Subgroups — #43252853

Order 28544.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383982	C₂	1	1	yes	maximal, characteristic
2	#28383401	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297296	C₄	1	2	yes	—
5	#20297244	—	2	1	no	—
6	#20297238	—	2	1	no	—
7	#20297236	—	2	1	no	—
8	M₃₂×C₂₂₃	—	2	1	no	—
9	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₂₃	—	2	2	no	—
11	Dic₂₂₃⋊C₄	—	4	2	no	—
12	C₂₂₃⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
15	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₂₃	—	4	1	no	—
17	C₂×Dic₂₂₃	—	4	2	no	—
18	C₂×C₄×C₂₂₃	—	2	1	no	—
19	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
21	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
22	C₈×C₂₂₃	Q₁₆	1	1	yes	characteristic
23	C₈×C₂₂₃	Q₁₆	1	1	yes	characteristic
24	C₂²×C₂₂₃	#100	1	1	yes	characteristic
25	C₄×C₂₂₃	—	4	1	no	—
26	C₄×C₂₂₃	#103	1	1	yes	characteristic
27	C₄×C₂₂₃	#103	1	1	yes	characteristic
28	Dic₂₂₃	—	8	2	no	—
29	C₂×C₂₂₃	—	2	1	no	—
30	C₂×C₂₂₃	#333	1	1	yes	characteristic
31	C₂₂₃	#1366	1	1	yes	characteristic, complemented, Sylow
32	#1366	—	223	1	no	maximal, Sylow
33	#499	—	223	1	no	—
34	#364	—	223	1	no	—
35	#364	D₂₂₃	1	1	yes	characteristic
36	#125	—	223	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	446	1	no	—
39	#107	—	446	1	no	—
40	#106	—	446	1	no	—
41	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
43	C₂×C₈	D₈₉₂	1	1	yes	characteristic
44	C₄⋊C₄	—	892	2	no	—
45	C₄⋊C₄	—	446	2	no	—
46	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	892	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	892	2	no	—
51	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
52	C₈	#16813158	1	1	yes	characteristic
53	C₈	#16813159	1	1	yes	characteristic
54	C₈	#16813160	1	1	yes	characteristic
55	C₈	Dic₈₉₂	1	1	yes	characteristic
56	C₂²	#20297220	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	1784	2	no	—
59	C₄	#20297227	1	1	yes	characteristic
60	C₄	#20297229	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#28383328	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383328
commutator	C₄²×C₂₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	#364×C₂₂₃	C₂
socle	C₂×C₂₂₃	#333
prime radical	C₂²×C₂₂₃	#100
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	#364	D₂₂₃
hypocenter	C₂₂₃	#1366
LCS	G ▷ C₄²×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
31. C₂₂₃	#1366	semidirect

Maximal subgroups

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

subgroup	index
#28383982	2
#28383401	2
#364×C₂₂₃	2
#1366	223