Group #21456166 — subgroups

Subgroups — #21456166

Order 7808.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16840318	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840313	C₂	1	1	yes	maximal, characteristic
3	#16840130	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840128	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840127	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840123	C₂	1	4	yes	maximal, complemented
7	#16839757	C₂	1	4	yes	maximal, complemented
8	#97×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11758226	C₂²	1	2	yes	—
11	C₂×C₈×D₆₁	C₂²	1	2	yes	—
12	#14519776	C₂²	1	2	yes	—
13	#14519774	C₂²	1	1	yes	characteristic
14	#14519769	C₂²	1	1	yes	characteristic
15	#14519766	C₂²	1	2	yes	—
16	#14519755	—	2	2	no	—
17	#14519752	—	2	2	no	—
18	#14519742	—	2	2	no	—
19	#14519739	—	2	2	no	—
20	#14519735	C₂²	1	2	yes	—
21	C₄²×D₆₁	C₂²	1	1	yes	characteristic
22	#14519701	C₂²	1	4	yes	—
23	#14519699	C₂²	1	4	yes	—
24	#14519697	C₂²	1	4	yes	—
25	#14519688	C₂²	1	1	yes	characteristic
26	#14519687	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
28	#125×C₆₁	C₂²	1	1	yes	characteristic
29	#103×C₆₁	C₂²	1	4	yes	—
30	#97×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
32	#11758226	—	2	8	no	—
33	C₂×D₂₄₄	—	2	2	no	—
34	C₂×D₂₄₄	—	2	2	no	—
35	C₂×D₂₄₄	C₂³	1	2	yes	—
36	C₂×C₄×D₆₁	—	2	2	no	—
37	C₂×C₄×D₆₁	D₄	1	2	yes	—
38	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₂₂	—	2	2	no	—
40	C₂×C₈×C₆₁	C₂³	1	2	yes	—
41	C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
44	C₄²×C₆₁	C₂³	1	1	yes	characteristic
45	#11758198	—	2	2	no	—
46	#11758198	—	2	4	no	—
47	C₄⋊Dic₆₁	C₂³	1	2	yes	—
48	#11758196	—	2	2	no	—
49	#11758196	—	2	2	no	—
50	C₄×Dic₆₁	—	2	2	no	—
51	C₄×Dic₆₁	D₄	1	2	yes	—
52	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
53	C₂×#92418	C₂³	1	2	yes	—
54	C₈×D₆₁	—	2	4	no	—
55	C₂²×D₆₁	—	2	2	no	—
56	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₆₁	—	2	4	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₆₁	—	2	2	no	—
63	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
64	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
65	D₂₄₄	—	2	4	no	—
66	D₂₄₄	—	4	4	no	—
67	D₂₄₄	—	4	4	no	—
68	C₄×D₆₁	—	4	2	no	—
69	C₄×D₆₁	—	4	4	no	—
70	C₄×D₆₁	#41	1	4	yes	—
71	Dic₁₂₂	—	4	4	no	—
72	C₈×C₆₁	—	2	2	no	—
73	#92418	—	2	2	no	—
74	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
75	C₂×D₆₁	—	2	2	no	—
76	C₂×D₆₁	—	4	2	no	—
77	C₂×D₆₁	—	4	4	no	—
78	C₂×D₆₁	#124	1	2	yes	—
79	C₄×C₆₁	—	2	2	no	—
80	C₄×C₆₁	—	4	2	no	—
81	C₄×C₆₁	—	4	2	no	—
82	C₄×C₆₁	C₂×#41	1	2	yes	—
83	Dic₆₁	—	2	2	no	—
84	Dic₆₁	—	4	2	no	—
85	Dic₆₁	#124	1	2	yes	—
86	#3135	—	61	1	no	maximal, Sylow
87	C₂×C₆₁	C₂×#124	1	1	yes	characteristic
88	C₂×C₆₁	#575	1	2	yes	—
89	D₆₁	—	2	2	no	—
90	D₆₁	—	8	2	no	—
91	#533	—	61	1	no	—
92	#532	—	61	1	no	—
93	#491	—	61	1	no	—
94	#491	D₆₁	1	1	yes	characteristic, complemented
95	#490	—	61	1	no	—
96	#490	—	61	1	no	—
97	#489	—	61	4	no	—
98	#416	—	61	4	no	—
99	C₂×#97	—	61	1	no	—
100	C₆₁	#3135	1	1	yes	characteristic, complemented, Sylow
101	C₂×#41	—	61	2	no	—
102	C₂²×C₈	—	61	2	no	—
103	C₄⋊Q₈	—	61	2	no	—
104	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
105	#127	—	61	1	no	—
106	#125	—	61	1	no	—
107	#125	C₂×D₆₁	1	1	yes	characteristic
108	#124	—	61	2	no	—
109	C₂²⋊Q₈	—	122	2	no	—
110	#121	—	122	2	no	—
111	C₄×Q₈	—	122	2	no	—
112	C₄×D₄	—	122	2	no	—
113	#117	—	61	2	no	—
114	C₂×C₄²	—	61	1	no	—
115	#103	—	61	4	no	—
116	#103	C₂×D₆₁	1	4	yes	—
117	#102	—	61	4	no	—
118	#102	—	61	4	no	—
119	#97	—	61	2	no	—
120	#97	—	61	1	no	—
121	#97	C₂×D₆₁	1	1	yes	characteristic
122	#41	—	122	8	no	—
123	C₂×Q₈	—	122	2	no	—
124	C₂×Q₈	C₂²×D₆₁	1	2	yes	—
125	C₂×D₄	—	61	2	no	—
126	C₂×D₄	—	122	2	no	—
127	C₂×D₄	—	122	2	no	—
128	C₂²×C₄	—	61	1	no	—
129	C₂²×C₄	—	61	2	no	—
130	C₂²×C₄	—	122	2	no	—
131	C₂×C₈	—	61	2	no	—
132	C₂×C₈	—	122	4	no	—
133	C₂×C₈	C₂²×D₆₁	1	2	yes	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	61	2	no	—
137	C₄⋊C₄	—	122	2	no	—
138	C₄⋊C₄	—	122	2	no	—
139	C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
140	C₂²⋊C₄	—	122	4	no	—
141	C₂²⋊C₄	—	122	2	no	—
142	C₄²	—	61	2	no	—
143	C₄²	—	61	1	no	—
144	C₄²	—	122	2	no	—
145	C₄²	C₂²×D₆₁	1	1	yes	characteristic
146	C₂³	—	61	1	no	—
147	C₂³	—	122	2	no	—
148	Q₈	—	2	4	no	—
149	Q₈	—	244	4	no	—
150	D₄	—	122	4	no	—
151	D₄	—	244	4	no	—
152	D₄	—	244	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	61	1	no	—
156	C₂×C₄	—	61	4	no	—
157	C₂×C₄	—	61	2	no	—
158	C₂×C₄	—	122	2	no	—
159	C₂×C₄	—	244	2	no	—
160	C₂×C₄	—	244	4	no	—
161	C₂×C₄	D₄×D₆₁	1	2	yes	—
162	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	122	2	no	—
165	C₂²	—	61	2	no	—
166	C₂²	—	122	2	no	—
167	C₂²	—	244	2	no	—
168	C₂²	—	244	4	no	—
169	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	61	2	no	—
174	C₄	—	122	2	no	—
175	C₄	—	244	2	no	—
176	C₄	#41×D₆₁	1	2	yes	—
177	C₂	—	122	2	no	—
178	C₂	—	488	2	no	—
179	C₂	#124×D₆₁	1	1	yes	characteristic
180	C₂	#16840561	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840318

C₂

yes

maximal, characteristic, complemented

#16840313

C₂

yes

maximal, characteristic

#16840130

C₂

yes

maximal, characteristic, complemented

#16840128

C₂

yes

maximal, characteristic, complemented

#16840127

C₂

yes

maximal, characteristic, complemented

#16840123

C₂

yes

maximal, complemented

#16839757

C₂

yes

maximal, complemented

#97×D₆₁

C₂

yes

maximal, characteristic, complemented

C₆₁×#491

C₂

yes

maximal, characteristic, complemented

C₂×#11758226

C₂²

yes

—

C₂×C₈×D₆₁

C₂²

yes

—

#14519776

C₂²

yes

—

#14519774

C₂²

yes

characteristic

#14519769

C₂²

yes

characteristic

#14519766

C₂²

yes

—

—

—

—

—

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

#14519701

C₂²

yes

—

#14519699

C₂²

yes

—

#14519697

C₂²

yes

—

#14519688

C₂²

yes

characteristic

#14519687

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#125×C₆₁

C₂²

yes

characteristic

#103×C₆₁

C₂²

yes

—

#97×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

C₂³

yes

—

#11758226

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

#11758198

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

C₈×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

#41

yes

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#124

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#41

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#124

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

—

D₆₁

—

—

—

—

D₆₁

yes

characteristic, complemented

—

—

—

—

C₂×#97

—

100

C₆₁

#3135

yes

characteristic, complemented, Sylow

101

C₂×#41

—

102

C₂²×C₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

105

#127

—

106

#125

—

107

#125

C₂×D₆₁

yes

characteristic

108

#124

—

109

C₂²⋊Q₈

—

122

—

110

#121

—

122

—

111

C₄×Q₈

—

122

—

112

C₄×D₄

—

122

—

113

#117

—

114

C₂×C₄²

—

115

#103

—

116

#103

C₂×D₆₁

yes

—

117

#102

—

118

#102

—

119

#97

—

120

#97

—

121

#97

C₂×D₆₁

yes

characteristic

122

#41

—

122

—

123

C₂×Q₈

—

122

—

124

C₂×Q₈

C₂²×D₆₁

yes

—

125

C₂×D₄

—

126

C₂×D₄

—

122

—

127

C₂×D₄

—

122

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

122

—

131

C₂×C₈

—

132

C₂×C₈

—

122

—

133

C₂×C₈

C₂²×D₆₁

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

122

—

138

C₄⋊C₄

—

122

—

139

C₄⋊C₄

C₂²×D₆₁

yes

—

140

C₂²⋊C₄

—

122

—

141

C₂²⋊C₄

—

122

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

122

—

145

C₄²

C₂²×D₆₁

yes

characteristic

146

C₂³

—

147

C₂³

—

122

—

148

Q₈

—

149

Q₈

—

244

—

150

D₄

—

122

—

151

D₄

—

244

—

152

D₄

—

244

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

122

—

159

C₂×C₄

—

244

—

160

C₂×C₄

—

244

—

161

C₂×C₄

D₄×D₆₁

yes

—

162

C₂×C₄

C₂³×D₆₁

yes

characteristic

163

C₈

—

164

C₈

—

122

—

165

C₂²

—

166

C₂²

—

122

—

167

C₂²

—

244

—

168

C₂²

—

244

—

169

C₂²

C₂×D₄×D₆₁

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

122

—

175

C₄

—

244

—

176

C₄

#41×D₆₁

yes

—

177

C₂

—

122

—

178

C₂

—

488

—

179

C₂

#124×D₆₁

yes

characteristic

180

C₂

#16840561

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#491	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#491	D₆₁
hypocenter	C₆₁	#3135
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#491

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
1. #16840318	C₂	semidirect
3. #16840130	C₂	semidirect
4. #16840128	C₂	semidirect
5. #16840127	C₂	semidirect
6. #16840123	C₂	semidirect
7. #16839757	C₂	semidirect
8. #97×D₆₁	C₂	semidirect
9. C₆₁×#491	C₂	semidirect
94. #491	D₆₁	semidirect
100. C₆₁	#3135	semidirect

H ≅ G/N

type

1. #16840318

C₂

semidirect

3. #16840130

C₂

semidirect

4. #16840128

C₂

semidirect

5. #16840127

C₂

semidirect

6. #16840123

C₂

semidirect

7. #16839757

C₂

semidirect

8. #97×D₆₁

C₂

semidirect

9. C₆₁×#491

C₂

semidirect

94. #491

D₆₁

semidirect

100. C₆₁

#3135

semidirect

Subgroups — #21456166

Conjugacy classes of subgroups

Complements