Subgroups — #13096559

Order 1568.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11716133	C₂	1	2	yes	maximal, complemented
2	C₇²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716073	C₂	1	2	yes	maximal, complemented
4	#11716063	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇²×Q₈	C₂²	1	2	yes	—
7	C₄⋊(C₇⋊D₇)	—	2	2	no	—
8	C₄⋊(C₇⋊D₇)	C₂²	1	2	yes	—
9	C₄×C₇⋊D₇	—	2	2	no	—
10	C₄×C₇⋊D₇	C₂²	1	1	yes	characteristic
11	C₇²×C₈	C₂²	1	1	yes	characteristic
12	C₇⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#6482	—	7	8	no	maximal
14	C₂×C₇⋊D₇	—	2	2	no	—
15	C₂×C₇⋊D₇	D₄	1	1	yes	characteristic
16	C₄×C₇²	—	2	2	no	—
17	C₄×C₇²	C₂³	1	1	yes	characteristic
18	C₇⋊Dic₇	D₄	1	1	yes	characteristic
19	#1161	—	7	16	no	—
20	C₇×Q₁₆	D₇	1	8	yes	complemented
21	#1143	—	7	16	no	—
22	D₅₆	—	7	8	no	—
23	C₈×D₇	—	7	8	no	—
24	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
25	C₇⋊D₇	—	2	1	no	—
26	C₇⋊D₇	—	4	2	no	—
27	C₇×Q₈	C₂×D₇	1	16	yes	—
28	D₂₈	—	7	16	no	—
29	D₂₈	—	14	16	no	—
30	C₄×D₇	—	7	8	no	—
31	C₄×D₇	—	14	16	no	—
32	C₇×C₈	C₂×D₇	1	8	yes	—
33	C₇⋊C₈	—	7	8	no	—
34	C₇²	#135	1	1	yes	characteristic, complemented, Sylow
35	#135	—	49	1	no	Sylow
36	C₂×D₇	—	7	8	no	—
37	C₂×D₇	—	14	16	no	—
38	C₄×C₇	—	2	16	no	—
39	C₄×C₇	C₂²×D₇	1	8	yes	—
40	Dic₇	—	7	8	no	—
41	#41	—	49	2	no	—
42	Q₁₆	C₇⋊D₇	1	1	yes	characteristic, complemented
43	SD₁₆	—	49	2	no	—
44	D₈	—	49	1	no	—
45	C₂×C₈	—	49	1	no	—
46	C₂×C₇	D₄×D₇	1	8	yes	—
47	D₇	—	14	8	no	—
48	D₇	—	28	16	no	—
49	Q₈	C₂×C₇⋊D₇	1	2	yes	—
50	D₄	—	49	2	no	—
51	D₄	—	98	2	no	—
52	C₂×C₄	—	49	1	no	—
53	C₂×C₄	—	98	2	no	—
54	C₈	—	49	1	no	—
55	C₈	C₂×C₇⋊D₇	1	1	yes	characteristic
56	C₇	#6482	1	8	yes	complemented
57	C₂²	—	49	1	no	—
58	C₂²	—	98	2	no	—
59	C₄	—	2	2	no	—
60	C₄	—	49	1	no	—
61	C₄	C₂²×C₇⋊D₇	1	1	yes	characteristic
62	C₂	—	98	1	no	—
63	C₂	—	196	2	no	—
64	C₂	D₄×C₇⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×C₇⋊D₇
commutator, cosocle	C₄×C₇²	C₂³
Frattini	C₄	C₂²×C₇⋊D₇
Fitting	C₇²×Q₁₆	C₂
socle	C₂×C₇²	C₂×D₄
hypercenter	Q₁₆	C₇⋊D₇
hypocenter	C₇²	#135
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #11716133	C₂	semidirect
2. C₇²×Q₁₆	C₂	semidirect
3. #11716073	C₂	semidirect
4. #11716063	C₂	semidirect
5. C₈×C₇⋊D₇	C₂	semidirect
20. C₇×Q₁₆	D₇	semidirect
34. C₇²	#135	semidirect
42. Q₁₆	C₇⋊D₇	semidirect
56. C₇	#6482	semidirect

Maximal subgroups

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

subgroup	index
#11716133	2
C₇²×Q₁₆	2
#11716073	2
#11716063	2
C₈×C₇⋊D₇	2
#6482	7