Subgroups — #19212012

Order 5888.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15671608	C₂	1	1	yes	maximal, characteristic
2	#15671530	C₂	1	1	yes	maximal, characteristic
3	C₂₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃⋊C₃₂	C₄	1	1	yes	characteristic
5	C₁₆×Dic₂₃	C₄	1	1	yes	characteristic
6	C₂₃⋊C₆₄	C₄	1	2	yes	complemented
7	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
8	C₂₃×C₆₄	C₄	1	2	yes	complemented
9	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
10	C₈×Dic₂₃	C₈	1	1	yes	characteristic
11	C₂×C₂₃⋊C₁₆	C₈	1	1	yes	characteristic
12	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
13	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
14	C₂₃⋊C₃₂	C₈	1	2	yes	—
15	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
16	C₄×Dic₂₃	C₁₆	1	1	yes	characteristic
17	C₂×C₂₃⋊C₈	C₁₆	1	1	yes	characteristic
18	C₁₆×C₂₃	C₄²	1	1	yes	characteristic
19	C₁₆×C₂₃	C₂×C₈	1	1	yes	characteristic
20	C₂₃⋊C₁₆	C₁₆	1	2	yes	—
21	#7511	—	23	1	no	maximal, Sylow
22	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
23	C₂×Dic₂₃	—	2	1	no	—
24	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
25	C₈×C₂₃	C₂×C₁₆	1	1	yes	characteristic
26	C₂₃⋊C₈	—	2	1	no	—
27	M₁₂₈	—	23	1	no	—
28	M₁₂₈	D₂₃	1	1	yes	characteristic
29	#1397	—	23	1	no	—
30	C₂²×C₂₃	M₆₄	1	1	yes	characteristic
31	C₄×C₂₃	C₄×C₁₆	1	1	yes	characteristic
32	C₄×C₂₃	M₆₄	1	1	yes	characteristic
33	Dic₂₃	—	4	1	no	—
34	C₂×C₃₂	—	23	1	no	—
35	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
36	C₄×C₁₆	—	23	1	no	—
37	C₆₄	—	23	2	no	—
38	C₆₄	Dic₂₃	1	2	yes	complemented
39	C₂×C₂₃	—	2	1	no	—
40	C₂×C₂₃	#1397	1	1	yes	characteristic
41	C₂×C₁₆	—	23	1	no	—
42	C₂×C₁₆	C₄×D₂₃	1	1	yes	characteristic
43	C₄×C₈	—	23	1	no	—
44	C₃₂	—	23	2	no	—
45	C₃₂	C₄×D₂₃	1	1	yes	characteristic
46	C₃₂	C₂×Dic₂₃	1	1	yes	characteristic
47	C₂₃	#7511	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₈	—	23	1	no	—
49	C₂×C₈	C₈×D₂₃	1	1	yes	characteristic
50	C₄²	—	23	1	no	—
51	C₁₆	—	23	2	no	—
52	C₁₆	C₈×D₂₃	1	1	yes	characteristic
53	C₁₆	C₄×Dic₂₃	1	1	yes	characteristic
54	C₂×C₄	—	46	1	no	—
55	C₂×C₄	C₁₆×D₂₃	1	1	yes	characteristic
56	C₈	—	46	1	no	—
57	C₈	C₁₆×D₂₃	1	1	yes	characteristic
58	C₈	C₈×Dic₂₃	1	1	yes	characteristic
59	C₂²	#13092250	1	1	yes	characteristic
60	C₄	—	92	1	no	—
61	C₄	C₁₆×Dic₂₃	1	1	yes	characteristic
62	C₄	#13092250	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#15671530	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₂₃
commutator	C₄×C₂₃	C₄×C₁₆
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂₃×M₁₂₈	C₂
socle	C₂×C₂₃	#1397
hypercenter	M₁₂₈	D₂₃
hypocenter	C₂₃	#7511
LCS	G ▷ C₄×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₁₆ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
6. C₂₃⋊C₆₄	C₄	semidirect
8. C₂₃×C₆₄	C₄	semidirect
38. C₆₄	Dic₂₃	semidirect
47. C₂₃	#7511	semidirect