Group #28364638 — subgroups

Subgroups — C₂²⋊C₁₆×D₁₀₉ #28364638

Order 13952.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×D₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#13182004	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294805	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₀₉	C₄	1	1	yes	characteristic
7	C₂²×#11724784	C₄	1	1	yes	characteristic
8	C₂×C₈×D₁₀₉	—	2	4	no	—
9	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
10	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×#13182004	C₂²	1	2	yes	complemented
14	C₁₆×D₁₀₉	—	2	4	no	—
15	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	complemented
17	C₂³×D₁₀₉	C₈	1	1	yes	characteristic
18	C₂²×Dic₁₀₉	C₈	1	1	yes	characteristic
19	C₂×C₄×D₁₀₉	—	2	2	no	—
20	C₂×C₄×D₁₀₉	—	2	2	no	—
21	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
22	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×#11724784	—	2	2	no	—
26	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
27	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
28	C₈×D₁₀₉	—	2	2	no	—
29	C₈×D₁₀₉	—	2	2	no	—
30	C₈×D₁₀₉	—	2	2	no	—
31	C₈×D₁₀₉	D₄	1	4	yes	—
32	#13182004	—	2	2	no	—
33	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	—	2	2	no	—
35	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₁₆×C₁₀₉	—	2	2	no	—
38	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
39	C₂²×D₁₀₉	—	2	2	no	—
40	C₂²×D₁₀₉	—	2	2	no	—
41	C₂²×D₁₀₉	C₁₆	1	4	yes	complemented
42	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
43	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	—	2	1	no	—
45	C₂×C₄×C₁₀₉	—	2	1	no	—
46	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₀₉	—	2	2	no	—
49	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
50	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
51	C₄×D₁₀₉	—	2	2	no	—
52	C₄×D₁₀₉	—	2	1	no	—
53	C₄×D₁₀₉	—	2	1	no	—
54	C₄×D₁₀₉	—	2	2	no	—
55	C₄×D₁₀₉	C₂²⋊C₄	1	2	yes	—
56	C₄×D₁₀₉	M₁₆	1	2	yes	—
57	C₈×C₁₀₉	—	2	1	no	—
58	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
59	#11724784	—	2	1	no	—
60	#11724784	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₁₀₉	—	2	1	no	—
62	C₂²×C₁₀₉	—	2	1	no	—
63	C₂²×C₁₀₉	C₂×C₁₆	1	2	yes	complemented
64	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×D₁₀₉	—	2	1	no	—
66	C₂×D₁₀₉	—	2	1	no	—
67	C₂×D₁₀₉	—	2	8	no	—
68	C₂×D₁₀₉	C₂²⋊C₈	1	2	yes	—
69	C₂×D₁₀₉	C₂×C₁₆	1	2	yes	—
70	C₂×D₁₀₉	C₂×C₁₆	1	2	yes	complemented
71	C₂×D₁₀₉	M₃₂	1	2	yes	—
72	C₄×C₁₀₉	—	2	1	no	—
73	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
75	Dic₁₀₉	—	2	1	no	—
76	Dic₁₀₉	C₂²⋊C₈	1	2	yes	—
77	C₂×C₁₀₉	—	2	2	no	—
78	C₂×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
80	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
81	D₁₀₉	—	2	2	no	—
82	D₁₀₉	C₂²⋊C₁₆	1	4	yes	direct product factor
83	C₂×C₂²⋊C₁₆	—	109	1	no	maximal, Sylow
84	C₁₀₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₈	—	109	1	no	—
86	C₂²×C₁₆	—	109	2	no	—
87	C₂²⋊C₁₆	—	109	1	no	—
88	C₂²⋊C₁₆	—	109	2	no	—
89	C₂²⋊C₁₆	D₁₀₉	1	1	yes	characteristic, direct product factor
90	C₂³×C₄	—	109	1	no	—
91	C₂²×C₈	—	109	1	no	—
92	C₂²×C₈	—	109	1	no	—
93	C₂²×C₈	—	109	1	no	—
94	C₂²×C₈	—	109	1	no	—
95	C₂²×C₈	—	109	1	no	—
96	C₂²×C₈	—	218	4	no	—
97	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
98	C₂×C₁₆	—	109	2	no	—
99	C₂×C₁₆	—	218	4	no	—
100	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	complemented
101	C₂⁴	—	109	1	no	—
102	C₂²×C₄	—	109	1	no	—
103	C₂²×C₄	—	109	1	no	—
104	C₂²×C₄	—	109	1	no	—
105	C₂²×C₄	—	109	1	no	—
106	C₂²×C₄	—	109	1	no	—
107	C₂²×C₄	—	218	2	no	—
108	C₂²×C₄	—	218	2	no	—
109	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	109	1	no	—
112	C₂×C₈	—	109	4	no	—
113	C₂×C₈	—	109	1	no	—
114	C₂×C₈	—	218	2	no	—
115	C₂×C₈	—	218	2	no	—
116	C₂×C₈	—	218	2	no	—
117	C₂×C₈	—	218	2	no	—
118	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
119	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	218	2	no	—
122	C₂³	—	109	1	no	—
123	C₂³	—	109	4	no	—
124	C₂³	—	109	1	no	—
125	C₂³	—	218	2	no	—
126	C₂³	—	218	2	no	—
127	C₂³	C₈×D₁₀₉	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	109	1	no	—
131	C₂×C₄	—	109	2	no	—
132	C₂×C₄	—	109	2	no	—
133	C₂×C₄	—	109	1	no	—
134	C₂×C₄	—	218	2	no	—
135	C₂×C₄	—	218	2	no	—
136	C₂×C₄	—	218	2	no	—
137	C₂×C₄	—	218	1	no	—
138	C₂×C₄	—	218	1	no	—
139	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	109	2	no	—
143	C₈	—	218	1	no	—
144	C₈	D₄×D₁₀₉	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	109	2	no	—
148	C₂²	—	109	2	no	—
149	C₂²	—	109	2	no	—
150	C₂²	—	109	2	no	—
151	C₂²	—	218	8	no	—
152	C₂²	—	218	1	no	—
153	C₂²	—	218	1	no	—
154	C₂²	C₁₆×D₁₀₉	1	2	yes	complemented
155	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	109	2	no	—
158	C₄	—	218	1	no	—
159	C₄	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
160	C₄	M₁₆×D₁₀₉	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	—	109	4	no	—
163	C₂	—	218	2	no	—
164	C₂	C₂²⋊C₈×D₁₀₉	1	1	yes	characteristic
165	C₂	C₂×C₁₆×D₁₀₉	1	1	yes	characteristic
166	C₂	M₃₂×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×C₁₆×D₁₀₉

C₂

yes

maximal, complemented

C₂²⋊#13182004

C₂

yes

maximal, characteristic, complemented

#20294805

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₀₉

C₄

yes

characteristic

C₂²×#11724784

C₄

yes

characteristic

C₂×C₈×D₁₀₉

—

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×#13182004

C₂²

yes

complemented

C₁₆×D₁₀₉

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

complemented

C₂³×D₁₀₉

C₈

yes

characteristic

C₂²×Dic₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₈×D₁₀₉

D₄

yes

—

#13182004

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₁₆

yes

complemented

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂²⋊C₄

yes

—

C₄×D₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₄

yes

—

#11724784

—

#11724784

C₂²⋊C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₁₆

yes

complemented

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₈

yes

—

C₂×D₁₀₉

C₂×C₁₆

yes

—

C₂×D₁₀₉

C₂×C₁₆

yes

complemented

C₂×D₁₀₉

M₃₂

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₈

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₀₉

C₂²×C₁₆

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

D₁₀₉

—

D₁₀₉

C₂²⋊C₁₆

yes

direct product factor

C₂×C₂²⋊C₁₆

—

109

maximal, Sylow

C₁₀₉

C₂×C₂²⋊C₁₆

yes

characteristic, complemented, Sylow

C₂³×C₈

—

109

—

C₂²×C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

C₂²⋊C₁₆

D₁₀₉

yes

characteristic, direct product factor

C₂³×C₄

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

100

C₂×C₁₆

C₂×D₁₀₉

yes

complemented

101

C₂⁴

—

109

—

102

C₂²×C₄

—

109

—

103

C₂²×C₄

—

109

—

104

C₂²×C₄

—

109

—

105

C₂²×C₄

—

109

—

106

C₂²×C₄

—

109

—

107

C₂²×C₄

—

218

—

108

C₂²×C₄

—

218

—

109

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

109

—

112

C₂×C₈

—

109

—

113

C₂×C₈

—

109

—

114

C₂×C₈

—

218

—

115

C₂×C₈

—

218

—

116

C₂×C₈

—

218

—

117

C₂×C₈

—

218

—

118

C₂×C₈

C₄×D₁₀₉

yes

characteristic

119

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

120

C₁₆

—

121

C₁₆

—

218

—

122

C₂³

—

109

—

123

C₂³

—

109

—

124

C₂³

—

109

—

125

C₂³

—

218

—

126

C₂³

—

218

—

127

C₂³

C₈×D₁₀₉

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

109

—

132

C₂×C₄

—

109

—

133

C₂×C₄

—

109

—

134

C₂×C₄

—

218

—

135

C₂×C₄

—

218

—

136

C₂×C₄

—

218

—

137

C₂×C₄

—

218

—

138

C₂×C₄

—

218

—

139

C₂×C₄

C₈×D₁₀₉

yes

characteristic

140

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

141

C₈

—

142

C₈

—

109

—

143

C₈

—

218

—

144

C₈

D₄×D₁₀₉

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

109

—

148

C₂²

—

109

—

149

C₂²

—

109

—

150

C₂²

—

109

—

151

C₂²

—

218

—

152

C₂²

—

218

—

153

C₂²

—

218

—

154

C₂²

C₁₆×D₁₀₉

yes

complemented

155

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

156

C₄

—

157

C₄

—

109

—

158

C₄

—

218

—

159

C₄

C₂²⋊C₄×D₁₀₉

yes

characteristic

160

C₄

M₁₆×D₁₀₉

yes

characteristic

161

C₂

—

162

C₂

—

109

—

163

C₂

—

218

—

164

C₂

C₂²⋊C₈×D₁₀₉

yes

characteristic

165

C₂

C₂×C₁₆×D₁₀₉

yes

characteristic

166

C₂

M₃₂×D₁₀₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₁₆
Fitting	C₂²⋊C₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂³×D₁₀₉	C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₂²⋊C₁₆	D₁₀₉
hypocenter	C₁₀₉	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator

C₂×C₁₀₉

C₂²×C₁₆

Fitting

C₂²⋊C₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×C₈

prime radical

C₂³×D₁₀₉

C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

C₂²⋊C₁₆

D₁₀₉

hypocenter

C₁₀₉

C₂×C₂²⋊C₁₆

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. C₂×C₁₆×D₁₀₉	C₂	semidirect
3. C₂²⋊#13182004	C₂	semidirect
4. #20294805	C₂	semidirect
5. C₂²⋊C₁₆×C₁₀₉	C₂	semidirect
13. C₂×#13182004	C₂²	semidirect
16. C₂×C₁₆×C₁₀₉	C₂²	semidirect
41. C₂²×D₁₀₉	C₁₆	semidirect
63. C₂²×C₁₀₉	C₂×C₁₆	semidirect
70. C₂×D₁₀₉	C₂×C₁₆	semidirect
82. D₁₀₉	C₂²⋊C₁₆	direct (N × H)
84. C₁₀₉	C₂×C₂²⋊C₁₆	semidirect
89. C₂²⋊C₁₆	D₁₀₉	direct (N × H)
100. C₂×C₁₆	C₂×D₁₀₉	semidirect
154. C₂²	C₁₆×D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×C₁₆×D₁₀₉

C₂

semidirect

3. C₂²⋊#13182004

C₂

semidirect

4. #20294805

C₂

semidirect

5. C₂²⋊C₁₆×C₁₀₉

C₂

semidirect

13. C₂×#13182004

C₂²

semidirect

16. C₂×C₁₆×C₁₀₉

C₂²

semidirect

41. C₂²×D₁₀₉

C₁₆

semidirect

63. C₂²×C₁₀₉

C₂×C₁₆

semidirect

70. C₂×D₁₀₉

C₂×C₁₆

semidirect

82. D₁₀₉

C₂²⋊C₁₆

direct (N × H)

84. C₁₀₉

C₂×C₂²⋊C₁₆

semidirect

89. C₂²⋊C₁₆

D₁₀₉

direct (N × H)

100. C₂×C₁₆

C₂×D₁₀₉

semidirect

154. C₂²

C₁₆×D₁₀₉

semidirect

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

subgroup	index
C₂²×C₈×D₁₀₉	2
C₂×C₁₆×D₁₀₉	2
C₂²⋊#13182004	2
#20294805	2
C₂²⋊C₁₆×C₁₀₉	2
C₂×C₂²⋊C₁₆	109

subgroup

index

C₂²×C₈×D₁₀₉

C₂×C₁₆×D₁₀₉

C₂²⋊#13182004

#20294805

C₂²⋊C₁₆×C₁₀₉

C₂×C₂²⋊C₁₆

109

Subgroups — C₂²⋊C₁₆×D₁₀₉ #28364638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22⋊C16×D109 #28364638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₁₆×D₁₀₉ #28364638