Subgroups — #14519802

Order 1952.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#11758201	C₂	1	2	yes	maximal, complemented
4	#11758199	C₂	1	2	yes	maximal, complemented
5	#11758194	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₁	—	2	1	no	—
7	Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
8	D₄×C₆₁	—	2	1	no	—
9	D₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
10	C₂×C₄×C₆₁	—	2	1	no	—
11	C₂×C₄×C₆₁	C₂²	1	1	yes	characteristic
12	D₂₄₄	—	2	1	no	—
13	D₂₄₄	C₂²	1	2	yes	—
14	#92418	C₂²	1	2	yes	complemented
15	C₂²×C₆₁	—	2	1	no	—
16	C₂²×C₆₁	D₄	1	1	yes	characteristic
17	C₂×D₆₁	—	2	2	no	—
18	C₂×D₆₁	—	4	2	no	—
19	C₄×C₆₁	—	2	1	no	—
20	C₄×C₆₁	D₄	1	1	yes	characteristic
21	C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×C₆₁	—	2	1	no	—
23	C₂×C₆₁	—	4	1	no	—
24	C₂×C₆₁	C₂×D₄	1	1	yes	characteristic
25	D₆₁	—	4	2	no	—
26	C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	61	1	no	maximal, Sylow
28	#41	D₆₁	1	1	yes	characteristic, complemented
29	C₂×D₄	—	61	1	no	—
30	SD₁₆	—	61	2	no	—
31	D₈	—	61	2	no	—
32	M₁₆	—	61	1	no	—
33	C₂³	—	122	1	no	—
34	Q₈	C₂×D₆₁	1	1	yes	characteristic, complemented
35	D₄	—	2	1	no	—
36	D₄	—	61	2	no	—
37	D₄	—	122	1	no	—
38	D₄	C₂×D₆₁	1	1	yes	characteristic, complemented
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	C₂×D₆₁	1	1	yes	characteristic
41	C₈	—	61	2	no	—
42	C₂²	—	2	1	no	—
43	C₂²	—	122	2	no	—
44	C₂²	—	244	2	no	—
45	C₂²	C₂²⋊D₆₁	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	C₂²⋊D₆₁	1	1	yes	characteristic
48	C₄	C₂²×D₆₁	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	1	no	—
51	C₂	—	244	2	no	—
52	C₂	C₂×C₂²⋊D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₆₁
commutator, cosocle	C₄×C₆₁	C₂³
Frattini	C₄	C₂²×D₆₁
Fitting	#41×C₆₁	C₂
socle	C₂×C₆₁	C₂×D₄
hypercenter	#41	D₆₁
hypocenter	C₆₁	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #41×C₆₁	C₂	semidirect
2. C₂×D₂₄₄	C₂	semidirect
3. #11758201	C₂	semidirect
4. #11758199	C₂	semidirect
5. #11758194	C₂	semidirect
7. Q₈×C₆₁	C₂²	semidirect
9. D₄×C₆₁	C₂²	semidirect
14. #92418	C₂²	semidirect
26. C₆₁	AGL(1,ℤ/8)	semidirect
28. #41	D₆₁	semidirect
34. Q₈	C₂×D₆₁	semidirect
38. D₄	C₂×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#41×C₆₁	2
C₂×D₂₄₄	2
#11758201	2
#11758199	2
#11758194	2
AGL(1,ℤ/8)	61