Subgroups — #28383396

Order 14272.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297244	C₂	1	1	yes	maximal, characteristic
2	#20297240	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#16813154	—	2	1	no	—
5	C₂₂₃⋊C₁₆	—	2	1	no	—
6	C₂×Q₈×C₂₂₃	—	2	1	no	—
7	Q₁₆×C₂₂₃	—	2	1	no	—
8	Q₁₆×C₂₂₃	C₄	1	2	yes	—
9	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂₂₃⋊C₈	—	4	1	no	—
11	Q₈×C₂₂₃	—	2	2	no	—
12	Q₈×C₂₂₃	—	4	1	no	—
13	C₂×C₄×C₂₂₃	—	4	1	no	—
14	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
15	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₂₂₃	SD₁₆	1	1	yes	characteristic
18	C₄×C₂₂₃	—	4	2	no	—
19	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₂₂₃	D₈	1	1	yes	characteristic
21	C₂×C₂₂₃	—	2	1	no	—
22	C₂×C₂₂₃	#102	1	1	yes	characteristic
23	C₂₂₃	#362	1	1	yes	characteristic, complemented, Sylow
24	#362	—	223	1	no	maximal, Sylow
25	C₂×Q₁₆	D₂₂₃	1	1	yes	characteristic
26	M₃₂	—	223	1	no	—
27	#108	—	223	1	no	—
28	C₂×Q₈	—	2	1	no	—
29	Q₁₆	—	2	1	no	—
30	Q₁₆	Dic₂₂₃	1	2	yes	—
31	M₁₆	—	446	1	no	—
32	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
33	C₁₆	—	446	1	no	—
34	Q₈	—	2	2	no	—
35	Q₈	—	4	1	no	—
36	C₂×C₄	—	4	1	no	—
37	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
38	C₈	—	892	1	no	—
39	C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
40	C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
41	C₂²	#16813158	1	1	yes	characteristic
42	C₄	—	4	2	no	—
43	C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
44	C₄	#16813155	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#20297225	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20297225
commutator	C₈×C₂₂₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₂₃
Fitting	C₂×Q₁₆×C₂₂₃	C₂
socle	C₂×C₂₂₃	#102
prime radical	C₂²×C₂₂₃	SD₁₆
cosocle	C₂×C₈×C₂₂₃	C₂²
hypercenter	C₂×Q₁₆	D₂₂₃
hypocenter	C₂₂₃	#362
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
23. C₂₂₃	#362	semidirect

Maximal subgroups

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

subgroup	index
#20297244	2
#20297240	2
C₂×Q₁₆×C₂₂₃	2
#362	223