Group #28364278 — subgroups

Subgroups — #28364278

Order 13952.

298 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 93 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×#16812699	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812622	C₂	1	2	yes	maximal
3	C₂×#16812622	C₂	1	2	yes	maximal
4	#20294781	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	2	yes	—
8	#16812699	—	2	4	no	—
9	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
11	#16812622	—	2	4	no	—
12	#16812622	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	#95×C₁₀₉	—	2	1	no	—
18	C₂×Dic₂₁₈	—	2	2	no	—
19	C₂×Dic₂₁₈	—	2	2	no	—
20	C₂×Dic₂₁₈	—	2	2	no	—
21	C₂×Dic₂₁₈	—	2	2	no	—
22	C₂×Dic₂₁₈	—	2	2	no	—
23	C₂×Dic₂₁₈	D₄	1	4	yes	—
24	C₂²×Dic₁₀₉	—	2	2	no	—
25	C₂×#11724784	—	2	2	no	—
26	C₂×#11724784	—	2	2	no	—
27	C₂×#11724784	—	2	2	no	—
28	C₄⋊Dic₁₀₉	—	2	4	no	—
29	C₄⋊Dic₁₀₉	—	2	4	no	—
30	C₂²×C₄×C₁₀₉	—	2	1	no	—
31	C₂²×C₄×C₁₀₉	—	2	1	no	—
32	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₄⋊C₄×C₁₀₉	—	2	2	no	—
37	C₄⋊C₄×C₁₀₉	—	4	2	no	—
38	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
39	C₄²×C₁₀₉	—	2	2	no	—
40	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	—	2	1	no	—
42	C₂×C₄×C₁₀₉	—	2	1	no	—
43	C₂×C₄×C₁₀₉	—	2	1	no	—
44	C₂×C₄×C₁₀₉	—	2	1	no	—
45	C₂×C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄×C₁₀₉	—	4	1	no	—
47	C₂×C₄×C₁₀₉	—	4	1	no	—
48	C₂×C₄×C₁₀₉	—	4	1	no	—
49	C₂×C₄×C₁₀₉	—	4	1	no	—
50	C₂×C₄×C₁₀₉	—	4	1	no	—
51	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
52	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
53	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	#41	1	2	yes	—
57	C₂×Dic₁₀₉	—	2	4	no	—
58	C₂×Dic₁₀₉	—	4	2	no	—
59	C₂×Dic₁₀₉	—	4	2	no	—
60	Dic₂₁₈	—	2	4	no	—
61	Dic₂₁₈	—	2	4	no	—
62	Dic₂₁₈	—	4	2	no	—
63	Dic₂₁₈	—	4	2	no	—
64	#11724784	—	4	2	no	—
65	C₂²×C₁₀₉	#120	1	1	yes	characteristic
66	C₂²×C₁₀₉	#121	1	1	yes	characteristic
67	C₂²×C₁₀₉	#124	1	1	yes	characteristic
68	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₁₀₉	#137	1	1	yes	characteristic
71	C₂²×C₁₀₉	#137	1	1	yes	characteristic
72	C₄×C₁₀₉	—	2	2	no	—
73	C₄×C₁₀₉	—	2	2	no	—
74	C₄×C₁₀₉	—	4	2	no	—
75	C₄×C₁₀₉	—	8	1	no	—
76	C₄×C₁₀₉	#121	1	2	yes	—
77	C₄×C₁₀₉	#123	1	2	yes	—
78	Dic₁₀₉	—	4	4	no	—
79	C₂×C₁₀₉	#394	1	1	yes	characteristic
80	C₂×C₁₀₉	#448	1	1	yes	characteristic
81	C₂×C₁₀₉	#451	1	1	yes	characteristic
82	C₂×C₁₀₉	#461	1	1	yes	characteristic
83	C₂×C₁₀₉	#462	1	1	yes	characteristic
84	C₂×C₁₀₉	#487	1	1	yes	characteristic
85	C₂×C₁₀₉	#491	1	1	yes	characteristic
86	#2043	—	109	1	no	maximal, Sylow
87	C₁₀₉	#2043	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊Q₈	—	109	1	no	—
89	C₂×#103	—	109	2	no	—
90	C₂×#103	—	109	2	no	—
91	#389	D₁₀₉	1	1	yes	characteristic
92	#336	—	109	1	no	—
93	C₂²×Q₈	—	109	2	no	—
94	C₂²×C₈	—	109	2	no	—
95	C₄⋊Q₈	—	218	4	no	—
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	—	218	1	no	—
98	C₂×C₄⋊C₄	—	218	1	no	—
99	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
102	#103	—	218	4	no	—
103	#103	—	218	4	no	—
104	#95	—	2	1	no	—
105	C₂×Q₈	—	109	4	no	—
106	C₂×Q₈	—	218	2	no	—
107	C₂×Q₈	—	218	2	no	—
108	C₂×Q₈	—	218	2	no	—
109	C₂×Q₈	—	218	2	no	—
110	C₂×Q₈	—	218	2	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	218	2	no	—
114	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
117	C₂×C₈	—	218	2	no	—
118	C₂×C₈	—	218	2	no	—
119	C₂×C₈	—	218	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	218	4	no	—
124	C₄⋊C₄	—	218	4	no	—
125	C₄⋊C₄	D₄₃₆	1	2	yes	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
128	Q₈	—	436	2	no	—
129	Q₈	—	436	2	no	—
130	Q₈	—	218	4	no	—
131	Q₈	—	218	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	436	2	no	—
143	C₂×C₄	—	436	2	no	—
144	C₂×C₄	—	218	4	no	—
145	C₂×C₄	#13182017	1	2	yes	—
146	C₂×C₄	#13182019	1	2	yes	—
147	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
148	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
149	C₂×C₄	#13182029	1	1	yes	characteristic
150	C₂×C₄	#13182030	1	2	yes	—
151	C₈	—	436	2	no	—
152	C₂²	#16812667	1	1	yes	characteristic
153	C₂²	#16812673	1	1	yes	characteristic
154	C₂²	#16812688	1	1	yes	characteristic
155	C₂²	C₂×#13182017	1	1	yes	characteristic
156	C₂²	C₂×#13182019	1	1	yes	characteristic
157	C₂²	#16812729	1	1	yes	characteristic
158	C₂²	#16812729	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	—	8	1	no	—
163	C₄	—	436	4	no	—
164	C₄	#16812674	1	2	yes	—
165	C₄	#16812684	1	2	yes	—
166	C₂	#20294943	1	1	yes	characteristic
167	C₂	#20295164	1	1	yes	characteristic
168	C₂	#20295181	1	1	yes	characteristic
169	C₂	#20295231	1	1	yes	characteristic
170	C₂	#20295236	1	1	yes	characteristic
171	C₂	#20295378	1	1	yes	characteristic
172	C₂	#20295396	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

C₂×#16812622

C₂

yes

maximal

C₂×#16812622

C₂

yes

maximal

#20294781

C₂

yes

maximal, characteristic

#389×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

#16812699

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

—

#16812622

—

#16812622

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#95×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

D₄

yes

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×Q₁₆

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

#123

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2043

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×#103

—

109

—

C₂×#103

—

109

—

#389

D₁₀₉

yes

characteristic

#336

—

109

—

C₂²×Q₈

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

100

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

101

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

102

#103

—

218

—

103

#103

—

218

—

104

#95

—

105

C₂×Q₈

—

109

—

106

C₂×Q₈

—

218

—

107

C₂×Q₈

—

218

—

108

C₂×Q₈

—

218

—

109

C₂×Q₈

—

218

—

110

C₂×Q₈

—

218

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

218

—

114

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

115

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

116

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

117

C₂×C₈

—

218

—

118

C₂×C₈

—

218

—

119

C₂×C₈

—

218

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

218

—

124

C₄⋊C₄

—

218

—

125

C₄⋊C₄

D₄₃₆

yes

—

126

C₄²

—

127

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

128

Q₈

—

436

—

129

Q₈

—

436

—

130

Q₈

—

218

—

131

Q₈

—

218

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

436

—

143

C₂×C₄

—

436

—

144

C₂×C₄

—

218

—

145

C₂×C₄

#13182017

yes

—

146

C₂×C₄

#13182019

yes

—

147

C₂×C₄

D₄×D₁₀₉

yes

—

148

C₂×C₄

C₂×D₄₃₆

yes

characteristic

149

C₂×C₄

#13182029

yes

characteristic

150

C₂×C₄

#13182030

yes

—

151

C₈

—

436

—

152

C₂²

#16812667

yes

characteristic

153

C₂²

#16812673

yes

characteristic

154

C₂²

#16812688

yes

characteristic

155

C₂²

C₂×#13182017

yes

characteristic

156

C₂²

C₂×#13182019

yes

characteristic

157

C₂²

#16812729

yes

characteristic

158

C₂²

#16812729

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

436

—

164

C₄

#16812674

yes

—

165

C₄

#16812684

yes

—

166

C₂

#20294943

yes

characteristic

167

C₂

#20295164

yes

characteristic

168

C₂

#20295181

yes

characteristic

169

C₂

#20295231

yes

characteristic

170

C₂

#20295236

yes

characteristic

171

C₂

#20295378

yes

characteristic

172

C₂

#20295396

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#389×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#389	D₁₀₉
hypocenter	C₁₀₉	#2043
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#389×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
87. C₁₀₉	#2043	semidirect

H ≅ G/N

type

87. C₁₀₉

#2043

semidirect

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

subgroup	index
C₂×#16812699	2
C₂×#16812622	2
#20294781	2
#389×C₁₀₉	2
#2043	109

subgroup

index

#389×C₁₀₉

109

Subgroups — #28364278

Conjugacy classes of subgroups

Complements

Maximal subgroups