Subgroups — C₃₁×#20661 #21477420

Order 7936.

1590 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#428	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#380	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1809	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1808	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1807	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1736	C₂	1	2	yes	maximal, complemented
9	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	—	2	8	no	—
11	C₂²×M₁₆×C₃₁	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
19	C₃₁×#428	—	2	8	no	—
20	C₂×C₃₁×#107	C₂²	1	4	yes	complemented
21	C₂×C₃₁×#106	C₂²	1	4	yes	complemented
22	C₃₁×#381	—	2	1	no	—
23	C₃₁×#380	—	2	1	no	—
24	C₃₁×#380	—	2	1	no	—
25	C₃₁×#380	C₂²	1	1	yes	characteristic
26	C₃₁×#380	C₂²	1	2	yes	—
27	C₃₁×#380	C₂²	1	1	yes	characteristic
28	C₃₁×#380	C₂²	1	2	yes	—
29	C₃₁×#380	C₂²	1	2	yes	—
30	C₃₁×#379	—	2	1	no	—
31	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
32	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
33	C₂×C₃₁×#95	—	2	1	no	—
34	C₃₁×#340	C₂²	1	4	yes	complemented
35	C₃₁×#340	C₂²	1	4	yes	complemented
36	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₁	—	2	4	no	—
38	C₂³×C₄×C₃₁	—	2	4	no	—
39	C₂³×C₄×C₃₁	—	2	2	no	—
40	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
41	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	Q₈	1	4	yes	—
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₃₁	—	2	4	no	—
49	C₂×M₁₆×C₃₁	—	2	32	no	—
50	C₂×M₁₆×C₃₁	—	2	8	no	—
51	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
54	C₂×M₁₆×C₃₁	D₄	1	8	yes	—
55	C₂²×C₈×C₃₁	—	2	8	no	—
56	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
57	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
59	C₃₁×#117	—	2	4	no	—
60	C₃₁×#117	—	2	4	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
74	C₂×C₄²×C₃₁	—	2	2	no	—
75	C₃₁×#107	—	2	8	no	—
76	C₃₁×#106	—	2	8	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	1	no	—
80	C₃₁×#95	—	2	1	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₂⁴×C₃₁	—	2	4	no	—
83	C₂⁴×C₃₁	—	2	2	no	—
84	C₂⁴×C₃₁	—	2	2	no	—
85	C₂⁴×C₃₁	—	4	2	no	—
86	C₂⁴×C₃₁	C₄⋊C₄	1	4	yes	—
87	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	—	2	4	no	—
91	C₂²×C₄×C₃₁	—	2	8	no	—
92	C₂²×C₄×C₃₁	—	2	8	no	—
93	C₂²×C₄×C₃₁	—	2	4	no	—
94	C₂²×C₄×C₃₁	—	2	4	no	—
95	C₂²×C₄×C₃₁	—	2	16	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	4	4	no	—
104	C₂²×C₄×C₃₁	—	4	4	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
107	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
108	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
109	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
119	C₂²×C₄×C₃₁	#41	1	4	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
121	M₁₆×C₃₁	—	2	8	no	—
122	M₁₆×C₃₁	—	2	16	no	—
123	M₁₆×C₃₁	C₂²⋊C₄	1	16	yes	complemented
124	C₂×C₈×C₃₁	—	2	4	no	—
125	C₂×C₈×C₃₁	—	2	16	no	—
126	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
128	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	4	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	4	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	4	4	no	—
136	C₂²⋊C₄×C₃₁	—	4	4	no	—
137	C₂²⋊C₄×C₃₁	—	4	4	no	—
138	C₂²⋊C₄×C₃₁	—	4	4	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₂²⋊C₄×C₃₁	—	4	4	no	—
141	C₂²⋊C₄×C₃₁	—	4	2	no	—
142	C₄²×C₃₁	—	4	4	no	—
143	#20661	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	—	2	2	no	—
149	C₂³×C₃₁	—	2	2	no	—
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	8	no	—
154	C₂³×C₃₁	—	2	8	no	—
155	C₂³×C₃₁	—	4	4	no	—
156	C₂³×C₃₁	—	4	8	no	—
157	C₂³×C₃₁	—	4	8	no	—
158	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
159	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
160	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
161	C₂³×C₃₁	#121	1	4	yes	—
162	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
163	C₂³×C₃₁	#137	1	2	yes	—
164	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	8	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	8	no	—
175	C₂×C₄×C₃₁	—	2	8	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	4	no	—
179	C₂×C₄×C₃₁	—	4	4	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	4	no	—
185	C₂×C₄×C₃₁	—	8	8	no	—
186	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
187	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
188	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
189	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
190	C₂×C₄×C₃₁	#117	1	4	yes	—
191	C₂×C₄×C₃₁	#121	1	4	yes	—
192	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
193	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
195	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
197	C₈×C₃₁	—	2	4	no	—
198	C₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	complemented
199	C₂³×M₁₆	C₂×C₃₁	1	1	yes	characteristic
200	C₂×#428	C₂×C₃₁	1	2	yes	complemented
201	#2280	C₂×C₃₁	1	1	yes	characteristic
202	C₂×#380	C₂×C₃₁	1	1	yes	characteristic
203	#1809	C₂×C₃₁	1	4	yes	complemented
204	#1808	C₂×C₃₁	1	2	yes	complemented
205	#1807	C₂×C₃₁	1	2	yes	complemented
206	#1736	C₂×C₃₁	1	2	yes	complemented
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	2	2	no	—
209	C₂²×C₃₁	—	2	2	no	—
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	2	4	no	—
212	C₂²×C₃₁	—	2	4	no	—
213	C₂²×C₃₁	—	2	4	no	—
214	C₂²×C₃₁	—	2	4	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	2	2	no	—
217	C₂²×C₃₁	—	2	2	no	—
218	C₂²×C₃₁	—	4	16	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	—	4	2	no	—
221	C₂²×C₃₁	#380	1	4	yes	—
222	C₂²×C₃₁	#428	1	4	yes	complemented
223	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
224	C₂²×C₃₁	C₂×#121	1	2	yes	—
225	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
227	C₂²×C₃₁	#575	1	2	yes	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	8	2	no	—
231	C₄×C₃₁	—	8	4	no	—
232	C₄×C₃₁	—	8	2	no	—
233	C₄×C₃₁	#380	1	4	yes	—
234	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
235	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
236	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
237	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
238	C₂²×M₁₆	—	2	8	no	—
239	C₂²×M₁₆	C₄×C₃₁	1	8	yes	complemented
240	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
241	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
242	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
243	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
244	C₂×#117	C₂²×C₃₁	1	2	yes	—
245	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
246	C₂²×C₂²⋊C₄	—	2	1	no	—
247	#428	—	2	8	no	—
248	C₂×#107	C₂²×C₃₁	1	4	yes	complemented
249	C₂×#106	C₂²×C₃₁	1	4	yes	complemented
250	#381	—	2	1	no	—
251	#380	—	2	1	no	—
252	#380	—	2	1	no	—
253	#380	C₂²×C₃₁	1	1	yes	characteristic
254	#380	C₂²×C₃₁	1	2	yes	—
255	#380	C₂²×C₃₁	1	1	yes	characteristic
256	#380	C₂²×C₃₁	1	2	yes	—
257	#380	C₂²×C₃₁	1	2	yes	—
258	#379	—	2	1	no	—
259	C₄×C₂²⋊C₄	—	2	1	no	—
260	C₄×C₂²⋊C₄	—	2	1	no	—
261	C₂×#95	—	2	1	no	—
262	#340	C₂²×C₃₁	1	4	yes	complemented
263	#340	C₂²×C₃₁	1	4	yes	complemented
264	C₂×C₃₁	—	2	4	no	—
265	C₂×C₃₁	—	2	4	no	—
266	C₂×C₃₁	—	4	2	no	—
267	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
268	C₂×C₃₁	C₂×#428	1	1	yes	characteristic
269	C₂×C₃₁	#2911	1	1	yes	characteristic
270	C₂×C₃₁	#3055	1	2	yes	—
271	C₂×C₃₁	#3056	1	2	yes	—
272	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
273	C₂³×C₄	—	2	4	no	—
274	C₂³×C₄	—	2	4	no	—
275	C₂³×C₄	—	2	2	no	—
276	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
277	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
278	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
279	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
280	C₂³×C₄	Q₈×C₃₁	1	4	yes	—
281	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
284	C₂×M₁₆	—	2	4	no	—
285	C₂×M₁₆	—	2	32	no	—
286	C₂×M₁₆	—	2	8	no	—
287	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
288	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
289	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
290	C₂×M₁₆	D₄×C₃₁	1	8	yes	—
291	C₂²×C₈	—	2	8	no	—
292	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
293	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
294	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
295	#117	—	2	4	no	—
296	#117	—	2	4	no	—
297	C₂×C₄⋊C₄	—	2	4	no	—
298	C₂×C₄⋊C₄	—	2	4	no	—
299	C₂×C₄⋊C₄	—	2	2	no	—
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	—	4	4	no	—
310	C₂×C₄²	—	2	2	no	—
311	#107	—	2	8	no	—
312	#106	—	2	8	no	—
313	#95	—	2	2	no	—
314	#95	—	2	2	no	—
315	#95	—	2	1	no	—
316	#95	—	2	1	no	—
317	#95	—	2	2	no	—
318	C₃₁	#20661	1	1	yes	characteristic, direct product factor, Sylow
319	C₂⁴	—	2	4	no	—
320	C₂⁴	—	2	2	no	—
321	C₂⁴	—	2	2	no	—
322	C₂⁴	—	4	2	no	—
323	C₂⁴	C₄⋊C₄×C₃₁	1	4	yes	—
324	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	8	no	—
329	C₂²×C₄	—	2	8	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	16	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	4	4	no	—
341	C₂²×C₄	—	4	4	no	—
342	C₂²×C₄	—	4	4	no	—
343	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
344	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
345	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
346	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
347	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
352	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
354	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
355	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
356	C₂²×C₄	#41×C₃₁	1	4	yes	—
357	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
358	M₁₆	—	2	8	no	—
359	M₁₆	—	2	16	no	—
360	M₁₆	C₂²⋊C₄×C₃₁	1	16	yes	complemented
361	C₂×C₈	—	2	4	no	—
362	C₂×C₈	—	2	16	no	—
363	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
364	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
365	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	4	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	4	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	4	4	no	—
373	C₂²⋊C₄	—	4	4	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₂²⋊C₄	—	4	2	no	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₂²⋊C₄	—	4	2	no	—
379	C₄²	—	4	4	no	—
380	C₂³	—	2	2	no	—
381	C₂³	—	2	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	8	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	2	1	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	2	8	no	—
391	C₂³	—	4	4	no	—
392	C₂³	—	4	8	no	—
393	C₂³	—	4	8	no	—
394	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
395	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
396	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
397	C₂³	C₃₁×#121	1	4	yes	—
398	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
399	C₂³	C₃₁×#137	1	2	yes	—
400	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	8	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	8	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	4	4	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	8	8	no	—
422	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
423	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
424	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
425	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
426	C₂×C₄	C₃₁×#117	1	4	yes	—
427	C₂×C₄	C₃₁×#121	1	4	yes	—
428	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
429	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
431	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
433	C₈	—	2	4	no	—
434	C₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	complemented
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	2	4	no	—
440	C₂²	—	2	4	no	—
441	C₂²	—	2	4	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	2	2	no	—
444	C₂²	—	2	2	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	4	16	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	C₃₁×#380	1	4	yes	—
450	C₂²	C₃₁×#428	1	4	yes	complemented
451	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
452	C₂²	C₂×C₃₁×#121	1	2	yes	—
453	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
454	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
455	C₂²	C₃₁×#575	1	2	yes	—
456	C₄	—	2	2	no	—
457	C₄	—	2	2	no	—
458	C₄	—	8	4	no	—
459	C₄	—	8	2	no	—
460	C₄	—	8	2	no	—
461	C₄	C₃₁×#380	1	4	yes	—
462	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
463	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
464	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
465	C₂	—	2	4	no	—
466	C₂	—	2	4	no	—
467	C₂	—	4	2	no	—
468	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
469	C₂	C₂×C₃₁×#428	1	1	yes	characteristic
470	C₂	C₃₁×#2911	1	1	yes	characteristic
471	C₂	C₃₁×#3055	1	2	yes	—
472	C₂	C₃₁×#3056	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	C₂⁵×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#428	C₂	semidirect
5. C₃₁×#1809	C₂	semidirect
6. C₃₁×#1808	C₂	semidirect
7. C₃₁×#1807	C₂	semidirect
8. C₃₁×#1736	C₂	semidirect
11. C₂²×M₁₆×C₃₁	C₄	semidirect
20. C₂×C₃₁×#107	C₂²	semidirect
21. C₂×C₃₁×#106	C₂²	semidirect
34. C₃₁×#340	C₂²	semidirect
35. C₃₁×#340	C₂²	semidirect
53. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
56. C₂²×C₈×C₃₁	C₂×C₄	semidirect
123. M₁₆×C₃₁	C₂²⋊C₄	semidirect
143. #20661	C₃₁	direct (N × H)
198. C₈×C₃₁	C₂×C₂²⋊C₄	semidirect
200. C₂×#428	C₂×C₃₁	semidirect
203. #1809	C₂×C₃₁	semidirect
204. #1808	C₂×C₃₁	semidirect
205. #1807	C₂×C₃₁	semidirect
206. #1736	C₂×C₃₁	semidirect
222. C₂²×C₃₁	#428	semidirect
239. C₂²×M₁₆	C₄×C₃₁	semidirect
248. C₂×#107	C₂²×C₃₁	semidirect
249. C₂×#106	C₂²×C₃₁	semidirect
262. #340	C₂²×C₃₁	semidirect
263. #340	C₂²×C₃₁	semidirect
289. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
292. C₂²×C₈	C₂×C₄×C₃₁	semidirect
318. C₃₁	#20661	direct (N × H)
360. M₁₆	C₂²⋊C₄×C₃₁	semidirect
434. C₈	C₂×C₂²⋊C₄×C₃₁	semidirect
450. C₂²	C₃₁×#428	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₃₁	2
C₂×C₃₁×#428	2
C₃₁×#2280	2
C₂×C₃₁×#380	2
C₃₁×#1809	2
C₃₁×#1808	2
C₃₁×#1807	2
C₃₁×#1736	2
#20661	31

Subgroups — C31×#20661 #21477420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20661 #21477420