Subgroups — #30594010

Order 15616.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460732	C₂	1	2	yes	maximal
2	#21443735	C₂	1	1	yes	maximal, characteristic
3	C₄×#11758187	C₄	1	4	yes	—
4	C₄×#11758185	C₄	1	1	yes	characteristic
5	C₈×#92418	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
7	C₂×#11758187	—	2	4	no	—
8	C₂×#11758185	C₈	1	2	yes	—
9	C₄×#92418	C₂×C₄	1	2	yes	—
10	C₂×C₁₆×C₆₁	C₈	1	2	yes	—
11	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
13	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
14	C₄²×C₆₁	C₄²	1	1	yes	characteristic
15	C₂×#92418	M₁₆	1	4	yes	—
16	#11758187	—	4	4	no	—
17	C₁₆×C₆₁	C₁₆	1	4	yes	complemented
18	#11758185	—	2	2	no	—
19	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
20	C₂×C₄×C₆₁	#97	1	2	yes	—
21	C₈×C₆₁	C₂×C₁₆	1	2	yes	—
22	C₈×C₆₁	M₃₂	1	2	yes	—
23	#92418	—	2	4	no	—
24	#7053	—	61	1	no	maximal, Sylow
25	C₂²×C₆₁	#322	1	1	yes	characteristic
26	C₄×C₆₁	C₄×C₁₆	1	1	yes	characteristic
27	C₄×C₆₁	#346	1	1	yes	characteristic
28	C₄×C₆₁	#347	1	4	yes	—
29	#1312	—	61	2	no	—
30	#1311	—	61	1	no	—
31	C₂×C₆₁	#1312	1	1	yes	characteristic
32	C₂×C₆₁	#1313	1	2	yes	—
33	C₄×C₁₆	—	61	4	no	—
34	C₄×C₁₆	—	61	1	no	—
35	C₄×C₁₆	#6923	1	1	yes	characteristic
36	C₈²	—	61	1	no	—
37	C₆₁	#7053	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	—	61	2	no	—
39	C₂×C₁₆	—	122	4	no	—
40	C₂×C₁₆	#92420	1	2	yes	—
41	C₄×C₈	—	61	2	no	—
42	C₄×C₈	C₂×#6923	1	1	yes	characteristic
43	C₂×C₈	—	61	4	no	—
44	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
45	C₂×C₈	C₂×#92420	1	1	yes	characteristic
46	C₄²	C₄×#6923	1	1	yes	characteristic
47	C₁₆	—	122	2	no	—
48	C₁₆	—	244	4	no	—
49	C₁₆	#11758187	1	4	yes	complemented
50	C₂×C₄	C₄×#92420	1	1	yes	characteristic
51	C₂×C₄	#14519831	1	2	yes	—
52	C₈	—	122	4	no	—
53	C₈	C₂×#11758187	1	2	yes	—
54	C₈	#14519852	1	2	yes	—
55	C₂²	#16840603	1	1	yes	characteristic
56	C₄	C₄×#11758187	1	1	yes	characteristic
57	C₄	#16840651	1	1	yes	characteristic
58	C₄	#16840653	1	4	yes	—
59	C₂	#21460732	1	1	yes	characteristic
60	C₂	#21460736	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×#6923
commutator	C₄×C₆₁	C₄×C₁₆
Frattini	C₄×C₈	C₂×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle, prime radical	C₂²×C₆₁	#322
cosocle	C₈×#92418	C₂²
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#7053
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
17. C₁₆×C₆₁	C₁₆	semidirect
37. C₆₁	#7053	semidirect
49. C₁₆	#11758187	semidirect

Maximal subgroups

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

subgroup	index
#21460732	2
#21443735	2
#7053	61