Subgroups — #28383984

Order 14272.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297303	C₂	1	2	yes	maximal
2	#20297238	C₂	1	1	yes	maximal, characteristic
3	#20297237	C₂	1	2	yes	maximal
4	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
5	#107×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₄₆	—	2	2	no	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
9	Dic₂₂₃⋊C₄	—	2	2	no	—
10	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
13	Dic₄₄₆	—	4	4	no	—
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
16	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
17	C₂₂₃⋊C₈	Q₈	1	2	yes	complemented
18	C₂×C₄×C₂₂₃	—	2	2	no	—
19	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₈×C₂₂₃	Q₈	1	2	yes	complemented
21	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
22	C₄×C₂₂₃	—	4	2	no	—
23	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
24	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
25	Dic₂₂₃	—	4	2	no	—
26	Dic₂₂₃	D₈	1	2	yes	—
27	Dic₂₂₃	Q₁₆	1	2	yes	—
28	C₂×C₂₂₃	C₄⋊Q₈	1	1	yes	characteristic
29	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
30	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
31	C₂₂₃	#500	1	1	yes	characteristic, complemented, Sylow
32	#500	—	223	1	no	maximal, Sylow
33	C₄⋊Q₈	—	223	2	no	—
34	#107	—	223	1	no	—
35	#107	—	223	2	no	—
36	#107	D₂₂₃	1	1	yes	characteristic
37	C₄×C₈	—	223	1	no	—
38	C₂×Q₈	—	446	2	no	—
39	C₂×C₈	—	223	1	no	—
40	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
41	C₄⋊C₄	—	446	2	no	—
42	C₄⋊C₄	—	223	2	no	—
43	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
44	C₄²	—	223	1	no	—
45	Q₈	—	892	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	446	2	no	—
48	C₂×C₄	—	223	1	no	—
49	C₂×C₄	—	223	1	no	—
50	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
51	C₈	—	223	2	no	—
52	C₈	Dic₄₄₆	1	2	yes	complemented
53	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
54	C₄	—	4	2	no	—
55	C₄	—	892	2	no	—
56	C₄	—	223	2	no	—
57	C₄	—	223	2	no	—
58	C₄	Q₈×D₂₂₃	1	1	yes	characteristic
59	C₄	C₂×Dic₄₄₆	1	1	yes	characteristic
60	C₂	#20297303	1	1	yes	characteristic
61	C₂	D₈×D₂₂₃	1	1	yes	characteristic
62	C₂	Q₁₆×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#107×C₂₂₃	C₂
socle, prime radical	C₂²×C₂₂₃	C₂×D₄
hypercenter	#107	D₂₂₃
hypocenter	C₂₂₃	#500
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
17. C₂₂₃⋊C₈	Q₈	semidirect
20. C₈×C₂₂₃	Q₈	semidirect
31. C₂₂₃	#500	semidirect
52. C₈	Dic₄₄₆	semidirect

Maximal subgroups

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

subgroup	index
#20297303	2
#20297238	2
#20297237	2
C₈×Dic₂₂₃	2
#107×C₂₂₃	2
#500	223