Subgroups — #17986407

Order 4544.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14556127	C₂	1	1	yes	maximal, characteristic
2	#14556119	C₂	1	1	yes	maximal, characteristic
3	#14556061	C₂	1	1	yes	maximal, characteristic
4	#14556060	C₂	1	1	yes	maximal, characteristic
5	#14556059	C₂	1	1	yes	maximal, characteristic
6	#14556038	C₂	1	1	yes	maximal, characteristic
7	#107×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₄₂	—	2	1	no	—
9	C₂×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₇₁	—	2	1	no	—
11	C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
13	Dic₇₁⋊C₄	—	2	1	no	—
14	Dic₇₁⋊C₄	—	2	1	no	—
15	C₄×Dic₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇₁	—	2	1	no	—
20	C₂×C₄×C₇₁	—	2	1	no	—
21	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
22	C₂×Dic₇₁	—	2	1	no	—
23	C₂×Dic₇₁	—	2	1	no	—
24	C₂×Dic₇₁	D₄	1	1	yes	characteristic
25	C₂×Dic₇₁	D₄	1	1	yes	characteristic
26	Dic₁₄₂	—	4	2	no	—
27	C₈×C₇₁	Q₈	1	2	yes	complemented
28	C₇₁⋊C₈	Q₈	1	2	yes	complemented
29	C₂²×C₇₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₇₁	—	4	1	no	—
31	C₄×C₇₁	—	4	1	no	—
32	C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
34	Dic₇₁	—	2	1	no	—
35	Dic₇₁	—	2	1	no	—
36	Dic₇₁	—	4	1	no	—
37	Dic₇₁	—	4	1	no	—
38	C₂×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₇₁	#137	1	1	yes	characteristic
41	C₇₁	#501	1	1	yes	characteristic, complemented, Sylow
42	#501	—	71	1	no	maximal, Sylow
43	C₄⋊Q₈	—	71	1	no	—
44	#125	—	71	1	no	—
45	#107	—	71	1	no	—
46	#107	D₇₁	1	1	yes	characteristic
47	#106	—	71	1	no	—
48	#106	—	71	1	no	—
49	#97	—	71	1	no	—
50	C₂×Q₈	—	142	1	no	—
51	C₂×C₈	—	71	1	no	—
52	C₂×C₈	C₂×D₇₁	1	1	yes	characteristic
53	C₄⋊C₄	—	71	1	no	—
54	C₄⋊C₄	—	71	1	no	—
55	C₄⋊C₄	—	142	1	no	—
56	C₄⋊C₄	—	142	1	no	—
57	C₄⋊C₄	—	142	1	no	—
58	C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
60	C₄²	—	71	1	no	—
61	Q₈	—	284	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	71	1	no	—
65	C₂×C₄	—	71	1	no	—
66	C₂×C₄	—	142	1	no	—
67	C₂×C₄	—	142	1	no	—
68	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
69	C₈	—	71	2	no	—
70	C₈	Dic₁₄₂	1	2	yes	complemented
71	C₂²	D₄×D₇₁	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	284	1	no	—
75	C₄	—	284	1	no	—
76	C₄	—	142	1	no	—
77	C₄	—	142	1	no	—
78	C₄	Q₈×D₇₁	1	1	yes	characteristic
79	C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
80	C₂	#14556127	1	1	yes	characteristic
81	C₂	#14556152	1	1	yes	characteristic
82	C₂	#14556160	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇₁
commutator, cosocle	C₂×C₄×C₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₇₁
Fitting	#107×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₄
hypercenter	#107	D₇₁
hypocenter	C₇₁	#501
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
27. C₈×C₇₁	Q₈	semidirect
28. C₇₁⋊C₈	Q₈	semidirect
41. C₇₁	#501	semidirect
70. C₈	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#14556127	2
#14556119	2
#14556061	2
#14556060	2
#14556059	2
#14556038	2
#107×C₇₁	2
#501	71