Group #29583137 — subgroups

Subgroups — #29583137

Order 15616.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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	#21454206	C₂	1	1	yes	maximal, characteristic
2	#21446796	C₂	1	1	yes	maximal, characteristic
3	#21444384	C₂	1	2	yes	maximal
4	#21443678	C₂	1	2	yes	maximal
5	C₆₁×#1276	C₂	1	1	yes	maximal, characteristic
6	#16840319	C₂²	1	1	yes	characteristic
7	#16839786	C₂²	1	1	yes	characteristic
8	#16839660	—	2	2	no	—
9	#16839646	—	2	4	no	—
10	#16839628	C₂²	1	2	yes	—
11	#16839522	—	2	2	no	—
12	#16839500	—	2	4	no	—
13	C₆₁×#431	C₂²	1	2	yes	—
14	C₄³×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#11758194	—	2	1	no	—
16	#14519777	Q₈	1	2	yes	—
17	#14519769	—	2	1	no	—
18	#14519756	—	2	2	no	—
19	C₄×Dic₁₂₂	—	2	2	no	—
20	#14519736	—	2	1	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×#11758196	—	2	4	no	—
23	C₄⋊#92418	—	2	2	no	—
24	#14519684	—	2	4	no	—
25	#14519684	—	2	2	no	—
26	#14519684	D₄	1	4	yes	—
27	C₂×C₄²×C₆₁	—	2	2	no	—
28	C₂×C₄²×C₆₁	D₄	1	2	yes	—
29	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₁	—	2	2	no	—
31	C₂²⋊C₈×C₆₁	—	2	2	no	—
32	C₄×C₈×C₆₁	—	2	2	no	—
33	C₂²×C₄×C₆₁	—	2	2	no	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₆₁	—	2	4	no	—
37	C₂×Dic₁₂₂	—	2	2	no	—
38	C₂×C₈×C₆₁	—	4	2	no	—
39	C₄²×C₆₁	—	2	2	no	—
40	C₄²×C₆₁	—	2	2	no	—
41	C₄²×C₆₁	—	4	4	no	—
42	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₆₁	#41	1	2	yes	—
44	C₄²×C₆₁	#41	1	1	yes	characteristic
45	C₂²⋊Dic₆₁	—	4	1	no	—
46	C₄⋊Dic₆₁	—	2	2	no	—
47	C₄⋊Dic₆₁	—	2	2	no	—
48	C₄⋊Dic₆₁	—	4	2	no	—
49	C₄⋊Dic₆₁	—	4	2	no	—
50	#11758196	—	4	8	no	—
51	#11758196	—	4	2	no	—
52	C₄×Dic₆₁	—	4	1	no	—
53	#11758194	—	4	2	no	—
54	C₂×#92418	—	4	1	no	—
55	C₂³×C₆₁	#120	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	2	4	no	—
57	C₂×C₄×C₆₁	—	4	2	no	—
58	C₂×C₄×C₆₁	—	4	2	no	—
59	C₂×C₄×C₆₁	—	4	2	no	—
60	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#124	1	2	yes	—
63	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#135	1	2	yes	—
65	C₂×C₄×C₆₁	#137	1	2	yes	—
66	C₂×Dic₆₁	—	4	8	no	—
67	C₂×Dic₆₁	—	4	2	no	—
68	C₂×Dic₆₁	—	4	4	no	—
69	Dic₁₂₂	—	8	2	no	—
70	C₈×C₆₁	—	8	2	no	—
71	#92418	—	8	1	no	—
72	#10369	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	—	2	2	no	—
74	C₂²×C₆₁	#393	1	1	yes	characteristic
75	C₂²×C₆₁	#452	1	2	yes	—
76	C₄×C₆₁	—	2	2	no	—
77	C₄×C₆₁	—	2	2	no	—
78	C₄×C₆₁	—	4	4	no	—
79	C₄×C₆₁	#479	1	2	yes	—
80	C₄×C₆₁	#488	1	1	yes	characteristic
81	C₄×C₆₁	#491	1	1	yes	characteristic
82	Dic₆₁	—	8	4	no	—
83	Dic₆₁	—	8	2	no	—
84	#2864	—	61	1	no	—
85	#1994	—	61	1	no	—
86	#1518	—	61	2	no	—
87	#1286	—	61	2	no	—
88	#1276	D₆₁	1	1	yes	characteristic
89	C₂×C₆₁	—	2	2	no	—
90	C₂×C₆₁	#2026	1	1	yes	characteristic
91	C₂×C₆₁	#2031	1	2	yes	—
92	#533	—	61	1	no	—
93	#431	C₂×D₆₁	1	2	yes	—
94	#423	—	61	1	no	—
95	#389	—	122	2	no	—
96	#387	—	122	4	no	—
97	#382	—	61	2	no	—
98	C₄³	C₂×D₆₁	1	1	yes	characteristic
99	#339	—	122	2	no	—
100	#328	—	122	4	no	—
101	C₆₁	#10369	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	122	1	no	—
103	C₄⋊Q₈	—	61	2	no	—
104	#125	—	122	1	no	—
105	C₂²⋊Q₈	—	122	2	no	—
106	C₄×Q₈	—	122	2	no	—
107	#117	—	122	1	no	—
108	C₂×C₄⋊C₄	—	122	4	no	—
109	C₂×C₄⋊C₄	—	122	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	D₂₄₄	1	2	yes	—
112	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	122	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	61	4	no	—
118	#95	—	122	4	no	—
119	#95	—	122	2	no	—
120	C₂×Q₈	—	122	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	122	4	no	—
123	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
124	C₂²×C₄	D₄×D₆₁	1	2	yes	—
125	M₁₆	—	244	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	244	1	no	—
128	C₄⋊C₄	—	122	2	no	—
129	C₄⋊C₄	—	122	2	no	—
130	C₄⋊C₄	—	244	8	no	—
131	C₄⋊C₄	—	244	2	no	—
132	C₄⋊C₄	—	244	2	no	—
133	C₄⋊C₄	—	244	2	no	—
134	C₂²⋊C₄	—	244	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	244	1	no	—
139	C₄²	#11758222	1	2	yes	—
140	C₄²	#11758224	1	1	yes	characteristic
141	C₄²	Q₈×D₆₁	1	1	yes	characteristic
142	C₂³	#14519747	1	1	yes	characteristic
143	Q₈	—	488	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	244	8	no	—
149	C₂×C₄	—	244	4	no	—
150	C₂×C₄	—	244	2	no	—
151	C₂×C₄	#14519755	1	2	yes	—
152	C₂×C₄	#14519757	1	1	yes	characteristic
153	C₂×C₄	#14519763	1	2	yes	—
154	C₂×C₄	#14519765	1	1	yes	characteristic
155	C₂×C₄	#14519799	1	2	yes	—
156	C₂×C₄	#14519810	1	2	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	488	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#16839675	1	1	yes	characteristic
161	C₂²	#16839926	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	488	4	no	—
166	C₄	—	488	2	no	—
167	C₄	#16840070	1	2	yes	—
168	C₄	#16840117	1	1	yes	characteristic
169	C₄	#16840131	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#21446971	1	1	yes	characteristic
172	C₂	#21446992	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21454206

C₂

yes

maximal, characteristic

#21446796

C₂

yes

maximal, characteristic

#21444384

C₂

yes

maximal

#21443678

C₂

yes

maximal

C₆₁×#1276

C₂

yes

maximal, characteristic

#16840319

C₂²

yes

characteristic

#16839786

C₂²

yes

—

—

C₂²

yes

—

#16839522

—

#16839500

—

C₆₁×#431

C₂²

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₂×#11758194

—

#14519777

Q₈

yes

—

#14519769

—

#14519756

—

C₄×Dic₁₂₂

—

#14519736

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

D₄

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄²×C₆₁

#41

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

#120

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

#124

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

#10369

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#393

yes

characteristic

C₂²×C₆₁

#452

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#479

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#2026

yes

characteristic

C₂×C₆₁

#2031

yes

—

#533

—

#431

C₂×D₆₁

yes

—

#423

—

#389

—

122

—

#387

—

122

—

#382

—

C₄³

C₂×D₆₁

yes

characteristic

#339

—

122

—

100

#328

—

122

—

101

C₆₁

#10369

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

122

—

103

C₄⋊Q₈

—

104

#125

—

122

—

105

C₂²⋊Q₈

—

122

—

106

C₄×Q₈

—

122

—

107

#117

—

122

—

108

C₂×C₄⋊C₄

—

122

—

109

C₂×C₄⋊C₄

—

122

—

110

C₂×C₄²

—

111

C₂×C₄²

D₂₄₄

yes

—

112

C₂×C₄²

C₂²×D₆₁

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

122

—

115

C₂²⋊C₈

—

116

C₄×C₈

—

117

#95

—

118

#95

—

122

—

119

#95

—

122

—

120

C₂×Q₈

—

122

—

121

C₂²×C₄

—

122

C₂²×C₄

—

122

—

123

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

124

C₂²×C₄

D₄×D₆₁

yes

—

125

M₁₆

—

244

—

126

C₂×C₈

—

127

C₂×C₈

—

244

—

128

C₄⋊C₄

—

122

—

129

C₄⋊C₄

—

122

—

130

C₄⋊C₄

—

244

—

131

C₄⋊C₄

—

244

—

132

C₄⋊C₄

—

244

—

133

C₄⋊C₄

—

244

—

134

C₂²⋊C₄

—

244

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

244

—

139

C₄²

#11758222

yes

—

140

C₄²

#11758224

yes

characteristic

141

C₄²

Q₈×D₆₁

yes

characteristic

142

C₂³

#14519747

yes

characteristic

143

Q₈

—

488

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

244

—

149

C₂×C₄

—

244

—

150

C₂×C₄

—

244

—

151

C₂×C₄

#14519755

yes

—

152

C₂×C₄

#14519757

yes

characteristic

153

C₂×C₄

#14519763

yes

—

154

C₂×C₄

#14519765

yes

characteristic

155

C₂×C₄

#14519799

yes

—

156

C₂×C₄

#14519810

yes

—

157

C₈

—

158

C₈

—

488

—

159

C₂²

—

160

C₂²

#16839675

yes

characteristic

161

C₂²

#16839926

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

488

—

166

C₄

—

488

—

167

C₄

#16840070

yes

—

168

C₄

#16840117

yes

characteristic

169

C₄

#16840131

yes

characteristic

170

C₂

—

171

C₂

#21446971

yes

characteristic

172

C₂

#21446992

yes

—

	subgroup	G/H
center	C₂²	#16839675
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1276	C₂
socle	C₂²×C₆₁	#393
prime radical	C₂³×C₆₁	#120
hypercenter	#1276	D₆₁
hypocenter	C₆₁	#10369
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#16839675

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1276

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
101. C₆₁	#10369	semidirect

H ≅ G/N

type

101. C₆₁

#10369

semidirect

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

subgroup	index
#21454206	2
#21446796	2
#21444384	2
#21443678	2
C₆₁×#1276	2
#10369	61

subgroup

index

C₆₁×#1276

Subgroups — #29583137

Conjugacy classes of subgroups

Complements

Maximal subgroups