Subgroups — #28383944

Order 14272.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297296	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297225	C₂	1	4	yes	maximal, complemented
3	#20297215	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
6	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
7	Dic₂₂₃⋊C₄	—	2	2	no	—
8	C₂×D₄×C₂₂₃	—	2	2	no	—
9	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
10	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₂₃	—	2	2	no	—
12	C₂₂₃⋊C₈	—	2	2	no	—
13	C₂³×C₂₂₃	—	2	2	no	—
14	D₄×C₂₂₃	—	2	4	no	—
15	D₄×C₂₂₃	—	4	4	no	—
16	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
17	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
18	C₂²×C₂₂₃	—	4	2	no	—
19	C₂²×C₂₂₃	—	4	4	no	—
20	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
21	C₄×C₂₂₃	—	2	2	no	—
22	C₄×C₂₂₃	#41	1	2	yes	—
23	Dic₂₂₃	—	4	2	no	—
24	C₂×C₂₂₃	—	8	2	no	—
25	C₂×C₂₂₃	#124	1	1	yes	characteristic
26	C₂×C₂₂₃	AGL(1,ℤ/8)	1	2	yes	—
27	C₂₂₃	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	223	1	no	maximal, Sylow
29	#127	D₂₂₃	1	1	yes	characteristic
30	#125	—	223	1	no	—
31	#102	—	223	4	no	—
32	#97	—	223	1	no	—
33	C₂×D₄	—	2	2	no	—
34	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
35	C₂×C₈	—	223	2	no	—
36	C₄⋊C₄	—	446	2	no	—
37	C₄⋊C₄	—	223	2	no	—
38	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
39	C₂³	—	2	2	no	—
40	D₄	—	2	4	no	—
41	D₄	—	4	4	no	—
42	C₂×C₄	—	446	2	no	—
43	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
44	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
45	C₈	—	446	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	—	4	4	no	—
48	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	892	2	no	—
51	C₄	#16813169	1	2	yes	—
52	C₂	—	8	2	no	—
53	C₂	#20297291	1	1	yes	characteristic
54	C₂	#20297330	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	#490
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. #20297296	C₂	semidirect
2. #20297225	C₂	semidirect
3. #20297215	C₂	semidirect
27. C₂₂₃	#490	semidirect

Maximal subgroups

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

subgroup	index
#20297296	2
#20297225	2
#20297215	2
#127×C₂₂₃	2
#490	223