Subgroups — C₂³×D₂₁₅₂ #46825601

Order 34432.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₀₇₆	C₂	1	2	yes	maximal, complemented
2	C₂²×D₂₁₅₂	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₂₆₉	—	2	2	no	—
5	C₂²×D₁₀₇₆	—	2	14	no	—
6	C₂²×D₁₀₇₆	C₂²	1	28	yes	complemented
7	C₂×D₂₁₅₂	C₂²	1	112	yes	direct product factor
8	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₆₉	C₂²	1	14	yes	complemented
10	C₂³×D₂₆₉	—	2	28	no	—
11	C₂³×D₂₆₉	—	4	16	no	—
12	C₂×D₁₀₇₆	—	2	84	no	—
13	C₂×D₁₀₇₆	C₂³	1	56	yes	complemented
14	D₂₁₅₂	C₂³	1	64	yes	direct product factor
15	C₂⁴×C₂₆₉	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₆₉	D₄	1	7	yes	—
17	C₂²×C₄×C₂₆₉	C₂³	1	7	yes	—
18	C₂×C₈×C₂₆₉	C₂³	1	28	yes	complemented
19	C₂²×D₂₆₉	—	2	56	no	—
20	C₂²×D₂₆₉	—	4	112	no	—
21	D₁₀₇₆	—	2	56	no	—
22	D₁₀₇₆	C₂⁴	1	16	yes	complemented
23	C₂³×C₂₆₉	D₈	1	8	yes	complemented
24	C₂³×C₂₆₉	C₂×D₄	1	7	yes	—
25	C₂×C₄×C₂₆₉	C₂×D₄	1	21	yes	—
26	C₂×C₄×C₂₆₉	C₂⁴	1	7	yes	—
27	C₈×C₂₆₉	C₂⁴	1	8	yes	complemented
28	C₂²×C₂₆₉	C₂×D₈	1	28	yes	complemented
29	C₂²×C₂₆₉	C₂²×D₄	1	7	yes	—
30	C₂×D₂₆₉	—	2	16	no	—
31	C₂×D₂₆₉	—	4	112	no	—
32	C₄×C₂₆₉	C₂²×D₄	1	7	yes	—
33	C₄×C₂₆₉	C₂⁵	1	1	yes	characteristic
34	C₂×C₂₆₉	C₂²×D₈	1	14	yes	complemented
35	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
36	D₂₆₉	—	4	16	no	—
37	C₂₆₉	C₂³×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₂³×D₈	—	269	1	no	maximal, Sylow
39	C₂³×D₄	—	269	2	no	—
40	C₂²×D₈	—	269	28	no	—
41	C₂³×C₈	D₂₆₉	1	1	yes	characteristic, complemented
42	C₂⁵	—	538	2	no	—
43	C₂²×D₄	—	269	28	no	—
44	C₂²×D₄	—	538	14	no	—
45	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
46	C₂×D₈	—	269	112	no	—
47	C₂²×C₈	C₂×D₂₆₉	1	14	yes	complemented
48	C₂⁴	—	538	28	no	—
49	C₂⁴	—	1076	16	no	—
50	C₂⁴	D₁₀₇₆	1	1	yes	characteristic
51	C₂×D₄	—	269	56	no	—
52	C₂×D₄	—	538	84	no	—
53	C₂²×C₄	D₁₀₇₆	1	7	yes	—
54	C₂²×C₄	C₂²×D₂₆₉	1	7	yes	—
55	D₈	—	269	64	no	—
56	C₂×C₈	C₂²×D₂₆₉	1	28	yes	complemented
57	C₂³	—	538	56	no	—
58	C₂³	—	1076	112	no	—
59	C₂³	D₂₁₅₂	1	8	yes	direct product factor
60	C₂³	C₂×D₁₀₇₆	1	7	yes	—
61	D₄	—	269	16	no	—
62	D₄	—	538	56	no	—
63	C₂×C₄	C₂×D₁₀₇₆	1	21	yes	—
64	C₂×C₄	C₂³×D₂₆₉	1	7	yes	—
65	C₈	C₂³×D₂₆₉	1	8	yes	complemented
66	C₂²	—	538	16	no	—
67	C₂²	—	1076	112	no	—
68	C₂²	C₂×D₂₁₅₂	1	28	yes	direct product factor
69	C₂²	C₂²×D₁₀₇₆	1	7	yes	—
70	C₄	C₂²×D₁₀₇₆	1	7	yes	—
71	C₄	C₂⁴×D₂₆₉	1	1	yes	characteristic
72	C₂	—	1076	16	no	—
73	C₂	C₂²×D₂₁₅₂	1	14	yes	direct product factor
74	C₂	C₂³×D₁₀₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₁₀₇₆
commutator, cosocle	C₄×C₂₆₉	C₂⁵
Frattini	C₄	C₂⁴×D₂₆₉
Fitting	C₂³×C₈×C₂₆₉	C₂
socle	C₂⁴×C₂₆₉	D₄
hypercenter	C₂³×C₈	D₂₆₉
hypocenter	C₂₆₉	C₂³×D₈
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×D₁₀₇₆	C₂	semidirect
2. C₂²×D₂₁₅₂	C₂	direct (N × H)
3. C₂³×C₈×C₂₆₉	C₂	semidirect
6. C₂²×D₁₀₇₆	C₂²	semidirect
7. C₂×D₂₁₅₂	C₂²	direct (N × H)
9. C₂²×C₈×C₂₆₉	C₂²	semidirect
13. C₂×D₁₀₇₆	C₂³	semidirect
14. D₂₁₅₂	C₂³	direct (N × H)
18. C₂×C₈×C₂₆₉	C₂³	semidirect
22. D₁₀₇₆	C₂⁴	semidirect
23. C₂³×C₂₆₉	D₈	semidirect
27. C₈×C₂₆₉	C₂⁴	semidirect
28. C₂²×C₂₆₉	C₂×D₈	semidirect
34. C₂×C₂₆₉	C₂²×D₈	semidirect
37. C₂₆₉	C₂³×D₈	semidirect
41. C₂³×C₈	D₂₆₉	semidirect
47. C₂²×C₈	C₂×D₂₆₉	semidirect
56. C₂×C₈	C₂²×D₂₆₉	semidirect
59. C₂³	D₂₁₅₂	direct (N × H)
65. C₈	C₂³×D₂₆₉	semidirect
68. C₂²	C₂×D₂₁₅₂	direct (N × H)
73. C₂	C₂²×D₂₁₅₂	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×D₁₀₇₆	2
C₂²×D₂₁₅₂	2
C₂³×C₈×C₂₆₉	2
C₂³×D₈	269

Subgroups — C23×D2152 #46825601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₂₁₅₂ #46825601