Subgroups — #15671467

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	2	yes	maximal
2	C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
4	C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
5	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃⋊C₁₆	—	2	2	no	—
7	C₂×C₈×C₂₃	—	2	1	no	—
8	C₂×C₈×C₂₃	C₈	1	1	yes	characteristic
9	C₂×C₈×C₂₃	C₈	1	1	yes	characteristic
10	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
11	C₂×C₈×C₂₃	Q₈	1	1	yes	characteristic
12	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂₃⋊C₁₆	—	4	2	no	—
14	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₂₃	M₁₆	1	1	yes	characteristic
17	C₈×C₂₃	—	2	2	no	—
18	C₈×C₂₃	—	2	1	no	—
19	C₈×C₂₃	—	2	1	no	—
20	C₈×C₂₃	C₁₆	1	2	yes	complemented
21	C₈×C₂₃	D₈	1	1	yes	characteristic
22	C₈×C₂₃	Q₁₆	1	1	yes	characteristic
23	#1370	—	23	1	no	maximal, Sylow
24	C₂²×C₂₃	C₄⋊C₈	1	1	yes	characteristic
25	C₄×C₂₃	—	2	1	no	—
26	C₄×C₂₃	#107	1	1	yes	characteristic
27	C₄×C₂₃	#108	1	1	yes	characteristic
28	C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
29	C₄×C₂₃	M₃₂	1	1	yes	characteristic
30	C₄⋊C₁₆	—	23	2	no	—
31	C₈²	D₂₃	1	1	yes	characteristic
32	C₂×C₂₃	#335	1	1	yes	characteristic
33	C₂×C₂₃	C₄⋊C₁₆	1	1	yes	characteristic
34	C₂×C₂₃	#364	1	1	yes	characteristic
35	C₂×C₁₆	—	46	2	no	—
36	C₄×C₈	Dic₂₃	1	1	yes	characteristic
37	C₄×C₈	Dic₂₃	1	1	yes	characteristic
38	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
39	C₂₃	#1370	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	C₂₃⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₂₃⋊C₈	1	1	yes	characteristic
43	C₂×C₈	Dic₄₆	1	1	yes	characteristic
44	C₂×C₈	D₉₂	1	1	yes	characteristic
45	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
46	C₁₆	—	92	2	no	—
47	C₂×C₄	C₂×C₂₃⋊C₈	1	1	yes	characteristic
48	C₂×C₄	#67267	1	1	yes	characteristic
49	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	2	1	no	—
52	C₈	—	2	1	no	—
53	C₈	C₂₃⋊C₁₆	1	2	yes	complemented
54	C₈	D₁₈₄	1	1	yes	characteristic
55	C₈	Dic₉₂	1	1	yes	characteristic
56	C₂²	C₄⋊(C₂₃⋊C₈)	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	C₂×C₂₃⋊C₁₆	1	1	yes	characteristic
59	C₄	#10624994	1	1	yes	characteristic
60	C₄	#10625000	1	1	yes	characteristic
61	C₄	#10625001	1	1	yes	characteristic
62	C₂	#13092170	1	1	yes	characteristic
63	C₂	C₄⋊(C₂₃⋊C₁₆)	1	1	yes	characteristic
64	C₂	#13092238	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₉₂
commutator	C₄×C₂₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₈²×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	C₄⋊C₈
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	C₈²	D₂₃
hypocenter	C₂₃	#1370
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₂₃	C₁₆	semidirect
39. C₂₃	#1370	semidirect
53. C₈	C₂₃⋊C₁₆	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
C₈²×C₂₃	2
#1370	23