Group #42126624 — subgroups

Subgroups — C₁₀₉×#8630 #42126624

Order 27904.

388 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1568	C₂	1	2	yes	maximal, complemented
3	C₁₀₉×#1536	C₂	1	2	yes	maximal
4	C₁₀₉×#1307	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₁₀₉	C₄	1	2	yes	—
6	#532×C₁₀₉	C₂²	1	1	yes	characteristic
7	#436×C₁₀₉	—	2	2	no	—
8	C₈×D₄×C₁₀₉	—	2	2	no	—
9	#432×C₁₀₉	C₂²	1	4	yes	—
10	#423×C₁₀₉	C₂²	1	1	yes	characteristic
11	#408×C₁₀₉	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#335×C₁₀₉	—	2	2	no	—
14	#334×C₁₀₉	—	2	2	no	—
15	#329×C₁₀₉	—	2	4	no	—
16	C₂×#41×C₁₀₉	C₂×C₄	1	2	yes	—
17	#131×C₁₀₉	—	4	4	no	—
18	C₂×M₁₆×C₁₀₉	—	2	2	no	—
19	C₂×M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₀₉	—	2	2	no	—
21	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
24	#127×C₁₀₉	Q₈	1	1	yes	characteristic
25	#124×C₁₀₉	D₄	1	2	yes	—
26	#121×C₁₀₉	—	2	2	no	—
27	C₄×D₄×C₁₀₉	—	2	2	no	—
28	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₁₀₉	—	2	2	no	—
30	C₄⋊C₈×C₁₀₉	—	2	4	no	—
31	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
32	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
33	#97×C₁₀₉	—	2	4	no	—
34	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
35	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
36	#41×C₁₀₉	—	4	2	no	—
37	#41×C₁₀₉	—	4	2	no	—
38	C₂×Q₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
39	C₂×D₄×C₁₀₉	—	2	2	no	—
40	C₂×D₄×C₁₀₉	—	2	2	no	—
41	C₂×D₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₀₉	—	2	2	no	—
43	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
45	M₁₆×C₁₀₉	—	4	4	no	—
46	M₁₆×C₁₀₉	—	4	2	no	—
47	C₂×C₈×C₁₀₉	—	2	2	no	—
48	C₂×C₈×C₁₀₉	—	2	2	no	—
49	C₂×C₈×C₁₀₉	—	4	1	no	—
50	C₂×C₈×C₁₀₉	—	4	4	no	—
51	C₂×C₈×C₁₀₉	—	4	4	no	—
52	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₁₀₉	—	2	2	no	—
56	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
57	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
58	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
60	C₂³×C₁₀₉	—	2	2	no	—
61	C₂³×C₁₀₉	#117	1	1	yes	characteristic
62	Q₈×C₁₀₉	—	4	2	no	—
63	D₄×C₁₀₉	—	4	2	no	—
64	D₄×C₁₀₉	—	4	4	no	—
65	D₄×C₁₀₉	—	4	4	no	—
66	C₂×C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₄×C₁₀₉	—	2	1	no	—
68	C₂×C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄×C₁₀₉	—	4	2	no	—
70	C₂×C₄×C₁₀₉	—	4	2	no	—
71	C₂×C₄×C₁₀₉	—	4	2	no	—
72	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₀₉	#121	1	2	yes	—
75	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
76	C₈×C₁₀₉	—	2	1	no	—
77	C₈×C₁₀₉	—	2	2	no	—
78	C₈×C₁₀₉	—	4	2	no	—
79	C₈×C₁₀₉	—	8	4	no	—
80	C₈×C₁₀₉	#100	1	2	yes	—
81	C₂²×C₁₀₉	—	2	1	no	—
82	C₂²×C₁₀₉	—	2	2	no	—
83	C₂²×C₁₀₉	—	4	2	no	—
84	C₂²×C₁₀₉	—	4	4	no	—
85	C₂²×C₁₀₉	#380	1	1	yes	characteristic
86	C₄×C₁₀₉	—	2	1	no	—
87	C₄×C₁₀₉	—	2	2	no	—
88	C₄×C₁₀₉	—	2	2	no	—
89	C₄×C₁₀₉	—	4	2	no	—
90	C₄×C₁₀₉	C₂×#100	1	1	yes	characteristic
91	C₄×C₁₀₉	#413	1	1	yes	characteristic
92	#8630	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×C₁₀₉	—	4	1	no	—
94	C₂×C₁₀₉	—	8	2	no	—
95	C₂×C₁₀₉	#1800	1	1	yes	characteristic
96	C₂×C₁₀₉	#1814	1	2	yes	—
97	#2931	C₂×C₁₀₉	1	1	yes	characteristic
98	#1568	C₂×C₁₀₉	1	2	yes	complemented
99	#1536	C₂×C₁₀₉	1	2	yes	—
100	#1307	C₂×C₁₀₉	1	2	yes	complemented
101	C₁₀₉	#8630	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×#131	C₄×C₁₀₉	1	2	yes	—
103	#532	C₂²×C₁₀₉	1	1	yes	characteristic
104	#436	—	2	2	no	—
105	C₈×D₄	—	2	2	no	—
106	#432	C₂²×C₁₀₉	1	4	yes	—
107	#423	C₂²×C₁₀₉	1	1	yes	characteristic
108	#408	C₄×C₁₀₉	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
110	#335	—	2	2	no	—
111	#334	—	2	2	no	—
112	#329	—	2	4	no	—
113	C₂×#41	C₂×C₄×C₁₀₉	1	2	yes	—
114	#131	—	4	4	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
120	C₄⋊Q₈	Q₈×C₁₀₉	1	1	yes	characteristic
121	#127	Q₈×C₁₀₉	1	1	yes	characteristic
122	#124	D₄×C₁₀₉	1	2	yes	—
123	#121	—	2	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	4	no	—
128	C₂²⋊C₈	—	2	4	no	—
129	C₂²⋊C₈	—	2	4	no	—
130	#97	—	2	4	no	—
131	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
132	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
133	#41	—	4	2	no	—
134	#41	—	4	2	no	—
135	C₂×Q₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
141	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
142	M₁₆	—	4	4	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	—	4	4	no	—
149	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	2	yes	—
150	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	4	no	—
154	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
155	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
156	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
157	C₂³	—	2	2	no	—
158	C₂³	#117×C₁₀₉	1	1	yes	characteristic
159	Q₈	—	4	2	no	—
160	D₄	—	4	2	no	—
161	D₄	—	4	4	no	—
162	D₄	—	4	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
171	C₂×C₄	#121×C₁₀₉	1	2	yes	—
172	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	2	no	—
175	C₈	—	4	2	no	—
176	C₈	—	8	4	no	—
177	C₈	#100×C₁₀₉	1	2	yes	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	2	no	—
181	C₂²	—	4	4	no	—
182	C₂²	#380×C₁₀₉	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂×#100×C₁₀₉	1	1	yes	characteristic
188	C₄	#413×C₁₀₉	1	1	yes	characteristic
189	C₂	—	4	1	no	—
190	C₂	—	8	2	no	—
191	C₂	C₁₀₉×#1800	1	1	yes	characteristic
192	C₂	C₁₀₉×#1814	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2931

C₂

yes

maximal, characteristic

C₁₀₉×#1568

C₂

yes

maximal, complemented

C₁₀₉×#1536

C₂

yes

maximal

C₁₀₉×#1307

C₂

yes

maximal, complemented

C₂×#131×C₁₀₉

C₄

yes

—

#532×C₁₀₉

C₂²

yes

characteristic

#436×C₁₀₉

—

C₈×D₄×C₁₀₉

—

#432×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

C₂²

yes

characteristic

#408×C₁₀₉

C₄

yes

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#335×C₁₀₉

—

#334×C₁₀₉

—

#329×C₁₀₉

—

C₂×#41×C₁₀₉

C₂×C₄

yes

—

#131×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂×C₄

yes

—

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₁₀₉

Q₈

yes

characteristic

#127×C₁₀₉

Q₈

yes

characteristic

#124×C₁₀₉

D₄

yes

—

#121×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

D₄

yes

—

C₄×C₈×C₁₀₉

D₄

yes

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₄⋊C₄

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₄⋊C₄

yes

—

C₂²×C₄×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

—

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₄²×C₁₀₉

C₂×Q₈

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#117

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×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

#100

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#380

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#1800

yes

characteristic

C₂×C₁₀₉

#1814

yes

—

#2931

C₂×C₁₀₉

yes

characteristic

#1568

C₂×C₁₀₉

yes

complemented

#1536

C₂×C₁₀₉

yes

—

100

#1307

C₂×C₁₀₉

yes

complemented

101

C₁₀₉

#8630

yes

characteristic, direct product factor, Sylow

102

C₂×#131

C₄×C₁₀₉

yes

—

103

#532

C₂²×C₁₀₉

yes

characteristic

104

#436

—

105

C₈×D₄

—

106

#432

C₂²×C₁₀₉

yes

—

107

#423

C₂²×C₁₀₉

yes

characteristic

108

#408

C₄×C₁₀₉

yes

—

109

C₂×C₄×C₈

C₂²×C₁₀₉

yes

characteristic

110

#335

—

111

#334

—

112

#329

—

113

C₂×#41

C₂×C₄×C₁₀₉

yes

—

114

#131

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₂×C₄×C₁₀₉

yes

—

117

C₂²×C₈

—

118

C₂²×C₈

C₂×C₄×C₁₀₉

yes

characteristic

119

C₂²×C₈

C₂×C₄×C₁₀₉

yes

characteristic

120

C₄⋊Q₈

Q₈×C₁₀₉

yes

characteristic

121

#127

Q₈×C₁₀₉

yes

characteristic

122

#124

D₄×C₁₀₉

yes

—

123

#121

—

124

C₄×D₄

—

125

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₂²⋊C₈

—

129

C₂²⋊C₈

—

130

#97

—

131

C₄×C₈

D₄×C₁₀₉

yes

—

132

C₄×C₈

D₄×C₁₀₉

yes

—

133

#41

—

134

#41

—

135

C₂×Q₈

C₄⋊C₄×C₁₀₉

yes

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

C₄⋊C₄×C₁₀₉

yes

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

141

C₂²×C₄

#41×C₁₀₉

yes

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

—

150

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

characteristic

151

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

characteristic

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

154

C₄²

C₂×D₄×C₁₀₉

yes

—

155

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

156

C₄²

C₂×Q₈×C₁₀₉

yes

characteristic

157

C₂³

—

158

C₂³

#117×C₁₀₉

yes

characteristic

159

Q₈

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

170

C₂×C₄

C₂×C₄⋊C₄×C₁₀₉

yes

characteristic

171

C₂×C₄

#121×C₁₀₉

yes

—

172

C₂×C₄

C₂²⋊Q₈×C₁₀₉

yes

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

#100×C₁₀₉

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

#380×C₁₀₉

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₂×#100×C₁₀₉

yes

characteristic

188

C₄

#413×C₁₀₉

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₁₀₉×#1800

yes

characteristic

192

C₂

C₁₀₉×#1814

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	#532×C₁₀₉	C₂²
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

#532×C₁₀₉

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. C₁₀₉×#1568	C₂	semidirect
4. C₁₀₉×#1307	C₂	semidirect
92. #8630	C₁₀₉	direct (N × H)
98. #1568	C₂×C₁₀₉	semidirect
100. #1307	C₂×C₁₀₉	semidirect
101. C₁₀₉	#8630	direct (N × H)

H ≅ G/N

type

2. C₁₀₉×#1568

C₂

semidirect

4. C₁₀₉×#1307

C₂

semidirect

92. #8630

C₁₀₉

direct (N × H)

98. #1568

C₂×C₁₀₉

semidirect

100. #1307

C₂×C₁₀₉

semidirect

101. C₁₀₉

#8630

direct (N × H)

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

subgroup	index
C₁₀₉×#2931	2
C₁₀₉×#1568	2
C₁₀₉×#1536	2
C₁₀₉×#1307	2
#8630	109

subgroup

index

C₁₀₉×#2931

C₁₀₉×#1568

C₁₀₉×#1536

C₁₀₉×#1307

#8630

109

Subgroups — C₁₀₉×#8630 #42126624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#8630 #42126624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#8630 #42126624