Subgroups — #15671462

Order 2944.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092193	C₂	1	1	yes	maximal, characteristic
2	#13092168	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂₃×C₄⋊C₈	—	2	1	no	—
5	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₂₃⋊C₁₆	C₄	1	1	yes	characteristic
8	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
9	C₂×C₈×C₂₃	—	4	1	no	—
10	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₂₃	Q₈	1	1	yes	characteristic
12	C₄²×C₂₃	D₄	1	1	yes	characteristic
13	C₂×C₂₃⋊C₈	—	4	1	no	—
14	C₂₃⋊C₁₆	—	2	1	no	—
15	C₂₃⋊C₁₆	C₈	1	2	yes	complemented
16	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
18	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
19	C₈×C₂₃	—	2	1	no	—
20	C₈×C₂₃	—	8	1	no	—
21	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
22	C₈×C₂₃	M₁₆	1	1	yes	characteristic
23	C₂₃⋊C₈	—	8	1	no	—
24	#1368	—	23	1	no	maximal, Sylow
25	C₂²×C₂₃	#106	1	1	yes	characteristic
26	C₄×C₂₃	—	2	1	no	—
27	C₄×C₂₃	—	2	1	no	—
28	C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
29	C₄×C₂₃	#108	1	1	yes	characteristic
30	#346	—	23	1	no	—
31	#334	—	23	1	no	—
32	#334	D₂₃	1	1	yes	characteristic
33	C₂×C₂₃	#334	1	1	yes	characteristic
34	C₂×C₂₃	#365	1	1	yes	characteristic
35	C₂×C₂₃	#365	1	1	yes	characteristic
36	C₂×C₁₆	—	23	1	no	—
37	C₂×C₁₆	—	23	1	no	—
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	—	46	1	no	—
40	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
41	C₂₃	#1368	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	92	1	no	—
44	C₂×C₈	Dic₄₆	1	1	yes	characteristic
45	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
46	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
47	C₁₆	—	23	2	no	—
48	C₁₆	—	46	1	no	—
49	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
50	C₂×C₄	#67273	1	1	yes	characteristic
51	C₂×C₄	#67274	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	8	1	no	—
54	C₈	—	184	1	no	—
55	C₈	C₈×D₂₃	1	1	yes	characteristic
56	C₈	#67262	1	1	yes	characteristic
57	C₂²	#10624990	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	Dic₂₃⋊C₈	1	1	yes	characteristic
61	C₄	#10625004	1	1	yes	characteristic
62	C₂	#13092167	1	1	yes	characteristic
63	C₂	#13092239	1	1	yes	characteristic
64	C₂	#13092241	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10624990
commutator	C₈×C₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#334	C₂
socle, prime radical	C₂²×C₂₃	#106
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#334	D₂₃
hypocenter	C₂₃	#1368
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
15. C₂₃⋊C₁₆	C₈	semidirect
41. C₂₃	#1368	semidirect

Maximal subgroups

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

subgroup	index
#13092193	2
#13092168	2
C₂₃×#334	2
#1368	23