Subgroups — #13096558

Order 1568.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11716133	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716074	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716073	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716062	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716061	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇²×Q₈	C₂²	1	1	yes	characteristic
9	C₇²×Q₈	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇⋊D₇)	—	2	1	no	—
11	C₄⋊(C₇⋊D₇)	C₂²	1	1	yes	characteristic
12	C₄×C₇⋊D₇	—	2	1	no	—
13	C₄×C₇⋊D₇	—	2	1	no	—
14	C₄×C₇⋊D₇	C₂²	1	1	yes	characteristic
15	#87648	—	2	1	no	—
16	#87648	C₂²	1	1	yes	characteristic
17	C₇²×C₈	C₂²	1	1	yes	characteristic
18	C₇⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
19	#6481	—	7	8	no	maximal
20	C₂×C₇⋊D₇	—	2	1	no	—
21	C₂×C₇⋊D₇	D₄	1	1	yes	characteristic
22	C₄×C₇²	—	2	1	no	—
23	C₄×C₇²	—	2	1	no	—
24	C₄×C₇²	C₂³	1	1	yes	characteristic
25	C₇⋊Dic₇	—	2	1	no	—
26	C₇⋊Dic₇	D₄	1	1	yes	characteristic
27	#1161	—	7	8	no	—
28	Q₈×D₇	—	7	8	no	—
29	C₇×Q₁₆	D₇	1	8	yes	complemented
30	#1144	—	7	8	no	—
31	#1143	—	7	8	no	—
32	#1132	—	7	8	no	—
33	#1131	—	7	8	no	—
34	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
35	C₇⋊D₇	—	2	1	no	—
36	C₇⋊D₇	—	4	1	no	—
37	C₇×Q₈	C₂×D₇	1	8	yes	—
38	C₇×Q₈	C₂×D₇	1	8	yes	—
39	D₂₈	—	7	8	no	—
40	D₂₈	—	14	8	no	—
41	C₄×D₇	—	7	8	no	—
42	C₄×D₇	—	14	8	no	—
43	C₄×D₇	—	14	8	no	—
44	Dic₁₄	—	7	8	no	—
45	Dic₁₄	—	14	8	no	—
46	C₇×C₈	C₂×D₇	1	8	yes	—
47	C₇⋊C₈	—	7	8	no	—
48	C₇²	#137	1	1	yes	characteristic, complemented, Sylow
49	#137	—	49	1	no	Sylow
50	C₂×D₇	—	7	8	no	—
51	C₂×D₇	—	14	8	no	—
52	C₄×C₇	—	2	8	no	—
53	C₄×C₇	—	2	8	no	—
54	C₄×C₇	C₂²×D₇	1	8	yes	—
55	Dic₇	—	7	8	no	—
56	Dic₇	—	14	8	no	—
57	#41	—	49	1	no	—
58	C₂×Q₈	—	49	1	no	—
59	Q₁₆	—	49	1	no	—
60	Q₁₆	C₇⋊D₇	1	1	yes	characteristic, complemented
61	SD₁₆	—	49	1	no	—
62	SD₁₆	—	49	1	no	—
63	M₁₆	—	49	1	no	—
64	C₂×C₇	D₄×D₇	1	8	yes	—
65	D₇	—	14	8	no	—
66	D₇	—	28	8	no	—
67	Q₈	—	49	1	no	—
68	Q₈	—	98	1	no	—
69	Q₈	C₂×C₇⋊D₇	1	1	yes	characteristic
70	Q₈	C₂×C₇⋊D₇	1	1	yes	characteristic
71	D₄	—	49	1	no	—
72	D₄	—	98	1	no	—
73	C₂×C₄	—	49	1	no	—
74	C₂×C₄	—	98	1	no	—
75	C₂×C₄	—	98	1	no	—
76	C₈	—	49	1	no	—
77	C₈	C₂×C₇⋊D₇	1	1	yes	characteristic
78	C₇	#6481	1	8	yes	complemented
79	C₂²	—	49	1	no	—
80	C₂²	—	98	1	no	—
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	49	1	no	—
84	C₄	—	98	1	no	—
85	C₄	C₂²×C₇⋊D₇	1	1	yes	characteristic
86	C₂	—	98	1	no	—
87	C₂	—	196	1	no	—
88	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
prime radical	#11716133	C₂
hypercenter	Q₁₆	C₇⋊D₇
hypocenter	C₇²	#137
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
2. Q₈×C₇⋊D₇	C₂	semidirect
3. C₇²×Q₁₆	C₂	semidirect
4. #11716074	C₂	semidirect
5. #11716073	C₂	semidirect
6. #11716062	C₂	semidirect
7. #11716061	C₂	semidirect
29. C₇×Q₁₆	D₇	semidirect
48. C₇²	#137	semidirect
60. Q₁₆	C₇⋊D₇	semidirect
78. C₇	#6481	semidirect

Maximal subgroups

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

subgroup	index
#11716133	2
Q₈×C₇⋊D₇	2
C₇²×Q₁₆	2
#11716074	2
#11716073	2
#11716062	2
#11716061	2
#6481	7