Group #28372607 — subgroups

Subgroups — C₂×#20295268 #28372607

Order 13952.

512 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂²×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812679	C₂	1	2	yes	maximal
4	#20295268	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16812622	C₂	1	2	yes	maximal, complemented
6	C₂×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
8	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₄₃₆	—	2	4	no	—
10	C₂×Dic₄₃₆	C₂²	1	4	yes	complemented
11	C₂²×#11724784	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₁₀₉	—	2	4	no	—
13	C₂×C₈×D₁₀₉	C₂²	1	4	yes	—
14	#16812679	—	2	4	no	—
15	#16812679	C₂²	1	8	yes	—
16	C₂×C₄⋊Dic₁₀₉	—	2	2	no	—
17	C₂×#13182006	—	2	2	no	—
18	#16812622	C₂²	1	8	yes	complemented
19	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
21	#107×C₁₀₉	C₂²	1	4	yes	complemented
22	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
23	C₂×Dic₂₁₈	—	2	4	no	—
24	C₂×Dic₂₁₈	—	2	4	no	—
25	C₂×Dic₂₁₈	—	2	2	no	—
26	C₂×Dic₂₁₈	C₂³	1	4	yes	—
27	C₂²×Dic₁₀₉	—	2	2	no	—
28	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
29	C₂×C₄×D₁₀₉	—	2	2	no	—
30	C₂×C₄×D₁₀₉	—	2	2	no	—
31	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
32	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
33	Dic₄₃₆	—	2	8	no	—
34	C₂×#11724784	—	2	2	no	—
35	C₂×#11724784	C₂³	1	2	yes	—
36	C₈×D₁₀₉	—	2	8	no	—
37	C₄⋊Dic₁₀₉	—	2	8	no	—
38	#13182006	—	2	8	no	—
39	C₂²×C₄×C₁₀₉	—	2	2	no	—
40	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	D₄	1	4	yes	—
42	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
43	C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
45	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₁₀₉	—	2	2	no	—
47	C₂²×D₁₀₉	—	2	2	no	—
48	C₂²×D₁₀₉	Q₁₆	1	4	yes	complemented
49	C₂²×D₁₀₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₀₉	—	2	4	no	—
51	C₂×C₄×C₁₀₉	—	4	4	no	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₀₉	#41	1	2	yes	—
55	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₁₀₉	—	2	2	no	—
57	C₂×Dic₁₀₉	—	2	4	no	—
58	C₂×Dic₁₀₉	—	4	4	no	—
59	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
60	C₄×D₁₀₉	—	2	4	no	—
61	C₄×D₁₀₉	—	2	2	no	—
62	C₄×D₁₀₉	—	2	2	no	—
63	Dic₂₁₈	—	2	4	no	—
64	Dic₂₁₈	—	2	4	no	—
65	Dic₂₁₈	—	4	4	no	—
66	C₈×C₁₀₉	C₂×D₄	1	4	yes	—
67	#11724784	—	2	2	no	—
68	C₂²×C₁₀₉	#121	1	2	yes	—
69	C₂²×C₁₀₉	C₂×Q₁₆	1	2	yes	—
70	C₂²×C₁₀₉	#135	1	2	yes	—
71	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
72	C₂×D₁₀₉	—	2	2	no	—
73	C₂×D₁₀₉	—	2	2	no	—
74	C₂×D₁₀₉	—	2	8	no	—
75	C₂×D₁₀₉	C₂×Q₁₆	1	4	yes	complemented
76	C₄×C₁₀₉	—	4	4	no	—
77	C₄×C₁₀₉	#121	1	2	yes	—
78	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
80	Dic₁₀₉	—	2	2	no	—
81	Dic₁₀₉	—	4	4	no	—
82	C₂×C₁₀₉	#467	1	4	yes	complemented
83	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
84	C₂×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
85	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
86	D₁₀₉	—	2	4	no	—
87	C₂×#467	—	109	1	no	maximal, Sylow
88	C₁₀₉	C₂×#467	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₁₆	—	109	1	no	—
90	C₂³×C₈	—	109	1	no	—
91	C₂×C₂²⋊Q₈	—	109	2	no	—
92	#467	—	109	8	no	—
93	C₂×#107	D₁₀₉	1	1	yes	characteristic, complemented
94	C₂×#103	—	109	2	no	—
95	C₂²×Q₈	—	109	2	no	—
96	C₂³×C₄	—	109	1	no	—
97	C₂×Q₁₆	—	109	4	no	—
98	C₂×Q₁₆	—	218	4	no	—
99	C₂²×C₈	—	109	1	no	—
100	C₂²×C₈	—	109	4	no	—
101	C₂²×C₈	—	218	4	no	—
102	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
103	C₂²⋊Q₈	—	109	8	no	—
104	C₂²⋊Q₈	—	218	4	no	—
105	C₂×C₄⋊C₄	—	218	2	no	—
106	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
107	C₂×C₂²⋊C₄	—	218	2	no	—
108	#107	C₂×D₁₀₉	1	4	yes	complemented
109	#103	—	109	8	no	—
110	C₂⁴	—	109	1	no	—
111	C₂×Q₈	—	109	4	no	—
112	C₂×Q₈	—	218	4	no	—
113	C₂×Q₈	—	218	4	no	—
114	C₂×Q₈	—	218	2	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	109	1	no	—
117	C₂²×C₄	—	109	2	no	—
118	C₂²×C₄	—	109	2	no	—
119	C₂²×C₄	—	218	2	no	—
120	C₂²×C₄	—	218	2	no	—
121	C₂²×C₄	—	218	2	no	—
122	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
123	Q₁₆	—	218	8	no	—
124	C₂×C₈	—	109	2	no	—
125	C₂×C₈	—	218	2	no	—
126	C₂×C₈	—	218	8	no	—
127	C₂×C₈	D₄₃₆	1	4	yes	—
128	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	218	8	no	—
131	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
132	C₂²⋊C₄	—	218	8	no	—
133	C₂³	—	109	4	no	—
134	C₂³	—	109	2	no	—
135	C₂³	—	218	2	no	—
136	C₂³	—	218	2	no	—
137	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
138	Q₈	—	436	4	no	—
139	Q₈	—	218	4	no	—
140	Q₈	—	218	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	109	2	no	—
144	C₂×C₄	—	436	4	no	—
145	C₂×C₄	—	218	4	no	—
146	C₂×C₄	—	218	2	no	—
147	C₂×C₄	—	218	4	no	—
148	C₂×C₄	—	218	2	no	—
149	C₂×C₄	—	218	2	no	—
150	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
152	C₂×C₄	#13182030	1	2	yes	—
153	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
154	C₈	—	218	2	no	—
155	C₈	C₂×D₄₃₆	1	4	yes	—
156	C₂²	—	109	4	no	—
157	C₂²	—	218	2	no	—
158	C₂²	—	218	8	no	—
159	C₂²	—	218	2	no	—
160	C₂²	#16812674	1	2	yes	—
161	C₂²	Q₁₆×D₁₀₉	1	2	yes	—
162	C₂²	#16812718	1	2	yes	—
163	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
164	C₄	—	4	4	no	—
165	C₄	—	436	4	no	—
166	C₄	—	218	2	no	—
167	C₄	#16812674	1	2	yes	—
168	C₄	C₂²×D₄₃₆	1	1	yes	characteristic
169	C₄	C₂×#13182030	1	1	yes	characteristic
170	C₂	—	218	4	no	—
171	C₂	#20295268	1	4	yes	direct product factor
172	C₂	C₂×#16812674	1	1	yes	characteristic
173	C₂	C₂×Q₁₆×D₁₀₉	1	1	yes	characteristic
174	C₂	C₂×#16812718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₃₆

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#16812679

C₂

yes

maximal

#20295268

C₂

yes

maximal, direct product factor

C₂×#16812622

C₂

yes

maximal, complemented

C₂×#107×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₂₁₈

C₂²

yes

—

C₂²×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×Dic₄₃₆

—

C₂×Dic₄₃₆

C₂²

yes

complemented

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

—

C₂×C₈×D₁₀₉

C₂²

yes

—

#16812679

—

#16812679

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

—

C₂×#13182006

—

#16812622

C₂²

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₂²

yes

complemented

C₂³×D₁₀₉

D₄

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

D₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

—

Dic₄₃₆

—

C₂×#11724784

—

C₂×#11724784

C₂³

yes

—

C₈×D₁₀₉

—

C₄⋊Dic₁₀₉

—

#13182006

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

Q₁₆

yes

complemented

C₂²×D₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×D₄

yes

—

#11724784

—

C₂²×C₁₀₉

#121

yes

—

C₂²×C₁₀₉

C₂×Q₁₆

yes

—

C₂²×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×Q₁₆

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

C₂²×Q₁₆

yes

characteristic

C₂×C₁₀₉

C₂×#135

yes

characteristic

D₁₀₉

—

C₂×#467

—

109

maximal, Sylow

C₁₀₉

C₂×#467

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

109

—

C₂³×C₈

—

109

—

C₂×C₂²⋊Q₈

—

109

—

#467

—

109

—

C₂×#107

D₁₀₉

yes

characteristic, complemented

C₂×#103

—

109

—

C₂²×Q₈

—

109

—

C₂³×C₄

—

109

—

C₂×Q₁₆

—

109

—

C₂×Q₁₆

—

218

—

C₂²×C₈

—

109

—

100

C₂²×C₈

—

109

—

101

C₂²×C₈

—

218

—

102

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

103

C₂²⋊Q₈

—

109

—

104

C₂²⋊Q₈

—

218

—

105

C₂×C₄⋊C₄

—

218

—

106

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

107

C₂×C₂²⋊C₄

—

218

—

108

#107

C₂×D₁₀₉

yes

complemented

109

#103

—

109

—

110

C₂⁴

—

109

—

111

C₂×Q₈

—

109

—

112

C₂×Q₈

—

218

—

113

C₂×Q₈

—

218

—

114

C₂×Q₈

—

218

—

115

C₂²×C₄

—

116

C₂²×C₄

—

109

—

117

C₂²×C₄

—

109

—

118

C₂²×C₄

—

109

—

119

C₂²×C₄

—

218

—

120

C₂²×C₄

—

218

—

121

C₂²×C₄

—

218

—

122

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

123

Q₁₆

—

218

—

124

C₂×C₈

—

109

—

125

C₂×C₈

—

218

—

126

C₂×C₈

—

218

—

127

C₂×C₈

D₄₃₆

yes

—

128

C₂×C₈

C₂²×D₁₀₉

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

218

—

131

C₄⋊C₄

C₂²×D₁₀₉

yes

—

132

C₂²⋊C₄

—

218

—

133

C₂³

—

109

—

134

C₂³

—

109

—

135

C₂³

—

218

—

136

C₂³

—

218

—

137

C₂³

D₄×D₁₀₉

yes

characteristic

138

Q₈

—

436

—

139

Q₈

—

218

—

140

Q₈

—

218

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

109

—

144

C₂×C₄

—

436

—

145

C₂×C₄

—

218

—

146

C₂×C₄

—

218

—

147

C₂×C₄

—

218

—

148

C₂×C₄

—

218

—

149

C₂×C₄

—

218

—

150

C₂×C₄

D₄×D₁₀₉

yes

characteristic

151

C₂×C₄

C₂×D₄₃₆

yes

—

152

C₂×C₄

#13182030

yes

—

153

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

154

C₈

—

218

—

155

C₈

C₂×D₄₃₆

yes

—

156

C₂²

—

109

—

157

C₂²

—

218

—

158

C₂²

—

218

—

159

C₂²

—

218

—

160

C₂²

#16812674

yes

—

161

C₂²

Q₁₆×D₁₀₉

yes

—

162

C₂²

#16812718

yes

—

163

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

164

C₄

—

165

C₄

—

436

—

166

C₄

—

218

—

167

C₄

#16812674

yes

—

168

C₄

C₂²×D₄₃₆

yes

characteristic

169

C₄

C₂×#13182030

yes

characteristic

170

C₂

—

218

—

171

C₂

#20295268

yes

direct product factor

172

C₂

C₂×#16812674

yes

characteristic

173

C₂

C₂×Q₁₆×D₁₀₉

yes

characteristic

174

C₂

C₂×#16812718

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#107×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂³×D₁₀₉	D₄
hypercenter	C₂×#107	D₁₀₉
hypocenter	C₁₀₉	C₂×#467
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#107×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×D₄

prime radical

C₂³×D₁₀₉

D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
1. C₂²×Dic₄₃₆	C₂	semidirect
4. #20295268	C₂	direct (N × H)
5. C₂×#16812622	C₂	semidirect
6. C₂×#107×C₁₀₉	C₂	semidirect
10. C₂×Dic₄₃₆	C₂²	semidirect
18. #16812622	C₂²	semidirect
21. #107×C₁₀₉	C₂²	semidirect
48. C₂²×D₁₀₉	Q₁₆	semidirect
75. C₂×D₁₀₉	C₂×Q₁₆	semidirect
82. C₂×C₁₀₉	#467	semidirect
88. C₁₀₉	C₂×#467	semidirect
93. C₂×#107	D₁₀₉	semidirect
108. #107	C₂×D₁₀₉	semidirect
171. C₂	#20295268	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₄₃₆

C₂

semidirect

4. #20295268

C₂

direct (N × H)

5. C₂×#16812622

C₂

semidirect

6. C₂×#107×C₁₀₉

C₂

semidirect

10. C₂×Dic₄₃₆

C₂²

semidirect

18. #16812622

C₂²

semidirect

21. #107×C₁₀₉

C₂²

semidirect

48. C₂²×D₁₀₉

Q₁₆

semidirect

75. C₂×D₁₀₉

C₂×Q₁₆

semidirect

82. C₂×C₁₀₉

#467

semidirect

88. C₁₀₉

C₂×#467

semidirect

93. C₂×#107

D₁₀₉

semidirect

108. #107

C₂×D₁₀₉

semidirect

171. C₂

#20295268

direct (N × H)

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

subgroup	index
C₂²×Dic₄₃₆	2
C₂²×C₈×D₁₀₉	2
C₂×#16812679	2
#20295268	2
C₂×#16812622	2
C₂×#107×C₁₀₉	2
C₂×#467	109

subgroup

index

C₂²×Dic₄₃₆

C₂²×C₈×D₁₀₉

C₂×#107×C₁₀₉

109

Subgroups — C₂×#20295268 #28372607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295268 #28372607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295268 #28372607