Group #21459109 — subgroups

Subgroups — C₂×#16840318 #21459109

Order 7808.

1168 conjugacy classes of subgroups; only the 541 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 541 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	16	yes	maximal, direct product factor
2	C₂×#14519776	C₂	1	2	yes	maximal, complemented
3	C₂×#14519769	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14519755	C₂	1	4	yes	maximal
5	C₂×#14519742	C₂	1	4	yes	maximal, complemented
6	C₂×#14519735	C₂	1	2	yes	maximal
7	C₂×C₄²×D₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₁₂₂	C₂²	1	4	yes	—
10	C₂²×C₄×D₆₁	C₂²	1	2	yes	—
11	C₂²×C₄×D₆₁	C₂²	1	1	yes	characteristic
12	#14519776	C₂²	1	16	yes	complemented
13	#14519769	C₂²	1	8	yes	complemented
14	#14519755	C₂²	1	32	yes	—
15	#14519742	C₂²	1	32	yes	complemented
16	#14519735	C₂²	1	16	yes	—
17	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
18	C₂×#11758196	C₂²	1	4	yes	—
19	C₂×#11758196	C₂²	1	4	yes	—
20	C₂×#11758198	C₂²	1	4	yes	—
21	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₆₁	C₂²	1	4	yes	—
24	C₄²×D₆₁	C₂²	1	8	yes	—
25	#125×C₆₁	C₂²	1	8	yes	complemented
26	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
28	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
29	C₂³×D₆₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
31	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
33	C₂²×Dic₆₁	C₂³	1	4	yes	—
34	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
35	C₂²×Dic₆₁	C₂³	1	2	yes	—
36	C₂×C₄×D₆₁	—	2	8	no	—
37	C₂×C₄×D₆₁	Q₈	1	8	yes	complemented
38	C₂×C₄×D₆₁	C₂³	1	8	yes	—
39	C₂×C₄×D₆₁	C₂³	1	4	yes	—
40	C₂×Dic₁₂₂	—	2	16	no	—
41	C₂×Dic₁₂₂	C₂³	1	16	yes	—
42	C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
43	C₄⋊C₄×C₆₁	C₂³	1	16	yes	—
44	C₄²×C₆₁	C₂³	1	4	yes	—
45	#11758198	C₂³	1	16	yes	—
46	C₄⋊Dic₆₁	C₂³	1	8	yes	—
47	#11758196	C₂³	1	16	yes	—
48	#11758196	C₂³	1	16	yes	—
49	C₄×Dic₆₁	C₂³	1	8	yes	—
50	C₄×Dic₆₁	C₂³	1	4	yes	—
51	C₄×Dic₆₁	C₂³	1	16	yes	—
52	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
53	C₂²×D₆₁	—	2	4	no	—
54	C₂²×D₆₁	C₂×Q₈	1	4	yes	—
55	C₂²×D₆₁	C₂⁴	1	2	yes	—
56	C₂×C₄×C₆₁	—	2	8	no	—
57	C₂×C₄×C₆₁	—	2	4	no	—
58	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
59	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
60	C₂×C₄×C₆₁	C₂⁴	1	8	yes	—
61	C₂×C₄×C₆₁	C₂⁴	1	2	yes	—
62	C₂×Dic₆₁	—	2	8	no	—
63	C₂×Dic₆₁	C₂×Q₈	1	4	yes	—
64	C₂×Dic₆₁	#41	1	8	yes	—
65	C₂×Dic₆₁	C₂⁴	1	8	yes	—
66	C₂×Dic₆₁	C₂⁴	1	2	yes	—
67	C₂×Dic₆₁	C₂⁴	1	4	yes	—
68	C₄×D₆₁	—	2	16	no	—
69	C₄×D₆₁	C₂×Q₈	1	16	yes	complemented
70	Dic₁₂₂	—	2	32	no	—
71	C₂²×C₆₁	C₂²×Q₈	1	2	yes	—
72	C₂²×C₆₁	C₂×#41	1	4	yes	—
73	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
74	C₂×D₆₁	—	2	8	no	—
75	C₂×D₆₁	—	2	4	no	—
76	C₂×D₆₁	C₂²×Q₈	1	4	yes	—
77	C₄×C₆₁	—	2	8	no	—
78	C₄×C₆₁	—	2	4	no	—
79	C₄×C₆₁	C₂²×Q₈	1	4	yes	—
80	Dic₆₁	—	2	8	no	—
81	Dic₆₁	C₂²×Q₈	1	4	yes	—
82	Dic₆₁	C₂×#41	1	8	yes	—
83	C₂×#533	—	61	1	no	maximal, Sylow
84	C₂×C₆₁	#533	1	4	yes	complemented
85	C₂×C₆₁	C₂³×Q₈	1	1	yes	characteristic
86	C₂×C₆₁	C₂²×#41	1	2	yes	—
87	D₆₁	—	2	4	no	—
88	#533	—	61	16	no	—
89	C₂×C₄⋊Q₈	—	61	2	no	—
90	C₂×#125	—	61	1	no	—
91	C₂×#125	D₆₁	1	1	yes	characteristic, complemented
92	C₂×C₂²⋊Q₈	—	61	4	no	—
93	C₂×C₄×Q₈	—	61	4	no	—
94	C₂×#117	—	61	2	no	—
95	C₂²×C₄²	—	61	1	no	—
96	C₆₁	C₂×#533	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₈	—	61	4	no	—
98	C₂³×C₄	—	61	1	no	—
99	C₂³×C₄	—	61	2	no	—
100	C₄⋊Q₈	—	61	16	no	—
101	#125	—	61	8	no	—
102	#125	C₂×D₆₁	1	8	yes	complemented
103	C₂²⋊Q₈	—	61	32	no	—
104	C₄×Q₈	—	61	32	no	—
105	#117	—	61	16	no	—
106	C₂×C₄⋊C₄	—	61	2	no	—
107	C₂×C₄⋊C₄	—	61	4	no	—
108	C₂×C₄⋊C₄	—	61	4	no	—
109	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
110	C₂×C₄⋊C₄	C₂×D₆₁	1	4	yes	—
111	C₂×C₂²⋊C₄	—	61	4	no	—
112	C₂×C₄²	—	61	4	no	—
113	C₂×C₄²	—	61	2	no	—
114	C₂×C₄²	—	61	1	no	—
115	C₂×C₄²	—	61	8	no	—
116	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
117	C₂⁴	—	61	1	no	—
118	C₂×Q₈	—	61	16	no	—
119	C₂×Q₈	—	122	16	no	—
120	C₂²×C₄	—	61	1	no	—
121	C₂²×C₄	—	61	2	no	—
122	C₂²×C₄	—	61	8	no	—
123	C₂²×C₄	—	61	4	no	—
124	C₂²×C₄	—	61	4	no	—
125	C₂²×C₄	—	61	8	no	—
126	C₂²×C₄	—	122	8	no	—
127	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
128	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
129	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
130	C₄⋊C₄	—	61	16	no	—
131	C₄⋊C₄	—	61	16	no	—
132	C₄⋊C₄	—	61	8	no	—
133	C₄⋊C₄	C₂²×D₆₁	1	8	yes	—
134	C₄⋊C₄	C₂²×D₆₁	1	16	yes	—
135	C₂²⋊C₄	—	61	16	no	—
136	C₄²	—	61	8	no	—
137	C₄²	—	61	4	no	—
138	C₄²	—	61	16	no	—
139	C₄²	C₂²×D₆₁	1	4	yes	—
140	C₂³	—	61	4	no	—
141	C₂³	—	61	2	no	—
142	C₂³	—	122	4	no	—
143	C₂³	C₂³×D₆₁	1	1	yes	characteristic
144	Q₈	—	122	32	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	61	4	no	—
148	C₂×C₄	—	61	8	no	—
149	C₂×C₄	—	61	2	no	—
150	C₂×C₄	—	61	4	no	—
151	C₂×C₄	—	61	16	no	—
152	C₂×C₄	—	61	8	no	—
153	C₂×C₄	—	122	16	no	—
154	C₂×C₄	—	122	8	no	—
155	C₂×C₄	Q₈×D₆₁	1	4	yes	—
156	C₂×C₄	C₂³×D₆₁	1	4	yes	—
157	C₂×C₄	C₂³×D₆₁	1	8	yes	—
158	C₂×C₄	C₂³×D₆₁	1	2	yes	—
159	C₂²	—	61	4	no	—
160	C₂²	—	122	8	no	—
161	C₂²	—	122	4	no	—
162	C₂²	C₂×Q₈×D₆₁	1	2	yes	—
163	C₂²	#41×D₆₁	1	4	yes	—
164	C₂²	C₂⁴×D₆₁	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	2	8	no	—
167	C₄	—	61	4	no	—
168	C₄	—	61	8	no	—
169	C₄	—	122	8	no	—
170	C₄	C₂×Q₈×D₆₁	1	4	yes	—
171	C₂	—	122	4	no	—
172	C₂	#16840318	1	4	yes	direct product factor
173	C₂	C₂²×Q₈×D₆₁	1	1	yes	characteristic
174	C₂	C₂×#41×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840318

C₂

yes

maximal, direct product factor

C₂×#14519776

C₂

yes

maximal, complemented

C₂×#14519769

C₂

yes

maximal, characteristic, complemented

C₂×#14519755

C₂

yes

maximal

C₂×#14519742

C₂

yes

maximal, complemented

C₂×#14519735

C₂

yes

maximal

C₂×C₄²×D₆₁

C₂

yes

maximal, characteristic

C₂×#125×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₂₂

C₂²

yes

—

C₂²×C₄×D₆₁

C₂²

yes

—

C₂²×C₄×D₆₁

C₂²

yes

characteristic

#14519776

C₂²

yes

complemented

#14519769

C₂²

yes

complemented

#14519755

C₂²

yes

—

#14519742

C₂²

yes

complemented

#14519735

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×#11758198

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

—

#125×C₆₁

C₂²

yes

complemented

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂³×D₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

Q₈

yes

complemented

C₂×C₄×D₆₁

C₂³

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758198

C₂³

yes

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

C₂³

yes

—

#11758196

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

C₂×Q₈

yes

—

C₂²×D₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

#41

yes

—

C₂×Dic₆₁

C₂⁴

yes

—

C₂×Dic₆₁

C₂⁴

yes

—

C₂×Dic₆₁

C₂⁴

yes

—

C₄×D₆₁

—

C₄×D₆₁

C₂×Q₈

yes

complemented

Dic₁₂₂

—

C₂²×C₆₁

C₂²×Q₈

yes

—

C₂²×C₆₁

C₂×#41

yes

—

C₂²×C₆₁

C₂⁵

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂²×Q₈

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²×Q₈

yes

—

Dic₆₁

—

Dic₆₁

C₂²×Q₈

yes

—

Dic₆₁

C₂×#41

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂³×Q₈

yes

characteristic

C₂×C₆₁

C₂²×#41

yes

—

D₆₁

—

#533

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×#125

D₆₁

yes

characteristic, complemented

C₂×C₂²⋊Q₈

—

C₂×C₄×Q₈

—

C₂×#117

—

C₂²×C₄²

—

C₆₁

C₂×#533

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

—

100

C₄⋊Q₈

—

101

#125

—

102

#125

C₂×D₆₁

yes

complemented

103

C₂²⋊Q₈

—

104

C₄×Q₈

—

105

#117

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

110

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

C₂×D₆₁

yes

characteristic

117

C₂⁴

—

118

C₂×Q₈

—

119

C₂×Q₈

—

122

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

122

—

127

C₂²×C₄

C₂²×D₆₁

yes

—

128

C₂²×C₄

C₂²×D₆₁

yes

—

129

C₂²×C₄

C₂²×D₆₁

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₂²×D₆₁

yes

—

134

C₄⋊C₄

C₂²×D₆₁

yes

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂²×D₆₁

yes

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

122

—

143

C₂³

C₂³×D₆₁

yes

characteristic

144

Q₈

—

122

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

122

—

154

C₂×C₄

—

122

—

155

C₂×C₄

Q₈×D₆₁

yes

—

156

C₂×C₄

C₂³×D₆₁

yes

—

157

C₂×C₄

C₂³×D₆₁

yes

—

158

C₂×C₄

C₂³×D₆₁

yes

—

159

C₂²

—

160

C₂²

—

122

—

161

C₂²

—

122

—

162

C₂²

C₂×Q₈×D₆₁

yes

—

163

C₂²

#41×D₆₁

yes

—

164

C₂²

C₂⁴×D₆₁

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

122

—

170

C₄

C₂×Q₈×D₆₁

yes

—

171

C₂

—

122

—

172

C₂

#16840318

yes

direct product factor

173

C₂

C₂²×Q₈×D₆₁

yes

characteristic

174

C₂

C₂×#41×D₆₁

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₆₁
commutator	C₂²×C₆₁	C₂⁵
Frattini	C₂²	C₂⁴×D₆₁
Fitting	C₂×#125×C₆₁	C₂
socle	C₂³×C₆₁	C₂⁴
hypercenter	C₂×#125	D₆₁
hypocenter	C₆₁	C₂×#533
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂³×D₆₁

commutator

C₂²×C₆₁

C₂⁵

Frattini

C₂²

C₂⁴×D₆₁

Fitting

C₂×#125×C₆₁

C₂

socle

C₂³×C₆₁

C₂⁴

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
1. #16840318	C₂	direct (N × H)
2. C₂×#14519776	C₂	semidirect
3. C₂×#14519769	C₂	semidirect
5. C₂×#14519742	C₂	semidirect
8. C₂×#125×C₆₁	C₂	semidirect
12. #14519776	C₂²	semidirect
13. #14519769	C₂²	semidirect
15. #14519742	C₂²	semidirect
25. #125×C₆₁	C₂²	semidirect
37. C₂×C₄×D₆₁	Q₈	semidirect
69. C₄×D₆₁	C₂×Q₈	semidirect
84. C₂×C₆₁	#533	semidirect
91. C₂×#125	D₆₁	semidirect
96. C₆₁	C₂×#533	semidirect
102. #125	C₂×D₆₁	semidirect
172. C₂	#16840318	direct (N × H)

H ≅ G/N

type

1. #16840318

C₂

direct (N × H)

2. C₂×#14519776

C₂

semidirect

3. C₂×#14519769

C₂

semidirect

5. C₂×#14519742

C₂

semidirect

8. C₂×#125×C₆₁

C₂

semidirect

12. #14519776

C₂²

semidirect

13. #14519769

C₂²

semidirect

15. #14519742

C₂²

semidirect

25. #125×C₆₁

C₂²

semidirect

37. C₂×C₄×D₆₁

Q₈

semidirect

69. C₄×D₆₁

C₂×Q₈

semidirect

84. C₂×C₆₁

#533

semidirect

91. C₂×#125

D₆₁

semidirect

96. C₆₁

C₂×#533

semidirect

102. #125

C₂×D₆₁

semidirect

172. C₂

#16840318

direct (N × H)

Subgroups — C₂×#16840318 #21459109

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840318 #21459109

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840318 #21459109