Group #29572469 — subgroups

Subgroups — #29572469

Order 15616.

656 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 147 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	#21446789	C₂	1	3	yes	maximal
2	C₂×#16839504	C₂	1	3	yes	maximal
3	C₂×C₆₁×#564	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519688	C₂²	1	3	yes	—
5	#16839632	C₂²	1	1	yes	characteristic
6	#16839524	—	2	6	no	—
7	#16839504	—	2	6	no	—
8	C₆₁×#564	—	2	6	no	—
9	C₆₁×#564	C₄	1	4	yes	complemented
10	C₂×#125×C₆₁	—	2	6	no	—
11	C₂×#125×C₆₁	C₂²	1	3	yes	—
12	C₂²×#92418	—	2	3	no	—
13	C₂×C₄⋊Dic₆₁	—	2	3	no	—
14	#14519688	—	2	6	no	—
15	#125×C₆₁	—	2	24	no	—
16	#125×C₆₁	—	2	24	no	—
17	#125×C₆₁	—	2	6	no	—
18	#125×C₆₁	C₂×C₄	1	6	yes	—
19	#125×C₆₁	D₄	1	6	yes	—
20	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
21	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
22	C₂×C₄⋊C₄×C₆₁	D₄	1	6	yes	—
23	C₂×C₄²×C₆₁	—	2	2	no	—
24	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	—	2	6	no	—
26	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
27	C₂²×Dic₆₁	—	4	1	no	—
28	C₄⋊C₄×C₆₁	—	2	24	no	—
29	C₄⋊C₄×C₆₁	—	2	12	no	—
30	C₄⋊C₄×C₆₁	—	2	12	no	—
31	C₄⋊C₄×C₆₁	—	2	6	no	—
32	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	12	yes	—
33	C₄²×C₆₁	—	2	8	no	—
34	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₆₁	C₂×D₄	1	3	yes	—
36	C₄⋊Dic₆₁	—	4	6	no	—
37	C₂×#92418	—	2	6	no	—
38	C₂×#92418	—	4	3	no	—
39	C₂×#92418	—	4	3	no	—
40	C₂³×C₆₁	#120	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	3	no	—
42	C₂×C₄×C₆₁	—	2	3	no	—
43	C₂×C₄×C₆₁	—	2	12	no	—
44	C₂×C₄×C₆₁	—	4	6	no	—
45	C₂×C₄×C₆₁	—	4	6	no	—
46	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
47	C₂×C₄×C₆₁	#120	1	3	yes	—
48	C₂×Dic₆₁	—	8	3	no	—
49	#92418	—	4	6	no	—
50	#8509	—	61	1	no	maximal, Sylow
51	C₂²×C₆₁	#379	1	1	yes	characteristic
52	C₂²×C₆₁	#454	1	3	yes	—
53	C₂²×C₆₁	#456	1	3	yes	—
54	C₄×C₆₁	—	2	3	no	—
55	C₄×C₆₁	—	2	3	no	—
56	C₄×C₆₁	—	4	12	no	—
57	Dic₆₁	—	16	1	no	—
58	C₂×#564	D₆₁	1	1	yes	characteristic
59	#1992	—	61	3	no	—
60	C₂×#330	—	61	3	no	—
61	C₂×C₆₁	#1658	1	1	yes	characteristic
62	C₂×C₆₁	#1663	1	3	yes	—
63	C₂×C₆₁	#1793	1	3	yes	—
64	#564	—	2	6	no	—
65	#564	Dic₆₁	1	4	yes	complemented
66	C₂×#125	—	2	6	no	—
67	C₂×#125	C₂×D₆₁	1	3	yes	—
68	C₂×#97	—	61	3	no	—
69	#384	—	61	1	no	—
70	#340	—	122	6	no	—
71	#330	—	122	6	no	—
72	C₆₁	#8509	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	122	3	no	—
74	#125	—	2	24	no	—
75	#125	—	2	24	no	—
76	#125	—	2	6	no	—
77	#125	C₂×Dic₆₁	1	6	yes	—
78	#125	C₂²⋊D₆₁	1	6	yes	—
79	C₂×C₄⋊C₄	—	2	6	no	—
80	C₂×C₄⋊C₄	—	2	6	no	—
81	C₂×C₄⋊C₄	—	122	3	no	—
82	C₂×C₄⋊C₄	C₂²⋊D₆₁	1	6	yes	—
83	C₂×C₄²	—	2	2	no	—
84	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
85	#97	—	122	6	no	—
86	C₂²×C₄	—	2	6	no	—
87	C₂²×C₄	—	244	1	no	—
88	C₂²×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
89	C₂×C₈	—	122	6	no	—
90	C₂×C₈	—	244	3	no	—
91	C₂×C₈	—	244	3	no	—
92	C₄⋊C₄	—	2	24	no	—
93	C₄⋊C₄	—	2	12	no	—
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	—	244	6	no	—
97	C₄⋊C₄	C₂²⋊Dic₆₁	1	12	yes	—
98	C₄²	—	2	8	no	—
99	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
100	C₄²	C₂×C₂²⋊D₆₁	1	3	yes	—
101	C₂³	#14519746	1	1	yes	characteristic
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	—	488	3	no	—
108	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	3	yes	—
109	C₂×C₄	#14519746	1	3	yes	—
110	C₈	—	244	6	no	—
111	C₂²	#16839618	1	1	yes	characteristic
112	C₂²	#16839935	1	3	yes	—
113	C₂²	#16839945	1	3	yes	—
114	C₄	—	2	3	no	—
115	C₄	—	2	3	no	—
116	C₄	—	4	12	no	—
117	C₄	—	976	1	no	—
118	C₂	#21445121	1	1	yes	characteristic
119	C₂	#21445147	1	3	yes	—
120	C₂	#21445778	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446789

C₂

yes

maximal

C₂×#16839504

C₂

yes

maximal

C₂×C₆₁×#564

C₂

yes

maximal, characteristic

C₂×#14519688

C₂²

yes

—

#16839632

C₂²

yes

characteristic

#16839524

—

#16839504

—

C₆₁×#564

—

C₆₁×#564

C₄

yes

complemented

C₂×#125×C₆₁

—

C₂×#125×C₆₁

C₂²

yes

—

C₂²×#92418

—

C₂×C₄⋊Dic₆₁

—

#14519688

—

#125×C₆₁

—

#125×C₆₁

—

#125×C₆₁

—

#125×C₆₁

C₂×C₄

yes

—

#125×C₆₁

D₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#120

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

#120

yes

—

C₂×Dic₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#456

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

C₂×#564

D₆₁

yes

characteristic

#1992

—

C₂×#330

—

C₂×C₆₁

#1658

yes

characteristic

C₂×C₆₁

#1663

yes

—

C₂×C₆₁

#1793

yes

—

#564

—

#564

Dic₆₁

yes

complemented

C₂×#125

—

C₂×#125

C₂×D₆₁

yes

—

C₂×#97

—

#384

—

#340

—

122

—

#330

—

122

—

C₆₁

#8509

yes

characteristic, complemented, Sylow

C₂²×C₈

—

122

—

—

—

—

C₂×Dic₆₁

yes

—

#125

C₂²⋊D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂²⋊D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#97

—

122

—

C₂²×C₄

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

—

244

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

244

—

C₄⋊C₄

C₂²⋊Dic₆₁

yes

—

C₄²

—

C₄²

C₂²×Dic₆₁

yes

characteristic

100

C₄²

C₂×C₂²⋊D₆₁

yes

—

101

C₂³

#14519746

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

488

—

108

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

—

109

C₂×C₄

#14519746

yes

—

110

C₈

—

244

—

111

C₂²

#16839618

yes

characteristic

112

C₂²

#16839935

yes

—

113

C₂²

#16839945

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

976

—

118

C₂

#21445121

yes

characteristic

119

C₂

#21445147

yes

—

120

C₂

#21445778

yes

—

	subgroup	G/H
center	C₂³	#14519746
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#564	C₂
socle, prime radical	C₂³×C₆₁	#120
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#564	D₆₁
hypocenter	C₆₁	#8509
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

subgroup

G/H

center

C₂³

#14519746

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₆₁×#564

C₂

socle, prime radical

C₂³×C₆₁

#120

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

N	H ≅ G/N	type
9. C₆₁×#564	C₄	semidirect
65. #564	Dic₆₁	semidirect
72. C₆₁	#8509	semidirect

H ≅ G/N

type

9. C₆₁×#564

C₄

semidirect

65. #564

Dic₆₁

semidirect

72. C₆₁

#8509

semidirect

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

subgroup	index
#21446789	2
C₂×#16839504	2
C₂×C₆₁×#564	2
#8509	61

subgroup

index

#21446789

C₂×#16839504

C₂×C₆₁×#564

#8509

Subgroups — #29572469

Conjugacy classes of subgroups

Complements

Maximal subgroups