Subgroups — #560×C₂₂₃ #28383272

Order 14272.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#127×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#126×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	#123×C₂₂₃	C₂	1	3	yes	maximal, complemented
4	#121×C₂₂₃	C₂	1	6	yes	maximal, complemented
5	#120×C₂₂₃	C₂	1	3	yes	maximal, complemented
6	C₂⁴×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₂₂₃	C₂²	1	6	yes	—
8	C₂×D₄×C₂₂₃	C₂²	1	3	yes	—
9	C₂×D₄×C₂₂₃	C₂²	1	3	yes	complemented
10	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	complemented
11	C₄⋊C₄×C₂₂₃	C₂²	1	6	yes	complemented
12	C₂²⋊C₄×C₂₂₃	C₂²	1	6	yes	—
13	C₂²⋊C₄×C₂₂₃	C₂²	1	3	yes	—
14	C₂²⋊C₄×C₂₂₃	C₂²	1	3	yes	complemented
15	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂³×C₂₂₃	—	4	3	no	—
17	C₂³×C₂₂₃	C₂³	1	2	yes	—
18	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
19	C₂³×C₂₂₃	C₂³	1	3	yes	—
20	D₄×C₂₂₃	—	4	6	no	—
21	D₄×C₂₂₃	—	4	3	no	—
22	D₄×C₂₂₃	—	4	3	no	—
23	C₂×C₄×C₂₂₃	—	4	3	no	—
24	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
25	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
26	C₂²×C₂₂₃	—	2	3	no	—
27	C₂²×C₂₂₃	—	2	6	no	—
28	C₂²×C₂₂₃	—	2	6	no	—
29	C₂²×C₂₂₃	—	2	3	no	—
30	C₂²×C₂₂₃	—	4	4	no	—
31	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
32	C₄×C₂₂₃	—	2	6	no	—
33	C₄×C₂₂₃	—	2	3	no	—
34	C₂×C₂₂₃	—	4	3	no	—
35	C₂×C₂₂₃	—	4	2	no	—
36	C₂×C₂₂₃	—	4	1	no	—
37	C₂×C₂₂₃	H(5,𝔽₂)	1	3	yes	—
38	C₂₂₃	#560	1	1	yes	characteristic, direct product factor, Sylow
39	#560	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	#127	C₂×C₂₂₃	1	1	yes	characteristic, complemented
41	#126	C₂×C₂₂₃	1	2	yes	complemented
42	#123	C₂×C₂₂₃	1	3	yes	complemented
43	#121	C₂×C₂₂₃	1	6	yes	complemented
44	#120	C₂×C₂₂₃	1	3	yes	complemented
45	C₂⁴	C₂²×C₂₂₃	1	1	yes	characteristic
46	C₂×D₄	C₂²×C₂₂₃	1	6	yes	—
47	C₂×D₄	C₂²×C₂₂₃	1	3	yes	—
48	C₂×D₄	C₂²×C₂₂₃	1	3	yes	complemented
49	C₂²×C₄	C₂²×C₂₂₃	1	3	yes	complemented
50	C₄⋊C₄	C₂²×C₂₂₃	1	6	yes	complemented
51	C₂²⋊C₄	C₂²×C₂₂₃	1	6	yes	—
52	C₂²⋊C₄	C₂²×C₂₂₃	1	3	yes	—
53	C₂²⋊C₄	C₂²×C₂₂₃	1	3	yes	complemented
54	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic, complemented
55	C₂³	—	4	3	no	—
56	C₂³	C₂³×C₂₂₃	1	2	yes	—
57	C₂³	C₂³×C₂₂₃	1	1	yes	characteristic
58	C₂³	C₂³×C₂₂₃	1	3	yes	—
59	D₄	—	4	6	no	—
60	D₄	—	4	3	no	—
61	D₄	—	4	3	no	—
62	C₂×C₄	—	4	3	no	—
63	C₂×C₄	C₂³×C₂₂₃	1	3	yes	—
64	C₂×C₄	C₂³×C₂₂₃	1	6	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	6	no	—
68	C₂²	—	2	3	no	—
69	C₂²	—	4	4	no	—
70	C₂²	C₂⁴×C₂₂₃	1	1	yes	characteristic
71	C₄	—	2	6	no	—
72	C₄	—	2	3	no	—
73	C₂	—	4	2	no	—
74	C₂	—	4	1	no	—
75	C₂	—	4	3	no	—
76	C₂	H(5,𝔽₂)×C₂₂₃	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #127×C₂₂₃	C₂	semidirect
2. #126×C₂₂₃	C₂	semidirect
3. #123×C₂₂₃	C₂	semidirect
4. #121×C₂₂₃	C₂	semidirect
5. #120×C₂₂₃	C₂	semidirect
9. C₂×D₄×C₂₂₃	C₂²	semidirect
10. C₂²×C₄×C₂₂₃	C₂²	semidirect
11. C₄⋊C₄×C₂₂₃	C₂²	semidirect
14. C₂²⋊C₄×C₂₂₃	C₂²	semidirect
15. C₄²×C₂₂₃	C₂²	semidirect
38. C₂₂₃	#560	direct (N × H)
39. #560	C₂₂₃	direct (N × H)
40. #127	C₂×C₂₂₃	semidirect
41. #126	C₂×C₂₂₃	semidirect
42. #123	C₂×C₂₂₃	semidirect
43. #121	C₂×C₂₂₃	semidirect
44. #120	C₂×C₂₂₃	semidirect
48. C₂×D₄	C₂²×C₂₂₃	semidirect
49. C₂²×C₄	C₂²×C₂₂₃	semidirect
50. C₄⋊C₄	C₂²×C₂₂₃	semidirect
53. C₂²⋊C₄	C₂²×C₂₂₃	semidirect
54. C₄²	C₂²×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#127×C₂₂₃	2
#126×C₂₂₃	2
#123×C₂₂₃	2
#121×C₂₂₃	2
#120×C₂₂₃	2
#560	223

Subgroups — #560×C223 #28383272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #560×C₂₂₃ #28383272