Subgroups — #13092827

Order 1472.

90 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	#10625040	C₂	1	1	yes	maximal, characteristic
3	#10624990	C₂	1	2	yes	maximal
4	#10624989	C₂	1	2	yes	maximal
5	#10624985	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₆	—	2	1	no	—
7	C₄⋊C₄×C₂₃	—	2	1	no	—
8	C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₃	—	2	1	no	—
13	C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₂₃	—	2	1	no	—
16	C₂×C₄×C₂₃	—	2	1	no	—
17	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×Dic₂₃	—	2	2	no	—
21	Dic₄₆	—	4	2	no	—
22	C₂₃⋊C₈	Q₈	1	4	yes	complemented
23	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₃	—	2	1	no	—
25	C₄×C₂₃	—	2	1	no	—
26	C₄×C₂₃	—	4	1	no	—
27	C₄×C₂₃	—	4	1	no	—
28	C₄×C₂₃	C₂×Q₈	1	2	yes	—
29	Dic₂₃	—	4	2	no	—
30	#501	—	23	1	no	maximal, Sylow
31	C₂×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
32	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
33	C₂×C₂₃	#137	1	1	yes	characteristic
34	C₄⋊Q₈	—	23	1	no	—
35	#125	D₂₃	1	1	yes	characteristic
36	#107	—	23	2	no	—
37	#106	—	23	2	no	—
38	#97	—	23	1	no	—
39	C₂₃	#501	1	1	yes	characteristic, complemented, Sylow
40	C₂×Q₈	—	46	1	no	—
41	C₂×C₈	—	23	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	23	2	no	—
45	C₄⋊C₄	—	46	1	no	—
46	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
47	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
48	C₄²	C₂×D₂₃	1	1	yes	characteristic
49	Q₈	—	92	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	46	2	no	—
53	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
56	C₈	—	23	4	no	—
57	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	92	2	no	—
63	C₄	Q₈×D₂₃	1	2	yes	—
64	C₂	#10625115	1	1	yes	characteristic
65	C₂	#10625120	1	1	yes	characteristic
66	C₂	#10625122	1	1	yes	characteristic

	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	#501
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
22. C₂₃⋊C₈	Q₈	semidirect
39. C₂₃	#501	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
#10625040	2
#10624990	2
#10624989	2
#10624985	2
#501	23