Subgroups — #15671463

Order 2944.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092168	C₂	1	2	yes	maximal
2	C₂₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₂₃	C₄	1	1	yes	characteristic
4	C₂×C₁₆×C₂₃	C₄	1	1	yes	characteristic
5	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
7	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
8	C₂×C₈×C₂₃	Q₈	1	1	yes	characteristic
9	C₄²×C₂₃	D₄	1	1	yes	characteristic
10	C₂×C₂₃⋊C₈	—	4	2	no	—
11	C₁₆×C₂₃	—	2	1	no	—
12	C₁₆×C₂₃	C₈	1	2	yes	complemented
13	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
15	C₈×C₂₃	—	2	1	no	—
16	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
17	C₈×C₂₃	M₁₆	1	1	yes	characteristic
18	C₂₃⋊C₈	—	8	2	no	—
19	#1368	—	23	1	no	maximal, Sylow
20	C₂²×C₂₃	#106	1	1	yes	characteristic
21	C₄×C₂₃	—	2	2	no	—
22	C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₃	#108	1	1	yes	characteristic
24	#346	D₂₃	1	1	yes	characteristic
25	#334	—	23	2	no	—
26	C₂×C₂₃	#334	1	1	yes	characteristic
27	C₂×C₂₃	#365	1	2	yes	—
28	C₂×C₁₆	Dic₂₃	1	1	yes	characteristic
29	C₂×C₁₆	Dic₂₃	1	1	yes	characteristic
30	C₄⋊C₈	—	46	2	no	—
31	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
32	C₂₃	#1368	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₈	—	92	2	no	—
34	C₂×C₈	Dic₄₆	1	1	yes	characteristic
35	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
36	C₄²	D₉₂	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	C₂₃⋊C₈	1	2	yes	complemented
39	C₂×C₄	#67263	1	2	yes	—
40	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
41	C₈	—	2	1	no	—
42	C₈	—	184	2	no	—
43	C₈	C₂×C₂₃⋊C₈	1	1	yes	characteristic
44	C₈	#67267	1	1	yes	characteristic
45	C₂²	#10624999	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄⋊(C₂₃⋊C₈)	1	1	yes	characteristic
48	C₄	#10625001	1	1	yes	characteristic
49	C₂	#13092168	1	1	yes	characteristic
50	C₂	#13092240	1	2	yes	—

	subgroup	G/H
center	C₂²	#10624999
commutator	C₈×C₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#346	C₂
socle, prime radical	C₂²×C₂₃	#106
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#346	D₂₃
hypocenter	C₂₃	#1368
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₂₃	C₈	semidirect
32. C₂₃	#1368	semidirect
38. 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
#13092168	2
C₂₃×#346	2
#1368	23