Group #29589051 — subgroups

Subgroups — #29589051

Order 15616.

260 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447616	C₂	1	2	yes	maximal
2	#21444101	C₂	1	1	yes	maximal, characteristic
3	#21443738	C₂	1	2	yes	maximal, complemented
4	C₈×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839813	C₄	1	2	yes	—
7	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
8	#16839580	C₄	1	4	yes	—
9	#16839541	C₄	1	4	yes	—
10	C₄×#11758185	C₂²	1	2	yes	—
11	#16839484	C₄	1	2	yes	—
12	C₈×#92418	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₈²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
16	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂×#11758185	C₂×C₄	1	4	yes	—
18	#14519706	C₈	1	4	yes	—
19	#14519689	C₈	1	4	yes	—
20	C₄×#92418	C₂×C₄	1	1	yes	characteristic
21	C₄×#92418	C₂×C₄	1	2	yes	—
22	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₆₁	C₄²	1	1	yes	characteristic
27	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
29	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
30	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
32	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
33	C₂×#92418	C₄²	1	2	yes	—
34	C₂×#92418	C₂×C₈	1	4	yes	—
35	C₈×D₆₁	—	2	4	no	—
36	C₁₆×C₆₁	—	2	4	no	—
37	#11758185	—	2	4	no	—
38	C₂²×D₆₁	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
41	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
42	C₂×Dic₆₁	C₄×C₈	1	1	yes	characteristic
43	C₄×D₆₁	—	2	2	no	—
44	C₄×D₆₁	—	2	4	no	—
45	C₈×C₆₁	—	2	4	no	—
46	C₈×C₆₁	#131	1	4	yes	—
47	#92418	—	2	2	no	—
48	#92418	—	2	4	no	—
49	#11412	—	61	1	no	maximal, Sylow
50	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
51	C₂×D₆₁	—	2	2	no	—
52	C₂×D₆₁	—	2	1	no	—
53	C₄×C₆₁	#405	1	2	yes	—
54	C₄×C₆₁	#504	1	4	yes	—
55	Dic₆₁	—	2	2	no	—
56	Dic₆₁	—	2	1	no	—
57	#2162	—	61	2	no	—
58	#1450	—	61	1	no	—
59	#1312	—	61	2	no	—
60	#1311	D₆₁	1	1	yes	characteristic, complemented
61	C₈×C₁₆	—	61	1	no	—
62	C₂×C₆₁	#1450	1	1	yes	characteristic
63	C₂×C₆₁	#2108	1	2	yes	—
64	D₆₁	—	4	1	no	—
65	#431	—	61	2	no	—
66	C₂×C₄×C₈	—	61	1	no	—
67	C₄⋊C₁₆	—	61	4	no	—
68	C₂²⋊C₁₆	—	61	4	no	—
69	C₄×C₁₆	—	61	2	no	—
70	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
71	#322	—	61	2	no	—
72	C₈²	—	61	1	no	—
73	C₈²	C₂×D₆₁	1	1	yes	characteristic
74	C₆₁	#11412	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	61	2	no	—
76	C₂×C₄²	—	61	1	no	—
77	C₂×C₁₆	—	61	4	no	—
78	C₂×C₁₆	C₄×D₆₁	1	4	yes	—
79	C₄⋊C₈	—	61	4	no	—
80	C₂²⋊C₈	—	61	4	no	—
81	C₄×C₈	—	61	2	no	—
82	C₄×C₈	—	61	2	no	—
83	C₄×C₈	—	61	1	no	—
84	C₄×C₈	C₄×D₆₁	1	2	yes	—
85	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
86	C₂²×C₄	—	61	1	no	—
87	C₂²×C₄	—	61	2	no	—
88	C₂×C₈	—	61	2	no	—
89	C₂×C₈	—	61	4	no	—
90	C₂×C₈	—	122	4	no	—
91	C₂×C₈	C₈×D₆₁	1	4	yes	—
92	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
93	C₄²	—	61	1	no	—
94	C₄²	—	61	2	no	—
95	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
96	C₁₆	—	2	4	no	—
97	C₁₆	—	122	4	no	—
98	C₂³	—	61	1	no	—
99	C₂×C₄	—	61	1	no	—
100	C₂×C₄	—	61	2	no	—
101	C₂×C₄	—	122	2	no	—
102	C₂×C₄	—	122	4	no	—
103	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
104	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	122	2	no	—
107	C₈	—	122	4	no	—
108	C₈	#14519783	1	4	yes	—
109	C₂²	—	122	1	no	—
110	C₂²	—	122	2	no	—
111	C₂²	C₄×C₈×D₆₁	1	1	yes	characteristic
112	C₄	—	122	1	no	—
113	C₄	—	122	2	no	—
114	C₄	#16839716	1	2	yes	—
115	C₄	#16840180	1	4	yes	—
116	C₂	—	244	1	no	—
117	C₂	#21444101	1	1	yes	characteristic
118	C₂	#21447417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447616

C₂

yes

maximal

#21444101

C₂

yes

maximal, characteristic

#21443738

C₂

yes

maximal, complemented

C₈×#11758185

C₂

yes

maximal, characteristic, complemented

C₆₁×#1311

C₂

yes

maximal, characteristic, complemented

#16839813

C₄

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839580

C₄

yes

—

#16839541

C₄

yes

—

C₄×#11758185

C₂²

yes

—

#16839484

C₄

yes

—

C₈×#92418

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₄²×D₆₁

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

—

#14519706

C₈

yes

—

#14519689

C₈

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂×C₄

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄×D₆₁

C₄²

yes

characteristic

C₂×C₄×D₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₄²

yes

characteristic

C₄×Dic₆₁

C₂×C₈

yes

—

C₂×#92418

C₄²

yes

—

C₂×#92418

C₂×C₈

yes

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

—

C₂²×D₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×Dic₆₁

C₄×C₈

yes

—

C₂×Dic₆₁

C₄×C₈

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#131

yes

—

—

—

—

C₂²×C₆₁

C₂×C₄×C₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#405

yes

—

C₄×C₆₁

#504

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

D₆₁

yes

characteristic, complemented

C₈×C₁₆

—

C₂×C₆₁

#1450

yes

characteristic

C₂×C₆₁

#2108

yes

—

D₆₁

—

#431

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₁

yes

—

#322

—

C₈²

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#11412

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₆₁

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₆₁

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₈×D₆₁

yes

—

C₂×C₈

C₂×C₄×D₆₁

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×D₆₁

yes

characteristic

C₁₆

—

C₁₆

—

122

—

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

122

—

102

C₂×C₄

—

122

—

103

C₂×C₄

C₄²×D₆₁

yes

characteristic

104

C₂×C₄

C₂×C₈×D₆₁

yes

—

105

C₈

—

106

C₈

—

122

—

107

C₈

—

122

—

108

C₈

#14519783

yes

—

109

C₂²

—

122

—

110

C₂²

—

122

—

111

C₂²

C₄×C₈×D₆₁

yes

characteristic

112

C₄

—

122

—

113

C₄

—

122

—

114

C₄

#16839716

yes

—

115

C₄

#16840180

yes

—

116

C₂

—

244

—

117

C₂

#21444101

yes

characteristic

118

C₂

#21447417

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₁
commutator, socle	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₆₁×#1311	C₂
prime radical	C₂²×D₆₁	C₄×C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1311	D₆₁
hypocenter	C₆₁	#11412
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄×C₈ ◁ #1311

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₆₁

commutator, socle

C₂²×C₆₁

C₂×C₄×C₈

Fitting

C₆₁×#1311

C₂

prime radical

C₂²×D₆₁

C₄×C₈

cosocle

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₄×C₈ ◁ #1311

N	H ≅ G/N	type
3. #21443738	C₂	semidirect
4. C₈×#11758185	C₂	semidirect
5. C₆₁×#1311	C₂	semidirect
60. #1311	D₆₁	semidirect
74. C₆₁	#11412	semidirect

H ≅ G/N

type

3. #21443738

C₂

semidirect

4. C₈×#11758185

C₂

semidirect

5. C₆₁×#1311

C₂

semidirect

60. #1311

D₆₁

semidirect

74. C₆₁

#11412

semidirect

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

subgroup	index
#21447616	2
#21444101	2
#21443738	2
C₈×#11758185	2
C₆₁×#1311	2
#11412	61

subgroup

index

C₆₁×#1311

Subgroups — #29589051

Conjugacy classes of subgroups

Complements

Maximal subgroups