Subgroups — #29567919

Order 15616.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21444023	C₂	1	1	yes	maximal, characteristic
2	#21444021	C₂	1	1	yes	maximal, characteristic
3	C₆₁×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	#16839592	—	2	1	no	—
5	#16839588	—	2	1	no	—
6	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
7	C₆₁×C₆₄	C₄	1	2	yes	complemented
8	#14519713	—	4	1	no	—
9	#14519711	—	4	1	no	—
10	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
11	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₆₁	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
14	C₄⋊Dic₆₁	—	8	1	no	—
15	#11758194	—	8	1	no	—
16	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₆₁	Q₁₆	1	1	yes	characteristic
18	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
19	C₂×Dic₆₁	—	16	1	no	—
20	C₈×C₆₁	#107	1	1	yes	characteristic
21	C₈×C₆₁	Q₃₂	1	1	yes	characteristic
22	#92418	—	16	1	no	—
23	#7548	—	61	1	no	maximal, Sylow
24	C₂²×C₆₁	SD₆₄	1	1	yes	characteristic
25	C₄×C₆₁	#366	1	1	yes	characteristic
26	C₄×C₆₁	SD₆₄	1	1	yes	characteristic
27	Dic₆₁	—	32	1	no	—
28	M₁₂₈	D₆₁	1	1	yes	characteristic
29	#1425	—	61	1	no	—
30	#1424	—	61	1	no	—
31	C₂×C₆₁	—	2	1	no	—
32	C₂×C₆₁	#1424	1	1	yes	characteristic
33	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
34	#368	—	122	1	no	—
35	#366	—	122	1	no	—
36	C₆₄	Dic₆₁	1	2	yes	complemented
37	C₆₁	#7548	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
39	#108	—	244	1	no	—
40	#107	—	244	1	no	—
41	C₃₂	Dic₁₂₂	1	1	yes	characteristic
42	C₃₂	C₂×Dic₆₁	1	1	yes	characteristic
43	M₁₆	—	488	1	no	—
44	C₂×C₈	D₄₈₈	1	1	yes	characteristic
45	C₄⋊C₄	—	488	1	no	—
46	C₁₆	Dic₂₄₄	1	1	yes	characteristic
47	C₁₆	C₄⋊Dic₆₁	1	1	yes	characteristic
48	C₂×C₄	—	976	1	no	—
49	C₂×C₄	D₉₇₆	1	1	yes	characteristic
50	C₈	—	976	1	no	—
51	C₈	#14519711	1	1	yes	characteristic
52	C₈	Dic₄₈₈	1	1	yes	characteristic
53	C₂²	#16839601	1	1	yes	characteristic
54	C₄	—	1952	1	no	—
55	C₄	#16839588	1	1	yes	characteristic
56	C₄	#16839601	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#21444021	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21444021
commutator	C₃₂×C₆₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₆₁
Fitting	C₆₁×M₁₂₈	C₂
socle	C₂×C₆₁	#1424
prime radical	C₂²×C₆₁	SD₆₄
cosocle	C₂×C₃₂×C₆₁	C₂²
hypercenter	M₁₂₈	D₆₁
hypocenter	C₆₁	#7548
LCS	G ▷ C₃₂×C₆₁ ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
7. C₆₁×C₆₄	C₄	semidirect
36. C₆₄	Dic₆₁	semidirect
37. C₆₁	#7548	semidirect

Maximal subgroups

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

subgroup	index
#21444023	2
#21444021	2
C₆₁×M₁₂₈	2
#7548	61