Subgroups — #43252999

Order 28544.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383410	C₂	1	1	yes	maximal, characteristic
2	#28383408	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297240	—	2	1	no	—
5	#20297238	—	2	1	no	—
6	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₃₂×C₂₂₃	C₄	1	2	yes	complemented
8	#16813154	—	4	1	no	—
9	C₄⋊Dic₂₂₃	—	4	1	no	—
10	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
11	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₂₂₃	Q₈	1	1	yes	characteristic
13	C₂×Dic₂₂₃	—	8	1	no	—
14	C₂₂₃⋊C₈	—	8	1	no	—
15	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
16	C₈×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₂₂₃	Q₁₆	1	1	yes	characteristic
18	C₂²×C₂₂₃	SD₃₂	1	1	yes	characteristic
19	C₄×C₂₂₃	#107	1	1	yes	characteristic
20	C₄×C₂₂₃	SD₃₂	1	1	yes	characteristic
21	Dic₂₂₃	—	16	1	no	—
22	C₂×C₂₂₃	—	2	1	no	—
23	C₂×C₂₂₃	#367	1	1	yes	characteristic
24	C₂₂₃	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	223	1	no	maximal, Sylow
26	M₆₄	D₂₂₃	1	1	yes	characteristic
27	#368	—	223	1	no	—
28	#367	—	223	1	no	—
29	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
30	#108	—	446	1	no	—
31	#107	—	446	1	no	—
32	C₃₂	Dic₂₂₃	1	2	yes	complemented
33	M₁₆	—	892	1	no	—
34	C₂×C₈	D₈₉₂	1	1	yes	characteristic
35	C₄⋊C₄	—	892	1	no	—
36	C₁₆	C₂×Dic₂₂₃	1	1	yes	characteristic
37	C₁₆	Dic₄₄₆	1	1	yes	characteristic
38	C₂×C₄	—	1784	1	no	—
39	C₂×C₄	D₁₇₈₄	1	1	yes	characteristic
40	C₈	—	1784	1	no	—
41	C₈	C₄⋊Dic₂₂₃	1	1	yes	characteristic
42	C₈	Dic₈₉₂	1	1	yes	characteristic
43	C₂²	#20297249	1	1	yes	characteristic
44	C₄	—	3568	1	no	—
45	C₄	#20297238	1	1	yes	characteristic
46	C₄	#20297249	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#28383408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383408
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	M₆₄×C₂₂₃	C₂
socle	C₂×C₂₂₃	#367
prime radical	C₂²×C₂₂₃	SD₃₂
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	M₆₄	D₂₂₃
hypocenter	C₂₂₃	#1426
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₂₂₃	C₄	semidirect
24. C₂₂₃	#1426	semidirect
32. C₃₂	Dic₂₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#28383410	2
#28383408	2
M₆₄×C₂₂₃	2
#1426	223