Group #42126633 — subgroups

Subgroups — C₁₀₉×#8639 #42126633

Order 27904.

336 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 138 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	C₁₀₉×#2960	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1567	C₂	1	2	yes	maximal
3	C₁₀₉×#1561	C₂	1	2	yes	maximal
4	C₁₀₉×#1309	C₂	1	2	yes	maximal
5	#533×C₁₀₉	C₂²	1	1	yes	characteristic
6	#446×C₁₀₉	—	2	2	no	—
7	C₈×Q₈×C₁₀₉	—	2	2	no	—
8	#424×C₁₀₉	C₄	1	2	yes	—
9	#423×C₁₀₉	C₂²	1	4	yes	—
10	#423×C₁₀₉	C₂²	1	1	yes	characteristic
11	#405×C₁₀₉	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#335×C₁₀₉	—	2	2	no	—
14	#335×C₁₀₉	—	2	2	no	—
15	#332×C₁₀₉	—	2	4	no	—
16	C₂×M₁₆×C₁₀₉	—	2	4	no	—
17	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₀₉	D₄	1	2	yes	—
21	#125×C₁₀₉	Q₈	1	2	yes	—
22	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
23	C₄×Q₈×C₁₀₉	—	2	2	no	—
24	#117×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₁₀₉	—	2	4	no	—
27	C₄⋊C₈×C₁₀₉	—	2	2	no	—
28	C₄⋊C₈×C₁₀₉	—	2	4	no	—
29	C₄⋊C₈×C₁₀₉	—	2	4	no	—
30	#97×C₁₀₉	—	2	2	no	—
31	C₄×C₈×C₁₀₉	—	2	2	no	—
32	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
33	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
34	C₂×Q₈×C₁₀₉	—	2	2	no	—
35	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
37	M₁₆×C₁₀₉	—	4	2	no	—
38	M₁₆×C₁₀₉	—	8	4	no	—
39	C₂×C₈×C₁₀₉	—	2	2	no	—
40	C₂×C₈×C₁₀₉	—	2	2	no	—
41	C₂×C₈×C₁₀₉	—	4	1	no	—
42	C₂×C₈×C₁₀₉	—	4	4	no	—
43	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₁₀₉	—	2	2	no	—
47	C₄⋊C₄×C₁₀₉	—	2	4	no	—
48	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
49	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
50	C₄²×C₁₀₉	—	2	2	no	—
51	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
52	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₁₀₉	#117	1	1	yes	characteristic
55	Q₈×C₁₀₉	—	4	2	no	—
56	Q₈×C₁₀₉	—	4	2	no	—
57	C₂×C₄×C₁₀₉	—	2	1	no	—
58	C₂×C₄×C₁₀₉	—	2	1	no	—
59	C₂×C₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	—	2	4	no	—
62	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₀₉	#121	1	2	yes	—
65	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
66	C₈×C₁₀₉	—	2	1	no	—
67	C₈×C₁₀₉	—	2	2	no	—
68	C₈×C₁₀₉	—	4	2	no	—
69	C₈×C₁₀₉	—	8	4	no	—
70	C₈×C₁₀₉	#101	1	2	yes	—
71	C₂²×C₁₀₉	—	2	1	no	—
72	C₂²×C₁₀₉	—	2	2	no	—
73	C₂²×C₁₀₉	#380	1	1	yes	characteristic
74	C₄×C₁₀₉	—	2	1	no	—
75	C₄×C₁₀₉	—	2	2	no	—
76	C₄×C₁₀₉	—	4	4	no	—
77	C₄×C₁₀₉	C₂×#101	1	1	yes	characteristic
78	C₄×C₁₀₉	#413	1	1	yes	characteristic
79	C₄×C₁₀₉	#472	1	2	yes	—
80	C₄×C₁₀₉	#473	1	2	yes	—
81	#8639	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₀₉	—	4	1	no	—
83	C₂×C₁₀₉	#1801	1	1	yes	characteristic
84	C₂×C₁₀₉	#1816	1	2	yes	—
85	#2960	C₂×C₁₀₉	1	1	yes	characteristic
86	#1567	C₂×C₁₀₉	1	2	yes	—
87	#1561	C₂×C₁₀₉	1	2	yes	—
88	#1309	C₂×C₁₀₉	1	2	yes	—
89	C₁₀₉	#8639	1	1	yes	characteristic, direct product factor, Sylow
90	#533	C₂²×C₁₀₉	1	1	yes	characteristic
91	#446	—	2	2	no	—
92	C₈×Q₈	—	2	2	no	—
93	#424	C₄×C₁₀₉	1	2	yes	—
94	#423	C₂²×C₁₀₉	1	4	yes	—
95	#423	C₂²×C₁₀₉	1	1	yes	characteristic
96	#405	C₄×C₁₀₉	1	2	yes	—
97	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
98	#335	—	2	2	no	—
99	#335	—	2	2	no	—
100	#332	—	2	4	no	—
101	C₂×M₁₆	—	2	4	no	—
102	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
103	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
104	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
105	C₄⋊Q₈	D₄×C₁₀₉	1	2	yes	—
106	#125	Q₈×C₁₀₉	1	2	yes	—
107	C₂²⋊Q₈	—	2	2	no	—
108	C₄×Q₈	—	2	2	no	—
109	#117	C₂×C₄×C₁₀₉	1	2	yes	—
110	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	2	4	no	—
115	#97	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
118	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
119	C₂×Q₈	—	2	2	no	—
120	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
121	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	8	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	4	no	—
128	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
129	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
130	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
137	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
138	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
139	C₂³	#117×C₁₀₉	1	1	yes	characteristic
140	Q₈	—	4	2	no	—
141	Q₈	—	4	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
149	C₂×C₄	#121×C₁₀₉	1	2	yes	—
150	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	4	2	no	—
154	C₈	—	8	4	no	—
155	C₈	#101×C₁₀₉	1	2	yes	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	2	no	—
158	C₂²	#380×C₁₀₉	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	C₂×#101×C₁₀₉	1	1	yes	characteristic
163	C₄	#413×C₁₀₉	1	1	yes	characteristic
164	C₄	#472×C₁₀₉	1	2	yes	—
165	C₄	#473×C₁₀₉	1	2	yes	—
166	C₂	—	4	1	no	—
167	C₂	C₁₀₉×#1801	1	1	yes	characteristic
168	C₂	C₁₀₉×#1816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2960

C₂

yes

maximal, characteristic

C₁₀₉×#1567

C₂

yes

maximal

C₁₀₉×#1561

C₂

yes

maximal

C₁₀₉×#1309

C₂

yes

maximal

#533×C₁₀₉

C₂²

yes

characteristic

#446×C₁₀₉

—

C₈×Q₈×C₁₀₉

—

#424×C₁₀₉

C₄

yes

—

#423×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

C₂²

yes

characteristic

#405×C₁₀₉

C₄

yes

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#335×C₁₀₉

—

#335×C₁₀₉

—

#332×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂×C₄

yes

—

C₂²×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₁₀₉

D₄

yes

—

#125×C₁₀₉

Q₈

yes

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

C₂×C₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

D₄

yes

—

C₄×C₈×C₁₀₉

D₄

yes

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

#41

yes

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₄²×C₁₀₉

C₂×Q₈

yes

characteristic

C₂³×C₁₀₉

#117

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

#121

yes

—

C₂×C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#101

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#380

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

C₄×C₁₀₉

#473

yes

—

#8639

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#1801

yes

characteristic

C₂×C₁₀₉

#1816

yes

—

#2960

C₂×C₁₀₉

yes

characteristic

#1567

C₂×C₁₀₉

yes

—

#1561

C₂×C₁₀₉

yes

—

#1309

C₂×C₁₀₉

yes

—

C₁₀₉

#8639

yes

characteristic, direct product factor, Sylow

#533

C₂²×C₁₀₉

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₄×C₁₀₉

yes

—

#423

C₂²×C₁₀₉

yes

—

#423

C₂²×C₁₀₉

yes

characteristic

#405

C₄×C₁₀₉

yes

—

C₂×C₄×C₈

C₂²×C₁₀₉

yes

characteristic

#335

—

#335

—

100

#332

—

101

C₂×M₁₆

—

102

C₂×M₁₆

C₂×C₄×C₁₀₉

yes

—

103

C₂²×C₈

C₂×C₄×C₁₀₉

yes

characteristic

104

C₂²×C₈

C₂×C₄×C₁₀₉

yes

characteristic

105

C₄⋊Q₈

D₄×C₁₀₉

yes

—

106

#125

Q₈×C₁₀₉

yes

—

107

C₂²⋊Q₈

—

108

C₄×Q₈

—

109

#117

C₂×C₄×C₁₀₉

yes

—

110

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

#97

—

116

C₄×C₈

—

117

C₄×C₈

D₄×C₁₀₉

yes

—

118

C₄×C₈

D₄×C₁₀₉

yes

—

119

C₂×Q₈

—

120

C₂²×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

121

C₂²×C₄

#41×C₁₀₉

yes

—

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

characteristic

129

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

—

130

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₄⋊C₄×C₁₀₉

yes

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

C₂×D₄×C₁₀₉

yes

—

137

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

138

C₄²

C₂×Q₈×C₁₀₉

yes

characteristic

139

C₂³

#117×C₁₀₉

yes

characteristic

140

Q₈

—

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

148

C₂×C₄

C₂×C₄⋊C₄×C₁₀₉

yes

characteristic

149

C₂×C₄

#121×C₁₀₉

yes

—

150

C₂×C₄

C₂²⋊Q₈×C₁₀₉

yes

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

#101×C₁₀₉

yes

—

156

C₂²

—

157

C₂²

—

158

C₂²

#380×C₁₀₉

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

C₂×#101×C₁₀₉

yes

characteristic

163

C₄

#413×C₁₀₉

yes

characteristic

164

C₄

#472×C₁₀₉

yes

—

165

C₄

#473×C₁₀₉

yes

—

166

C₂

—

167

C₂

C₁₀₉×#1801

yes

characteristic

168

C₂

C₁₀₉×#1816

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#380

commutator

C₂²×C₄

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

prime radical

C₂³×C₁₀₉

#117

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
81. #8639	C₁₀₉	direct (N × H)
89. C₁₀₉	#8639	direct (N × H)

H ≅ G/N

type

81. #8639

C₁₀₉

direct (N × H)

89. C₁₀₉

#8639

direct (N × H)

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

subgroup	index
C₁₀₉×#2960	2
C₁₀₉×#1567	2
C₁₀₉×#1561	2
C₁₀₉×#1309	2
#8639	109

subgroup

index

C₁₀₉×#2960

C₁₀₉×#1567

C₁₀₉×#1561

C₁₀₉×#1309

#8639

109

Subgroups — C₁₀₉×#8639 #42126633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#8639 #42126633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8639 #42126633