Subgroups — #11764679

Order 1120.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇⋊#170	C₂	1	2	yes	maximal, complemented
3	C₂²×C₇×D₅	C₄	1	1	yes	characteristic
4	C₂×C₄×C₅×C₇	C₄	1	1	yes	characteristic
5	C₂×C₇×Dic₅	C₂²	1	1	yes	characteristic
6	C₄×C₇×D₅	—	2	2	no	—
7	C₇⋊#170	—	2	2	no	—
8	C₂²⋊(C₇⋊C₈)	—	5	1	no	maximal, Hall
9	#4003	—	7	1	no	maximal, Hall
10	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₇×D₅	—	2	1	no	—
12	C₂×C₇×D₅	—	2	1	no	—
13	C₂×C₇×D₅	C₈	1	2	yes	complemented
14	C₄×C₅×C₇	—	2	1	no	—
15	C₇×Dic₅	D₄	1	2	yes	—
16	C₂²×C₄×C₇	—	5	1	no	—
17	C₂×C₇⋊C₈	—	5	2	no	—
18	C₂×C₄×D₅	D₇	1	1	yes	characteristic
19	C₂×#170	—	7	2	no	—
20	C₂×C₅×C₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₅×C₇	C₂×C₈	1	1	yes	characteristic
22	C₂×C₅×C₇	M₁₆	1	1	yes	characteristic
23	C₇×D₅	—	2	2	no	—
24	C₂³×C₇	—	5	1	no	—
25	C₂×C₄×C₇	—	5	1	no	—
26	C₂×C₄×C₇	—	10	2	no	—
27	C₂×C₄×C₇	AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₇⋊C₈	—	10	2	no	—
29	C₂²×D₅	Dic₇	1	1	yes	characteristic
30	C₂×C₄×C₅	Dic₇	1	1	yes	characteristic
31	C₂×Dic₅	C₂×D₇	1	1	yes	characteristic
32	C₄×D₅	—	2	2	no	—
33	#170	—	14	2	no	—
34	C₅×C₇	C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
35	C₂²⋊C₈	—	35	1	no	Sylow
36	C₂²×C₇	—	5	2	no	—
37	C₂²×C₇	—	10	1	no	—
38	C₂²×C₇	—	10	1	no	—
39	C₂²×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₄×C₇	—	2	1	no	—
41	C₄×C₇	—	5	2	no	—
42	C₂²×C₅	C₂×Dic₇	1	1	yes	characteristic
43	C₂×D₅	—	2	1	no	—
44	C₂×D₅	—	2	1	no	—
45	C₂×D₅	C₇⋊C₈	1	2	yes	complemented
46	C₄×C₅	—	2	1	no	—
47	Dic₅	C₂²⋊D₇	1	2	yes	—
48	C₂²×C₄	—	5	1	no	—
49	C₂×C₈	—	35	2	no	—
50	C₂×C₇	—	10	2	no	—
51	C₂×C₇	D₅⋊C₈	1	1	yes	characteristic
52	C₂×C₇	#700	1	1	yes	characteristic
53	C₂×C₇	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₂×C₅	C₂×C₇⋊C₈	1	1	yes	characteristic
55	C₂×C₅	#1136	1	1	yes	characteristic
56	C₂×C₅	C₂²⋊Dic₇	1	1	yes	characteristic
57	D₅	—	2	2	no	—
58	C₂³	—	5	1	no	—
59	C₂×C₄	—	5	1	no	—
60	C₂×C₄	—	10	2	no	—
61	C₂×C₄	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₈	—	70	2	no	—
63	C₇	#4003	1	1	yes	characteristic, complemented, Sylow
64	C₅	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	5	2	no	—
66	C₂²	—	10	1	no	—
67	C₂²	—	10	1	no	—
68	C₂²	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	5	2	no	—
71	C₂	—	10	2	no	—
72	C₂	D₅⋊(C₇⋊C₈)	1	1	yes	characteristic
73	C₂	#10587931	1	1	yes	characteristic
74	C₂	C₂²⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×C₇⋊AGL(1,𝔽₅)
commutator	C₂×C₅×C₇	C₂×C₈
Fitting	C₂×C₄×C₅×C₇	C₄
socle	C₂²×C₅×C₇	C₂×C₄
prime radical	C₂²×C₇×D₅	C₄
cosocle	C₂×C₇×Dic₅	C₂²
hypercenter	C₂×C₄	C₇⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	C₂²⋊C₈
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₇⋊#170	C₂	semidirect
13. C₂×C₇×D₅	C₈	semidirect
34. C₅×C₇	C₂²⋊C₈	semidirect
45. C₂×D₅	C₇⋊C₈	semidirect
63. C₇	#4003	semidirect
64. C₅	C₂²⋊(C₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×D₅	2
C₂×C₇⋊#170	2
C₂²⋊(C₇⋊C₈)	5
#4003	7