Group #21456396 — subgroups

Subgroups — #21456396

Order 7808.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16840310	C₂	1	2	yes	maximal, complemented
2	#16840167	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840108	C₂	1	2	yes	maximal, complemented
4	#16839995	C₂	1	4	yes	maximal, complemented
5	D₈×Dic₆₁	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758224	—	2	2	no	—
9	C₂×C₈×D₆₁	C₂²	1	2	yes	—
10	#14519777	C₂²	1	2	yes	—
11	#14519764	—	2	2	no	—
12	#14519748	—	2	2	no	—
13	#14519748	C₂²	1	4	yes	—
14	D₄×Dic₆₁	—	2	2	no	—
15	D₄×Dic₆₁	C₂²	1	4	yes	—
16	C₄²×D₆₁	C₂²	1	1	yes	characteristic
17	#14519711	C₂²	1	4	yes	—
18	#14519698	C₂²	1	8	yes	—
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	C₈×Dic₆₁	C₂²	1	2	yes	—
21	C₂×D₈×C₆₁	C₂²	1	4	yes	—
22	#127×C₆₁	C₂²	1	2	yes	—
23	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×D₄×C₆₁	—	2	2	no	—
25	C₂×D₄×C₆₁	—	2	2	no	—
26	C₂×D₄×C₆₁	C₂³	1	4	yes	—
27	C₂×C₂²⋊D₆₁	—	2	4	no	—
28	C₂²×Dic₆₁	—	2	4	no	—
29	#11758224	—	4	8	no	—
30	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
31	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
32	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₂₂	—	2	2	no	—
34	D₈×C₆₁	—	2	8	no	—
35	C₂×C₈×C₆₁	C₂³	1	2	yes	—
36	C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₆₁	—	2	4	no	—
38	C₂²⋊Dic₆₁	—	2	4	no	—
39	C₄⋊Dic₆₁	—	2	2	no	—
40	C₄⋊Dic₆₁	C₂³	1	4	yes	—
41	C₄×Dic₆₁	D₄	1	1	yes	characteristic
42	C₄×Dic₆₁	D₄	1	1	yes	characteristic
43	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
44	C₂×#92418	C₂³	1	2	yes	—
45	C₈×D₆₁	—	2	4	no	—
46	C₂³×C₆₁	—	2	4	no	—
47	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
48	D₄×C₆₁	—	2	8	no	—
49	D₄×C₆₁	—	4	4	no	—
50	D₄×C₆₁	—	4	4	no	—
51	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₆₁	—	4	8	no	—
55	C₂×Dic₆₁	—	2	4	no	—
56	C₂×Dic₆₁	—	4	8	no	—
57	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
60	C₄×D₆₁	—	2	2	no	—
61	C₄×D₆₁	—	2	2	no	—
62	C₄×D₆₁	D₈	1	4	yes	complemented
63	Dic₁₂₂	—	4	4	no	—
64	C₈×C₆₁	#41	1	4	yes	complemented
65	#92418	—	2	2	no	—
66	C₂²×C₆₁	—	4	4	no	—
67	C₂²×C₆₁	—	4	4	no	—
68	C₂²×C₆₁	—	4	4	no	—
69	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
70	C₂×D₆₁	—	2	1	no	—
71	C₂×D₆₁	—	2	1	no	—
72	C₂×D₆₁	C₂×D₈	1	2	yes	—
73	C₄×C₆₁	C₂×D₈	1	2	yes	—
74	C₄×C₆₁	#135	1	2	yes	—
75	C₄×C₆₁	C₂×#41	1	2	yes	—
76	Dic₆₁	—	2	1	no	—
77	Dic₆₁	—	2	1	no	—
78	Dic₆₁	—	4	4	no	—
79	Dic₆₁	C₂×D₈	1	2	yes	—
80	#3169	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	—	8	4	no	—
82	C₂×C₆₁	#532	1	1	yes	characteristic
83	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
84	C₂×C₆₁	C₂×#135	1	1	yes	characteristic
85	D₆₁	—	2	2	no	—
86	#532	—	61	2	no	—
87	#500	—	61	1	no	—
88	#493	D₆₁	1	1	yes	characteristic, complemented
89	#486	—	61	2	no	—
90	#466	—	61	4	no	—
91	C₄×D₈	—	61	4	no	—
92	C₂×C₄×C₈	—	61	1	no	—
93	C₆₁	#3169	1	1	yes	characteristic, complemented, Sylow
94	C₂×#41	—	122	2	no	—
95	C₂×D₈	C₂×D₆₁	1	4	yes	—
96	C₂²×C₈	—	61	2	no	—
97	C₄⋊Q₈	—	61	2	no	—
98	#127	C₂×D₆₁	1	2	yes	—
99	#124	—	122	2	no	—
100	#121	—	61	4	no	—
101	#121	—	122	2	no	—
102	C₄×D₄	—	61	4	no	—
103	C₄×D₄	—	122	2	no	—
104	C₂×C₄²	—	61	1	no	—
105	#107	—	61	4	no	—
106	#102	—	61	8	no	—
107	C₄×C₈	—	61	2	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
110	#41	—	244	8	no	—
111	C₂×Q₈	—	122	2	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	122	4	no	—
115	C₂×D₄	C₂²×D₆₁	1	4	yes	—
116	C₂²×C₄	—	61	1	no	—
117	C₂²×C₄	—	61	1	no	—
118	C₂²×C₄	—	61	1	no	—
119	C₂²×C₄	—	122	4	no	—
120	D₈	—	2	8	no	—
121	C₂×C₈	—	61	2	no	—
122	C₂×C₈	—	122	4	no	—
123	C₂×C₈	C₂²×D₆₁	1	2	yes	—
124	C₄⋊C₄	—	61	4	no	—
125	C₄⋊C₄	—	122	2	no	—
126	C₂²⋊C₄	—	122	4	no	—
127	C₂²⋊C₄	—	122	4	no	—
128	C₄²	—	61	1	no	—
129	C₄²	—	61	1	no	—
130	C₄²	—	61	1	no	—
131	C₄²	C₂²×D₆₁	1	1	yes	characteristic
132	C₂³	—	2	4	no	—
133	C₂³	—	61	1	no	—
134	Q₈	—	244	4	no	—
135	D₄	—	2	8	no	—
136	D₄	—	4	4	no	—
137	D₄	—	4	4	no	—
138	D₄	—	244	8	no	—
139	C₂×C₄	—	61	1	no	—
140	C₂×C₄	—	61	4	no	—
141	C₂×C₄	—	61	1	no	—
142	C₂×C₄	—	61	1	no	—
143	C₂×C₄	—	122	2	no	—
144	C₂×C₄	—	122	2	no	—
145	C₂×C₄	—	122	4	no	—
146	C₂×C₄	—	244	8	no	—
147	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
148	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
150	C₈	—	122	2	no	—
151	C₈	#11758224	1	4	yes	complemented
152	C₂²	—	4	4	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	61	2	no	—
156	C₂²	—	122	1	no	—
157	C₂²	—	122	1	no	—
158	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
159	C₄	—	61	2	no	—
160	C₄	—	122	1	no	—
161	C₄	—	122	1	no	—
162	C₄	—	244	4	no	—
163	C₄	D₈×D₆₁	1	2	yes	—
164	C₄	#14519796	1	2	yes	—
165	C₄	C₂×#11758224	1	2	yes	—
166	C₂	—	8	4	no	—
167	C₂	—	122	2	no	—
168	C₂	#16840310	1	1	yes	characteristic
169	C₂	C₂×D₈×D₆₁	1	1	yes	characteristic
170	C₂	C₂×#14519796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840310

C₂

yes

maximal, complemented

#16840167

C₂

yes

maximal, characteristic, complemented

#16840108

C₂

yes

maximal, complemented

#16839995

C₂

yes

maximal, complemented

D₈×Dic₆₁

C₂

yes

maximal, complemented

C₄×C₈×D₆₁

C₂

yes

maximal, characteristic, complemented

C₆₁×#493

C₂

yes

maximal, characteristic, complemented

C₂×#11758224

—

C₂×C₈×D₆₁

C₂²

yes

—

#14519777

C₂²

yes

—

#14519764

—

#14519748

—

#14519748

C₂²

yes

—

D₄×Dic₆₁

—

D₄×Dic₆₁

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519698

C₂²

yes

—

C₄×#92418

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

—

C₂×D₈×C₆₁

C₂²

yes

—

#127×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂³

yes

—

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

—

#11758224

—

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

D₈×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

C₈×D₆₁

—

C₂³×C₆₁

—

C₂²×D₆₁

C₂×D₄

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

D₈

yes

complemented

Dic₁₂₂

—

C₈×C₆₁

#41

yes

complemented

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

C₂×#41

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×D₈

yes

—

#3169

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#532

yes

characteristic

C₂×C₆₁

C₂²×D₈

yes

characteristic

C₂×C₆₁

C₂×#135

yes

characteristic

D₆₁

—

#532

—

#500

—

#493

D₆₁

yes

characteristic, complemented

#486

—

#466

—

C₄×D₈

—

C₂×C₄×C₈

—

C₆₁

#3169

yes

characteristic, complemented, Sylow

C₂×#41

—

122

—

C₂×D₈

C₂×D₆₁

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

C₂×D₆₁

yes

—

#124

—

122

—

100

#121

—

101

#121

—

122

—

102

C₄×D₄

—

103

C₄×D₄

—

122

—

104

C₂×C₄²

—

105

#107

—

106

#102

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₂×D₆₁

yes

characteristic

110

#41

—

244

—

111

C₂×Q₈

—

122

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

122

—

115

C₂×D₄

C₂²×D₆₁

yes

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

122

—

120

D₈

—

121

C₂×C₈

—

122

C₂×C₈

—

122

—

123

C₂×C₈

C₂²×D₆₁

yes

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

122

—

126

C₂²⋊C₄

—

122

—

127

C₂²⋊C₄

—

122

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₆₁

yes

characteristic

132

C₂³

—

133

C₂³

—

134

Q₈

—

244

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

244

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

122

—

144

C₂×C₄

—

122

—

145

C₂×C₄

—

122

—

146

C₂×C₄

—

244

—

147

C₂×C₄

D₄×D₆₁

yes

characteristic

148

C₂×C₄

D₄×D₆₁

yes

characteristic

149

C₂×C₄

C₂³×D₆₁

yes

characteristic

150

C₈

—

122

—

151

C₈

#11758224

yes

complemented

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

122

—

157

C₂²

—

122

—

158

C₂²

C₂×D₄×D₆₁

yes

characteristic

159

C₄

—

160

C₄

—

122

—

161

C₄

—

122

—

162

C₄

—

244

—

163

C₄

D₈×D₆₁

yes

—

164

C₄

#14519796

yes

—

165

C₄

C₂×#11758224

yes

—

166

C₂

—

167

C₂

—

122

—

168

C₂

#16840310

yes

characteristic

169

C₂

C₂×D₈×D₆₁

yes

characteristic

170

C₂

C₂×#14519796

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#493	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#493	D₆₁
hypocenter	C₆₁	#3169
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#493

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #16840310	C₂	semidirect
2. #16840167	C₂	semidirect
3. #16840108	C₂	semidirect
4. #16839995	C₂	semidirect
5. D₈×Dic₆₁	C₂	semidirect
6. C₄×C₈×D₆₁	C₂	semidirect
7. C₆₁×#493	C₂	semidirect
62. C₄×D₆₁	D₈	semidirect
64. C₈×C₆₁	#41	semidirect
88. #493	D₆₁	semidirect
93. C₆₁	#3169	semidirect
151. C₈	#11758224	semidirect

H ≅ G/N

type

1. #16840310

C₂

semidirect

2. #16840167

C₂

semidirect

3. #16840108

C₂

semidirect

4. #16839995

C₂

semidirect

5. D₈×Dic₆₁

C₂

semidirect

6. C₄×C₈×D₆₁

C₂

semidirect

7. C₆₁×#493

C₂

semidirect

62. C₄×D₆₁

D₈

semidirect

64. C₈×C₆₁

#41

semidirect

88. #493

D₆₁

semidirect

93. C₆₁

#3169

semidirect

151. C₈

#11758224

semidirect

Subgroups — #21456396

Conjugacy classes of subgroups

Complements