Subgroups — #37566946

Order 24352.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27131650	C₂	1	2	yes	maximal, complemented
2	#27131649	C₂	1	3	yes	maximal, complemented
3	C₂×D₃₀₄₄	C₂	1	3	yes	maximal, complemented
4	D₄×D₇₆₁	C₂	1	6	yes	maximal, complemented
5	#41×C₇₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₆₁	C₂²	1	6	yes	complemented
7	Dic₁₅₂₂	C₂²	1	1	yes	characteristic, complemented
8	D₃₀₄₄	C₂²	1	3	yes	complemented
9	D₃₀₄₄	C₂²	1	6	yes	complemented
10	C₂²⋊D₇₆₁	C₂²	1	6	yes	complemented
11	C₄×D₇₆₁	C₂²	1	6	yes	complemented
12	Q₈×C₇₆₁	C₂²	1	1	yes	characteristic, complemented
13	D₄×C₇₆₁	C₂²	1	3	yes	complemented
14	C₂×C₄×C₇₆₁	C₂²	1	3	yes	complemented
15	C₂²×C₇₆₁	C₂³	1	3	yes	—
16	C₂×D₇₆₁	—	4	6	no	—
17	C₂×D₇₆₁	C₂³	1	6	yes	—
18	C₄×C₇₆₁	C₂³	1	1	yes	characteristic
19	C₄×C₇₆₁	C₂³	1	3	yes	—
20	Dic₇₆₁	C₂³	1	2	yes	—
21	C₂×C₇₆₁	—	2	3	no	—
22	C₂×C₇₆₁	C₂⁴	1	1	yes	characteristic
23	D₇₆₁	—	2	6	no	—
24	C₇₆₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
25	H(5,𝔽₂)	—	761	1	no	maximal, Sylow
26	#41	—	761	2	no	—
27	#41	—	761	3	no	—
28	#41	D₇₆₁	1	1	yes	characteristic, complemented
29	C₂×D₄	—	761	6	no	—
30	C₂×D₄	—	761	3	no	—
31	C₂³	—	761	6	no	—
32	Q₈	—	761	1	no	—
33	Q₈	C₂×D₇₆₁	1	1	yes	characteristic, complemented
34	D₄	—	761	6	no	—
35	D₄	—	761	6	no	—
36	D₄	—	761	3	no	—
37	D₄	C₂×D₇₆₁	1	3	yes	complemented
38	C₂×C₄	—	761	6	no	—
39	C₂×C₄	C₂×D₇₆₁	1	3	yes	complemented
40	C₂²	—	3044	6	no	—
41	C₂²	—	761	6	no	—
42	C₂²	C₂²×D₇₆₁	1	3	yes	—
43	C₄	—	761	2	no	—
44	C₄	C₂²×D₇₆₁	1	3	yes	—
45	C₄	C₂²×D₇₆₁	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	1522	6	no	—
48	C₂	C₂³×D₇₆₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₇₆₁
commutator, socle, cosocle	C₂×C₇₆₁	C₂⁴
Fitting	#41×C₇₆₁	C₂
hypercenter	#41	D₇₆₁
hypocenter	C₇₆₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₇₆₁ ▷ C₇₆₁
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #27131650	C₂	semidirect
2. #27131649	C₂	semidirect
3. C₂×D₃₀₄₄	C₂	semidirect
4. D₄×D₇₆₁	C₂	semidirect
5. #41×C₇₆₁	C₂	semidirect
6. C₂²×D₇₆₁	C₂²	semidirect
7. Dic₁₅₂₂	C₂²	semidirect
8. D₃₀₄₄	C₂²	semidirect
9. D₃₀₄₄	C₂²	semidirect
10. C₂²⋊D₇₆₁	C₂²	semidirect
11. C₄×D₇₆₁	C₂²	semidirect
12. Q₈×C₇₆₁	C₂²	semidirect
13. D₄×C₇₆₁	C₂²	semidirect
14. C₂×C₄×C₇₆₁	C₂²	semidirect
24. C₇₆₁	H(5,𝔽₂)	semidirect
28. #41	D₇₆₁	semidirect
33. Q₈	C₂×D₇₆₁	semidirect
37. D₄	C₂×D₇₆₁	semidirect
39. C₂×C₄	C₂×D₇₆₁	semidirect

Maximal subgroups

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

subgroup	index
#27131650	2
#27131649	2
C₂×D₃₀₄₄	2
D₄×D₇₆₁	2
#41×C₇₆₁	2
H(5,𝔽₂)	761