Subgroups — #30594012

Order 15616.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460732	C₂	1	1	yes	maximal, characteristic
2	#21460732	C₂	1	1	yes	maximal, characteristic
3	#21443735	C₂	1	1	yes	maximal, characteristic
4	C₄×#11758187	C₄	1	2	yes	—
5	C₄×#11758187	C₄	1	2	yes	—
6	C₄×#11758185	C₄	1	1	yes	characteristic
7	C₈×#92418	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
9	C₂×#11758187	—	2	2	no	—
10	C₂×#11758187	—	2	2	no	—
11	C₂×#11758185	C₈	1	2	yes	—
12	C₄×#92418	C₂×C₄	1	1	yes	characteristic
13	C₄×#92418	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₆₁	C₈	1	2	yes	—
15	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
17	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
18	C₄²×C₆₁	C₄²	1	1	yes	characteristic
19	C₂×#92418	M₁₆	1	2	yes	—
20	C₂×#92418	M₁₆	1	2	yes	—
21	#11758187	—	4	2	no	—
22	#11758187	—	4	2	no	—
23	C₁₆×C₆₁	—	2	2	no	—
24	#11758185	C₁₆	1	4	yes	complemented
25	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
27	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
28	C₈×C₆₁	C₂×C₁₆	1	2	yes	—
29	C₈×C₆₁	M₃₂	1	2	yes	—
30	#92418	—	2	2	no	—
31	#92418	—	2	2	no	—
32	#7053	—	61	1	no	maximal, Sylow
33	C₂²×C₆₁	#322	1	1	yes	characteristic
34	C₄×C₆₁	C₄×C₁₆	1	1	yes	characteristic
35	C₄×C₆₁	#346	1	1	yes	characteristic
36	C₄×C₆₁	#347	1	2	yes	—
37	C₄×C₆₁	#347	1	2	yes	—
38	#1312	—	61	1	no	—
39	#1312	—	61	1	no	—
40	#1311	—	61	1	no	—
41	C₂×C₆₁	#1312	1	1	yes	characteristic
42	C₂×C₆₁	#1313	1	1	yes	characteristic
43	C₂×C₆₁	#1313	1	1	yes	characteristic
44	C₄×C₁₆	—	61	2	no	—
45	C₄×C₁₆	—	61	2	no	—
46	C₄×C₁₆	—	61	1	no	—
47	C₄×C₁₆	#6923	1	1	yes	characteristic
48	C₈²	—	61	1	no	—
49	C₆₁	#7053	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	—	61	2	no	—
51	C₂×C₁₆	—	122	2	no	—
52	C₂×C₁₆	—	122	2	no	—
53	C₂×C₁₆	#92420	1	2	yes	—
54	C₄×C₈	—	61	1	no	—
55	C₄×C₈	—	61	1	no	—
56	C₄×C₈	C₂×#6923	1	1	yes	characteristic
57	C₂×C₈	—	61	2	no	—
58	C₂×C₈	—	61	2	no	—
59	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
60	C₂×C₈	C₂×#92420	1	1	yes	characteristic
61	C₄²	C₄×#6923	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	C₁₆	—	61	4	no	—
64	C₁₆	—	244	2	no	—
65	C₁₆	—	244	2	no	—
66	C₂×C₄	C₄×#92420	1	1	yes	characteristic
67	C₂×C₄	#14519831	1	1	yes	characteristic
68	C₂×C₄	#14519831	1	1	yes	characteristic
69	C₈	—	122	2	no	—
70	C₈	—	122	2	no	—
71	C₈	D₆₁⋊C₁₆	1	2	yes	—
72	C₈	#14519851	1	2	yes	—
73	C₂²	#16840603	1	1	yes	characteristic
74	C₄	#16840648	1	1	yes	characteristic
75	C₄	#16840650	1	1	yes	characteristic
76	C₄	#16840652	1	2	yes	—
77	C₄	#16840653	1	2	yes	—
78	C₂	#21460734	1	1	yes	characteristic
79	C₂	#21460735	1	1	yes	characteristic
80	C₂	#21460736	1	1	yes	characteristic

	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
24. #11758185	C₁₆	semidirect
49. C₆₁	#7053	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