Subgroups — #43252994

Order 28544.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#28383406	C₂	1	1	yes	maximal, characteristic
3	#366×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297238	—	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	C₄⋊Dic₂₂₃	—	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	C₂×Dic₂₂₃	—	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	D₁₆	1	1	yes	characteristic
19	C₄×C₂₂₃	—	16	1	no	—
20	C₄×C₂₂₃	#107	1	1	yes	characteristic
21	C₄×C₂₂₃	Q₃₂	1	1	yes	characteristic
22	Dic₂₂₃	—	16	1	no	—
23	C₂×C₂₂₃	#366	1	1	yes	characteristic
24	C₂×C₂₂₃	SD₆₄	1	1	yes	characteristic
25	C₂×C₂₂₃	SD₆₄	1	1	yes	characteristic
26	C₂₂₃	#1424	1	1	yes	characteristic, complemented, Sylow
27	#1424	—	223	1	no	maximal, Sylow
28	C₂×C₃₂	—	223	1	no	—
29	#366	—	223	1	no	—
30	#366	D₂₂₃	1	1	yes	characteristic
31	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	#107	—	446	1	no	—
34	C₃₂	—	223	2	no	—
35	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
36	C₄⋊C₄	—	4	1	no	—
37	C₄⋊C₄	—	892	1	no	—
38	C₁₆	C₄×D₂₂₃	1	1	yes	characteristic
39	C₁₆	Dic₄₄₆	1	1	yes	characteristic
40	C₂×C₄	—	8	1	no	—
41	C₂×C₄	—	1784	1	no	—
42	C₂×C₄	#16813155	1	1	yes	characteristic
43	C₈	Dic₂₂₃⋊C₄	1	1	yes	characteristic
44	C₈	#16813160	1	1	yes	characteristic
45	C₂²	#20297245	1	1	yes	characteristic
46	C₄	—	16	1	no	—
47	C₄	—	3568	1	no	—
48	C₄	#20297237	1	1	yes	characteristic
49	C₄	#20297250	1	1	yes	characteristic
50	C₂	#28383405	1	1	yes	characteristic
51	C₂	#28383417	1	1	yes	characteristic
52	C₂	#28383418	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20297245
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	#366×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	D₁₆
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	#366	D₂₂₃
hypocenter	C₂₂₃	#1424
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
5. C₂₂₃⋊C₃₂	C₄	semidirect
26. C₂₂₃	#1424	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₂₃⋊C₃₂	2
#28383406	2
#366×C₂₂₃	2
#1424	223