Subgroups — #28383956

Order 14272.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813155	C₂	1	4	yes	maximal, complemented
2	#20297301	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, complemented
5	C₂×D₈₉₂	—	2	2	no	—
6	C₂×D₈₉₂	C₂²	1	2	yes	—
7	#16813155	—	2	8	no	—
8	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
9	C₂×D₄×C₂₂₃	—	2	2	no	—
10	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
11	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂²×D₂₂₃	—	2	2	no	—
13	D₈₉₂	—	2	4	no	—
14	D₈₉₂	—	4	4	no	—
15	C₂₂₃⋊C₈	D₄	1	4	yes	complemented
16	C₂³×C₂₂₃	—	2	2	no	—
17	D₄×C₂₂₃	—	2	4	no	—
18	D₄×C₂₂₃	—	4	4	no	—
19	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
20	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₂²×C₂₂₃	—	4	2	no	—
22	C₂²×C₂₂₃	—	4	4	no	—
23	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₂₃	—	4	2	no	—
25	C₂×D₂₂₃	—	4	4	no	—
26	C₄×C₂₂₃	D₈	1	4	yes	complemented
27	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
28	C₂×C₂₂₃	—	8	2	no	—
29	C₂×C₂₂₃	#127	1	1	yes	characteristic
30	C₂×C₂₂₃	C₂×D₈	1	2	yes	—
31	D₂₂₃	—	8	2	no	—
32	C₂₂₃	#493	1	1	yes	characteristic, complemented, Sylow
33	#493	—	223	1	no	maximal, Sylow
34	C₂×D₈	—	223	4	no	—
35	#127	—	223	1	no	—
36	#127	D₂₂₃	1	1	yes	characteristic, complemented
37	C₄×C₈	—	223	1	no	—
38	C₂×D₄	—	2	2	no	—
39	C₂×D₄	—	446	2	no	—
40	C₂×D₄	—	223	2	no	—
41	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
42	D₈	—	446	8	no	—
43	C₂×C₈	—	223	2	no	—
44	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	—	446	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	4	4	no	—
49	D₄	—	892	4	no	—
50	D₄	—	446	4	no	—
51	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
52	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
53	C₈	—	223	4	no	—
54	C₂²	—	4	2	no	—
55	C₂²	—	4	4	no	—
56	C₂²	—	892	2	no	—
57	C₂²	—	892	4	no	—
58	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
59	C₄	#16813155	1	4	yes	complemented
60	C₄	D₄×D₂₂₃	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	1784	2	no	—
63	C₂	#20297300	1	1	yes	characteristic
64	C₂	C₂×#16813155	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#127×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
hypercenter	#127	D₂₂₃
hypocenter	C₂₂₃	#493
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×#16813155	C₂	semidirect
2. #20297301	C₂	semidirect
3. C₄×C₂₂₃⋊C₈	C₂	semidirect
4. #127×C₂₂₃	C₂	semidirect
15. C₂₂₃⋊C₈	D₄	semidirect
26. C₄×C₂₂₃	D₈	semidirect
32. C₂₂₃	#493	semidirect
36. #127	D₂₂₃	semidirect
59. C₄	#16813155	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813155	2
#20297301	2
C₄×C₂₂₃⋊C₈	2
#127×C₂₂₃	2
#493	223