Subgroups — Dic₇⋊(C₇⋊C₈) #13096063

Order 1568.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
5	C₂×C₇×Dic₇	C₄	1	1	yes	characteristic
6	C₂×C₇×Dic₇	C₄	1	1	yes	characteristic
7	C₇⋊(C₇⋊C₈)	—	2	1	no	—
8	C₇×C₇⋊C₈	—	2	1	no	—
9	Dic₇⋊C₈	—	7	1	no	maximal
10	C₄⋊(C₇⋊C₈)	—	7	1	no	maximal
11	C₂²×C₇²	C₂×C₄	1	1	yes	characteristic
12	C₄×C₇²	D₄	1	1	yes	characteristic
13	C₄×C₇²	Q₈	1	1	yes	characteristic
14	C₇×Dic₇	—	2	1	no	—
15	C₇×Dic₇	C₈	1	2	yes	complemented
16	C₂×C₇×C₈	—	7	1	no	—
17	C₄²×C₇	—	7	1	no	—
18	C₄×Dic₇	D₇	1	1	yes	characteristic
19	C₂×C₇⋊C₈	—	7	1	no	—
20	C₂×C₇⋊C₈	—	7	1	no	—
21	C₂×C₇⋊C₈	—	14	3	no	—
22	C₂×C₇⋊C₈	D₇	1	1	yes	characteristic
23	C₂×C₇²	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₇²	C₂×C₈	1	1	yes	characteristic
25	C₂×C₇²	M₁₆	1	1	yes	characteristic
26	C₂×C₄×C₇	—	2	3	no	—
27	C₂×C₄×C₇	—	7	1	no	—
28	C₂×C₄×C₇	—	7	1	no	—
29	C₂×C₄×C₇	C₂×D₇	1	1	yes	characteristic
30	C₂×C₄×C₇	C₂×D₇	1	1	yes	characteristic
31	C₂×Dic₇	Dic₇	1	1	yes	characteristic
32	C₂×Dic₇	Dic₇	1	1	yes	characteristic
33	C₇×C₈	—	14	1	no	—
34	C₇⋊C₈	—	2	1	no	—
35	C₇⋊C₈	—	14	6	no	—
36	C₇⋊C₈	—	14	1	no	—
37	C₇⋊C₈	—	14	1	no	—
38	C₇²	C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
39	C₄⋊C₈	—	49	1	no	Sylow
40	C₂²×C₇	—	2	3	no	—
41	C₂²×C₇	C₄×D₇	1	1	yes	characteristic
42	C₂²×C₇	C₂×Dic₇	1	1	yes	characteristic
43	C₄×C₇	—	2	3	no	—
44	C₄×C₇	—	2	3	no	—
45	C₄×C₇	—	7	2	no	—
46	C₄×C₇	—	14	1	no	—
47	C₄×C₇	Dic₁₄	1	1	yes	characteristic
48	C₄×C₇	Dic₁₄	1	1	yes	characteristic
49	C₄×C₇	D₂₈	1	1	yes	characteristic
50	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
51	Dic₇	—	2	1	no	—
52	Dic₇	C₇⋊C₈	1	2	yes	complemented
53	C₂×C₈	—	7	1	no	—
54	C₂×C₈	—	49	1	no	—
55	C₄²	—	7	1	no	—
56	C₂×C₇	—	2	3	no	—
57	C₂×C₇	—	2	3	no	—
58	C₂×C₇	—	2	3	no	—
59	C₂×C₇	C₈×D₇	1	1	yes	characteristic
60	C₂×C₇	#1131	1	1	yes	characteristic
61	C₂×C₇	C₂×C₇⋊C₈	1	1	yes	characteristic
62	C₂×C₇	#1136	1	1	yes	characteristic
63	C₂×C₇	Dic₇⋊C₄	1	1	yes	characteristic
64	C₂×C₇	C₄⋊Dic₇	1	1	yes	characteristic
65	C₂×C₄	—	7	1	no	—
66	C₂×C₄	—	7	1	no	—
67	C₂×C₄	D₇²	1	1	yes	characteristic
68	C₈	—	14	1	no	—
69	C₈	—	98	1	no	—
70	C₇	—	2	3	no	—
71	C₇	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic, complemented
72	C₇	Dic₇⋊C₈	1	1	yes	characteristic, complemented
73	C₂²	D₇×Dic₇	1	1	yes	characteristic
74	C₄	—	7	2	no	—
75	C₄	—	14	1	no	—
76	C₄	#87641	1	1	yes	characteristic
77	C₄	#87642	1	1	yes	characteristic
78	C₂	D₇×C₇⋊C₈	1	1	yes	characteristic
79	C₂	#11716030	1	1	yes	characteristic
80	C₂	Dic₇⋊Dic₇	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, prime radical	C₂²×C₇²	C₂×C₄
hypocenter	C₇²	C₄⋊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
15. C₇×Dic₇	C₈	semidirect
38. C₇²	C₄⋊C₈	semidirect
52. Dic₇	C₇⋊C₈	semidirect
71. C₇	C₄⋊(C₇⋊C₈)	semidirect
72. C₇	Dic₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇⋊(C₇⋊C₈)	2
C₄×C₇×Dic₇	2
C₂×C₇×C₇⋊C₈	2
Dic₇⋊C₈	7
C₄⋊(C₇⋊C₈)	7

Subgroups — Dic7⋊(C7⋊C8) #13096063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇⋊(C₇⋊C₈) #13096063