Group #42130840 — subgroups

Subgroups — C₁₀₉×#12846 #42130840

Order 27904.

478 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 174 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×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2112	C₂	1	2	yes	maximal
3	C₁₀₉×#1377	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₁₀₉	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₁₀₉	C₂²	1	4	yes	—
8	C₂×#97×C₁₀₉	—	2	4	no	—
9	C₂×#97×C₁₀₉	C₄	1	2	yes	—
10	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
11	C₂²⋊C₁₆×C₁₀₉	—	2	4	no	—
12	C₂³×C₄×C₁₀₉	C₈	1	2	yes	—
13	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	—	2	2	no	—
15	C₂²×C₈×C₁₀₉	—	2	4	no	—
16	C₂²×C₈×C₁₀₉	—	2	4	no	—
17	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₀₉	D₄	1	4	yes	—
20	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₀₉	—	2	2	no	—
22	C₂×C₄²×C₁₀₉	—	2	2	no	—
23	C₂×C₄²×C₁₀₉	C₈	1	2	yes	—
24	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
25	M₃₂×C₁₀₉	—	2	8	no	—
26	C₂×C₁₆×C₁₀₉	—	2	4	no	—
27	#97×C₁₀₉	—	2	4	no	—
28	#97×C₁₀₉	—	2	4	no	—
29	C₂⁴×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	—	2	2	no	—
32	C₂²×C₄×C₁₀₉	—	2	2	no	—
33	C₂²×C₄×C₁₀₉	—	2	2	no	—
34	C₂²×C₄×C₁₀₉	—	4	4	no	—
35	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₁₀₉	M₁₆	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	—	2	8	no	—
41	C₂×C₈×C₁₀₉	—	2	2	no	—
42	C₂×C₈×C₁₀₉	—	2	2	no	—
43	C₂×C₈×C₁₀₉	—	2	2	no	—
44	C₂×C₈×C₁₀₉	—	2	8	no	—
45	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₀₉	M₁₆	1	4	yes	—
47	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₁₀₉	#41	1	4	yes	—
49	C₄²×C₁₀₉	—	2	4	no	—
50	C₄²×C₁₀₉	—	2	2	no	—
51	C₄²×C₁₀₉	—	2	2	no	—
52	C₁₆×C₁₀₉	—	4	4	no	—
53	C₂³×C₁₀₉	—	2	4	no	—
54	C₂³×C₁₀₉	—	2	2	no	—
55	C₂³×C₁₀₉	C₂²⋊C₈	1	2	yes	—
56	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	—	2	2	no	—
58	C₂×C₄×C₁₀₉	—	2	2	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	2	4	no	—
61	C₂×C₄×C₁₀₉	—	2	4	no	—
62	C₂×C₄×C₁₀₉	—	4	4	no	—
63	C₂×C₄×C₁₀₉	—	4	4	no	—
64	C₂×C₄×C₁₀₉	—	4	4	no	—
65	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	2	yes	—
66	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	#117	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₁₀₉	—	2	2	no	—
71	C₈×C₁₀₉	—	2	4	no	—
72	C₈×C₁₀₉	#123	1	4	yes	—
73	C₂²×C₁₀₉	—	2	8	no	—
74	C₂²×C₁₀₉	—	2	2	no	—
75	C₂²×C₁₀₉	—	2	2	no	—
76	C₂²×C₁₀₉	—	2	2	no	—
77	C₂²×C₁₀₉	#349	1	4	yes	—
78	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₂²×C₁₀₉	#431	1	2	yes	—
80	C₄×C₁₀₉	—	2	2	no	—
81	C₄×C₁₀₉	—	4	4	no	—
82	C₄×C₁₀₉	#381	1	1	yes	characteristic
83	C₄×C₁₀₉	#407	1	1	yes	characteristic
84	C₄×C₁₀₉	#432	1	2	yes	—
85	#12846	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₁₀₉	—	2	4	no	—
87	C₂×C₁₀₉	—	2	2	no	—
88	C₂×C₁₀₉	#1817	1	1	yes	characteristic
89	C₂×C₁₀₉	C₂×#349	1	2	yes	—
90	C₂²×#97	C₂×C₁₀₉	1	1	yes	characteristic
91	#2112	C₂×C₁₀₉	1	2	yes	—
92	#1377	C₂×C₁₀₉	1	4	yes	complemented
93	C₁₀₉	#12846	1	1	yes	characteristic, direct product factor, Sylow
94	C₂³×C₈	C₄×C₁₀₉	1	1	yes	characteristic
95	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
96	C₂²×C₄²	C₄×C₁₀₉	1	1	yes	characteristic
97	C₂×M₃₂	C₂²×C₁₀₉	1	4	yes	—
98	C₂×#97	—	2	4	no	—
99	C₂×#97	C₄×C₁₀₉	1	2	yes	—
100	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
101	C₂²⋊C₁₆	—	2	4	no	—
102	C₂³×C₄	C₈×C₁₀₉	1	2	yes	—
103	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
104	C₂²×C₈	—	2	2	no	—
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	2	4	no	—
107	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
109	C₂²×C₈	D₄×C₁₀₉	1	4	yes	—
110	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	C₈×C₁₀₉	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
115	M₃₂	—	2	8	no	—
116	C₂×C₁₆	—	2	4	no	—
117	#97	—	2	4	no	—
118	#97	—	2	4	no	—
119	C₂⁴	C₂×C₈×C₁₀₉	1	1	yes	characteristic
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	4	4	no	—
125	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
126	C₂²×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₈×C₁₀₉	1	4	yes	—
128	C₂²×C₄	M₁₆×C₁₀₉	1	2	yes	—
129	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
130	C₂×C₈	—	2	8	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
136	C₂×C₈	M₁₆×C₁₀₉	1	4	yes	—
137	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
138	C₂×C₈	#41×C₁₀₉	1	4	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₁₆	—	4	4	no	—
143	C₂³	—	2	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	C₂²⋊C₈×C₁₀₉	1	2	yes	—
146	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
157	C₂×C₄	#117×C₁₀₉	1	2	yes	—
158	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
159	C₂×C₄	C₂×M₁₆×C₁₀₉	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	4	no	—
162	C₈	#123×C₁₀₉	1	4	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	#349×C₁₀₉	1	4	yes	—
168	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
169	C₂²	#431×C₁₀₉	1	2	yes	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	#381×C₁₀₉	1	1	yes	characteristic
173	C₄	#407×C₁₀₉	1	1	yes	characteristic
174	C₄	#432×C₁₀₉	1	2	yes	—
175	C₂	—	2	4	no	—
176	C₂	—	2	2	no	—
177	C₂	C₁₀₉×#1817	1	1	yes	characteristic
178	C₂	C₂×#349×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#97×C₁₀₉

C₂

yes

maximal, characteristic

C₁₀₉×#2112

C₂

yes

maximal

C₁₀₉×#1377

C₂

yes

maximal, complemented

C₂³×C₈×C₁₀₉

C₄

yes

characteristic

C₂³×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄²×C₁₀₉

C₄

yes

characteristic

C₂×M₃₂×C₁₀₉

C₂²

yes

—

C₂×#97×C₁₀₉

—

C₂×#97×C₁₀₉

C₄

yes

—

C₂×#97×C₁₀₉

C₂²

yes

—

C₂²⋊C₁₆×C₁₀₉

—

C₂³×C₄×C₁₀₉

C₈

yes

—

C₂³×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₈×C₁₀₉

D₄

yes

—

C₂²×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₈

yes

—

C₂×C₄²×C₁₀₉

C₂×C₄

yes

—

M₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

#97×C₁₀₉

—

#97×C₁₀₉

—

C₂⁴×C₁₀₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₈

yes

—

C₂²×C₄×C₁₀₉

M₁₆

yes

—

C₂²×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₀₉

M₁₆

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

—

C₂×C₈×C₁₀₉

#41

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₈

yes

—

C₂³×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₈

yes

—

C₂×C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

#117

yes

—

C₂×C₄×C₁₀₉

C₂×M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#123

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#349

yes

—

C₂²×C₁₀₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₁₀₉

#431

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

#12846

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#1817

yes

characteristic

C₂×C₁₀₉

C₂×#349

yes

—

C₂²×#97

C₂×C₁₀₉

yes

characteristic

#2112

C₂×C₁₀₉

yes

—

#1377

C₂×C₁₀₉

yes

complemented

C₁₀₉

#12846

yes

characteristic, direct product factor, Sylow

C₂³×C₈

C₄×C₁₀₉

yes

characteristic

C₂³×C₈

C₂²×C₁₀₉

yes

characteristic

C₂²×C₄²

C₄×C₁₀₉

yes

characteristic

C₂×M₃₂

C₂²×C₁₀₉

yes

—

C₂×#97

—

C₂×#97

C₄×C₁₀₉

yes

—

100

C₂×#97

C₂²×C₁₀₉

yes

—

101

C₂²⋊C₁₆

—

102

C₂³×C₄

C₈×C₁₀₉

yes

—

103

C₂³×C₄

C₂×C₄×C₁₀₉

yes

characteristic

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂²×C₈

C₂×C₄×C₁₀₉

yes

characteristic

108

C₂²×C₈

C₂×C₄×C₁₀₉

yes

—

109

C₂²×C₈

D₄×C₁₀₉

yes

—

110

C₂²×C₈

C₂³×C₁₀₉

yes

characteristic

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

C₈×C₁₀₉

yes

—

114

C₂×C₄²

C₂×C₄×C₁₀₉

yes

—

115

M₃₂

—

116

C₂×C₁₆

—

117

#97

—

118

#97

—

119

C₂⁴

C₂×C₈×C₁₀₉

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂²⋊C₄×C₁₀₉

yes

—

126

C₂²×C₄

C₂×C₈×C₁₀₉

yes

characteristic

127

C₂²×C₄

C₂×C₈×C₁₀₉

yes

—

128

C₂²×C₄

M₁₆×C₁₀₉

yes

—

129

C₂²×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

—

136

C₂×C₈

M₁₆×C₁₀₉

yes

—

137

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

138

C₂×C₈

#41×C₁₀₉

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₁₆

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂²⋊C₈×C₁₀₉

yes

—

146

C₂³

C₂²×C₈×C₁₀₉

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂²⋊C₈×C₁₀₉

yes

—

156

C₂×C₄

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

157

C₂×C₄

#117×C₁₀₉

yes

—

158

C₂×C₄

C₂×M₁₆×C₁₀₉

yes

—

159

C₂×C₄

C₂×M₁₆×C₁₀₉

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

#123×C₁₀₉

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

#349×C₁₀₉

yes

—

168

C₂²

C₂×C₂²⋊C₈×C₁₀₉

yes

characteristic

169

C₂²

#431×C₁₀₉

yes

—

170

C₄

—

171

C₄

—

172

C₄

#381×C₁₀₉

yes

characteristic

173

C₄

#407×C₁₀₉

yes

characteristic

174

C₄

#432×C₁₀₉

yes

—

175

C₂

—

176

C₂

—

177

C₂

C₁₀₉×#1817

yes

characteristic

178

C₂

C₂×#349×C₁₀₉

yes

—

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

subgroup

G/H

center

C₂×C₄×C₁₀₉

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₈×C₁₀₉

Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₉

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₁₀₉

C₂×C₈

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
3. C₁₀₉×#1377	C₂	semidirect
85. #12846	C₁₀₉	direct (N × H)
92. #1377	C₂×C₁₀₉	semidirect
93. C₁₀₉	#12846	direct (N × H)

H ≅ G/N

type

3. C₁₀₉×#1377

C₂

semidirect

85. #12846

C₁₀₉

direct (N × H)

92. #1377

C₂×C₁₀₉

semidirect

93. C₁₀₉

#12846

direct (N × H)

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

subgroup	index
C₂²×#97×C₁₀₉	2
C₁₀₉×#2112	2
C₁₀₉×#1377	2
#12846	109

subgroup

index

C₂²×#97×C₁₀₉

C₁₀₉×#2112

C₁₀₉×#1377

#12846

109

Subgroups — C₁₀₉×#12846 #42130840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12846 #42130840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12846 #42130840