Group #42130832 — subgroups

Subgroups — C₁₀₉×#12838 #42130832

Order 27904.

374 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 158 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	C₂²×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#368×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#1810	C₂	1	2	yes	maximal
4	C₁₀₉×#1381	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂×M₃₂×C₁₀₉	—	2	4	no	—
7	C₂×M₃₂×C₁₀₉	C₄	1	4	yes	—
8	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#108×C₁₀₉	C₂²	1	4	yes	—
11	#407×C₁₀₉	—	2	2	no	—
12	#368×C₁₀₉	—	2	4	no	—
13	#341×C₁₀₉	—	2	2	no	—
14	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
15	C₂×M₁₆×C₁₀₉	—	2	4	no	—
16	C₂²×C₈×C₁₀₉	—	2	2	no	—
17	C₂²×C₈×C₁₀₉	—	2	2	no	—
18	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₉	Q₈	1	2	yes	—
21	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
22	M₃₂×C₁₀₉	—	2	8	no	—
23	C₂×C₁₆×C₁₀₉	—	2	2	no	—
24	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
27	#108×C₁₀₉	—	2	4	no	—
28	#108×C₁₀₉	—	2	4	no	—
29	C₂²⋊C₈×C₁₀₉	—	4	4	no	—
30	C₂⁴×C₁₀₉	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	—	2	2	no	—
32	C₂²×C₄×C₁₀₉	—	2	2	no	—
33	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₀₉	D₈	1	2	yes	—
35	C₂²×C₄×C₁₀₉	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	M₁₆×C₁₀₉	—	4	4	no	—
38	M₁₆×C₁₀₉	—	4	4	no	—
39	C₂×C₈×C₁₀₉	—	2	2	no	—
40	C₂×C₈×C₁₀₉	—	2	2	no	—
41	C₂×C₈×C₁₀₉	—	2	2	no	—
42	C₂×C₈×C₁₀₉	—	2	4	no	—
43	C₂×C₈×C₁₀₉	—	4	4	no	—
44	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	#41	1	2	yes	—
49	C₁₆×C₁₀₉	—	2	2	no	—
50	C₁₆×C₁₀₉	C₂²⋊C₄	1	4	yes	—
51	C₂³×C₁₀₉	—	2	2	no	—
52	C₂³×C₁₀₉	—	4	2	no	—
53	C₂³×C₁₀₉	#107	1	2	yes	—
54	C₂³×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	—	2	2	no	—
56	C₂×C₄×C₁₀₉	—	2	2	no	—
57	C₂×C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	2	4	no	—
59	C₂×C₄×C₁₀₉	#107	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	#121	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	#135	1	2	yes	—
64	C₈×C₁₀₉	—	2	2	no	—
65	C₈×C₁₀₉	—	8	4	no	—
66	C₈×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₈×C₁₀₉	#117	1	1	yes	characteristic
68	C₈×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
69	C₂²×C₁₀₉	—	2	2	no	—
70	C₂²×C₁₀₉	—	2	2	no	—
71	C₂²×C₁₀₉	—	2	4	no	—
72	C₂²×C₁₀₉	—	4	4	no	—
73	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
74	C₂²×C₁₀₉	#467	1	2	yes	—
75	C₄×C₁₀₉	—	2	2	no	—
76	C₄×C₁₀₉	#380	1	1	yes	characteristic
77	C₄×C₁₀₉	#427	1	1	yes	characteristic
78	C₄×C₁₀₉	#466	1	2	yes	—
79	#12838	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₁₀₉	—	2	2	no	—
81	C₂×C₁₀₉	—	4	2	no	—
82	C₂×C₁₀₉	#1808	1	1	yes	characteristic
83	C₂×C₁₀₉	#2161	1	2	yes	—
84	C₂²×M₃₂	C₂×C₁₀₉	1	1	yes	characteristic
85	C₂×#368	C₂×C₁₀₉	1	2	yes	complemented
86	#1810	C₂×C₁₀₉	1	2	yes	—
87	#1381	C₂×C₁₀₉	1	2	yes	complemented
88	C₁₀₉	#12838	1	1	yes	characteristic, direct product factor, Sylow
89	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
90	C₂×M₃₂	—	2	4	no	—
91	C₂×M₃₂	C₄×C₁₀₉	1	4	yes	—
92	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
93	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
94	C₂×#108	C₂²×C₁₀₉	1	4	yes	—
95	#407	—	2	2	no	—
96	#368	—	2	4	no	—
97	#341	—	2	2	no	—
98	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
99	C₂×M₁₆	—	2	4	no	—
100	C₂²×C₈	—	2	2	no	—
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
103	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
104	C₂²×C₈	Q₈×C₁₀₉	1	2	yes	—
105	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
106	M₃₂	—	2	8	no	—
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
109	C₂×C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
110	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
111	#108	—	2	4	no	—
112	#108	—	2	4	no	—
113	C₂²⋊C₈	—	4	4	no	—
114	C₂⁴	Q₁₆×C₁₀₉	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
118	C₂²×C₄	D₈×C₁₀₉	1	2	yes	—
119	C₂²×C₄	Q₁₆×C₁₀₉	1	1	yes	characteristic
120	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
121	M₁₆	—	4	4	no	—
122	M₁₆	—	4	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	4	4	no	—
128	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
130	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
131	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
132	C₂×C₈	#41×C₁₀₉	1	2	yes	—
133	C₁₆	—	2	2	no	—
134	C₁₆	C₂²⋊C₄×C₁₀₉	1	4	yes	—
135	C₂³	—	2	2	no	—
136	C₂³	—	4	2	no	—
137	C₂³	#107×C₁₀₉	1	2	yes	—
138	C₂³	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	#107×C₁₀₉	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
145	C₂×C₄	#121×C₁₀₉	1	2	yes	—
146	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
147	C₂×C₄	#135×C₁₀₉	1	2	yes	—
148	C₈	—	2	2	no	—
149	C₈	—	8	4	no	—
150	C₈	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
151	C₈	#117×C₁₀₉	1	1	yes	characteristic
152	C₈	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
158	C₂²	#467×C₁₀₉	1	2	yes	—
159	C₄	—	2	2	no	—
160	C₄	#380×C₁₀₉	1	1	yes	characteristic
161	C₄	#427×C₁₀₉	1	1	yes	characteristic
162	C₄	#466×C₁₀₉	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	4	2	no	—
165	C₂	C₁₀₉×#1808	1	1	yes	characteristic
166	C₂	C₁₀₉×#2161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₃₂×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#368×C₁₀₉

C₂

yes

maximal, complemented

C₁₀₉×#1810

C₂

yes

maximal

C₁₀₉×#1381

C₂

yes

maximal, complemented

C₂³×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×M₃₂×C₁₀₉

—

C₂×M₃₂×C₁₀₉

C₄

yes

—

C₂²×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×#108×C₁₀₉

C₂²

yes

—

#407×C₁₀₉

—

#368×C₁₀₉

—

#341×C₁₀₉

—

C₂³×C₄×C₁₀₉

D₄

yes

characteristic

C₂×M₁₆×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₈×C₁₀₉

D₄

yes

characteristic

C₂²×C₈×C₁₀₉

Q₈

yes

—

C₂²×C₈×C₁₀₉

C₂³

yes

characteristic

M₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

#108×C₁₀₉

—

#108×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

C₂⁴×C₁₀₉

Q₁₆

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₀₉

D₈

yes

—

C₂²×C₄×C₁₀₉

Q₁₆

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

—

C₂×C₈×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₈×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×D₄

yes

—

C₂×C₈×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₀₉

#41

yes

—

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

C₂²⋊C₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#107

yes

—

C₂³×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#107

yes

—

C₂×C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

#121

yes

—

C₂×C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₉

#135

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₀₉

#117

yes

characteristic

C₈×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×#107

yes

characteristic

C₂²×C₁₀₉

#467

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

#12838

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#1808

yes

characteristic

C₂×C₁₀₉

#2161

yes

—

C₂²×M₃₂

C₂×C₁₀₉

yes

characteristic

C₂×#368

C₂×C₁₀₉

yes

complemented

#1810

C₂×C₁₀₉

yes

—

#1381

C₂×C₁₀₉

yes

complemented

C₁₀₉

#12838

yes

characteristic, direct product factor, Sylow

C₂³×C₈

C₂²×C₁₀₉

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₄×C₁₀₉

yes

—

C₂²×C₁₆

C₂²×C₁₀₉

yes

characteristic

C₂²×C₁₆

C₂²×C₁₀₉

yes

characteristic

C₂×#108

C₂²×C₁₀₉

yes

—

#407

—

#368

—

#341

—

C₂³×C₄

D₄×C₁₀₉

yes

characteristic

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₂×C₄×C₁₀₉

yes

—

103

C₂²×C₈

D₄×C₁₀₉

yes

characteristic

104

C₂²×C₈

Q₈×C₁₀₉

yes

—

105

C₂²×C₈

C₂³×C₁₀₉

yes

characteristic

106

M₃₂

—

107

C₂×C₁₆

—

108

C₂×C₁₆

C₂×C₄×C₁₀₉

yes

—

109

C₂×C₁₆

C₂×C₄×C₁₀₉

yes

—

110

C₂×C₁₆

D₄×C₁₀₉

yes

—

111

#108

—

112

#108

—

113

C₂²⋊C₈

—

114

C₂⁴

Q₁₆×C₁₀₉

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₄⋊C₄×C₁₀₉

yes

—

118

C₂²×C₄

D₈×C₁₀₉

yes

—

119

C₂²×C₄

Q₁₆×C₁₀₉

yes

characteristic

120

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₄⋊C₄×C₁₀₉

yes

—

129

C₂×C₈

C₂²×C₄×C₁₀₉

yes

characteristic

130

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

131

C₂×C₈

C₂×Q₈×C₁₀₉

yes

characteristic

132

C₂×C₈

#41×C₁₀₉

yes

—

133

C₁₆

—

134

C₁₆

C₂²⋊C₄×C₁₀₉

yes

—

135

C₂³

—

136

C₂³

—

137

C₂³

#107×C₁₀₉

yes

—

138

C₂³

C₂×Q₁₆×C₁₀₉

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#107×C₁₀₉

yes

—

144

C₂×C₄

C₂×C₄⋊C₄×C₁₀₉

yes

characteristic

145

C₂×C₄

#121×C₁₀₉

yes

—

146

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

147

C₂×C₄

#135×C₁₀₉

yes

—

148

C₈

—

149

C₈

—

150

C₈

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

151

C₈

#117×C₁₀₉

yes

characteristic

152

C₈

C₂²⋊Q₈×C₁₀₉

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×#107×C₁₀₉

yes

characteristic

158

C₂²

#467×C₁₀₉

yes

—

159

C₄

—

160

C₄

#380×C₁₀₉

yes

characteristic

161

C₄

#427×C₁₀₉

yes

characteristic

162

C₄

#466×C₁₀₉

yes

—

163

C₂

—

164

C₂

—

165

C₂

C₁₀₉×#1808

yes

characteristic

166

C₂

C₁₀₉×#2161

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#107
prime radical	C₂⁴×C₁₀₉	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₀₉

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₉

socle

C₂²×C₁₀₉

C₂×#107

prime radical

C₂⁴×C₁₀₉

Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. C₂×#368×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1381	C₂	semidirect
79. #12838	C₁₀₉	direct (N × H)
85. C₂×#368	C₂×C₁₀₉	semidirect
87. #1381	C₂×C₁₀₉	semidirect
88. C₁₀₉	#12838	direct (N × H)

H ≅ G/N

type

2. C₂×#368×C₁₀₉

C₂

semidirect

4. C₁₀₉×#1381

C₂

semidirect

79. #12838

C₁₀₉

direct (N × H)

85. C₂×#368

C₂×C₁₀₉

semidirect

87. #1381

C₂×C₁₀₉

semidirect

88. C₁₀₉

#12838

direct (N × H)

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

subgroup	index
C₂²×M₃₂×C₁₀₉	2
C₂×#368×C₁₀₉	2
C₁₀₉×#1810	2
C₁₀₉×#1381	2
#12838	109

subgroup

index

C₂²×M₃₂×C₁₀₉

C₂×#368×C₁₀₉

C₁₀₉×#1810

C₁₀₉×#1381

#12838

109

Subgroups — C₁₀₉×#12838 #42130832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12838 #42130832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12838 #42130832