Subgroups — #31886824

Order 18176.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588085	C₂	1	3	yes	maximal
2	C₂×#17985721	C₂	1	3	yes	maximal
3	C₂×C₈²×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
5	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
6	#17985753	C₄	1	12	yes	—
7	#17985721	C₄	1	12	yes	complemented
8	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
9	C₈²×C₇₁	C₄	1	4	yes	complemented
10	C₂²×C₇₁⋊C₈	C₂×C₄	1	6	yes	—
11	C₂×C₄×Dic₇₁	C₂×C₄	1	6	yes	—
12	C₄×C₇₁⋊C₈	C₂×C₄	1	12	yes	—
13	C₂²×C₈×C₇₁	C₂×C₄	1	6	yes	—
14	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
15	C₄×C₈×C₇₁	C₂×C₄	1	6	yes	—
16	C₄×C₈×C₇₁	C₂×C₄	1	6	yes	—
17	C₂²×Dic₇₁	C₄²	1	4	yes	—
18	C₂×C₇₁⋊C₈	—	2	12	no	—
19	C₂×C₇₁⋊C₈	C₄²	1	12	yes	—
20	C₄×Dic₇₁	—	2	12	no	—
21	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
22	C₂×C₈×C₇₁	—	2	12	no	—
23	C₂×C₈×C₇₁	C₄²	1	12	yes	—
24	C₄²×C₇₁	C₂²×C₄	1	3	yes	—
25	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
26	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
27	C₂×C₄×C₇₁	C₂×C₄²	1	3	yes	—
28	C₂×C₄×C₇₁	C₂×C₄²	1	3	yes	—
29	C₂×C₄×C₇₁	#131	1	12	yes	—
30	C₂×Dic₇₁	—	2	12	no	—
31	C₈×C₇₁	—	2	12	no	—
32	C₇₁⋊C₈	—	2	12	no	—
33	C₂²×C₇₁	C₄³	1	1	yes	characteristic
34	C₂²×C₇₁	#405	1	6	yes	—
35	C₄×C₇₁	#405	1	6	yes	—
36	C₄×C₇₁	#405	1	6	yes	—
37	Dic₇₁	—	4	4	no	—
38	#7845	—	71	1	no	maximal, Sylow
39	C₂×C₇₁	#1454	1	4	yes	—
40	C₂×C₇₁	#1726	1	3	yes	—
41	#1744	—	71	3	no	—
42	C₂×#322	—	71	3	no	—
43	C₂×C₈²	D₇₁	1	1	yes	characteristic
44	C₇₁	#7845	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×C₈	—	71	3	no	—
46	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
47	C₄³	—	71	1	no	—
48	#336	—	71	12	no	—
49	#322	—	71	12	no	—
50	C₈²	Dic₇₁	1	4	yes	complemented
51	C₂²×C₈	—	71	6	no	—
52	C₂²×C₈	C₄×D₇₁	1	6	yes	—
53	C₂×C₄²	—	71	6	no	—
54	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
55	C₄×C₈	—	71	12	no	—
56	C₄×C₈	C₄×D₇₁	1	6	yes	—
57	C₄×C₈	C₂×Dic₇₁	1	6	yes	—
58	C₂²×C₄	—	71	4	no	—
59	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
60	C₂×C₈	—	2	12	no	—
61	C₂×C₈	—	71	12	no	—
62	C₂×C₈	—	142	12	no	—
63	C₂×C₈	C₄×Dic₇₁	1	12	yes	—
64	C₄²	—	142	12	no	—
65	C₄²	C₂×C₄×D₇₁	1	3	yes	—
66	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
67	C₂³	C₄²×D₇₁	1	1	yes	characteristic
68	C₂×C₄	—	142	12	no	—
69	C₂×C₄	C₄²×D₇₁	1	3	yes	—
70	C₂×C₄	C₂×C₄×Dic₇₁	1	3	yes	—
71	C₂×C₄	#14556134	1	12	yes	—
72	C₈	—	2	12	no	—
73	C₈	—	142	12	no	—
74	C₂²	C₄²×Dic₇₁	1	1	yes	characteristic
75	C₂²	#17985953	1	6	yes	—
76	C₄	—	284	4	no	—
77	C₄	#17985953	1	6	yes	—
78	C₄	#17985954	1	6	yes	—
79	C₂	#22586639	1	4	yes	—
80	C₂	#22588026	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator	C₂²×C₇₁	C₄³
Fitting	C₂×C₈²×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×C₈²	D₇₁
hypocenter	C₇₁	#7845
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
7. #17985721	C₄	semidirect
9. C₈²×C₇₁	C₄	semidirect
44. C₇₁	#7845	semidirect
50. C₈²	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22588085	2
C₂×#17985721	2
C₂×C₈²×C₇₁	2
#7845	71