Subgroups — #13092816

Order 1472.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#125	C₂	1	1	yes	maximal, characteristic
2	#10625041	C₂	1	1	yes	maximal, characteristic
3	#10624990	C₂	1	2	yes	maximal
4	#10624989	C₂	1	2	yes	maximal
5	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×C₂₃	—	2	2	no	—
7	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₃	C₂²	1	2	yes	—
10	Dic₂₃⋊C₄	—	2	2	no	—
11	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₂₃	—	2	2	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	Q₈	1	2	yes	—
18	C₂₃⋊C₈	Q₈	1	2	yes	—
19	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
20	C₄×C₂₃	—	2	2	no	—
21	C₄×C₂₃	—	4	2	no	—
22	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
24	Dic₂₃	—	4	2	no	—
25	#499	—	23	1	no	maximal, Sylow
26	C₂×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₂₃	#135	1	2	yes	—
28	#125	—	23	1	no	—
29	#125	D₂₃	1	1	yes	characteristic
30	#107	—	23	2	no	—
31	#106	—	23	2	no	—
32	C₄×C₈	—	23	1	no	—
33	C₂₃	#499	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₈	—	23	1	no	—
35	C₂×C₈	—	23	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	23	2	no	—
38	C₄⋊C₄	—	46	2	no	—
39	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
40	C₄²	C₂×D₂₃	1	1	yes	characteristic
41	C₂×C₄	—	2	2	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	—	23	2	no	—
46	C₈	—	23	2	no	—
47	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	92	2	no	—
51	C₄	Q₈×D₂₃	1	1	yes	characteristic
52	C₄	Q₈×D₂₃	1	1	yes	characteristic
53	C₂	#10625115	1	1	yes	characteristic
54	C₂	#10625121	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#125	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₄
hypercenter	#125	D₂₃
hypocenter	C₂₃	#499
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
33. C₂₃	#499	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#125	2
#10625041	2
#10624990	2
#10624989	2
C₄×C₂₃⋊C₈	2
#499	23