Subgroups — #28383985

Order 14272.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297304	C₂	1	2	yes	maximal
2	#20297238	C₂	1	4	yes	maximal
3	C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₄₆	—	2	2	no	—
5	C₄⋊Dic₂₂₃	—	2	2	no	—
6	C₄⋊Dic₂₂₃	C₂²	1	4	yes	—
7	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
8	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
9	Dic₄₄₆	—	4	4	no	—
10	C₂×Dic₂₂₃	—	2	4	no	—
11	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
12	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
14	C₈×C₂₂₃	Q₈	1	4	yes	complemented
15	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
16	C₄×C₂₂₃	D₈	1	2	yes	—
17	C₄×C₂₂₃	Q₁₆	1	2	yes	—
18	C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
19	Dic₂₂₃	—	4	4	no	—
20	C₂×C₂₂₃	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
22	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
23	C₂₂₃	#500	1	1	yes	characteristic, complemented, Sylow
24	#500	—	223	1	no	maximal, Sylow
25	C₄⋊Q₈	—	223	2	no	—
26	#107	—	223	4	no	—
27	C₄×C₈	D₂₂₃	1	1	yes	characteristic
28	C₂×Q₈	—	446	2	no	—
29	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
30	C₄⋊C₄	—	446	2	no	—
31	C₄⋊C₄	—	223	4	no	—
32	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
33	Q₈	—	892	4	no	—
34	C₂×C₄	—	446	4	no	—
35	C₂×C₄	D₈₉₂	1	1	yes	characteristic
36	C₂×C₄	D₈₉₂	1	1	yes	characteristic
37	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
38	C₈	Dic₄₄₆	1	4	yes	complemented
39	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
40	C₄	—	892	4	no	—
41	C₄	D₁₇₈₄	1	2	yes	—
42	C₄	Dic₈₉₂	1	2	yes	—
43	C₄	C₂×Dic₄₄₆	1	2	yes	—
44	C₂	#20297304	1	1	yes	characteristic
45	C₂	C₂×D₁₇₈₄	1	1	yes	characteristic
46	C₂	C₂×Dic₈₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₉₂
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄×C₈×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×D₄
hypercenter	C₄×C₈	D₂₂₃
hypocenter	C₂₂₃	#500
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
14. C₈×C₂₂₃	Q₈	semidirect
23. C₂₂₃	#500	semidirect
38. C₈	Dic₄₄₆	semidirect

Maximal subgroups

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

subgroup	index
#20297304	2
#20297238	2
C₄×C₈×C₂₂₃	2
#500	223