Group #29585000 — subgroups

Subgroups — #29585000

Order 15616.

320 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 101 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	#21446759	C₂	1	1	yes	maximal, characteristic
2	#21445994	C₂	1	1	yes	maximal, characteristic
3	#21444594	C₂	1	2	yes	maximal
4	#21444249	C₂	1	2	yes	maximal
5	C₆₁×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	C₂×#14519713	—	2	2	no	—
8	#16839786	C₂²	1	2	yes	—
9	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
10	#16839516	—	2	2	no	—
11	#16839515	—	2	4	no	—
12	#16839494	—	2	4	no	—
13	C₆₁×#431	C₂²	1	2	yes	—
14	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₂₂	Q₈	1	1	yes	characteristic
16	C₂×#11758194	—	2	2	no	—
17	C₂²×#92418	—	2	1	no	—
18	C₄×Dic₁₂₂	—	2	2	no	—
19	C₄×Dic₁₂₂	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	Q₈	1	1	yes	characteristic
21	C₂×#11758196	—	2	1	no	—
22	C₂×C₄×Dic₆₁	—	2	1	no	—
23	#14519713	—	4	4	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	—	2	2	no	—
26	C₄⋊#92418	—	2	2	no	—
27	C₂²×C₈×C₆₁	D₄	1	2	yes	—
28	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₆₁	—	2	2	no	—
30	C₂²⋊C₈×C₆₁	—	2	2	no	—
31	#97×C₆₁	—	2	2	no	—
32	C₄×C₈×C₆₁	D₄	1	4	yes	—
33	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
36	C₂²×Dic₆₁	—	2	1	no	—
37	C₂²×Dic₆₁	—	2	1	no	—
38	C₂×Dic₁₂₂	—	2	2	no	—
39	C₂×Dic₁₂₂	—	4	2	no	—
40	C₂×C₈×C₆₁	—	2	2	no	—
41	C₂×C₈×C₆₁	—	2	2	no	—
42	C₂×C₈×C₆₁	—	2	4	no	—
43	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
44	C₄²×C₆₁	C₂×D₄	1	2	yes	—
45	C₄²×C₆₁	#41	1	2	yes	—
46	C₄⋊Dic₆₁	—	2	2	no	—
47	#11758196	—	4	2	no	—
48	C₄×Dic₆₁	—	4	1	no	—
49	C₄×Dic₆₁	—	4	1	no	—
50	#11758194	—	4	4	no	—
51	C₂×#92418	—	4	1	no	—
52	C₂×#92418	—	4	2	no	—
53	C₂×#92418	—	4	2	no	—
54	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	4	1	no	—
56	C₂×C₄×C₆₁	—	4	1	no	—
57	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₆₁	#123	1	2	yes	—
59	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
62	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
63	C₂×Dic₆₁	—	2	2	no	—
64	C₂×Dic₆₁	—	4	1	no	—
65	C₂×Dic₆₁	—	4	1	no	—
66	C₂×Dic₆₁	—	4	2	no	—
67	C₂×Dic₆₁	—	4	1	no	—
68	Dic₁₂₂	—	2	4	no	—
69	Dic₁₂₂	—	4	2	no	—
70	C₈×C₆₁	—	2	4	no	—
71	C₈×C₆₁	—	4	4	no	—
72	#92418	—	8	1	no	—
73	#92418	—	8	2	no	—
74	#10662	—	61	1	no	maximal, Sylow
75	C₂²×C₆₁	—	2	1	no	—
76	C₂²×C₆₁	—	2	1	no	—
77	C₂²×C₆₁	#399	1	1	yes	characteristic
78	C₂²×C₆₁	#477	1	2	yes	—
79	C₄×C₆₁	—	2	2	no	—
80	C₄×C₆₁	—	2	2	no	—
81	C₄×C₆₁	#484	1	2	yes	—
82	C₄×C₆₁	#494	1	1	yes	characteristic
83	C₄×C₆₁	#496	1	1	yes	characteristic
84	Dic₆₁	—	4	2	no	—
85	Dic₆₁	—	8	1	no	—
86	#1985	—	61	1	no	—
87	#1849	—	61	1	no	—
88	#1571	—	61	2	no	—
89	#1487	—	61	2	no	—
90	#1275	D₆₁	1	1	yes	characteristic
91	C₂×C₆₁	—	2	2	no	—
92	C₂×C₆₁	#2091	1	1	yes	characteristic
93	C₂×C₆₁	#2095	1	1	yes	characteristic
94	C₂×C₆₁	#2098	1	1	yes	characteristic
95	C₂×C₄×Q₈	—	61	1	no	—
96	#431	C₂×D₆₁	1	2	yes	—
97	C₂×#108	—	122	2	no	—
98	#423	—	61	2	no	—
99	C₂×C₄⋊C₈	—	61	1	no	—
100	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
101	#336	—	122	2	no	—
102	#335	—	122	4	no	—
103	Q₈⋊C₈	—	122	4	no	—
104	C₆₁	#10662	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	61	1	no	—
106	C₂×M₁₆	—	122	2	no	—
107	C₂²×C₈	—	122	1	no	—
108	C₂²×C₈	D₂₄₄	1	2	yes	—
109	C₄×Q₈	—	122	2	no	—
110	C₄×Q₈	—	122	2	no	—
111	C₂×C₄⋊C₄	—	61	1	no	—
112	C₂×C₄⋊C₄	—	122	1	no	—
113	C₂×C₄²	—	122	1	no	—
114	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
115	#108	—	244	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	122	2	no	—
118	C₄⋊C₈	—	122	2	no	—
119	C₄⋊C₈	—	122	2	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	D₂₄₄	1	4	yes	—
123	C₂×Q₈	—	122	2	no	—
124	C₂×Q₈	—	244	2	no	—
125	C₂²×C₄	—	122	1	no	—
126	C₂²×C₄	—	122	1	no	—
127	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
128	C₂²×C₄	#11758224	1	1	yes	characteristic
129	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
130	M₁₆	—	244	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	244	1	no	—
135	C₂×C₈	—	244	2	no	—
136	C₂×C₈	—	244	2	no	—
137	C₂×C₈	Dic₂₄₄	1	4	yes	—
138	C₄⋊C₄	—	122	2	no	—
139	C₄⋊C₄	—	244	2	no	—
140	C₄²	—	244	1	no	—
141	C₄²	—	244	1	no	—
142	C₄²	C₂×D₂₄₄	1	2	yes	—
143	C₄²	#11758224	1	2	yes	—
144	C₂³	#14519757	1	1	yes	characteristic
145	Q₈	—	122	4	no	—
146	Q₈	—	244	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	122	2	no	—
150	C₂×C₄	—	244	2	no	—
151	C₂×C₄	—	244	1	no	—
152	C₂×C₄	—	244	1	no	—
153	C₂×C₄	—	244	1	no	—
154	C₂×C₄	#14519757	1	2	yes	—
155	C₂×C₄	#14519761	1	2	yes	—
156	C₂×C₄	#14519761	1	1	yes	characteristic
157	C₂×C₄	#14519774	1	1	yes	characteristic
158	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
159	C₂×C₄	#14519805	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	4	4	no	—
162	C₈	—	488	1	no	—
163	C₈	—	488	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#16839699	1	1	yes	characteristic
167	C₂²	#16840058	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	488	1	no	—
171	C₄	—	244	2	no	—
172	C₄	#16840098	1	2	yes	—
173	C₄	#16840142	1	1	yes	characteristic
174	C₄	#16840151	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#21447336	1	1	yes	characteristic
177	C₂	#21447356	1	1	yes	characteristic
178	C₂	#21447373	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21446759

C₂

yes

maximal, characteristic

#21445994

C₂

yes

maximal, characteristic

#21444594

C₂

yes

maximal

#21444249

C₂

yes

maximal

C₆₁×#1275

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₂₂

C₂²

yes

characteristic

C₂×#14519713

—

#16839786

C₂²

yes

—

C₂×C₄⋊#92418

C₂²

yes

—

—

—

C₆₁×#431

C₂²

yes

—

C₂×#97×C₆₁

C₂²

yes

characteristic

C₂²×Dic₁₂₂

Q₈

yes

characteristic

C₂×#11758194

—

C₂²×#92418

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

—

C₂×C₄⋊Dic₆₁

Q₈

yes

characteristic

C₂×#11758196

—

C₂×C₄×Dic₆₁

—

—

—

—

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

#97×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

Q₁₆

yes

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

—

—

—

—

C₂³×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#399

yes

characteristic

C₂²×C₆₁

#477

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#484

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×Q₈

—

#431

C₂×D₆₁

yes

—

C₂×#108

—

122

—

#423

—

C₂×C₄⋊C₈

—

100

C₂×#97

C₂×D₆₁

yes

characteristic

101

#336

—

122

—

102

#335

—

122

—

103

Q₈⋊C₈

—

122

—

104

C₆₁

#10662

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×M₁₆

—

122

—

107

C₂²×C₈

—

122

—

108

C₂²×C₈

D₂₄₄

yes

—

109

C₄×Q₈

—

122

—

110

C₄×Q₈

—

122

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

122

—

113

C₂×C₄²

—

122

—

114

C₂×C₄²

C₂²×D₆₁

yes

characteristic

115

#108

—

244

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

122

—

118

C₄⋊C₈

—

122

—

119

C₄⋊C₈

—

122

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

D₂₄₄

yes

—

123

C₂×Q₈

—

122

—

124

C₂×Q₈

—

244

—

125

C₂²×C₄

—

122

—

126

C₂²×C₄

—

122

—

127

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

128

C₂²×C₄

#11758224

yes

characteristic

129

C₂²×C₄

Q₈×D₆₁

yes

characteristic

130

M₁₆

—

244

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

244

—

135

C₂×C₈

—

244

—

136

C₂×C₈

—

244

—

137

C₂×C₈

Dic₂₄₄

yes

—

138

C₄⋊C₄

—

122

—

139

C₄⋊C₄

—

244

—

140

C₄²

—

244

—

141

C₄²

—

244

—

142

C₄²

C₂×D₂₄₄

yes

—

143

C₄²

#11758224

yes

—

144

C₂³

#14519757

yes

characteristic

145

Q₈

—

122

—

146

Q₈

—

244

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

122

—

150

C₂×C₄

—

244

—

151

C₂×C₄

—

244

—

152

C₂×C₄

—

244

—

153

C₂×C₄

—

244

—

154

C₂×C₄

#14519757

yes

—

155

C₂×C₄

#14519761

yes

—

156

C₂×C₄

#14519761

yes

characteristic

157

C₂×C₄

#14519774

yes

characteristic

158

C₂×C₄

C₂×Dic₂₄₄

yes

—

159

C₂×C₄

#14519805

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

488

—

163

C₈

—

488

—

164

C₂²

—

165

C₂²

—

166

C₂²

#16839699

yes

characteristic

167

C₂²

#16840058

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

488

—

171

C₄

—

244

—

172

C₄

#16840098

yes

—

173

C₄

#16840142

yes

characteristic

174

C₄

#16840151

yes

characteristic

175

C₂

—

176

C₂

#21447336

yes

characteristic

177

C₂

#21447356

yes

characteristic

178

C₂

#21447373

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14519774
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1275	C₂
socle	C₂²×C₆₁	#399
prime radical	C₂³×C₆₁	C₂²⋊Q₈
hypercenter	#1275	D₆₁
hypocenter	C₆₁	#10662
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#14519774

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1275

C₂

socle

C₂²×C₆₁

#399

prime radical

C₂³×C₆₁

C₂²⋊Q₈

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
104. C₆₁	#10662	semidirect

H ≅ G/N

type

104. C₆₁

#10662

semidirect

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

subgroup	index
#21446759	2
#21445994	2
#21444594	2
#21444249	2
C₆₁×#1275	2
#10662	61

subgroup

index

C₆₁×#1275

Subgroups — #29585000

Conjugacy classes of subgroups

Complements

Maximal subgroups