Subgroups — #15671532

Order 2944.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092251	C₂	1	1	yes	maximal, characteristic
2	#13092192	C₂	1	1	yes	maximal, characteristic
3	C₂₃×M₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
5	C₈×Dic₂₃	C₄	1	1	yes	characteristic
6	C₂×C₂₃⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂₃×C₃₂	C₄	1	2	yes	complemented
8	C₂₃⋊C₃₂	C₄	1	2	yes	complemented
9	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
10	C₄×Dic₂₃	C₈	1	1	yes	characteristic
11	C₂×C₂₃⋊C₈	C₈	1	1	yes	characteristic
12	C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂₃⋊C₁₆	C₈	1	2	yes	—
15	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
16	C₂×Dic₂₃	—	2	1	no	—
17	C₈×C₂₃	C₄²	1	1	yes	characteristic
18	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
19	C₂₃⋊C₈	—	2	1	no	—
20	#1398	—	23	1	no	maximal, Sylow
21	C₂²×C₂₃	M₃₂	1	1	yes	characteristic
22	C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
23	C₄×C₂₃	M₃₂	1	1	yes	characteristic
24	Dic₂₃	—	4	1	no	—
25	M₆₄	—	23	1	no	—
26	M₆₄	D₂₃	1	1	yes	characteristic
27	#346	—	23	1	no	—
28	C₂×C₂₃	—	2	1	no	—
29	C₂×C₂₃	#346	1	1	yes	characteristic
30	C₂×C₁₆	—	23	1	no	—
31	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
32	C₄×C₈	—	23	1	no	—
33	C₃₂	—	23	2	no	—
34	C₃₂	Dic₂₃	1	2	yes	complemented
35	C₂₃	#1398	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₈	—	23	1	no	—
37	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
38	C₄²	—	23	1	no	—
39	C₁₆	—	23	2	no	—
40	C₁₆	C₄×D₂₃	1	1	yes	characteristic
41	C₁₆	C₂×Dic₂₃	1	1	yes	characteristic
42	C₂×C₄	—	46	1	no	—
43	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
44	C₈	—	46	1	no	—
45	C₈	C₈×D₂₃	1	1	yes	characteristic
46	C₈	C₄×Dic₂₃	1	1	yes	characteristic
47	C₂²	#10624980	1	1	yes	characteristic
48	C₄	—	92	1	no	—
49	C₄	#10624980	1	1	yes	characteristic
50	C₄	C₈×Dic₂₃	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#13092192	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₂₃
commutator	C₄×C₂₃	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₂₃
Fitting	C₂₃×M₆₄	C₂
socle	C₂×C₂₃	#346
prime radical	C₂²×C₂₃	M₃₂
cosocle	C₂×C₁₆×C₂₃	C₂²
hypercenter	M₆₄	D₂₃
hypocenter	C₂₃	#1398
LCS	G ▷ C₄×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₈ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₂₃×C₃₂	C₄	semidirect
8. C₂₃⋊C₃₂	C₄	semidirect
34. C₃₂	Dic₂₃	semidirect
35. C₂₃	#1398	semidirect

Maximal subgroups

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

subgroup	index
#13092251	2
#13092192	2
C₂₃×M₆₄	2
#1398	23