Subgroups — #13092210

Order 1472.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#127	C₂	1	1	yes	maximal, characteristic
2	#10625016	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×C₂₃	—	2	2	no	—
4	C₂×D₄×C₂₃	C₄	1	1	yes	characteristic, complemented
5	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₄²×C₂₃	C₄	1	1	yes	characteristic, complemented
7	C₂²⋊Dic₂₃	—	2	2	no	—
8	C₂³×C₂₃	—	2	1	no	—
9	C₂³×C₂₃	D₄	1	2	yes	—
10	D₄×C₂₃	—	4	1	no	—
11	D₄×C₂₃	—	4	1	no	—
12	D₄×C₂₃	—	4	4	no	—
13	C₂×C₄×C₂₃	—	2	1	no	—
14	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₂₃	—	4	2	no	—
16	C₂²×C₂₃	—	2	2	no	—
17	C₂²×C₂₃	—	4	2	no	—
18	C₂²×C₂₃	—	4	2	no	—
19	C₂²×C₂₃	—	4	1	no	—
20	C₂²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₃	—	2	2	no	—
22	C₄×C₂₃	—	2	1	no	—
23	Dic₂₃	—	8	2	no	—
24	#353	—	23	1	no	maximal, Sylow
25	C₂×C₂₃	—	2	1	no	—
26	C₂×C₂₃	—	4	2	no	—
27	C₂×C₂₃	—	8	1	no	—
28	C₂×C₂₃	#99	1	1	yes	characteristic
29	#127	D₂₃	1	1	yes	characteristic
30	#99	—	23	2	no	—
31	C₂₃	#353	1	1	yes	characteristic, complemented, Sylow
32	C₂×D₄	—	2	2	no	—
33	C₂×D₄	Dic₂₃	1	1	yes	characteristic, complemented
34	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
35	C₂²⋊C₄	—	46	2	no	—
36	C₄²	Dic₂₃	1	1	yes	characteristic, complemented
37	C₂³	—	2	1	no	—
38	C₂³	C₂²⋊D₂₃	1	2	yes	—
39	D₄	—	4	1	no	—
40	D₄	—	4	1	no	—
41	D₄	—	4	4	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	92	2	no	—
44	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
45	C₂²	—	2	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	—	4	2	no	—
48	C₂²	—	4	1	no	—
49	C₂²	C₂²⋊Dic₂₃	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	2	1	no	—
52	C₄	—	184	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	4	2	no	—
55	C₂	—	8	1	no	—
56	C₂	#10625016	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10625016
commutator	C₂×C₄×C₂₃	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₃
Fitting, prime radical	C₂₃×#127	C₂
socle	C₂×C₂₃	#99
cosocle	C₂×D₄×C₂₃	C₂²
hypercenter	#127	D₂₃
hypocenter	C₂₃	#353
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #127

Complements

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

N	H ≅ G/N	type
2. #10625016	C₂	semidirect
4. C₂×D₄×C₂₃	C₄	semidirect
6. C₄²×C₂₃	C₄	semidirect
31. C₂₃	#353	semidirect
33. C₂×D₄	Dic₂₃	semidirect
36. C₄²	Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#127	2
#10625016	2
#353	23