Subgroups — C₂²×C₇×#168 #11765164

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×C₇×#168	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	C₇×#168	C₂²	1	4	yes	direct product factor
7	C₂²×C₇×C₈	—	5	1	no	maximal, Hall
8	C₂²×#168	C₇	1	1	yes	maximal, characteristic, direct product factor, 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	D₅	1	1	yes	characteristic
14	C₂×C₇×C₈	—	5	6	no	—
15	C₂²×C₄×C₅	C₂×C₇	1	1	yes	characteristic
16	C₂×#168	C₂×C₇	1	6	yes	direct product factor
17	C₂×C₅×C₇	C₂×C₈	1	6	yes	complemented
18	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
19	C₂³×C₇	Dic₅	1	1	yes	characteristic
20	C₂×C₄×C₇	Dic₅	1	3	yes	—
21	C₂×C₄×C₇	C₂×D₅	1	3	yes	—
22	C₇×C₈	—	5	4	no	—
23	C₂³×C₅	C₄×C₇	1	1	yes	characteristic
24	C₂×C₄×C₅	C₄×C₇	1	3	yes	—
25	C₂×C₄×C₅	C₂²×C₇	1	3	yes	—
26	#168	C₂²×C₇	1	4	yes	direct product factor
27	C₅×C₇	C₂²×C₈	1	1	yes	characteristic, complemented, Hall
28	C₂²×C₈	—	5	1	no	Sylow
29	C₂²×C₇	#168	1	4	yes	direct product factor
30	C₂²×C₇	C₂×Dic₅	1	3	yes	—
31	C₄×C₇	C₂×Dic₅	1	3	yes	—
32	C₄×C₇	C₂²×D₅	1	1	yes	characteristic
33	C₂²×C₅	C₇×C₈	1	4	yes	complemented
34	C₂²×C₅	C₂×C₄×C₇	1	3	yes	—
35	C₄×C₅	C₂×C₄×C₇	1	3	yes	—
36	C₄×C₅	C₂³×C₇	1	1	yes	characteristic
37	C₂²×C₄	C₇×D₅	1	1	yes	characteristic
38	C₂×C₈	—	5	6	no	—
39	C₂×C₇	C₂×#168	1	6	yes	direct product factor
40	C₂×C₇	C₂²×Dic₅	1	1	yes	characteristic
41	C₂×C₅	C₂×C₇×C₈	1	6	yes	complemented
42	C₂×C₅	C₂²×C₄×C₇	1	1	yes	characteristic
43	C₂³	C₇×Dic₅	1	1	yes	characteristic
44	C₂×C₄	C₇×Dic₅	1	3	yes	—
45	C₂×C₄	C₂×C₇×D₅	1	3	yes	—
46	C₈	—	5	4	no	—
47	C₇	C₂²×#168	1	1	yes	characteristic, direct product factor, Sylow
48	C₅	C₂²×C₇×C₈	1	1	yes	characteristic, complemented, Sylow
49	C₂²	C₇×#168	1	4	yes	direct product factor
50	C₂²	C₂×C₇×Dic₅	1	3	yes	—
51	C₄	C₂×C₇×Dic₅	1	3	yes	—
52	C₄	C₂²×C₇×D₅	1	1	yes	characteristic
53	C₂	C₂×C₇×#168	1	6	yes	direct product factor
54	C₂	C₂²×C₇×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₇	D₅
commutator, hypocenter	C₅	C₂²×C₇×C₈
Frattini	C₄	C₂²×C₇×D₅
Fitting	C₂²×C₄×C₅×C₇	C₂
socle, prime radical	C₂³×C₅×C₇	C₄
cosocle	C₄×C₅	C₂³×C₇
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#168	C₂	direct (N × H)
6. C₇×#168	C₂²	direct (N × H)
8. C₂²×#168	C₇	direct (N × H)
9. C₂²×C₅×C₇	C₈	semidirect
16. C₂×#168	C₂×C₇	direct (N × H)
17. C₂×C₅×C₇	C₂×C₈	semidirect
26. #168	C₂²×C₇	direct (N × H)
27. C₅×C₇	C₂²×C₈	semidirect
29. C₂²×C₇	#168	direct (N × H)
33. C₂²×C₅	C₇×C₈	semidirect
39. C₂×C₇	C₂×#168	direct (N × H)
41. C₂×C₅	C₂×C₇×C₈	semidirect
47. C₇	C₂²×#168	direct (N × H)
48. C₅	C₂²×C₇×C₈	semidirect
49. C₂²	C₇×#168	direct (N × H)
53. C₂	C₂×C₇×#168	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×C₇×#168	2
C₂²×C₇×C₈	5
C₂²×#168	7

Subgroups — C22×C7×#168 #11765164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#168 #11765164