Group #28337707 — subgroups

Subgroups — #28337707

Order 13696.

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×#16812109	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812032	C₂	1	2	yes	maximal
3	C₂×#16812032	C₂	1	2	yes	maximal
4	#20292382	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₄	C₂²	1	2	yes	—
7	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
8	#16812109	—	2	4	no	—
9	C₂×C₄⋊Dic₁₀₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₇	—	2	1	no	—
11	#16812032	—	2	4	no	—
12	#16812032	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
17	#95×C₁₀₇	—	2	1	no	—
18	C₂×Dic₂₁₄	—	2	2	no	—
19	C₂×Dic₂₁₄	—	2	2	no	—
20	C₂×Dic₂₁₄	—	2	2	no	—
21	C₂×Dic₂₁₄	—	2	2	no	—
22	C₂×Dic₂₁₄	—	2	2	no	—
23	C₂×Dic₂₁₄	D₄	1	4	yes	—
24	C₂²×Dic₁₀₇	—	2	2	no	—
25	C₂×C₁₀₇⋊C₈	—	2	2	no	—
26	C₂×C₁₀₇⋊C₈	—	2	2	no	—
27	C₂×C₁₀₇⋊C₈	—	2	2	no	—
28	C₄⋊Dic₁₀₇	—	2	4	no	—
29	C₄⋊Dic₁₀₇	—	2	4	no	—
30	C₂²×C₄×C₁₀₇	—	2	1	no	—
31	C₂²×C₄×C₁₀₇	—	2	1	no	—
32	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₀₇	—	2	1	no	—
36	C₄⋊C₄×C₁₀₇	—	2	2	no	—
37	C₄⋊C₄×C₁₀₇	—	4	2	no	—
38	C₄⋊C₄×C₁₀₇	D₄	1	2	yes	—
39	C₄²×C₁₀₇	—	2	2	no	—
40	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₇	—	2	1	no	—
42	C₂×C₄×C₁₀₇	—	2	1	no	—
43	C₂×C₄×C₁₀₇	—	2	1	no	—
44	C₂×C₄×C₁₀₇	—	2	1	no	—
45	C₂×C₄×C₁₀₇	—	2	2	no	—
46	C₂×C₄×C₁₀₇	—	4	1	no	—
47	C₂×C₄×C₁₀₇	—	4	1	no	—
48	C₂×C₄×C₁₀₇	—	4	1	no	—
49	C₂×C₄×C₁₀₇	—	4	1	no	—
50	C₂×C₄×C₁₀₇	—	4	1	no	—
51	C₂×C₄×C₁₀₇	SD₁₆	1	2	yes	—
52	C₂×C₄×C₁₀₇	Q₁₆	1	2	yes	—
53	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₀₇	#41	1	2	yes	—
57	C₂×Dic₁₀₇	—	2	4	no	—
58	C₂×Dic₁₀₇	—	4	2	no	—
59	C₂×Dic₁₀₇	—	4	2	no	—
60	Dic₂₁₄	—	2	4	no	—
61	Dic₂₁₄	—	2	4	no	—
62	Dic₂₁₄	—	4	2	no	—
63	Dic₂₁₄	—	4	2	no	—
64	C₁₀₇⋊C₈	—	4	2	no	—
65	C₂²×C₁₀₇	#120	1	1	yes	characteristic
66	C₂²×C₁₀₇	#121	1	1	yes	characteristic
67	C₂²×C₁₀₇	#124	1	1	yes	characteristic
68	C₂²×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₁₀₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₁₀₇	#137	1	1	yes	characteristic
71	C₂²×C₁₀₇	#137	1	1	yes	characteristic
72	C₄×C₁₀₇	—	2	2	no	—
73	C₄×C₁₀₇	—	2	2	no	—
74	C₄×C₁₀₇	—	4	2	no	—
75	C₄×C₁₀₇	—	8	1	no	—
76	C₄×C₁₀₇	#121	1	2	yes	—
77	C₄×C₁₀₇	#123	1	2	yes	—
78	Dic₁₀₇	—	4	4	no	—
79	C₂×C₁₀₇	#394	1	1	yes	characteristic
80	C₂×C₁₀₇	#448	1	1	yes	characteristic
81	C₂×C₁₀₇	#451	1	1	yes	characteristic
82	C₂×C₁₀₇	#461	1	1	yes	characteristic
83	C₂×C₁₀₇	#462	1	1	yes	characteristic
84	C₂×C₁₀₇	#487	1	1	yes	characteristic
85	C₂×C₁₀₇	#491	1	1	yes	characteristic
86	#2043	—	107	1	no	maximal, Sylow
87	C₁₀₇	#2043	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊Q₈	—	107	1	no	—
89	C₂×#103	—	107	2	no	—
90	C₂×#103	—	107	2	no	—
91	#389	D₁₀₇	1	1	yes	characteristic
92	#336	—	107	1	no	—
93	C₂²×Q₈	—	107	2	no	—
94	C₂²×C₈	—	107	2	no	—
95	C₄⋊Q₈	—	214	4	no	—
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	—	214	1	no	—
98	C₂×C₄⋊C₄	—	214	1	no	—
99	C₂×C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
102	#103	—	214	4	no	—
103	#103	—	214	4	no	—
104	#95	—	2	1	no	—
105	C₂×Q₈	—	107	4	no	—
106	C₂×Q₈	—	214	2	no	—
107	C₂×Q₈	—	214	2	no	—
108	C₂×Q₈	—	214	2	no	—
109	C₂×Q₈	—	214	2	no	—
110	C₂×Q₈	—	214	2	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	214	2	no	—
114	C₂²×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
117	C₂×C₈	—	214	2	no	—
118	C₂×C₈	—	214	2	no	—
119	C₂×C₈	—	214	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	214	4	no	—
124	C₄⋊C₄	—	214	4	no	—
125	C₄⋊C₄	D₄₂₈	1	2	yes	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
128	Q₈	—	428	2	no	—
129	Q₈	—	428	2	no	—
130	Q₈	—	214	4	no	—
131	Q₈	—	214	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	—	428	2	no	—
143	C₂×C₄	—	428	2	no	—
144	C₂×C₄	—	214	4	no	—
145	C₂×C₄	#13133806	1	2	yes	—
146	C₂×C₄	#13133808	1	2	yes	—
147	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
148	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
149	C₂×C₄	#13133818	1	1	yes	characteristic
150	C₂×C₄	#13133819	1	2	yes	—
151	C₈	—	428	2	no	—
152	C₂²	#16812077	1	1	yes	characteristic
153	C₂²	#16812083	1	1	yes	characteristic
154	C₂²	#16812098	1	1	yes	characteristic
155	C₂²	C₂×#13133806	1	1	yes	characteristic
156	C₂²	C₂×#13133808	1	1	yes	characteristic
157	C₂²	#16812139	1	1	yes	characteristic
158	C₂²	#16812139	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	—	428	4	no	—
164	C₄	#16812084	1	2	yes	—
165	C₄	#16812094	1	2	yes	—
166	C₂	#20292544	1	1	yes	characteristic
167	C₂	#20292765	1	1	yes	characteristic
168	C₂	#20292782	1	1	yes	characteristic
169	C₂	#20292832	1	1	yes	characteristic
170	C₂	#20292837	1	1	yes	characteristic
171	C₂	#20292979	1	1	yes	characteristic
172	C₂	#20292997	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812109

C₂

yes

maximal, characteristic

C₂×#16812032

C₂

yes

maximal

C₂×#16812032

C₂

yes

maximal

#20292382

C₂

yes

maximal, characteristic

#389×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

#16812109

—

C₂×C₄⋊Dic₁₀₇

—

C₂×C₄⋊Dic₁₀₇

—

#16812032

—

#16812032

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#95×C₁₀₇

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

D₄

yes

—

C₂²×Dic₁₀₇

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

characteristic

C₂²×C₄×C₁₀₇

D₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

SD₁₆

yes

—

C₂×C₄×C₁₀₇

Q₁₆

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

C₂×C₄×C₁₀₇

yes

C₂×C₄×C₁₀₇

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₇

C₂×Q₁₆

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#121

yes

—

C₄×C₁₀₇

#123

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#2043

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

107

—

C₂×#103

—

107

—

C₂×#103

—

107

—

#389

D₁₀₇

yes

characteristic

#336

—

107

—

C₂²×Q₈

—

107

—

C₂²×C₈

—

107

—

C₄⋊Q₈

—

214

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

214

—

C₂×C₄⋊C₄

—

214

—

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

100

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

101

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

102

#103

—

214

—

103

#103

—

214

—

104

#95

—

105

C₂×Q₈

—

107

—

106

C₂×Q₈

—

214

—

107

C₂×Q₈

—

214

—

108

C₂×Q₈

—

214

—

109

C₂×Q₈

—

214

—

110

C₂×Q₈

—

214

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

214

—

114

C₂²×C₄

C₂²⋊D₁₀₇

yes

characteristic

115

C₂²×C₄

C₂²⋊D₁₀₇

yes

characteristic

116

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

117

C₂×C₈

—

214

—

118

C₂×C₈

—

214

—

119

C₂×C₈

—

214

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

214

—

124

C₄⋊C₄

—

214

—

125

C₄⋊C₄

D₄₂₈

yes

—

126

C₄²

—

127

C₂³

C₂×C₂²⋊D₁₀₇

yes

characteristic

128

Q₈

—

428

—

129

Q₈

—

428

—

130

Q₈

—

214

—

131

Q₈

—

214

—

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

—

428

—

143

C₂×C₄

—

428

—

144

C₂×C₄

—

214

—

145

C₂×C₄

#13133806

yes

—

146

C₂×C₄

#13133808

yes

—

147

C₂×C₄

D₄×D₁₀₇

yes

—

148

C₂×C₄

C₂×D₄₂₈

yes

characteristic

149

C₂×C₄

#13133818

yes

characteristic

150

C₂×C₄

#13133819

yes

—

151

C₈

—

428

—

152

C₂²

#16812077

yes

characteristic

153

C₂²

#16812083

yes

characteristic

154

C₂²

#16812098

yes

characteristic

155

C₂²

C₂×#13133806

yes

characteristic

156

C₂²

C₂×#13133808

yes

characteristic

157

C₂²

#16812139

yes

characteristic

158

C₂²

#16812139

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

428

—

164

C₄

#16812084

yes

—

165

C₄

#16812094

yes

—

166

C₂

#20292544

yes

characteristic

167

C₂

#20292765

yes

characteristic

168

C₂

#20292782

yes

characteristic

169

C₂

#20292832

yes

characteristic

170

C₂

#20292837

yes

characteristic

171

C₂

#20292979

yes

characteristic

172

C₂

#20292997

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₇
commutator, cosocle	C₂²×C₄×C₁₀₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	#389×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂×D₄
hypercenter	#389	D₁₀₇
hypocenter	C₁₀₇	#2043
LCS	G ▷ C₂²×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₇

commutator, cosocle

C₂²×C₄×C₁₀₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

#389×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂×D₄

D₁₀₇

C₁₀₇

G ▷ C₂²×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
87. C₁₀₇	#2043	semidirect

H ≅ G/N

type

87. C₁₀₇

#2043

semidirect

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

subgroup	index
C₂×#16812109	2
C₂×#16812032	2
#20292382	2
#389×C₁₀₇	2
#2043	107

subgroup

index

#389×C₁₀₇

107

Subgroups — #28337707

Conjugacy classes of subgroups

Complements

Maximal subgroups