Group #28377911 — subgroups

Subgroups — #28377911

Order 13952.

188 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 70 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	#20295972	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×D₁₀₉⋊C₈	—	2	1	no	—
5	C₂×C₄⋊#89137	—	2	1	no	—
6	C₂×C₄⋊#89137	—	2	1	no	—
7	C₂×C₄⋊#89137	C₄	1	2	yes	—
8	#16812766	C₄	1	2	yes	complemented
9	C₄⋊#11724786	—	2	1	no	—
10	C₂×C₈×D₁₀₉	—	2	1	no	—
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊#11724784	C₄	1	1	yes	characteristic, complemented
13	#16812628	—	2	1	no	—
14	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×#89137	—	2	2	no	—
16	D₁₀₉⋊C₈	—	4	1	no	—
17	D₁₀₉⋊C₈	—	4	1	no	—
18	C₂×#11724786	—	2	2	no	—
19	C₄⋊#89137	—	4	2	no	—
20	C₄⋊#89137	—	4	1	no	—
21	C₄⋊#89137	—	4	1	no	—
22	C₄⋊#89137	—	4	1	no	—
23	C₄⋊#89137	—	4	1	no	—
24	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
27	C₂×#11724784	—	2	1	no	—
28	C₈×D₁₀₉	—	4	1	no	—
29	C₈×D₁₀₉	—	4	1	no	—
30	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
32	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	—	2	1	no	—
34	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×#89137	—	4	2	no	—
37	C₂×#89137	—	4	2	no	—
38	C₂×#89137	—	4	2	no	—
39	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄×D₁₀₉	—	2	1	no	—
46	C₄×D₁₀₉	—	2	1	no	—
47	C₄×D₁₀₉	D₈	1	1	yes	characteristic
48	C₄×D₁₀₉	D₈	1	1	yes	characteristic
49	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
50	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
51	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
52	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
53	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
54	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
55	#11724786	—	4	2	no	—
56	C₈×C₁₀₉	—	4	1	no	—
57	#11724784	—	4	1	no	—
58	C₂²×C₁₀₉	#95	1	1	yes	characteristic
59	C₂×D₁₀₉	#100	1	1	yes	characteristic
60	C₂×D₁₀₉	#101	1	1	yes	characteristic
61	C₂×D₁₀₉	#102	1	1	yes	characteristic
62	C₂×D₁₀₉	#102	1	1	yes	characteristic
63	C₂×D₁₀₉	#103	1	1	yes	characteristic
64	C₂×D₁₀₉	#103	1	1	yes	characteristic
65	#89137	—	8	2	no	—
66	C₄×C₁₀₉	—	2	1	no	—
67	C₄×C₁₀₉	#102	1	1	yes	characteristic
68	C₄×C₁₀₉	#103	1	1	yes	characteristic
69	C₄×C₁₀₉	#106	1	1	yes	characteristic
70	C₄×C₁₀₉	#107	1	1	yes	characteristic
71	Dic₁₀₉	—	2	1	no	—
72	Dic₁₀₉	#102	1	1	yes	characteristic
73	Dic₁₀₉	#103	1	1	yes	characteristic
74	Dic₁₀₉	#106	1	1	yes	characteristic
75	Dic₁₀₉	#107	1	1	yes	characteristic
76	C₂×C₁₀₉	#340	1	1	yes	characteristic
77	C₂×C₁₀₉	#340	1	1	yes	characteristic
78	C₂×C₁₀₉	#343	1	1	yes	characteristic
79	D₁₀₉	#331	1	1	yes	characteristic
80	D₁₀₉	#332	1	1	yes	characteristic
81	D₁₀₉	#332	1	1	yes	characteristic
82	D₁₀₉	#333	1	1	yes	characteristic
83	#1296	—	109	1	no	maximal, Sylow
84	C₁₀₉	#1296	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊C₈	—	109	1	no	—
86	C₂×C₄⋊C₈	—	109	1	no	—
87	#384	—	109	1	no	—
88	C₂²×C₈	—	218	1	no	—
89	C₂²×C₈	—	218	1	no	—
90	C₂×C₄⋊C₄	—	109	2	no	—
91	C₂×C₄⋊C₄	—	218	1	no	—
92	C₂×C₄⋊C₄	—	218	1	no	—
93	C₂×C₄²	—	109	1	no	—
94	C₄⋊C₈	—	109	2	no	—
95	C₄⋊C₈	—	109	1	no	—
96	C₄⋊C₈	—	218	1	no	—
97	C₄⋊C₈	—	218	1	no	—
98	C₄⋊C₈	#89137	1	1	yes	characteristic, complemented
99	C₂²×C₄	—	109	1	no	—
100	C₂²×C₄	—	109	1	no	—
101	C₂²×C₄	—	109	1	no	—
102	C₂²×C₄	—	218	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	436	1	no	—
105	C₂×C₈	—	436	1	no	—
106	C₂×C₈	—	436	1	no	—
107	C₂×C₈	—	436	1	no	—
108	C₂×C₈	—	218	2	no	—
109	C₂×C₈	—	218	1	no	—
110	C₄⋊C₄	—	436	1	no	—
111	C₄⋊C₄	—	436	1	no	—
112	C₄⋊C₄	—	436	1	no	—
113	C₄⋊C₄	—	436	1	no	—
114	C₄⋊C₄	—	436	2	no	—
115	C₄²	—	109	1	no	—
116	C₄²	—	109	1	no	—
117	C₄²	—	109	1	no	—
118	C₄²	C₂×#89137	1	1	yes	characteristic
119	C₂³	—	109	1	no	—
120	C₂×C₄	—	109	1	no	—
121	C₂×C₄	—	109	1	no	—
122	C₂×C₄	—	109	1	no	—
123	C₂×C₄	—	109	1	no	—
124	C₂×C₄	—	109	1	no	—
125	C₂×C₄	—	109	1	no	—
126	C₂×C₄	—	109	1	no	—
127	C₂×C₄	—	109	1	no	—
128	C₂×C₄	—	109	1	no	—
129	C₂×C₄	—	109	1	no	—
130	C₂×C₄	—	109	1	no	—
131	C₂×C₄	—	436	2	no	—
132	C₂×C₄	—	436	2	no	—
133	C₂×C₄	—	436	2	no	—
134	C₂×C₄	—	218	1	no	—
135	C₂×C₄	—	218	1	no	—
136	C₂×C₄	C₄×#89137	1	1	yes	characteristic
137	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
138	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
139	C₈	—	4	1	no	—
140	C₈	—	436	2	no	—
141	C₈	—	436	1	no	—
142	C₂²	—	109	1	no	—
143	C₂²	—	109	1	no	—
144	C₂²	—	109	1	no	—
145	C₂²	—	109	1	no	—
146	C₂²	—	109	1	no	—
147	C₂²	—	109	1	no	—
148	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	872	2	no	—
151	C₄	—	109	1	no	—
152	C₄	—	109	1	no	—
153	C₄	—	109	1	no	—
154	C₄	—	109	1	no	—
155	C₄	—	218	1	no	—
156	C₄	#16812761	1	1	yes	characteristic
157	C₄	#16812762	1	1	yes	characteristic
158	C₄	#16812767	1	1	yes	characteristic
159	C₄	#16812768	1	1	yes	characteristic
160	C₂	—	109	1	no	—
161	C₂	—	109	1	no	—
162	C₂	—	109	1	no	—
163	C₂	—	109	1	no	—
164	C₂	#20295899	1	1	yes	characteristic
165	C₂	#20295900	1	1	yes	characteristic
166	C₂	#20295905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₉⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

#20295972

C₂

yes

maximal, characteristic

C₄⋊C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

—

C₂×C₄⋊#89137

—

C₂×C₄⋊#89137

—

C₂×C₄⋊#89137

C₄

yes

—

#16812766

C₄

yes

complemented

C₄⋊#11724786

—

C₂×C₈×D₁₀₉

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic, complemented

#16812628

—

C₄⋊C₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

—

—

—

—

—

—

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×#11724784

—

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

—

—

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

yes

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1296

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄⋊C₈

—

109

—

#384

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₄⋊C₈

#89137

yes

characteristic, complemented

C₂²×C₄

—

109

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

—

109

—

102

C₂²×C₄

—

218

—

103

C₂×C₈

—

104

C₂×C₈

—

436

—

105

C₂×C₈

—

436

—

106

C₂×C₈

—

436

—

107

C₂×C₈

—

436

—

108

C₂×C₈

—

218

—

109

C₂×C₈

—

218

—

110

C₄⋊C₄

—

436

—

111

C₄⋊C₄

—

436

—

112

C₄⋊C₄

—

436

—

113

C₄⋊C₄

—

436

—

114

C₄⋊C₄

—

436

—

115

C₄²

—

109

—

116

C₄²

—

109

—

117

C₄²

—

109

—

118

C₄²

C₂×#89137

yes

characteristic

119

C₂³

—

109

—

120

C₂×C₄

—

109

—

121

C₂×C₄

—

109

—

122

C₂×C₄

—

109

—

123

C₂×C₄

—

109

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

109

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

109

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

436

—

132

C₂×C₄

—

436

—

133

C₂×C₄

—

436

—

134

C₂×C₄

—

218

—

135

C₂×C₄

—

218

—

136

C₂×C₄

C₄×#89137

yes

characteristic

137

C₂×C₄

C₂²⋊#89137

yes

characteristic

138

C₂×C₄

C₄⋊#89137

yes

characteristic

139

C₈

—

140

C₈

—

436

—

141

C₈

—

436

—

142

C₂²

—

109

—

143

C₂²

—

109

—

144

C₂²

—

109

—

145

C₂²

—

109

—

146

C₂²

—

109

—

147

C₂²

—

109

—

148

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

149

C₄

—

150

C₄

—

872

—

151

C₄

—

109

—

152

C₄

—

109

—

153

C₄

—

109

—

154

C₄

—

109

—

155

C₄

—

218

—

156

C₄

#16812761

yes

characteristic

157

C₄

#16812762

yes

characteristic

158

C₄

#16812767

yes

characteristic

159

C₄

#16812768

yes

characteristic

160

C₂

—

109

—

161

C₂

—

109

—

162

C₂

—

109

—

163

C₂

—

109

—

164

C₂

#20295899

yes

characteristic

165

C₂

#20295900

yes

characteristic

166

C₂

#20295905

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#89137)⋊C₄
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂²×D₁₀₉	C₂²⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#1296
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

(C₂×#89137)⋊C₄

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₄²

C₂×#89137

Fitting

C₄⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂²×D₁₀₉

C₂²⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₄⋊C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
8. #16812766	C₄	semidirect
12. C₄⋊#11724784	C₄	semidirect
14. C₄⋊C₈×C₁₀₉	C₄	semidirect
84. C₁₀₉	#1296	semidirect
98. C₄⋊C₈	#89137	semidirect

H ≅ G/N

type

8. #16812766

C₄

semidirect

12. C₄⋊#11724784

C₄

semidirect

14. C₄⋊C₈×C₁₀₉

C₄

semidirect

84. C₁₀₉

#1296

semidirect

98. C₄⋊C₈

#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
#20295972	2
C₄⋊C₈×D₁₀₉	2
#1296	109

subgroup

index

D₁₀₉⋊(C₄⋊C₈)

#20295972

C₄⋊C₈×D₁₀₉

#1296

109

Subgroups — #28377911

Conjugacy classes of subgroups

Complements

Maximal subgroups