Subgroups — #43252860

Order 28544.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383401	C₂	1	1	yes	maximal, characteristic
2	#28383356	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297244	—	2	1	no	—
5	C₂×C₂₂₃⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₂₂₃⋊C₁₆	C₄	1	1	yes	characteristic
7	M₃₂×C₂₂₃	—	2	1	no	—
8	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂₂₃⋊C₁₆	—	2	1	no	—
10	C₂₂₃⋊C₁₆	—	4	1	no	—
11	C₂₂₃⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
14	C₄²×C₂₂₃	Q₈	1	1	yes	characteristic
15	C₁₆×C₂₂₃	—	4	1	no	—
16	C₂×C₄×C₂₂₃	—	2	1	no	—
17	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
20	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
21	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
22	C₂²×C₂₂₃	#108	1	1	yes	characteristic
23	C₄×C₂₂₃	—	4	1	no	—
24	C₄×C₂₂₃	C₄⋊C₈	1	1	yes	characteristic
25	C₄×C₂₂₃	#106	1	1	yes	characteristic
26	C₂×C₂₂₃	—	2	1	no	—
27	C₂×C₂₂₃	#334	1	1	yes	characteristic
28	C₂₂₃	#1369	1	1	yes	characteristic, complemented, Sylow
29	#1369	—	223	1	no	maximal, Sylow
30	#364	—	223	1	no	—
31	#364	D₂₂₃	1	1	yes	characteristic
32	#346	—	223	1	no	—
33	M₃₂	—	2	1	no	—
34	M₃₂	—	446	1	no	—
35	C₂×C₁₆	—	223	1	no	—
36	C₂×C₁₆	—	223	1	no	—
37	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
38	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
40	C₄²	Dic₄₄₆	1	1	yes	characteristic
41	C₁₆	—	4	1	no	—
42	C₁₆	—	892	1	no	—
43	C₁₆	—	446	1	no	—
44	C₁₆	—	223	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
47	C₈	C₈×D₂₂₃	1	1	yes	characteristic
48	C₈	#16813153	1	1	yes	characteristic
49	C₈	#16813157	1	1	yes	characteristic
50	C₈	#16813158	1	1	yes	characteristic
51	C₂²	#20297239	1	1	yes	characteristic
52	C₄	—	4	1	no	—
53	C₄	Dic₂₂₃⋊C₈	1	1	yes	characteristic
54	C₄	#20297235	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	#28383330	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20297235
commutator	C₈×C₂₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	#364×C₂₂₃	C₂
socle	C₂×C₂₂₃	#334
prime radical	C₂²×C₂₂₃	#108
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	#364	D₂₂₃
hypocenter	C₂₂₃	#1369
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
28. C₂₂₃	#1369	semidirect

Maximal subgroups

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

subgroup	index
#28383401	2
#28383356	2
#364×C₂₂₃	2
#1369	223