Subgroups — #19212013

Order 5888.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15671608	C₂	1	2	yes	maximal
2	C₂₃×#1397	C₂	1	1	yes	maximal, characteristic
3	C₂₃⋊C₆₄	C₄	1	4	yes	complemented
4	C₂×C₂₃×C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₃	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₂₃	C₈	1	1	yes	characteristic
10	C₂₃×C₃₂	C₈	1	2	yes	—
11	C₂₃×C₃₂	C₂×C₄	1	2	yes	—
12	C₂×C₈×C₂₃	C₁₆	1	1	yes	characteristic
13	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
14	C₄²×C₂₃	C₁₆	1	1	yes	characteristic
15	C₁₆×C₂₃	C₁₆	1	2	yes	—
16	C₁₆×C₂₃	C₄²	1	1	yes	characteristic
17	C₁₆×C₂₃	C₂×C₈	1	1	yes	characteristic
18	#7511	—	23	1	no	maximal, Sylow
19	C₂×C₄×C₂₃	—	2	1	no	—
20	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
21	C₈×C₂₃	—	2	1	no	—
22	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
23	C₈×C₂₃	C₂×C₁₆	1	1	yes	characteristic
24	M₁₂₈	—	23	2	no	—
25	#1397	D₂₃	1	1	yes	characteristic
26	C₂²×C₂₃	M₆₄	1	1	yes	characteristic
27	C₄×C₂₃	—	4	1	no	—
28	C₄×C₂₃	C₄×C₁₆	1	1	yes	characteristic
29	C₄×C₂₃	M₆₄	1	1	yes	characteristic
30	C₂×C₃₂	Dic₂₃	1	1	yes	characteristic
31	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
32	C₄×C₁₆	Dic₂₃	1	1	yes	characteristic
33	C₆₄	—	23	4	no	—
34	C₂×C₂₃	—	2	1	no	—
35	C₂×C₂₃	#1397	1	1	yes	characteristic
36	C₂×C₁₆	C₂₃⋊C₈	1	1	yes	characteristic
37	C₂×C₁₆	C₂×Dic₂₃	1	1	yes	characteristic
38	C₄×C₈	C₂₃⋊C₈	1	1	yes	characteristic
39	C₃₂	C₂₃⋊C₈	1	2	yes	—
40	C₃₂	C₄×D₂₃	1	2	yes	—
41	C₂₃	#7511	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₈	C₂₃⋊C₁₆	1	1	yes	characteristic
43	C₂×C₈	C₂×C₂₃⋊C₈	1	1	yes	characteristic
44	C₄²	C₂₃⋊C₁₆	1	1	yes	characteristic
45	C₁₆	C₂₃⋊C₁₆	1	2	yes	—
46	C₁₆	C₂×C₂₃⋊C₈	1	1	yes	characteristic
47	C₁₆	C₄×Dic₂₃	1	1	yes	characteristic
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	C₂×C₂₃⋊C₁₆	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	C₄×C₂₃⋊C₈	1	1	yes	characteristic
52	C₈	C₂×C₂₃⋊C₁₆	1	1	yes	characteristic
53	C₂²	#13092251	1	1	yes	characteristic
54	C₄	—	4	1	no	—
55	C₄	C₄×C₂₃⋊C₁₆	1	1	yes	characteristic
56	C₄	#13092251	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#15671531	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂₃⋊C₆₄	C₄	semidirect
41. C₂₃	#7511	semidirect