Group #28379210 — subgroups

Subgroups — #28379210

Order 13952.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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	D₁₀₉⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic
3	#20295899	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#89137	C₂²	1	2	yes	—
7	C₂×C₄×#89137	—	2	1	no	—
8	#16812766	—	2	1	no	—
9	C₄⋊#11724786	—	2	1	no	—
10	(C₂×#89137)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₁₀₉	—	2	1	no	—
12	#16812699	C₄	1	1	yes	characteristic, complemented
13	#16812698	—	2	1	no	—
14	#16812698	C₄	1	2	yes	complemented
15	#16812697	—	2	1	no	—
16	C₄⋊C₄×D₁₀₉	—	2	1	no	—
17	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
18	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
20	C₂²×#89137	—	2	2	no	—
21	D₁₀₉⋊C₈	—	2	4	no	—
22	C₂×#11724786	—	2	2	no	—
23	C₄⋊#89137	—	2	2	no	—
24	C₄⋊#89137	Q₈	1	4	yes	complemented
25	C₄×#89137	—	4	2	no	—
26	C₂×Dic₂₁₈	—	2	1	no	—
27	C₂×Dic₂₁₈	—	2	2	no	—
28	Q₈×D₁₀₉	—	4	4	no	—
29	C₂×C₄×D₁₀₉	—	2	2	no	—
30	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
31	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
32	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₁₀₉	—	2	1	no	—
34	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
35	#13182008	—	2	2	no	—
36	#13182008	—	2	2	no	—
37	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
39	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
40	C₂×Q₈×C₁₀₉	—	2	1	no	—
41	C₄⋊C₄×C₁₀₉	—	2	1	no	—
42	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×#89137	—	2	4	no	—
46	C₂×#89137	—	4	2	no	—
47	C₂×#89137	—	4	2	no	—
48	Q₈×C₁₀₉	—	4	2	no	—
49	C₂×C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₁₀₉	—	2	2	no	—
54	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₄×D₁₀₉	—	2	2	no	—
58	C₄×D₁₀₉	—	4	4	no	—
59	C₄×D₁₀₉	SD₁₆	1	2	yes	—
60	C₄×D₁₀₉	Q₁₆	1	2	yes	—
61	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
62	C₄×D₁₀₉	#41	1	2	yes	—
63	Dic₂₁₈	—	4	4	no	—
64	Dic₂₁₈	—	4	2	no	—
65	#11724786	—	4	2	no	—
66	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
68	C₂×D₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×D₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×D₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
71	#89137	—	4	4	no	—
72	C₄×C₁₀₉	—	2	1	no	—
73	C₄×C₁₀₉	—	4	2	no	—
74	C₄×C₁₀₉	#103	1	2	yes	—
75	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
76	Dic₁₀₉	—	2	1	no	—
77	Dic₁₀₉	—	4	2	no	—
78	Dic₁₀₉	#103	1	2	yes	—
79	Dic₁₀₉	#124	1	1	yes	characteristic
80	Dic₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₁₀₉	#391	1	1	yes	characteristic
82	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
83	C₂×C₁₀₉	#418	1	1	yes	characteristic
84	D₁₀₉	#476	1	2	yes	—
85	D₁₀₉	#477	1	2	yes	—
86	#1987	—	109	1	no	maximal, Sylow
87	C₁₀₉	#1987	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊Q₈	—	109	1	no	—
89	C₂×C₄⋊C₈	—	109	1	no	—
90	C₄×C₄⋊C₄	—	109	1	no	—
91	#340	—	109	4	no	—
92	C₂²×Q₈	—	218	1	no	—
93	C₂²×C₈	—	109	2	no	—
94	C₄⋊Q₈	—	109	2	no	—
95	C₄⋊Q₈	—	109	1	no	—
96	C₄⋊Q₈	—	218	1	no	—
97	C₄⋊Q₈	—	218	1	no	—
98	C₄⋊Q₈	#89137	1	1	yes	characteristic, complemented
99	C₂×C₄⋊C₄	—	109	2	no	—
100	C₂×C₄⋊C₄	—	109	2	no	—
101	C₂×C₄⋊C₄	—	218	1	no	—
102	C₂×C₄²	—	109	1	no	—
103	C₂×C₄²	—	218	1	no	—
104	C₄⋊C₈	—	218	1	no	—
105	C₄⋊C₈	—	218	1	no	—
106	#95	—	218	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	436	4	no	—
109	C₂×Q₈	—	218	2	no	—
110	C₂×Q₈	—	218	1	no	—
111	C₂²×C₄	—	109	1	no	—
112	C₂²×C₄	—	109	1	no	—
113	C₂²×C₄	—	109	1	no	—
114	C₂²×C₄	—	218	2	no	—
115	C₂²×C₄	—	218	2	no	—
116	C₂×C₈	—	218	2	no	—
117	C₂×C₈	—	218	4	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	109	4	no	—
120	C₄⋊C₄	—	109	2	no	—
121	C₄⋊C₄	—	218	2	no	—
122	C₄⋊C₄	—	218	2	no	—
123	C₄⋊C₄	—	218	2	no	—
124	C₄⋊C₄	—	218	1	no	—
125	C₄⋊C₄	C₂×#89137	1	2	yes	—
126	C₄²	—	109	1	no	—
127	C₄²	—	109	1	no	—
128	C₄²	—	109	1	no	—
129	C₄²	—	436	2	no	—
130	C₄²	C₂×#89137	1	1	yes	characteristic
131	C₂³	—	109	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	436	2	no	—
134	Q₈	—	436	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	109	1	no	—
137	C₂×C₄	—	109	1	no	—
138	C₂×C₄	—	109	2	no	—
139	C₂×C₄	—	109	2	no	—
140	C₂×C₄	—	109	2	no	—
141	C₂×C₄	—	109	2	no	—
142	C₂×C₄	—	109	1	no	—
143	C₂×C₄	—	436	4	no	—
144	C₂×C₄	—	436	2	no	—
145	C₂×C₄	—	436	2	no	—
146	C₂×C₄	—	218	2	no	—
147	C₂×C₄	—	218	4	no	—
148	C₂×C₄	—	218	2	no	—
149	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
150	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
151	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
152	C₈	—	436	2	no	—
153	C₂²	—	109	2	no	—
154	C₂²	—	109	1	no	—
155	C₂²	—	109	1	no	—
156	C₂²	—	109	2	no	—
157	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	109	1	no	—
161	C₄	—	109	1	no	—
162	C₄	—	109	2	no	—
163	C₄	—	436	4	no	—
164	C₄	—	436	2	no	—
165	C₄	—	218	1	no	—
166	C₄	#16812762	1	2	yes	—
167	C₄	Q₈×#89137	1	2	yes	—
168	C₂	—	109	2	no	—
169	C₂	—	109	2	no	—
170	C₂	(C₂×#89137)⋊Q₈	1	1	yes	characteristic
171	C₂	C₂×#16812762	1	1	yes	characteristic
172	C₂	#20296018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₉⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊#89137

C₂

yes

maximal, characteristic

#20295899

C₂

yes

maximal

C₄⋊Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

C₂²

yes

—

C₂×C₄⋊#89137

C₂²

yes

—

C₂×C₄×#89137

—

#16812766

—

C₄⋊#11724786

—

(C₂×#89137)⋊C₄

—

C₂×Q₈×D₁₀₉

—

#16812699

C₄

yes

characteristic, complemented

#16812698

—

#16812698

C₄

yes

complemented

#16812697

—

C₄⋊C₄×D₁₀₉

—

C₄⋊C₄×D₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

D₁₀₉⋊C₈

—

C₂×#11724786

—

C₄⋊#89137

—

C₄⋊#89137

Q₈

yes

complemented

C₄×#89137

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×D₄

yes

—

—

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

SD₁₆

yes

—

C₄×D₁₀₉

Q₁₆

yes

—

C₄×D₁₀₉

C₂×Q₈

yes

—

C₄×D₁₀₉

#41

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

C₂²⋊Q₈

yes

—

C₂×D₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×D₁₀₉

C₂×Q₁₆

yes

characteristic

C₂×D₁₀₉

AGL(1,ℤ/8)

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#103

yes

—

C₄×C₁₀₉

C₄×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#103

yes

—

Dic₁₀₉

#124

yes

characteristic

Dic₁₀₉

C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

yes

—

D₁₀₉

#477

yes

—

#1987

—

109

maximal, Sylow

C₁₀₉

#1987

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×C₄⋊C₈

—

109

—

C₄×C₄⋊C₄

—

109

—

#340

—

109

—

C₂²×Q₈

—

218

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

#89137

yes

characteristic, complemented

C₂×C₄⋊C₄

—

109

—

100

C₂×C₄⋊C₄

—

109

—

101

C₂×C₄⋊C₄

—

218

—

102

C₂×C₄²

—

109

—

103

C₂×C₄²

—

218

—

104

C₄⋊C₈

—

218

—

105

C₄⋊C₈

—

218

—

106

#95

—

218

—

107

C₂×Q₈

—

108

C₂×Q₈

—

436

—

109

C₂×Q₈

—

218

—

110

C₂×Q₈

—

218

—

111

C₂²×C₄

—

109

—

112

C₂²×C₄

—

109

—

113

C₂²×C₄

—

109

—

114

C₂²×C₄

—

218

—

115

C₂²×C₄

—

218

—

116

C₂×C₈

—

218

—

117

C₂×C₈

—

218

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

109

—

120

C₄⋊C₄

—

109

—

121

C₄⋊C₄

—

218

—

122

C₄⋊C₄

—

218

—

123

C₄⋊C₄

—

218

—

124

C₄⋊C₄

—

218

—

125

C₄⋊C₄

C₂×#89137

yes

—

126

C₄²

—

109

—

127

C₄²

—

109

—

128

C₄²

—

109

—

129

C₄²

—

436

—

130

C₄²

C₂×#89137

yes

characteristic

131

C₂³

—

109

—

132

Q₈

—

133

Q₈

—

436

—

134

Q₈

—

436

—

135

C₂×C₄

—

136

C₂×C₄

—

109

—

137

C₂×C₄

—

109

—

138

C₂×C₄

—

109

—

139

C₂×C₄

—

109

—

140

C₂×C₄

—

109

—

141

C₂×C₄

—

109

—

142

C₂×C₄

—

109

—

143

C₂×C₄

—

436

—

144

C₂×C₄

—

436

—

145

C₂×C₄

—

436

—

146

C₂×C₄

—

218

—

147

C₂×C₄

—

218

—

148

C₂×C₄

—

218

—

149

C₂×C₄

C₂²⋊#89137

yes

characteristic

150

C₂×C₄

C₂²⋊#89137

yes

characteristic

151

C₂×C₄

C₂²×#89137

yes

characteristic

152

C₈

—

436

—

153

C₂²

—

109

—

154

C₂²

—

109

—

155

C₂²

—

109

—

156

C₂²

—

109

—

157

C₂²

C₂×C₂²⋊#89137

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

109

—

161

C₄

—

109

—

162

C₄

—

109

—

163

C₄

—

436

—

164

C₄

—

436

—

165

C₄

—

218

—

166

C₄

#16812762

yes

—

167

C₄

Q₈×#89137

yes

—

168

C₂

—

109

—

169

C₂

—

109

—

170

C₂

(C₂×#89137)⋊Q₈

yes

characteristic

171

C₂

C₂×#16812762

yes

characteristic

172

C₂

#20296018

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄⋊Q₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₉	C₂×D₄
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1987
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄⋊Q₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₉

C₂×D₄

cosocle

C₂×C₄×D₁₀₉

C₂³

C₄⋊Q₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #16812699	C₄	semidirect
14. #16812698	C₄	semidirect
19. C₄⋊Q₈×C₁₀₉	C₄	semidirect
24. C₄⋊#89137	Q₈	semidirect
87. C₁₀₉	#1987	semidirect
98. C₄⋊Q₈	#89137	semidirect

H ≅ G/N

type

12. #16812699

C₄

semidirect

14. #16812698

C₄

semidirect

19. C₄⋊Q₈×C₁₀₉

C₄

semidirect

24. C₄⋊#89137

Q₈

semidirect

87. C₁₀₉

#1987

semidirect

98. C₄⋊Q₈

#89137

semidirect

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

subgroup	index
D₁₀₉⋊(C₄⋊C₈)	2
C₄×C₄⋊#89137	2
#20295899	2
C₄⋊Q₈×D₁₀₉	2
#1987	109

subgroup

index

D₁₀₉⋊(C₄⋊C₈)

C₄×C₄⋊#89137

#20295899

C₄⋊Q₈×D₁₀₉

#1987

109

Subgroups — #28379210

Conjugacy classes of subgroups

Complements

Maximal subgroups