Subgroups — C₃₁×#32783 #21489542

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2629	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2620	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2095	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2093	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2083	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2042	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2036	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2033	C₂	1	2	yes	maximal
9	C₃₁×#2026	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2025	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1818	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₃₁×#398	—	2	2	no	—
25	C₃₁×#396	—	2	2	no	—
26	C₃₁×#396	—	2	2	no	—
27	C₃₁×#394	—	2	2	no	—
28	C₃₁×#391	C₂²	1	1	yes	characteristic
29	C₃₁×#390	C₂²	1	1	yes	characteristic
30	C₃₁×#389	C₂²	1	2	yes	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#380	C₂²	1	2	yes	—
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×D₄×C₃₁	—	2	1	no	—
41	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₃₁×#125	—	4	2	no	—
46	C₃₁×#124	—	4	2	no	—
47	C₃₁×C₂²⋊Q₈	—	4	2	no	—
48	C₃₁×#121	—	4	2	no	—
49	C₃₁×#117	—	4	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#107	—	4	2	no	—
69	C₃₁×#107	—	4	2	no	—
70	C₃₁×#103	—	4	2	no	—
71	C₃₁×#103	—	4	2	no	—
72	C₃₁×#102	—	4	2	no	—
73	C₃₁×#102	—	4	2	no	—
74	C₃₁×C₂²⋊C₈	—	4	2	no	—
75	C₃₁×#95	—	2	2	no	—
76	C₃₁×#95	—	2	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₂⁴×C₃₁	—	2	1	no	—
79	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂×Q₈×C₃₁	—	2	2	no	—
81	C₂×Q₈×C₃₁	—	2	2	no	—
82	C₂×Q₈×C₃₁	—	2	1	no	—
83	C₂×Q₈×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	1	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	8	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	4	2	no	—
116	C₄⋊C₄×C₃₁	—	4	2	no	—
117	C₄⋊C₄×C₃₁	—	4	4	no	—
118	C₄⋊C₄×C₃₁	—	8	2	no	—
119	C₄⋊C₄×C₃₁	—	8	2	no	—
120	C₄⋊C₄×C₃₁	—	8	2	no	—
121	C₂²⋊C₄×C₃₁	—	4	2	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₂²⋊C₄×C₃₁	—	4	4	no	—
124	C₂²⋊C₄×C₃₁	—	4	2	no	—
125	C₂²⋊C₄×C₃₁	—	8	2	no	—
126	C₂²⋊C₄×C₃₁	—	8	2	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	#32783	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	2	1	no	—
131	C₂³×C₃₁	—	2	1	no	—
132	C₂³×C₃₁	—	2	1	no	—
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	—	2	2	no	—
135	C₂³×C₃₁	—	4	1	no	—
136	C₂³×C₃₁	—	4	1	no	—
137	C₂³×C₃₁	—	4	1	no	—
138	C₂³×C₃₁	—	4	2	no	—
139	C₂³×C₃₁	—	4	2	no	—
140	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	Q₈×C₃₁	—	4	2	no	—
142	Q₈×C₃₁	—	4	2	no	—
143	D₄×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	8	4	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	4	1	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	1	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	1	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
171	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
173	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
174	C₈×C₃₁	—	4	2	no	—
175	C₈×C₃₁	—	4	2	no	—
176	#2629	C₂×C₃₁	1	1	yes	characteristic, complemented
177	#2620	C₂×C₃₁	1	1	yes	characteristic
178	#2095	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#2093	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#2083	C₂×C₃₁	1	2	yes	complemented
181	#2042	C₂×C₃₁	1	2	yes	complemented
182	#2036	C₂×C₃₁	1	2	yes	complemented
183	#2033	C₂×C₃₁	1	2	yes	—
184	#2026	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#2025	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1818	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₂²×C₃₁	—	4	1	no	—
188	C₂²×C₃₁	—	4	2	no	—
189	C₂²×C₃₁	—	4	1	no	—
190	C₂²×C₃₁	—	4	1	no	—
191	C₂²×C₃₁	—	4	2	no	—
192	C₂²×C₃₁	—	8	1	no	—
193	C₂²×C₃₁	—	8	2	no	—
194	C₂²×C₃₁	—	8	1	no	—
195	C₂²×C₃₁	—	8	1	no	—
196	C₂²×C₃₁	—	8	2	no	—
197	C₂²×C₃₁	#534	1	1	yes	characteristic
198	C₂²×C₃₁	#535	1	1	yes	characteristic
199	C₂²×C₃₁	#536	1	1	yes	characteristic
200	C₂²×C₃₁	#576	1	2	yes	—
201	C₂²×C₃₁	#578	1	2	yes	—
202	C₄×C₃₁	—	4	1	no	—
203	C₄×C₃₁	—	4	2	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	—	8	1	no	—
206	C₄×C₃₁	—	8	2	no	—
207	C₄×C₃₁	—	8	1	no	—
208	C₄×C₃₁	—	8	2	no	—
209	C₄×C₃₁	—	8	1	no	—
210	C₄×C₃₁	#562	1	2	yes	—
211	C₄×C₃₁	#562	1	2	yes	—
212	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
214	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×#107	C₂²×C₃₁	1	2	yes	—
218	C₂×#107	C₂²×C₃₁	1	2	yes	—
219	C₂×#103	C₂²×C₃₁	1	2	yes	—
220	C₂×#103	C₂²×C₃₁	1	2	yes	—
221	C₂×#102	C₂²×C₃₁	1	2	yes	—
222	C₂×#102	C₂²×C₃₁	1	2	yes	—
223	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
224	#398	—	2	2	no	—
225	#396	—	2	2	no	—
226	#396	—	2	2	no	—
227	#394	—	2	2	no	—
228	#391	C₂²×C₃₁	1	1	yes	characteristic
229	#390	C₂²×C₃₁	1	1	yes	characteristic
230	#389	C₂²×C₃₁	1	2	yes	—
231	#384	C₂²×C₃₁	1	1	yes	characteristic
232	#384	C₂²×C₃₁	1	1	yes	characteristic
233	#380	C₂²×C₃₁	1	2	yes	—
234	#340	C₂²×C₃₁	1	2	yes	—
235	#340	C₂²×C₃₁	1	2	yes	—
236	#336	C₂²×C₃₁	1	2	yes	—
237	#336	C₂²×C₃₁	1	1	yes	characteristic
238	#336	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×C₃₁	—	8	1	no	—
240	C₂×C₃₁	—	16	1	no	—
241	C₂×C₃₁	#2816	1	1	yes	characteristic
242	C₂×C₃₁	#3203	1	2	yes	—
243	C₂×C₃₁	#3220	1	1	yes	characteristic
244	C₂×C₃₁	#3223	1	1	yes	characteristic
245	C₂×C₃₁	#3243	1	2	yes	—
246	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
247	C₂²×D₄	—	2	1	no	—
248	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
249	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
252	#125	—	4	2	no	—
253	#124	—	4	2	no	—
254	C₂²⋊Q₈	—	4	2	no	—
255	#121	—	4	2	no	—
256	#117	—	4	2	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
263	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
264	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
265	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
266	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
267	C₂×C₂²⋊C₄	—	2	1	no	—
268	C₂×C₂²⋊C₄	—	2	1	no	—
269	C₂×C₂²⋊C₄	—	2	2	no	—
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
273	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
275	#107	—	4	2	no	—
276	#107	—	4	2	no	—
277	#103	—	4	2	no	—
278	#103	—	4	2	no	—
279	#102	—	4	2	no	—
280	#102	—	4	2	no	—
281	C₂²⋊C₈	—	4	2	no	—
282	#95	—	2	2	no	—
283	#95	—	2	2	no	—
284	#95	—	2	2	no	—
285	C₃₁	#32783	1	1	yes	characteristic, direct product factor, Sylow
286	C₂⁴	—	2	1	no	—
287	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	1	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	4	2	no	—
297	C₂×D₄	—	8	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
310	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	2	no	—
324	C₄⋊C₄	—	4	2	no	—
325	C₄⋊C₄	—	4	4	no	—
326	C₄⋊C₄	—	8	2	no	—
327	C₄⋊C₄	—	8	2	no	—
328	C₄⋊C₄	—	8	2	no	—
329	C₂²⋊C₄	—	4	2	no	—
330	C₂²⋊C₄	—	4	2	no	—
331	C₂²⋊C₄	—	4	4	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₂²⋊C₄	—	8	2	no	—
334	C₂²⋊C₄	—	8	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	no	—
337	C₂³	—	2	1	no	—
338	C₂³	—	2	1	no	—
339	C₂³	—	2	1	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	2	2	no	—
342	C₂³	—	4	1	no	—
343	C₂³	—	4	1	no	—
344	C₂³	—	4	1	no	—
345	C₂³	—	4	2	no	—
346	C₂³	—	4	2	no	—
347	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	D₄	—	4	2	no	—
351	D₄	—	4	2	no	—
352	D₄	—	8	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
378	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
379	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
380	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
381	C₈	—	4	2	no	—
382	C₈	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	8	1	no	—
391	C₂²	—	8	2	no	—
392	C₂²	—	8	1	no	—
393	C₂²	C₃₁×#534	1	1	yes	characteristic
394	C₂²	C₃₁×#535	1	1	yes	characteristic
395	C₂²	C₃₁×#536	1	1	yes	characteristic
396	C₂²	C₃₁×#576	1	2	yes	—
397	C₂²	C₃₁×#578	1	2	yes	—
398	C₄	—	4	2	no	—
399	C₄	—	4	2	no	—
400	C₄	—	4	1	no	—
401	C₄	—	8	2	no	—
402	C₄	—	8	1	no	—
403	C₄	—	8	1	no	—
404	C₄	—	8	2	no	—
405	C₄	—	8	1	no	—
406	C₄	C₃₁×#562	1	2	yes	—
407	C₄	C₃₁×#562	1	2	yes	—
408	C₂	—	8	1	no	—
409	C₂	—	16	1	no	—
410	C₂	C₃₁×#2816	1	1	yes	characteristic
411	C₂	C₃₁×#3203	1	2	yes	—
412	C₂	C₃₁×#3220	1	1	yes	characteristic
413	C₂	C₃₁×#3223	1	1	yes	characteristic
414	C₂	C₃₁×#3243	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2629	C₂	semidirect
3. C₃₁×#2095	C₂	semidirect
4. C₃₁×#2093	C₂	semidirect
5. C₃₁×#2083	C₂	semidirect
6. C₃₁×#2042	C₂	semidirect
7. C₃₁×#2036	C₂	semidirect
9. C₃₁×#2026	C₂	semidirect
10. C₃₁×#2025	C₂	semidirect
11. C₃₁×#1818	C₂	semidirect
129. #32783	C₃₁	direct (N × H)
176. #2629	C₂×C₃₁	semidirect
178. #2095	C₂×C₃₁	semidirect
179. #2093	C₂×C₃₁	semidirect
180. #2083	C₂×C₃₁	semidirect
181. #2042	C₂×C₃₁	semidirect
182. #2036	C₂×C₃₁	semidirect
184. #2026	C₂×C₃₁	semidirect
185. #2025	C₂×C₃₁	semidirect
186. #1818	C₂×C₃₁	semidirect
285. C₃₁	#32783	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2629	2
C₃₁×#2620	2
C₃₁×#2095	2
C₃₁×#2093	2
C₃₁×#2083	2
C₃₁×#2042	2
C₃₁×#2036	2
C₃₁×#2033	2
C₃₁×#2026	2
C₃₁×#2025	2
C₃₁×#1818	2
#32783	31

Subgroups — C31×#32783 #21489542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32783 #21489542