Subgroups — #13092218

Order 1472.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625000	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄×C₂₃	—	2	1	no	—
5	D₈×C₂₃	—	2	1	no	—
6	D₈×C₂₃	C₄	1	2	yes	complemented
7	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₃	—	2	1	no	—
9	C₂₃⋊C₁₆	—	2	1	no	—
10	C₂³×C₂₃	—	4	1	no	—
11	D₄×C₂₃	—	2	2	no	—
12	D₄×C₂₃	—	4	1	no	—
13	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
14	C₂×Dic₂₃	—	4	1	no	—
15	C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₂₃	—	4	2	no	—
18	C₂²×C₂₃	—	8	1	no	—
19	C₂²×C₂₃	—	8	1	no	—
20	C₂²×C₂₃	D₈	1	1	yes	characteristic
21	C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₂₃	SD₁₆	1	1	yes	characteristic
23	Dic₂₃	—	8	1	no	—
24	#357	—	23	1	no	maximal, Sylow
25	C₂×C₂₃	—	8	2	no	—
26	C₂×C₂₃	#102	1	1	yes	characteristic
27	C₂×C₂₃	D₁₆	1	1	yes	characteristic
28	C₂×C₂₃	SD₃₂	1	1	yes	characteristic
29	C₂×D₈	D₂₃	1	1	yes	characteristic
30	C₂×C₁₆	—	23	1	no	—
31	#107	—	23	1	no	—
32	C₂₃	#357	1	1	yes	characteristic, complemented, Sylow
33	C₂×D₄	—	2	1	no	—
34	D₈	—	2	1	no	—
35	D₈	Dic₂₃	1	2	yes	complemented
36	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
37	C₄⋊C₄	—	46	1	no	—
38	C₁₆	—	46	1	no	—
39	C₂³	—	4	1	no	—
40	D₄	—	2	2	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	92	1	no	—
43	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
44	C₈	C₂×Dic₂₃	1	1	yes	characteristic
45	C₈	C₂²⋊D₂₃	1	1	yes	characteristic
46	C₂²	—	4	2	no	—
47	C₂²	—	8	1	no	—
48	C₂²	—	8	1	no	—
49	C₂²	#67272	1	1	yes	characteristic
50	C₄	—	184	1	no	—
51	C₄	#67273	1	1	yes	characteristic
52	C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
53	C₂	—	8	2	no	—
54	C₂	#10625008	1	1	yes	characteristic
55	C₂	#10625009	1	1	yes	characteristic
56	C₂	#10625014	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#67272
commutator	C₈×C₂₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₃
Fitting, prime radical	C₂×D₈×C₂₃	C₂
socle	C₂²×C₂₃	D₈
cosocle	C₂×C₈×C₂₃	C₂²
hypercenter	C₂×D₈	D₂₃
hypocenter	C₂₃	#357
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
2. #10625000	C₂	semidirect
3. C₂×C₂₃⋊C₁₆	C₂	semidirect
6. D₈×C₂₃	C₄	semidirect
32. C₂₃	#357	semidirect
35. D₈	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×C₂₃	2
#10625000	2
C₂×C₂₃⋊C₁₆	2
#357	23