Subgroups — C₂×C₄⋊(C₂₃⋊C₈) #13092438

Order 1472.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃⋊C₈	C₂	1	2	yes	maximal
3	C₄⋊(C₂₃⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₄²×C₂₃	C₄	1	2	yes	—
8	C₄²×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃⋊C₈	—	2	4	no	—
10	C₂×C₂₃⋊C₈	C₂²	1	4	yes	—
11	C₂³×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₂₃	—	2	2	no	—
13	C₂×C₄×C₂₃	C₈	1	4	yes	complemented
14	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₂₃	D₄	1	2	yes	—
18	C₂×C₄×C₂₃	Q₈	1	2	yes	—
19	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂₃⋊C₈	—	2	4	no	—
21	C₂²×C₂₃	C₄⋊C₄	1	2	yes	—
22	C₂²×C₂₃	C₂×C₈	1	2	yes	—
23	C₂²×C₂₃	M₁₆	1	2	yes	—
24	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₂₃	—	2	2	no	—
26	C₄×C₂₃	C₄⋊C₄	1	2	yes	—
27	C₄×C₂₃	C₂×C₈	1	4	yes	complemented
28	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄⋊C₈	—	23	1	no	maximal, Sylow
31	C₂×C₂₃	C₄⋊C₈	1	4	yes	complemented
32	C₂×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₂₃	C₂²×C₈	1	1	yes	characteristic
34	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
35	C₂²×C₈	—	23	2	no	—
36	C₂×C₄²	D₂₃	1	1	yes	characteristic
37	C₄⋊C₈	—	23	4	no	—
38	C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₄	Dic₂₃	1	1	yes	characteristic
40	C₂²×C₄	Dic₂₃	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
42	C₂×C₈	—	23	4	no	—
43	C₂×C₈	—	46	4	no	—
44	C₄²	Dic₂₃	1	2	yes	—
45	C₄²	C₂×D₂₃	1	2	yes	—
46	C₂³	C₂×Dic₂₃	1	1	yes	characteristic
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	C₂₃⋊C₈	1	4	yes	complemented
49	C₂×C₄	Dic₄₆	1	2	yes	—
50	C₂×C₄	D₉₂	1	2	yes	—
51	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
52	C₂×C₄	C₂×Dic₂₃	1	2	yes	—
53	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
54	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
55	C₈	—	46	4	no	—
56	C₂²	C₂×C₂₃⋊C₈	1	2	yes	—
57	C₂²	#67267	1	2	yes	—
58	C₂²	C₄⋊Dic₂₃	1	2	yes	—
59	C₂²	C₂²×Dic₂₃	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₂×C₂₃⋊C₈	1	4	yes	complemented
62	C₄	C₄⋊Dic₂₃	1	2	yes	—
63	C₄	C₂×Dic₄₆	1	1	yes	characteristic
64	C₄	C₂×D₉₂	1	1	yes	characteristic
65	C₂	C₄⋊(C₂₃⋊C₈)	1	4	yes	direct product factor
66	C₂	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
67	C₂	C₂×#67267	1	1	yes	characteristic
68	C₂	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃
commutator	C₂×C₂₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂×C₄²×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂×C₄²	D₂₃
hypocenter	C₂₃	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₂₃⋊C₈)	C₂	direct (N × H)
13. C₂×C₄×C₂₃	C₈	semidirect
27. C₄×C₂₃	C₂×C₈	semidirect
31. C₂×C₂₃	C₄⋊C₈	semidirect
38. C₂₃	C₂×C₄⋊C₈	semidirect
48. C₂×C₄	C₂₃⋊C₈	semidirect
61. C₄	C₂×C₂₃⋊C₈	semidirect
65. C₂	C₄⋊(C₂₃⋊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
C₂×C₄²×C₂₃	2
C₂²×C₂₃⋊C₈	2
C₄⋊(C₂₃⋊C₈)	2
C₂×C₄⋊C₈	23

Subgroups — C2×C4⋊(C23⋊C8) #13092438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₂₃⋊C₈) #13092438