Subgroups — C₂²×C₇⋊(C₇⋊C₈) #13096560

Order 1568.

380 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇⋊(C₇⋊C₈)	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₇²	C₄	1	1	yes	characteristic
4	C₂×C₄×C₇²	C₄	1	3	yes	—
5	C₂×C₄×C₇²	C₂²	1	3	yes	—
6	C₇⋊(C₇⋊C₈)	C₂²	1	4	yes	direct product factor
7	C₂²×C₇⋊C₈	—	7	8	no	maximal
8	C₂²×C₇²	C₈	1	4	yes	complemented
9	C₂²×C₇²	C₂×C₄	1	3	yes	—
10	C₄×C₇²	C₂×C₄	1	3	yes	—
11	C₄×C₇²	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₇	D₇	1	8	yes	—
13	C₂×C₇⋊C₈	—	7	48	no	—
14	C₂×C₇²	C₂×C₈	1	6	yes	complemented
15	C₂×C₇²	C₂²×C₄	1	1	yes	characteristic
16	C₂³×C₇	Dic₇	1	8	yes	—
17	C₂×C₄×C₇	Dic₇	1	24	yes	—
18	C₂×C₄×C₇	C₂×D₇	1	24	yes	—
19	C₇⋊C₈	—	7	32	no	—
20	C₇²	C₂²×C₈	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₈	—	49	1	no	Sylow
22	C₂²×C₇	C₇⋊C₈	1	32	yes	complemented
23	C₂²×C₇	C₂×Dic₇	1	24	yes	—
24	C₄×C₇	C₂×Dic₇	1	24	yes	—
25	C₄×C₇	C₂²×D₇	1	8	yes	—
26	C₂²×C₄	C₇⋊D₇	1	1	yes	characteristic
27	C₂×C₈	—	49	6	no	—
28	C₂×C₇	C₂×C₇⋊C₈	1	48	yes	complemented
29	C₂×C₇	C₂²×Dic₇	1	8	yes	—
30	C₂³	C₇⋊Dic₇	1	1	yes	characteristic
31	C₂×C₄	C₇⋊Dic₇	1	3	yes	—
32	C₂×C₄	C₂×C₇⋊D₇	1	3	yes	—
33	C₈	—	49	4	no	—
34	C₇	C₂²×C₇⋊C₈	1	8	yes	complemented
35	C₂²	C₇⋊(C₇⋊C₈)	1	4	yes	direct product factor
36	C₂²	C₂×C₇⋊Dic₇	1	3	yes	—
37	C₄	C₂×C₇⋊Dic₇	1	3	yes	—
38	C₄	C₂²×C₇⋊D₇	1	1	yes	characteristic
39	C₂	C₂×C₇⋊(C₇⋊C₈)	1	6	yes	direct product factor
40	C₂	C₂²×C₇⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	C₇⋊D₇
commutator, hypocenter	C₇²	C₂²×C₈
Frattini	C₄	C₂²×C₇⋊D₇
Fitting	C₂²×C₄×C₇²	C₂
socle, prime radical	C₂³×C₇²	C₄
cosocle	C₄×C₇²	C₂³
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₇⋊(C₇⋊C₈)	C₂	direct (N × H)
6. C₇⋊(C₇⋊C₈)	C₂²	direct (N × H)
8. C₂²×C₇²	C₈	semidirect
14. C₂×C₇²	C₂×C₈	semidirect
20. C₇²	C₂²×C₈	semidirect
22. C₂²×C₇	C₇⋊C₈	semidirect
28. C₂×C₇	C₂×C₇⋊C₈	semidirect
34. C₇	C₂²×C₇⋊C₈	semidirect
35. C₂²	C₇⋊(C₇⋊C₈)	direct (N × H)
39. C₂	C₂×C₇⋊(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×C₇²	2
C₂×C₇⋊(C₇⋊C₈)	2
C₂²×C₇⋊C₈	7

Subgroups — C22×C7⋊(C7⋊C8) #13096560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇⋊(C₇⋊C₈) #13096560