Subgroups — C₇×#135 #6538

Order 224.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41	C₂	1	2	yes	maximal, complemented
2	C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
4	C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×Q₈	C₂²	1	2	yes	—
7	C₇×D₄	—	2	2	no	—
8	C₇×D₄	C₂²	1	2	yes	—
9	C₂×C₄×C₇	—	2	2	no	—
10	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×C₈	C₂²	1	1	yes	characteristic
13	#135	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₇	—	2	2	no	—
15	C₂²×C₇	D₄	1	1	yes	characteristic
16	C₄×C₇	—	2	2	no	—
17	C₄×C₇	D₄	1	1	yes	characteristic
18	C₄×C₇	C₂³	1	1	yes	characteristic
19	#41	C₂×C₇	1	2	yes	complemented
20	Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
21	SD₁₆	C₂×C₇	1	2	yes	complemented
22	D₈	C₂×C₇	1	1	yes	characteristic, complemented
23	C₂×C₈	C₂×C₇	1	1	yes	characteristic, complemented
24	C₂×C₇	—	2	1	no	—
25	C₂×C₇	—	4	2	no	—
26	C₂×C₇	C₂×D₄	1	1	yes	characteristic
27	Q₈	C₂²×C₇	1	2	yes	—
28	D₄	—	2	2	no	—
29	D₄	C₂²×C₇	1	2	yes	—
30	C₂×C₄	—	2	2	no	—
31	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
32	C₈	C₂²×C₇	1	1	yes	characteristic
33	C₈	C₂²×C₇	1	1	yes	characteristic
34	C₇	#135	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²	—	2	2	no	—
36	C₂²	C₇×D₄	1	1	yes	characteristic
37	C₄	—	2	2	no	—
38	C₄	C₇×D₄	1	1	yes	characteristic
39	C₄	C₂³×C₇	1	1	yes	characteristic
40	C₂	—	2	1	no	—
41	C₂	—	4	2	no	—
42	C₂	C₂×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₇
socle	C₂×C₇	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#41	C₂	semidirect
2. C₇×Q₁₆	C₂	semidirect
3. C₇×SD₁₆	C₂	semidirect
4. C₇×D₈	C₂	semidirect
5. C₂×C₇×C₈	C₂	semidirect
13. #135	C₇	direct (N × H)
19. #41	C₂×C₇	semidirect
20. Q₁₆	C₂×C₇	semidirect
21. SD₁₆	C₂×C₇	semidirect
22. D₈	C₂×C₇	semidirect
23. C₂×C₈	C₂×C₇	semidirect
34. C₇	#135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#41	2
C₇×Q₁₆	2
C₇×SD₁₆	2
C₇×D₈	2
C₂×C₇×C₈	2
#135	7

Subgroups — C7×#135 #6538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#135 #6538