Subgroups — C₂×D₄×D₇₆₁ #37566936

Order 24352.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₇₆₁	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₀₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₇₆₁	C₂	1	2	yes	maximal, complemented
4	D₄×D₇₆₁	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×D₇₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₇₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₇₆₁	—	2	8	no	—
8	C₂²×D₇₆₁	C₂²	1	1	yes	characteristic
9	C₂²×D₇₆₁	C₂²	1	8	yes	complemented
10	C₂²×D₇₆₁	C₂²	1	2	yes	complemented
11	D₃₀₄₄	C₂²	1	4	yes	complemented
12	C₂²⋊D₇₆₁	C₂²	1	8	yes	complemented
13	C₂×Dic₇₆₁	C₂²	1	1	yes	characteristic, complemented
14	C₄×D₇₆₁	C₂²	1	4	yes	complemented
15	C₂³×C₇₆₁	C₂²	1	2	yes	complemented
16	D₄×C₇₆₁	C₂²	1	4	yes	complemented
17	C₂×C₄×C₇₆₁	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₇₆₁	—	2	4	no	—
19	C₂²×C₇₆₁	C₂³	1	1	yes	characteristic
20	C₂²×C₇₆₁	C₂³	1	4	yes	complemented
21	C₂×D₇₆₁	—	2	4	no	—
22	C₂×D₇₆₁	—	2	16	no	—
23	C₂×D₇₆₁	D₄	1	4	yes	direct product factor
24	C₂×D₇₆₁	C₂³	1	2	yes	—
25	C₂×D₇₆₁	C₂³	1	4	yes	complemented
26	C₄×C₇₆₁	C₂³	1	2	yes	complemented
27	Dic₇₆₁	C₂³	1	2	yes	complemented
28	C₂×C₇₆₁	—	2	4	no	—
29	C₂×C₇₆₁	C₂×D₄	1	2	yes	complemented
30	C₂×C₇₆₁	C₂⁴	1	1	yes	characteristic
31	D₇₆₁	—	2	4	no	—
32	D₇₆₁	C₂×D₄	1	4	yes	direct product factor
33	C₇₆₁	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
34	C₂²×D₄	—	761	1	no	maximal, Sylow
35	C₂⁴	—	761	2	no	—
36	C₂×D₄	—	761	2	no	—
37	C₂×D₄	—	761	8	no	—
38	C₂×D₄	—	761	1	no	—
39	C₂×D₄	D₇₆₁	1	1	yes	characteristic, direct product factor
40	C₂²×C₄	—	761	1	no	—
41	C₂³	—	1522	8	no	—
42	C₂³	—	761	2	no	—
43	C₂³	—	761	8	no	—
44	C₂³	—	761	1	no	—
45	C₂³	C₂×D₇₆₁	1	2	yes	complemented
46	D₄	—	761	8	no	—
47	D₄	—	761	4	no	—
48	D₄	C₂×D₇₆₁	1	4	yes	direct product factor
49	C₂×C₄	—	761	1	no	—
50	C₂×C₄	—	761	4	no	—
51	C₂×C₄	C₂×D₇₆₁	1	1	yes	characteristic, complemented
52	C₂²	—	2	4	no	—
53	C₂²	—	1522	16	no	—
54	C₂²	—	1522	4	no	—
55	C₂²	—	761	4	no	—
56	C₂²	—	761	2	no	—
57	C₂²	—	761	4	no	—
58	C₂²	C₂²×D₇₆₁	1	1	yes	characteristic
59	C₂²	C₂²×D₇₆₁	1	4	yes	complemented
60	C₄	—	761	2	no	—
61	C₄	C₂²×D₇₆₁	1	2	yes	complemented
62	C₂	—	2	4	no	—
63	C₂	—	1522	4	no	—
64	C₂	—	761	4	no	—
65	C₂	D₄×D₇₆₁	1	2	yes	direct product factor
66	C₂	C₂³×D₇₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆₁
commutator, cosocle	C₂×C₇₆₁	C₂⁴
Frattini	C₂	C₂³×D₇₆₁
Fitting	C₂×D₄×C₇₆₁	C₂
socle	C₂²×C₇₆₁	C₂³
hypercenter	C₂×D₄	D₇₆₁
hypocenter	C₇₆₁	C₂²×D₄
LCS	G ▷ C₂×C₇₆₁ ▷ C₇₆₁
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₇₆₁	C₂	semidirect
2. C₂×D₃₀₄₄	C₂	semidirect
3. C₂×C₂²⋊D₇₆₁	C₂	semidirect
4. D₄×D₇₆₁	C₂	direct (N × H)
5. C₂×C₄×D₇₆₁	C₂	semidirect
6. C₂×D₄×C₇₆₁	C₂	semidirect
9. C₂²×D₇₆₁	C₂²	semidirect
10. C₂²×D₇₆₁	C₂²	semidirect
11. D₃₀₄₄	C₂²	semidirect
12. C₂²⋊D₇₆₁	C₂²	semidirect
13. C₂×Dic₇₆₁	C₂²	semidirect
14. C₄×D₇₆₁	C₂²	semidirect
15. C₂³×C₇₆₁	C₂²	semidirect
16. D₄×C₇₆₁	C₂²	semidirect
17. C₂×C₄×C₇₆₁	C₂²	semidirect
20. C₂²×C₇₆₁	C₂³	semidirect
23. C₂×D₇₆₁	D₄	direct (N × H)
25. C₂×D₇₆₁	C₂³	semidirect
26. C₄×C₇₆₁	C₂³	semidirect
27. Dic₇₆₁	C₂³	semidirect
29. C₂×C₇₆₁	C₂×D₄	semidirect
32. D₇₆₁	C₂×D₄	direct (N × H)
33. C₇₆₁	C₂²×D₄	semidirect
39. C₂×D₄	D₇₆₁	direct (N × H)
45. C₂³	C₂×D₇₆₁	semidirect
48. D₄	C₂×D₇₆₁	direct (N × H)
51. C₂×C₄	C₂×D₇₆₁	semidirect
59. C₂²	C₂²×D₇₆₁	semidirect
61. C₄	C₂²×D₇₆₁	semidirect
65. C₂	D₄×D₇₆₁	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×D₇₆₁	2
C₂×D₃₀₄₄	2
C₂×C₂²⋊D₇₆₁	2
D₄×D₇₆₁	2
C₂×C₄×D₇₆₁	2
C₂×D₄×C₇₆₁	2
C₂²×D₄	761

Subgroups — C2×D4×D761 #37566936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×D₇₆₁ #37566936