Subgroups — #16840119

Order 3904.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519765	C₂	1	1	yes	maximal, characteristic
2	#14519700	C₂	1	2	yes	maximal, complemented
3	#14519697	C₂	1	2	yes	maximal
4	C₄×#92418	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₄₄	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₆₁	—	2	2	no	—
9	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
10	C₄²×C₆₁	C₂²	1	1	yes	characteristic
11	#11758198	—	2	2	no	—
12	C₂×#92418	C₂²	1	2	yes	—
13	C₂²×D₆₁	—	2	1	no	—
14	C₂×C₄×C₆₁	—	2	2	no	—
15	C₂×C₄×C₆₁	D₄	1	2	yes	—
16	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
17	C₂×Dic₆₁	—	2	1	no	—
18	D₂₄₄	—	2	2	no	—
19	Dic₁₂₂	—	2	2	no	—
20	#92418	—	2	2	no	—
21	C₂²×C₆₁	C₂×D₄	1	1	yes	characteristic
22	C₂×D₆₁	—	4	1	no	—
23	C₂×D₆₁	—	4	2	no	—
24	C₄×C₆₁	—	2	2	no	—
25	C₄×C₆₁	—	4	2	no	—
26	C₄×C₆₁	#41	1	2	yes	—
27	Dic₆₁	—	4	1	no	—
28	C₂×C₆₁	#124	1	1	yes	characteristic
29	C₂×C₆₁	#135	1	2	yes	—
30	D₆₁	—	8	1	no	—
31	#488	—	61	1	no	maximal, Sylow
32	C₆₁	#488	1	1	yes	characteristic, complemented, Sylow
33	#125	D₆₁	1	1	yes	characteristic, complemented
34	#124	—	61	1	no	—
35	#103	—	61	2	no	—
36	#102	—	61	2	no	—
37	C₄×C₈	—	61	1	no	—
38	C₂×Q₈	—	61	1	no	—
39	C₂×D₄	—	61	1	no	—
40	C₂×C₈	—	61	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
43	C₂²⋊C₄	—	122	2	no	—
44	C₄²	C₂×D₆₁	1	1	yes	characteristic
45	C₂³	—	122	1	no	—
46	Q₈	—	122	2	no	—
47	D₄	—	122	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	122	1	no	—
50	C₂×C₄	C₂²⋊D₆₁	1	2	yes	—
51	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
52	C₈	—	122	2	no	—
53	C₂²	—	244	1	no	—
54	C₂²	—	244	2	no	—
55	C₂²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	244	1	no	—
59	C₄	#11758226	1	2	yes	—
60	C₂	—	488	1	no	—
61	C₂	#14519766	1	1	yes	characteristic
62	C₂	#14519797	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₆₁
commutator, cosocle	C₂×C₄×C₆₁	C₂³
Frattini	C₂×C₄	C₂²×D₆₁
Fitting	#125×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₄
prime radical	C₂×D₂₄₄	C₂²
hypercenter	#125	D₆₁
hypocenter	C₆₁	#488
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #14519700	C₂	semidirect
4. C₄×#92418	C₂	semidirect
5. #125×C₆₁	C₂	semidirect
32. C₆₁	#488	semidirect
33. #125	D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#14519765	2
#14519700	2
#14519697	2
C₄×#92418	2
#125×C₆₁	2
#488	61