Subgroups — D₄⋊(C₂₂₃⋊C₈) #28383308

Order 14272.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₈	—	2	1	no	—
5	C₂×C₂₂₃⋊C₈	—	2	1	no	—
6	C₂×D₄×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₂₃	—	2	1	no	—
8	C₄⋊C₄×C₂₂₃	C₄	1	1	yes	characteristic
9	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
10	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂₂₃⋊C₈	—	2	2	no	—
12	C₂₂₃⋊C₈	—	4	1	no	—
13	C₂³×C₂₂₃	—	2	1	no	—
14	D₄×C₂₂₃	—	2	1	no	—
15	D₄×C₂₂₃	C₈	1	2	yes	complemented
16	C₂×C₄×C₂₂₃	—	2	1	no	—
17	C₂×C₄×C₂₂₃	—	4	1	no	—
18	C₂×C₄×C₂₂₃	—	4	1	no	—
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₂₂₃	—	2	2	no	—
23	C₂²×C₂₂₃	—	4	1	no	—
24	C₂²×C₂₂₃	—	4	1	no	—
25	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₂₂₃	—	2	1	no	—
27	C₄×C₂₂₃	—	4	1	no	—
28	C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
29	C₄×C₂₂₃	M₁₆	1	1	yes	characteristic
30	C₄×C₂₂₃	D₈	1	1	yes	characteristic
31	C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
32	C₂×C₂₂₃	—	4	2	no	—
33	C₂×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₂₂₃	#102	1	1	yes	characteristic
35	C₂×C₂₂₃	#104	1	1	yes	characteristic
36	C₂₂₃	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
37	D₄⋊C₈	—	223	1	no	maximal, Sylow
38	C₄×D₄	D₂₂₃	1	1	yes	characteristic
39	C₄⋊C₈	—	223	1	no	—
40	C₄×C₈	—	223	1	no	—
41	C₂×D₄	Dic₂₂₃	1	1	yes	characteristic
42	C₂²×C₄	—	2	1	no	—
43	C₂×C₈	—	446	1	no	—
44	C₂×C₈	—	446	1	no	—
45	C₄⋊C₄	Dic₂₂₃	1	1	yes	characteristic
46	C₂²⋊C₄	—	2	1	no	—
47	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
48	C₂³	—	2	1	no	—
49	D₄	—	2	1	no	—
50	D₄	C₂₂₃⋊C₈	1	2	yes	complemented
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
57	C₈	—	892	1	no	—
58	C₈	—	446	2	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	1	no	—
61	C₂²	—	4	1	no	—
62	C₂²	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	1	no	—
65	C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
66	C₄	#16813154	1	1	yes	characteristic
67	C₄	#16813155	1	1	yes	characteristic
68	C₄	#16813158	1	1	yes	characteristic
69	C₂	—	4	2	no	—
70	C₂	C₂²⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
71	C₂	#20297225	1	1	yes	characteristic
72	C₂	#20297231	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂×C₈
Frattini	C₄²	C₂×D₂₂₃
Fitting	C₄×D₄×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
prime radical	C₂×D₄×C₂₂₃	C₄
cosocle	C₄²×C₂₂₃	C₂²
hypercenter	C₄×D₄	D₂₂₃
hypocenter	C₂₂₃	D₄⋊C₈
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₂₂₃⋊C₈)	C₂	semidirect
2. C₄×C₂₂₃⋊C₈	C₂	semidirect
15. D₄×C₂₂₃	C₈	semidirect
36. C₂₂₃	D₄⋊C₈	semidirect
50. D₄	C₂₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₂₃⋊C₈)	2
C₄×C₂₂₃⋊C₈	2
C₄×D₄×C₂₂₃	2
D₄⋊C₈	223

Subgroups — D4⋊(C223⋊C8) #28383308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄⋊(C₂₂₃⋊C₈) #28383308