Subgroups — C₃₁×#23093 #21479852

Order 7936.

1042 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1970	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
9	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#102	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	—	2	1	no	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#117	—	2	1	no	—
18	C₃₁×#459	C₄	1	8	yes	complemented
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#390	—	2	1	no	—
29	C₃₁×#390	C₂²	1	2	yes	—
30	C₃₁×#390	C₂²	1	2	yes	—
31	C₃₁×#388	—	2	2	no	—
32	C₃₁×#386	—	2	2	no	—
33	C₃₁×#384	C₂²	1	1	yes	characteristic
34	C₃₁×#380	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×D₄×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	—	4	2	no	—
49	C₂×D₈×C₃₁	—	4	2	no	—
50	C₂×D₈×C₃₁	C₂×C₄	1	8	yes	complemented
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#127	—	2	2	no	—
55	C₃₁×#127	C₂×C₄	1	4	yes	—
56	C₄×D₄×C₃₁	—	2	4	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	4	2	no	—
59	C₄×D₄×C₃₁	—	4	4	no	—
60	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
61	C₃₁×#117	—	4	4	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄²×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#107	—	4	2	no	—
81	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
82	C₃₁×#102	—	2	4	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	4	2	no	—
86	C₃₁×#102	C₂×C₄	1	8	yes	complemented
87	C₄×C₈×C₃₁	—	2	2	no	—
88	C₄×C₈×C₃₁	—	4	2	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	D₄	1	2	yes	—
92	C₂⁴×C₃₁	—	2	1	no	—
93	C₂⁴×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	4	no	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	—	4	2	no	—
107	C₂×D₄×C₃₁	—	4	2	no	—
108	C₂×D₄×C₃₁	—	8	1	no	—
109	C₂×D₄×C₃₁	—	8	1	no	—
110	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
111	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	4	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	4	4	no	—
118	C₂²×C₄×C₃₁	—	4	2	no	—
119	C₂²×C₄×C₃₁	—	4	2	no	—
120	C₂²×C₄×C₃₁	—	4	4	no	—
121	C₂²×C₄×C₃₁	—	8	2	no	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
127	D₈×C₃₁	—	4	8	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
136	C₂×C₈×C₃₁	#41	1	2	yes	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	4	2	no	—
145	C₄⋊C₄×C₃₁	—	8	1	no	—
146	C₄⋊C₄×C₃₁	D₈	1	4	yes	complemented
147	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
148	C₄⋊C₄×C₃₁	#41	1	2	yes	—
149	C₂²⋊C₄×C₃₁	—	2	4	no	—
150	C₂²⋊C₄×C₃₁	—	4	4	no	—
151	C₂²⋊C₄×C₃₁	—	4	4	no	—
152	C₂²⋊C₄×C₃₁	—	4	4	no	—
153	C₂²⋊C₄×C₃₁	—	8	2	no	—
154	C₂²⋊C₄×C₃₁	—	8	2	no	—
155	C₄²×C₃₁	—	2	2	no	—
156	C₄²×C₃₁	—	2	2	no	—
157	C₄²×C₃₁	—	4	4	no	—
158	C₄²×C₃₁	—	4	2	no	—
159	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
160	C₄²×C₃₁	#41	1	2	yes	—
161	#23093	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	2	4	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	1	no	—
166	C₂³×C₃₁	—	4	2	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	1	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	D₄×C₃₁	—	2	2	no	—
178	D₄×C₃₁	—	2	4	no	—
179	D₄×C₃₁	—	2	2	no	—
180	D₄×C₃₁	—	2	4	no	—
181	D₄×C₃₁	—	4	1	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	4	1	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	4	8	no	—
186	D₄×C₃₁	—	8	4	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	8	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	8	8	no	—
208	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
210	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
211	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
212	C₂×C₄×C₃₁	#135	1	2	yes	—
213	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
219	C₈×C₃₁	—	2	4	no	—
220	C₈×C₃₁	—	2	2	no	—
221	C₈×C₃₁	—	4	1	no	—
222	C₂×#459	C₂×C₃₁	1	1	yes	characteristic
223	#2366	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2354	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1970	C₂×C₃₁	1	2	yes	complemented
226	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#1925	C₂×C₃₁	1	2	yes	complemented
229	#1879	C₂×C₃₁	1	2	yes	complemented
230	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₄×#102	C₂×C₃₁	1	2	yes	complemented
232	C₄×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	4	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	4	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	8	1	no	—
240	C₂²×C₃₁	—	8	2	no	—
241	C₂²×C₃₁	—	8	1	no	—
242	C₂²×C₃₁	—	8	2	no	—
243	C₂²×C₃₁	—	8	2	no	—
244	C₂²×C₃₁	—	8	1	no	—
245	C₂²×C₃₁	—	8	2	no	—
246	C₂²×C₃₁	—	8	1	no	—
247	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
248	C₂²×C₃₁	#546	1	1	yes	characteristic
249	C₂²×C₃₁	#550	1	1	yes	characteristic
250	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
251	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
252	C₂²×C₃₁	#576	1	1	yes	characteristic
253	C₂²×C₃₁	#577	1	1	yes	characteristic
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	4	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	8	1	no	—
260	C₄×C₃₁	—	8	2	no	—
261	C₄×C₃₁	C₄×D₈	1	4	yes	—
262	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
263	C₄×C₃₁	#518	1	1	yes	characteristic
264	C₄×C₃₁	#553	1	1	yes	characteristic
265	C₄×C₃₁	#555	1	1	yes	characteristic
266	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
267	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄×D₄	—	2	1	no	—
269	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
270	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#117	—	2	1	no	—
272	#459	C₄×C₃₁	1	8	yes	complemented
273	C₂×#107	C₂²×C₃₁	1	2	yes	—
274	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×#102	C₂²×C₃₁	1	2	yes	—
277	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#102	C₂²×C₃₁	1	2	yes	—
279	C₂×#102	C₂²×C₃₁	1	2	yes	—
280	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
282	#390	—	2	1	no	—
283	#390	C₂²×C₃₁	1	2	yes	—
284	#390	C₂²×C₃₁	1	2	yes	—
285	#388	—	2	2	no	—
286	#386	—	2	2	no	—
287	#384	C₂²×C₃₁	1	1	yes	characteristic
288	#380	—	2	1	no	—
289	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
290	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
291	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
292	#340	C₂²×C₃₁	1	2	yes	—
293	#340	C₂²×C₃₁	1	1	yes	characteristic
294	#340	C₂²×C₃₁	1	1	yes	characteristic
295	#336	C₂²×C₃₁	1	2	yes	—
296	C₂×C₃₁	—	8	2	no	—
297	C₂×C₃₁	—	8	4	no	—
298	C₂×C₃₁	#2363	1	1	yes	characteristic
299	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
300	C₂×C₃₁	#2946	1	1	yes	characteristic
301	C₂×C₃₁	#3294	1	1	yes	characteristic
302	C₂×C₃₁	#3305	1	1	yes	characteristic
303	C₂×C₃₁	#3391	1	1	yes	characteristic
304	C₂×C₃₁	#3394	1	1	yes	characteristic
305	C₂²×D₄	—	2	1	no	—
306	C₂²×D₄	—	2	1	no	—
307	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
308	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂³×C₄	—	2	1	no	—
310	C₂³×C₄	—	2	2	no	—
311	C₂×D₈	—	4	2	no	—
312	C₂×D₈	—	4	2	no	—
313	C₂×D₈	C₂×C₄×C₃₁	1	8	yes	complemented
314	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
316	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	#127	—	2	2	no	—
318	#127	C₂×C₄×C₃₁	1	4	yes	—
319	C₄×D₄	—	2	4	no	—
320	C₄×D₄	—	2	2	no	—
321	C₄×D₄	—	4	2	no	—
322	C₄×D₄	—	4	4	no	—
323	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
324	#117	—	4	4	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₄²	—	2	2	no	—
340	C₂×C₄²	—	2	1	no	—
341	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
342	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
343	#107	—	4	2	no	—
344	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
345	#102	—	2	4	no	—
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	#102	—	4	2	no	—
349	#102	C₂×C₄×C₃₁	1	8	yes	complemented
350	C₄×C₈	—	2	2	no	—
351	C₄×C₈	—	4	2	no	—
352	#95	—	2	2	no	—
353	#95	—	2	1	no	—
354	#95	D₄×C₃₁	1	2	yes	—
355	C₃₁	#23093	1	1	yes	characteristic, direct product factor, Sylow
356	C₂⁴	—	2	1	no	—
357	C₂⁴	—	2	2	no	—
358	C₂×D₄	—	2	4	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	4	2	no	—
371	C₂×D₄	—	4	2	no	—
372	C₂×D₄	—	8	1	no	—
373	C₂×D₄	—	8	1	no	—
374	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
375	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	2	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	8	2	no	—
386	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
391	D₈	—	4	8	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
400	C₂×C₈	#41×C₃₁	1	2	yes	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	4	2	no	—
409	C₄⋊C₄	—	8	1	no	—
410	C₄⋊C₄	D₈×C₃₁	1	4	yes	complemented
411	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
412	C₄⋊C₄	#41×C₃₁	1	2	yes	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	4	4	no	—
415	C₂²⋊C₄	—	4	4	no	—
416	C₂²⋊C₄	—	4	4	no	—
417	C₂²⋊C₄	—	8	2	no	—
418	C₂²⋊C₄	—	8	2	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	4	4	no	—
422	C₄²	—	4	2	no	—
423	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
424	C₄²	#41×C₃₁	1	2	yes	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	4	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	1	no	—
431	C₂³	—	4	1	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	1	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	1	no	—
439	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
440	D₄	—	2	2	no	—
441	D₄	—	2	4	no	—
442	D₄	—	2	2	no	—
443	D₄	—	2	4	no	—
444	D₄	—	4	8	no	—
445	D₄	—	4	1	no	—
446	D₄	—	4	2	no	—
447	D₄	—	4	1	no	—
448	D₄	—	4	2	no	—
449	D₄	—	8	4	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	8	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	8	8	no	—
471	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
472	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
473	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
474	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
475	C₂×C₄	C₃₁×#135	1	2	yes	—
476	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
478	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
481	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
482	C₈	—	2	4	no	—
483	C₈	—	2	2	no	—
484	C₈	—	4	1	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	4	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	8	1	no	—
492	C₂²	—	8	2	no	—
493	C₂²	—	8	1	no	—
494	C₂²	—	8	2	no	—
495	C₂²	—	8	2	no	—
496	C₂²	—	8	1	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	8	1	no	—
499	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
500	C₂²	C₃₁×#546	1	1	yes	characteristic
501	C₂²	C₃₁×#550	1	1	yes	characteristic
502	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
503	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
504	C₂²	C₃₁×#576	1	1	yes	characteristic
505	C₂²	C₃₁×#577	1	1	yes	characteristic
506	C₄	—	2	2	no	—
507	C₄	—	2	4	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	8	1	no	—
512	C₄	—	8	2	no	—
513	C₄	C₄×D₈×C₃₁	1	4	yes	—
514	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
515	C₄	C₃₁×#518	1	1	yes	characteristic
516	C₄	C₃₁×#553	1	1	yes	characteristic
517	C₄	C₃₁×#555	1	1	yes	characteristic
518	C₂	—	8	2	no	—
519	C₂	—	8	4	no	—
520	C₂	C₃₁×#2363	1	1	yes	characteristic
521	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
522	C₂	C₃₁×#2946	1	1	yes	characteristic
523	C₂	C₃₁×#3294	1	1	yes	characteristic
524	C₂	C₃₁×#3305	1	1	yes	characteristic
525	C₂	C₃₁×#3391	1	1	yes	characteristic
526	C₂	C₃₁×#3394	1	1	yes	characteristic

	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×#459	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2366	C₂	semidirect
3. C₃₁×#2354	C₂	semidirect
4. C₃₁×#1970	C₂	semidirect
5. C₃₁×#1957	C₂	semidirect
6. C₃₁×#1947	C₂	semidirect
7. C₃₁×#1925	C₂	semidirect
8. C₃₁×#1879	C₂	semidirect
9. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
10. C₄×C₃₁×#102	C₂	semidirect
11. C₄×C₃₁×#102	C₂	semidirect
18. C₃₁×#459	C₄	semidirect
50. C₂×D₈×C₃₁	C₂×C₄	semidirect
60. C₄×D₄×C₃₁	C₂×C₄	semidirect
81. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
86. C₃₁×#102	C₂×C₄	semidirect
146. C₄⋊C₄×C₃₁	D₈	semidirect
161. #23093	C₃₁	direct (N × H)
223. #2366	C₂×C₃₁	semidirect
224. #2354	C₂×C₃₁	semidirect
225. #1970	C₂×C₃₁	semidirect
226. #1957	C₂×C₃₁	semidirect
227. #1947	C₂×C₃₁	semidirect
228. #1925	C₂×C₃₁	semidirect
229. #1879	C₂×C₃₁	semidirect
230. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
231. C₄×#102	C₂×C₃₁	semidirect
232. C₄×#102	C₂×C₃₁	semidirect
272. #459	C₄×C₃₁	semidirect
313. C₂×D₈	C₂×C₄×C₃₁	semidirect
323. C₄×D₄	C₂×C₄×C₃₁	semidirect
344. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
349. #102	C₂×C₄×C₃₁	semidirect
355. C₃₁	#23093	direct (N × H)
410. C₄⋊C₄	D₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#459	2
C₃₁×#2366	2
C₃₁×#2354	2
C₃₁×#1970	2
C₃₁×#1957	2
C₃₁×#1947	2
C₃₁×#1925	2
C₃₁×#1879	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×#102	2
#23093	31

Subgroups — C31×#23093 #21479852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23093 #21479852