Subgroups — #15671466

Order 2944.

76 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	C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×C₂₃	—	2	1	no	—
5	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃⋊C₁₆	—	2	1	no	—
7	C₄×C₂₃⋊C₈	C₄	1	1	yes	characteristic
8	C₄×C₂₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
10	C₂×C₈×C₂₃	Q₈	1	1	yes	characteristic
11	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₂₃⋊C₈	—	2	1	no	—
13	C₂×C₂₃⋊C₈	C₈	1	1	yes	characteristic
14	C₂×C₂₃⋊C₈	C₈	1	1	yes	characteristic
15	C₁₆×C₂₃	—	4	1	no	—
16	C₂₃⋊C₁₆	—	4	1	no	—
17	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄×C₂₃	M₁₆	1	1	yes	characteristic
20	C₈×C₂₃	—	2	1	no	—
21	C₈×C₂₃	D₈	1	1	yes	characteristic
22	C₈×C₂₃	Q₁₆	1	1	yes	characteristic
23	C₂₃⋊C₈	—	2	2	no	—
24	C₂₃⋊C₈	—	2	1	no	—
25	C₂₃⋊C₈	C₁₆	1	2	yes	complemented
26	#1370	—	23	1	no	maximal, Sylow
27	C₂²×C₂₃	C₄⋊C₈	1	1	yes	characteristic
28	C₄×C₂₃	—	2	1	no	—
29	C₄×C₂₃	#107	1	1	yes	characteristic
30	C₄×C₂₃	#108	1	1	yes	characteristic
31	C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
32	C₄×C₂₃	M₃₂	1	1	yes	characteristic
33	C₄⋊C₁₆	—	23	1	no	—
34	C₄⋊C₁₆	D₂₃	1	1	yes	characteristic
35	C₈²	—	23	1	no	—
36	C₂×C₂₃	#335	1	1	yes	characteristic
37	C₂×C₂₃	C₄⋊C₁₆	1	1	yes	characteristic
38	C₂×C₂₃	#364	1	1	yes	characteristic
39	C₂×C₁₆	—	2	1	no	—
40	C₂×C₁₆	—	46	1	no	—
41	C₄×C₈	—	23	1	no	—
42	C₄×C₈	—	23	1	no	—
43	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
44	C₂₃	#1370	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₈	—	23	1	no	—
46	C₂×C₈	—	23	1	no	—
47	C₂×C₈	—	46	1	no	—
48	C₂×C₈	Dic₄₆	1	1	yes	characteristic
49	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
50	C₄²	C₄×D₂₃	1	1	yes	characteristic
51	C₁₆	—	4	1	no	—
52	C₁₆	—	92	1	no	—
53	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
54	C₂×C₄	#67262	1	1	yes	characteristic
55	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	—	23	2	no	—
58	C₈	—	46	2	no	—
59	C₈	—	46	1	no	—
60	C₈	#67272	1	1	yes	characteristic
61	C₈	#67275	1	1	yes	characteristic
62	C₂²	Dic₂₃⋊C₈	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	C₁₆×D₂₃	1	1	yes	characteristic
65	C₄	#10624980	1	1	yes	characteristic
66	C₄	#10624989	1	1	yes	characteristic
67	C₄	#10625004	1	1	yes	characteristic
68	C₂	#13092169	1	1	yes	characteristic
69	C₂	Dic₂₃⋊C₁₆	1	1	yes	characteristic
70	C₂	#13092237	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₂₃	C₄⋊C₈
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	C₄⋊C₁₆	D₂₃
hypocenter	C₂₃	#1370
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
25. C₂₃⋊C₈	C₁₆	semidirect
44. C₂₃	#1370	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
C₂₃×C₄⋊C₁₆	2
#1370	23