Subgroups — #15671313

Order 2944.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×C₂₃	C₄	1	2	yes	—
5	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃⋊C₁₆	C₄	1	2	yes	—
7	C₄×C₂₃⋊C₈	C₄	1	2	yes	—
8	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
10	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₂₃⋊C₈	C₈	1	4	yes	—
12	C₁₆×C₂₃	C₈	1	4	yes	complemented
13	C₂₃⋊C₁₆	C₈	1	4	yes	complemented
14	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
15	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
16	C₈×C₂₃	C₂×C₈	1	2	yes	—
17	C₈×C₂₃	C₂×C₈	1	2	yes	—
18	C₂₃⋊C₈	—	2	4	no	—
19	#1311	—	23	1	no	maximal, Sylow
20	C₂²×C₂₃	C₄×C₈	1	1	yes	characteristic
21	C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
22	C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
23	C₄×C₂₃	M₃₂	1	4	yes	—
24	C₄×C₁₆	—	23	1	no	—
25	C₄×C₁₆	D₂₃	1	1	yes	characteristic
26	C₈²	—	23	1	no	—
27	C₂×C₂₃	C₈²	1	1	yes	characteristic
28	C₂×C₂₃	#346	1	2	yes	—
29	C₂×C₁₆	—	23	2	no	—
30	C₂×C₁₆	Dic₂₃	1	2	yes	—
31	C₄×C₈	—	23	2	no	—
32	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
33	C₂₃	#1311	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₈	—	23	4	no	—
35	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
37	C₄²	C₄×D₂₃	1	1	yes	characteristic
38	C₁₆	—	23	4	no	—
39	C₁₆	C₂₃⋊C₈	1	4	yes	complemented
40	C₂×C₄	C₈×D₂₃	1	2	yes	—
41	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
42	C₈	—	46	4	no	—
43	C₈	C₈×D₂₃	1	2	yes	—
44	C₈	C₂×C₂₃⋊C₈	1	2	yes	—
45	C₂²	C₈×Dic₂₃	1	1	yes	characteristic
46	C₄	#10624980	1	4	yes	—
47	C₄	C₄×C₂₃⋊C₈	1	1	yes	characteristic
48	C₄	C₈×Dic₂₃	1	1	yes	characteristic
49	C₂	C₈×C₂₃⋊C₈	1	1	yes	characteristic
50	C₂	#13092192	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₂₃
commutator	C₂×C₂₃	C₈²
Fitting	C₄×C₁₆×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	C₄×C₈
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	C₄×C₁₆	D₂₃
hypocenter	C₂₃	#1311
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₂₃	C₈	semidirect
13. C₂₃⋊C₁₆	C₈	semidirect
33. C₂₃	#1311	semidirect
39. C₁₆	C₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂₃⋊C₁₆	2
C₈×C₂₃⋊C₈	2
C₄×C₁₆×C₂₃	2
#1311	23