Group #28363147 — subgroups

Subgroups — #28363147

Order 13952.

260 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16812628	C₂	1	2	yes	maximal
2	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294781	C₂	1	1	yes	maximal, characteristic
4	#20294780	C₂	1	2	yes	maximal
5	#336×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	#16812628	—	2	4	no	—
14	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×#11724784	—	2	2	no	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	C₄×Dic₁₀₉	—	2	2	no	—
25	C₄×Dic₁₀₉	—	2	2	no	—
26	C₄×Dic₁₀₉	—	2	2	no	—
27	C₄×Dic₁₀₉	C₈	1	4	yes	complemented
28	C₄×Dic₁₀₉	D₄	1	2	yes	—
29	C₄×Dic₁₀₉	Q₈	1	2	yes	—
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	—	2	2	no	—
34	C₂×C₈×C₁₀₉	—	2	2	no	—
35	C₂×C₈×C₁₀₉	—	2	2	no	—
36	C₄²×C₁₀₉	—	2	2	no	—
37	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	—	2	1	no	—
39	C₂×C₄×C₁₀₉	—	2	1	no	—
40	C₂×C₄×C₁₀₉	—	2	1	no	—
41	C₂×C₄×C₁₀₉	—	2	1	no	—
42	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	#41	1	2	yes	—
47	C₂×C₄×C₁₀₉	#41	1	2	yes	—
48	C₂×Dic₁₀₉	—	2	1	no	—
49	C₂×Dic₁₀₉	—	2	1	no	—
50	C₂×Dic₁₀₉	—	2	1	no	—
51	C₂×Dic₁₀₉	—	2	1	no	—
52	C₂×Dic₁₀₉	—	2	1	no	—
53	C₂×Dic₁₀₉	—	2	1	no	—
54	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
55	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
56	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
57	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
58	C₂×Dic₁₀₉	M₁₆	1	2	yes	—
59	C₂×Dic₁₀₉	M₁₆	1	2	yes	—
60	C₈×C₁₀₉	—	4	2	no	—
61	#11724784	—	4	2	no	—
62	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₀₉	#117	1	1	yes	characteristic
64	C₂²×C₁₀₉	#117	1	1	yes	characteristic
65	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
66	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
67	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
69	C₄×C₁₀₉	—	2	2	no	—
70	C₄×C₁₀₉	—	2	2	no	—
71	C₄×C₁₀₉	#124	1	2	yes	—
72	C₄×C₁₀₉	#125	1	2	yes	—
73	Dic₁₀₉	—	2	2	no	—
74	Dic₁₀₉	—	2	2	no	—
75	Dic₁₀₉	—	2	2	no	—
76	Dic₁₀₉	C₄⋊C₈	1	4	yes	—
77	C₂×C₁₀₉	#382	1	1	yes	characteristic
78	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₁₀₉	#423	1	1	yes	characteristic
80	C₂×C₁₀₉	#431	1	1	yes	characteristic
81	C₂×C₁₀₉	#431	1	1	yes	characteristic
82	C₂×C₁₀₉	#432	1	1	yes	characteristic
83	C₂×C₁₀₉	#432	1	1	yes	characteristic
84	#1831	—	109	1	no	maximal, Sylow
85	C₁₀₉	#1831	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₄⋊C₈	—	109	2	no	—
87	C₄³	—	109	1	no	—
88	#336	—	109	2	no	—
89	#336	—	109	1	no	—
90	#336	D₁₀₉	1	1	yes	characteristic
91	C₂²×C₈	—	109	2	no	—
92	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
93	C₂×C₄²	—	109	1	no	—
94	C₂×C₄²	—	109	1	no	—
95	C₂×C₄²	—	109	1	no	—
96	C₂×C₄²	—	109	1	no	—
97	C₂×C₄²	—	109	1	no	—
98	C₂×C₄²	—	109	1	no	—
99	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
100	C₄⋊C₈	—	218	4	no	—
101	C₂²×C₄	—	109	1	no	—
102	C₂²×C₄	—	109	1	no	—
103	C₂²×C₄	—	109	1	no	—
104	C₂²×C₄	—	109	1	no	—
105	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
106	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	218	2	no	—
112	C₂×C₈	—	218	2	no	—
113	C₂×C₈	—	218	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	109	4	no	—
116	C₄²	—	109	2	no	—
117	C₄²	—	109	2	no	—
118	C₄²	—	218	2	no	—
119	C₄²	—	218	2	no	—
120	C₄²	—	218	2	no	—
121	C₄²	—	218	2	no	—
122	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	109	2	no	—
128	C₂×C₄	—	109	2	no	—
129	C₂×C₄	—	109	2	no	—
130	C₂×C₄	—	109	2	no	—
131	C₂×C₄	—	109	2	no	—
132	C₂×C₄	—	109	2	no	—
133	C₂×C₄	—	218	1	no	—
134	C₂×C₄	—	218	1	no	—
135	C₂×C₄	—	218	1	no	—
136	C₂×C₄	—	218	1	no	—
137	C₂×C₄	—	218	1	no	—
138	C₂×C₄	—	218	1	no	—
139	C₂×C₄	C₈×D₁₀₉	1	2	yes	—
140	C₂×C₄	#13182012	1	2	yes	—
141	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
143	C₂×C₄	#13182028	1	2	yes	—
144	C₂×C₄	#13182029	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₈	—	436	2	no	—
147	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
148	C₂²	#16812655	1	1	yes	characteristic
149	C₂²	#16812656	1	1	yes	characteristic
150	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
151	C₂²	M₁₆×D₁₀₉	1	1	yes	characteristic
152	C₂²	M₁₆×D₁₀₉	1	1	yes	characteristic
153	C₂²	C₂×#13182012	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	109	4	no	—
157	C₄	—	218	2	no	—
158	C₄	—	218	2	no	—
159	C₄	—	218	2	no	—
160	C₄	#16812685	1	2	yes	—
161	C₄	#16812689	1	2	yes	—
162	C₂	#20294890	1	1	yes	characteristic
163	C₂	C₄⋊C₈×D₁₀₉	1	1	yes	characteristic
164	C₂	#20295048	1	1	yes	characteristic
165	C₂	#20295076	1	1	yes	characteristic
166	C₂	#20295077	1	1	yes	characteristic
167	C₂	#20295080	1	1	yes	characteristic
168	C₂	#20295081	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812628

C₂

yes

maximal

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294781

C₂

yes

maximal, characteristic

#20294780

C₂

yes

maximal

#336×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812628

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

—

—

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₈

yes

complemented

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

M₁₆

yes

—

C₂×Dic₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

#125

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₄⋊C₈

yes

—

C₂×C₁₀₉

#382

yes

characteristic

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1831

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₄³

—

109

—

#336

—

109

—

#336

—

109

—

#336

D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

100

C₄⋊C₈

—

218

—

101

C₂²×C₄

—

109

—

102

C₂²×C₄

—

109

—

103

C₂²×C₄

—

109

—

104

C₂²×C₄

—

109

—

105

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

106

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

107

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

218

—

112

C₂×C₈

—

218

—

113

C₂×C₈

—

218

—

114

C₄²

—

115

C₄²

—

109

—

116

C₄²

—

109

—

117

C₄²

—

109

—

118

C₄²

—

218

—

119

C₄²

—

218

—

120

C₄²

—

218

—

121

C₄²

—

218

—

122

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

109

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

109

—

132

C₂×C₄

—

109

—

133

C₂×C₄

—

218

—

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

—

140

C₂×C₄

#13182012

yes

—

141

C₂×C₄

D₄×D₁₀₉

yes

characteristic

142

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

143

C₂×C₄

#13182028

yes

—

144

C₂×C₄

#13182029

yes

—

145

C₈

—

146

C₈

—

436

—

147

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

148

C₂²

#16812655

yes

characteristic

149

C₂²

#16812656

yes

characteristic

150

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

151

C₂²

M₁₆×D₁₀₉

yes

characteristic

152

C₂²

M₁₆×D₁₀₉

yes

characteristic

153

C₂²

C₂×#13182012

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

109

—

157

C₄

—

218

—

158

C₄

—

218

—

159

C₄

—

218

—

160

C₄

#16812685

yes

—

161

C₄

#16812689

yes

—

162

C₂

#20294890

yes

characteristic

163

C₂

C₄⋊C₈×D₁₀₉

yes

characteristic

164

C₂

#20295048

yes

characteristic

165

C₂

#20295076

yes

characteristic

166

C₂

#20295077

yes

characteristic

167

C₂

#20295080

yes

characteristic

168

C₂

#20295081

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂²×C₈
Fitting	#336×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#336	D₁₀₉
hypocenter	C₁₀₉	#1831
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂²×C₈

Fitting

#336×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
27. C₄×Dic₁₀₉	C₈	semidirect
85. C₁₀₉	#1831	semidirect

H ≅ G/N

type

27. C₄×Dic₁₀₉

C₈

semidirect

85. C₁₀₉

#1831

semidirect

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

subgroup	index
C₂×#16812628	2
C₄²×Dic₁₀₉	2
#20294781	2
#20294780	2
#336×C₁₀₉	2
#1831	109

subgroup

index

C₄²×Dic₁₀₉

#336×C₁₀₉

109

Subgroups — #28363147

Conjugacy classes of subgroups

Complements

Maximal subgroups