Group #42130866 — subgroups

Subgroups — C₁₀₉×#12872 #42130866

Order 27904.

398 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 142 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×#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2112	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1810	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1387	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂×#117×C₁₀₉	—	2	1	no	—
7	C₂×M₃₂×C₁₀₉	C₂²	1	2	yes	—
8	C₂×#108×C₁₀₉	C₂²	1	2	yes	—
9	#427×C₁₀₉	—	2	2	no	—
10	#427×C₁₀₉	C₄	1	4	yes	—
11	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
12	C₂×#106×C₁₀₉	—	2	1	no	—
13	#407×C₁₀₉	—	2	1	no	—
14	C₂²⋊C₁₆×C₁₀₉	—	2	2	no	—
15	#341×C₁₀₉	—	2	1	no	—
16	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂×M₁₆×C₁₀₉	—	2	2	no	—
18	C₂²×C₈×C₁₀₉	—	2	4	no	—
19	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
22	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
23	#117×C₁₀₉	—	2	4	no	—
24	#117×C₁₀₉	—	4	2	no	—
25	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
26	C₂×C₂²⋊C₄×C₁₀₉	—	4	1	no	—
27	C₂×C₄²×C₁₀₉	—	4	1	no	—
28	M₃₂×C₁₀₉	—	2	4	no	—
29	C₂×C₁₆×C₁₀₉	—	2	2	no	—
30	#108×C₁₀₉	—	2	4	no	—
31	#107×C₁₀₉	—	2	2	no	—
32	#107×C₁₀₉	C₂×C₄	1	4	yes	—
33	#106×C₁₀₉	—	2	4	no	—
34	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
35	C₂⁴×C₁₀₉	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	—	2	2	no	—
37	C₂²×C₄×C₁₀₉	—	2	2	no	—
38	C₂²×C₄×C₁₀₉	—	4	2	no	—
39	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₀₉	D₈	1	2	yes	—
41	C₂²×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	M₁₆×C₁₀₉	—	4	2	no	—
44	M₁₆×C₁₀₉	—	4	2	no	—
45	C₂×C₈×C₁₀₉	—	2	2	no	—
46	C₂×C₈×C₁₀₉	—	2	4	no	—
47	C₂×C₈×C₁₀₉	—	2	4	no	—
48	C₂×C₈×C₁₀₉	—	4	2	no	—
49	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₀₉	#41	1	4	yes	—
53	C₄⋊C₄×C₁₀₉	—	2	4	no	—
54	C₄⋊C₄×C₁₀₉	—	4	2	no	—
55	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
56	C₄²×C₁₀₉	—	4	4	no	—
57	C₁₆×C₁₀₉	—	4	2	no	—
58	C₂³×C₁₀₉	—	2	2	no	—
59	C₂³×C₁₀₉	—	4	2	no	—
60	C₂³×C₁₀₉	#102	1	2	yes	—
61	C₂³×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	—	2	2	no	—
64	C₂×C₄×C₁₀₉	—	2	2	no	—
65	C₂×C₄×C₁₀₉	—	2	4	no	—
66	C₂×C₄×C₁₀₉	—	4	4	no	—
67	C₂×C₄×C₁₀₉	—	8	4	no	—
68	C₂×C₄×C₁₀₉	#102	1	2	yes	—
69	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	#123	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₁₀₉	#137	1	2	yes	—
73	C₈×C₁₀₉	—	2	2	no	—
74	C₈×C₁₀₉	—	8	2	no	—
75	C₈×C₁₀₉	#117	1	2	yes	—
76	C₈×C₁₀₉	#123	1	2	yes	—
77	C₂²×C₁₀₉	—	2	2	no	—
78	C₂²×C₁₀₉	—	2	2	no	—
79	C₂²×C₁₀₉	—	2	4	no	—
80	C₂²×C₁₀₉	—	4	4	no	—
81	C₂²×C₁₀₉	C₂×#102	1	1	yes	characteristic
82	C₂²×C₁₀₉	#483	1	2	yes	—
83	C₄×C₁₀₉	—	2	2	no	—
84	C₄×C₁₀₉	—	8	4	no	—
85	C₄×C₁₀₉	#381	1	1	yes	characteristic
86	C₄×C₁₀₉	#419	1	1	yes	characteristic
87	C₄×C₁₀₉	#480	1	2	yes	—
88	#12872	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₁₀₉	—	2	2	no	—
90	C₂×C₁₀₉	—	4	2	no	—
91	C₂×C₁₀₉	#1823	1	1	yes	characteristic
92	C₂×C₁₀₉	#2148	1	2	yes	—
93	C₂×#427	C₂×C₁₀₉	1	1	yes	characteristic
94	#2112	C₂×C₁₀₉	1	1	yes	characteristic
95	#1810	C₂×C₁₀₉	1	1	yes	characteristic
96	#1387	C₂×C₁₀₉	1	4	yes	complemented
97	C₁₀₉	#12872	1	1	yes	characteristic, direct product factor, Sylow
98	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
99	C₂×#117	—	2	1	no	—
100	C₂×M₃₂	C₂²×C₁₀₉	1	2	yes	—
101	C₂×#108	C₂²×C₁₀₉	1	2	yes	—
102	#427	—	2	2	no	—
103	#427	C₄×C₁₀₉	1	4	yes	—
104	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
105	C₂×#106	—	2	1	no	—
106	#407	—	2	1	no	—
107	C₂²⋊C₁₆	—	2	2	no	—
108	#341	—	2	1	no	—
109	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
110	C₂×M₁₆	—	2	2	no	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
113	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
114	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
115	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
116	#117	—	2	4	no	—
117	#117	—	4	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₂²⋊C₄	—	4	1	no	—
120	C₂×C₄²	—	4	1	no	—
121	M₃₂	—	2	4	no	—
122	C₂×C₁₆	—	2	2	no	—
123	#108	—	2	4	no	—
124	#107	—	2	2	no	—
125	#107	C₂×C₄×C₁₀₉	1	4	yes	—
126	#106	—	2	4	no	—
127	C₂²⋊C₈	—	4	2	no	—
128	C₂⁴	SD₁₆×C₁₀₉	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
133	C₂²×C₄	D₈×C₁₀₉	1	2	yes	—
134	C₂²×C₄	SD₁₆×C₁₀₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
143	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
144	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
145	C₂×C₈	#41×C₁₀₉	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₂²⋊C₄	—	4	4	no	—
149	C₄²	—	4	4	no	—
150	C₁₆	—	4	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	4	2	no	—
153	C₂³	#102×C₁₀₉	1	2	yes	—
154	C₂³	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	#102×C₁₀₉	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
163	C₂×C₄	#123×C₁₀₉	1	2	yes	—
164	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
165	C₂×C₄	#137×C₁₀₉	1	2	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	8	2	no	—
168	C₈	#117×C₁₀₉	1	2	yes	—
169	C₈	#123×C₁₀₉	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	4	4	no	—
174	C₂²	C₂×#102×C₁₀₉	1	1	yes	characteristic
175	C₂²	#483×C₁₀₉	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	8	4	no	—
178	C₄	#381×C₁₀₉	1	1	yes	characteristic
179	C₄	#419×C₁₀₉	1	1	yes	characteristic
180	C₄	#480×C₁₀₉	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	4	2	no	—
183	C₂	C₁₀₉×#1823	1	1	yes	characteristic
184	C₂	C₁₀₉×#2148	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#427×C₁₀₉

C₂

yes

maximal, characteristic

C₁₀₉×#2112

C₂

yes

maximal, characteristic

C₁₀₉×#1810

C₂

yes

maximal, characteristic

C₁₀₉×#1387

C₂

yes

maximal, complemented

C₂³×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×#117×C₁₀₉

—

C₂×M₃₂×C₁₀₉

C₂²

yes

—

C₂×#108×C₁₀₉

C₂²

yes

—

#427×C₁₀₉

—

#427×C₁₀₉

C₄

yes

—

C₂×#107×C₁₀₉

C₂²

yes

—

C₂×#106×C₁₀₉

—

#407×C₁₀₉

—

C₂²⋊C₁₆×C₁₀₉

—

#341×C₁₀₉

—

C₂³×C₄×C₁₀₉

D₄

yes

characteristic

C₂×M₁₆×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₈×C₁₀₉

D₄

yes

characteristic

C₂²×C₈×C₁₀₉

D₄

yes

—

C₂²×C₈×C₁₀₉

C₂³

yes

characteristic

#117×C₁₀₉

—

#117×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₂²⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

—

M₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

#108×C₁₀₉

—

#107×C₁₀₉

—

#107×C₁₀₉

C₂×C₄

yes

—

#106×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

C₂⁴×C₁₀₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₉

D₈

yes

—

C₂²×C₄×C₁₀₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₉

#41

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#102

yes

—

C₂³×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#102

yes

—

C₂×C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

#123

yes

—

C₂×C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₉

#137

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#117

yes

—

C₈×C₁₀₉

#123

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×#102

yes

characteristic

C₂²×C₁₀₉

#483

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

#12872

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#1823

yes

characteristic

C₂×C₁₀₉

#2148

yes

—

C₂×#427

C₂×C₁₀₉

yes

characteristic

#2112

C₂×C₁₀₉

yes

characteristic

#1810

C₂×C₁₀₉

yes

characteristic

#1387

C₂×C₁₀₉

yes

complemented

C₁₀₉

#12872

yes

characteristic, direct product factor, Sylow

C₂³×C₈

C₂²×C₁₀₉

yes

characteristic

C₂×#117

—

100

C₂×M₃₂

C₂²×C₁₀₉

yes

—

101

C₂×#108

C₂²×C₁₀₉

yes

—

102

#427

—

103

#427

C₄×C₁₀₉

yes

—

104

C₂×#107

C₂²×C₁₀₉

yes

—

105

C₂×#106

—

106

#407

—

107

C₂²⋊C₁₆

—

108

#341

—

109

C₂³×C₄

D₄×C₁₀₉

yes

characteristic

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₂²×C₈

C₂×C₄×C₁₀₉

yes

—

113

C₂²×C₈

D₄×C₁₀₉

yes

characteristic

114

C₂²×C₈

D₄×C₁₀₉

yes

—

115

C₂²×C₈

C₂³×C₁₀₉

yes

characteristic

116

#117

—

117

#117

—

118

C₂×C₄⋊C₄

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₄²

—

121

M₃₂

—

122

C₂×C₁₆

—

123

#108

—

124

#107

—

125

#107

C₂×C₄×C₁₀₉

yes

—

126

#106

—

127

C₂²⋊C₈

—

128

C₂⁴

SD₁₆×C₁₀₉

yes

characteristic

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂²⋊C₄×C₁₀₉

yes

—

133

C₂²×C₄

D₈×C₁₀₉

yes

—

134

C₂²×C₄

SD₁₆×C₁₀₉

yes

characteristic

135

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

—

143

C₂×C₈

C₂²×C₄×C₁₀₉

yes

characteristic

144

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

145

C₂×C₈

#41×C₁₀₉

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₁₆

—

151

C₂³

—

152

C₂³

—

153

C₂³

#102×C₁₀₉

yes

—

154

C₂³

C₂×SD₁₆×C₁₀₉

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

#102×C₁₀₉

yes

—

162

C₂×C₄

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

163

C₂×C₄

#123×C₁₀₉

yes

—

164

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

165

C₂×C₄

#137×C₁₀₉

yes

—

166

C₈

—

167

C₈

—

168

C₈

#117×C₁₀₉

yes

—

169

C₈

#123×C₁₀₉

yes

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₂×#102×C₁₀₉

yes

characteristic

175

C₂²

#483×C₁₀₉

yes

—

176

C₄

—

177

C₄

—

178

C₄

#381×C₁₀₉

yes

characteristic

179

C₄

#419×C₁₀₉

yes

characteristic

180

C₄

#480×C₁₀₉

yes

—

181

C₂

—

182

C₂

—

183

C₂

C₁₀₉×#1823

yes

characteristic

184

C₂

C₁₀₉×#2148

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
prime radical	C₂⁴×C₁₀₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

C₂×#102

commutator

C₂×C₈

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₉

prime radical

C₂⁴×C₁₀₉

SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
4. C₁₀₉×#1387	C₂	semidirect
88. #12872	C₁₀₉	direct (N × H)
96. #1387	C₂×C₁₀₉	semidirect
97. C₁₀₉	#12872	direct (N × H)

H ≅ G/N

type

4. C₁₀₉×#1387

C₂

semidirect

88. #12872

C₁₀₉

direct (N × H)

96. #1387

C₂×C₁₀₉

semidirect

97. C₁₀₉

#12872

direct (N × H)

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

subgroup	index
C₂×#427×C₁₀₉	2
C₁₀₉×#2112	2
C₁₀₉×#1810	2
C₁₀₉×#1387	2
#12872	109

subgroup

index

C₂×#427×C₁₀₉

C₁₀₉×#2112

C₁₀₉×#1810

C₁₀₉×#1387

#12872

109

Subgroups — C₁₀₉×#12872 #42130866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12872 #42130866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12872 #42130866