Subgroups — #13092374

Order 1472.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#67262	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
3	#10625043	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	#10624997	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	(C₂×D₂₃)⋊C₄	C₄	1	2	yes	—
13	Dic₂₃⋊C₄	C₄	1	2	yes	—
14	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
17	#67262	C₄	1	4	yes	complemented
18	C₈×D₂₃	C₄	1	4	yes	complemented
19	C₂²×D₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₂₃	C₂×C₄	1	2	yes	—
23	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
24	C₄×D₂₃	C₂×C₄	1	2	yes	—
25	C₄×D₂₃	C₂×C₄	1	2	yes	—
26	C₈×C₂₃	C₂×C₄	1	2	yes	—
27	C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₂₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₂₃	—	2	2	no	—
32	C₂×D₂₃	C₄²	1	2	yes	—
33	C₄×C₂₃	—	2	2	no	—
34	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	Dic₂₃	—	2	2	no	—
37	Dic₂₃	C₄²	1	2	yes	—
38	#405	—	23	1	no	maximal, Sylow
39	C₂×C₂₃	C₂×C₄²	1	1	yes	characteristic
40	C₂×C₂₃	#131	1	2	yes	—
41	D₂₃	—	2	2	no	—
42	C₂×M₁₆	—	23	1	no	—
43	C₂²×C₈	—	23	1	no	—
44	#117	—	23	1	no	—
45	#97	—	23	1	no	—
46	#97	D₂₃	1	1	yes	characteristic, complemented
47	C₄×C₈	—	23	1	no	—
48	C₄×C₈	—	23	1	no	—
49	C₂₃	#405	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₄	—	23	1	no	—
51	M₁₆	—	23	4	no	—
52	C₂×C₈	—	23	1	no	—
53	C₂×C₈	—	23	1	no	—
54	C₂×C₈	—	23	4	no	—
55	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
56	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
57	C₄⋊C₄	—	23	2	no	—
58	C₂²⋊C₄	—	23	2	no	—
59	C₄²	—	23	1	no	—
60	C₄²	C₂×D₂₃	1	1	yes	characteristic
61	C₂³	—	23	1	no	—
62	C₂×C₄	—	23	1	no	—
63	C₂×C₄	—	23	2	no	—
64	C₂×C₄	—	23	2	no	—
65	C₂×C₄	—	23	2	no	—
66	C₂×C₄	C₄×D₂₃	1	2	yes	—
67	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
68	C₈	—	23	2	no	—
69	C₈	—	23	2	no	—
70	C₈	C₄×D₂₃	1	2	yes	—
71	C₈	C₄×D₂₃	1	2	yes	—
72	C₂²	—	23	2	no	—
73	C₂²	—	46	2	no	—
74	C₂²	C₂×C₄×D₂₃	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	23	2	no	—
77	C₄	—	46	2	no	—
78	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
79	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
80	C₂	—	46	2	no	—
81	C₂	C₄²×D₂₃	1	1	yes	characteristic
82	C₂	#10625078	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator	C₂×C₂₃	C₂×C₄²
Fitting	C₂₃×#97	C₂
socle	C₂²×C₂₃	C₂²×C₄
prime radical	C₂²×D₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	#97	D₂₃
hypocenter	C₂₃	#405
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
4. C₂₃×#97	C₂	semidirect
5. #10624997	C₂	semidirect
6. C₈×Dic₂₃	C₂	semidirect
7. C₄×C₂₃⋊C₈	C₂	semidirect
17. #67262	C₄	semidirect
18. C₈×D₂₃	C₄	semidirect
46. #97	D₂₃	semidirect
49. C₂₃	#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×#67262	2
C₂×C₈×D₂₃	2
#10625043	2
C₂₃×#97	2
#10624997	2
C₈×Dic₂₃	2
C₄×C₂₃⋊C₈	2
#405	23