Subgroups — D₄⋊(C₇⋊C₈) #89329

Order 448.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×D₄	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₇	—	2	1	no	—
6	C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
7	C₇×C₂²⋊C₄	—	2	1	no	—
8	C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇⋊C₈	—	2	1	no	—
10	C₂×C₇⋊C₈	—	2	1	no	—
11	D₄⋊C₈	—	7	1	no	maximal, Sylow
12	C₂³×C₇	—	2	1	no	—
13	C₇×D₄	—	2	1	no	—
14	C₇×D₄	C₈	1	2	yes	complemented
15	C₂×C₄×C₇	—	2	1	no	—
16	C₂×C₄×C₇	—	4	1	no	—
17	C₂×C₄×C₇	—	4	1	no	—
18	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₇	D₄	1	1	yes	characteristic
21	C₇⋊C₈	—	2	2	no	—
22	C₇⋊C₈	—	4	1	no	—
23	C₄×D₄	D₇	1	1	yes	characteristic
24	C₄⋊C₈	—	7	1	no	—
25	C₄×C₈	—	7	1	no	—
26	C₂²×C₇	—	2	2	no	—
27	C₂²×C₇	—	4	1	no	—
28	C₂²×C₇	—	4	1	no	—
29	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₇	—	2	1	no	—
31	C₄×C₇	—	4	1	no	—
32	C₄×C₇	C₂×C₈	1	1	yes	characteristic
33	C₄×C₇	M₁₆	1	1	yes	characteristic
34	C₄×C₇	D₈	1	1	yes	characteristic
35	C₄×C₇	SD₁₆	1	1	yes	characteristic
36	C₂×D₄	Dic₇	1	1	yes	characteristic
37	C₂²×C₄	—	2	1	no	—
38	C₂×C₈	—	14	1	no	—
39	C₂×C₈	—	14	1	no	—
40	C₄⋊C₄	Dic₇	1	1	yes	characteristic
41	C₂²⋊C₄	—	2	1	no	—
42	C₄²	C₂×D₇	1	1	yes	characteristic
43	C₂×C₇	—	4	2	no	—
44	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₇	#102	1	1	yes	characteristic
46	C₂×C₇	#104	1	1	yes	characteristic
47	C₂³	—	2	1	no	—
48	D₄	—	2	1	no	—
49	D₄	C₇⋊C₈	1	2	yes	complemented
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
56	C₈	—	14	2	no	—
57	C₈	—	28	1	no	—
58	C₇	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	2	2	no	—
60	C₂²	—	4	1	no	—
61	C₂²	—	4	1	no	—
62	C₂²	C₂²⋊Dic₇	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	1	no	—
65	C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
66	C₄	#1136	1	1	yes	characteristic
67	C₄	#1141	1	1	yes	characteristic
68	C₄	#1142	1	1	yes	characteristic
69	C₂	—	4	2	no	—
70	C₂	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
71	C₂	#6406	1	1	yes	characteristic
72	C₂	#6411	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₈
Frattini	C₄²	C₂×D₇
Fitting	C₄×C₇×D₄	C₂
socle	C₂²×C₇	C₂²⋊C₄
prime radical	C₂×C₇×D₄	C₄
cosocle	C₄²×C₇	C₂²
hypercenter	C₄×D₄	D₇
hypocenter	C₇	D₄⋊C₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₇⋊C₈)	C₂	semidirect
3. C₄×C₇⋊C₈	C₂	semidirect
14. C₇×D₄	C₈	semidirect
49. D₄	C₇⋊C₈	semidirect
58. C₇	D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×D₄	2
C₄⋊(C₇⋊C₈)	2
C₄×C₇⋊C₈	2
D₄⋊C₈	7

Subgroups — D4⋊(C7⋊C8) #89329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄⋊(C₇⋊C₈) #89329