Group #42130486 — subgroups

Subgroups — C₁₀₉×#12492 #42130486

Order 27904.

268 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 82 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×#2207	C₂	1	2	yes	maximal, complemented
2	C₁₀₉×#1712	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1591	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×SD₃₂×C₁₀₉	—	2	2	no	—
7	#493×C₁₀₉	—	2	1	no	—
8	#493×C₁₀₉	—	2	1	no	—
9	#493×C₁₀₉	C₂²	1	2	yes	—
10	#446×C₁₀₉	—	2	1	no	—
11	C₄×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₁₀₉	C₂²	1	2	yes	—
13	#357×C₁₀₉	—	2	2	no	—
14	#335×C₁₀₉	C₂²	1	1	yes	characteristic
15	Q₈⋊C₈×C₁₀₉	—	2	1	no	—
16	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂×D₈×C₁₀₉	—	2	2	no	—
19	C₂×D₈×C₁₀₉	—	2	2	no	—
20	C₂×D₈×C₁₀₉	—	4	2	no	—
21	C₂×D₈×C₁₀₉	—	4	1	no	—
22	C₂×D₈×C₁₀₉	—	4	1	no	—
23	#127×C₁₀₉	—	2	2	no	—
24	C₄×Q₈×C₁₀₉	—	2	1	no	—
25	SD₃₂×C₁₀₉	—	4	4	no	—
26	C₂×C₁₆×C₁₀₉	—	2	2	no	—
27	#107×C₁₀₉	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₁₀₉	—	2	1	no	—
29	C₄⋊C₈×C₁₀₉	—	4	1	no	—
30	#103×C₁₀₉	—	2	1	no	—
31	#97×C₁₀₉	—	4	1	no	—
32	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂×Q₈×C₁₀₉	—	2	1	no	—
36	C₂×D₄×C₁₀₉	—	4	2	no	—
37	C₂×D₄×C₁₀₉	—	4	2	no	—
38	C₂×D₄×C₁₀₉	—	4	2	no	—
39	Q₁₆×C₁₀₉	—	2	2	no	—
40	D₈×C₁₀₉	—	4	2	no	—
41	D₈×C₁₀₉	—	4	2	no	—
42	D₈×C₁₀₉	—	8	2	no	—
43	D₈×C₁₀₉	—	8	4	no	—
44	D₈×C₁₀₉	—	8	2	no	—
45	D₈×C₁₀₉	—	8	2	no	—
46	C₂×C₈×C₁₀₉	—	2	1	no	—
47	C₂×C₈×C₁₀₉	—	4	1	no	—
48	C₂×C₈×C₁₀₉	D₈	1	1	yes	characteristic
49	C₂×C₈×C₁₀₉	D₈	1	1	yes	characteristic
50	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
52	C₄⋊C₄×C₁₀₉	—	2	1	no	—
53	C₄⋊C₄×C₁₀₉	—	4	1	no	—
54	C₄²×C₁₀₉	—	4	1	no	—
55	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₁₆×C₁₀₉	—	4	2	no	—
57	C₂³×C₁₀₉	—	8	2	no	—
58	Q₈×C₁₀₉	—	4	1	no	—
59	Q₈×C₁₀₉	—	4	1	no	—
60	D₄×C₁₀₉	—	8	4	no	—
61	D₄×C₁₀₉	—	8	2	no	—
62	D₄×C₁₀₉	—	8	2	no	—
63	D₄×C₁₀₉	—	8	2	no	—
64	D₄×C₁₀₉	—	8	2	no	—
65	C₂×C₄×C₁₀₉	—	4	1	no	—
66	C₂×C₄×C₁₀₉	—	4	1	no	—
67	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₈×C₁₀₉	—	2	2	no	—
71	C₈×C₁₀₉	—	2	1	no	—
72	C₈×C₁₀₉	—	2	1	no	—
73	C₈×C₁₀₉	—	8	1	no	—
74	C₈×C₁₀₉	SD₃₂	1	2	yes	complemented
75	C₈×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
76	C₂²×C₁₀₉	—	16	2	no	—
77	C₂²×C₁₀₉	—	16	2	no	—
78	C₂²×C₁₀₉	—	16	2	no	—
79	C₂²×C₁₀₉	#459	1	1	yes	characteristic
80	C₄×C₁₀₉	—	2	1	no	—
81	C₄×C₁₀₉	—	8	1	no	—
82	C₄×C₁₀₉	—	8	1	no	—
83	C₄×C₁₀₉	#469	1	1	yes	characteristic
84	C₄×C₁₀₉	#472	1	1	yes	characteristic
85	C₄×C₁₀₉	C₂×SD₃₂	1	1	yes	characteristic
86	C₄×C₁₀₉	#509	1	1	yes	characteristic
87	#12492	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₁₀₉	—	32	2	no	—
89	C₂×C₁₀₉	#1687	1	1	yes	characteristic
90	C₂×C₁₀₉	#2207	1	1	yes	characteristic
91	C₂×C₁₀₉	#2213	1	1	yes	characteristic
92	#2207	C₂×C₁₀₉	1	2	yes	complemented
93	#1712	C₂×C₁₀₉	1	1	yes	characteristic
94	#1591	C₂×C₁₀₉	1	1	yes	characteristic, complemented
95	#1367	C₂×C₁₀₉	1	1	yes	characteristic, complemented
96	#1361	C₂×C₁₀₉	1	2	yes	complemented
97	C₁₀₉	#12492	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×SD₃₂	—	2	2	no	—
99	#493	—	2	1	no	—
100	#493	—	2	1	no	—
101	#493	C₂²×C₁₀₉	1	2	yes	—
102	#446	—	2	1	no	—
103	C₄×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
104	C₄⋊C₁₆	C₂²×C₁₀₉	1	2	yes	—
105	#357	—	2	2	no	—
106	#335	C₂²×C₁₀₉	1	1	yes	characteristic
107	Q₈⋊C₈	—	2	1	no	—
108	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
109	C₂×Q₁₆	D₄×C₁₀₉	1	1	yes	characteristic
110	C₂×D₈	—	2	2	no	—
111	C₂×D₈	—	2	2	no	—
112	C₂×D₈	—	4	2	no	—
113	C₂×D₈	—	4	1	no	—
114	C₂×D₈	—	4	1	no	—
115	#127	—	2	2	no	—
116	C₄×Q₈	—	2	1	no	—
117	SD₃₂	—	4	4	no	—
118	C₂×C₁₆	—	2	2	no	—
119	#107	D₄×C₁₀₉	1	1	yes	characteristic
120	C₄⋊C₈	—	2	1	no	—
121	C₄⋊C₈	—	4	1	no	—
122	#103	—	2	1	no	—
123	#97	—	4	1	no	—
124	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
125	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
126	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
127	C₂×Q₈	—	2	1	no	—
128	C₂×D₄	—	4	2	no	—
129	C₂×D₄	—	4	2	no	—
130	C₂×D₄	—	4	2	no	—
131	Q₁₆	—	2	2	no	—
132	D₈	—	4	2	no	—
133	D₈	—	4	2	no	—
134	D₈	—	8	2	no	—
135	D₈	—	8	4	no	—
136	D₈	—	8	2	no	—
137	D₈	—	8	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	D₈×C₁₀₉	1	1	yes	characteristic
141	C₂×C₈	D₈×C₁₀₉	1	1	yes	characteristic
142	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
143	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄²	—	4	1	no	—
147	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
148	C₁₆	—	4	2	no	—
149	C₂³	—	8	2	no	—
150	Q₈	—	4	1	no	—
151	Q₈	—	4	1	no	—
152	D₄	—	8	4	no	—
153	D₄	—	8	2	no	—
154	D₄	—	8	2	no	—
155	D₄	—	8	2	no	—
156	D₄	—	8	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
160	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
161	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	1	no	—
166	C₈	SD₃₂×C₁₀₉	1	2	yes	complemented
167	C₈	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
168	C₂²	—	16	2	no	—
169	C₂²	—	16	2	no	—
170	C₂²	—	16	2	no	—
171	C₂²	#459×C₁₀₉	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	8	1	no	—
174	C₄	—	8	1	no	—
175	C₄	#469×C₁₀₉	1	1	yes	characteristic
176	C₄	#472×C₁₀₉	1	1	yes	characteristic
177	C₄	C₂×SD₃₂×C₁₀₉	1	1	yes	characteristic
178	C₄	#509×C₁₀₉	1	1	yes	characteristic
179	C₂	—	32	2	no	—
180	C₂	C₁₀₉×#1687	1	1	yes	characteristic
181	C₂	C₁₀₉×#2207	1	1	yes	characteristic
182	C₂	C₁₀₉×#2213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2207

C₂

yes

maximal, complemented

C₁₀₉×#1712

C₂

yes

maximal, characteristic

C₁₀₉×#1591

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1367

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1361

C₂

yes

maximal, complemented

C₂×SD₃₂×C₁₀₉

—

#493×C₁₀₉

—

#493×C₁₀₉

—

#493×C₁₀₉

C₂²

yes

—

#446×C₁₀₉

—

C₄×Q₁₆×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₀₉

C₂²

yes

—

#357×C₁₀₉

—

#335×C₁₀₉

C₂²

yes

characteristic

Q₈⋊C₈×C₁₀₉

—

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

D₄

yes

characteristic

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

#127×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

SD₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

#107×C₁₀₉

D₄

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

#103×C₁₀₉

—

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

D₄

yes

characteristic

C₄×C₈×C₁₀₉

D₄

yes

characteristic

C₄×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

Q₁₆×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₈

yes

characteristic

C₂×C₈×C₁₀₉

D₈

yes

characteristic

C₂×C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₉

#41

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#121

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

SD₃₂

yes

complemented

C₈×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#459

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

C₂×SD₃₂

yes

C₄×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

complemented

#1712

C₂×C₁₀₉

yes

characteristic

#1591

C₂×C₁₀₉

yes

characteristic, complemented

#1367

C₂×C₁₀₉

yes

characteristic, complemented

#1361

C₂×C₁₀₉

yes

complemented

C₁₀₉

#12492

yes

characteristic, direct product factor, Sylow

C₂×SD₃₂

—

#493

—

100

#493

—

101

#493

C₂²×C₁₀₉

yes

—

102

#446

—

103

C₄×Q₁₆

C₂²×C₁₀₉

yes

characteristic

104

C₄⋊C₁₆

C₂²×C₁₀₉

yes

—

105

#357

—

106

#335

C₂²×C₁₀₉

yes

characteristic

107

Q₈⋊C₈

—

108

C₈²

C₂²×C₁₀₉

yes

characteristic

109

C₂×Q₁₆

D₄×C₁₀₉

yes

characteristic

110

C₂×D₈

—

111

C₂×D₈

—

112

C₂×D₈

—

113

C₂×D₈

—

114

C₂×D₈

—

115

#127

—

116

C₄×Q₈

—

117

SD₃₂

—

118

C₂×C₁₆

—

119

#107

D₄×C₁₀₉

yes

characteristic

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

#103

—

123

#97

—

124

C₄×C₈

D₄×C₁₀₉

yes

characteristic

125

C₄×C₈

D₄×C₁₀₉

yes

characteristic

126

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

127

C₂×Q₈

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

Q₁₆

—

132

D₈

—

133

D₈

—

134

D₈

—

135

D₈

—

136

D₈

—

137

D₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

D₈×C₁₀₉

yes

characteristic

141

C₂×C₈

D₈×C₁₀₉

yes

characteristic

142

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

143

C₂×C₈

#41×C₁₀₉

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄²

—

147

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

148

C₁₆

—

149

C₂³

—

150

Q₈

—

151

Q₈

—

152

D₄

—

153

D₄

—

154

D₄

—

155

D₄

—

156

D₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

#121×C₁₀₉

yes

characteristic

160

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

161

C₂×C₄

AGL(1,ℤ/8)×C₁₀₉

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

SD₃₂×C₁₀₉

yes

complemented

167

C₈

AGL(1,ℤ/8)×C₁₀₉

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#459×C₁₀₉

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#469×C₁₀₉

yes

characteristic

176

C₄

#472×C₁₀₉

yes

characteristic

177

C₄

C₂×SD₃₂×C₁₀₉

yes

characteristic

178

C₄

#509×C₁₀₉

yes

characteristic

179

C₂

—

180

C₂

C₁₀₉×#1687

yes

characteristic

181

C₂

C₁₀₉×#2207

yes

characteristic

182

C₂

C₁₀₉×#2213

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
prime radical	C₁₀₉×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#459

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₀₉

prime radical

C₁₀₉×#1712

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. C₁₀₉×#2207	C₂	semidirect
3. C₁₀₉×#1591	C₂	semidirect
4. C₁₀₉×#1367	C₂	semidirect
5. C₁₀₉×#1361	C₂	semidirect
74. C₈×C₁₀₉	SD₃₂	semidirect
87. #12492	C₁₀₉	direct (N × H)
92. #2207	C₂×C₁₀₉	semidirect
94. #1591	C₂×C₁₀₉	semidirect
95. #1367	C₂×C₁₀₉	semidirect
96. #1361	C₂×C₁₀₉	semidirect
97. C₁₀₉	#12492	direct (N × H)
166. C₈	SD₃₂×C₁₀₉	semidirect

H ≅ G/N

type

1. C₁₀₉×#2207

C₂

semidirect

3. C₁₀₉×#1591

C₂

semidirect

4. C₁₀₉×#1367

C₂

semidirect

5. C₁₀₉×#1361

C₂

semidirect

74. C₈×C₁₀₉

SD₃₂

semidirect

87. #12492

C₁₀₉

direct (N × H)

92. #2207

C₂×C₁₀₉

semidirect

94. #1591

C₂×C₁₀₉

semidirect

95. #1367

C₂×C₁₀₉

semidirect

96. #1361

C₂×C₁₀₉

semidirect

97. C₁₀₉

#12492

direct (N × H)

166. C₈

SD₃₂×C₁₀₉

semidirect

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

subgroup	index
C₁₀₉×#2207	2
C₁₀₉×#1712	2
C₁₀₉×#1591	2
C₁₀₉×#1367	2
C₁₀₉×#1361	2
#12492	109

subgroup

index

C₁₀₉×#2207

C₁₀₉×#1712

C₁₀₉×#1591

C₁₀₉×#1367

C₁₀₉×#1361

#12492

109

Subgroups — C₁₀₉×#12492 #42130486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12492 #42130486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12492 #42130486