Subgroups — C₂×#120×C₂₂₃ #28383233

Order 14272.

662 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×C₂₂₃	C₂	1	3	yes	maximal, complemented
3	#120×C₂₂₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂²⋊C₄×C₂₂₃	C₂	1	3	yes	maximal, complemented
5	C₂⁴×C₂₂₃	—	2	12	no	—
6	C₂⁴×C₂₂₃	C₂²	1	3	yes	—
7	C₂⁴×C₂₂₃	C₂²	1	4	yes	complemented
8	C₂⁴×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
9	C₂×D₄×C₂₂₃	—	2	12	no	—
10	C₂×D₄×C₂₂₃	C₂²	1	12	yes	complemented
11	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	complemented
12	C₂²⋊C₄×C₂₂₃	C₂²	1	12	yes	complemented
13	C₂³×C₂₂₃	—	2	32	no	—
14	C₂³×C₂₂₃	—	2	12	no	—
15	C₂³×C₂₂₃	—	2	24	no	—
16	C₂³×C₂₂₃	—	2	6	no	—
17	C₂³×C₂₂₃	D₄	1	12	yes	complemented
18	C₂³×C₂₂₃	C₂³	1	6	yes	—
19	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
20	C₂³×C₂₂₃	C₂³	1	2	yes	complemented
21	D₄×C₂₂₃	—	2	24	no	—
22	C₂×C₄×C₂₂₃	—	2	6	no	—
23	C₂×C₄×C₂₂₃	C₂³	1	6	yes	complemented
24	C₂²×C₂₂₃	—	2	32	no	—
25	C₂²×C₂₂₃	—	2	24	no	—
26	C₂²×C₂₂₃	—	2	12	no	—
27	C₂²×C₂₂₃	—	2	6	no	—
28	C₂²×C₂₂₃	—	4	4	no	—
29	C₂²×C₂₂₃	C₂×D₄	1	6	yes	—
30	C₂²×C₂₂₃	C₂×D₄	1	12	yes	complemented
31	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
32	C₄×C₂₂₃	—	2	6	no	—
33	C₂×C₂₂₃	—	2	12	no	—
34	C₂×C₂₂₃	—	4	2	no	—
35	C₂×C₂₂₃	#120	1	4	yes	direct product factor
36	C₂×C₂₂₃	C₂²×D₄	1	3	yes	—
37	C₂₂₃	C₂×#120	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×#120	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂⁵	C₂×C₂₂₃	1	1	yes	characteristic, complemented
40	C₂²×D₄	C₂×C₂₂₃	1	3	yes	complemented
41	#120	C₂×C₂₂₃	1	8	yes	direct product factor
42	C₂×C₂²⋊C₄	C₂×C₂₂₃	1	3	yes	complemented
43	C₂⁴	—	2	12	no	—
44	C₂⁴	C₂²×C₂₂₃	1	3	yes	—
45	C₂⁴	C₂²×C₂₂₃	1	4	yes	complemented
46	C₂⁴	C₂²×C₂₂₃	1	1	yes	characteristic, complemented
47	C₂×D₄	—	2	12	no	—
48	C₂×D₄	C₂²×C₂₂₃	1	12	yes	complemented
49	C₂²×C₄	C₂²×C₂₂₃	1	3	yes	complemented
50	C₂²⋊C₄	C₂²×C₂₂₃	1	12	yes	complemented
51	C₂³	—	2	32	no	—
52	C₂³	—	2	12	no	—
53	C₂³	—	2	24	no	—
54	C₂³	—	2	6	no	—
55	C₂³	D₄×C₂₂₃	1	12	yes	complemented
56	C₂³	C₂³×C₂₂₃	1	6	yes	—
57	C₂³	C₂³×C₂₂₃	1	1	yes	characteristic
58	C₂³	C₂³×C₂₂₃	1	2	yes	complemented
59	D₄	—	2	24	no	—
60	C₂×C₄	—	2	6	no	—
61	C₂×C₄	C₂³×C₂₂₃	1	6	yes	complemented
62	C₂²	—	2	32	no	—
63	C₂²	—	2	24	no	—
64	C₂²	—	2	12	no	—
65	C₂²	—	2	6	no	—
66	C₂²	—	4	4	no	—
67	C₂²	C₂×D₄×C₂₂₃	1	6	yes	—
68	C₂²	C₂×D₄×C₂₂₃	1	12	yes	complemented
69	C₂²	C₂⁴×C₂₂₃	1	1	yes	characteristic
70	C₄	—	2	6	no	—
71	C₂	—	2	12	no	—
72	C₂	—	4	2	no	—
73	C₂	#120×C₂₂₃	1	4	yes	direct product factor
74	C₂	C₂²×D₄×C₂₂₃	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₂₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₂₂₃	C₂	semidirect
2. C₂²×D₄×C₂₂₃	C₂	semidirect
3. #120×C₂₂₃	C₂	direct (N × H)
4. C₂×C₂²⋊C₄×C₂₂₃	C₂	semidirect
7. C₂⁴×C₂₂₃	C₂²	semidirect
8. C₂⁴×C₂₂₃	C₂²	semidirect
10. C₂×D₄×C₂₂₃	C₂²	semidirect
11. C₂²×C₄×C₂₂₃	C₂²	semidirect
12. C₂²⋊C₄×C₂₂₃	C₂²	semidirect
17. C₂³×C₂₂₃	D₄	semidirect
20. C₂³×C₂₂₃	C₂³	semidirect
23. C₂×C₄×C₂₂₃	C₂³	semidirect
30. C₂²×C₂₂₃	C₂×D₄	semidirect
35. C₂×C₂₂₃	#120	direct (N × H)
37. C₂₂₃	C₂×#120	direct (N × H)
38. C₂×#120	C₂₂₃	direct (N × H)
39. C₂⁵	C₂×C₂₂₃	semidirect
40. C₂²×D₄	C₂×C₂₂₃	semidirect
41. #120	C₂×C₂₂₃	direct (N × H)
42. C₂×C₂²⋊C₄	C₂×C₂₂₃	semidirect
45. C₂⁴	C₂²×C₂₂₃	semidirect
46. C₂⁴	C₂²×C₂₂₃	semidirect
48. C₂×D₄	C₂²×C₂₂₃	semidirect
49. C₂²×C₄	C₂²×C₂₂₃	semidirect
50. C₂²⋊C₄	C₂²×C₂₂₃	semidirect
55. C₂³	D₄×C₂₂₃	semidirect
58. C₂³	C₂³×C₂₂₃	semidirect
61. C₂×C₄	C₂³×C₂₂₃	semidirect
68. C₂²	C₂×D₄×C₂₂₃	semidirect
73. C₂	#120×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₂₂₃	2
C₂²×D₄×C₂₂₃	2
#120×C₂₂₃	2
C₂×C₂²⋊C₄×C₂₂₃	2
C₂×#120	223

Subgroups — C2×#120×C223 #28383233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#120×C₂₂₃ #28383233