₈×C₃₁	—	4	1	no	—
222	Q₈×C₃₁	—	4	1	no	—
223	Q₈×C₃₁	—	8	2	no	—
224	Q₈×C₃₁	—	8	2	no	—
225	Q₈×C₃₁	—	8	1	no	—
226	Q₈×C₃₁	#135	1	2	yes	—
227	D₄×C₃₁	—	2	1	no	—
228	D₄×C₃₁	—	2	1	no	—
229	D₄×C₃₁	—	2	1	no	—
230	D₄×C₃₁	—	2	1	no	—
231	D₄×C₃₁	—	2	1	no	—
232	D₄×C₃₁	—	8	2	no	—
233	D₄×C₃₁	—	16	1	no	—
234	D₄×C₃₁	#137	1	2	yes	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	4	1	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	1	no	—
247	C₂×C₄×C₃₁	—	4	1	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	4	1	no	—
250	C₂×C₄×C₃₁	—	8	2	no	—
251	C₂×C₄×C₃₁	—	8	2	no	—
252	C₂×C₄×C₃₁	—	8	2	no	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
256	C₈×C₃₁	—	2	1	no	—
257	C₈×C₃₁	—	2	1	no	—
258	C₈×C₃₁	—	2	1	no	—
259	C₈×C₃₁	—	2	1	no	—
260	C₈×C₃₁	—	4	1	no	—
261	C₈×C₃₁	—	4	1	no	—
262	C₈×C₃₁	—	4	1	no	—
263	C₈×C₃₁	—	4	1	no	—
264	C₈×C₃₁	—	4	1	no	—
265	Q₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#3329	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#3303	C₂×C₃₁	1	1	yes	characteristic
268	#3298	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1718	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1699	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#1684	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#1683	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1673	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1654	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1651	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1646	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1637	C₂×C₃₁	1	1	yes	characteristic, complemented
278	#1581	C₂×C₃₁	1	1	yes	characteristic, complemented
279	SD₁₆⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂²×C₃₁	—	2	2	no	—
281	C₂²×C₃₁	—	4	1	no	—
282	C₂²×C₃₁	—	4	1	no	—
283	C₂²×C₃₁	—	4	1	no	—
284	C₂²×C₃₁	—	4	1	no	—
285	C₂²×C₃₁	—	4	1	no	—
286	C₂²×C₃₁	—	4	1	no	—
287	C₂²×C₃₁	—	4	1	no	—
288	C₂²×C₃₁	—	4	1	no	—
289	C₂²×C₃₁	D₄²	1	1	yes	characteristic
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	4	1	no	—
293	C₄×C₃₁	—	4	1	no	—
294	C₄×C₃₁	—	4	1	no	—
295	C₄×C₃₁	—	4	1	no	—
296	C₄×C₃₁	—	8	1	no	—
297	C₄×C₃₁	—	8	1	no	—
298	C₄×C₃₁	—	8	1	no	—
299	C₄×C₃₁	—	8	1	no	—
300	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
301	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
302	C₄×C₃₁	#576	1	1	yes	characteristic
303	C₄×C₃₁	#577	1	1	yes	characteristic
304	#555	C₂²×C₃₁	1	1	yes	characteristic
305	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
306	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
307	#547	C₂²×C₃₁	1	1	yes	characteristic
308	#500	—	2	1	no	—
309	#500	C₂²×C₃₁	1	1	yes	characteristic
310	#498	C₂²×C₃₁	1	1	yes	characteristic
311	#487	C₂²×C₃₁	1	1	yes	characteristic
312	#487	C₂²×C₃₁	1	1	yes	characteristic
313	#487	C₂²×C₃₁	1	1	yes	characteristic
314	#487	C₂²×C₃₁	1	1	yes	characteristic
315	#485	—	2	1	no	—
316	#485	—	2	1	no	—
317	#483	—	2	1	no	—
318	#477	C₂²×C₃₁	1	1	yes	characteristic
319	#476	C₂²×C₃₁	1	1	yes	characteristic
320	#475	C₂²×C₃₁	1	1	yes	characteristic
321	#474	—	2	1	no	—
322	#474	—	2	1	no	—
323	#474	C₂²×C₃₁	1	1	yes	characteristic
324	#474	C₂²×C₃₁	1	1	yes	characteristic
325	#468	—	2	1	no	—
326	#465	—	2	1	no	—
327	#465	—	2	1	no	—
328	#461	C₂²×C₃₁	1	1	yes	characteristic
329	#461	C₂²×C₃₁	1	1	yes	characteristic
330	#461	C₂²×C₃₁	1	1	yes	characteristic
331	#449	—	2	1	no	—
332	#448	—	2	1	no	—
333	#448	—	2	1	no	—
334	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
335	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
336	C₄×D₈	—	2	1	no	—
337	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
338	#436	C₂²×C₃₁	1	1	yes	characteristic
339	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
340	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
341	#419	—	2	1	no	—
342	#416	—	2	1	no	—
343	C₂×#103	—	2	1	no	—
344	#335	C₂²×C₃₁	1	1	yes	characteristic
345	#335	C₂²×C₃₁	1	1	yes	characteristic
346	#333	C₂²×C₃₁	1	1	yes	characteristic
347	#332	C₂²×C₃₁	1	1	yes	characteristic
348	#332	C₂²×C₃₁	1	1	yes	characteristic
349	#332	C₂²×C₃₁	1	1	yes	characteristic
350	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
351	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
352	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
353	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
354	#322	C₂²×C₃₁	1	1	yes	characteristic
355	C₂×C₃₁	—	4	2	no	—
356	C₂×C₃₁	—	8	1	no	—
357	C₂×C₃₁	—	8	1	no	—
358	C₂×C₃₁	#3266	1	1	yes	characteristic
359	C₂×C₃₁	#3280	1	1	yes	characteristic
360	C₂×C₃₁	#3289	1	1	yes	characteristic
361	C₂×#41	—	2	1	no	—
362	C₂²×Q₈	—	2	1	no	—
363	C₂²×Q₈	—	2	1	no	—
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×SD₁₆	—	2	1	no	—
366	C₂×SD₁₆	—	2	1	no	—
367	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
368	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
369	C₂×M₁₆	—	2	1	no	—
370	C₂²×C₈	—	2	1	no	—
371	C₂²×C₈	—	2	1	no	—
372	C₄⋊Q₈	—	2	1	no	—
373	C₄⋊Q₈	—	2	1	no	—
374	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
375	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
377	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
378	#124	—	4	1	no	—
379	#123	—	4	1	no	—
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	—	2	1	no	—
382	C₂²⋊Q₈	—	2	1	no	—
383	C₂²⋊Q₈	—	4	1	no	—
384	C₂²⋊Q₈	—	4	1	no	—
385	#121	—	2	1	no	—
386	#121	—	2	1	no	—
387	#121	—	2	1	no	—
388	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
392	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
393	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
394	#117	—	2	1	no	—
395	#117	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	#107	—	2	1	no	—
398	#107	—	2	1	no	—
399	#107	—	2	1	no	—
400	#107	—	2	1	no	—
401	#107	—	2	1	no	—
402	#107	D₄×C₃₁	1	1	yes	characteristic
403	#106	—	2	1	no	—
404	#106	—	2	1	no	—
405	#106	—	4	1	no	—
406	#106	D₄×C₃₁	1	1	yes	characteristic
407	C₄⋊C₈	—	2	1	no	—
408	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
409	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
410	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
411	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
412	#103	—	2	1	no	—
413	#103	—	2	1	no	—
414	#103	—	2	1	no	—
415	#103	—	2	1	no	—
416	#103	—	2	1	no	—
417	#103	—	2	1	no	—
418	#103	—	2	1	no	—
419	#103	—	2	1	no	—
420	#103	—	2	1	no	—
421	#103	—	2	1	no	—
422	#103	—	2	1	no	—
423	#103	—	4	1	no	—
424	#103	D₄×C₃₁	1	1	yes	characteristic
425	#102	—	2	1	no	—
426	#102	—	2	1	no	—
427	#102	—	2	1	no	—
428	#102	D₄×C₃₁	1	1	yes	characteristic
429	#102	D₄×C₃₁	1	1	yes	characteristic
430	#102	D₄×C₃₁	1	1	yes	characteristic
431	C₂²⋊C₈	—	2	1	no	—
432	C₂²⋊C₈	—	2	1	no	—
433	C₂²⋊C₈	—	2	1	no	—
434	C₄×C₈	—	2	1	no	—
435	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
436	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
437	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
438	C₃₁	#19780	1	1	yes	characteristic, direct product factor, Sylow
439	#41	—	4	2	no	—
440	#41	—	8	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	4	1	no	—
446	C₂×Q₈	—	4	1	no	—
447	C₂×Q₈	—	4	2	no	—
448	C₂×Q₈	—	4	2	no	—
449	C₂×Q₈	—	8	1	no	—
450	C₂×Q₈	—	8	1	no	—
451	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	—	4	1	no	—
454	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
456	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	4	1	no	—
461	C₂²×C₄	—	4	1	no	—
462	C₂²×C₄	—	4	1	no	—
463	SD₁₆	—	2	2	no	—
464	SD₁₆	—	8	1	no	—
465	SD₁₆	—	8	1	no	—
466	SD₁₆	—	8	1	no	—
467	D₈	—	2	2	no	—
468	M₁₆	—	8	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	8	1	no	—
475	C₂×C₈	—	8	1	no	—
476	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₄⋊C₄	—	4	1	no	—
492	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
493	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
494	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
495	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
496	C₂²⋊C₄	—	2	1	no	—
497	C₂²⋊C₄	—	2	1	no	—
498	C₂²⋊C₄	—	2	1	no	—
499	C₂²⋊C₄	—	4	1	no	—
500	C₂²⋊C₄	—	4	1	no	—
501	C₂²⋊C₄	—	4	1	no	—
502	C₂²⋊C₄	—	4	1	no	—
503	C₂²⋊C₄	—	4	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	4	1	no	—
506	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
507	C₂³	—	2	1	no	—
508	C₂³	—	2	1	no	—
509	C₂³	—	2	1	no	—
510	Q₈	—	2	1	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	4	1	no	—
513	Q₈	—	4	1	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	Q₈	—	4	1	no	—
517	Q₈	—	8	2	no	—
518	Q₈	—	8	2	no	—
519	Q₈	—	8	1	no	—
520	Q₈	C₃₁×#135	1	2	yes	—
521	D₄	—	2	1	no	—
522	D₄	—	2	1	no	—
523	D₄	—	2	1	no	—
524	D₄	—	2	1	no	—
525	D₄	—	2	1	no	—
526	D₄	—	8	2	no	—
527	D₄	—	16	1	no	—
528	D₄	C₃₁×#137	1	2	yes	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	8	2	no	—
545	C₂×C₄	—	8	2	no	—
546	C₂×C₄	—	8	2	no	—
547	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
550	C₈	—	2	1	no	—
551	C₈	—	2	1	no	—
552	C₈	—	2	1	no	—
553	C₈	—	2	1	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	1	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	4	1	no	—
562	C₂²	—	4	1	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	—	4	1	no	—
568	C₂²	D₄²×C₃₁	1	1	yes	characteristic
569	C₄	—	2	2	no	—
570	C₄	—	2	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	1	no	—
573	C₄	—	4	1	no	—
574	C₄	—	4	1	no	—
575	C₄	—	8	1	no	—
576	C₄	—	8	1	no	—
577	C₄	—	8	1	no	—
578	C₄	—	8	1	no	—
579	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
580	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
581	C₄	C₃₁×#576	1	1	yes	characteristic
582	C₄	C₃₁×#577	1	1	yes	characteristic
583	C₂	—	4	2	no	—
584	C₂	—	8	1	no	—
585	C₂	—	8	1	no	—
586	C₂	C₃₁×#3266	1	1	yes	characteristic
587	C₂	C₃₁×#3280	1	1	yes	characteristic
588	C₂	C₃₁×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3303	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×D₈×C₃₁	C₂	semidirect
2. C₃₁×#3329	C₂	semidirect
4. C₃₁×#3298	C₂	semidirect
5. C₃₁×#1718	C₂	semidirect
6. C₃₁×#1699	C₂	semidirect
7. C₃₁×#1684	C₂	semidirect
8. C₃₁×#1683	C₂	semidirect
9. C₃₁×#1673	C₂	semidirect
10. C₃₁×#1654	C₂	semidirect
11. C₃₁×#1651	C₂	semidirect
12. C₃₁×#1646	C₂	semidirect
13. C₃₁×#1637	C₂	semidirect
14. C₃₁×#1581	C₂	semidirect
15. C₃₁×SD₁₆⋊C₈	C₂	semidirect
212. #19780	C₃₁	direct (N × H)
265. Q₈×D₈	C₂×C₃₁	semidirect
266. #3329	C₂×C₃₁	semidirect
268. #3298	C₂×C₃₁	semidirect
269. #1718	C₂×C₃₁	semidirect
270. #1699	C₂×C₃₁	semidirect
271. #1684	C₂×C₃₁	semidirect
272. #1683	C₂×C₃₁	semidirect
273. #1673	C₂×C₃₁	semidirect
274. #1654	C₂×C₃₁	semidirect
275. #1651	C₂×C₃₁	semidirect
276. #1646	C₂×C₃₁	semidirect
277. #1637	C₂×C₃₁	semidirect
278. #1581	C₂×C₃₁	semidirect
279. SD₁₆⋊C₈	C₂×C₃₁	semidirect
438. C₃₁	#19780	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×D₈×C₃₁	2
C₃₁×#3329	2
C₃₁×#3303	2
C₃₁×#3298	2
C₃₁×#1718	2
C₃₁×#1699	2
C₃₁×#1684	2
C₃₁×#1683	2
C₃₁×#1673	2
C₃₁×#1654	2
C₃₁×#1651	2
C₃₁×#1646	2
C₃₁×#1637	2
C₃₁×#1581	2
C₃₁×SD₁₆⋊C₈	2
#19780	31

Subgroups — C31×#19780 #21476539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19780 #21476539