Subgroups — #43252995

Order 28544.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383406	C₂	1	2	yes	maximal
2	C₂×C₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297238	—	2	2	no	—
4	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
5	C₃₂×C₂₂₃	C₄	1	2	yes	complemented
6	C₄⋊Dic₂₂₃	—	4	2	no	—
7	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
8	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₂₂₃	Q₈	1	1	yes	characteristic
10	C₂×Dic₂₂₃	—	8	2	no	—
11	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
12	C₈×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₂₂₃	Q₁₆	1	1	yes	characteristic
14	C₂²×C₂₂₃	D₁₆	1	1	yes	characteristic
15	C₄×C₂₂₃	#107	1	1	yes	characteristic
16	C₄×C₂₂₃	Q₃₂	1	1	yes	characteristic
17	Dic₂₂₃	—	16	2	no	—
18	C₂×C₂₂₃	#366	1	1	yes	characteristic
19	C₂×C₂₂₃	SD₆₄	1	2	yes	—
20	C₂₂₃	#1424	1	1	yes	characteristic, complemented, Sylow
21	#1424	—	223	1	no	maximal, Sylow
22	C₂×C₃₂	D₂₂₃	1	1	yes	characteristic
23	#366	—	223	2	no	—
24	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
25	#107	—	446	2	no	—
26	C₃₂	Dic₂₂₃	1	2	yes	complemented
27	C₂×C₈	D₈₉₂	1	1	yes	characteristic
28	C₄⋊C₄	—	892	2	no	—
29	C₁₆	C₂×Dic₂₂₃	1	1	yes	characteristic
30	C₁₆	Dic₄₄₆	1	1	yes	characteristic
31	C₂×C₄	—	1784	2	no	—
32	C₂×C₄	D₁₇₈₄	1	1	yes	characteristic
33	C₈	C₄⋊Dic₂₂₃	1	1	yes	characteristic
34	C₈	Dic₈₉₂	1	1	yes	characteristic
35	C₂²	D₃₅₆₈	1	1	yes	characteristic
36	C₄	—	3568	2	no	—
37	C₄	#20297238	1	1	yes	characteristic
38	C₄	Dic₁₇₈₄	1	1	yes	characteristic
39	C₂	#28383406	1	1	yes	characteristic
40	C₂	#28383419	1	2	yes	—

	subgroup	G/H
center	C₂²	D₃₅₆₈
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	C₂×C₃₂×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	D₁₆
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	C₂×C₃₂	D₂₂₃
hypocenter	C₂₂₃	#1424
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₂₂₃	C₄	semidirect
20. C₂₂₃	#1424	semidirect
26. C₃₂	Dic₂₂₃	semidirect

Maximal subgroups

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

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