Subgroups — C₃₁×#19960 #21476719

Order 7936.

658 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3397	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3342	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×#1686	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1642	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1628	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×#556	C₂²	1	2	yes	—
14	C₃₁×#550	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#496	—	2	2	no	—
17	C₃₁×#493	C₂²	1	1	yes	characteristic
18	C₃₁×#492	C₂²	1	2	yes	—
19	C₃₁×#486	C₂²	1	2	yes	—
20	C₃₁×#485	—	2	2	no	—
21	C₃₁×#475	C₂²	1	2	yes	—
22	C₃₁×#467	—	2	1	no	—
23	C₃₁×#466	—	2	1	no	—
24	C₃₁×#463	—	2	2	no	—
25	C₃₁×#461	C₂²	1	2	yes	—
26	C₃₁×#460	—	2	2	no	—
27	C₃₁×#460	C₂²	1	2	yes	—
28	C₃₁×#459	C₂²	1	2	yes	—
29	C₃₁×#449	—	2	2	no	—
30	C₃₁×#446	C₂²	1	2	yes	—
31	C₃₁×#440	—	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	C₃₁×#427	—	2	1	no	—
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	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	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₂×D₈×C₃₁	—	2	1	no	—
55	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₃₁×#127	—	2	2	no	—
60	C₃₁×#127	C₂³	1	2	yes	—
61	C₃₁×#126	—	4	2	no	—
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×C₂²⋊Q₈	—	2	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₃₁×#121	—	4	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#107	—	2	1	no	—
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	D₄	1	1	yes	characteristic
75	C₃₁×#107	D₄	1	1	yes	characteristic
76	C₃₁×#106	—	2	2	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	D₄	1	2	yes	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	4	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	Q₁₆×C₃₁	—	2	2	no	—
110	SD₁₆×C₃₁	—	4	4	no	—
111	SD₁₆×C₃₁	—	4	4	no	—
112	SD₁₆×C₃₁	—	8	2	no	—
113	SD₁₆×C₃₁	—	8	2	no	—
114	D₈×C₃₁	—	2	2	no	—
115	D₈×C₃₁	—	4	2	no	—
116	D₈×C₃₁	—	8	1	no	—
117	D₈×C₃₁	—	8	2	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	—	2	1	no	—
122	C₂×C₈×C₃₁	—	4	2	no	—
123	C₂×C₈×C₃₁	—	4	2	no	—
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	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	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	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	4	2	no	—
142	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
143	#19960	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	1	no	—
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	Q₈×C₃₁	—	2	2	no	—
148	Q₈×C₃₁	—	2	2	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	4	2	no	—
151	D₄×C₃₁	—	2	1	no	—
152	D₄×C₃₁	—	2	1	no	—
153	D₄×C₃₁	—	2	1	no	—
154	D₄×C₃₁	—	2	1	no	—
155	D₄×C₃₁	—	4	2	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	8	2	no	—
160	D₄×C₃₁	—	8	4	no	—
161	D₄×C₃₁	—	16	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	8	4	no	—
172	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
175	C₈×C₃₁	—	2	1	no	—
176	C₈×C₃₁	—	2	1	no	—
177	C₈×C₃₁	—	4	1	no	—
178	C₈×C₃₁	—	4	1	no	—
179	C₈×C₃₁	—	4	1	no	—
180	C₈×C₃₁	—	4	2	no	—
181	C₈×C₃₁	#135	1	2	yes	complemented
182	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
183	#3397	C₂×C₃₁	1	2	yes	complemented
184	#3342	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#3287	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#1686	C₂×C₃₁	1	2	yes	complemented
188	#1673	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1642	C₂×C₃₁	1	2	yes	complemented
191	#1628	C₂×C₃₁	1	2	yes	complemented
192	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
193	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
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	—	4	1	no	—
199	C₂²×C₃₁	—	4	1	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	—	8	2	no	—
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	D₄²	1	1	yes	characteristic
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	4	1	no	—
207	C₄×C₃₁	—	4	2	no	—
208	C₄×C₃₁	—	4	2	no	—
209	C₄×C₃₁	—	8	2	no	—
210	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
211	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
212	C₄×C₃₁	#576	1	1	yes	characteristic
213	C₄×C₃₁	#577	1	1	yes	characteristic
214	C₂×#135	—	2	1	no	—
215	#556	C₂²×C₃₁	1	2	yes	—
216	#550	C₂²×C₃₁	1	2	yes	—
217	#500	C₂²×C₃₁	1	1	yes	characteristic
218	#496	—	2	2	no	—
219	#493	C₂²×C₃₁	1	1	yes	characteristic
220	#492	C₂²×C₃₁	1	2	yes	—
221	#486	C₂²×C₃₁	1	2	yes	—
222	#485	—	2	2	no	—
223	#475	C₂²×C₃₁	1	2	yes	—
224	#467	—	2	1	no	—
225	#466	—	2	1	no	—
226	#463	—	2	2	no	—
227	#461	C₂²×C₃₁	1	2	yes	—
228	#460	—	2	2	no	—
229	#460	C₂²×C₃₁	1	2	yes	—
230	#459	C₂²×C₃₁	1	2	yes	—
231	#449	—	2	2	no	—
232	#446	C₂²×C₃₁	1	2	yes	—
233	#440	—	2	2	no	—
234	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
235	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
237	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
238	#427	—	2	1	no	—
239	#335	C₂²×C₃₁	1	1	yes	characteristic
240	#335	C₂²×C₃₁	1	1	yes	characteristic
241	#332	C₂²×C₃₁	1	2	yes	—
242	#331	C₂²×C₃₁	1	2	yes	—
243	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
244	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
245	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₈²	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₃₁	—	8	1	no	—
248	C₂×C₃₁	—	8	1	no	—
249	C₂×C₃₁	—	16	2	no	—
250	C₂×C₃₁	#3272	1	1	yes	characteristic
251	C₂×C₃₁	#3287	1	1	yes	characteristic
252	C₂×C₃₁	#3290	1	1	yes	characteristic
253	C₂×#41	—	2	2	no	—
254	#135	—	4	4	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×SD₁₆	—	2	2	no	—
260	C₂×D₈	—	2	2	no	—
261	C₂×D₈	—	2	1	no	—
262	C₂×D₈	—	2	1	no	—
263	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
267	#127	—	2	2	no	—
268	#127	C₂³×C₃₁	1	2	yes	—
269	#126	—	4	2	no	—
270	#124	—	2	2	no	—
271	C₂²⋊Q₈	—	2	2	no	—
272	#121	—	2	2	no	—
273	#121	—	4	2	no	—
274	C₄×Q₈	—	2	2	no	—
275	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
276	C₄×D₄	—	2	2	no	—
277	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
278	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
279	#117	—	2	2	no	—
280	#107	—	2	1	no	—
281	#107	—	2	1	no	—
282	#107	D₄×C₃₁	1	1	yes	characteristic
283	#107	D₄×C₃₁	1	1	yes	characteristic
284	#106	—	2	2	no	—
285	C₄⋊C₈	—	2	2	no	—
286	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
289	#103	—	2	2	no	—
290	#103	—	2	2	no	—
291	#103	D₄×C₃₁	1	2	yes	—
292	#102	—	2	2	no	—
293	#102	—	2	2	no	—
294	#102	—	4	2	no	—
295	#102	D₄×C₃₁	1	1	yes	characteristic
296	#102	D₄×C₃₁	1	1	yes	characteristic
297	C₂²⋊C₈	—	2	1	no	—
298	C₂²⋊C₈	—	2	1	no	—
299	#97	—	2	2	no	—
300	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
301	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₃₁	#19960	1	1	yes	characteristic, direct product factor, Sylow
304	#41	—	4	4	no	—
305	#41	—	8	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	4	2	no	—
318	Q₁₆	—	2	2	no	—
319	SD₁₆	—	4	4	no	—
320	SD₁₆	—	4	4	no	—
321	SD₁₆	—	8	2	no	—
322	SD₁₆	—	8	2	no	—
323	D₈	—	2	2	no	—
324	D₈	—	4	2	no	—
325	D₈	—	8	1	no	—
326	D₈	—	8	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	—	4	2	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	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
343	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²⋊C₄	—	2	1	no	—
345	C₂²⋊C₄	—	2	1	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	4	2	no	—
348	C₂²⋊C₄	—	4	2	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	4	2	no	—
351	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	4	2	no	—
355	Q₈	—	2	2	no	—
356	Q₈	—	2	2	no	—
357	Q₈	—	4	2	no	—
358	Q₈	—	4	2	no	—
359	D₄	—	2	1	no	—
360	D₄	—	2	1	no	—
361	D₄	—	2	1	no	—
362	D₄	—	2	1	no	—
363	D₄	—	4	2	no	—
364	D₄	—	4	2	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	8	2	no	—
368	D₄	—	8	4	no	—
369	D₄	—	16	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	8	4	no	—
380	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
383	C₈	—	2	1	no	—
384	C₈	—	2	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	2	no	—
389	C₈	C₃₁×#135	1	2	yes	complemented
390	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	8	2	no	—
400	C₂²	D₄²×C₃₁	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	2	no	—
407	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
408	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
409	C₄	C₃₁×#576	1	1	yes	characteristic
410	C₄	C₃₁×#577	1	1	yes	characteristic
411	C₂	—	8	1	no	—
412	C₂	—	8	1	no	—
413	C₂	—	16	2	no	—
414	C₂	C₃₁×#3272	1	1	yes	characteristic
415	C₂	C₃₁×#3287	1	1	yes	characteristic
416	C₂	C₃₁×#3290	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×#3397	C₂	semidirect
2. C₃₁×#3342	C₂	semidirect
3. C₃₁×#3287	C₂	semidirect
4. C₃₁×#1706	C₂	semidirect
5. C₃₁×#1686	C₂	semidirect
6. C₃₁×#1673	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1642	C₂	semidirect
9. C₃₁×#1628	C₂	semidirect
10. C₃₁×#1591	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
143. #19960	C₃₁	direct (N × H)
181. C₈×C₃₁	#135	semidirect
183. #3397	C₂×C₃₁	semidirect
184. #3342	C₂×C₃₁	semidirect
185. #3287	C₂×C₃₁	semidirect
186. #1706	C₂×C₃₁	semidirect
187. #1686	C₂×C₃₁	semidirect
188. #1673	C₂×C₃₁	semidirect
189. #1667	C₂×C₃₁	semidirect
190. #1642	C₂×C₃₁	semidirect
191. #1628	C₂×C₃₁	semidirect
192. #1591	C₂×C₃₁	semidirect
193. C₈×D₈	C₂×C₃₁	semidirect
303. C₃₁	#19960	direct (N × H)
389. C₈	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×#3397	2
C₃₁×#3342	2
C₃₁×#3287	2
C₃₁×#1706	2
C₃₁×#1686	2
C₃₁×#1673	2
C₃₁×#1667	2
C₃₁×#1642	2
C₃₁×#1628	2
C₃₁×#1591	2
C₈×D₈×C₃₁	2
#19960	31

Subgroups — C31×#19960 #21476719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19960 #21476719