Subgroups — #43252998

Order 28544.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383414	C₂	1	1	yes	maximal, characteristic
2	#28383410	C₂	1	1	yes	maximal, characteristic
3	#367×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297240	—	2	1	no	—
5	C₂₂₃⋊C₃₂	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
7	#107×C₂₂₃	—	2	1	no	—
8	#16813154	—	4	1	no	—
9	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
10	C₄⋊C₄×C₂₂₃	—	4	1	no	—
11	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₂₂₃	Q₈	1	1	yes	characteristic
13	C₂₂₃⋊C₈	—	8	1	no	—
14	C₂×C₄×C₂₂₃	—	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	—	16	1	no	—
20	C₄×C₂₂₃	#107	1	1	yes	characteristic
21	C₄×C₂₂₃	SD₃₂	1	1	yes	characteristic
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	—	223	1	no	—
27	#368	—	223	1	no	—
28	#367	D₂₂₃	1	1	yes	characteristic
29	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
30	#108	—	446	1	no	—
31	#107	—	2	1	no	—
32	C₃₂	—	223	2	no	—
33	M₁₆	—	892	1	no	—
34	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
35	C₄⋊C₄	—	4	1	no	—
36	C₁₆	C₄×D₂₂₃	1	1	yes	characteristic
37	C₁₆	Dic₄₄₆	1	1	yes	characteristic
38	C₂×C₄	—	8	1	no	—
39	C₂×C₄	#16813155	1	1	yes	characteristic
40	C₈	—	1784	1	no	—
41	C₈	Dic₂₂₃⋊C₄	1	1	yes	characteristic
42	C₈	#16813160	1	1	yes	characteristic
43	C₂²	#20297248	1	1	yes	characteristic
44	C₄	—	16	1	no	—
45	C₄	#20297237	1	1	yes	characteristic
46	C₄	#20297247	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#28383407	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383407
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	#367×C₂₂₃	C₂
socle	C₂×C₂₂₃	#367
prime radical	C₂²×C₂₂₃	SD₃₂
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	#367	D₂₂₃
hypocenter	C₂₂₃	#1426
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #367

Complements

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

N	H ≅ G/N	type
5. C₂₂₃⋊C₃₂	C₄	semidirect
24. C₂₂₃	#1426	semidirect

Maximal subgroups

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

subgroup	index
#28383414	2
#28383410	2
#367×C₂₂₃	2
#1426	223