Subgroups — #29567913

Order 15616.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21444019	C₂	1	2	yes	maximal
2	C₂×C₆₁×C₆₄	C₂	1	1	yes	maximal, characteristic
3	#16839588	—	2	2	no	—
4	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
5	C₆₁×C₆₄	C₄	1	2	yes	complemented
6	#14519711	—	4	2	no	—
7	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
8	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
9	C₃₂×C₆₁	Q₈	1	1	yes	characteristic
10	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
11	C₄⋊Dic₆₁	—	8	2	no	—
12	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
13	C₁₆×C₆₁	Q₁₆	1	1	yes	characteristic
14	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
15	C₂×Dic₆₁	—	16	2	no	—
16	C₈×C₆₁	#107	1	1	yes	characteristic
17	C₈×C₆₁	Q₃₂	1	1	yes	characteristic
18	#7545	—	61	1	no	maximal, Sylow
19	C₂²×C₆₁	D₃₂	1	1	yes	characteristic
20	C₄×C₆₁	#366	1	1	yes	characteristic
21	C₄×C₆₁	Q₆₄	1	1	yes	characteristic
22	Dic₆₁	—	32	2	no	—
23	C₂×C₆₄	D₆₁	1	1	yes	characteristic
24	#1423	—	61	2	no	—
25	C₂×C₆₁	#1423	1	1	yes	characteristic
26	C₂×C₆₁	D₆₄	1	1	yes	characteristic
27	C₂×C₆₁	Q₁₂₈	1	1	yes	characteristic
28	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
29	#366	—	122	2	no	—
30	C₆₄	Dic₆₁	1	2	yes	complemented
31	C₆₁	#7545	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
33	#107	—	244	2	no	—
34	C₃₂	Dic₁₂₂	1	1	yes	characteristic
35	C₃₂	C₂×Dic₆₁	1	1	yes	characteristic
36	C₂×C₈	D₄₈₈	1	1	yes	characteristic
37	C₄⋊C₄	—	488	2	no	—
38	C₁₆	Dic₂₄₄	1	1	yes	characteristic
39	C₁₆	C₄⋊Dic₆₁	1	1	yes	characteristic
40	C₂×C₄	—	976	2	no	—
41	C₂×C₄	D₉₇₆	1	1	yes	characteristic
42	C₈	#14519711	1	1	yes	characteristic
43	C₈	Dic₄₈₈	1	1	yes	characteristic
44	C₂²	D₁₉₅₂	1	1	yes	characteristic
45	C₄	—	1952	2	no	—
46	C₄	#16839588	1	1	yes	characteristic
47	C₄	Dic₉₇₆	1	1	yes	characteristic
48	C₂	#21444019	1	1	yes	characteristic
49	C₂	D₃₉₀₄	1	1	yes	characteristic
50	C₂	Dic₁₉₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₉₅₂
commutator	C₃₂×C₆₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₆₁
Fitting	C₂×C₆₁×C₆₄	C₂
socle, prime radical	C₂²×C₆₁	D₃₂
cosocle	C₂×C₃₂×C₆₁	C₂²
hypercenter	C₂×C₆₄	D₆₁
hypocenter	C₆₁	#7545
LCS	G ▷ C₃₂×C₆₁ ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
5. C₆₁×C₆₄	C₄	semidirect
30. C₆₄	Dic₆₁	semidirect
31. C₆₁	#7545	semidirect

Maximal subgroups

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

subgroup	index
#21444019	2
C₂×C₆₁×C₆₄	2
#7545	61