Group #28364174 — subgroups

Subgroups — #28364174

Order 13952.

290 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 91 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×#16812624	C₂	1	2	yes	maximal
3	#20294932	C₂	1	1	yes	maximal, characteristic
4	#20294789	C₂	1	2	yes	maximal
5	#336×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	—	2	1	no	—
7	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
8	#16812699	—	2	4	no	—
9	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
12	#16812624	—	2	4	no	—
13	#16812606	—	2	1	no	—
14	#16812606	—	2	1	no	—
15	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₈	—	2	2	no	—
18	C₂×Dic₂₁₈	—	2	2	no	—
19	C₂×Dic₂₁₈	—	2	4	no	—
20	C₂×Dic₂₁₈	—	2	1	no	—
21	C₂×Dic₂₁₈	—	4	2	no	—
22	C₂×Dic₂₁₈	—	4	2	no	—
23	C₂×Dic₂₁₈	Q₈	1	2	yes	complemented
24	C₂²×Dic₁₀₉	—	2	1	no	—
25	C₂²×Dic₁₀₉	—	2	2	no	—
26	C₂²×Dic₁₀₉	—	2	1	no	—
27	C₄⋊Dic₁₀₉	—	2	4	no	—
28	C₄⋊Dic₁₀₉	—	2	2	no	—
29	C₄⋊Dic₁₀₉	—	2	2	no	—
30	C₄⋊Dic₁₀₉	D₄	1	4	yes	—
31	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	—	2	2	no	—
35	C₂×C₈×C₁₀₉	—	2	2	no	—
36	C₂×C₈×C₁₀₉	—	2	2	no	—
37	C₄²×C₁₀₉	—	2	2	no	—
38	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
43	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
44	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	#41	1	2	yes	—
49	C₂×Dic₁₀₉	—	2	4	no	—
50	C₂×Dic₁₀₉	—	2	2	no	—
51	C₂×Dic₁₀₉	—	4	1	no	—
52	C₂×Dic₁₀₉	—	4	1	no	—
53	C₂×Dic₁₀₉	—	4	2	no	—
54	C₂×Dic₁₀₉	—	4	2	no	—
55	C₂×Dic₁₀₉	—	4	1	no	—
56	C₂×Dic₁₀₉	—	4	1	no	—
57	C₂×Dic₁₀₉	—	4	1	no	—
58	Dic₂₁₈	—	2	4	no	—
59	Dic₂₁₈	—	4	8	no	—
60	Dic₂₁₈	—	4	2	no	—
61	C₈×C₁₀₉	—	4	2	no	—
62	C₂²×C₁₀₉	#120	1	1	yes	characteristic
63	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₀₉	#124	1	1	yes	characteristic
65	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₁₀₉	#137	1	1	yes	characteristic
69	C₄×C₁₀₉	—	2	2	no	—
70	C₄×C₁₀₉	—	2	2	no	—
71	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₁₀₉	#124	1	2	yes	—
73	Dic₁₀₉	—	4	2	no	—
74	Dic₁₀₉	—	4	4	no	—
75	Dic₁₀₉	—	8	1	no	—
76	C₂×C₁₀₉	#393	1	1	yes	characteristic
77	C₂×C₁₀₉	#450	1	1	yes	characteristic
78	C₂×C₁₀₉	#451	1	1	yes	characteristic
79	C₂×C₁₀₉	#475	1	1	yes	characteristic
80	C₂×C₁₀₉	#477	1	1	yes	characteristic
81	C₂×C₁₀₉	#487	1	1	yes	characteristic
82	C₂×C₁₀₉	#489	1	1	yes	characteristic
83	#2024	—	109	1	no	maximal, Sylow
84	C₁₀₉	#2024	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊Q₈	—	109	1	no	—
86	C₂×#103	—	109	2	no	—
87	#391	—	109	1	no	—
88	#340	—	109	2	no	—
89	#336	D₁₀₉	1	1	yes	characteristic
90	C₂²×Q₈	—	109	1	no	—
91	C₂²×Q₈	—	218	1	no	—
92	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
93	C₄⋊Q₈	—	218	4	no	—
94	C₂×C₄⋊C₄	—	109	2	no	—
95	C₂×C₄⋊C₄	—	109	1	no	—
96	C₂×C₄⋊C₄	—	218	1	no	—
97	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
98	#103	—	218	4	no	—
99	#95	—	218	1	no	—
100	#95	—	218	1	no	—
101	C₂×Q₈	—	109	2	no	—
102	C₂×Q₈	—	436	2	no	—
103	C₂×Q₈	—	436	2	no	—
104	C₂×Q₈	—	218	4	no	—
105	C₂×Q₈	—	218	2	no	—
106	C₂×Q₈	—	218	2	no	—
107	C₂×Q₈	—	218	1	no	—
108	C₂²×C₄	—	218	1	no	—
109	C₂²×C₄	—	218	1	no	—
110	C₂²×C₄	—	218	2	no	—
111	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
112	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₄⋊C₄	—	109	4	no	—
118	C₄⋊C₄	—	218	4	no	—
119	C₄⋊C₄	—	218	2	no	—
120	C₄⋊C₄	—	218	2	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
123	Q₈	—	436	8	no	—
124	Q₈	—	436	2	no	—
125	Q₈	—	218	4	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	436	1	no	—
131	C₂×C₄	—	436	1	no	—
132	C₂×C₄	—	436	1	no	—
133	C₂×C₄	—	436	1	no	—
134	C₂×C₄	—	436	1	no	—
135	C₂×C₄	—	436	2	no	—
136	C₂×C₄	—	436	2	no	—
137	C₂×C₄	—	218	4	no	—
138	C₂×C₄	—	218	2	no	—
139	C₂×C₄	#13182018	1	2	yes	—
140	C₂×C₄	Dic₄₃₆	1	2	yes	—
141	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
142	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
143	C₂×C₄	#13182028	1	1	yes	characteristic
144	C₂×C₄	#13182029	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₂²	#16812669	1	1	yes	characteristic
147	C₂²	#16812679	1	1	yes	characteristic
148	C₂²	#16812687	1	1	yes	characteristic
149	C₂²	C₂×#13182018	1	1	yes	characteristic
150	C₂²	C₂×Dic₄₃₆	1	1	yes	characteristic
151	C₂²	#16812727	1	1	yes	characteristic
152	C₂²	#16812732	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	872	1	no	—
156	C₄	—	436	4	no	—
157	C₄	—	436	2	no	—
158	C₄	#16812677	1	2	yes	—
159	C₄	#16812685	1	2	yes	—
160	C₂	#20294939	1	1	yes	characteristic
161	C₂	#20295179	1	1	yes	characteristic
162	C₂	#20295184	1	1	yes	characteristic
163	C₂	#20295312	1	1	yes	characteristic
164	C₂	#20295322	1	1	yes	characteristic
165	C₂	#20295377	1	1	yes	characteristic
166	C₂	#20295386	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

C₂×#16812624

C₂

yes

maximal

#20294932

C₂

yes

maximal, characteristic

#20294789

C₂

yes

maximal

#336×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

C₂²×Dic₂₁₈

C₂²

yes

characteristic

#16812699

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812624

—

#16812606

—

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

Q₈

yes

complemented

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

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×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

#120

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#124

yes

characteristic

C₂²×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×Q₁₆

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#124

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

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

#2024

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×#103

—

109

—

#391

—

109

—

#340

—

109

—

#336

D₁₀₉

yes

characteristic

C₂²×Q₈

—

109

—

C₂²×Q₈

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#103

—

218

—

#95

—

218

—

100

#95

—

218

—

101

C₂×Q₈

—

109

—

102

C₂×Q₈

—

436

—

103

C₂×Q₈

—

436

—

104

C₂×Q₈

—

218

—

105

C₂×Q₈

—

218

—

106

C₂×Q₈

—

218

—

107

C₂×Q₈

—

218

—

108

C₂²×C₄

—

218

—

109

C₂²×C₄

—

218

—

110

C₂²×C₄

—

218

—

111

C₂²×C₄

D₄₃₆

yes

characteristic

112

C₂²×C₄

D₄₃₆

yes

characteristic

113

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

109

—

118

C₄⋊C₄

—

218

—

119

C₄⋊C₄

—

218

—

120

C₄⋊C₄

—

218

—

121

C₄²

—

122

C₂³

C₂×D₄₃₆

yes

characteristic

123

Q₈

—

436

—

124

Q₈

—

436

—

125

Q₈

—

218

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

436

—

131

C₂×C₄

—

436

—

132

C₂×C₄

—

436

—

133

C₂×C₄

—

436

—

134

C₂×C₄

—

436

—

135

C₂×C₄

—

436

—

136

C₂×C₄

—

436

—

137

C₂×C₄

—

218

—

138

C₂×C₄

—

218

—

139

C₂×C₄

#13182018

yes

—

140

C₂×C₄

Dic₄₃₆

yes

—

141

C₂×C₄

D₄×D₁₀₉

yes

—

142

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

143

C₂×C₄

#13182028

yes

characteristic

144

C₂×C₄

#13182029

yes

—

145

C₈

—

146

C₂²

#16812669

yes

characteristic

147

C₂²

#16812679

yes

characteristic

148

C₂²

#16812687

yes

characteristic

149

C₂²

C₂×#13182018

yes

characteristic

150

C₂²

C₂×Dic₄₃₆

yes

characteristic

151

C₂²

#16812727

yes

characteristic

152

C₂²

#16812732

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

872

—

156

C₄

—

436

—

157

C₄

—

436

—

158

C₄

#16812677

yes

—

159

C₄

#16812685

yes

—

160

C₂

#20294939

yes

characteristic

161

C₂

#20295179

yes

characteristic

162

C₂

#20295184

yes

characteristic

163

C₂

#20295312

yes

characteristic

164

C₂

#20295322

yes

characteristic

165

C₂

#20295377

yes

characteristic

166

C₂

#20295386

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₃₆
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#336×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#336	D₁₀₉
hypocenter	C₁₀₉	#2024
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₄₃₆

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#336×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
23. C₂×Dic₂₁₈	Q₈	semidirect
84. C₁₀₉	#2024	semidirect

H ≅ G/N

type

23. C₂×Dic₂₁₈

Q₈

semidirect

84. C₁₀₉

#2024

semidirect

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

subgroup	index
C₂×#16812699	2
C₂×#16812624	2
#20294932	2
#20294789	2
#336×C₁₀₉	2
#2024	109

subgroup

index

#336×C₁₀₉

109

Subgroups — #28364174

Conjugacy classes of subgroups

Complements

Maximal subgroups