Subgroups — #29567818

Order 15616.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839481	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839481	C₄	1	4	yes	complemented
4	C₂×C₃₂×C₆₁	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₆₁	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₆₁	C₈	1	1	yes	characteristic
10	C₃₂×C₆₁	C₈	1	2	yes	—
11	C₃₂×C₆₁	C₂×C₄	1	2	yes	—
12	C₂×C₈×C₆₁	C₁₆	1	1	yes	characteristic
13	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
14	C₄²×C₆₁	C₁₆	1	1	yes	characteristic
15	C₁₆×C₆₁	C₁₆	1	2	yes	—
16	C₁₆×C₆₁	C₄²	1	1	yes	characteristic
17	C₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₆₁	C₃₂	1	2	yes	—
19	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
20	C₈×C₆₁	C₃₂	1	2	yes	—
21	C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
22	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
23	#7510	—	61	1	no	maximal, Sylow
24	C₂²×C₆₁	C₂×C₃₂	1	1	yes	characteristic
25	C₄×C₆₁	—	2	2	no	—
26	C₄×C₆₁	C₄×C₁₆	1	1	yes	characteristic
27	C₄×C₆₁	C₂×C₃₂	1	1	yes	characteristic
28	C₂×C₆₄	—	61	2	no	—
29	C₄×C₃₂	D₆₁	1	1	yes	characteristic
30	C₂×C₆₁	C₄×C₃₂	1	1	yes	characteristic
31	C₂×C₆₁	M₁₂₈	1	2	yes	—
32	C₂×C₃₂	Dic₆₁	1	1	yes	characteristic
33	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
34	C₄×C₁₆	Dic₆₁	1	1	yes	characteristic
35	C₆₄	—	61	4	no	—
36	C₆₁	#7510	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₁₆	#92418	1	1	yes	characteristic
38	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
39	C₄×C₈	#92418	1	1	yes	characteristic
40	C₃₂	#92418	1	2	yes	—
41	C₃₂	C₄×D₆₁	1	2	yes	—
42	C₂×C₈	#11758185	1	1	yes	characteristic
43	C₂×C₈	C₂×#92418	1	1	yes	characteristic
44	C₄²	#11758185	1	1	yes	characteristic
45	C₁₆	#11758185	1	2	yes	—
46	C₁₆	C₂×#92418	1	1	yes	characteristic
47	C₁₆	C₄×Dic₆₁	1	1	yes	characteristic
48	C₂×C₄	#14519683	1	2	yes	—
49	C₂×C₄	C₂×#11758185	1	1	yes	characteristic
50	C₈	#14519683	1	2	yes	—
51	C₈	C₄×#92418	1	1	yes	characteristic
52	C₈	C₂×#11758185	1	1	yes	characteristic
53	C₂²	C₂×#14519683	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	C₄×#11758185	1	1	yes	characteristic
56	C₄	C₂×#14519683	1	1	yes	characteristic
57	C₂	C₄×#14519683	1	1	yes	characteristic
58	C₂	#21444030	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₆₁
commutator	C₂×C₆₁	C₄×C₃₂
Fitting	C₄×C₃₂×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	C₂×C₃₂
cosocle	C₂×C₃₂×C₆₁	C₂²
hypercenter	C₄×C₃₂	D₆₁
hypocenter	C₆₁	#7510
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
3. #16839481	C₄	semidirect
36. C₆₁	#7510	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839481	2
C₄×C₃₂×C₆₁	2
#7510	61