Group #29579406 — subgroups

Subgroups — #29579406

Order 15616.

370 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 119 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	#21445950	C₂	1	1	yes	maximal, characteristic
2	#21444383	C₂	1	4	yes	maximal
3	#21444111	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#3160	C₂	1	1	yes	maximal, characteristic
5	#16839815	C₂²	1	2	yes	—
6	#16839628	C₂²	1	2	yes	—
7	#16839525	—	2	2	no	—
8	#16839499	—	2	8	no	—
9	#16839484	—	2	2	no	—
10	C₆₁×#533	C₂²	1	2	yes	—
11	C₆₁×#500	C₄	1	2	yes	complemented
12	C₆₁×#499	—	2	2	no	—
13	C₆₁×#498	C₄	1	2	yes	complemented
14	C₆₁×#427	—	2	2	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₁	—	2	2	no	—
17	C₄⋊#92418	—	2	2	no	—
18	C₂²⋊#92418	D₄	1	4	yes	—
19	C₄×#92418	—	2	2	no	—
20	#14519684	—	2	4	no	—
21	C₂²×C₈×C₆₁	D₄	1	2	yes	—
22	C₄⋊Q₈×C₆₁	C₂×C₄	1	4	yes	—
23	#125×C₆₁	—	2	2	no	—
24	C₂²⋊Q₈×C₆₁	—	2	4	no	—
25	C₄×Q₈×C₆₁	—	2	4	no	—
26	#117×C₆₁	—	2	2	no	—
27	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
28	#107×C₆₁	—	2	4	no	—
29	#106×C₆₁	—	2	4	no	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	D₄	1	2	yes	—
33	C₂×Q₈×C₆₁	—	2	4	no	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	—	2	4	no	—
38	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
40	C₄⋊C₄×C₆₁	—	2	4	no	—
41	C₄⋊C₄×C₆₁	—	2	4	no	—
42	C₄⋊C₄×C₆₁	—	4	4	no	—
43	C₂²⋊C₄×C₆₁	—	4	2	no	—
44	C₄²×C₆₁	—	4	2	no	—
45	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₆₁	#41	1	2	yes	—
48	C₄⋊Dic₆₁	—	4	4	no	—
49	C₂×#92418	—	2	4	no	—
50	C₂³×C₆₁	#127	1	1	yes	characteristic
51	Q₈×C₆₁	—	8	4	no	—
52	C₂×C₄×C₆₁	—	4	2	no	—
53	C₂×C₄×C₆₁	—	4	4	no	—
54	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₆₁	#121	1	2	yes	—
57	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#137	1	4	yes	—
59	C₂×Dic₆₁	—	8	4	no	—
60	C₂×Dic₆₁	—	8	2	no	—
61	C₈×C₆₁	—	2	2	no	—
62	C₈×C₆₁	—	2	2	no	—
63	#92418	—	4	4	no	—
64	#9759	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	—	2	2	no	—
66	C₂²×C₆₁	#390	1	1	yes	characteristic
67	C₂²×C₆₁	#497	1	2	yes	—
68	C₄×C₆₁	—	2	2	no	—
69	C₄×C₆₁	—	2	2	no	—
70	C₄×C₆₁	—	8	4	no	—
71	C₄×C₆₁	#419	1	2	yes	—
72	C₄×C₆₁	#491	1	2	yes	—
73	Dic₆₁	—	16	2	no	—
74	#3160	D₆₁	1	1	yes	characteristic
75	#1835	—	61	1	no	—
76	#1518	—	61	4	no	—
77	#1452	—	61	1	no	—
78	C₂×C₆₁	—	2	2	no	—
79	C₂×C₆₁	#1960	1	1	yes	characteristic
80	C₂×C₆₁	#1976	1	2	yes	—
81	#533	C₂×D₆₁	1	2	yes	—
82	#500	Dic₆₁	1	2	yes	complemented
83	#499	—	2	2	no	—
84	#498	Dic₆₁	1	2	yes	complemented
85	#431	—	61	2	no	—
86	#427	—	2	2	no	—
87	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	#382	—	61	2	no	—
89	#340	—	122	2	no	—
90	#328	—	122	8	no	—
91	#322	—	122	2	no	—
92	C₆₁	#9759	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
94	C₄⋊Q₈	C₂×Dic₆₁	1	4	yes	—
95	#125	—	2	2	no	—
96	C₂²⋊Q₈	—	2	4	no	—
97	C₄×Q₈	—	2	4	no	—
98	#117	—	2	2	no	—
99	C₂×C₄⋊C₄	—	122	2	no	—
100	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
101	#107	—	2	4	no	—
102	#106	—	2	4	no	—
103	C₄⋊C₈	—	122	2	no	—
104	C₂²⋊C₈	—	61	4	no	—
105	C₄×C₈	—	122	2	no	—
106	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
109	#95	—	122	4	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂²×C₄	—	244	2	no	—
112	C₂²×C₄	D₄×D₆₁	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	122	4	no	—
116	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
117	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	4	4	no	—
121	C₄⋊C₄	—	244	4	no	—
122	C₂²⋊C₄	—	4	2	no	—
123	C₄²	—	4	2	no	—
124	C₄²	#11758224	1	2	yes	—
125	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
126	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
127	C₂³	#14519773	1	1	yes	characteristic
128	Q₈	—	8	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	488	4	no	—
132	C₂×C₄	—	488	2	no	—
133	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
134	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
135	C₂×C₄	#14519748	1	2	yes	—
136	C₂×C₄	#14519764	1	1	yes	characteristic
137	C₂×C₄	#14519808	1	4	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	244	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#16839665	1	1	yes	characteristic
143	C₂²	#16840154	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	8	4	no	—
147	C₄	—	976	2	no	—
148	C₄	#16839771	1	2	yes	—
149	C₄	#16840130	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#21446640	1	1	yes	characteristic
152	C₂	#21446718	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445950

C₂

yes

maximal, characteristic

#21444383

C₂

yes

maximal

#21444111

C₂

yes

maximal, characteristic

C₆₁×#3160

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

#16839628

C₂²

yes

—

#16839525

—

#16839499

—

#16839484

—

C₆₁×#533

C₂²

yes

—

C₆₁×#500

C₄

yes

complemented

C₆₁×#499

—

C₆₁×#498

C₄

yes

complemented

C₆₁×#427

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

—

C₄⋊#92418

—

C₂²⋊#92418

D₄

yes

—

C₄×#92418

—

#14519684

—

C₂²×C₈×C₆₁

D₄

yes

—

C₄⋊Q₈×C₆₁

C₂×C₄

yes

—

#125×C₆₁

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#107×C₆₁

—

#106×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂³×C₆₁

#127

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#9759

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#390

yes

characteristic

C₂²×C₆₁

#497

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#419

yes

—

C₄×C₆₁

#491

yes

—

Dic₆₁

—

#3160

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#1960

yes

characteristic

C₂×C₆₁

#1976

yes

—

#533

C₂×D₆₁

yes

—

#500

Dic₆₁

yes

complemented

#499

—

#498

Dic₆₁

yes

complemented

#431

—

#427

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#382

—

#340

—

122

—

#328

—

122

—

#322

—

122

—

C₆₁

#9759

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₆₁

yes

—

C₄⋊Q₈

C₂×Dic₆₁

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

122

—

100

C₂×C₄²

C₂²×D₆₁

yes

characteristic

101

#107

—

102

#106

—

103

C₄⋊C₈

—

122

—

104

C₂²⋊C₈

—

105

C₄×C₈

—

122

—

106

C₄×C₈

C₂×Dic₆₁

yes

characteristic

107

C₄×C₈

C₂×Dic₆₁

yes

characteristic

108

C₄×C₈

C₂²⋊D₆₁

yes

—

109

#95

—

122

—

110

C₂×Q₈

—

111

C₂²×C₄

—

244

—

112

C₂²×C₄

D₄×D₆₁

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

122

—

116

C₂×C₈

C₂²⋊Dic₆₁

yes

—

117

C₂×C₈

C₂²⋊Dic₆₁

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

244

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

#11758224

yes

—

125

C₄²

C₂²×Dic₆₁

yes

characteristic

126

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

127

C₂³

#14519773

yes

characteristic

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

488

—

132

C₂×C₄

—

488

—

133

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

134

C₂×C₄

D₄×Dic₆₁

yes

—

135

C₂×C₄

#14519748

yes

—

136

C₂×C₄

#14519764

yes

characteristic

137

C₂×C₄

#14519808

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

244

—

141

C₂²

—

142

C₂²

#16839665

yes

characteristic

143

C₂²

#16840154

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

976

—

148

C₄

#16839771

yes

—

149

C₄

#16840130

yes

—

150

C₂

—

151

C₂

#21446640

yes

characteristic

152

C₂

#21446718

yes

—

	subgroup	G/H
center	C₂²	#16839665
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#3160	C₂
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#3160	D₆₁
hypocenter	C₆₁	#9759
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂²

#16839665

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#3160

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
11. C₆₁×#500	C₄	semidirect
13. C₆₁×#498	C₄	semidirect
82. #500	Dic₆₁	semidirect
84. #498	Dic₆₁	semidirect
92. C₆₁	#9759	semidirect

H ≅ G/N

type

11. C₆₁×#500

C₄

semidirect

13. C₆₁×#498

C₄

semidirect

82. #500

Dic₆₁

semidirect

84. #498

Dic₆₁

semidirect

92. C₆₁

#9759

semidirect

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

subgroup	index
#21445950	2
#21444383	2
#21444111	2
C₆₁×#3160	2
#9759	61

subgroup

index

C₆₁×#3160

Subgroups — #29579406

Conjugacy classes of subgroups

Complements

Maximal subgroups