Subgroups — D₈×Dic₇ #89921

Order 448.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₇	C₂	1	2	yes	maximal, complemented
3	#6406	C₂	1	2	yes	maximal, complemented
4	#6392	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	C₂²	1	2	yes	—
7	C₂²×Dic₇	—	2	2	no	—
8	C₇×D₈	C₄	1	4	yes	complemented
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂²⋊Dic₇	—	2	2	no	—
11	C₄⋊Dic₇	C₂²	1	2	yes	—
12	C₄×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×D₈	—	7	1	no	maximal, Sylow
15	C₂³×C₇	—	2	2	no	—
16	C₇×D₄	—	2	2	no	—
17	C₇×D₄	C₂×C₄	1	4	yes	complemented
18	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₇	—	2	2	no	—
20	C₂×Dic₇	—	4	4	no	—
21	C₂×Dic₇	D₄	1	1	yes	characteristic
22	C₂×Dic₇	D₄	1	1	yes	characteristic
23	C₇×C₈	C₂×C₄	1	2	yes	complemented
24	C₇⋊C₈	—	2	1	no	—
25	C₂×D₈	D₇	1	1	yes	characteristic
26	C₄×D₄	—	7	2	no	—
27	#107	—	7	1	no	—
28	#102	—	7	2	no	—
29	C₄×C₈	—	7	1	no	—
30	C₂²×C₇	—	2	4	no	—
31	C₂²×C₇	—	4	2	no	—
32	C₂²×C₇	—	4	2	no	—
33	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₇	#41	1	1	yes	characteristic
36	Dic₇	—	2	1	no	—
37	Dic₇	—	4	2	no	—
38	Dic₇	D₈	1	2	yes	direct product factor
39	C₂×D₄	C₂×D₇	1	2	yes	—
40	C₂²×C₄	—	14	2	no	—
41	D₈	Dic₇	1	4	yes	direct product factor
42	C₂×C₈	—	7	1	no	—
43	C₂×C₈	C₂×D₇	1	1	yes	characteristic
44	C₄⋊C₄	—	7	2	no	—
45	C₂²⋊C₄	—	14	2	no	—
46	C₄²	—	7	1	no	—
47	C₂×C₇	—	4	4	no	—
48	C₂×C₇	C₄×D₄	1	1	yes	characteristic
49	C₂×C₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₇	#135	1	1	yes	characteristic
51	C₂³	—	2	2	no	—
52	D₄	—	2	2	no	—
53	D₄	C₂×Dic₇	1	4	yes	complemented
54	C₂×C₄	—	7	1	no	—
55	C₂×C₄	—	7	1	no	—
56	C₂×C₄	—	14	2	no	—
57	C₂×C₄	—	28	4	no	—
58	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
59	C₈	—	14	1	no	—
60	C₈	C₂×Dic₇	1	2	yes	complemented
61	C₇	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	2	4	no	—
63	C₂²	—	4	2	no	—
64	C₂²	—	4	2	no	—
65	C₂²	D₄×D₇	1	1	yes	characteristic
66	C₄	—	7	2	no	—
67	C₄	—	14	1	no	—
68	C₄	—	28	2	no	—
69	C₄	#1159	1	1	yes	characteristic
70	C₄	C₂²×Dic₇	1	1	yes	characteristic
71	C₂	—	4	4	no	—
72	C₂	D₇×D₈	1	1	yes	characteristic
73	C₂	#6475	1	1	yes	characteristic
74	C₂	D₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₂×C₇×D₈	C₂
socle	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×D₈	D₇
hypocenter	C₇	C₄×D₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
2. D₄×Dic₇	C₂	semidirect
3. #6406	C₂	semidirect
4. #6392	C₂	semidirect
5. C₈×Dic₇	C₂	semidirect
8. C₇×D₈	C₄	semidirect
17. C₇×D₄	C₂×C₄	semidirect
23. C₇×C₈	C₂×C₄	semidirect
38. Dic₇	D₈	direct (N × H)
41. D₈	Dic₇	direct (N × H)
53. D₄	C₂×Dic₇	semidirect
60. C₈	C₂×Dic₇	semidirect
61. C₇	C₄×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₈	2
D₄×Dic₇	2
#6406	2
#6392	2
C₈×Dic₇	2
C₄×D₈	7

Subgroups — D8×Dic7 #89921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×Dic₇ #89921