Subgroups — C₃₁×#22232 #21478991

Order 7936.

838 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	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2330	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2328	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1991	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1847	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#490	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#390	C₂²	1	1	yes	characteristic
26	C₃₁×#390	C₂²	1	1	yes	characteristic
27	C₃₁×#389	—	2	2	no	—
28	C₃₁×#389	C₂²	1	2	yes	—
29	C₃₁×#387	—	2	2	no	—
30	C₃₁×#386	—	2	2	no	—
31	C₃₁×#386	—	2	2	no	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#382	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₃₁×#376	C₂²	1	1	yes	characteristic
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	—	2	1	no	—
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₃₁×#127	—	2	2	no	—
48	C₃₁×#127	C₂×C₄	1	4	yes	—
49	C₃₁×#125	—	2	2	no	—
50	C₃₁×#125	C₂×C₄	1	4	yes	complemented
51	C₄×D₄×C₃₁	—	4	2	no	—
52	C₄×D₄×C₃₁	—	4	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
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×#106	—	4	2	no	—
70	C₃₁×#102	—	4	2	no	—
71	C₃₁×#102	—	4	2	no	—
72	C₃₁×#102	C₂×C₄	1	8	yes	complemented
73	C₃₁×#102	C₂×C₄	1	8	yes	complemented
74	C₃₁×#97	C₂×C₄	1	4	yes	complemented
75	C₄×C₈×C₃₁	—	2	2	no	—
76	C₃₁×#95	—	2	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	D₄	1	2	yes	—
80	C₃₁×#95	D₄	1	2	yes	—
81	C₂⁴×C₃₁	—	2	1	no	—
82	C₂⁴×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	2	8	no	—
84	C₂×D₄×C₃₁	—	2	1	no	—
85	C₂×D₄×C₃₁	—	2	1	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	8	2	no	—
91	C₂×D₄×C₃₁	—	8	2	no	—
92	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	4	2	no	—
99	C₂²×C₄×C₃₁	—	4	2	no	—
100	C₂²×C₄×C₃₁	—	8	2	no	—
101	C₂²×C₄×C₃₁	—	8	2	no	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₃₁	—	2	8	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	8	2	no	—
120	C₄⋊C₄×C₃₁	—	8	2	no	—
121	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₂²⋊C₄×C₃₁	—	4	2	no	—
124	C₂²⋊C₄×C₃₁	—	8	2	no	—
125	C₂²⋊C₄×C₃₁	—	8	2	no	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	8	2	no	—
129	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
130	C₄²×C₃₁	#41	1	2	yes	—
131	#22232	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	2	2	no	—
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	—	4	1	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	—	4	2	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	D₄×C₃₁	—	2	4	no	—
144	D₄×C₃₁	—	2	4	no	—
145	D₄×C₃₁	—	4	2	no	—
146	D₄×C₃₁	—	4	2	no	—
147	D₄×C₃₁	—	8	8	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	4	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	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	2	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	—	8	4	no	—
168	C₂×C₄×C₃₁	—	8	4	no	—
169	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
170	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
172	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
173	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
174	C₈×C₃₁	—	2	4	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₂×#490	C₂×C₃₁	1	1	yes	characteristic
177	#2330	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#2328	C₂×C₃₁	1	1	yes	characteristic
179	#1991	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#1957	C₂×C₃₁	1	1	yes	characteristic
181	#1926	C₂×C₃₁	1	2	yes	complemented
182	#1925	C₂×C₃₁	1	2	yes	complemented
183	#1864	C₂×C₃₁	1	2	yes	complemented
184	#1862	C₂×C₃₁	1	2	yes	complemented
185	#1847	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₂²×C₃₁	—	4	2	no	—
188	C₂²×C₃₁	—	4	2	no	—
189	C₂²×C₃₁	—	4	4	no	—
190	C₂²×C₃₁	—	4	4	no	—
191	C₂²×C₃₁	—	8	1	no	—
192	C₂²×C₃₁	—	8	1	no	—
193	C₂²×C₃₁	—	8	1	no	—
194	C₂²×C₃₁	—	8	1	no	—
195	C₂²×C₃₁	—	8	2	no	—
196	C₂²×C₃₁	—	8	2	no	—
197	C₂²×C₃₁	C₂×C₄×D₄	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	#577	1	2	yes	—
202	C₄×C₃₁	—	2	4	no	—
203	C₄×C₃₁	—	4	4	no	—
204	C₄×C₃₁	—	4	1	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	—	8	2	no	—
208	C₄×C₃₁	—	8	1	no	—
209	C₄×C₃₁	—	8	1	no	—
210	C₄×C₃₁	#520	1	2	yes	—
211	C₄×C₃₁	#542	1	2	yes	—
212	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
214	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
216	#490	C₄×C₃₁	1	8	yes	complemented
217	C₂×#107	C₂²×C₃₁	1	2	yes	—
218	C₂×#106	C₂²×C₃₁	1	2	yes	—
219	C₂×#102	C₂²×C₃₁	1	2	yes	—
220	C₂×#102	C₂²×C₃₁	1	2	yes	—
221	C₂×#102	C₂²×C₃₁	1	2	yes	—
222	C₂×#102	C₂²×C₃₁	1	2	yes	—
223	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
225	#390	C₂²×C₃₁	1	1	yes	characteristic
226	#390	C₂²×C₃₁	1	1	yes	characteristic
227	#389	—	2	2	no	—
228	#389	C₂²×C₃₁	1	2	yes	—
229	#387	—	2	2	no	—
230	#386	—	2	2	no	—
231	#386	—	2	2	no	—
232	#384	C₂²×C₃₁	1	1	yes	characteristic
233	#382	C₂²×C₃₁	1	1	yes	characteristic
234	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
235	#376	C₂²×C₃₁	1	1	yes	characteristic
236	#340	C₂²×C₃₁	1	2	yes	—
237	#340	C₂²×C₃₁	1	2	yes	—
238	#336	C₂²×C₃₁	1	2	yes	—
239	#336	C₂²×C₃₁	1	2	yes	—
240	C₂×C₃₁	—	8	2	no	—
241	C₂×C₃₁	—	8	2	no	—
242	C₂×C₃₁	#2326	1	1	yes	characteristic
243	C₂×C₃₁	#2946	1	2	yes	—
244	C₂×C₃₁	#3220	1	1	yes	characteristic
245	C₂×C₃₁	#3222	1	1	yes	characteristic
246	C₂×C₃₁	#3241	1	2	yes	—
247	C₂²×D₄	—	2	2	no	—
248	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
249	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
250	C₂³×C₄	—	2	1	no	—
251	C₂³×C₄	—	2	1	no	—
252	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
253	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
254	#127	—	2	2	no	—
255	#127	C₂×C₄×C₃₁	1	4	yes	—
256	#125	—	2	2	no	—
257	#125	C₂×C₄×C₃₁	1	4	yes	complemented
258	C₄×D₄	—	4	2	no	—
259	C₄×D₄	—	4	2	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
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×C₄²	—	2	2	no	—
272	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
275	#107	—	4	2	no	—
276	#106	—	4	2	no	—
277	#102	—	4	2	no	—
278	#102	—	4	2	no	—
279	#102	C₂×C₄×C₃₁	1	8	yes	complemented
280	#102	C₂×C₄×C₃₁	1	8	yes	complemented
281	#97	C₂×C₄×C₃₁	1	4	yes	complemented
282	C₄×C₈	—	2	2	no	—
283	#95	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	—	2	2	no	—
286	#95	D₄×C₃₁	1	2	yes	—
287	#95	D₄×C₃₁	1	2	yes	—
288	C₃₁	#22232	1	1	yes	characteristic, direct product factor, Sylow
289	C₂⁴	—	2	1	no	—
290	C₂⁴	—	2	1	no	—
291	C₂×D₄	—	2	8	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	8	2	no	—
299	C₂×D₄	—	8	2	no	—
300	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
301	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	4	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	—	8	2	no	—
309	C₂²×C₄	—	8	2	no	—
310	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
314	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
321	C₄⋊C₄	—	2	8	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	8	2	no	—
328	C₄⋊C₄	—	8	2	no	—
329	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
330	C₂²⋊C₄	—	4	2	no	—
331	C₂²⋊C₄	—	4	2	no	—
332	C₂²⋊C₄	—	8	2	no	—
333	C₂²⋊C₄	—	8	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	8	2	no	—
337	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
338	C₄²	#41×C₃₁	1	2	yes	—
339	C₂³	—	2	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	4	1	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	—	4	2	no	—
348	C₂³	—	4	2	no	—
349	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
350	D₄	—	2	4	no	—
351	D₄	—	2	4	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	2	no	—
354	D₄	—	8	8	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	4	no	—
375	C₂×C₄	—	8	4	no	—
376	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
377	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
379	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
380	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
381	C₈	—	2	4	no	—
382	C₈	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	4	no	—
386	C₂²	—	4	4	no	—
387	C₂²	—	8	1	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	8	1	no	—
391	C₂²	—	8	2	no	—
392	C₂²	—	8	2	no	—
393	C₂²	C₂×C₄×D₄×C₃₁	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×#577	1	2	yes	—
398	C₄	—	2	4	no	—
399	C₄	—	4	4	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	2	no	—
403	C₄	—	8	1	no	—
404	C₄	—	8	2	no	—
405	C₄	—	8	1	no	—
406	C₄	C₃₁×#520	1	2	yes	—
407	C₄	C₃₁×#542	1	2	yes	—
408	C₂	—	8	2	no	—
409	C₂	—	8	2	no	—
410	C₂	C₃₁×#2326	1	1	yes	characteristic
411	C₂	C₃₁×#2946	1	2	yes	—
412	C₂	C₃₁×#3220	1	1	yes	characteristic
413	C₂	C₃₁×#3222	1	1	yes	characteristic
414	C₂	C₃₁×#3241	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2330	C₂	semidirect
4. C₃₁×#1991	C₂	semidirect
6. C₃₁×#1926	C₂	semidirect
7. C₃₁×#1925	C₂	semidirect
8. C₃₁×#1864	C₂	semidirect
9. C₃₁×#1862	C₂	semidirect
10. C₃₁×#1847	C₂	semidirect
11. C₃₁×#1746	C₂	semidirect
16. C₃₁×#490	C₄	semidirect
50. C₃₁×#125	C₂×C₄	semidirect
72. C₃₁×#102	C₂×C₄	semidirect
73. C₃₁×#102	C₂×C₄	semidirect
74. C₃₁×#97	C₂×C₄	semidirect
131. #22232	C₃₁	direct (N × H)
177. #2330	C₂×C₃₁	semidirect
179. #1991	C₂×C₃₁	semidirect
181. #1926	C₂×C₃₁	semidirect
182. #1925	C₂×C₃₁	semidirect
183. #1864	C₂×C₃₁	semidirect
184. #1862	C₂×C₃₁	semidirect
185. #1847	C₂×C₃₁	semidirect
186. #1746	C₂×C₃₁	semidirect
216. #490	C₄×C₃₁	semidirect
257. #125	C₂×C₄×C₃₁	semidirect
279. #102	C₂×C₄×C₃₁	semidirect
280. #102	C₂×C₄×C₃₁	semidirect
281. #97	C₂×C₄×C₃₁	semidirect
288. C₃₁	#22232	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#490	2
C₃₁×#2330	2
C₃₁×#2328	2
C₃₁×#1991	2
C₃₁×#1957	2
C₃₁×#1926	2
C₃₁×#1925	2
C₃₁×#1864	2
C₃₁×#1862	2
C₃₁×#1847	2
C₃₁×#1746	2
#22232	31

Subgroups — C31×#22232 #21478991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22232 #21478991