Subgroups — #13092224

Order 1472.

56 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	#10625001	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#41×C₂₃	—	2	1	no	—
5	Q₁₆×C₂₃	C₄	1	1	yes	characteristic
6	SD₁₆×C₂₃	—	2	1	no	—
7	D₈×C₂₃	C₄	1	1	yes	characteristic
8	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	#67267	—	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	C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂₃⋊C₈	—	4	1	no	—
19	C₂²×C₂₃	—	4	1	no	—
20	C₂²×C₂₃	SD₁₆	1	1	yes	characteristic
21	C₄×C₂₃	—	4	1	no	—
22	C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₂₃	D₈	1	1	yes	characteristic
24	#359	—	23	1	no	maximal, Sylow
25	C₂×C₂₃	—	2	1	no	—
26	C₂×C₂₃	—	8	1	no	—
27	C₂×C₂₃	#102	1	1	yes	characteristic
28	#135	D₂₃	1	1	yes	characteristic
29	C₂×C₁₆	—	23	1	no	—
30	#108	—	23	1	no	—
31	C₂₃	#359	1	1	yes	characteristic, complemented, Sylow
32	#41	—	2	1	no	—
33	Q₁₆	Dic₂₃	1	1	yes	characteristic
34	SD₁₆	—	2	1	no	—
35	D₈	Dic₂₃	1	1	yes	characteristic
36	M₁₆	—	46	1	no	—
37	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
38	C₁₆	—	46	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	4	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
44	C₈	—	92	1	no	—
45	C₈	C₂×Dic₂₃	1	1	yes	characteristic
46	C₈	C₂²⋊D₂₃	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	#67273	1	1	yes	characteristic
49	C₄	—	4	1	no	—
50	C₄	#67272	1	1	yes	characteristic
51	C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	8	1	no	—
54	C₂	#10625014	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#67272
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	#359
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #10625001	C₂	semidirect
3. C₂×C₂₃⋊C₁₆	C₂	semidirect
31. C₂₃	#359	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
#10625001	2
C₂×C₂₃⋊C₁₆	2
#359	23