Subgroups — #13096039

Order 1568.

67 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Dic₂₈	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	C₇×D₂₈	—	2	1	no	—
5	C₇×Dic₁₄	—	2	1	no	—
6	C₇²×C₈	C₂²	1	1	yes	characteristic
7	#6402	—	7	1	no	maximal
8	#6401	—	7	1	no	maximal
9	C₂×C₇×D₇	—	4	1	no	—
10	C₄×C₇²	D₄	1	1	yes	characteristic
11	C₇×Dic₇	—	4	1	no	—
12	C₇×Q₁₆	—	7	1	no	—
13	C₇×D₈	—	7	1	no	—
14	Dic₂₈	D₇	1	1	yes	characteristic, complemented
15	D₅₆	D₇	1	1	yes	characteristic
16	C₇⋊C₁₆	—	7	1	no	—
17	C₇⋊C₁₆	—	7	1	no	—
18	C₇⋊C₁₆	—	14	3	no	—
19	C₂×C₇²	D₈	1	1	yes	characteristic
20	C₇×D₇	—	8	1	no	—
21	C₇×Q₈	—	14	1	no	—
22	C₇×D₄	—	14	1	no	—
23	D₂₈	—	2	1	no	—
24	Dic₁₄	—	2	1	no	—
25	C₇×C₈	—	2	3	no	—
26	C₇×C₈	C₂×D₇	1	1	yes	characteristic
27	C₇×C₈	C₂×D₇	1	1	yes	characteristic
28	C₇²	SD₃₂	1	1	yes	characteristic, complemented, Sylow
29	SD₃₂	—	49	1	no	Sylow
30	C₂²×C₇	—	28	1	no	—
31	C₂×D₇	—	4	1	no	—
32	C₄×C₇	—	2	3	no	—
33	C₄×C₇	—	28	1	no	—
34	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
35	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
36	Dic₇	—	4	1	no	—
37	Q₁₆	—	7	1	no	—
38	D₈	—	7	1	no	—
39	C₁₆	—	49	1	no	—
40	C₂×C₇	—	2	3	no	—
41	C₂×C₇	—	56	1	no	—
42	C₂×C₇	#1141	1	1	yes	characteristic
43	C₂×C₇	#1141	1	1	yes	characteristic
44	D₇	—	8	1	no	—
45	Q₈	—	14	1	no	—
46	D₄	—	14	1	no	—
47	C₈	D₇²	1	1	yes	characteristic
48	C₇	—	2	3	no	—
49	C₇	#6401	1	1	yes	characteristic, complemented
50	C₇	#6402	1	1	yes	characteristic, complemented
51	C₂²	—	28	1	no	—
52	C₄	—	28	1	no	—
53	C₄	#87640	1	1	yes	characteristic
54	C₂	—	56	1	no	—
55	C₂	#11716032	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11716032
commutator, Fitting, cosocle	C₇²×C₈	C₂²
Frattini, hypercenter	C₈	D₇²
socle	C₂×C₇²	D₈
prime radical	C₇×D₅₆	C₂
hypocenter	C₇²	SD₃₂
LCS	G ▷ C₇²×C₈ ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. C₇×Dic₂₈	C₂	semidirect
3. C₇⋊(C₇⋊C₁₆)	C₂	semidirect
14. Dic₂₈	D₇	semidirect
28. C₇²	SD₃₂	semidirect
49. C₇	#6401	semidirect
50. C₇	#6402	semidirect

Maximal subgroups

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

subgroup	index
C₇×Dic₂₈	2
C₇×D₅₆	2
C₇⋊(C₇⋊C₁₆)	2
#6402	7
#6401	7