Group #28379194 — subgroups

Subgroups — #28379194

Order 13952.

276 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 68 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	D₁₀₉⋊#108	C₂	1	1	yes	maximal, characteristic
2	#20296019	C₂	1	2	yes	maximal
3	D₁₀₉⋊#101	C₂	1	2	yes	maximal
4	#20295989	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	D₁₀₉⋊M₁₆	C₂²	1	2	yes	—
7	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
8	#16812778	C₂²	1	1	yes	characteristic
9	#16812770	—	2	1	no	—
10	#16812769	—	2	1	no	—
11	#16812762	—	2	4	no	—
12	#16812760	—	2	4	no	—
13	#16812753	—	2	1	no	—
14	C₈×#89137	—	2	1	no	—
15	C₂×Q₈×D₁₀₉	C₂²	1	2	yes	—
16	C₂×Dic₄₃₆	C₄	1	1	yes	characteristic, complemented
17	C₂×#13182019	C₄	1	2	yes	complemented
18	Q₁₆×D₁₀₉	—	2	1	no	—
19	Q₁₆×D₁₀₉	—	2	1	no	—
20	Q₁₆×D₁₀₉	—	2	1	no	—
21	Q₁₆×D₁₀₉	—	2	1	no	—
22	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
24	#13182039	—	2	1	no	—
25	#13182039	—	2	2	no	—
26	#13182038	—	2	2	no	—
27	#13182038	D₄	1	2	yes	—
28	D₁₀₉⋊C₈	—	2	1	no	—
29	D₁₀₉⋊C₈	—	2	2	no	—
30	C₄⋊#89137	D₄	1	2	yes	—
31	C₂²⋊#89137	—	2	1	no	—
32	C₄×#89137	—	2	1	no	—
33	C₂×Dic₂₁₈	—	2	2	no	—
34	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
35	Q₈×D₁₀₉	—	2	2	no	—
36	Q₈×D₁₀₉	—	2	2	no	—
37	Q₈×D₁₀₉	—	2	2	no	—
38	Q₈×D₁₀₉	—	2	2	no	—
39	C₂×C₄×D₁₀₉	—	2	2	no	—
40	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
41	Dic₄₃₆	—	2	1	no	—
42	Dic₄₃₆	—	2	1	no	—
43	#13182019	—	2	2	no	—
44	#13182019	—	2	2	no	—
45	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
46	C₈×D₁₀₉	D₄	1	1	yes	characteristic
47	C₈×D₁₀₉	D₄	1	1	yes	characteristic
48	C₈×D₁₀₉	D₄	1	1	yes	characteristic
49	C₈×D₁₀₉	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
51	Q₁₆×C₁₀₉	—	2	1	no	—
52	Q₁₆×C₁₀₉	—	2	1	no	—
53	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×D₁₀₉	#41	1	1	yes	characteristic
55	C₂×#89137	—	2	2	no	—
56	Q₈×C₁₀₉	—	2	2	no	—
57	Q₈×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×Dic₁₀₉	—	2	2	no	—
61	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₄×D₁₀₉	—	4	2	no	—
63	C₄×D₁₀₉	—	4	2	no	—
64	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
65	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
66	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
67	C₄×D₁₀₉	#41	1	1	yes	characteristic
68	Dic₂₁₈	—	2	2	no	—
69	Dic₂₁₈	—	2	2	no	—
70	Dic₂₁₈	—	4	2	no	—
71	Dic₂₁₈	—	4	2	no	—
72	#11724786	—	2	2	no	—
73	#11724786	—	4	2	no	—
74	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
76	#11724784	C₂²⋊C₄	1	1	yes	characteristic
77	#11724784	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
79	C₂×D₁₀₉	—	2	2	no	—
80	C₂×D₁₀₉	#121	1	1	yes	characteristic
81	C₂×D₁₀₉	#124	1	1	yes	characteristic
82	#89137	—	4	2	no	—
83	C₄×C₁₀₉	—	4	2	no	—
84	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
86	Dic₁₀₉	—	4	2	no	—
87	Dic₁₀₉	#121	1	1	yes	characteristic
88	Dic₁₀₉	#127	1	1	yes	characteristic
89	C₂×C₁₀₉	—	2	1	no	—
90	C₂×C₁₀₉	#390	1	1	yes	characteristic
91	D₁₀₉	—	2	1	no	—
92	D₁₀₉	#473	1	2	yes	—
93	#1979	—	109	1	no	maximal, Sylow
94	C₁₀₉	#1979	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₁₆	—	109	1	no	—
96	C₂×#108	—	109	1	no	—
97	#419	—	109	2	no	—
98	C₂×#101	—	109	2	no	—
99	#405	—	109	1	no	—
100	C₂²×Q₈	—	109	2	no	—
101	C₂×Q₁₆	—	109	2	no	—
102	C₂×Q₁₆	—	109	1	no	—
103	C₂×Q₁₆	—	218	1	no	—
104	C₂×Q₁₆	—	218	1	no	—
105	C₂×Q₁₆	—	218	1	no	—
106	C₂×Q₁₆	—	218	1	no	—
107	C₂×Q₁₆	#89137	1	1	yes	characteristic, complemented
108	C₂×M₁₆	—	109	1	no	—
109	C₂×M₁₆	—	109	2	no	—
110	C₂²×C₈	—	109	1	no	—
111	#117	—	109	1	no	—
112	#108	—	218	1	no	—
113	#108	—	218	1	no	—
114	#103	—	218	4	no	—
115	#101	—	218	4	no	—
116	#97	—	218	1	no	—
117	C₄×C₈	—	218	1	no	—
118	C₂×Q₈	—	109	2	no	—
119	C₂×Q₈	—	218	2	no	—
120	C₂×Q₈	—	218	2	no	—
121	C₂×Q₈	—	218	2	no	—
122	C₂×Q₈	—	218	2	no	—
123	C₂×Q₈	—	218	2	no	—
124	C₂×Q₈	C₂×#89137	1	2	yes	—
125	C₂²×C₄	—	109	1	no	—
126	C₂²×C₄	—	218	2	no	—
127	Q₁₆	—	2	1	no	—
128	Q₁₆	—	2	1	no	—
129	Q₁₆	—	218	2	no	—
130	Q₁₆	—	218	2	no	—
131	Q₁₆	—	218	1	no	—
132	Q₁₆	—	218	1	no	—
133	M₁₆	—	109	2	no	—
134	M₁₆	—	218	2	no	—
135	M₁₆	—	218	2	no	—
136	M₁₆	—	218	1	no	—
137	C₂×C₈	—	109	1	no	—
138	C₂×C₈	—	109	1	no	—
139	C₂×C₈	—	109	1	no	—
140	C₂×C₈	—	109	1	no	—
141	C₂×C₈	—	109	1	no	—
142	C₂×C₈	—	218	2	no	—
143	C₂×C₈	—	218	1	no	—
144	C₂×C₈	C₂×#89137	1	1	yes	characteristic
145	C₄⋊C₄	—	109	2	no	—
146	C₂²⋊C₄	—	218	1	no	—
147	C₄²	—	218	1	no	—
148	C₂³	—	109	1	no	—
149	Q₈	—	2	2	no	—
150	Q₈	—	2	2	no	—
151	Q₈	—	436	2	no	—
152	Q₈	—	436	2	no	—
153	Q₈	—	218	2	no	—
154	Q₈	—	218	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	109	1	no	—
157	C₂×C₄	—	109	1	no	—
158	C₂×C₄	—	109	1	no	—
159	C₂×C₄	—	109	1	no	—
160	C₂×C₄	—	109	1	no	—
161	C₂×C₄	—	436	2	no	—
162	C₂×C₄	—	436	2	no	—
163	C₂×C₄	—	218	2	no	—
164	C₂×C₄	—	218	2	no	—
165	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
166	C₈	—	109	1	no	—
167	C₈	—	109	1	no	—
168	C₈	—	436	2	no	—
169	C₈	—	218	2	no	—
170	C₈	C₂²⋊#89137	1	1	yes	characteristic
171	C₈	C₂²⋊#89137	1	1	yes	characteristic
172	C₂²	—	109	1	no	—
173	C₂²	—	109	1	no	—
174	C₂²	—	218	2	no	—
175	C₂²	D₄×#89137	1	1	yes	characteristic
176	C₄	—	4	2	no	—
177	C₄	—	109	1	no	—
178	C₄	—	109	1	no	—
179	C₄	—	436	2	no	—
180	C₄	—	436	2	no	—
181	C₄	C₂×C₂²⋊#89137	1	1	yes	characteristic
182	C₄	D₄×#89137	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	109	2	no	—
185	C₂	—	218	1	no	—
186	C₂	#20295978	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₉⋊#108

C₂

yes

maximal, characteristic

#20296019

C₂

yes

maximal

D₁₀₉⋊#101

C₂

yes

maximal

#20295989

C₂

yes

maximal, characteristic

C₂×Q₁₆×D₁₀₉

C₂

yes

maximal, characteristic

D₁₀₉⋊M₁₆

C₂²

yes

—

D₁₀₉⋊M₁₆

C₂²

yes

characteristic

#16812778

C₂²

yes

—

—

—

—

—

—

C₂×Q₈×D₁₀₉

C₂²

yes

—

C₂×Dic₄₃₆

C₄

yes

characteristic, complemented

C₂×#13182019

C₄

yes

complemented

Q₁₆×D₁₀₉

—

Q₁₆×D₁₀₉

—

Q₁₆×D₁₀₉

—

Q₁₆×D₁₀₉

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

C₄

yes

characteristic, complemented

—

—

—

D₄

yes

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

—

C₄⋊#89137

D₄

yes

—

C₂²⋊#89137

—

C₄×#89137

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

Q₈×D₁₀₉

—

Q₈×D₁₀₉

—

Q₈×D₁₀₉

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

Dic₄₃₆

—

Dic₄₃₆

—

#13182019

—

#13182019

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

C₂²×D₁₀₉

yes

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

#41

yes

characteristic

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

#11724786

—

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

#11724784

C₂²⋊C₄

yes

characteristic

#11724784

C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

D₁₀₉

—

D₁₀₉

#473

yes

—

#1979

—

109

maximal, Sylow

C₁₀₉

#1979

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

109

—

C₂×#108

—

109

—

#419

—

109

—

C₂×#101

—

109

—

#405

—

109

—

100

C₂²×Q₈

—

109

—

101

C₂×Q₁₆

—

109

—

102

C₂×Q₁₆

—

109

—

103

C₂×Q₁₆

—

218

—

104

C₂×Q₁₆

—

218

—

105

C₂×Q₁₆

—

218

—

106

C₂×Q₁₆

—

218

—

107

C₂×Q₁₆

#89137

yes

characteristic, complemented

108

C₂×M₁₆

—

109

—

109

C₂×M₁₆

—

109

—

110

C₂²×C₈

—

109

—

111

#117

—

109

—

112

#108

—

218

—

113

#108

—

218

—

114

#103

—

218

—

115

#101

—

218

—

116

#97

—

218

—

117

C₄×C₈

—

218

—

118

C₂×Q₈

—

109

—

119

C₂×Q₈

—

218

—

120

C₂×Q₈

—

218

—

121

C₂×Q₈

—

218

—

122

C₂×Q₈

—

218

—

123

C₂×Q₈

—

218

—

124

C₂×Q₈

C₂×#89137

yes

—

125

C₂²×C₄

—

109

—

126

C₂²×C₄

—

218

—

127

Q₁₆

—

128

Q₁₆

—

129

Q₁₆

—

218

—

130

Q₁₆

—

218

—

131

Q₁₆

—

218

—

132

Q₁₆

—

218

—

133

M₁₆

—

109

—

134

M₁₆

—

218

—

135

M₁₆

—

218

—

136

M₁₆

—

218

—

137

C₂×C₈

—

109

—

138

C₂×C₈

—

109

—

139

C₂×C₈

—

109

—

140

C₂×C₈

—

109

—

141

C₂×C₈

—

109

—

142

C₂×C₈

—

218

—

143

C₂×C₈

—

218

—

144

C₂×C₈

C₂×#89137

yes

characteristic

145

C₄⋊C₄

—

109

—

146

C₂²⋊C₄

—

218

—

147

C₄²

—

218

—

148

C₂³

—

109

—

149

Q₈

—

150

Q₈

—

151

Q₈

—

436

—

152

Q₈

—

436

—

153

Q₈

—

218

—

154

Q₈

—

218

—

155

C₂×C₄

—

156

C₂×C₄

—

109

—

157

C₂×C₄

—

109

—

158

C₂×C₄

—

109

—

159

C₂×C₄

—

109

—

160

C₂×C₄

—

109

—

161

C₂×C₄

—

436

—

162

C₂×C₄

—

436

—

163

C₂×C₄

—

218

—

164

C₂×C₄

—

218

—

165

C₂×C₄

C₂²×#89137

yes

characteristic

166

C₈

—

109

—

167

C₈

—

109

—

168

C₈

—

436

—

169

C₈

—

218

—

170

C₈

C₂²⋊#89137

yes

characteristic

171

C₈

C₂²⋊#89137

yes

characteristic

172

C₂²

—

109

—

173

C₂²

—

109

—

174

C₂²

—

218

—

175

C₂²

D₄×#89137

yes

characteristic

176

C₄

—

177

C₄

—

109

—

178

C₄

—

109

—

179

C₄

—

436

—

180

C₄

—

436

—

181

C₄

C₂×C₂²⋊#89137

yes

characteristic

182

C₄

D₄×#89137

yes

characteristic

183

C₂

—

184

C₂

—

109

—

185

C₂

—

218

—

186

C₂

#20295978

yes

characteristic

	subgroup	G/H
center	C₂	#20295978
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×Q₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	#390
prime radical	C₂²×D₁₀₉	#41
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₂×Q₁₆	#89137
hypocenter	C₁₀₉	#1979
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂

#20295978

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂×Q₁₆×C₁₀₉

C₄

C₂×C₁₀₉

C₂²×D₁₀₉

C₂×C₄×D₁₀₉

C₂³

C₂×Q₁₆

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
16. C₂×Dic₄₃₆	C₄	semidirect
17. C₂×#13182019	C₄	semidirect
23. C₂×Q₁₆×C₁₀₉	C₄	semidirect
94. C₁₀₉	#1979	semidirect
107. C₂×Q₁₆	#89137	semidirect

H ≅ G/N

type

16. C₂×Dic₄₃₆

C₄

semidirect

17. C₂×#13182019

C₄

semidirect

23. C₂×Q₁₆×C₁₀₉

C₄

semidirect

94. C₁₀₉

#1979

semidirect

107. C₂×Q₁₆

#89137

semidirect

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

subgroup	index
D₁₀₉⋊#108	2
#20296019	2
D₁₀₉⋊#101	2
#20295989	2
C₂×Q₁₆×D₁₀₉	2
#1979	109

subgroup

index

D₁₀₉⋊#108

#20296019

D₁₀₉⋊#101

#20295989

C₂×Q₁₆×D₁₀₉

#1979

109

Subgroups — #28379194

Conjugacy classes of subgroups

Complements

Maximal subgroups