Subgroups — #43252989

Order 28544.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₂₃⋊C₃₂	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₂₃⋊C₃₂	—	2	2	no	—
4	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂×C₈×C₂₂₃	C₈	1	1	yes	characteristic
8	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₂₂₃	C₈	1	1	yes	characteristic
10	C₁₆×C₂₂₃	—	2	1	no	—
11	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
12	C₁₆×C₂₂₃	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	C₁₆	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	C₁₆	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
16	C₈×C₂₂₃	—	2	1	no	—
17	C₈×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
19	C₂²×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
20	C₄×C₂₂₃	—	2	1	no	—
21	C₄×C₂₂₃	C₃₂	1	2	yes	complemented
22	C₄×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₂₃	M₃₂	1	1	yes	characteristic
24	C₂×C₂₂₃	C₄⋊C₁₆	1	1	yes	characteristic
25	C₂×C₂₂₃	C₂×C₃₂	1	1	yes	characteristic
26	C₂×C₂₂₃	M₆₄	1	1	yes	characteristic
27	C₂₂₃	C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₄⋊C₃₂	—	223	1	no	maximal, Sylow
29	C₂×C₃₂	—	223	2	no	—
30	C₄×C₁₆	D₂₂₃	1	1	yes	characteristic
31	C₂×C₁₆	Dic₂₂₃	1	1	yes	characteristic
32	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
33	C₄×C₈	Dic₂₂₃	1	1	yes	characteristic
34	C₃₂	—	446	2	no	—
35	C₂×C₈	C₂₂₃⋊C₈	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
37	C₄²	C₂₂₃⋊C₈	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	C₁₆	D₈₉₂	1	1	yes	characteristic
40	C₁₆	Dic₄₄₆	1	1	yes	characteristic
41	C₂×C₄	C₂₂₃⋊C₁₆	1	1	yes	characteristic
42	C₂×C₄	C₂₂₃⋊C₁₆	1	1	yes	characteristic
43	C₂×C₄	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	C₄⋊Dic₂₂₃	1	1	yes	characteristic
46	C₈	#16813154	1	1	yes	characteristic
47	C₂²	C₂×C₂₂₃⋊C₁₆	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	C₂₂₃⋊C₃₂	1	2	yes	complemented
50	C₄	C₄⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
51	C₄	#20297244	1	1	yes	characteristic
52	C₂	C₄⋊(C₂₂₃⋊C₁₆)	1	1	yes	characteristic
53	C₂	C₂×C₂₂₃⋊C₃₂	1	1	yes	characteristic
54	C₂	#28383414	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
21. C₄×C₂₂₃	C₃₂	semidirect
27. C₂₂₃	C₄⋊C₃₂	semidirect
49. C₄	C₂₂₃⋊C₃₂	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
C₄×C₁₆×C₂₂₃	2
C₄⋊C₃₂	223