Subgroups — C₂×C₂²⋊(C₂₂₃⋊C₈) #28383541

Order 14272.

202 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, complemented
2	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₂₃⋊C₈	—	2	4	no	—
5	C₂×C₂₂₃⋊C₈	C₂²	1	4	yes	complemented
6	C₂⁴×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₂₃	—	2	4	no	—
8	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₂₂₃	C₄	1	2	yes	—
10	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂₂₃⋊C₈	—	2	4	no	—
13	C₂³×C₂₂₃	—	2	2	no	—
14	C₂³×C₂₂₃	—	2	2	no	—
15	C₂³×C₂₂₃	C₈	1	4	yes	complemented
16	C₂³×C₂₂₃	C₂×C₄	1	2	yes	—
17	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	—	2	2	no	—
19	C₂×C₄×C₂₂₃	—	2	4	no	—
20	C₂×C₄×C₂₂₃	—	2	4	no	—
21	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	D₄	1	4	yes	—
24	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₂₂₃	—	2	8	no	—
26	C₂²×C₂₂₃	—	2	2	no	—
27	C₂²×C₂₂₃	—	2	2	no	—
28	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
29	C₂²×C₂₂₃	C₂×C₈	1	2	yes	—
30	C₂²×C₂₂₃	C₂×C₈	1	4	yes	complemented
31	C₂²×C₂₂₃	M₁₆	1	2	yes	—
32	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₂₂₃	—	2	2	no	—
34	C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
35	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
36	C₂×C₂₂₃	—	2	4	no	—
37	C₂×C₂₂₃	C₂²⋊C₈	1	4	yes	complemented
38	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
41	C₂₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₂²⋊C₈	—	223	1	no	maximal, Sylow
43	C₂³×C₄	D₂₂₃	1	1	yes	characteristic
44	C₂²×C₈	—	223	2	no	—
45	C₂²⋊C₈	—	223	4	no	—
46	C₂⁴	Dic₂₂₃	1	1	yes	characteristic
47	C₂²×C₄	—	2	4	no	—
48	C₂²×C₄	Dic₂₂₃	1	1	yes	characteristic
49	C₂²×C₄	Dic₂₂₃	1	2	yes	—
50	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
51	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
52	C₂×C₈	—	446	4	no	—
53	C₂×C₈	—	223	4	no	—
54	C₂³	—	2	2	no	—
55	C₂³	—	2	2	no	—
56	C₂³	C₂₂₃⋊C₈	1	4	yes	complemented
57	C₂³	C₂×Dic₂₂₃	1	2	yes	—
58	C₂³	C₂×Dic₂₂₃	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
63	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₂₂₃	1	4	yes	—
65	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
66	C₈	—	446	4	no	—
67	C₂²	—	2	8	no	—
68	C₂²	—	2	2	no	—
69	C₂²	—	2	2	no	—
70	C₂²	C₂²⋊Dic₂₂₃	1	2	yes	—
71	C₂²	C₂×C₂₂₃⋊C₈	1	2	yes	—
72	C₂²	C₂×C₂₂₃⋊C₈	1	4	yes	complemented
73	C₂²	#16813154	1	2	yes	—
74	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	C₂²⋊Dic₂₂₃	1	2	yes	—
77	C₄	C₂×C₂²⋊D₂₂₃	1	2	yes	—
78	C₂	—	2	4	no	—
79	C₂	C₂²⋊(C₂₂₃⋊C₈)	1	4	yes	direct product factor
80	C₂	C₂×C₂²⋊Dic₂₂₃	1	1	yes	characteristic
81	C₂	C₂²×C₂₂₃⋊C₈	1	1	yes	characteristic
82	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	C₂³×C₄×C₂₂₃	C₂
socle	C₂³×C₂₂₃	C₂×C₄
prime radical	C₂⁴×C₂₂₃	C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂³×C₄	D₂₂₃
hypocenter	C₂₂₃	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₂₃⋊C₈	C₂	semidirect
2. C₂²⋊(C₂₂₃⋊C₈)	C₂	direct (N × H)
5. C₂×C₂₂₃⋊C₈	C₂²	semidirect
15. C₂³×C₂₂₃	C₈	semidirect
30. C₂²×C₂₂₃	C₂×C₈	semidirect
37. C₂×C₂₂₃	C₂²⋊C₈	semidirect
41. C₂₂₃	C₂×C₂²⋊C₈	semidirect
56. C₂³	C₂₂₃⋊C₈	semidirect
72. C₂²	C₂×C₂₂₃⋊C₈	semidirect
79. C₂	C₂²⋊(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
C₂²⋊(C₂₂₃⋊C₈)	2
C₂³×C₄×C₂₂₃	2
C₂×C₂²⋊C₈	223

Subgroups — C2×C22⋊(C223⋊C8) #28383541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊(C₂₂₃⋊C₈) #28383541