Group #28373360 — subgroups

Subgroups — C₂×#20295385 #28373360

Order 13952.

436 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×#16812699	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295385	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812623	C₂	1	2	yes	maximal, complemented
4	C₂×#16812620	C₂	1	2	yes	maximal
5	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₈	—	2	1	no	—
8	C₂²×#11724784	C₂²	1	2	yes	—
9	#16812699	—	2	2	no	—
10	#16812699	C₂²	1	4	yes	complemented
11	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
12	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
13	#16812623	C₂²	1	8	yes	complemented
14	#16812620	C₂²	1	8	yes	—
15	#16812610	C₂²	1	4	yes	complemented
16	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	#124×C₁₀₉	—	2	2	no	—
19	#124×C₁₀₉	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
21	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
22	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₁₈	—	2	4	no	—
24	C₂×Dic₂₁₈	—	4	4	no	—
25	C₂²×Dic₁₀₉	—	2	2	no	—
26	C₂×#11724784	—	2	4	no	—
27	C₂×#11724784	C₂³	1	4	yes	—
28	C₄⋊Dic₁₀₉	—	2	4	no	—
29	C₄⋊Dic₁₀₉	—	2	2	no	—
30	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
31	C₂⁴×C₁₀₉	—	2	1	no	—
32	C₂×Q₈×C₁₀₉	—	2	1	no	—
33	C₂×Q₈×C₁₀₉	—	2	4	no	—
34	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
35	C₂×D₄×C₁₀₉	—	2	1	no	—
36	C₂×D₄×C₁₀₉	—	2	4	no	—
37	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₀₉	—	2	1	no	—
39	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
43	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
44	C₄²×C₁₀₉	D₄	1	2	yes	—
45	C₄²×C₁₀₉	C₂³	1	2	yes	—
46	C₂³×C₁₀₉	—	2	2	no	—
47	C₂³×C₁₀₉	—	4	2	no	—
48	C₂³×C₁₀₉	—	4	2	no	—
49	C₂³×C₁₀₉	—	4	2	no	—
50	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₀₉	—	2	4	no	—
52	Q₈×C₁₀₉	—	4	2	no	—
53	D₄×C₁₀₉	—	2	4	no	—
54	D₄×C₁₀₉	—	4	2	no	—
55	C₂×C₄×C₁₀₉	—	2	2	no	—
56	C₂×C₄×C₁₀₉	—	2	2	no	—
57	C₂×C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	4	2	no	—
59	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	#41	1	4	yes	—
63	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₁₀₉	—	2	4	no	—
65	C₂×Dic₁₀₉	—	4	4	no	—
66	Dic₂₁₈	—	4	8	no	—
67	#11724784	—	2	4	no	—
68	C₂²×C₁₀₉	—	4	2	no	—
69	C₂²×C₁₀₉	—	4	2	no	—
70	C₂²×C₁₀₉	—	4	2	no	—
71	C₂²×C₁₀₉	—	8	4	no	—
72	C₂²×C₁₀₉	#124	1	2	yes	—
73	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂²×C₁₀₉	#137	1	2	yes	—
75	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₁₀₉	—	2	2	no	—
77	C₄×C₁₀₉	—	2	2	no	—
78	C₄×C₁₀₉	—	4	2	no	—
79	C₄×C₁₀₉	#124	1	2	yes	—
80	C₄×C₁₀₉	C₂×#41	1	2	yes	—
81	Dic₁₀₉	—	4	4	no	—
82	C₂×C₁₀₉	—	8	2	no	—
83	C₂×C₁₀₉	#489	1	4	yes	complemented
84	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
85	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
87	C₂×#489	—	109	1	no	maximal, Sylow
88	C₁₀₉	C₂×#489	1	1	yes	characteristic, complemented, Sylow
89	C₂×C₄⋊Q₈	—	109	1	no	—
90	C₂×#124	D₁₀₉	1	1	yes	characteristic
91	#489	—	109	8	no	—
92	C₂×#103	—	109	2	no	—
93	C₂×#102	—	109	2	no	—
94	C₂×#97	—	109	1	no	—
95	C₂²×Q₈	—	218	1	no	—
96	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
97	C₂²×D₄	C₂×D₁₀₉	1	1	yes	characteristic
98	C₂²×C₈	—	109	2	no	—
99	C₄⋊Q₈	—	109	4	no	—
100	C₄⋊Q₈	—	218	2	no	—
101	#124	—	2	2	no	—
102	#124	C₂×D₁₀₉	1	4	yes	—
103	C₂×C₄⋊C₄	—	109	2	no	—
104	C₂×C₄⋊C₄	—	218	1	no	—
105	C₂×C₂²⋊C₄	—	2	1	no	—
106	C₂×C₂²⋊C₄	—	2	1	no	—
107	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
108	#103	—	109	8	no	—
109	#102	—	109	8	no	—
110	#97	—	109	4	no	—
111	C₂⁴	—	2	1	no	—
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	—	436	4	no	—
115	C₂×Q₈	—	218	4	no	—
116	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	4	no	—
119	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	218	2	no	—
122	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
125	C₂×C₈	—	109	4	no	—
126	C₂×C₈	—	218	4	no	—
127	C₄⋊C₄	—	109	4	no	—
128	C₄⋊C₄	—	218	2	no	—
129	C₄⋊C₄	—	218	4	no	—
130	C₂²⋊C₄	—	2	4	no	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
133	C₄²	C₂²×D₁₀₉	1	2	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	4	2	no	—
136	C₂³	—	4	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
139	Q₈	—	2	4	no	—
140	Q₈	—	4	2	no	—
141	Q₈	—	436	8	no	—
142	D₄	—	2	4	no	—
143	D₄	—	4	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	436	4	no	—
149	C₂×C₄	—	218	4	no	—
150	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
151	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
153	C₂×C₄	#13182028	1	4	yes	—
154	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
155	C₈	—	218	4	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	8	4	no	—
160	C₂²	#16812686	1	2	yes	—
161	C₂²	#16812724	1	2	yes	—
162	C₂²	#16812729	1	2	yes	—
163	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	436	4	no	—
168	C₄	#16812686	1	2	yes	—
169	C₄	C₂×#13182028	1	2	yes	—
170	C₂	—	8	2	no	—
171	C₂	#20295385	1	4	yes	direct product factor
172	C₂	C₂×#16812686	1	1	yes	characteristic
173	C₂	C₂×#16812724	1	1	yes	characteristic
174	C₂	C₂×#16812729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic, complemented

#20295385

C₂

yes

maximal, direct product factor

C₂×#16812623

C₂

yes

maximal, complemented

C₂×#16812620

C₂

yes

maximal

C₂×#16812610

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

C₂²×#11724784

C₂²

yes

—

#16812699

—

#16812699

C₂²

yes

complemented

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812623

C₂²

yes

complemented

#16812620

C₂²

yes

—

#16812610

C₂²

yes

complemented

C₂²×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂²×D₄×C₁₀₉

C₂²

yes

characteristic

#124×C₁₀₉

—

#124×C₁₀₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₉

—

C₂×C₂²⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₂⁴×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#124

yes

—

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

#137

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

C₂×#489

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×#124

D₁₀₉

yes

characteristic

#489

—

109

—

C₂×#103

—

109

—

C₂×#102

—

109

—

C₂×#97

—

109

—

C₂²×Q₈

—

218

—

C₂²×Q₈

C₂×D₁₀₉

yes

characteristic

C₂²×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

100

C₄⋊Q₈

—

218

—

101

#124

—

102

#124

C₂×D₁₀₉

yes

—

103

C₂×C₄⋊C₄

—

109

—

104

C₂×C₄⋊C₄

—

218

—

105

C₂×C₂²⋊C₄

—

106

C₂×C₂²⋊C₄

—

107

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

108

#103

—

109

—

109

#102

—

109

—

110

#97

—

109

—

111

C₂⁴

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

436

—

115

C₂×Q₈

—

218

—

116

C₂×Q₈

C₂²×D₁₀₉

yes

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

C₂²×D₁₀₉

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

218

—

122

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

123

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

124

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

125

C₂×C₈

—

109

—

126

C₂×C₈

—

218

—

127

C₄⋊C₄

—

109

—

128

C₄⋊C₄

—

218

—

129

C₄⋊C₄

—

218

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₄²

C₂²⋊D₁₀₉

yes

—

133

C₄²

C₂²×D₁₀₉

yes

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

139

Q₈

—

140

Q₈

—

141

Q₈

—

436

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

436

—

149

C₂×C₄

—

218

—

150

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

151

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

152

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

153

C₂×C₄

#13182028

yes

—

154

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

155

C₈

—

218

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

#16812686

yes

—

161

C₂²

#16812724

yes

—

162

C₂²

#16812729

yes

—

163

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

436

—

168

C₄

#16812686

yes

—

169

C₄

C₂×#13182028

yes

—

170

C₂

—

171

C₂

#20295385

yes

direct product factor

172

C₂

C₂×#16812686

yes

characteristic

173

C₂

C₂×#16812724

yes

characteristic

174

C₂

C₂×#16812729

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#124×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂²×D₄×C₁₀₉	C₂²
hypercenter	C₂×#124	D₁₀₉
hypocenter	C₁₀₉	C₂×#489
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#124×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×D₄

prime radical

C₂²×D₄×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#16812699	C₂	semidirect
2. #20295385	C₂	direct (N × H)
3. C₂×#16812623	C₂	semidirect
5. C₂×#16812610	C₂	semidirect
10. #16812699	C₂²	semidirect
13. #16812623	C₂²	semidirect
15. #16812610	C₂²	semidirect
83. C₂×C₁₀₉	#489	semidirect
88. C₁₀₉	C₂×#489	semidirect
171. C₂	#20295385	direct (N × H)

H ≅ G/N

type

1. C₂×#16812699

C₂

semidirect

2. #20295385

C₂

direct (N × H)

3. C₂×#16812623

C₂

semidirect

5. C₂×#16812610

C₂

semidirect

10. #16812699

C₂²

semidirect

13. #16812623

C₂²

semidirect

15. #16812610

C₂²

semidirect

83. C₂×C₁₀₉

#489

semidirect

88. C₁₀₉

C₂×#489

semidirect

171. C₂

#20295385

direct (N × H)

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

subgroup	index
C₂×#16812699	2
#20295385	2
C₂×#16812623	2
C₂×#16812620	2
C₂×#16812610	2
C₂×#124×C₁₀₉	2
C₂×#489	109

subgroup

index

C₂×#124×C₁₀₉

109

Subgroups — C₂×#20295385 #28373360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295385 #28373360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295385 #28373360