Subgroups — C₇×#6402 #13096105

Order 1568.

79 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₅₆	C₂	1	1	yes	maximal, characteristic
3	C₇×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇²×Q₈	—	2	1	no	—
5	C₇×D₂₈	—	2	1	no	—
6	C₇²×C₈	C₂²	1	1	yes	characteristic
7	C₇×SD₃₂	—	7	1	no	maximal
8	#6402	C₇	1	1	yes	maximal, characteristic, direct product factor
9	C₂×C₇×D₇	—	4	1	no	—
10	C₄×C₇²	—	4	1	no	—
11	C₄×C₇²	D₄	1	1	yes	characteristic
12	C₇×Q₁₆	—	2	3	no	—
13	C₇×Q₁₆	D₇	1	1	yes	characteristic, complemented
14	C₇×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
15	C₇×D₈	—	7	1	no	—
16	D₅₆	C₂×C₇	1	1	yes	characteristic
17	C₇×C₁₆	—	7	1	no	—
18	C₇⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
19	C₂×C₇²	D₈	1	1	yes	characteristic
20	C₇×D₇	—	8	1	no	—
21	C₇×Q₈	—	2	6	no	—
22	C₇×Q₈	—	2	1	no	—
23	C₇×Q₈	—	2	1	no	—
24	C₇×D₄	—	14	1	no	—
25	D₂₈	—	2	1	no	—
26	C₇×C₈	—	2	3	no	—
27	C₇×C₈	C₂×D₇	1	1	yes	characteristic
28	C₇×C₈	C₂²×C₇	1	1	yes	characteristic
29	C₇²	SD₃₂	1	1	yes	characteristic, complemented, Sylow
30	SD₃₂	—	7	1	no	Sylow
31	C₂²×C₇	—	28	1	no	—
32	C₂×D₇	—	4	1	no	—
33	C₄×C₇	—	2	3	no	—
34	C₄×C₇	—	4	6	no	—
35	C₄×C₇	—	4	1	no	—
36	C₄×C₇	—	4	1	no	—
37	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
38	C₄×C₇	C₇×D₄	1	1	yes	characteristic
39	Q₁₆	C₇×D₇	1	1	yes	characteristic, complemented
40	D₈	—	7	1	no	—
41	C₁₆	—	7	1	no	—
42	C₂×C₇	—	2	3	no	—
43	C₂×C₇	—	56	1	no	—
44	C₂×C₇	#1141	1	1	yes	characteristic
45	C₂×C₇	C₇×D₈	1	1	yes	characteristic
46	D₇	—	8	1	no	—
47	Q₈	—	2	1	no	—
48	D₄	—	14	1	no	—
49	C₈	C₂×C₇×D₇	1	1	yes	characteristic
50	C₇	—	2	3	no	—
51	C₇	#6402	1	1	yes	characteristic, direct product factor
52	C₇	C₇×SD₃₂	1	1	yes	characteristic, complemented
53	C₂²	—	28	1	no	—
54	C₄	—	4	1	no	—
55	C₄	C₇×C₂²⋊D₇	1	1	yes	characteristic
56	C₂	—	56	1	no	—
57	C₂	C₇×#1141	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	#1141
commutator, cosocle	C₇×C₈	C₂²×C₇
Frattini	C₈	C₂×C₇×D₇
Fitting	C₇²×Q₁₆	C₂
socle	C₂×C₇²	D₈
prime radical	C₇×D₅₆	C₂
hypercenter	C₇×Q₁₆	D₇
hypocenter	C₇	C₇×SD₃₂
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈ ◁ C₇×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₇²×Q₁₆	C₂	semidirect
3. C₇×C₇⋊C₁₆	C₂	semidirect
8. #6402	C₇	direct (N × H)
13. C₇×Q₁₆	D₇	semidirect
14. C₇×Q₁₆	C₂×C₇	semidirect
18. C₇⋊C₁₆	C₂×C₇	semidirect
29. C₇²	SD₃₂	semidirect
39. Q₁₆	C₇×D₇	semidirect
51. C₇	#6402	direct (N × H)
52. C₇	C₇×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₇²×Q₁₆	2
C₇×D₅₆	2
C₇×C₇⋊C₁₆	2
C₇×SD₃₂	7
#6402	7

Subgroups — C7×#6402 #13096105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6402 #13096105