Subgroups — #43252996

Order 28544.

50 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	#28383410	C₂	1	1	yes	maximal, characteristic
3	#368×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297240	—	2	1	no	—
5	C₂₂₃⋊C₃₂	C₄	1	2	yes	—
6	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
7	#108×C₂₂₃	—	2	1	no	—
8	#16813154	—	4	1	no	—
9	M₁₆×C₂₂₃	—	4	1	no	—
10	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
11	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₂₂₃	Q₈	1	1	yes	characteristic
13	C₂₂₃⋊C₈	—	8	1	no	—
14	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
15	C₈×C₂₂₃	—	8	1	no	—
16	C₈×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₂₂₃	Q₁₆	1	1	yes	characteristic
18	C₂²×C₂₂₃	Q₃₂	1	1	yes	characteristic
19	C₄×C₂₂₃	#107	1	1	yes	characteristic
20	C₄×C₂₂₃	D₁₆	1	1	yes	characteristic
21	C₂×C₂₂₃	—	2	1	no	—
22	C₂×C₂₂₃	#366	1	1	yes	characteristic
23	C₂₂₃	#1425	1	1	yes	characteristic, complemented, Sylow
24	#1425	—	223	1	no	maximal, Sylow
25	C₂×C₃₂	—	223	1	no	—
26	#368	—	223	1	no	—
27	#368	D₂₂₃	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
29	#108	—	2	1	no	—
30	#108	—	446	1	no	—
31	C₃₂	—	223	2	no	—
32	M₁₆	—	4	1	no	—
33	M₁₆	—	892	1	no	—
34	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
35	C₁₆	C₄×D₂₂₃	1	1	yes	characteristic
36	C₁₆	Dic₄₄₆	1	1	yes	characteristic
37	C₂×C₄	#16813155	1	1	yes	characteristic
38	C₈	—	8	1	no	—
39	C₈	—	1784	1	no	—
40	C₈	Dic₂₂₃⋊C₄	1	1	yes	characteristic
41	C₈	#16813160	1	1	yes	characteristic
42	C₂²	#20297250	1	1	yes	characteristic
43	C₄	#20297237	1	1	yes	characteristic
44	C₄	#20297245	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#28383405	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20297245
commutator	C₁₆×C₂₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	#368×C₂₂₃	C₂
socle	C₂×C₂₂₃	#366
prime radical	C₂²×C₂₂₃	Q₃₂
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	#368	D₂₂₃
hypocenter	C₂₂₃	#1425
LCS	G ▷ C₁₆×C₂₂₃ ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
23. C₂₂₃	#1425	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
#28383410	2
#368×C₂₂₃	2
#1425	223