Subgroups — #15675691

Order 2944.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092251	C₂	1	3	yes	maximal, complemented
2	C₂×C₂₃⋊C₃₂	C₂	1	3	yes	maximal, complemented
3	C₂₃×#504	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#131	C₄	1	1	yes	characteristic
5	C₂₃×M₃₂	C₄	1	3	yes	—
6	C₂×C₁₆×C₂₃	C₂²	1	3	yes	—
7	C₂₃⋊C₃₂	C₂²	1	1	yes	characteristic
8	C₂₃⋊C₃₂	C₂²	1	3	yes	—
9	#41×C₂₃	C₈	1	1	yes	characteristic
10	M₁₆×C₂₃	C₈	1	3	yes	—
11	C₂×C₈×C₂₃	C₂×C₄	1	3	yes	—
12	C₁₆×C₂₃	C₂×C₄	1	3	yes	—
13	C₁₆×C₂₃	C₂³	1	1	yes	characteristic
14	Q₈×C₂₃	C₁₆	1	1	yes	characteristic
15	D₄×C₂₃	C₁₆	1	3	yes	—
16	C₂×C₄×C₂₃	C₂×C₈	1	3	yes	—
17	C₈×C₂₃	C₂×C₈	1	3	yes	—
18	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
19	#2258	—	23	1	no	maximal, Sylow
20	C₂²×C₂₃	C₂×C₁₆	1	3	yes	—
21	C₄×C₂₃	C₂×C₁₆	1	3	yes	—
22	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
23	#504	D₂₃	1	1	yes	characteristic
24	M₆₄	—	23	3	no	—
25	C₂×C₃₂	—	23	3	no	—
26	C₂×C₂₃	—	2	3	no	—
27	C₂×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
28	#131	Dic₂₃	1	1	yes	characteristic
29	M₃₂	Dic₂₃	1	3	yes	—
30	C₂×C₁₆	C₂×D₂₃	1	3	yes	—
31	C₃₂	—	23	1	no	—
32	C₃₂	—	23	3	no	—
33	C₂₃	#2258	1	1	yes	characteristic, complemented, Sylow
34	#41	C₂₃⋊C₈	1	1	yes	characteristic
35	M₁₆	C₂₃⋊C₈	1	3	yes	—
36	C₂×C₈	C₂×Dic₂₃	1	3	yes	—
37	C₁₆	C₂×Dic₂₃	1	3	yes	—
38	C₁₆	C₂²×D₂₃	1	1	yes	characteristic
39	Q₈	C₂₃⋊C₁₆	1	1	yes	characteristic
40	D₄	C₂₃⋊C₁₆	1	3	yes	—
41	C₂×C₄	C₂×C₂₃⋊C₈	1	3	yes	—
42	C₈	C₂×C₂₃⋊C₈	1	3	yes	—
43	C₈	C₂²×Dic₂₃	1	1	yes	characteristic
44	C₂²	C₂×C₂₃⋊C₁₆	1	3	yes	—
45	C₄	C₂×C₂₃⋊C₁₆	1	3	yes	—
46	C₄	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
47	C₂	—	2	3	no	—
48	C₂	C₂²×C₂₃⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₂₃
commutator, socle	C₂×C₂₃	C₂²×C₁₆
Fitting	C₂₃×#504	C₂
prime radical	#41×C₂₃	C₈
cosocle	C₁₆×C₂₃	C₂³
hypercenter	#504	D₂₃
hypocenter	C₂₃	#2258
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #13092251	C₂	semidirect
2. C₂×C₂₃⋊C₃₂	C₂	semidirect
33. C₂₃	#2258	semidirect

Maximal subgroups

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

subgroup	index
#13092251	2
C₂×C₂₃⋊C₃₂	2
C₂₃×#504	2
#2258	23