Subgroups — #13092781

Order 1472.

110 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	#10625041	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625014	C₂	1	4	yes	maximal, complemented
4	#10624985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×C₂₃	—	2	2	no	—
6	C₂×D₄×C₂₃	C₂²	1	2	yes	—
7	C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₃	C₂²	1	2	yes	—
9	Dic₂₃⋊C₄	—	2	2	no	—
10	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
11	C₂³×C₂₃	—	2	2	no	—
12	D₄×C₂₃	—	2	4	no	—
13	D₄×C₂₃	—	4	4	no	—
14	C₂×C₄×C₂₃	D₄	1	2	yes	—
15	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×Dic₂₃	—	2	2	no	—
17	C₂₃⋊C₈	—	2	2	no	—
18	C₂²×C₂₃	—	4	2	no	—
19	C₂²×C₂₃	—	4	4	no	—
20	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
21	C₄×C₂₃	—	2	2	no	—
22	C₄×C₂₃	#41	1	2	yes	—
23	Dic₂₃	—	4	2	no	—
24	#490	—	23	1	no	maximal, Sylow
25	C₂×C₂₃	—	8	2	no	—
26	C₂×C₂₃	#124	1	1	yes	characteristic
27	C₂×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
28	#127	D₂₃	1	1	yes	characteristic
29	#125	—	23	1	no	—
30	#102	—	23	4	no	—
31	#97	—	23	1	no	—
32	C₂₃	#490	1	1	yes	characteristic, complemented, Sylow
33	C₂×D₄	—	2	2	no	—
34	C₂×D₄	C₂×D₂₃	1	2	yes	—
35	C₂×C₈	—	23	2	no	—
36	C₄⋊C₄	—	23	2	no	—
37	C₄⋊C₄	—	46	2	no	—
38	C₄²	C₂×D₂₃	1	1	yes	characteristic
39	C₂³	—	2	2	no	—
40	D₄	—	2	4	no	—
41	D₄	—	4	4	no	—
42	C₂×C₄	—	46	2	no	—
43	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
44	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
45	C₈	—	46	2	no	—
46	C₂²	—	4	4	no	—
47	C₂²	—	4	2	no	—
48	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	92	2	no	—
51	C₄	#67290	1	2	yes	—
52	C₂	—	8	2	no	—
53	C₂	#10625103	1	2	yes	—
54	C₂	#10625107	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting, prime radical	C₂₃×#127	C₂
socle	C₂²×C₂₃	C₂×D₄
hypercenter	#127	D₂₃
hypocenter	C₂₃	#490
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. #10625041	C₂	semidirect
3. #10625014	C₂	semidirect
4. #10624985	C₂	semidirect
32. C₂₃	#490	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
#10625041	2
#10625014	2
#10624985	2
#490	23