Subgroups — #28383926

Order 14272.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297304	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297225	C₂	1	4	yes	maximal, complemented
3	C₄×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₄₆	—	2	1	no	—
6	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₂₂₃	—	2	1	no	—
8	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
9	C₂×D₄×C₂₂₃	—	2	1	no	—
10	C₂×D₄×C₂₂₃	—	2	1	no	—
11	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
12	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
13	Dic₄₄₆	—	4	2	no	—
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂₂₃⋊C₈	—	2	2	no	—
16	C₂³×C₂₂₃	—	2	2	no	—
17	D₄×C₂₂₃	—	2	4	no	—
18	D₄×C₂₂₃	—	4	2	no	—
19	D₄×C₂₂₃	—	4	2	no	—
20	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₂₂₃	—	4	2	no	—
24	C₂²×C₂₂₃	—	4	2	no	—
25	C₂²×C₂₂₃	—	4	2	no	—
26	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₂₃	D₈	1	2	yes	—
28	C₄×C₂₂₃	SD₁₆	1	2	yes	—
29	C₄×C₂₂₃	#41	1	2	yes	—
30	Dic₂₂₃	—	4	2	no	—
31	C₂×C₂₂₃	—	8	2	no	—
32	C₂×C₂₂₃	#124	1	1	yes	characteristic
33	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
34	C₂×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
35	C₂₂₃	#486	1	1	yes	characteristic, complemented, Sylow
36	#486	—	223	1	no	maximal, Sylow
37	C₄⋊Q₈	—	223	1	no	—
38	#127	D₂₂₃	1	1	yes	characteristic
39	#102	—	223	4	no	—
40	C₄×C₈	—	223	1	no	—
41	C₂×Q₈	—	446	1	no	—
42	C₂×D₄	—	2	1	no	—
43	C₂×D₄	—	2	1	no	—
44	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
45	C₂×C₈	—	223	2	no	—
46	C₄⋊C₄	—	446	1	no	—
47	C₄⋊C₄	—	223	2	no	—
48	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	Q₈	—	892	2	no	—
51	D₄	—	2	4	no	—
52	D₄	—	4	2	no	—
53	D₄	—	4	2	no	—
54	C₂×C₄	—	446	2	no	—
55	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
58	C₈	—	446	2	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
63	C₄	—	892	2	no	—
64	C₄	#16813155	1	2	yes	—
65	C₄	#16813158	1	2	yes	—
66	C₄	#16813169	1	2	yes	—
67	C₂	—	8	2	no	—
68	C₂	#20297291	1	1	yes	characteristic
69	C₂	C₂×#16813155	1	1	yes	characteristic
70	C₂	C₂×#16813158	1	1	yes	characteristic

	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	#486
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. #20297304	C₂	semidirect
2. #20297225	C₂	semidirect
3. C₄×C₂₂₃⋊C₈	C₂	semidirect
35. C₂₂₃	#486	semidirect

Maximal subgroups

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

subgroup	index
#20297304	2
#20297225	2
C₄×C₂₂₃⋊C₈	2
#127×C₂₂₃	2
#486	223