Subgroups — #13092529

Order 1472.

132 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	D₄×Dic₂₃	C₂	1	2	yes	maximal, complemented
3	#10625014	C₂	1	2	yes	maximal, complemented
4	#10624999	C₂	1	1	yes	maximal, characteristic, complemented
5	#10624997	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₂₃	C₂²	1	2	yes	—
7	C₂²×Dic₂₃	—	2	2	no	—
8	D₈×C₂₃	C₄	1	4	yes	complemented
9	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²⋊Dic₂₃	—	2	2	no	—
11	C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂³×C₂₃	—	2	2	no	—
15	D₄×C₂₃	—	2	2	no	—
16	D₄×C₂₃	C₂×C₄	1	4	yes	complemented
17	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×Dic₂₃	—	2	2	no	—
19	C₂×Dic₂₃	—	4	4	no	—
20	C₂×Dic₂₃	D₄	1	2	yes	—
21	C₈×C₂₃	C₂×C₄	1	2	yes	complemented
22	C₂₃⋊C₈	—	2	1	no	—
23	C₂²×C₂₃	—	2	4	no	—
24	C₂²×C₂₃	—	4	2	no	—
25	C₂²×C₂₃	—	4	2	no	—
26	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₂₃	#41	1	1	yes	characteristic
29	Dic₂₃	—	2	2	no	—
30	Dic₂₃	—	4	2	no	—
31	#442	—	23	1	no	maximal, Sylow
32	C₂×C₂₃	—	4	4	no	—
33	C₂×C₂₃	C₄×D₄	1	1	yes	characteristic
34	C₂×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
35	C₂×D₈	D₂₃	1	1	yes	characteristic
36	C₄×D₄	—	23	2	no	—
37	#106	—	23	1	no	—
38	#102	—	23	2	no	—
39	#97	—	23	1	no	—
40	C₂₃	#442	1	1	yes	characteristic, complemented, Sylow
41	C₂×D₄	C₂×D₂₃	1	2	yes	—
42	C₂²×C₄	—	46	2	no	—
43	D₈	Dic₂₃	1	4	yes	complemented
44	C₂×C₈	—	23	1	no	—
45	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
46	C₄⋊C₄	—	23	2	no	—
47	C₂²⋊C₄	—	46	2	no	—
48	C₄²	—	23	1	no	—
49	C₂³	—	2	2	no	—
50	D₄	—	2	2	no	—
51	D₄	C₂×Dic₂₃	1	4	yes	complemented
52	C₂×C₄	—	23	2	no	—
53	C₂×C₄	—	46	2	no	—
54	C₂×C₄	—	92	4	no	—
55	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
56	C₈	—	46	1	no	—
57	C₈	C₂×Dic₂₃	1	2	yes	complemented
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	D₄×D₂₃	1	1	yes	characteristic
62	C₄	—	46	2	no	—
63	C₄	—	92	2	no	—
64	C₄	#67290	1	1	yes	characteristic
65	C₄	C₂²×Dic₂₃	1	1	yes	characteristic
66	C₂	—	4	4	no	—
67	C₂	#10625082	1	2	yes	—
68	C₂	D₄×Dic₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃
commutator	C₄×C₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃
Fitting, prime radical	C₂×D₈×C₂₃	C₂
socle	C₂²×C₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂×D₈	D₂₃
hypocenter	C₂₃	#442
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
2. D₄×Dic₂₃	C₂	semidirect
3. #10625014	C₂	semidirect
4. #10624999	C₂	semidirect
5. #10624997	C₂	semidirect
8. D₈×C₂₃	C₄	semidirect
16. D₄×C₂₃	C₂×C₄	semidirect
21. C₈×C₂₃	C₂×C₄	semidirect
40. C₂₃	#442	semidirect
43. D₈	Dic₂₃	semidirect
51. D₄	C₂×Dic₂₃	semidirect
57. C₈	C₂×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
D₄×Dic₂₃	2
#10625014	2
#10624999	2
#10624997	2
#442	23