Group #28379260 — subgroups

Subgroups — #28379260

Order 13952.

260 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 82 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	2	yes	maximal, complemented
2	#20295921	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#13182032	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812774	—	2	4	no	—
6	C₂×#13182032	C₂²	1	4	yes	complemented
7	C₂²×C₄×D₁₀₉	C₄	1	1	yes	characteristic
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₁₀₉	—	2	4	no	—
10	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
12	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	#13182032	—	2	4	no	—
16	C₂³×D₁₀₉	C₈	1	1	yes	characteristic
17	C₂²×Dic₁₀₉	C₈	1	1	yes	characteristic
18	C₂×C₄×D₁₀₉	—	2	2	no	—
19	C₂×C₄×D₁₀₉	—	2	2	no	—
20	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
21	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
22	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×#11724784	—	2	2	no	—
25	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
26	C₂×#11724784	C₂³	1	1	yes	characteristic
27	C₈×D₁₀₉	—	2	2	no	—
28	C₈×D₁₀₉	—	2	2	no	—
29	C₈×D₁₀₉	—	2	2	no	—
30	C₈×D₁₀₉	D₄	1	4	yes	—
31	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	—	2	2	no	—
33	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₂²×D₁₀₉	—	2	2	no	—
37	C₂²×D₁₀₉	—	2	2	no	—
38	C₂²×D₁₀₉	—	2	2	no	—
39	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
40	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	—	2	1	no	—
42	C₂×C₄×C₁₀₉	—	2	1	no	—
43	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₀₉	—	2	2	no	—
46	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
47	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
48	C₄×D₁₀₉	—	2	2	no	—
49	C₄×D₁₀₉	—	2	1	no	—
50	C₄×D₁₀₉	—	2	1	no	—
51	C₄×D₁₀₉	—	2	2	no	—
52	C₄×D₁₀₉	C₂²⋊C₄	1	2	yes	—
53	C₄×D₁₀₉	M₁₆	1	2	yes	—
54	C₈×C₁₀₉	—	2	1	no	—
55	C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
56	#11724784	—	2	1	no	—
57	#11724784	C₂×D₄	1	2	yes	—
58	C₂²×C₁₀₉	—	2	1	no	—
59	C₂²×C₁₀₉	—	2	1	no	—
60	C₂²×C₁₀₉	M₃₂	1	2	yes	complemented
61	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
62	C₂×D₁₀₉	—	2	8	no	—
63	C₂×D₁₀₉	—	2	1	no	—
64	C₂×D₁₀₉	—	2	1	no	—
65	C₂×D₁₀₉	—	2	1	no	—
66	C₂×D₁₀₉	—	2	1	no	—
67	C₂×D₁₀₉	C₂²⋊C₈	1	2	yes	—
68	C₂×D₁₀₉	M₃₂	1	2	yes	complemented
69	C₄×C₁₀₉	—	2	1	no	—
70	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
72	Dic₁₀₉	—	2	1	no	—
73	Dic₁₀₉	C₂²⋊C₈	1	2	yes	—
74	C₂×C₁₀₉	—	2	2	no	—
75	C₂×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
77	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
78	D₁₀₉	—	2	2	no	—
79	D₁₀₉	—	2	2	no	—
80	#2112	—	109	1	no	maximal, Sylow
81	C₁₀₉	#2112	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	109	1	no	—
83	C₂×M₃₂	—	109	2	no	—
84	C₂²⋊C₁₆	—	109	2	no	—
85	C₂²⋊C₁₆	—	109	2	no	—
86	C₂³×C₄	—	109	1	no	—
87	C₂²×C₈	—	109	1	no	—
88	C₂²×C₈	—	109	1	no	—
89	C₂²×C₈	—	109	1	no	—
90	C₂²×C₈	—	109	1	no	—
91	C₂²×C₈	—	109	1	no	—
92	C₂²×C₈	—	218	4	no	—
93	C₂²×C₈	#89137	1	1	yes	characteristic
94	M₃₂	—	218	4	no	—
95	C₂×C₁₆	—	109	4	no	—
96	C₂⁴	—	109	1	no	—
97	C₂²×C₄	—	109	1	no	—
98	C₂²×C₄	—	109	1	no	—
99	C₂²×C₄	—	109	1	no	—
100	C₂²×C₄	—	109	1	no	—
101	C₂²×C₄	—	109	1	no	—
102	C₂²×C₄	—	218	2	no	—
103	C₂²×C₄	—	218	2	no	—
104	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	109	1	no	—
107	C₂×C₈	—	109	4	no	—
108	C₂×C₈	—	109	1	no	—
109	C₂×C₈	—	218	2	no	—
110	C₂×C₈	—	218	2	no	—
111	C₂×C₈	—	218	2	no	—
112	C₂×C₈	—	218	2	no	—
113	C₂×C₈	C₂×#89137	1	1	yes	characteristic
114	C₂×C₈	C₂×#89137	1	1	yes	characteristic
115	C₁₆	—	218	4	no	—
116	C₂³	—	109	1	no	—
117	C₂³	—	109	1	no	—
118	C₂³	—	218	2	no	—
119	C₂³	—	218	2	no	—
120	C₂³	—	218	2	no	—
121	C₂³	D₁₀₉⋊C₈	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	109	1	no	—
125	C₂×C₄	—	109	2	no	—
126	C₂×C₄	—	109	2	no	—
127	C₂×C₄	—	109	1	no	—
128	C₂×C₄	—	218	2	no	—
129	C₂×C₄	—	218	2	no	—
130	C₂×C₄	—	218	2	no	—
131	C₂×C₄	—	218	1	no	—
132	C₂×C₄	—	218	1	no	—
133	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
134	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	—	109	2	no	—
137	C₈	—	218	1	no	—
138	C₈	C₂²⋊#89137	1	2	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	109	2	no	—
142	C₂²	—	109	2	no	—
143	C₂²	—	218	8	no	—
144	C₂²	—	218	1	no	—
145	C₂²	—	218	1	no	—
146	C₂²	—	218	1	no	—
147	C₂²	—	218	1	no	—
148	C₂²	#16812774	1	2	yes	complemented
149	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	109	2	no	—
152	C₄	—	218	1	no	—
153	C₄	C₂×C₂²⋊#89137	1	1	yes	characteristic
154	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	218	2	no	—
157	C₂	—	218	2	no	—
158	C₂	D₁₀₉⋊(C₂²⋊C₈)	1	1	yes	characteristic
159	C₂	D₁₀₉⋊M₃₂	1	1	yes	characteristic
160	C₂	C₂×#16812774	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812774

C₂

yes

maximal, complemented

#20295921

C₂

yes

maximal, complemented

C₂²⋊#13182032

C₂

yes

maximal, complemented

C₂²×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#16812774

—

C₂×#13182032

C₂²

yes

complemented

C₂²×C₄×D₁₀₉

C₄

yes

characteristic

C₂²×#11724784

C₂²

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

C₂²×C₈×C₁₀₉

C₄

yes

characteristic

#13182032

—

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

C₂³

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₈×D₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

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

C₂²⋊C₄

yes

—

#11724784

—

#11724784

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

M₃₂

yes

complemented

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₈

yes

—

C₂×D₁₀₉

M₃₂

yes

complemented

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

C₂×M₃₂

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

D₁₀₉

—

D₁₀₉

—

#2112

—

109

maximal, Sylow

C₁₀₉

#2112

yes

characteristic, complemented, Sylow

C₂³×C₈

—

109

—

C₂×M₃₂

—

109

—

C₂²⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

C₂³×C₄

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

#89137

yes

characteristic

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

—

109

—

102

C₂²×C₄

—

218

—

103

C₂²×C₄

—

218

—

104

C₂²×C₄

C₂×#89137

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

109

—

107

C₂×C₈

—

109

—

108

C₂×C₈

—

109

—

109

C₂×C₈

—

218

—

110

C₂×C₈

—

218

—

111

C₂×C₈

—

218

—

112

C₂×C₈

—

218

—

113

C₂×C₈

C₂×#89137

yes

characteristic

114

C₂×C₈

C₂×#89137

yes

characteristic

115

C₁₆

—

218

—

116

C₂³

—

109

—

117

C₂³

—

109

—

118

C₂³

—

218

—

119

C₂³

—

218

—

120

C₂³

—

218

—

121

C₂³

D₁₀₉⋊C₈

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

109

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

218

—

129

C₂×C₄

—

218

—

130

C₂×C₄

—

218

—

131

C₂×C₄

—

218

—

132

C₂×C₄

—

218

—

133

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

134

C₂×C₄

C₂²×#89137

yes

characteristic

135

C₈

—

136

C₈

—

109

—

137

C₈

—

218

—

138

C₈

C₂²⋊#89137

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

109

—

142

C₂²

—

109

—

143

C₂²

—

218

—

144

C₂²

—

218

—

145

C₂²

—

218

—

146

C₂²

—

218

—

147

C₂²

—

218

—

148

C₂²

#16812774

yes

complemented

149

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

150

C₄

—

151

C₄

—

109

—

152

C₄

—

218

—

153

C₄

C₂×C₂²⋊#89137

yes

characteristic

154

C₄

D₁₀₉⋊M₁₆

yes

characteristic

155

C₂

—

156

C₂

—

218

—

157

C₂

—

218

—

158

C₂

D₁₀₉⋊(C₂²⋊C₈)

yes

characteristic

159

C₂

D₁₀₉⋊M₃₂

yes

characteristic

160

C₂

C₂×#16812774

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₂²×C₈×C₁₀₉	C₄
prime radical	C₂³×D₁₀₉	C₈
cosocle	C₂×#11724784	C₂³
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	#2112
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₂²×C₈×C₁₀₉

C₄

prime radical

C₂³×D₁₀₉

C₈

C₂³

C₂²×C₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
1. C₂×#16812774	C₂	semidirect
2. #20295921	C₂	semidirect
3. C₂²⋊#13182032	C₂	semidirect
6. C₂×#13182032	C₂²	semidirect
60. C₂²×C₁₀₉	M₃₂	semidirect
68. C₂×D₁₀₉	M₃₂	semidirect
81. C₁₀₉	#2112	semidirect
148. C₂²	#16812774	semidirect

H ≅ G/N

type

1. C₂×#16812774

C₂

semidirect

2. #20295921

C₂

semidirect

3. C₂²⋊#13182032

C₂

semidirect

6. C₂×#13182032

C₂²

semidirect

60. C₂²×C₁₀₉

M₃₂

semidirect

68. C₂×D₁₀₉

M₃₂

semidirect

81. C₁₀₉

#2112

semidirect

148. C₂²

#16812774

semidirect

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

subgroup	index
C₂×#16812774	2
#20295921	2
C₂²⋊#13182032	2
C₂²×C₈×D₁₀₉	2
#2112	109

subgroup

index

C₂²×C₈×D₁₀₉

109

Subgroups — #28379260

Conjugacy classes of subgroups

Complements

Maximal subgroups