Subgroups — C₂²×D₁₅₆ #11929302

Order 1248.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₉	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×C₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₅₆	C₂	1	12	yes	maximal, direct product factor
4	C₂²×D₅₂	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×D₃₉	—	2	8	no	—
7	C₂²×D₃₉	C₂²	1	12	yes	complemented
8	C₂×C₃×C₄×C₁₃	C₂²	1	6	yes	complemented
9	D₁₅₆	C₂²	1	16	yes	direct product factor
10	C₂³×D₁₃	—	3	2	no	—
11	C₂²×C₄×C₁₃	D₃	1	1	yes	characteristic, complemented
12	C₂×D₅₂	—	3	12	no	—
13	C₂²×C₃×C₁₃	D₄	1	4	yes	complemented
14	C₂²×C₃×C₁₃	C₂³	1	3	yes	—
15	C₂×D₃₉	—	2	24	no	—
16	C₂×D₃₉	C₂³	1	8	yes	complemented
17	C₃×C₄×C₁₃	C₂³	1	4	yes	complemented
18	C₂³×C₁₃	C₂×D₃	1	1	yes	characteristic
19	C₂²×D₁₃	—	3	12	no	—
20	C₂²×D₁₃	—	6	8	no	—
21	C₂×C₄×C₁₃	C₂×D₃	1	6	yes	complemented
22	D₅₂	—	3	16	no	—
23	C₂²×D₁₂	—	13	1	no	maximal, Hall
24	C₂×C₃×C₁₃	C₂×D₄	1	6	yes	complemented
25	C₂×C₃×C₁₃	C₂⁴	1	1	yes	characteristic
26	D₃₉	—	2	8	no	—
27	C₂²×C₁₃	D₁₂	1	4	yes	complemented
28	C₂²×C₁₃	C₂²×D₃	1	3	yes	—
29	C₂×D₁₃	—	3	8	no	—
30	C₂×D₁₃	—	6	24	no	—
31	C₄×C₁₃	C₂²×D₃	1	4	yes	complemented
32	C₂³×D₃	—	13	2	no	—
33	C₂²×C₃×C₄	D₁₃	1	1	yes	characteristic, complemented
34	C₂×D₁₂	—	13	12	no	—
35	C₃×C₁₃	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
36	C₂²×D₄	—	39	1	no	Sylow
37	C₂×C₁₃	C₂×D₁₂	1	6	yes	complemented
38	C₂×C₁₃	C₂³×D₃	1	1	yes	characteristic
39	D₁₃	—	6	8	no	—
40	C₂³×C₃	C₂×D₁₃	1	1	yes	characteristic
41	C₂²×D₃	—	13	12	no	—
42	C₂²×D₃	—	26	8	no	—
43	C₂×C₃×C₄	C₂×D₁₃	1	6	yes	complemented
44	D₁₂	—	13	16	no	—
45	C₂⁴	—	39	2	no	—
46	C₂×D₄	—	39	12	no	—
47	C₂²×C₄	D₃₉	1	1	yes	characteristic, complemented
48	C₁₃	C₂²×D₁₂	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₃	D₅₂	1	4	yes	complemented
50	C₂²×C₃	C₂²×D₁₃	1	3	yes	—
51	C₂×D₃	—	13	8	no	—
52	C₂×D₃	—	26	24	no	—
53	C₃×C₄	C₂²×D₁₃	1	4	yes	complemented
54	C₂³	—	39	12	no	—
55	C₂³	—	78	8	no	—
56	C₂³	C₂×D₃₉	1	1	yes	characteristic
57	D₄	—	39	16	no	—
58	C₂×C₄	C₂×D₃₉	1	6	yes	complemented
59	C₂×C₃	C₂×D₅₂	1	6	yes	complemented
60	C₂×C₃	C₂³×D₁₃	1	1	yes	characteristic
61	D₃	—	26	8	no	—
62	C₂²	—	39	8	no	—
63	C₂²	—	78	24	no	—
64	C₂²	D₁₅₆	1	4	yes	direct product factor
65	C₂²	C₂²×D₃₉	1	3	yes	—
66	C₄	C₂²×D₃₉	1	4	yes	complemented
67	C₃	C₂²×D₅₂	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	78	8	no	—
69	C₂	C₂×D₁₅₆	1	6	yes	direct product factor
70	C₂	C₂³×D₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₉
commutator, cosocle	C₂×C₃×C₁₃	C₂⁴
Frattini	C₂	C₂³×D₃₉
Fitting	C₂²×C₃×C₄×C₁₃	C₂
socle	C₂³×C₃×C₁₃	C₂²
hypercenter	C₂²×C₄	D₃₉
hypocenter	C₃×C₁₃	C₂²×D₄
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃₉	C₂	semidirect
2. C₂²×C₃×C₄×C₁₃	C₂	semidirect
3. C₂×D₁₅₆	C₂	direct (N × H)
7. C₂²×D₃₉	C₂²	semidirect
8. C₂×C₃×C₄×C₁₃	C₂²	semidirect
9. D₁₅₆	C₂²	direct (N × H)
11. C₂²×C₄×C₁₃	D₃	semidirect
13. C₂²×C₃×C₁₃	D₄	semidirect
16. C₂×D₃₉	C₂³	semidirect
17. C₃×C₄×C₁₃	C₂³	semidirect
21. C₂×C₄×C₁₃	C₂×D₃	semidirect
24. C₂×C₃×C₁₃	C₂×D₄	semidirect
27. C₂²×C₁₃	D₁₂	semidirect
31. C₄×C₁₃	C₂²×D₃	semidirect
33. C₂²×C₃×C₄	D₁₃	semidirect
35. C₃×C₁₃	C₂²×D₄	semidirect
37. C₂×C₁₃	C₂×D₁₂	semidirect
43. C₂×C₃×C₄	C₂×D₁₃	semidirect
47. C₂²×C₄	D₃₉	semidirect
48. C₁₃	C₂²×D₁₂	semidirect
49. C₂²×C₃	D₅₂	semidirect
53. C₃×C₄	C₂²×D₁₃	semidirect
58. C₂×C₄	C₂×D₃₉	semidirect
59. C₂×C₃	C₂×D₅₂	semidirect
64. C₂²	D₁₅₆	direct (N × H)
66. C₄	C₂²×D₃₉	semidirect
67. C₃	C₂²×D₅₂	semidirect
69. C₂	C₂×D₁₅₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₃₉	2
C₂²×C₃×C₄×C₁₃	2
C₂×D₁₅₆	2
C₂²×D₅₂	3
C₂²×D₁₂	13

Subgroups — C22×D156 #11929302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₁₅₆ #11929302