Group #21456466 — subgroups

Subgroups — #21456466

Order 7808.

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

normal subgroups only

This group has 386 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	#16840366	C₂	1	2	yes	maximal, complemented
2	#16840164	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840126	C₂	1	2	yes	maximal, complemented
4	#16840012	C₂	1	2	yes	maximal, complemented
5	#16839995	C₂	1	2	yes	maximal, complemented
6	#16839874	C₂	1	2	yes	maximal, complemented
7	D₈×Dic₆₁	C₂	1	2	yes	maximal, complemented
8	#16839716	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11758189	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
12	#14519769	C₂²	1	2	yes	—
13	#14519758	—	2	4	no	—
14	#14519749	—	2	4	no	—
15	#14519748	C₂²	1	4	yes	—
16	D₄×Dic₆₁	C₂²	1	4	yes	—
17	#14519734	C₂²	1	1	yes	characteristic
18	#14519711	C₂²	1	2	yes	—
19	#14519709	C₂²	1	2	yes	—
20	#14519698	C₂²	1	4	yes	—
21	#14519698	C₂²	1	4	yes	—
22	#14519688	C₂²	1	1	yes	characteristic
23	#14519687	C₂²	1	1	yes	characteristic
24	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
25	C₂×D₈×C₆₁	C₂²	1	2	yes	—
26	C₂×D₈×C₆₁	C₂²	1	2	yes	—
27	#127×C₆₁	C₂²	1	2	yes	—
28	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×D₄×C₆₁	—	2	4	no	—
30	C₂×D₄×C₆₁	C₂³	1	4	yes	—
31	C₂×C₂²⋊D₆₁	—	2	4	no	—
32	C₂²×Dic₆₁	—	2	4	no	—
33	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
34	D₈×C₆₁	—	2	4	no	—
35	D₈×C₆₁	—	2	4	no	—
36	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
38	C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₆₁	—	2	4	no	—
40	C₂²⋊Dic₆₁	—	2	4	no	—
41	#11758198	D₄	1	2	yes	—
42	C₄⋊Dic₆₁	C₂³	1	4	yes	—
43	#11758196	—	2	4	no	—
44	#11758196	D₄	1	2	yes	—
45	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
46	C₂×#92418	C₂³	1	1	yes	characteristic
47	C₂×#92418	C₂³	1	1	yes	characteristic
48	#11758189	—	2	2	no	—
49	C₈×D₆₁	—	2	2	no	—
50	C₂³×C₆₁	—	2	4	no	—
51	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
52	D₄×C₆₁	—	2	4	no	—
53	D₄×C₆₁	—	2	4	no	—
54	D₄×C₆₁	—	8	4	no	—
55	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
57	C₂²⋊D₆₁	—	8	4	no	—
58	C₂×Dic₆₁	—	2	4	no	—
59	C₂×Dic₆₁	—	8	4	no	—
60	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
61	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
62	C₄×D₆₁	—	2	1	no	—
63	C₄×D₆₁	—	2	1	no	—
64	C₈×C₆₁	#41	1	2	yes	—
65	C₈×C₆₁	#41	1	2	yes	—
66	#92418	—	2	1	no	—
67	#92418	—	2	1	no	—
68	C₂²×C₆₁	—	4	4	no	—
69	C₂²×C₆₁	—	4	4	no	—
70	C₂²×C₆₁	—	4	4	no	—
71	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
72	C₂×D₆₁	—	2	1	no	—
73	C₂×D₆₁	—	2	2	no	—
74	C₄×C₆₁	—	2	2	no	—
75	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
76	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
77	Dic₆₁	—	2	1	no	—
78	Dic₆₁	—	2	2	no	—
79	Dic₆₁	—	4	4	no	—
80	#3176	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	—	8	4	no	—
82	C₂×C₆₁	#532	1	1	yes	characteristic
83	C₂×C₆₁	#576	1	2	yes	—
84	D₆₁	—	4	1	no	—
85	#540	—	61	2	no	—
86	#499	—	61	1	no	—
87	#493	D₆₁	1	1	yes	characteristic, complemented
88	#490	—	61	2	no	—
89	#469	—	61	2	no	—
90	#466	—	61	2	no	—
91	#442	—	61	2	no	—
92	C₄×D₈	—	61	2	no	—
93	#405	—	61	1	no	—
94	C₆₁	#3176	1	1	yes	characteristic, complemented, Sylow
95	C₂×D₈	C₂×D₆₁	1	2	yes	—
96	C₂×D₈	C₂×D₆₁	1	2	yes	—
97	C₂×M₁₆	—	61	1	no	—
98	C₂²×C₈	—	61	1	no	—
99	#127	C₂×D₆₁	1	2	yes	—
100	#125	—	61	2	no	—
101	#123	—	122	4	no	—
102	#121	—	61	4	no	—
103	#121	—	122	4	no	—
104	C₄×D₄	—	61	4	no	—
105	#117	—	61	1	no	—
106	#107	—	61	2	no	—
107	#106	—	61	2	no	—
108	#102	—	61	4	no	—
109	#102	—	61	4	no	—
110	#97	—	61	1	no	—
111	#97	—	61	1	no	—
112	C₄×C₈	—	61	1	no	—
113	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	122	4	no	—
116	C₂×D₄	C₂²×D₆₁	1	4	yes	—
117	C₂²×C₄	—	61	1	no	—
118	C₂²×C₄	—	122	4	no	—
119	D₈	—	2	4	no	—
120	D₈	—	2	4	no	—
121	M₁₆	—	122	2	no	—
122	C₂×C₈	—	61	1	no	—
123	C₂×C₈	—	61	1	no	—
124	C₂×C₈	—	122	2	no	—
125	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	C₄⋊C₄	—	61	2	no	—
128	C₄⋊C₄	—	61	4	no	—
129	C₄⋊C₄	—	122	4	no	—
130	C₂²⋊C₄	—	61	2	no	—
131	C₂²⋊C₄	—	122	4	no	—
132	C₂²⋊C₄	—	122	4	no	—
133	C₄²	—	61	1	no	—
134	C₄²	C₂²×D₆₁	1	1	yes	characteristic
135	C₂³	—	2	4	no	—
136	C₂³	—	61	1	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	8	4	no	—
140	D₄	—	488	4	no	—
141	C₂×C₄	—	61	2	no	—
142	C₂×C₄	—	61	1	no	—
143	C₂×C₄	—	122	1	no	—
144	C₂×C₄	—	122	1	no	—
145	C₂×C₄	—	122	4	no	—
146	C₂×C₄	—	488	4	no	—
147	C₂×C₄	D₄×D₆₁	1	2	yes	—
148	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
149	C₈	—	122	1	no	—
150	C₈	—	122	1	no	—
151	C₈	#11758224	1	2	yes	—
152	C₈	#11758224	1	2	yes	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	122	2	no	—
157	C₂²	—	122	1	no	—
158	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	122	2	no	—
161	C₄	—	122	1	no	—
162	C₄	—	244	4	no	—
163	C₄	C₂×#11758224	1	1	yes	characteristic
164	C₄	C₂×#11758224	1	1	yes	characteristic
165	C₂	—	8	4	no	—
166	C₂	—	244	1	no	—
167	C₂	#16840310	1	1	yes	characteristic
168	C₂	#16840563	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840366

C₂

yes

maximal, complemented

#16840164

C₂

yes

maximal, characteristic, complemented

#16840126

C₂

yes

maximal, complemented

#16840012

C₂

yes

maximal, complemented

#16839995

C₂

yes

maximal, complemented

#16839874

C₂

yes

maximal, complemented

D₈×Dic₆₁

C₂

yes

maximal, complemented

#16839716

C₂

yes

maximal, characteristic, complemented

C₆₁×#493

C₂

yes

maximal, characteristic, complemented

C₂×#11758189

C₂²

yes

characteristic

C₂×C₈×D₆₁

C₂²

yes

characteristic

#14519769

C₂²

yes

—

#14519758

—

#14519749

—

#14519748

C₂²

yes

—

D₄×Dic₆₁

C₂²

yes

—

#14519734

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519709

C₂²

yes

—

#14519698

C₂²

yes

—

#14519698

C₂²

yes

—

#14519688

C₂²

yes

characteristic

#14519687

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

characteristic

C₂×D₈×C₆₁

C₂²

yes

—

C₂×D₈×C₆₁

C₂²

yes

—

#127×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₂³

yes

—

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

D₈×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

—

#11758198

D₄

yes

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

characteristic

#11758189

—

C₈×D₆₁

—

C₂³×C₆₁

—

C₂²×D₆₁

C₂×D₄

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

#41

yes

—

C₈×C₆₁

#41

yes

—

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#3176

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#532

yes

characteristic

C₂×C₆₁

#576

yes

—

D₆₁

—

#540

—

#499

—

#493

D₆₁

yes

characteristic, complemented

—

—

—

—

C₄×D₈

—

#405

—

C₆₁

#3176

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₆₁

yes

—

C₂×D₈

C₂×D₆₁

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#127

C₂×D₆₁

yes

—

100

#125

—

101

#123

—

122

—

102

#121

—

103

#121

—

122

—

104

C₄×D₄

—

105

#117

—

106

#107

—

107

#106

—

108

#102

—

109

#102

—

110

#97

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

C₂×D₆₁

yes

characteristic

114

C₂×D₄

—

115

C₂×D₄

—

122

—

116

C₂×D₄

C₂²×D₆₁

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

122

—

119

D₈

—

120

D₈

—

121

M₁₆

—

122

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

122

—

125

C₂×C₈

C₂²×D₆₁

yes

characteristic

126

C₂×C₈

C₂²×D₆₁

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

122

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

122

—

132

C₂²⋊C₄

—

122

—

133

C₄²

—

134

C₄²

C₂²×D₆₁

yes

characteristic

135

C₂³

—

136

C₂³

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

488

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

122

—

144

C₂×C₄

—

122

—

145

C₂×C₄

—

122

—

146

C₂×C₄

—

488

—

147

C₂×C₄

D₄×D₆₁

yes

—

148

C₂×C₄

C₂³×D₆₁

yes

characteristic

149

C₈

—

122

—

150

C₈

—

122

—

151

C₈

#11758224

yes

—

152

C₈

#11758224

yes

—

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

C₂×#11758224

yes

characteristic

164

C₄

C₂×#11758224

yes

characteristic

165

C₂

—

166

C₂

—

244

—

167

C₂

#16840310

yes

characteristic

168

C₂

#16840563

yes

—

	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	#3176
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. #16840366	C₂	semidirect
2. #16840164	C₂	semidirect
3. #16840126	C₂	semidirect
4. #16840012	C₂	semidirect
5. #16839995	C₂	semidirect
6. #16839874	C₂	semidirect
7. D₈×Dic₆₁	C₂	semidirect
8. #16839716	C₂	semidirect
9. C₆₁×#493	C₂	semidirect
87. #493	D₆₁	semidirect
94. C₆₁	#3176	semidirect

H ≅ G/N

type

1. #16840366

C₂

semidirect

2. #16840164

C₂

semidirect

3. #16840126

C₂

semidirect

4. #16840012

C₂

semidirect

5. #16839995

C₂

semidirect

6. #16839874

C₂

semidirect

7. D₈×Dic₆₁

C₂

semidirect

8. #16839716

C₂

semidirect

9. C₆₁×#493

C₂

semidirect

87. #493

D₆₁

semidirect

94. C₆₁

#3176

semidirect

Subgroups — #21456466

Conjugacy classes of subgroups

Complements