Subgroups — #29601051

Order 15616.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448112	C₂	1	1	yes	maximal, characteristic
2	#21444030	C₂	1	1	yes	maximal, characteristic, complemented
3	#21444029	C₂	1	2	yes	maximal, complemented
4	C₆₄×D₆₁	C₂	1	2	yes	maximal, complemented
5	C₂×C₆₁×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840178	C₄	1	1	yes	characteristic
7	#16839596	C₄	1	1	yes	characteristic
8	#16839595	C₄	1	2	yes	—
9	C₃₂×D₆₁	C₂²	1	2	yes	—
10	#16839481	C₂²	1	2	yes	—
11	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×C₆₄	C₂²	1	1	yes	characteristic
13	C₆₁×C₆₄	C₂²	1	1	yes	characteristic
14	#14519783	C₈	1	1	yes	characteristic
15	#14519717	C₈	1	1	yes	characteristic
16	#14519716	C₈	1	2	yes	—
17	C₁₆×D₆₁	C₂×C₄	1	2	yes	—
18	#14519683	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₃₂×C₆₁	C₂³	1	1	yes	characteristic
22	#11758222	C₁₆	1	1	yes	characteristic
23	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
24	#11758194	C₁₆	1	1	yes	characteristic
25	#11758189	C₁₆	1	2	yes	—
26	C₈×D₆₁	C₂×C₈	1	2	yes	—
27	C₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
28	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	#11758185	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
31	C₂²⋊D₆₁	C₃₂	1	2	yes	—
32	D₂₄₄	C₃₂	1	1	yes	characteristic
33	C₄×D₆₁	C₂×C₁₆	1	2	yes	—
34	Dic₁₂₂	C₃₂	1	1	yes	characteristic
35	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
36	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
37	#92418	C₂×C₁₆	1	2	yes	—
38	#13737	—	61	1	no	maximal, Sylow
39	C₂²×C₆₁	C₂×C₃₂	1	1	yes	characteristic
40	C₂×D₆₁	C₂×C₃₂	1	2	yes	—
41	C₄×C₆₁	C₂×C₃₂	1	1	yes	characteristic
42	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
43	Dic₆₁	C₂×C₃₂	1	2	yes	—
44	#2258	—	61	1	no	—
45	M₁₂₈	—	61	1	no	—
46	M₁₂₈	—	61	2	no	—
47	C₂×C₆₄	—	61	2	no	—
48	C₂×C₆₄	D₆₁	1	1	yes	characteristic, complemented
49	C₂×C₆₁	—	2	1	no	—
50	C₂×C₆₁	C₂²×C₃₂	1	1	yes	characteristic
51	D₆₁	—	2	2	no	—
52	#504	—	61	1	no	—
53	M₆₄	—	61	1	no	—
54	M₆₄	—	61	2	no	—
55	C₂×C₃₂	—	61	2	no	—
56	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
57	C₆₄	—	61	2	no	—
58	C₆₄	C₂×D₆₁	1	1	yes	characteristic
59	C₆₄	C₂×D₆₁	1	1	yes	characteristic
60	C₆₁	#13737	1	1	yes	characteristic, complemented, Sylow
61	#131	—	61	1	no	—
62	M₃₂	—	61	1	no	—
63	M₃₂	—	61	2	no	—
64	C₂×C₁₆	—	61	2	no	—
65	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
66	C₃₂	—	61	2	no	—
67	C₃₂	C₄×D₆₁	1	1	yes	characteristic
68	C₃₂	C₂²×D₆₁	1	1	yes	characteristic
69	#41	—	61	1	no	—
70	M₁₆	—	61	1	no	—
71	M₁₆	—	61	2	no	—
72	C₂×C₈	—	61	2	no	—
73	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
74	C₁₆	—	61	2	no	—
75	C₁₆	C₈×D₆₁	1	1	yes	characteristic
76	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
77	Q₈	—	61	1	no	—
78	D₄	—	61	2	no	—
79	D₄	—	61	1	no	—
80	C₂×C₄	—	61	2	no	—
81	C₂×C₄	C₁₆×D₆₁	1	1	yes	characteristic
82	C₈	—	61	2	no	—
83	C₈	C₁₆×D₆₁	1	1	yes	characteristic
84	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
85	C₂²	—	61	2	no	—
86	C₂²	C₃₂×D₆₁	1	1	yes	characteristic
87	C₄	—	61	2	no	—
88	C₄	C₃₂×D₆₁	1	1	yes	characteristic
89	C₄	C₂×C₁₆×D₆₁	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	122	2	no	—
92	C₂	C₂×C₃₂×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₆₄	C₂×D₆₁
commutator, socle	C₂×C₆₁	C₂²×C₃₂
Frattini	C₃₂	C₂²×D₆₁
Fitting	C₂×C₆₁×C₆₄	C₂
prime radical	#11758222	C₁₆
cosocle	C₃₂×C₆₁	C₂³
hypercenter	C₂×C₆₄	D₆₁
hypocenter	C₆₁	#13737
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₆₄ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #21444030	C₂	semidirect
3. #21444029	C₂	semidirect
4. C₆₄×D₆₁	C₂	semidirect
5. C₂×C₆₁×C₆₄	C₂	semidirect
48. C₂×C₆₄	D₆₁	semidirect
60. C₆₁	#13737	semidirect

Maximal subgroups

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

subgroup	index
#21448112	2
#21444030	2
#21444029	2
C₆₄×D₆₁	2
C₂×C₆₁×C₆₄	2
#13737	61