Subgroups — #17986409

Order 4544.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14556128	C₂	1	1	yes	maximal, characteristic
2	#14556061	C₂	1	2	yes	maximal
3	#14556059	C₂	1	2	yes	maximal
4	#14556039	C₂	1	1	yes	maximal, characteristic
5	#125×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₄₂	—	2	1	no	—
7	C₂×C₇₁⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₇₁	—	2	1	no	—
9	C₄⋊Dic₇₁	C₂²	1	2	yes	—
10	C₄⋊C₄×C₇₁	—	2	1	no	—
11	C₄⋊C₄×C₇₁	—	2	1	no	—
12	C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
14	C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₁	—	2	1	no	—
16	C₂×C₄×C₇₁	—	2	1	no	—
17	C₂×C₄×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₇₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
20	C₂×Dic₇₁	—	2	2	no	—
21	Dic₁₄₂	—	4	2	no	—
22	C₇₁⋊C₈	Q₈	1	4	yes	complemented
23	C₂²×C₇₁	C₂×D₄	1	1	yes	characteristic
24	C₄×C₇₁	—	2	1	no	—
25	C₄×C₇₁	—	2	1	no	—
26	C₄×C₇₁	—	4	1	no	—
27	C₄×C₇₁	—	4	1	no	—
28	C₄×C₇₁	C₂×Q₈	1	2	yes	—
29	Dic₇₁	—	4	2	no	—
30	C₂×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₇₁	#137	1	1	yes	characteristic
33	C₇₁	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	71	1	no	maximal, Sylow
35	C₄⋊Q₈	—	71	1	no	—
36	#125	D₇₁	1	1	yes	characteristic
37	#107	—	71	2	no	—
38	#106	—	71	2	no	—
39	#97	—	71	1	no	—
40	C₂×Q₈	—	142	1	no	—
41	C₂×C₈	—	71	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	71	2	no	—
45	C₄⋊C₄	—	142	1	no	—
46	C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
47	C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
48	C₄²	C₂×D₇₁	1	1	yes	characteristic
49	Q₈	—	284	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	142	2	no	—
53	C₂×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
56	C₈	—	71	4	no	—
57	C₂²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	284	2	no	—
63	C₄	Q₈×D₇₁	1	2	yes	—
64	C₂	#14556126	1	1	yes	characteristic
65	C₂	#14556153	1	1	yes	characteristic
66	C₂	#14556158	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇₁
commutator, cosocle	C₂×C₄×C₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₇₁
Fitting	#125×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₄
hypercenter	#125	D₇₁
hypocenter	C₇₁	#501
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
22. C₇₁⋊C₈	Q₈	semidirect
33. C₇₁	#501	semidirect

Maximal subgroups

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

subgroup	index
#14556128	2
#14556061	2
#14556059	2
#14556039	2
#125×C₇₁	2
#501	71