Subgroups — #13092227

Order 1472.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#135	C₂	1	1	yes	maximal, characteristic
2	#10624999	C₂	1	1	yes	maximal, characteristic, complemented
3	#10624994	C₂	1	1	yes	maximal, characteristic, complemented
4	#41×C₂₃	—	2	1	no	—
5	Q₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
6	SD₁₆×C₂₃	—	2	1	no	—
7	D₈×C₂₃	C₄	1	1	yes	characteristic, complemented
8	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₃	—	2	1	no	—
10	C₂₃⋊C₁₆	—	2	1	no	—
11	Q₈×C₂₃	—	2	1	no	—
12	D₄×C₂₃	—	2	1	no	—
13	D₄×C₂₃	—	4	1	no	—
14	C₂×C₄×C₂₃	—	4	1	no	—
15	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂×Dic₂₃	—	4	1	no	—
17	C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₈×C₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₂₃	—	4	1	no	—
20	C₂²×C₂₃	D₈	1	1	yes	characteristic
21	C₄×C₂₃	—	4	1	no	—
22	C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₃	SD₁₆	1	1	yes	characteristic
24	Dic₂₃	—	8	1	no	—
25	#360	—	23	1	no	maximal, Sylow
26	C₂×C₂₃	—	2	1	no	—
27	C₂×C₂₃	—	8	1	no	—
28	C₂×C₂₃	#102	1	1	yes	characteristic
29	#135	D₂₃	1	1	yes	characteristic
30	M₃₂	—	23	1	no	—
31	#106	—	23	1	no	—
32	C₂₃	#360	1	1	yes	characteristic, complemented, Sylow
33	#41	—	2	1	no	—
34	Q₁₆	Dic₂₃	1	1	yes	characteristic, complemented
35	SD₁₆	—	2	1	no	—
36	D₈	Dic₂₃	1	1	yes	characteristic, complemented
37	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
38	C₄⋊C₄	—	46	1	no	—
39	C₁₆	—	46	1	no	—
40	Q₈	—	2	1	no	—
41	D₄	—	2	1	no	—
42	D₄	—	4	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	92	1	no	—
45	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
46	C₈	C₂×Dic₂₃	1	1	yes	characteristic
47	C₈	C₂²⋊D₂₃	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	#67272	1	1	yes	characteristic
50	C₄	—	4	1	no	—
51	C₄	—	184	1	no	—
52	C₄	#67273	1	1	yes	characteristic
53	C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	8	1	no	—
56	C₂	#10625014	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10625014
commutator	C₈×C₂₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₃
Fitting, prime radical	C₂₃×#135	C₂
socle	C₂×C₂₃	#102
cosocle	C₂×C₈×C₂₃	C₂²
hypercenter	#135	D₂₃
hypocenter	C₂₃	#360
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #10624999	C₂	semidirect
3. #10624994	C₂	semidirect
5. Q₁₆×C₂₃	C₄	semidirect
7. D₈×C₂₃	C₄	semidirect
32. C₂₃	#360	semidirect
34. Q₁₆	Dic₂₃	semidirect
36. D₈	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#135	2
#10624999	2
#10624994	2
#360	23