Subgroups — #29567687

Order 15616.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21444017	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#92418	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1422	C₂	1	1	yes	maximal, characteristic
4	#16839596	—	2	1	no	—
5	C₄×#11758185	C₄	1	1	yes	characteristic
6	C₈×#92418	C₄	1	1	yes	characteristic
7	C₆₁×M₆₄	—	2	1	no	—
8	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×#11758185	C₈	1	1	yes	characteristic
10	C₂×#11758185	C₈	1	1	yes	characteristic
11	C₄×#92418	—	2	1	no	—
12	#14519683	—	4	1	no	—
13	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₆₁	—	4	1	no	—
17	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
19	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
20	C₂×#92418	—	2	2	no	—
21	C₁₆×C₆₁	—	2	1	no	—
22	C₁₆×C₆₁	D₈	1	1	yes	characteristic
23	C₁₆×C₆₁	Q₁₆	1	1	yes	characteristic
24	#11758185	—	2	1	no	—
25	#11758185	C₁₆	1	2	yes	—
26	C₂×C₄×C₆₁	—	2	1	no	—
27	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₆₁	#107	1	1	yes	characteristic
29	C₈×C₆₁	#108	1	1	yes	characteristic
30	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₆₁	M₃₂	1	1	yes	characteristic
32	#92418	—	2	4	no	—
33	#7459	—	61	1	no	maximal, Sylow
34	C₂²×C₆₁	#364	1	1	yes	characteristic
35	C₄×C₆₁	—	2	2	no	—
36	C₄×C₆₁	#335	1	1	yes	characteristic
37	C₄×C₆₁	C₄⋊C₁₆	1	1	yes	characteristic
38	#1422	—	61	1	no	—
39	#1422	D₆₁	1	1	yes	characteristic
40	C₈×C₁₆	—	61	1	no	—
41	C₂×C₆₁	—	2	1	no	—
42	C₂×C₆₁	#1370	1	1	yes	characteristic
43	M₆₄	—	2	1	no	—
44	M₆₄	—	122	1	no	—
45	C₄×C₁₆	—	61	1	no	—
46	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
47	C₈²	—	61	1	no	—
48	C₆₁	#7459	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₁₆	—	61	1	no	—
50	C₂×C₁₆	—	61	1	no	—
51	C₂×C₁₆	Dic₁₂₂	1	1	yes	characteristic
52	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
53	C₄×C₈	—	122	1	no	—
54	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
55	C₃₂	—	4	1	no	—
56	C₃₂	—	244	1	no	—
57	C₂×C₈	—	122	2	no	—
58	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
59	C₂×C₈	#11758196	1	1	yes	characteristic
60	C₄²	#11758189	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	—	61	2	no	—
63	C₁₆	—	122	1	no	—
64	C₁₆	#11758199	1	1	yes	characteristic
65	C₁₆	#11758202	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	#14519706	1	1	yes	characteristic
68	C₈	—	122	4	no	—
69	C₈	#14519710	1	1	yes	characteristic
70	C₈	#14519712	1	1	yes	characteristic
71	C₈	C₁₆×D₆₁	1	1	yes	characteristic
72	C₈	#14519716	1	1	yes	characteristic
73	C₂²	#16839582	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	#16839514	1	1	yes	characteristic
76	C₄	#16839580	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	#21443888	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#11758199
commutator	C₈×C₆₁	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₆₁
Fitting	C₆₁×#1422	C₂
socle	C₂×C₆₁	#1370
prime radical	C₂²×C₆₁	#364
cosocle	C₄×C₁₆×C₆₁	C₂²
hypercenter	#1422	D₆₁
hypocenter	C₆₁	#7459
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1422

Complements

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

N	H ≅ G/N	type
48. C₆₁	#7459	semidirect

Maximal subgroups

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

subgroup	index
#21444017	2
C₁₆×#92418	2
C₆₁×#1422	2
#7459	61