Subgroups — #29567821

Order 15616.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₂×D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839481	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×D₆₁	C₄	1	1	yes	characteristic
5	C₂×#14519683	C₄	1	1	yes	characteristic
6	C₃₂×D₆₁	—	2	1	no	—
7	C₃₂×D₆₁	—	2	1	no	—
8	#16839481	—	2	1	no	—
9	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×C₆₄	—	2	1	no	—
11	C₂×C₈×D₆₁	C₈	1	1	yes	characteristic
12	C₂×#11758185	C₈	1	1	yes	characteristic
13	C₁₆×D₆₁	—	2	1	no	—
14	C₁₆×D₆₁	—	2	1	no	—
15	#14519683	—	2	1	no	—
16	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₆₁	D₄	1	1	yes	characteristic
18	C₃₂×C₆₁	D₄	1	1	yes	characteristic
19	C₂×C₄×D₆₁	C₁₆	1	1	yes	characteristic
20	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
21	C₂×#92418	C₁₆	1	1	yes	characteristic
22	C₈×D₆₁	—	2	1	no	—
23	C₈×D₆₁	—	2	1	no	—
24	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₁₆×C₆₁	M₁₆	1	1	yes	characteristic
26	#11758185	—	2	1	no	—
27	C₂²×D₆₁	C₃₂	1	1	yes	characteristic
28	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
29	C₂×Dic₆₁	C₃₂	1	1	yes	characteristic
30	C₄×D₆₁	—	2	1	no	—
31	C₄×D₆₁	—	2	1	no	—
32	C₈×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₆₁	M₃₂	1	1	yes	characteristic
34	#92418	—	2	1	no	—
35	C₂²⋊C₆₄	—	61	1	no	maximal, Sylow
36	C₂²×C₆₁	C₂×C₃₂	1	1	yes	characteristic
37	C₂×D₆₁	—	2	1	no	—
38	C₂×D₆₁	—	2	1	no	—
39	C₂×D₆₁	C₆₄	1	2	yes	complemented
40	C₄×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
41	C₄×C₆₁	M₆₄	1	1	yes	characteristic
42	Dic₆₁	—	2	1	no	—
43	C₂²×C₃₂	—	61	1	no	—
44	C₂×C₆₄	—	61	1	no	—
45	C₂×C₆₄	D₆₁	1	1	yes	characteristic, complemented
46	C₂×C₆₁	C₂²⋊C₃₂	1	1	yes	characteristic
47	C₂×C₆₁	C₂×C₆₄	1	1	yes	characteristic
48	C₂×C₆₁	M₁₂₈	1	1	yes	characteristic
49	D₆₁	—	2	2	no	—
50	C₂²×C₁₆	—	61	1	no	—
51	C₂×C₃₂	—	61	1	no	—
52	C₂×C₃₂	—	122	1	no	—
53	C₂×C₃₂	—	122	1	no	—
54	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
55	C₆₄	—	2	1	no	—
56	C₆₄	—	122	1	no	—
57	C₆₁	C₂²⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	—	61	1	no	—
59	C₂×C₁₆	—	61	1	no	—
60	C₂×C₁₆	—	122	1	no	—
61	C₂×C₁₆	—	122	1	no	—
62	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
63	C₃₂	—	122	1	no	—
64	C₃₂	D₂₄₄	1	1	yes	characteristic
65	C₃₂	C₂²⋊D₆₁	1	1	yes	characteristic
66	C₂²×C₄	—	61	1	no	—
67	C₂×C₈	—	61	1	no	—
68	C₂×C₈	—	122	1	no	—
69	C₂×C₈	—	122	1	no	—
70	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
71	C₁₆	—	122	1	no	—
72	C₁₆	#11758189	1	1	yes	characteristic
73	C₁₆	#11758198	1	1	yes	characteristic
74	C₂³	—	61	1	no	—
75	C₂×C₄	—	61	1	no	—
76	C₂×C₄	—	122	1	no	—
77	C₂×C₄	—	122	1	no	—
78	C₂×C₄	C₁₆×D₆₁	1	1	yes	characteristic
79	C₈	—	122	1	no	—
80	C₈	#14519689	1	1	yes	characteristic
81	C₈	#14519716	1	1	yes	characteristic
82	C₂²	—	61	2	no	—
83	C₂²	—	122	1	no	—
84	C₂²	—	122	1	no	—
85	C₂²	C₃₂×D₆₁	1	1	yes	characteristic
86	C₄	—	122	1	no	—
87	C₄	#16839541	1	1	yes	characteristic
88	C₄	#16839595	1	1	yes	characteristic
89	C₂	—	122	2	no	—
90	C₂	#21443956	1	1	yes	characteristic
91	C₂	C₆₄×D₆₁	1	1	yes	characteristic
92	C₂	#21444029	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#16839481	C₂	semidirect
3. C₂×C₆₁×C₆₄	C₂	semidirect
39. C₂×D₆₁	C₆₄	semidirect
45. C₂×C₆₄	D₆₁	semidirect
57. C₆₁	C₂²⋊C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₂×D₆₁	2
C₂×#16839481	2
C₂×C₆₁×C₆₄	2
C₂²⋊C₆₄	61