Subgroups — #30595061

Order 15616.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460846	C₂	1	1	yes	maximal, characteristic
2	#21460844	C₂	1	1	yes	maximal, characteristic
3	#21443954	C₂	1	1	yes	maximal, characteristic
4	#16840684	—	2	1	no	—
5	#16840623	—	2	1	no	—
6	#16839596	C₄	1	1	yes	characteristic
7	#16839537	C₂²	1	1	yes	characteristic
8	C₆₁×M₆₄	C₄	1	1	yes	characteristic
9	#14519851	—	4	1	no	—
10	C₄⋊#92420	—	4	1	no	—
11	C₂×#11758185	Q₈	1	1	yes	characteristic
12	C₈×Dic₆₁	D₄	1	1	yes	characteristic
13	#14519683	—	2	1	no	—
14	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₆₁	C₈	1	2	yes	complemented
16	C₂×#92420	—	8	1	no	—
17	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
18	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
19	C₂×#92418	SD₁₆	1	1	yes	characteristic
20	#11758187	—	8	1	no	—
21	C₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
22	C₁₆×C₆₁	M₁₆	1	1	yes	characteristic
23	#11758185	—	2	1	no	—
24	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
25	C₂×Dic₆₁	—	2	1	no	—
26	#92420	—	16	1	no	—
27	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₆₁	#108	1	1	yes	characteristic
29	#92418	—	2	1	no	—
30	#7457	—	61	1	no	maximal, Sylow
31	C₂²×C₆₁	#365	1	1	yes	characteristic
32	C₄×C₆₁	#334	1	1	yes	characteristic
33	C₄×C₆₁	#365	1	1	yes	characteristic
34	Dic₆₁	—	4	1	no	—
35	#1398	—	61	1	no	—
36	#1369	—	61	1	no	—
37	#1368	—	61	1	no	—
38	C₂×C₆₁	—	2	1	no	—
39	C₂×C₆₁	#1368	1	1	yes	characteristic
40	M₆₄	—	61	1	no	—
41	M₆₄	#6923	1	1	yes	characteristic
42	#364	—	122	1	no	—
43	#346	—	61	1	no	—
44	#334	—	122	1	no	—
45	C₆₁	#7457	1	1	yes	characteristic, complemented, Sylow
46	M₃₂	—	244	1	no	—
47	C₂×C₁₆	—	61	1	no	—
48	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
49	C₄⋊C₈	—	244	1	no	—
50	C₄×C₈	—	61	1	no	—
51	C₃₂	—	122	1	no	—
52	C₃₂	#92420	1	2	yes	complemented
53	C₂×C₈	—	61	1	no	—
54	C₂×C₈	—	488	1	no	—
55	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
56	C₄²	—	61	1	no	—
57	C₁₆	—	122	1	no	—
58	C₁₆	—	488	1	no	—
59	C₁₆	#11758213	1	1	yes	characteristic
60	C₁₆	C₂×#92420	1	1	yes	characteristic
61	C₂×C₄	—	122	1	no	—
62	C₂×C₄	#14519845	1	1	yes	characteristic
63	C₈	—	122	1	no	—
64	C₈	—	976	1	no	—
65	C₈	C₄⋊#92420	1	1	yes	characteristic
66	C₈	#14519847	1	1	yes	characteristic
67	C₂²	#16840687	1	1	yes	characteristic
68	C₄	—	244	1	no	—
69	C₄	#16840623	1	1	yes	characteristic
70	C₄	#16840687	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#21460844	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21460844
commutator	C₁₆×C₆₁	C₂×C₈
Frattini	C₂×C₁₆	C₂×#6923
Fitting	C₆₁×M₆₄	C₄
socle	C₂×C₆₁	#1368
prime radical	C₂²×C₆₁	#365
cosocle	#16839537	C₂²
hypercenter	M₆₄	#6923
hypocenter	C₆₁	#7457
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₆₁	C₈	semidirect
45. C₆₁	#7457	semidirect
52. C₃₂	#92420	semidirect

Maximal subgroups

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

subgroup	index
#21460846	2
#21460844	2
#21443954	2
#7457	61