Subgroups — #13096049

Order 1568.

181 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₇⋊D₇	C₄	1	1	yes	characteristic
4	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
5	C₂×C₇⋊Dic₇	C₄	1	1	yes	characteristic
6	C₄×C₇⋊D₇	—	2	2	no	—
7	C₇×C₇⋊C₈	—	2	2	no	—
8	(C₂×D₇)⋊C₈	—	7	2	no	maximal
9	C₂²×C₇²	C₂×C₄	1	1	yes	characteristic
10	C₂×C₇⋊D₇	—	2	1	no	—
11	C₂×C₇⋊D₇	—	2	1	no	—
12	C₂×C₇⋊D₇	C₈	1	2	yes	complemented
13	C₄×C₇²	D₄	1	2	yes	—
14	C₇⋊Dic₇	—	2	1	no	—
15	C₂×C₄×D₇	—	7	2	no	—
16	C₂×C₄×D₇	—	14	3	no	—
17	C₂×C₇×C₈	—	7	2	no	—
18	C₂×C₇⋊C₈	D₇	1	2	yes	complemented
19	C₂×C₇²	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₇²	C₂×C₈	1	1	yes	characteristic
21	C₂×C₇²	M₁₆	1	1	yes	characteristic
22	C₇⋊D₇	—	2	2	no	—
23	C₂²×D₇	—	7	2	no	—
24	C₂²×D₇	—	14	3	no	—
25	C₂×C₄×C₇	—	2	3	no	—
26	C₂×C₄×C₇	C₂×D₇	1	2	yes	—
27	C₂×Dic₇	—	7	2	no	—
28	C₂×Dic₇	—	14	3	no	—
29	C₄×D₇	—	14	12	no	—
30	C₄×D₇	—	14	2	no	—
31	C₄×D₇	—	14	2	no	—
32	C₇×C₈	—	14	2	no	—
33	C₇⋊C₈	—	2	2	no	—
34	C₇²	C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
35	C₂²⋊C₈	—	49	1	no	Sylow
36	C₂²×C₇	—	2	3	no	—
37	C₂²×C₇	C₄×D₇	1	2	yes	—
38	C₂×D₇	—	7	4	no	—
39	C₂×D₇	—	14	6	no	—
40	C₂×D₇	—	14	6	no	—
41	C₂×D₇	—	14	6	no	—
42	C₂×D₇	—	14	2	no	—
43	C₂×D₇	—	14	2	no	—
44	C₄×C₇	—	2	6	no	—
45	C₄×C₇	D₂₈	1	2	yes	—
46	C₄×C₇	C₂²⋊D₇	1	2	yes	—
47	Dic₇	—	14	6	no	—
48	Dic₇	—	14	2	no	—
49	C₂²×C₄	—	49	1	no	—
50	C₂×C₈	—	7	2	no	—
51	C₂×C₇	—	2	3	no	—
52	C₂×C₇	—	2	3	no	—
53	C₂×C₇	—	2	3	no	—
54	C₂×C₇	C₈×D₇	1	2	yes	—
55	C₂×C₇	#1131	1	2	yes	—
56	C₂×C₇	(C₂×D₇)⋊C₄	1	2	yes	—
57	D₇	—	14	12	no	—
58	D₇	—	14	4	no	—
59	C₂³	—	49	1	no	—
60	C₂×C₄	—	49	1	no	—
61	C₂×C₄	—	98	2	no	—
62	C₂×C₄	D₇²	1	1	yes	characteristic
63	C₈	—	14	2	no	—
64	C₇	—	2	3	no	—
65	C₇	(C₂×D₇)⋊C₈	1	2	yes	complemented
66	C₂²	—	49	2	no	—
67	C₂²	—	98	1	no	—
68	C₂²	—	98	1	no	—
69	C₂²	#87639	1	1	yes	characteristic
70	C₄	—	98	1	no	—
71	C₄	#87641	1	2	yes	—
72	C₂	—	98	2	no	—
73	C₂	#11716029	1	1	yes	characteristic
74	C₂	#11716031	1	1	yes	characteristic
75	C₂	#11716041	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	D₇²
commutator	C₂×C₇²	C₂×C₈
Fitting, cosocle	C₂×C₄×C₇²	C₂²
socle	C₂²×C₇²	C₂×C₄
prime radical	C₂²×C₇⋊D₇	C₄
hypocenter	C₇²	C₂²⋊C₈
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₇⋊C₈	C₂	semidirect
12. C₂×C₇⋊D₇	C₈	semidirect
18. C₂×C₇⋊C₈	D₇	semidirect
34. C₇²	C₂²⋊C₈	semidirect
65. C₇	(C₂×D₇)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇⋊D₇	2
C₂×C₇×C₇⋊C₈	2
(C₂×D₇)⋊C₈	7