Subgroups — #43252859

Order 28544.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383331	C₂	1	2	yes	maximal
2	#346×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₂₃⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₂₃⋊C₈	—	4	2	no	—
8	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
10	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
11	C₁₆×C₂₂₃	—	2	1	no	—
12	C₁₆×C₂₂₃	C₈	1	2	yes	complemented
13	C₂₂₃⋊C₈	—	8	2	no	—
14	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	SD₁₆	1	2	yes	—
16	C₈×C₂₂₃	—	2	1	no	—
17	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
18	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
19	C₂²×C₂₂₃	#106	1	1	yes	characteristic
20	C₄×C₂₂₃	—	2	2	no	—
21	C₄×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₂₂₃	#108	1	1	yes	characteristic
23	C₂×C₂₂₃	#334	1	1	yes	characteristic
24	C₂×C₂₂₃	#365	1	2	yes	—
25	C₂₂₃	#1368	1	1	yes	characteristic, complemented, Sylow
26	#1368	—	223	1	no	maximal, Sylow
27	#346	D₂₂₃	1	1	yes	characteristic
28	#334	—	223	2	no	—
29	C₂×C₁₆	Dic₂₂₃	1	1	yes	characteristic
30	C₂×C₁₆	Dic₂₂₃	1	1	yes	characteristic
31	C₄⋊C₈	—	446	2	no	—
32	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
33	C₂×C₈	—	892	2	no	—
34	C₂×C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
35	C₂×C₈	Dic₄₄₆	1	1	yes	characteristic
36	C₄²	D₈₉₂	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	C₂₂₃⋊C₈	1	2	yes	complemented
39	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
40	C₂×C₄	#16813159	1	2	yes	—
41	C₈	—	2	1	no	—
42	C₈	—	1784	2	no	—
43	C₈	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
44	C₈	#16813154	1	1	yes	characteristic
45	C₂²	#20297236	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
48	C₄	#20297240	1	1	yes	characteristic
49	C₂	#28383331	1	1	yes	characteristic
50	C₂	#28383403	1	2	yes	—

	subgroup	G/H
center	C₂²	#20297236
commutator	C₈×C₂₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	#346×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	#106
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	#346	D₂₂₃
hypocenter	C₂₂₃	#1368
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₂₂₃	C₈	semidirect
25. C₂₂₃	#1368	semidirect
38. C₁₆	C₂₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#28383331	2
#346×C₂₂₃	2
#1368	223