Subgroups — #19212106

Order 5888.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15671597	C₂	1	2	yes	maximal
2	C₂×C₂₃×C₆₄	C₂	1	1	yes	maximal, characteristic
3	#13092243	—	2	2	no	—
4	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
5	C₂₃×C₆₄	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
7	#10625000	—	4	2	no	—
8	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
9	C₂₃×C₃₂	Q₈	1	1	yes	characteristic
10	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
11	C₄⋊Dic₂₃	—	8	2	no	—
12	C₁₆×C₂₃	C₄⋊C₄	1	1	yes	characteristic
13	C₁₆×C₂₃	Q₁₆	1	1	yes	characteristic
14	#7545	—	23	1	no	maximal, Sylow
15	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
16	C₂×Dic₂₃	—	16	2	no	—
17	C₈×C₂₃	#107	1	1	yes	characteristic
18	C₈×C₂₃	Q₃₂	1	1	yes	characteristic
19	C₂×C₆₄	D₂₃	1	1	yes	characteristic
20	#1423	—	23	2	no	—
21	C₂²×C₂₃	D₃₂	1	1	yes	characteristic
22	C₄×C₂₃	#366	1	1	yes	characteristic
23	C₄×C₂₃	Q₆₄	1	1	yes	characteristic
24	Dic₂₃	—	32	2	no	—
25	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
26	#366	—	46	2	no	—
27	C₆₄	Dic₂₃	1	2	yes	complemented
28	C₂×C₂₃	#1423	1	1	yes	characteristic
29	C₂×C₂₃	D₆₄	1	1	yes	characteristic
30	C₂×C₂₃	Q₁₂₈	1	1	yes	characteristic
31	C₂×C₁₆	D₉₂	1	1	yes	characteristic
32	#107	—	92	2	no	—
33	C₃₂	Dic₄₆	1	1	yes	characteristic
34	C₃₂	C₂×Dic₂₃	1	1	yes	characteristic
35	C₂₃	#7545	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₈	D₁₈₄	1	1	yes	characteristic
37	C₄⋊C₄	—	184	2	no	—
38	C₁₆	Dic₉₂	1	1	yes	characteristic
39	C₁₆	C₄⋊Dic₂₃	1	1	yes	characteristic
40	C₂×C₄	—	368	2	no	—
41	C₂×C₄	D₃₆₈	1	1	yes	characteristic
42	C₈	Dic₁₈₄	1	1	yes	characteristic
43	C₈	#10625000	1	1	yes	characteristic
44	C₂²	D₇₃₆	1	1	yes	characteristic
45	C₄	—	736	2	no	—
46	C₄	#13092243	1	1	yes	characteristic
47	C₄	Dic₃₆₈	1	1	yes	characteristic
48	C₂	#15671597	1	1	yes	characteristic
49	C₂	D₁₄₇₂	1	1	yes	characteristic
50	C₂	Dic₇₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₇₃₆
commutator	C₂₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂×C₂₃×C₆₄	C₂
socle	C₂²×C₂₃	D₃₂
hypercenter	C₂×C₆₄	D₂₃
hypocenter	C₂₃	#7545
LCS	G ▷ C₂₃×C₃₂ ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
5. C₂₃×C₆₄	C₄	semidirect
27. C₆₄	Dic₂₃	semidirect
35. C₂₃	#7545	semidirect