Subgroups — C₂×C₄⋊(C₂₂₃⋊C₈) #28383601

Order 14272.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₂₃⋊C₈	C₂	1	2	yes	maximal
2	C₄⋊(C₂₂₃⋊C₈)	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₈	—	2	4	no	—
5	C₂×C₂₂₃⋊C₈	C₂²	1	4	yes	—
6	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₄²×C₂₂₃	C₄	1	2	yes	—
10	C₄²×C₂₂₃	C₂²	1	2	yes	—
11	C₂₂₃⋊C₈	—	2	4	no	—
12	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	—	2	2	no	—
14	C₂×C₄×C₂₂₃	C₈	1	4	yes	complemented
15	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
19	C₂×C₄×C₂₂₃	Q₈	1	2	yes	—
20	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₂²×C₂₂₃	C₄⋊C₄	1	2	yes	—
22	C₂²×C₂₂₃	C₂×C₈	1	2	yes	—
23	C₂²×C₂₂₃	M₁₆	1	2	yes	—
24	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₂₃	—	2	2	no	—
26	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
27	C₄×C₂₂₃	C₂×C₈	1	4	yes	complemented
28	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₂₂₃	C₄⋊C₈	1	4	yes	complemented
31	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
34	C₂₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₄⋊C₈	—	223	1	no	maximal, Sylow
36	C₂²×C₈	—	223	2	no	—
37	C₂×C₄²	D₂₂₃	1	1	yes	characteristic
38	C₄⋊C₈	—	223	4	no	—
39	C₂²×C₄	Dic₂₂₃	1	1	yes	characteristic
40	C₂²×C₄	Dic₂₂₃	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
42	C₂×C₈	—	446	4	no	—
43	C₂×C₈	—	223	4	no	—
44	C₄²	Dic₂₂₃	1	2	yes	—
45	C₄²	C₂×D₂₂₃	1	2	yes	—
46	C₂³	C₂×Dic₂₂₃	1	1	yes	characteristic
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	C₂₂₃⋊C₈	1	4	yes	complemented
49	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
50	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
51	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
52	C₂×C₄	D₈₉₂	1	2	yes	—
53	C₂×C₄	Dic₄₄₆	1	2	yes	—
54	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
55	C₈	—	446	4	no	—
56	C₂²	C₄⋊Dic₂₂₃	1	2	yes	—
57	C₂²	C₂×C₂₂₃⋊C₈	1	2	yes	—
58	C₂²	#16813154	1	2	yes	—
59	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₄⋊Dic₂₂₃	1	2	yes	—
62	C₄	C₂×C₂₂₃⋊C₈	1	4	yes	complemented
63	C₄	C₂×D₈₉₂	1	1	yes	characteristic
64	C₄	C₂×Dic₄₄₆	1	1	yes	characteristic
65	C₂	C₄⋊(C₂₂₃⋊C₈)	1	4	yes	direct product factor
66	C₂	C₂×C₄⋊Dic₂₂₃	1	1	yes	characteristic
67	C₂	C₂²×C₂₂₃⋊C₈	1	1	yes	characteristic
68	C₂	C₂×#16813154	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂×C₄²×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×C₄²	D₂₂₃
hypocenter	C₂₂₃	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₂₂₃⋊C₈)	C₂	direct (N × H)
14. C₂×C₄×C₂₂₃	C₈	semidirect
27. C₄×C₂₂₃	C₂×C₈	semidirect
30. C₂×C₂₂₃	C₄⋊C₈	semidirect
34. C₂₂₃	C₂×C₄⋊C₈	semidirect
48. C₂×C₄	C₂₂₃⋊C₈	semidirect
62. C₄	C₂×C₂₂₃⋊C₈	semidirect
65. C₂	C₄⋊(C₂₂₃⋊C₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₂₃⋊C₈	2
C₄⋊(C₂₂₃⋊C₈)	2
C₂×C₄²×C₂₂₃	2
C₂×C₄⋊C₈	223

Subgroups — C2×C4⋊(C223⋊C8) #28383601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₂₂₃⋊C₈) #28383601