Subgroups — #19212112

Order 5888.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15671601	C₂	1	1	yes	maximal, characteristic
2	#15671599	C₂	1	1	yes	maximal, characteristic
3	C₂₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	#13092247	—	2	1	no	—
5	#13092243	—	2	1	no	—
6	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
7	C₂₃×C₆₄	C₄	1	2	yes	complemented
8	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
9	#10625001	—	4	1	no	—
10	#10625000	—	4	1	no	—
11	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
12	C₂₃×C₃₂	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
14	C₄⋊Dic₂₃	—	8	1	no	—
15	#67267	—	8	1	no	—
16	C₁₆×C₂₃	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₂₃	Q₁₆	1	1	yes	characteristic
18	#7548	—	23	1	no	maximal, Sylow
19	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
20	C₂×Dic₂₃	—	16	1	no	—
21	C₈×C₂₃	#107	1	1	yes	characteristic
22	C₈×C₂₃	Q₃₂	1	1	yes	characteristic
23	C₂₃⋊C₈	—	16	1	no	—
24	M₁₂₈	D₂₃	1	1	yes	characteristic
25	#1425	—	23	1	no	—
26	#1424	—	23	1	no	—
27	C₂²×C₂₃	SD₆₄	1	1	yes	characteristic
28	C₄×C₂₃	#366	1	1	yes	characteristic
29	C₄×C₂₃	SD₆₄	1	1	yes	characteristic
30	Dic₂₃	—	32	1	no	—
31	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
32	#368	—	46	1	no	—
33	#366	—	46	1	no	—
34	C₆₄	Dic₂₃	1	2	yes	complemented
35	C₂×C₂₃	—	2	1	no	—
36	C₂×C₂₃	#1424	1	1	yes	characteristic
37	C₂×C₁₆	D₉₂	1	1	yes	characteristic
38	#108	—	92	1	no	—
39	#107	—	92	1	no	—
40	C₃₂	Dic₄₆	1	1	yes	characteristic
41	C₃₂	C₂×Dic₂₃	1	1	yes	characteristic
42	C₂₃	#7548	1	1	yes	characteristic, complemented, Sylow
43	M₁₆	—	184	1	no	—
44	C₂×C₈	D₁₈₄	1	1	yes	characteristic
45	C₄⋊C₄	—	184	1	no	—
46	C₁₆	Dic₉₂	1	1	yes	characteristic
47	C₁₆	C₄⋊Dic₂₃	1	1	yes	characteristic
48	C₂×C₄	—	368	1	no	—
49	C₂×C₄	D₃₆₈	1	1	yes	characteristic
50	C₈	—	368	1	no	—
51	C₈	Dic₁₈₄	1	1	yes	characteristic
52	C₈	#10625000	1	1	yes	characteristic
53	C₂²	#13092256	1	1	yes	characteristic
54	C₄	—	736	1	no	—
55	C₄	#13092243	1	1	yes	characteristic
56	C₄	#13092256	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#15671599	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#15671599
commutator	C₂₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂₃×M₁₂₈	C₂
socle	C₂×C₂₃	#1424
hypercenter	M₁₂₈	D₂₃
hypocenter	C₂₃	#7548
LCS	G ▷ C₂₃×C₃₂ ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
7. C₂₃×C₆₄	C₄	semidirect
34. C₆₄	Dic₂₃	semidirect
42. C₂₃	#7548	semidirect