Subgroups — C₂²×C₅×C₇⋊C₈ #11765078

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₇⋊C₈	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₅×C₇	C₄	1	1	yes	characteristic
4	C₂×C₄×C₅×C₇	C₄	1	3	yes	—
5	C₂×C₄×C₅×C₇	C₂²	1	3	yes	—
6	C₅×C₇⋊C₈	C₂²	1	4	yes	direct product factor
7	C₂²×C₇⋊C₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₅×C₈	—	7	1	no	maximal, Hall
9	C₂²×C₅×C₇	C₈	1	4	yes	complemented
10	C₂²×C₅×C₇	C₂×C₄	1	3	yes	—
11	C₄×C₅×C₇	C₂×C₄	1	3	yes	—
12	C₄×C₅×C₇	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₇	C₂×C₅	1	1	yes	characteristic
14	C₂×C₇⋊C₈	C₂×C₅	1	6	yes	direct product factor
15	C₂²×C₄×C₅	D₇	1	1	yes	characteristic
16	C₂×C₅×C₈	—	7	6	no	—
17	C₂×C₅×C₇	C₂×C₈	1	6	yes	complemented
18	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
19	C₂³×C₇	C₄×C₅	1	1	yes	characteristic
20	C₂×C₄×C₇	C₄×C₅	1	3	yes	—
21	C₂×C₄×C₇	C₂²×C₅	1	3	yes	—
22	C₇⋊C₈	C₂²×C₅	1	4	yes	direct product factor
23	C₂³×C₅	Dic₇	1	1	yes	characteristic
24	C₂×C₄×C₅	Dic₇	1	3	yes	—
25	C₂×C₄×C₅	C₂×D₇	1	3	yes	—
26	C₅×C₈	—	7	4	no	—
27	C₅×C₇	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
28	C₂²×C₈	—	7	1	no	Sylow
29	C₂²×C₇	C₅×C₈	1	4	yes	complemented
30	C₂²×C₇	C₂×C₄×C₅	1	3	yes	—
31	C₄×C₇	C₂×C₄×C₅	1	3	yes	—
32	C₄×C₇	C₂³×C₅	1	1	yes	characteristic
33	C₂²×C₅	C₇⋊C₈	1	4	yes	direct product factor
34	C₂²×C₅	C₂×Dic₇	1	3	yes	—
35	C₄×C₅	C₂×Dic₇	1	3	yes	—
36	C₄×C₅	C₂²×D₇	1	1	yes	characteristic
37	C₂²×C₄	C₅×D₇	1	1	yes	characteristic
38	C₂×C₈	—	7	6	no	—
39	C₂×C₇	C₂×C₅×C₈	1	6	yes	complemented
40	C₂×C₇	C₂²×C₄×C₅	1	1	yes	characteristic
41	C₂×C₅	C₂×C₇⋊C₈	1	6	yes	direct product factor
42	C₂×C₅	C₂²×Dic₇	1	1	yes	characteristic
43	C₂³	C₅×Dic₇	1	1	yes	characteristic
44	C₂×C₄	C₅×Dic₇	1	3	yes	—
45	C₂×C₄	C₂×C₅×D₇	1	3	yes	—
46	C₈	—	7	4	no	—
47	C₇	C₂²×C₅×C₈	1	1	yes	characteristic, complemented, Sylow
48	C₅	C₂²×C₇⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²	C₅×C₇⋊C₈	1	4	yes	direct product factor
50	C₂²	C₂×C₅×Dic₇	1	3	yes	—
51	C₄	C₂×C₅×Dic₇	1	3	yes	—
52	C₄	C₂²×C₅×D₇	1	1	yes	characteristic
53	C₂	C₂×C₅×C₇⋊C₈	1	6	yes	direct product factor
54	C₂	C₂²×C₅×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₅	D₇
commutator, hypocenter	C₇	C₂²×C₅×C₈
Frattini	C₄	C₂²×C₅×D₇
Fitting	C₂²×C₄×C₅×C₇	C₂
socle, prime radical	C₂³×C₅×C₇	C₄
cosocle	C₄×C₇	C₂³×C₅
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₇⋊C₈	C₂	direct (N × H)
6. C₅×C₇⋊C₈	C₂²	direct (N × H)
7. C₂²×C₇⋊C₈	C₅	direct (N × H)
9. C₂²×C₅×C₇	C₈	semidirect
14. C₂×C₇⋊C₈	C₂×C₅	direct (N × H)
17. C₂×C₅×C₇	C₂×C₈	semidirect
22. C₇⋊C₈	C₂²×C₅	direct (N × H)
27. C₅×C₇	C₂²×C₈	semidirect
29. C₂²×C₇	C₅×C₈	semidirect
33. C₂²×C₅	C₇⋊C₈	direct (N × H)
39. C₂×C₇	C₂×C₅×C₈	semidirect
41. C₂×C₅	C₂×C₇⋊C₈	direct (N × H)
47. C₇	C₂²×C₅×C₈	semidirect
48. C₅	C₂²×C₇⋊C₈	direct (N × H)
49. C₂²	C₅×C₇⋊C₈	direct (N × H)
53. C₂	C₂×C₅×C₇⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₇	2
C₂×C₅×C₇⋊C₈	2
C₂²×C₇⋊C₈	5
C₂²×C₅×C₈	7

Subgroups — C22×C5×C7⋊C8 #11765078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₇⋊C₈ #11765078