Subgroups — C₃₁×#19937 #21476696

Order 7936.

660 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×#3366	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3338	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3287	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1698	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1631	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1622	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₃₁×#554	C₂²	1	2	yes	—
14	C₃₁×#550	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#493	C₂²	1	1	yes	characteristic
17	C₃₁×#492	C₂²	1	2	yes	—
18	C₃₁×#490	—	2	2	no	—
19	C₃₁×#486	C₂²	1	2	yes	—
20	C₃₁×#484	—	2	2	no	—
21	C₃₁×#478	—	2	2	no	—
22	C₃₁×#477	C₂²	1	2	yes	—
23	C₃₁×#476	C₂²	1	2	yes	—
24	C₃₁×#467	—	2	1	no	—
25	C₃₁×#466	—	2	1	no	—
26	C₃₁×#460	C₂²	1	2	yes	—
27	C₃₁×#459	—	2	2	no	—
28	C₃₁×#459	C₂²	1	2	yes	—
29	C₃₁×#449	—	2	2	no	—
30	C₃₁×#446	C₂²	1	2	yes	—
31	C₃₁×#442	—	2	2	no	—
32	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#107	—	2	1	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#335	C₂²	1	1	yes	characteristic
39	C₃₁×#333	C₂²	1	2	yes	—
40	C₃₁×#331	C₂²	1	2	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₃₁×#135	—	4	4	no	—
47	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	4	2	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₃₁×#127	—	2	2	no	—
58	C₃₁×#127	C₂³	1	2	yes	—
59	C₃₁×#125	—	2	2	no	—
60	C₃₁×C₂²⋊Q₈	—	2	2	no	—
61	C₃₁×C₂²⋊Q₈	—	4	2	no	—
62	C₃₁×#121	—	2	2	no	—
63	C₃₁×#121	—	4	2	no	—
64	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₃₁×#107	—	2	1	no	—
71	C₃₁×#107	—	2	1	no	—
72	C₃₁×#107	—	2	2	no	—
73	C₃₁×#107	D₄	1	1	yes	characteristic
74	C₃₁×#107	D₄	1	1	yes	characteristic
75	C₃₁×#106	—	2	2	no	—
76	C₃₁×C₄⋊C₈	—	2	2	no	—
77	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#103	D₄	1	2	yes	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	D₄	1	1	yes	characteristic
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×#97	—	2	2	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	C₂×D₄	1	2	yes	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	4	2	no	—
103	C₂×D₄×C₃₁	—	4	2	no	—
104	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	4	2	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	Q₁₆×C₃₁	—	2	2	no	—
111	SD₁₆×C₃₁	—	4	4	no	—
112	SD₁₆×C₃₁	—	8	4	no	—
113	D₈×C₃₁	—	2	2	no	—
114	D₈×C₃₁	—	4	4	no	—
115	D₈×C₃₁	—	4	2	no	—
116	D₈×C₃₁	—	8	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	4	2	no	—
122	C₂×C₈×C₃₁	—	4	2	no	—
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	4	2	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²⋊C₄×C₃₁	—	2	1	no	—
136	C₂²⋊C₄×C₃₁	—	2	1	no	—
137	C₂²⋊C₄×C₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
141	#19937	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	Q₈×C₃₁	—	2	2	no	—
146	Q₈×C₃₁	—	2	2	no	—
147	Q₈×C₃₁	—	8	2	no	—
148	D₄×C₃₁	—	2	1	no	—
149	D₄×C₃₁	—	2	1	no	—
150	D₄×C₃₁	—	2	1	no	—
151	D₄×C₃₁	—	4	2	no	—
152	D₄×C₃₁	—	4	2	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	2	no	—
155	D₄×C₃₁	—	8	2	no	—
156	D₄×C₃₁	—	8	4	no	—
157	D₄×C₃₁	—	16	2	no	—
158	D₄×C₃₁	#135	1	2	yes	complemented
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	8	4	no	—
169	C₂×C₄×C₃₁	—	16	2	no	—
170	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
173	C₈×C₃₁	—	2	2	no	—
174	C₈×C₃₁	—	2	1	no	—
175	C₈×C₃₁	—	2	1	no	—
176	C₈×C₃₁	—	2	1	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	—	4	1	no	—
179	C₈×C₃₁	—	4	1	no	—
180	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
181	#3366	C₂×C₃₁	1	2	yes	complemented
182	#3338	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#3287	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1698	C₂×C₃₁	1	2	yes	complemented
186	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1631	C₂×C₃₁	1	2	yes	complemented
189	#1622	C₂×C₃₁	1	2	yes	complemented
190	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
191	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
192	C₂²×C₃₁	—	2	2	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	—	4	1	no	—
196	C₂²×C₃₁	—	4	1	no	—
197	C₂²×C₃₁	—	4	1	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	1	no	—
203	C₄×C₃₁	—	4	1	no	—
204	C₄×C₃₁	—	4	1	no	—
205	C₄×C₃₁	—	4	2	no	—
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	—	8	2	no	—
208	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
209	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₄×C₃₁	#577	1	1	yes	characteristic
211	C₄×C₃₁	#578	1	1	yes	characteristic
212	C₂×#135	—	2	1	no	—
213	#554	C₂²×C₃₁	1	2	yes	—
214	#550	C₂²×C₃₁	1	2	yes	—
215	#500	C₂²×C₃₁	1	1	yes	characteristic
216	#493	C₂²×C₃₁	1	1	yes	characteristic
217	#492	C₂²×C₃₁	1	2	yes	—
218	#490	—	2	2	no	—
219	#486	C₂²×C₃₁	1	2	yes	—
220	#484	—	2	2	no	—
221	#478	—	2	2	no	—
222	#477	C₂²×C₃₁	1	2	yes	—
223	#476	C₂²×C₃₁	1	2	yes	—
224	#467	—	2	1	no	—
225	#466	—	2	1	no	—
226	#460	C₂²×C₃₁	1	2	yes	—
227	#459	—	2	2	no	—
228	#459	C₂²×C₃₁	1	2	yes	—
229	#449	—	2	2	no	—
230	#446	C₂²×C₃₁	1	2	yes	—
231	#442	—	2	2	no	—
232	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
233	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
235	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#107	—	2	1	no	—
237	#335	C₂²×C₃₁	1	1	yes	characteristic
238	#335	C₂²×C₃₁	1	1	yes	characteristic
239	#333	C₂²×C₃₁	1	2	yes	—
240	#331	C₂²×C₃₁	1	2	yes	—
241	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
242	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
243	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
244	C₈²	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₃₁	—	4	2	no	—
246	C₂×C₃₁	—	8	1	no	—
247	C₂×C₃₁	—	16	2	no	—
248	C₂×C₃₁	#3272	1	1	yes	characteristic
249	C₂×C₃₁	#3285	1	1	yes	characteristic
250	C₂×C₃₁	#3292	1	1	yes	characteristic
251	C₂×#41	—	2	2	no	—
252	#135	—	4	4	no	—
253	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
254	C₂×SD₁₆	—	2	2	no	—
255	C₂×SD₁₆	—	4	2	no	—
256	C₂×D₈	—	2	2	no	—
257	C₂×D₈	—	2	1	no	—
258	C₂×D₈	—	2	1	no	—
259	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
260	C₂²×C₈	—	2	1	no	—
261	C₂²×C₈	—	2	1	no	—
262	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
263	#127	—	2	2	no	—
264	#127	C₂³×C₃₁	1	2	yes	—
265	#125	—	2	2	no	—
266	C₂²⋊Q₈	—	2	2	no	—
267	C₂²⋊Q₈	—	4	2	no	—
268	#121	—	2	2	no	—
269	#121	—	4	2	no	—
270	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
271	C₄×D₄	—	2	2	no	—
272	C₄×D₄	—	2	2	no	—
273	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
274	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
275	C₂×C₄⋊C₄	—	2	2	no	—
276	#107	—	2	1	no	—
277	#107	—	2	1	no	—
278	#107	—	2	2	no	—
279	#107	D₄×C₃₁	1	1	yes	characteristic
280	#107	D₄×C₃₁	1	1	yes	characteristic
281	#106	—	2	2	no	—
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
284	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
286	#103	—	2	2	no	—
287	#103	D₄×C₃₁	1	2	yes	—
288	#102	—	2	2	no	—
289	#102	—	2	2	no	—
290	#102	—	2	2	no	—
291	#102	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	D₄×C₃₁	1	1	yes	characteristic
294	#102	D₄×C₃₁	1	1	yes	characteristic
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	1	no	—
297	#97	—	2	2	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	#19937	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	C₂×D₄×C₃₁	1	2	yes	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂×D₄	—	4	2	no	—
311	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	4	2	no	—
316	C₂²×C₄	—	4	2	no	—
317	Q₁₆	—	2	2	no	—
318	SD₁₆	—	4	4	no	—
319	SD₁₆	—	8	4	no	—
320	D₈	—	2	2	no	—
321	D₈	—	4	4	no	—
322	D₈	—	4	2	no	—
323	D₈	—	8	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	4	2	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
341	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂²⋊C₄	—	2	1	no	—
343	C₂²⋊C₄	—	2	1	no	—
344	C₂²⋊C₄	—	4	2	no	—
345	C₂²⋊C₄	—	4	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
348	C₂³	—	2	1	no	—
349	C₂³	—	2	1	no	—
350	C₂³	—	4	2	no	—
351	Q₈	—	2	2	no	—
352	Q₈	—	2	2	no	—
353	Q₈	—	8	2	no	—
354	D₄	—	2	1	no	—
355	D₄	—	2	1	no	—
356	D₄	—	2	1	no	—
357	D₄	—	4	2	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	4	2	no	—
361	D₄	—	8	2	no	—
362	D₄	—	8	4	no	—
363	D₄	—	16	2	no	—
364	D₄	C₃₁×#135	1	2	yes	complemented
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	4	no	—
375	C₂×C₄	—	16	2	no	—
376	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
379	C₈	—	2	2	no	—
380	C₈	—	2	1	no	—
381	C₈	—	2	1	no	—
382	C₈	—	2	1	no	—
383	C₈	—	4	2	no	—
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	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	1	no	—
398	C₄	—	4	1	no	—
399	C₄	—	4	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	2	no	—
403	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
404	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
405	C₄	C₃₁×#577	1	1	yes	characteristic
406	C₄	C₃₁×#578	1	1	yes	characteristic
407	C₂	—	4	2	no	—
408	C₂	—	8	1	no	—
409	C₂	—	16	2	no	—
410	C₂	C₃₁×#3272	1	1	yes	characteristic
411	C₂	C₃₁×#3285	1	1	yes	characteristic
412	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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. C₃₁×#3366	C₂	semidirect
2. C₃₁×#3338	C₂	semidirect
3. C₃₁×#3287	C₂	semidirect
4. C₃₁×#1706	C₂	semidirect
5. C₃₁×#1698	C₂	semidirect
6. C₃₁×#1675	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1631	C₂	semidirect
9. C₃₁×#1622	C₂	semidirect
10. C₃₁×#1591	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
141. #19937	C₃₁	direct (N × H)
158. D₄×C₃₁	#135	semidirect
181. #3366	C₂×C₃₁	semidirect
182. #3338	C₂×C₃₁	semidirect
183. #3287	C₂×C₃₁	semidirect
184. #1706	C₂×C₃₁	semidirect
185. #1698	C₂×C₃₁	semidirect
186. #1675	C₂×C₃₁	semidirect
187. #1667	C₂×C₃₁	semidirect
188. #1631	C₂×C₃₁	semidirect
189. #1622	C₂×C₃₁	semidirect
190. #1591	C₂×C₃₁	semidirect
191. C₈×D₈	C₂×C₃₁	semidirect
301. C₃₁	#19937	direct (N × H)
364. D₄	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3366	2
C₃₁×#3338	2
C₃₁×#3287	2
C₃₁×#1706	2
C₃₁×#1698	2
C₃₁×#1675	2
C₃₁×#1667	2
C₃₁×#1631	2
C₃₁×#1622	2
C₃₁×#1591	2
C₈×D₈×C₃₁	2
#19937	31

Subgroups — C31×#19937 #21476696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19937 #21476696