Subgroups — C₃₁×#19889 #21476648

Order 7936.

710 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3391	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3278	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1685	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1641	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1627	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	—	2	2	no	—
13	C₃₁×#550	C₂²	1	2	yes	—
14	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#500	—	2	1	no	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#493	—	2	1	no	—
18	C₃₁×#493	C₂²	1	1	yes	characteristic
19	C₃₁×#492	C₂²	1	2	yes	—
20	C₃₁×#486	C₂²	1	2	yes	—
21	C₃₁×#474	—	2	2	no	—
22	C₃₁×#474	C₂²	1	2	yes	—
23	C₃₁×#468	—	2	2	no	—
24	C₃₁×#462	C₂²	1	2	yes	—
25	C₃₁×#460	C₂²	1	2	yes	—
26	C₃₁×#459	—	2	2	no	—
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#452	—	2	2	no	—
29	C₃₁×#448	—	2	2	no	—
30	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₃₁	—	2	1	no	—
34	C₄×D₈×C₃₁	—	2	1	no	—
35	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#436	C₂²	1	2	yes	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#335	C₂²	1	1	yes	characteristic
39	C₃₁×#332	C₂²	1	2	yes	—
40	C₃₁×#331	C₂²	1	2	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂²×Q₈×C₃₁	—	2	2	no	—
47	C₃₁×#137	—	4	8	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	1	no	—
55	C₂×D₈×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₃₁×#127	—	2	2	no	—
61	C₃₁×#127	C₂³	1	2	yes	—
62	C₃₁×C₂²⋊Q₈	—	2	2	no	—
63	C₃₁×C₂²⋊Q₈	—	4	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₃₁×#121	—	4	2	no	—
66	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	C₂³	1	2	yes	—
71	C₃₁×#107	—	2	1	no	—
72	C₃₁×#107	—	2	1	no	—
73	C₃₁×#107	—	2	2	no	—
74	C₃₁×#107	D₄	1	1	yes	characteristic
75	C₃₁×#107	D₄	1	1	yes	characteristic
76	C₃₁×C₄⋊C₈	—	2	1	no	—
77	C₃₁×C₄⋊C₈	—	2	1	no	—
78	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	D₄	1	1	yes	characteristic
83	C₃₁×#103	D₄	1	1	yes	characteristic
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	4	2	no	—
88	C₃₁×#102	D₄	1	2	yes	—
89	C₃₁×C₂²⋊C₈	—	2	2	no	—
90	C₄×C₈×C₃₁	—	2	1	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	#41×C₃₁	—	4	4	no	—
96	#41×C₃₁	—	8	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	4	2	no	—
99	C₂×Q₈×C₃₁	—	4	4	no	—
100	C₂×Q₈×C₃₁	—	8	2	no	—
101	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	4	2	no	—
107	C₂×D₄×C₃₁	—	4	2	no	—
108	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	Q₁₆×C₃₁	—	4	4	no	—
113	Q₁₆×C₃₁	D₈	1	4	yes	complemented
114	SD₁₆×C₃₁	—	4	4	no	—
115	SD₁₆×C₃₁	—	4	4	no	—
116	SD₁₆×C₃₁	—	8	2	no	—
117	D₈×C₃₁	—	2	2	no	—
118	D₈×C₃₁	—	4	2	no	—
119	D₈×C₃₁	—	8	1	no	—
120	D₈×C₃₁	—	8	2	no	—
121	M₁₆×C₃₁	—	4	4	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₂²⋊C₄×C₃₁	—	4	2	no	—
141	C₄²×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
144	#19889	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	Q₈×C₃₁	—	2	1	no	—
148	Q₈×C₃₁	—	2	1	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	4	2	no	—
151	Q₈×C₃₁	—	8	4	no	—
152	Q₈×C₃₁	C₂×D₈	1	2	yes	—
153	Q₈×C₃₁	C₂×D₈	1	2	yes	—
154	D₄×C₃₁	—	2	2	no	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	8	4	no	—
161	D₄×C₃₁	—	8	2	no	—
162	D₄×C₃₁	—	16	2	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	8	4	no	—
172	C₂×C₄×C₃₁	—	8	4	no	—
173	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
176	C₈×C₃₁	—	2	1	no	—
177	C₈×C₃₁	—	2	1	no	—
178	C₈×C₃₁	—	4	1	no	—
179	C₈×C₃₁	—	4	2	no	—
180	C₈×C₃₁	—	4	1	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	C₂×D₈	1	2	yes	—
183	C₈×C₃₁	#137	1	2	yes	complemented
184	#3391	C₂×C₃₁	1	1	yes	characteristic
185	Q₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#3278	C₂×C₃₁	1	2	yes	complemented
187	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1685	C₂×C₃₁	1	2	yes	complemented
189	#1674	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1668	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1641	C₂×C₃₁	1	2	yes	complemented
192	#1627	C₂×C₃₁	1	2	yes	complemented
193	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
195	C₂²×C₃₁	—	4	2	no	—
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	D₄²	1	1	yes	characteristic
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	2	2	no	—
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	4	2	no	—
206	C₄×C₃₁	—	4	1	no	—
207	C₄×C₃₁	—	4	1	no	—
208	C₄×C₃₁	—	8	2	no	—
209	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
210	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
211	C₄×C₃₁	#576	1	1	yes	characteristic
212	C₄×C₃₁	#577	1	1	yes	characteristic
213	C₂×#137	—	2	2	no	—
214	#550	C₂²×C₃₁	1	2	yes	—
215	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
216	#500	—	2	1	no	—
217	#500	C₂²×C₃₁	1	1	yes	characteristic
218	#493	—	2	1	no	—
219	#493	C₂²×C₃₁	1	1	yes	characteristic
220	#492	C₂²×C₃₁	1	2	yes	—
221	#486	C₂²×C₃₁	1	2	yes	—
222	#474	—	2	2	no	—
223	#474	C₂²×C₃₁	1	2	yes	—
224	#468	—	2	2	no	—
225	#462	C₂²×C₃₁	1	2	yes	—
226	#460	C₂²×C₃₁	1	2	yes	—
227	#459	—	2	2	no	—
228	#459	C₂²×C₃₁	1	2	yes	—
229	#452	—	2	2	no	—
230	#448	—	2	2	no	—
231	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
234	C₄×D₈	—	2	1	no	—
235	C₄×D₈	—	2	1	no	—
236	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
237	#436	C₂²×C₃₁	1	2	yes	—
238	#335	C₂²×C₃₁	1	1	yes	characteristic
239	#335	C₂²×C₃₁	1	1	yes	characteristic
240	#332	C₂²×C₃₁	1	2	yes	—
241	#331	C₂²×C₃₁	1	2	yes	—
242	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
243	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
244	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
245	C₈²	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×C₃₁	—	8	2	no	—
247	C₂×C₃₁	—	16	2	no	—
248	C₂×C₃₁	#3278	1	1	yes	characteristic
249	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
250	C₂×C₃₁	#3290	1	1	yes	characteristic
251	C₂×#41	—	2	2	no	—
252	C₂²×Q₈	—	2	2	no	—
253	#137	—	4	8	no	—
254	C₂×Q₁₆	—	2	2	no	—
255	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
256	C₂×SD₁₆	—	2	2	no	—
257	C₂×SD₁₆	—	2	2	no	—
258	C₂×SD₁₆	—	2	2	no	—
259	C₂×D₈	—	2	2	no	—
260	C₂×D₈	—	2	1	no	—
261	C₂×D₈	—	2	1	no	—
262	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
263	C₂×M₁₆	—	2	2	no	—
264	C₄⋊Q₈	—	2	2	no	—
265	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
266	#127	—	2	2	no	—
267	#127	C₂³×C₃₁	1	2	yes	—
268	C₂²⋊Q₈	—	2	2	no	—
269	C₂²⋊Q₈	—	4	2	no	—
270	#121	—	2	2	no	—
271	#121	—	4	2	no	—
272	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₄×D₄	—	2	2	no	—
275	C₄×D₄	—	2	2	no	—
276	C₄×D₄	C₂³×C₃₁	1	2	yes	—
277	#107	—	2	1	no	—
278	#107	—	2	1	no	—
279	#107	—	2	2	no	—
280	#107	D₄×C₃₁	1	1	yes	characteristic
281	#107	D₄×C₃₁	1	1	yes	characteristic
282	C₄⋊C₈	—	2	1	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
285	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
287	#103	—	2	2	no	—
288	#103	D₄×C₃₁	1	1	yes	characteristic
289	#103	D₄×C₃₁	1	1	yes	characteristic
290	#102	—	2	2	no	—
291	#102	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	—	4	2	no	—
294	#102	D₄×C₃₁	1	2	yes	—
295	C₂²⋊C₈	—	2	2	no	—
296	C₄×C₈	—	2	1	no	—
297	C₄×C₈	—	2	1	no	—
298	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
300	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
301	C₃₁	#19889	1	1	yes	characteristic, direct product factor, Sylow
302	#41	—	4	4	no	—
303	#41	—	8	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	4	2	no	—
306	C₂×Q₈	—	4	4	no	—
307	C₂×Q₈	—	8	2	no	—
308	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	—	4	2	no	—
319	Q₁₆	—	4	4	no	—
320	Q₁₆	D₈×C₃₁	1	4	yes	complemented
321	SD₁₆	—	4	4	no	—
322	SD₁₆	—	4	4	no	—
323	SD₁₆	—	8	2	no	—
324	D₈	—	2	2	no	—
325	D₈	—	4	2	no	—
326	D₈	—	8	1	no	—
327	D₈	—	8	2	no	—
328	M₁₆	—	4	4	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
344	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	4	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	1	no	—
350	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
351	C₂³	—	2	2	no	—
352	C₂³	—	4	2	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	2	1	no	—
355	Q₈	—	4	2	no	—
356	Q₈	—	4	2	no	—
357	Q₈	—	8	4	no	—
358	Q₈	C₂×D₈×C₃₁	1	2	yes	—
359	Q₈	C₂×D₈×C₃₁	1	2	yes	—
360	D₄	—	2	2	no	—
361	D₄	—	2	2	no	—
362	D₄	—	4	2	no	—
363	D₄	—	4	2	no	—
364	D₄	—	4	2	no	—
365	D₄	—	4	2	no	—
366	D₄	—	8	4	no	—
367	D₄	—	8	2	no	—
368	D₄	—	16	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	8	4	no	—
378	C₂×C₄	—	8	4	no	—
379	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
382	C₈	—	2	1	no	—
383	C₈	—	2	1	no	—
384	C₈	—	4	1	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	1	no	—
387	C₈	—	4	1	no	—
388	C₈	C₂×D₈×C₃₁	1	2	yes	—
389	C₈	C₃₁×#137	1	2	yes	complemented
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	2	no	—
396	C₂²	D₄²×C₃₁	1	1	yes	characteristic
397	C₄	—	2	2	no	—
398	C₄	—	2	2	no	—
399	C₄	—	2	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	8	2	no	—
404	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
405	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
406	C₄	C₃₁×#576	1	1	yes	characteristic
407	C₄	C₃₁×#577	1	1	yes	characteristic
408	C₂	—	8	2	no	—
409	C₂	—	16	2	no	—
410	C₂	C₃₁×#3278	1	1	yes	characteristic
411	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
412	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3391	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
2. Q₈×D₈×C₃₁	C₂	semidirect
3. C₃₁×#3278	C₂	semidirect
4. C₃₁×#1706	C₂	semidirect
5. C₃₁×#1685	C₂	semidirect
6. C₃₁×#1674	C₂	semidirect
7. C₃₁×#1668	C₂	semidirect
8. C₃₁×#1641	C₂	semidirect
9. C₃₁×#1627	C₂	semidirect
10. C₃₁×#1589	C₂	semidirect
11. C₈×Q₁₆×C₃₁	C₂	semidirect
113. Q₁₆×C₃₁	D₈	semidirect
144. #19889	C₃₁	direct (N × H)
183. C₈×C₃₁	#137	semidirect
185. Q₈×D₈	C₂×C₃₁	semidirect
186. #3278	C₂×C₃₁	semidirect
187. #1706	C₂×C₃₁	semidirect
188. #1685	C₂×C₃₁	semidirect
189. #1674	C₂×C₃₁	semidirect
190. #1668	C₂×C₃₁	semidirect
191. #1641	C₂×C₃₁	semidirect
192. #1627	C₂×C₃₁	semidirect
193. #1589	C₂×C₃₁	semidirect
194. C₈×Q₁₆	C₂×C₃₁	semidirect
301. C₃₁	#19889	direct (N × H)
320. Q₁₆	D₈×C₃₁	semidirect
389. C₈	C₃₁×#137	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3391	2
Q₈×D₈×C₃₁	2
C₃₁×#3278	2
C₃₁×#1706	2
C₃₁×#1685	2
C₃₁×#1674	2
C₃₁×#1668	2
C₃₁×#1641	2
C₃₁×#1627	2
C₃₁×#1589	2
C₈×Q₁₆×C₃₁	2
#19889	31

Subgroups — C31×#19889 #21476648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19889 #21476648