Group #28379263 — subgroups

Subgroups — #28379263

Order 13952.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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×#16812774	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₀₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295921	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#13182032	C₂	1	2	yes	maximal, complemented
5	C₂×#16812707	C₂	1	1	yes	maximal, characteristic
6	#16812774	—	2	2	no	—
7	D₁₀₉⋊C₁₆	—	2	2	no	—
8	C₂×#13182032	C₂²	1	2	yes	—
9	C₂×#13182032	C₂²	1	2	yes	—
10	C₂×#13182029	C₄	1	1	yes	characteristic
11	#16812707	—	2	2	no	—
12	#16812707	—	2	2	no	—
13	C₂×#13182012	C₄	1	1	yes	characteristic
14	C₂×#13182012	C₄	1	1	yes	characteristic
15	C₂²×#11724784	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
19	#13182032	—	2	2	no	—
20	#13182032	—	2	2	no	—
21	#13182029	—	2	2	no	—
22	#13182029	—	2	2	no	—
23	C₂×Dic₂₁₈	C₈	1	1	yes	characteristic
24	C₂×D₄₃₆	C₈	1	1	yes	characteristic
25	C₂×C₂²⋊D₁₀₉	C₈	1	1	yes	characteristic
26	C₂×C₂²⋊D₁₀₉	C₈	1	1	yes	characteristic
27	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
29	#13182012	—	2	2	no	—
30	#13182012	—	2	2	no	—
31	C₂×#11724784	—	2	1	no	—
32	C₂×#11724784	—	2	1	no	—
33	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
34	C₂×#11724784	C₂³	1	1	yes	characteristic
35	C₈×D₁₀₉	—	2	1	no	—
36	C₈×D₁₀₉	—	2	1	no	—
37	C₈×D₁₀₉	D₄	1	2	yes	—
38	C₈×D₁₀₉	D₄	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₁₀₉	—	2	2	no	—
41	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
44	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	—	2	1	no	—
47	C₂×C₄×C₁₀₉	—	2	1	no	—
48	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊D₁₀₉	—	2	2	no	—
51	C₂²⋊D₁₀₉	—	2	2	no	—
52	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
53	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
54	D₄₃₆	—	2	2	no	—
55	C₄×D₁₀₉	—	2	1	no	—
56	C₄×D₁₀₉	—	2	1	no	—
57	C₄×D₁₀₉	C₂²⋊C₄	1	2	yes	—
58	C₄×D₁₀₉	M₁₆	1	2	yes	—
59	Dic₂₁₈	—	2	2	no	—
60	C₈×C₁₀₉	—	2	1	no	—
61	C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
62	#11724784	—	2	1	no	—
63	#11724784	C₂×D₄	1	1	yes	characteristic
64	#11724784	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₁₀₉	—	2	1	no	—
66	C₂²×C₁₀₉	—	2	2	no	—
67	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
68	C₂×D₁₀₉	—	2	2	no	—
69	C₂×D₁₀₉	—	2	1	no	—
70	C₂×D₁₀₉	—	2	2	no	—
71	C₂×D₁₀₉	C₂²⋊C₈	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	—	4	1	no	—
78	C₂×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₁₀₉	#504	1	2	yes	—
80	D₁₀₉	—	2	2	no	—
81	D₁₀₉	—	4	1	no	—
82	#2113	—	109	1	no	maximal, Sylow
83	C₁₀₉	#2113	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	—	109	1	no	—
85	C₂×M₃₂	—	109	1	no	—
86	C₂²×C₁₆	—	109	1	no	—
87	C₂²⋊C₁₆	—	109	2	no	—
88	C₂²⋊C₁₆	—	109	2	no	—
89	C₂×#41	—	109	1	no	—
90	#131	—	218	2	no	—
91	#131	—	218	2	no	—
92	C₂×M₁₆	—	109	1	no	—
93	C₂×M₁₆	—	109	1	no	—
94	C₂×M₁₆	#89137	1	1	yes	characteristic
95	C₂²×C₈	—	109	1	no	—
96	C₂²×C₈	—	109	1	no	—
97	C₂²×C₈	—	109	1	no	—
98	M₃₂	—	218	2	no	—
99	C₂×C₁₆	—	109	2	no	—
100	C₂×C₁₆	—	109	2	no	—
101	C₂×C₁₆	—	218	2	no	—
102	#41	—	218	2	no	—
103	#41	—	218	2	no	—
104	C₂×Q₈	—	109	1	no	—
105	C₂×D₄	—	109	1	no	—
106	C₂×D₄	—	109	1	no	—
107	C₂×D₄	—	109	1	no	—
108	C₂²×C₄	—	109	1	no	—
109	C₂²×C₄	—	109	1	no	—
110	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
111	M₁₆	—	2	2	no	—
112	M₁₆	—	218	2	no	—
113	M₁₆	—	218	2	no	—
114	C₂×C₈	—	109	1	no	—
115	C₂×C₈	—	109	2	no	—
116	C₂×C₈	—	109	2	no	—
117	C₂×C₈	—	109	1	no	—
118	C₂×C₈	—	218	1	no	—
119	C₂×C₈	—	218	1	no	—
120	C₂×C₈	—	218	1	no	—
121	C₂×C₈	—	218	1	no	—
122	C₂×C₈	C₂×#89137	1	1	yes	characteristic
123	C₂×C₈	C₂×#89137	1	1	yes	characteristic
124	C₁₆	—	218	2	no	—
125	C₁₆	—	218	2	no	—
126	C₂³	—	109	1	no	—
127	C₂³	—	109	1	no	—
128	C₂³	D₁₀₉⋊C₈	1	1	yes	characteristic
129	Q₈	—	218	2	no	—
130	D₄	—	218	2	no	—
131	D₄	—	218	2	no	—
132	D₄	—	218	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	109	1	no	—
136	C₂×C₄	—	109	2	no	—
137	C₂×C₄	—	109	2	no	—
138	C₂×C₄	—	109	1	no	—
139	C₂×C₄	—	218	1	no	—
140	C₂×C₄	—	218	1	no	—
141	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
142	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	109	1	no	—
145	C₈	—	109	1	no	—
146	C₈	—	218	1	no	—
147	C₈	C₂²⋊#89137	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	109	2	no	—
151	C₂²	—	218	1	no	—
152	C₂²	—	218	2	no	—
153	C₂²	—	218	2	no	—
154	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	109	2	no	—
157	C₄	—	218	1	no	—
158	C₄	C₂×C₂²⋊#89137	1	1	yes	characteristic
159	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
160	C₂	—	4	1	no	—
161	C₂	—	436	1	no	—
162	C₂	—	218	2	no	—
163	C₂	D₁₀₉⋊(C₂²⋊C₈)	1	1	yes	characteristic
164	C₂	#20296085	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812774

C₂

yes

maximal, characteristic, complemented

C₂×D₁₀₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#20295921

C₂

yes

maximal, complemented

C₂²⋊#13182032

C₂

yes

maximal, complemented

C₂×#16812707

C₂

yes

maximal, characteristic

#16812774

—

D₁₀₉⋊C₁₆

—

C₂×#13182032

C₂²

yes

—

C₂×#13182032

C₂²

yes

—

C₂×#13182029

C₄

yes

—

—

C₄

yes

characteristic

C₂×#13182012

C₄

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₄

yes

—

—

—

—

C₂×Dic₂₁₈

C₈

yes

characteristic

C₂×D₄₃₆

C₈

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₈

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₈×D₁₀₉

D₄

yes

—

C₈×D₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

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

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂²⋊C₄

yes

—

C₄×D₁₀₉

M₁₆

yes

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂²⋊C₄

yes

—

#11724784

—

#11724784

C₂×D₄

yes

characteristic

#11724784

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₈

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

#504

yes

—

D₁₀₉

—

D₁₀₉

—

#2113

—

109

maximal, Sylow

C₁₀₉

#2113

yes

characteristic, complemented, Sylow

C₂×#131

—

109

—

C₂×M₃₂

—

109

—

C₂²×C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

C₂×#41

—

109

—

#131

—

218

—

#131

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

#89137

yes

characteristic

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

100

C₂×C₁₆

—

109

—

101

C₂×C₁₆

—

218

—

102

#41

—

218

—

103

#41

—

218

—

104

C₂×Q₈

—

109

—

105

C₂×D₄

—

109

—

106

C₂×D₄

—

109

—

107

C₂×D₄

—

109

—

108

C₂²×C₄

—

109

—

109

C₂²×C₄

—

109

—

110

C₂²×C₄

C₂×#89137

yes

characteristic

111

M₁₆

—

112

M₁₆

—

218

—

113

M₁₆

—

218

—

114

C₂×C₈

—

109

—

115

C₂×C₈

—

109

—

116

C₂×C₈

—

109

—

117

C₂×C₈

—

109

—

118

C₂×C₈

—

218

—

119

C₂×C₈

—

218

—

120

C₂×C₈

—

218

—

121

C₂×C₈

—

218

—

122

C₂×C₈

C₂×#89137

yes

characteristic

123

C₂×C₈

C₂×#89137

yes

characteristic

124

C₁₆

—

218

—

125

C₁₆

—

218

—

126

C₂³

—

109

—

127

C₂³

—

109

—

128

C₂³

D₁₀₉⋊C₈

yes

characteristic

129

Q₈

—

218

—

130

D₄

—

218

—

131

D₄

—

218

—

132

D₄

—

218

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

109

—

136

C₂×C₄

—

109

—

137

C₂×C₄

—

109

—

138

C₂×C₄

—

109

—

139

C₂×C₄

—

218

—

140

C₂×C₄

—

218

—

141

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

142

C₂×C₄

C₂²×#89137

yes

characteristic

143

C₈

—

144

C₈

—

109

—

145

C₈

—

109

—

146

C₈

—

218

—

147

C₈

C₂²⋊#89137

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

109

—

151

C₂²

—

218

—

152

C₂²

—

218

—

153

C₂²

—

218

—

154

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

155

C₄

—

156

C₄

—

109

—

157

C₄

—

218

—

158

C₄

C₂×C₂²⋊#89137

yes

characteristic

159

C₄

D₁₀₉⋊M₁₆

yes

characteristic

160

C₂

—

161

C₂

—

436

—

162

C₂

—

218

—

163

C₂

D₁₀₉⋊(C₂²⋊C₈)

yes

characteristic

164

C₂

#20296085

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₂×M₁₆×C₁₀₉	C₄
prime radical	C₂×#13182029	C₄
cosocle	C₂×#11724784	C₂³
hypercenter	C₂×M₁₆	#89137
hypocenter	C₁₀₉	#2113
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₂×M₁₆×C₁₀₉

C₄

C₄

C₂³

C₂×M₁₆

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#16812774	C₂	semidirect
2. C₂×D₁₀₉⋊C₁₆	C₂	semidirect
3. #20295921	C₂	semidirect
4. C₂²⋊#13182032	C₂	semidirect
83. C₁₀₉	#2113	semidirect

H ≅ G/N

type

1. C₂×#16812774

C₂

semidirect

2. C₂×D₁₀₉⋊C₁₆

C₂

semidirect

3. #20295921

C₂

semidirect

4. C₂²⋊#13182032

C₂

semidirect

83. C₁₀₉

#2113

semidirect

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

subgroup	index
C₂×#16812774	2
C₂×D₁₀₉⋊C₁₆	2
#20295921	2
C₂²⋊#13182032	2
C₂×#16812707	2
#2113	109

subgroup

index

C₂×D₁₀₉⋊C₁₆

109

Subgroups — #28379263

Conjugacy classes of subgroups

Complements

Maximal subgroups