Subgroups — C₃₁×#22436 #21479195

Order 7936.

1818 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#442	C₂	1	2	yes	maximal, complemented
3	D₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1972	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1879	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1786	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
10	C₂²×D₈×C₃₁	—	2	8	no	—
11	C₂²×D₈×C₃₁	C₄	1	8	yes	complemented
12	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
17	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
18	C₃₁×#442	—	2	8	no	—
19	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
20	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
21	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
24	C₃₁×#390	C₂²	1	4	yes	complemented
25	C₃₁×#386	—	2	4	no	—
26	C₃₁×#380	C₂²	1	2	yes	—
27	C₃₁×#379	—	2	2	no	—
28	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#340	C₂²	1	2	yes	complemented
30	C₃₁×#336	C₂²	1	2	yes	complemented
31	C₂⁵×C₃₁	—	2	2	no	—
32	C₂²×D₄×C₃₁	—	2	8	no	—
33	C₂²×D₄×C₃₁	—	2	2	no	—
34	C₂²×D₄×C₃₁	—	2	4	no	—
35	C₂²×D₄×C₃₁	—	4	4	no	—
36	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
39	C₂³×C₄×C₃₁	—	2	4	no	—
40	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×D₈×C₃₁	—	2	8	no	—
42	C₂×D₈×C₃₁	—	2	32	no	—
43	C₂×D₈×C₃₁	—	2	4	no	—
44	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
45	C₂×D₈×C₃₁	C₂×C₄	1	8	yes	complemented
46	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
47	C₂×D₈×C₃₁	D₄	1	8	yes	—
48	C₂²×C₈×C₃₁	—	2	4	no	—
49	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₃₁	—	2	8	no	—
54	C₄×D₄×C₃₁	—	4	4	no	—
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
58	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
59	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
60	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
61	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
62	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
63	C₃₁×#106	—	2	4	no	—
64	C₃₁×#102	—	2	8	no	—
65	C₃₁×#102	—	4	4	no	—
66	C₃₁×C₂²⋊C₈	—	2	2	no	—
67	C₃₁×#97	—	2	4	no	—
68	C₃₁×#95	—	2	2	no	—
69	C₃₁×#95	D₄	1	2	yes	—
70	C₂⁴×C₃₁	—	2	2	no	—
71	C₂⁴×C₃₁	—	2	2	no	—
72	C₂⁴×C₃₁	—	2	8	no	—
73	C₂⁴×C₃₁	—	4	8	no	—
74	C₂⁴×C₃₁	—	4	4	no	—
75	C₂⁴×C₃₁	—	4	4	no	—
76	C₂⁴×C₃₁	—	8	4	no	—
77	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×D₄×C₃₁	—	2	16	no	—
79	C₂×D₄×C₃₁	—	2	8	no	—
80	C₂×D₄×C₃₁	—	2	4	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	2	2	no	—
83	C₂×D₄×C₃₁	—	2	4	no	—
84	C₂×D₄×C₃₁	—	2	4	no	—
85	C₂×D₄×C₃₁	—	4	8	no	—
86	C₂×D₄×C₃₁	—	4	16	no	—
87	C₂×D₄×C₃₁	—	4	8	no	—
88	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
89	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
90	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	4	no	—
94	C₂²×C₄×C₃₁	—	4	8	no	—
95	C₂²×C₄×C₃₁	—	4	8	no	—
96	C₂²×C₄×C₃₁	—	8	4	no	—
97	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	#41	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	D₈×C₃₁	—	2	16	no	—
103	D₈×C₃₁	—	2	8	no	—
104	D₈×C₃₁	C₂²⋊C₄	1	16	yes	complemented
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	8	no	—
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	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
113	C₄⋊C₄×C₃₁	—	2	4	no	—
114	C₄⋊C₄×C₃₁	—	2	4	no	—
115	C₂²⋊C₄×C₃₁	—	2	4	no	—
116	C₂²⋊C₄×C₃₁	—	4	8	no	—
117	C₂²⋊C₄×C₃₁	—	4	8	no	—
118	C₂²⋊C₄×C₃₁	—	4	4	no	—
119	C₂²⋊C₄×C₃₁	—	8	4	no	—
120	C₄²×C₃₁	—	2	2	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	#22436	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₃₁	—	2	2	no	—
124	C₂³×C₃₁	—	2	2	no	—
125	C₂³×C₃₁	—	2	4	no	—
126	C₂³×C₃₁	—	2	8	no	—
127	C₂³×C₃₁	—	2	4	no	—
128	C₂³×C₃₁	—	4	16	no	—
129	C₂³×C₃₁	—	4	2	no	—
130	C₂³×C₃₁	—	4	2	no	—
131	C₂³×C₃₁	—	4	4	no	—
132	C₂³×C₃₁	—	4	4	no	—
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	—	4	2	no	—
135	C₂³×C₃₁	—	4	4	no	—
136	C₂³×C₃₁	—	4	16	no	—
137	C₂³×C₃₁	—	8	8	no	—
138	C₂³×C₃₁	—	8	16	no	—
139	C₂³×C₃₁	—	8	16	no	—
140	C₂³×C₃₁	C₄×D₄	1	2	yes	—
141	C₂³×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
142	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	D₄×C₃₁	—	2	4	no	—
144	D₄×C₃₁	—	2	4	no	—
145	D₄×C₃₁	—	2	4	no	—
146	D₄×C₃₁	—	2	8	no	—
147	D₄×C₃₁	—	4	16	no	—
148	D₄×C₃₁	—	4	4	no	—
149	D₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	complemented
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	4	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	4	no	—
157	C₂×C₄×C₃₁	—	4	4	no	—
158	C₂×C₄×C₃₁	—	4	4	no	—
159	C₂×C₄×C₃₁	—	4	4	no	—
160	C₂×C₄×C₃₁	—	8	16	no	—
161	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
162	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
164	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
166	C₈×C₃₁	—	2	2	no	—
167	C₈×C₃₁	—	4	2	no	—
168	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	complemented
169	C₂³×D₈	C₂×C₃₁	1	1	yes	characteristic
170	C₂×#442	C₂×C₃₁	1	2	yes	complemented
171	D₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
172	#1972	C₂×C₃₁	1	2	yes	complemented
173	#1879	C₂×C₃₁	1	4	yes	complemented
174	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
175	#1786	C₂×C₃₁	1	2	yes	complemented
176	#1752	C₂×C₃₁	1	1	yes	characteristic, complemented
177	C₂²×C₃₁	—	2	2	no	—
178	C₂²×C₃₁	—	2	2	no	—
179	C₂²×C₃₁	—	2	8	no	—
180	C₂²×C₃₁	—	2	8	no	—
181	C₂²×C₃₁	—	4	4	no	—
182	C₂²×C₃₁	—	4	4	no	—
183	C₂²×C₃₁	—	4	8	no	—
184	C₂²×C₃₁	—	4	8	no	—
185	C₂²×C₃₁	—	4	4	no	—
186	C₂²×C₃₁	—	4	4	no	—
187	C₂²×C₃₁	—	4	4	no	—
188	C₂²×C₃₁	—	8	32	no	—
189	C₂²×C₃₁	—	8	4	no	—
190	C₂²×C₃₁	—	8	4	no	—
191	C₂²×C₃₁	#442	1	4	yes	complemented
192	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂²×C₃₁	D₄²	1	2	yes	—
194	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
195	C₂²×C₃₁	#576	1	2	yes	—
196	C₄×C₃₁	—	2	2	no	—
197	C₄×C₃₁	—	4	4	no	—
198	C₄×C₃₁	—	8	4	no	—
199	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
200	C₄×C₃₁	#518	1	1	yes	characteristic
201	C₄×C₃₁	#546	1	2	yes	—
202	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
203	C₂²×D₈	—	2	8	no	—
204	C₂²×D₈	C₄×C₃₁	1	8	yes	complemented
205	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
206	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
207	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
208	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
209	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
210	C₂²×C₂²⋊C₄	—	2	2	no	—
211	#442	—	2	8	no	—
212	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
213	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
214	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
215	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
217	#390	C₂²×C₃₁	1	4	yes	complemented
218	#386	—	2	4	no	—
219	#380	C₂²×C₃₁	1	2	yes	—
220	#379	—	2	2	no	—
221	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
222	#340	C₂²×C₃₁	1	2	yes	complemented
223	#336	C₂²×C₃₁	1	2	yes	complemented
224	C₂×C₃₁	—	2	4	no	—
225	C₂×C₃₁	—	4	8	no	—
226	C₂×C₃₁	—	8	4	no	—
227	C₂×C₃₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
229	C₂×C₃₁	#2948	1	1	yes	characteristic
230	C₂×C₃₁	#3264	1	4	yes	—
231	C₂⁵	—	2	2	no	—
232	C₂²×D₄	—	2	8	no	—
233	C₂²×D₄	—	2	2	no	—
234	C₂²×D₄	—	2	4	no	—
235	C₂²×D₄	—	4	4	no	—
236	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
237	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
238	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
239	C₂³×C₄	—	2	4	no	—
240	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
241	C₂×D₈	—	2	8	no	—
242	C₂×D₈	—	2	32	no	—
243	C₂×D₈	—	2	4	no	—
244	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
245	C₂×D₈	C₂×C₄×C₃₁	1	8	yes	complemented
246	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
247	C₂×D₈	D₄×C₃₁	1	8	yes	—
248	C₂²×C₈	—	2	4	no	—
249	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
250	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
253	C₄×D₄	—	2	8	no	—
254	C₄×D₄	—	4	4	no	—
255	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
256	C₂×C₂²⋊C₄	—	2	2	no	—
257	C₂×C₂²⋊C₄	—	2	4	no	—
258	C₂×C₂²⋊C₄	—	2	4	no	—
259	C₂×C₂²⋊C₄	—	2	4	no	—
260	C₂×C₂²⋊C₄	—	4	8	no	—
261	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
262	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
263	#106	—	2	4	no	—
264	#102	—	2	8	no	—
265	#102	—	4	4	no	—
266	C₂²⋊C₈	—	2	2	no	—
267	#97	—	2	4	no	—
268	#95	—	2	2	no	—
269	#95	D₄×C₃₁	1	2	yes	—
270	C₃₁	#22436	1	1	yes	characteristic, direct product factor, Sylow
271	C₂⁴	—	2	2	no	—
272	C₂⁴	—	2	2	no	—
273	C₂⁴	—	2	8	no	—
274	C₂⁴	—	4	8	no	—
275	C₂⁴	—	4	4	no	—
276	C₂⁴	—	4	4	no	—
277	C₂⁴	—	8	4	no	—
278	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₂×D₄	—	2	16	no	—
280	C₂×D₄	—	2	8	no	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	4	no	—
285	C₂×D₄	—	2	4	no	—
286	C₂×D₄	—	4	8	no	—
287	C₂×D₄	—	4	16	no	—
288	C₂×D₄	—	4	8	no	—
289	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
290	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
291	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	4	8	no	—
296	C₂²×C₄	—	4	8	no	—
297	C₂²×C₄	—	8	4	no	—
298	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
299	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
300	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	#41×C₃₁	1	2	yes	—
302	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
303	D₈	—	2	16	no	—
304	D₈	—	2	8	no	—
305	D₈	C₂²⋊C₄×C₃₁	1	16	yes	complemented
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	8	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
312	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
313	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
314	C₄⋊C₄	—	2	4	no	—
315	C₄⋊C₄	—	2	4	no	—
316	C₂²⋊C₄	—	2	4	no	—
317	C₂²⋊C₄	—	4	8	no	—
318	C₂²⋊C₄	—	4	8	no	—
319	C₂²⋊C₄	—	4	4	no	—
320	C₂²⋊C₄	—	8	4	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₂³	—	2	2	no	—
324	C₂³	—	2	2	no	—
325	C₂³	—	2	4	no	—
326	C₂³	—	2	8	no	—
327	C₂³	—	2	4	no	—
328	C₂³	—	4	16	no	—
329	C₂³	—	4	2	no	—
330	C₂³	—	4	2	no	—
331	C₂³	—	4	4	no	—
332	C₂³	—	4	16	no	—
333	C₂³	—	4	4	no	—
334	C₂³	—	4	2	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	4	no	—
337	C₂³	—	8	8	no	—
338	C₂³	—	8	16	no	—
339	C₂³	—	8	16	no	—
340	C₂³	C₄×D₄×C₃₁	1	2	yes	—
341	C₂³	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
342	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
343	D₄	—	2	4	no	—
344	D₄	—	2	4	no	—
345	D₄	—	2	4	no	—
346	D₄	—	2	8	no	—
347	D₄	—	4	16	no	—
348	D₄	—	4	4	no	—
349	D₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	complemented
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	—	4	4	no	—
360	C₂×C₄	—	8	16	no	—
361	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
362	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
364	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
365	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
366	C₈	—	2	2	no	—
367	C₈	—	4	2	no	—
368	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	complemented
369	C₂²	—	2	2	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	2	8	no	—
372	C₂²	—	2	8	no	—
373	C₂²	—	4	4	no	—
374	C₂²	—	4	4	no	—
375	C₂²	—	4	8	no	—
376	C₂²	—	4	8	no	—
377	C₂²	—	4	4	no	—
378	C₂²	—	4	4	no	—
379	C₂²	—	4	4	no	—
380	C₂²	—	8	32	no	—
381	C₂²	—	8	4	no	—
382	C₂²	—	8	4	no	—
383	C₂²	C₃₁×#442	1	4	yes	complemented
384	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
385	C₂²	D₄²×C₃₁	1	2	yes	—
386	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
387	C₂²	C₃₁×#576	1	2	yes	—
388	C₄	—	2	2	no	—
389	C₄	—	4	4	no	—
390	C₄	—	8	4	no	—
391	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
392	C₄	C₃₁×#518	1	1	yes	characteristic
393	C₄	C₃₁×#546	1	2	yes	—
394	C₂	—	2	4	no	—
395	C₂	—	4	8	no	—
396	C₂	—	8	4	no	—
397	C₂	D₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
398	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
399	C₂	C₃₁×#2948	1	1	yes	characteristic
400	C₂	C₃₁×#3264	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×D₈×C₃₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#442	C₂	semidirect
3. D₄×C₂²⋊C₄×C₃₁	C₂	semidirect
4. C₃₁×#1972	C₂	semidirect
5. C₃₁×#1879	C₂	semidirect
6. C₃₁×#1807	C₂	semidirect
7. C₃₁×#1786	C₂	semidirect
8. C₃₁×#1752	C₂	semidirect
11. C₂²×D₈×C₃₁	C₄	semidirect
16. C₂×C₄×D₄×C₃₁	C₂²	semidirect
19. C₂×C₃₁×#106	C₂²	semidirect
20. C₂×C₃₁×#102	C₂²	semidirect
21. C₂×C₃₁×#102	C₂²	semidirect
23. C₂×C₃₁×#97	C₂²	semidirect
24. C₃₁×#390	C₂²	semidirect
29. C₃₁×#340	C₂²	semidirect
30. C₃₁×#336	C₂²	semidirect
36. C₂²×D₄×C₃₁	C₂×C₄	semidirect
45. C₂×D₈×C₃₁	C₂×C₄	semidirect
46. C₂×D₈×C₃₁	C₂×C₄	semidirect
49. C₂²×C₈×C₃₁	C₂×C₄	semidirect
104. D₈×C₃₁	C₂²⋊C₄	semidirect
122. #22436	C₃₁	direct (N × H)
149. D₄×C₃₁	C₂×C₂²⋊C₄	semidirect
168. C₈×C₃₁	C₂×C₂²⋊C₄	semidirect
170. C₂×#442	C₂×C₃₁	semidirect
171. D₄×C₂²⋊C₄	C₂×C₃₁	semidirect
172. #1972	C₂×C₃₁	semidirect
173. #1879	C₂×C₃₁	semidirect
174. #1807	C₂×C₃₁	semidirect
175. #1786	C₂×C₃₁	semidirect
176. #1752	C₂×C₃₁	semidirect
191. C₂²×C₃₁	#442	semidirect
204. C₂²×D₈	C₄×C₃₁	semidirect
209. C₂×C₄×D₄	C₂²×C₃₁	semidirect
212. C₂×#106	C₂²×C₃₁	semidirect
213. C₂×#102	C₂²×C₃₁	semidirect
214. C₂×#102	C₂²×C₃₁	semidirect
216. C₂×#97	C₂²×C₃₁	semidirect
217. #390	C₂²×C₃₁	semidirect
222. #340	C₂²×C₃₁	semidirect
223. #336	C₂²×C₃₁	semidirect
236. C₂²×D₄	C₂×C₄×C₃₁	semidirect
245. C₂×D₈	C₂×C₄×C₃₁	semidirect
246. C₂×D₈	C₂×C₄×C₃₁	semidirect
249. C₂²×C₈	C₂×C₄×C₃₁	semidirect
270. C₃₁	#22436	direct (N × H)
305. D₈	C₂²⋊C₄×C₃₁	semidirect
349. D₄	C₂×C₂²⋊C₄×C₃₁	semidirect
368. C₈	C₂×C₂²⋊C₄×C₃₁	semidirect
383. C₂²	C₃₁×#442	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₃₁	2
C₂×C₃₁×#442	2
D₄×C₂²⋊C₄×C₃₁	2
C₃₁×#1972	2
C₃₁×#1879	2
C₃₁×#1807	2
C₃₁×#1786	2
C₃₁×#1752	2
#22436	31

Subgroups — C31×#22436 #21479195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22436 #21479195