Subgroups — C₃₁×#23001 #21479760

Order 7936.

1158 conjugacy classes of subgroups.

The subgroup lattice has 540 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	C₂×C₃₁×#442	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2366	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1936	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, 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	2	yes	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
19	C₃₁×#459	C₄	1	8	yes	complemented
20	C₃₁×#442	—	2	8	no	—
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
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×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#97	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#390	—	2	1	no	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	2	yes	—
33	C₃₁×#386	—	2	2	no	—
34	C₃₁×#385	—	2	2	no	—
35	C₃₁×#384	C₂²	1	1	yes	characteristic
36	C₃₁×#380	—	2	1	no	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#340	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
46	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂³×C₄×C₃₁	—	2	2	no	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	4	no	—
51	C₂×D₈×C₃₁	—	2	4	no	—
52	C₂×D₈×C₃₁	C₂×C₄	1	8	yes	complemented
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#127	—	2	2	no	—
57	C₃₁×#127	C₂×C₄	1	4	yes	complemented
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	—	2	4	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	4	2	no	—
62	C₄×D₄×C₃₁	—	4	2	no	—
63	C₄×D₄×C₃₁	—	4	4	no	—
64	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
69	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₄²×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#106	—	2	4	no	—
84	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	4	no	—
87	C₃₁×#102	—	4	2	no	—
88	C₃₁×#102	C₂×C₄	1	8	yes	complemented
89	C₃₁×#102	D₄	1	4	yes	complemented
90	C₃₁×#97	—	2	4	no	—
91	C₄×C₈×C₃₁	—	2	2	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	D₄	1	2	yes	—
95	C₂⁴×C₃₁	—	2	1	no	—
96	C₂⁴×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	2	4	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	1	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	4	2	no	—
108	C₂×D₄×C₃₁	—	4	2	no	—
109	C₂×D₄×C₃₁	—	8	1	no	—
110	C₂×D₄×C₃₁	—	8	1	no	—
111	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
112	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂×D₄×C₃₁	#41	1	2	yes	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	4	4	no	—
121	C₂²×C₄×C₃₁	—	4	4	no	—
122	C₂²×C₄×C₃₁	—	4	4	no	—
123	C₂²×C₄×C₃₁	—	4	4	no	—
124	C₂²×C₄×C₃₁	—	4	2	no	—
125	C₂²×C₄×C₃₁	—	4	2	no	—
126	C₂²×C₄×C₃₁	—	8	2	no	—
127	C₂²×C₄×C₃₁	—	8	2	no	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	D₈×C₃₁	—	2	16	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
142	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	4	4	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	—	8	1	no	—
154	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
155	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²⋊C₄×C₃₁	—	2	4	no	—
157	C₂²⋊C₄×C₃₁	—	4	4	no	—
158	C₂²⋊C₄×C₃₁	—	4	4	no	—
159	C₂²⋊C₄×C₃₁	—	4	4	no	—
160	C₂²⋊C₄×C₃₁	—	8	2	no	—
161	C₄²×C₃₁	—	2	2	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	4	2	no	—
164	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
165	C₄²×C₃₁	#41	1	2	yes	—
166	#23001	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	2	no	—
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	2	2	no	—
170	C₂³×C₃₁	—	2	4	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	—	4	1	no	—
173	C₂³×C₃₁	—	4	1	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	2	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	2	no	—
181	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	D₄×C₃₁	—	2	4	no	—
183	D₄×C₃₁	—	2	2	no	—
184	D₄×C₃₁	—	2	4	no	—
185	D₄×C₃₁	—	2	2	no	—
186	D₄×C₃₁	—	2	2	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	2	no	—
189	D₄×C₃₁	—	4	8	no	—
190	D₄×C₃₁	—	8	4	no	—
191	D₄×C₃₁	C₄×D₄	1	4	yes	complemented
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	8	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	8	8	no	—
214	C₂×C₄×C₃₁	—	8	8	no	—
215	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
216	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
217	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
218	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
219	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₃₁	—	2	4	no	—
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	4	1	no	—
228	C₂×#459	C₂×C₃₁	1	1	yes	characteristic
229	C₂×#442	C₂×C₃₁	1	2	yes	complemented
230	#2366	C₂×C₃₁	1	1	yes	characteristic, complemented
231	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1936	C₂×C₃₁	1	2	yes	complemented
235	#1879	C₂×C₃₁	1	2	yes	complemented
236	#1864	C₂×C₃₁	1	2	yes	complemented
237	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₄×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂²×C₃₁	—	2	4	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	—	4	2	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	4	4	no	—
247	C₂²×C₃₁	—	4	4	no	—
248	C₂²×C₃₁	—	4	4	no	—
249	C₂²×C₃₁	—	8	2	no	—
250	C₂²×C₃₁	—	8	2	no	—
251	C₂²×C₃₁	—	8	2	no	—
252	C₂²×C₃₁	—	8	2	no	—
253	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
254	C₂²×C₃₁	D₄²	1	1	yes	characteristic
255	C₂²×C₃₁	#548	1	1	yes	characteristic
256	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂²×C₃₁	#576	1	1	yes	characteristic
259	C₂²×C₃₁	#577	1	1	yes	characteristic
260	C₄×C₃₁	—	2	2	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	—	4	2	no	—
264	C₄×C₃₁	—	4	2	no	—
265	C₄×C₃₁	—	8	1	no	—
266	C₄×C₃₁	—	8	2	no	—
267	C₄×C₃₁	#442	1	4	yes	complemented
268	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
269	C₄×C₃₁	#520	1	1	yes	characteristic
270	C₄×C₃₁	#547	1	1	yes	characteristic
271	C₄×C₃₁	#553	1	1	yes	characteristic
272	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
273	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄×D₄	—	2	1	no	—
275	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
276	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
277	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
278	C₂²×C₄⋊C₄	—	2	1	no	—
279	#459	C₄×C₃₁	1	8	yes	complemented
280	#442	—	2	8	no	—
281	C₂×#106	C₂²×C₃₁	1	2	yes	—
282	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×#102	C₂²×C₃₁	1	2	yes	—
285	C₂×#102	C₂²×C₃₁	1	2	yes	—
286	C₂×#102	C₂²×C₃₁	1	2	yes	—
287	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#97	C₂²×C₃₁	1	2	yes	—
289	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
290	#390	—	2	1	no	—
291	#390	C₂²×C₃₁	1	2	yes	—
292	#389	C₂²×C₃₁	1	2	yes	—
293	#386	—	2	2	no	—
294	#385	—	2	2	no	—
295	#384	C₂²×C₃₁	1	1	yes	characteristic
296	#380	—	2	1	no	—
297	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
298	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
299	#340	C₂²×C₃₁	1	1	yes	characteristic
300	#340	C₂²×C₃₁	1	2	yes	—
301	#340	C₂²×C₃₁	1	1	yes	characteristic
302	#336	C₂²×C₃₁	1	2	yes	—
303	C₂×C₃₁	—	4	4	no	—
304	C₂×C₃₁	—	8	4	no	—
305	C₂×C₃₁	#2361	1	1	yes	characteristic
306	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
307	C₂×C₃₁	#2946	1	1	yes	characteristic
308	C₂×C₃₁	#3272	1	1	yes	characteristic
309	C₂×C₃₁	#3274	1	1	yes	characteristic
310	C₂×C₃₁	#3366	1	1	yes	characteristic
311	C₂×C₃₁	#3373	1	1	yes	characteristic
312	C₂²×D₄	—	2	1	no	—
313	C₂²×D₄	—	2	1	no	—
314	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
315	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
316	C₂³×C₄	—	2	1	no	—
317	C₂³×C₄	—	2	2	no	—
318	C₂³×C₄	—	2	2	no	—
319	C₂×D₈	—	2	4	no	—
320	C₂×D₈	—	2	4	no	—
321	C₂×D₈	C₂×C₄×C₃₁	1	8	yes	complemented
322	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
323	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
324	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
325	#127	—	2	2	no	—
326	#127	C₂×C₄×C₃₁	1	4	yes	complemented
327	C₄×D₄	—	2	4	no	—
328	C₄×D₄	—	2	4	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	—	4	2	no	—
331	C₄×D₄	—	4	2	no	—
332	C₄×D₄	—	4	4	no	—
333	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	4	4	no	—
338	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₂×C₂²⋊C₄	—	2	1	no	—
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
351	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
352	#106	—	2	4	no	—
353	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
354	#102	—	2	2	no	—
355	#102	—	2	4	no	—
356	#102	—	4	2	no	—
357	#102	C₂×C₄×C₃₁	1	8	yes	complemented
358	#102	D₄×C₃₁	1	4	yes	complemented
359	#97	—	2	4	no	—
360	C₄×C₈	—	2	2	no	—
361	#95	—	2	1	no	—
362	#95	—	2	2	no	—
363	#95	D₄×C₃₁	1	2	yes	—
364	C₃₁	#23001	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	—	2	1	no	—
366	C₂⁴	—	2	2	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂×D₄	—	8	1	no	—
380	C₂×D₄	—	8	1	no	—
381	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
382	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
383	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
384	C₂×D₄	#41×C₃₁	1	2	yes	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	4	4	no	—
391	C₂²×C₄	—	4	4	no	—
392	C₂²×C₄	—	4	4	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	8	2	no	—
397	C₂²×C₄	—	8	2	no	—
398	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
403	D₈	—	2	16	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
412	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	4	4	no	—
422	C₄⋊C₄	—	4	2	no	—
423	C₄⋊C₄	—	8	1	no	—
424	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
425	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	4	4	no	—
428	C₂²⋊C₄	—	4	4	no	—
429	C₂²⋊C₄	—	4	4	no	—
430	C₂²⋊C₄	—	8	2	no	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	2	no	—
433	C₄²	—	4	2	no	—
434	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
435	C₄²	#41×C₃₁	1	2	yes	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	2	2	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	4	1	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	4	1	no	—
443	C₂³	—	4	1	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
451	D₄	—	2	4	no	—
452	D₄	—	2	2	no	—
453	D₄	—	2	4	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	4	8	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	2	no	—
459	D₄	—	8	4	no	—
460	D₄	C₄×D₄×C₃₁	1	4	yes	complemented
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	8	8	no	—
483	C₂×C₄	—	8	8	no	—
484	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
485	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
486	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
487	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
488	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
489	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
493	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
494	C₈	—	2	4	no	—
495	C₈	—	2	2	no	—
496	C₈	—	4	1	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	2	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	4	no	—
505	C₂²	—	4	4	no	—
506	C₂²	—	4	4	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	2	no	—
509	C₂²	—	8	2	no	—
510	C₂²	—	8	2	no	—
511	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
512	C₂²	D₄²×C₃₁	1	1	yes	characteristic
513	C₂²	C₃₁×#548	1	1	yes	characteristic
514	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
515	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
516	C₂²	C₃₁×#576	1	1	yes	characteristic
517	C₂²	C₃₁×#577	1	1	yes	characteristic
518	C₄	—	2	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	2	no	—
522	C₄	—	4	2	no	—
523	C₄	—	8	1	no	—
524	C₄	—	8	2	no	—
525	C₄	C₃₁×#442	1	4	yes	complemented
526	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
527	C₄	C₃₁×#520	1	1	yes	characteristic
528	C₄	C₃₁×#547	1	1	yes	characteristic
529	C₄	C₃₁×#553	1	1	yes	characteristic
530	C₂	—	4	4	no	—
531	C₂	—	8	4	no	—
532	C₂	C₃₁×#2361	1	1	yes	characteristic
533	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
534	C₂	C₃₁×#2946	1	1	yes	characteristic
535	C₂	C₃₁×#3272	1	1	yes	characteristic
536	C₂	C₃₁×#3274	1	1	yes	characteristic
537	C₂	C₃₁×#3366	1	1	yes	characteristic
538	C₂	C₃₁×#3373	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. C₂×C₃₁×#442	C₂	semidirect
3. C₃₁×#2366	C₂	semidirect
4. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#1957	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
7. C₃₁×#1936	C₂	semidirect
8. C₃₁×#1879	C₂	semidirect
9. C₃₁×#1864	C₂	semidirect
10. C₃₁×#1770	C₂	semidirect
11. C₄×C₃₁×#102	C₂	semidirect
19. C₃₁×#459	C₄	semidirect
52. C₂×D₈×C₃₁	C₂×C₄	semidirect
57. C₃₁×#127	C₂×C₄	semidirect
64. C₄×D₄×C₃₁	C₂×C₄	semidirect
84. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
88. C₃₁×#102	C₂×C₄	semidirect
89. C₃₁×#102	D₄	semidirect
166. #23001	C₃₁	direct (N × H)
191. D₄×C₃₁	C₄×D₄	semidirect
229. C₂×#442	C₂×C₃₁	semidirect
230. #2366	C₂×C₃₁	semidirect
231. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
232. #1957	C₂×C₃₁	semidirect
233. #1948	C₂×C₃₁	semidirect
234. #1936	C₂×C₃₁	semidirect
235. #1879	C₂×C₃₁	semidirect
236. #1864	C₂×C₃₁	semidirect
237. #1770	C₂×C₃₁	semidirect
238. C₄×#102	C₂×C₃₁	semidirect
267. C₄×C₃₁	#442	semidirect
279. #459	C₄×C₃₁	semidirect
321. C₂×D₈	C₂×C₄×C₃₁	semidirect
326. #127	C₂×C₄×C₃₁	semidirect
333. C₄×D₄	C₂×C₄×C₃₁	semidirect
353. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
357. #102	C₂×C₄×C₃₁	semidirect
358. #102	D₄×C₃₁	semidirect
364. C₃₁	#23001	direct (N × H)
460. D₄	C₄×D₄×C₃₁	semidirect
525. C₄	C₃₁×#442	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
C₂×C₃₁×#442	2
C₃₁×#2366	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#1957	2
C₃₁×#1948	2
C₃₁×#1936	2
C₃₁×#1879	2
C₃₁×#1864	2
C₃₁×#1770	2
C₄×C₃₁×#102	2
#23001	31

Subgroups — C31×#23001 #21479760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23001 #21479760