Subgroups — #28383952

Order 14272.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813158	C₂	1	4	yes	maximal, complemented
2	#20297304	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₄₆	C₂²	1	2	yes	—
6	#16813158	—	2	8	no	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₂₃	—	2	2	no	—
9	C₂×D₄×C₂₂₃	—	2	2	no	—
10	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
11	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
12	Dic₄₄₆	—	2	4	no	—
13	C₂×Dic₂₂₃	—	2	2	no	—
14	C₂₂₃⋊C₈	D₄	1	4	yes	complemented
15	C₂³×C₂₂₃	—	2	2	no	—
16	D₄×C₂₂₃	—	2	4	no	—
17	D₄×C₂₂₃	—	4	4	no	—
18	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
19	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₂²×C₂₂₃	—	4	2	no	—
21	C₂²×C₂₂₃	—	4	4	no	—
22	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
23	C₄×C₂₂₃	SD₁₆	1	4	yes	complemented
24	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
25	Dic₂₂₃	—	4	2	no	—
26	C₂×C₂₂₃	—	8	2	no	—
27	C₂×C₂₂₃	#127	1	1	yes	characteristic
28	C₂×C₂₂₃	C₂×SD₁₆	1	2	yes	—
29	C₂₂₃	#492	1	1	yes	characteristic, complemented, Sylow
30	#492	—	223	1	no	maximal, Sylow
31	C₂×SD₁₆	—	223	4	no	—
32	C₄⋊Q₈	—	223	1	no	—
33	#127	D₂₂₃	1	1	yes	characteristic
34	C₄×C₈	—	223	1	no	—
35	C₂×Q₈	—	223	2	no	—
36	C₂×D₄	—	2	2	no	—
37	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
38	SD₁₆	—	446	8	no	—
39	C₂×C₈	—	223	2	no	—
40	C₄⋊C₄	—	446	2	no	—
41	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
42	C₂³	—	2	2	no	—
43	Q₈	—	446	4	no	—
44	D₄	—	2	4	no	—
45	D₄	—	4	4	no	—
46	C₂×C₄	—	446	2	no	—
47	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
48	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
49	C₈	—	223	4	no	—
50	C₂²	—	4	2	no	—
51	C₂²	—	4	4	no	—
52	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
53	C₄	—	892	2	no	—
54	C₄	#16813158	1	4	yes	complemented
55	C₄	D₄×D₂₂₃	1	2	yes	—
56	C₂	—	8	2	no	—
57	C₂	#20297300	1	1	yes	characteristic
58	C₂	C₂×#16813158	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting, prime radical	#127×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
hypercenter	#127	D₂₂₃
hypocenter	C₂₂₃	#492
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#16813158	C₂	semidirect
2. #20297304	C₂	semidirect
3. C₄×C₂₂₃⋊C₈	C₂	semidirect
14. C₂₂₃⋊C₈	D₄	semidirect
23. C₄×C₂₂₃	SD₁₆	semidirect
29. C₂₂₃	#492	semidirect
54. C₄	#16813158	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813158	2
#20297304	2
C₄×C₂₂₃⋊C₈	2
#127×C₂₂₃	2
#492	223