Subgroups — #15671469

Order 2944.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092170	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₂₃	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	D₈	1	1	yes	characteristic
15	C₂×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
16	C₈×C₂₃	—	2	1	no	—
17	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
18	C₈×C₂₃	M₁₆	1	1	yes	characteristic
19	C₂₃⋊C₈	—	8	2	no	—
20	#1371	—	23	1	no	maximal, Sylow
21	C₂²×C₂₃	#107	1	1	yes	characteristic
22	C₄×C₂₃	—	2	1	no	—
23	C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₂₃	#108	1	1	yes	characteristic
25	C₄×C₂₃	D₁₆	1	1	yes	characteristic
26	C₄×C₂₃	Q₃₂	1	1	yes	characteristic
27	C₄×C₁₆	D₂₃	1	1	yes	characteristic
28	#335	—	23	2	no	—
29	C₂×C₂₃	#335	1	1	yes	characteristic
30	C₂×C₂₃	#366	1	1	yes	characteristic
31	C₂×C₂₃	#368	1	1	yes	characteristic
32	C₂×C₁₆	Dic₂₃	1	1	yes	characteristic
33	C₂×C₁₆	Dic₂₃	1	1	yes	characteristic
34	C₄⋊C₈	—	46	2	no	—
35	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
36	C₂₃	#1371	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₈	—	92	2	no	—
38	C₂×C₈	Dic₄₆	1	1	yes	characteristic
39	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
40	C₄²	D₉₂	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	C₂₃⋊C₈	1	2	yes	complemented
43	C₂×C₄	D₁₈₄	1	1	yes	characteristic
44	C₂×C₄	Dic₉₂	1	1	yes	characteristic
45	C₂×C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	184	2	no	—
48	C₈	C₂×C₂₃⋊C₈	1	1	yes	characteristic
49	C₈	#67267	1	1	yes	characteristic
50	C₂²	#10625000	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	D₃₆₈	1	1	yes	characteristic
53	C₄	Dic₁₈₄	1	1	yes	characteristic
54	C₄	C₄⋊(C₂₃⋊C₈)	1	1	yes	characteristic
55	C₄	#10625001	1	1	yes	characteristic
56	C₂	#13092170	1	1	yes	characteristic
57	C₂	#13092243	1	1	yes	characteristic
58	C₂	#13092247	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₈₄
commutator	C₈×C₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₄×C₁₆×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	#107
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	C₄×C₁₆	D₂₃
hypocenter	C₂₃	#1371
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ 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
36. C₂₃	#1371	semidirect
42. 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
#13092170	2
C₄×C₁₆×C₂₃	2
#1371	23