Subgroups — C₃₁×#19902 #21476661

Order 7936.

672 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×#3334	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3330	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1699	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1654	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1654	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1589	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×#547	C₂²	1	2	yes	—
14	C₃₁×#547	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#498	C₂²	1	2	yes	—
17	C₃₁×#494	C₂²	1	1	yes	characteristic
18	C₃₁×#487	C₂²	1	2	yes	—
19	C₃₁×#484	—	2	2	no	—
20	C₃₁×#480	—	2	2	no	—
21	C₃₁×#474	C₂²	1	2	yes	—
22	C₃₁×#474	C₂²	1	2	yes	—
23	C₃₁×#468	—	2	2	no	—
24	C₃₁×#467	—	2	1	no	—
25	C₃₁×#466	—	2	1	no	—
26	C₃₁×#461	C₂²	1	2	yes	—
27	C₃₁×#461	C₂²	1	2	yes	—
28	C₃₁×#452	—	2	2	no	—
29	C₃₁×#449	—	2	2	no	—
30	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#436	C₂²	1	2	yes	—
33	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#107	—	2	1	no	—
36	C₃₁×#417	—	2	2	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×#332	C₂²	1	2	yes	—
41	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×D₄⋊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	C₂×#41×C₃₁	—	2	2	no	—
47	C₃₁×#135	—	4	4	no	—
48	C₂×Q₁₆×C₃₁	—	2	1	no	—
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₃₁×#123	—	4	2	no	—
60	C₃₁×#123	—	4	2	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	—	2	2	no	—
63	C₃₁×#121	—	2	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
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	—	2	2	no	—
74	C₃₁×#107	D₄	1	1	yes	characteristic
75	C₃₁×#107	D₄	1	1	yes	characteristic
76	C₃₁×#106	—	4	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	—	2	2	no	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	D₄	1	1	yes	characteristic
87	C₃₁×#102	D₄	1	2	yes	—
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	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	—	4	4	no	—
97	#41×C₃₁	—	8	2	no	—
98	#41×C₃₁	—	8	2	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂×D₄×C₃₁	—	4	2	no	—
103	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
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	2	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	Q₁₆×C₃₁	—	4	2	no	—
112	Q₁₆×C₃₁	—	8	1	no	—
113	SD₁₆×C₃₁	—	4	4	no	—
114	SD₁₆×C₃₁	—	8	2	no	—
115	D₈×C₃₁	—	2	2	no	—
116	D₈×C₃₁	—	2	2	no	—
117	M₁₆×C₃₁	—	8	2	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	2	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	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₃₁	—	2	1	no	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	2	1	no	—
140	C₂²⋊C₄×C₃₁	—	4	2	no	—
141	C₂²⋊C₄×C₃₁	—	4	2	no	—
142	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
143	#19902	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	—	2	2	no	—
147	Q₈×C₃₁	—	4	2	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	4	2	no	—
151	D₄×C₃₁	—	2	1	no	—
152	D₄×C₃₁	—	2	2	no	—
153	D₄×C₃₁	—	2	1	no	—
154	D₄×C₃₁	—	2	1	no	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	8	4	no	—
157	D₄×C₃₁	—	8	4	no	—
158	D₄×C₃₁	#135	1	2	yes	complemented
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	8	4	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	2	no	—
176	C₈×C₃₁	—	2	1	no	—
177	C₈×C₃₁	—	2	1	no	—
178	C₈×C₃₁	—	2	1	no	—
179	C₈×C₃₁	—	4	1	no	—
180	C₈×C₃₁	—	4	1	no	—
181	C₈×C₃₁	—	4	2	no	—
182	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
183	#3334	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#3330	C₂×C₃₁	1	2	yes	complemented
185	#3283	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1707	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#1699	C₂×C₃₁	1	2	yes	complemented
188	#1673	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1654	C₂×C₃₁	1	2	yes	complemented
191	#1654	C₂×C₃₁	1	2	yes	complemented
192	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
193	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂²×C₃₁	—	2	2	no	—
195	C₂²×C₃₁	—	4	1	no	—
196	C₂²×C₃₁	—	4	1	no	—
197	C₂²×C₃₁	—	4	1	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	4	2	no	—
200	C₂²×C₃₁	—	4	1	no	—
201	C₂²×C₃₁	—	4	1	no	—
202	C₂²×C₃₁	—	4	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	—	8	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	#547	C₂²×C₃₁	1	2	yes	—
216	#547	C₂²×C₃₁	1	2	yes	—
217	#500	C₂²×C₃₁	1	1	yes	characteristic
218	#498	C₂²×C₃₁	1	2	yes	—
219	#494	C₂²×C₃₁	1	1	yes	characteristic
220	#487	C₂²×C₃₁	1	2	yes	—
221	#484	—	2	2	no	—
222	#480	—	2	2	no	—
223	#474	C₂²×C₃₁	1	2	yes	—
224	#474	C₂²×C₃₁	1	2	yes	—
225	#468	—	2	2	no	—
226	#467	—	2	1	no	—
227	#466	—	2	1	no	—
228	#461	C₂²×C₃₁	1	2	yes	—
229	#461	C₂²×C₃₁	1	2	yes	—
230	#452	—	2	2	no	—
231	#449	—	2	2	no	—
232	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
234	#436	C₂²×C₃₁	1	2	yes	—
235	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
236	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×#107	—	2	1	no	—
238	#417	—	2	2	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	#332	C₂²×C₃₁	1	2	yes	—
243	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
244	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
245	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
246	C₈²	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₃₁	—	4	2	no	—
248	C₂×C₃₁	—	8	1	no	—
249	C₂×C₃₁	—	8	2	no	—
250	C₂×C₃₁	#3272	1	1	yes	characteristic
251	C₂×C₃₁	#3280	1	1	yes	characteristic
252	C₂×C₃₁	#3293	1	1	yes	characteristic
253	C₂×#41	—	2	2	no	—
254	C₂×#41	—	2	2	no	—
255	#135	—	4	4	no	—
256	C₂×Q₁₆	—	2	1	no	—
257	C₂×Q₁₆	—	2	1	no	—
258	C₂×SD₁₆	—	2	2	no	—
259	C₂×SD₁₆	—	2	2	no	—
260	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
261	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
262	C₂×M₁₆	—	2	2	no	—
263	C₂²×C₈	—	2	1	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
266	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
267	#123	—	4	2	no	—
268	#123	—	4	2	no	—
269	C₂²⋊Q₈	—	2	2	no	—
270	C₂²⋊Q₈	—	2	2	no	—
271	#121	—	2	2	no	—
272	#121	—	2	2	no	—
273	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
274	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
275	C₄×D₄	C₂³×C₃₁	1	2	yes	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	#107	—	2	1	no	—
279	#107	—	2	1	no	—
280	#107	—	2	2	no	—
281	#107	—	2	2	no	—
282	#107	D₄×C₃₁	1	1	yes	characteristic
283	#107	D₄×C₃₁	1	1	yes	characteristic
284	#106	—	4	2	no	—
285	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
288	#103	—	2	2	no	—
289	#103	—	2	2	no	—
290	#103	—	2	2	no	—
291	#103	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	—	2	2	no	—
294	#102	D₄×C₃₁	1	1	yes	characteristic
295	#102	D₄×C₃₁	1	2	yes	—
296	#102	D₄×C₃₁	1	1	yes	characteristic
297	C₂²⋊C₈	—	2	1	no	—
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	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	#19902	1	1	yes	characteristic, direct product factor, Sylow
304	#41	—	4	4	no	—
305	#41	—	4	4	no	—
306	#41	—	8	2	no	—
307	#41	—	8	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×D₄	—	4	2	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
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	2	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	—	4	2	no	—
320	Q₁₆	—	4	2	no	—
321	Q₁₆	—	8	1	no	—
322	SD₁₆	—	4	4	no	—
323	SD₁₆	—	8	2	no	—
324	D₈	—	2	2	no	—
325	D₈	—	2	2	no	—
326	M₁₆	—	8	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	2	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	—	2	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
345	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	4	2	no	—
350	C₂²⋊C₄	—	4	2	no	—
351	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	2	2	no	—
355	Q₈	—	4	2	no	—
356	Q₈	—	4	2	no	—
357	Q₈	—	4	2	no	—
358	Q₈	—	4	2	no	—
359	D₄	—	2	1	no	—
360	D₄	—	2	2	no	—
361	D₄	—	2	1	no	—
362	D₄	—	2	1	no	—
363	D₄	—	2	2	no	—
364	D₄	—	8	4	no	—
365	D₄	—	8	4	no	—
366	D₄	C₃₁×#135	1	2	yes	complemented
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	8	4	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	2	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	1	no	—
386	C₈	—	2	1	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	1	no	—
389	C₈	—	4	2	no	—
390	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	2	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	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	—	8	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	—	4	2	no	—
412	C₂	—	8	1	no	—
413	C₂	—	8	2	no	—
414	C₂	C₃₁×#3272	1	1	yes	characteristic
415	C₂	C₃₁×#3280	1	1	yes	characteristic
416	C₂	C₃₁×#3293	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×#3334	C₂	semidirect
2. C₃₁×#3330	C₂	semidirect
3. C₃₁×#3283	C₂	semidirect
4. C₃₁×#1707	C₂	semidirect
5. C₃₁×#1699	C₂	semidirect
6. C₃₁×#1673	C₂	semidirect
7. C₃₁×#1669	C₂	semidirect
8. C₃₁×#1654	C₂	semidirect
9. C₃₁×#1654	C₂	semidirect
10. C₃₁×#1589	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
143. #19902	C₃₁	direct (N × H)
158. D₄×C₃₁	#135	semidirect
183. #3334	C₂×C₃₁	semidirect
184. #3330	C₂×C₃₁	semidirect
185. #3283	C₂×C₃₁	semidirect
186. #1707	C₂×C₃₁	semidirect
187. #1699	C₂×C₃₁	semidirect
188. #1673	C₂×C₃₁	semidirect
189. #1669	C₂×C₃₁	semidirect
190. #1654	C₂×C₃₁	semidirect
191. #1654	C₂×C₃₁	semidirect
192. #1589	C₂×C₃₁	semidirect
193. C₈×D₈	C₂×C₃₁	semidirect
303. C₃₁	#19902	direct (N × H)
366. 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×#3334	2
C₃₁×#3330	2
C₃₁×#3283	2
C₃₁×#1707	2
C₃₁×#1699	2
C₃₁×#1673	2
C₃₁×#1669	2
C₃₁×#1654	2
C₃₁×#1589	2
C₈×D₈×C₃₁	2
#19902	31

Subgroups — C31×#19902 #21476661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19902 #21476661