Group #28360313 — subgroups

Subgroups — C₁₀₉×#3059 #28360313

Order 13952.

460 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 200 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×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#536×C₁₀₉	C₂	1	2	yes	maximal
4	#470×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	#467×C₁₀₉	C₂	1	4	yes	maximal, complemented
6	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#419×C₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₁₀₉	—	2	4	no	—
11	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
12	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
13	#131×C₁₀₉	—	2	4	no	—
14	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
17	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
19	#124×C₁₀₉	—	2	2	no	—
20	#123×C₁₀₉	—	2	4	no	—
21	C₂²⋊Q₈×C₁₀₉	C₂²	1	8	yes	—
22	#117×C₁₀₉	C₂²	1	2	yes	—
23	#107×C₁₀₉	C₂²	1	2	yes	—
24	#106×C₁₀₉	C₂²	1	2	yes	—
25	#103×C₁₀₉	C₂²	1	8	yes	—
26	#41×C₁₀₉	—	2	2	no	—
27	#41×C₁₀₉	—	2	2	no	—
28	C₂×Q₈×C₁₀₉	—	2	2	no	—
29	C₂×Q₈×C₁₀₉	—	2	4	no	—
30	C₂×Q₈×C₁₀₉	—	2	4	no	—
31	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
33	C₂×D₄×C₁₀₉	D₄	1	2	yes	—
34	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₉	—	2	2	no	—
36	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	Q₁₆×C₁₀₉	—	2	4	no	—
39	Q₁₆×C₁₀₉	—	2	4	no	—
40	M₁₆×C₁₀₉	—	2	2	no	—
41	M₁₆×C₁₀₉	—	2	4	no	—
42	C₂×C₈×C₁₀₉	—	2	4	no	—
43	C₂×C₈×C₁₀₉	D₄	1	4	yes	—
44	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
45	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₀₉	—	2	8	no	—
48	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
49	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
50	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
51	C₄²×C₁₀₉	—	2	2	no	—
52	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
53	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₁₀₉	—	2	2	no	—
55	Q₈×C₁₀₉	—	2	4	no	—
56	Q₈×C₁₀₉	—	2	4	no	—
57	Q₈×C₁₀₉	—	4	4	no	—
58	D₄×C₁₀₉	—	2	4	no	—
59	D₄×C₁₀₉	—	2	2	no	—
60	C₂×C₄×C₁₀₉	—	2	2	no	—
61	C₂×C₄×C₁₀₉	—	2	2	no	—
62	C₂×C₄×C₁₀₉	—	2	4	no	—
63	C₂×C₄×C₁₀₉	—	2	4	no	—
64	C₂×C₄×C₁₀₉	—	4	4	no	—
65	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₁₀₉	#41	1	2	yes	—
69	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
70	C₈×C₁₀₉	—	2	2	no	—
71	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
72	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
73	C₂²×C₁₀₉	—	2	2	no	—
74	C₂²×C₁₀₉	—	2	2	no	—
75	C₂²×C₁₀₉	—	2	4	no	—
76	C₂²×C₁₀₉	#121	1	2	yes	—
77	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₁₀₉	—	2	2	no	—
79	C₄×C₁₀₉	—	4	4	no	—
80	C₄×C₁₀₉	—	4	4	no	—
81	C₄×C₁₀₉	#121	1	2	yes	—
82	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
84	C₂×C₁₀₉	—	2	2	no	—
85	C₂×C₁₀₉	—	4	2	no	—
86	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
87	C₂×C₁₀₉	#578	1	2	yes	—
88	#3059	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₁₀₉	#3059	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
91	C₂×#131	C₂×C₁₀₉	1	1	yes	characteristic
92	#536	C₂×C₁₀₉	1	2	yes	—
93	#470	C₂×C₁₀₉	1	4	yes	complemented
94	#467	C₂×C₁₀₉	1	4	yes	complemented
95	#427	C₂×C₁₀₉	1	1	yes	characteristic, complemented
96	#419	C₂×C₁₀₉	1	2	yes	complemented
97	C₂×#41	C₂²×C₁₀₉	1	1	yes	characteristic
98	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
99	C₂×Q₁₆	—	2	4	no	—
100	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
101	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
102	#131	—	2	4	no	—
103	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
104	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
105	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	—
106	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
107	C₄⋊Q₈	—	2	2	no	—
108	#124	—	2	2	no	—
109	#123	—	2	4	no	—
110	C₂²⋊Q₈	C₂²×C₁₀₉	1	8	yes	—
111	#117	C₂²×C₁₀₉	1	2	yes	—
112	#107	C₂²×C₁₀₉	1	2	yes	—
113	#106	C₂²×C₁₀₉	1	2	yes	—
114	#103	C₂²×C₁₀₉	1	8	yes	—
115	#41	—	2	2	no	—
116	#41	—	2	2	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
121	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
122	C₂×D₄	D₄×C₁₀₉	1	2	yes	—
123	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
126	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
127	Q₁₆	—	2	4	no	—
128	Q₁₆	—	2	4	no	—
129	M₁₆	—	2	2	no	—
130	M₁₆	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	D₄×C₁₀₉	1	4	yes	—
133	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
134	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
135	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	2	8	no	—
140	C₄²	—	2	2	no	—
141	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
142	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
143	Q₈	—	2	2	no	—
144	Q₈	—	2	4	no	—
145	Q₈	—	2	4	no	—
146	Q₈	—	4	4	no	—
147	D₄	—	2	4	no	—
148	D₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
155	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
157	C₂×C₄	#41×C₁₀₉	1	2	yes	—
158	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	C₂×D₄×C₁₀₉	1	2	yes	—
161	C₈	C₂×D₄×C₁₀₉	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	#121×C₁₀₉	1	2	yes	—
166	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	4	4	no	—
169	C₄	—	4	4	no	—
170	C₄	#121×C₁₀₉	1	2	yes	—
171	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
172	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	4	2	no	—
175	C₂	C₂×#121×C₁₀₉	1	1	yes	characteristic
176	C₂	#578×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#131×C₁₀₉

C₂

yes

maximal, characteristic

#536×C₁₀₉

C₂

yes

maximal

#470×C₁₀₉

C₂

yes

maximal, complemented

#467×C₁₀₉

C₂

yes

maximal, complemented

#427×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#419×C₁₀₉

C₂

yes

maximal, complemented

C₂×#41×C₁₀₉

C₂²

yes

characteristic

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

C₂²

yes

—

C₂×Q₁₆×C₁₀₉

C₂²

yes

—

#131×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

—

#124×C₁₀₉

—

#123×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

C₂²

yes

—

#117×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

C₂²

yes

—

#103×C₁₀₉

C₂²

yes

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂×D₄×C₁₀₉

D₄

yes

—

C₂×D₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₄

yes

—

C₈×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#121

yes

characteristic

C₂×C₁₀₉

#578

yes

—

#3059

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#3059

yes

characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂×C₁₀₉

yes

characteristic, complemented

C₂×#131

C₂×C₁₀₉

yes

characteristic

#536

C₂×C₁₀₉

yes

—

#470

C₂×C₁₀₉

yes

complemented

#467

C₂×C₁₀₉

yes

complemented

#427

C₂×C₁₀₉

yes

characteristic, complemented

#419

C₂×C₁₀₉

yes

complemented

C₂×#41

C₂²×C₁₀₉

yes

characteristic

C₂²×Q₈

C₂²×C₁₀₉

yes

—

C₂×Q₁₆

—

100

C₂×Q₁₆

C₂²×C₁₀₉

yes

—

101

C₂×Q₁₆

C₂²×C₁₀₉

yes

—

102

#131

—

103

C₂×M₁₆

C₂²×C₁₀₉

yes

characteristic

104

C₂×M₁₆

C₂²×C₁₀₉

yes

—

105

C₂²×C₈

C₂²×C₁₀₉

yes

—

106

C₂²×C₈

C₂²×C₁₀₉

yes

characteristic

107

C₄⋊Q₈

—

108

#124

—

109

#123

—

110

C₂²⋊Q₈

C₂²×C₁₀₉

yes

—

111

#117

C₂²×C₁₀₉

yes

—

112

#107

C₂²×C₁₀₉

yes

—

113

#106

C₂²×C₁₀₉

yes

—

114

#103

C₂²×C₁₀₉

yes

—

115

#41

—

116

#41

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

D₄×C₁₀₉

yes

characteristic

121

C₂×Q₈

C₂³×C₁₀₉

yes

—

122

C₂×D₄

D₄×C₁₀₉

yes

—

123

C₂×D₄

D₄×C₁₀₉

yes

characteristic

124

C₂²×C₄

—

125

C₂²×C₄

C₂³×C₁₀₉

yes

—

126

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

127

Q₁₆

—

128

Q₁₆

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

D₄×C₁₀₉

yes

—

133

C₂×C₈

C₂³×C₁₀₉

yes

—

134

C₂×C₈

C₂³×C₁₀₉

yes

characteristic

135

C₂×C₈

C₂³×C₁₀₉

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

C₂³×C₁₀₉

yes

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

C₂×D₄×C₁₀₉

yes

—

142

C₂³

C₂×D₄×C₁₀₉

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

155

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

156

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

157

C₂×C₄

#41×C₁₀₉

yes

—

158

C₂×C₄

C₂⁴×C₁₀₉

yes

characteristic

159

C₈

—

160

C₈

C₂×D₄×C₁₀₉

yes

—

161

C₈

C₂×D₄×C₁₀₉

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#121×C₁₀₉

yes

—

166

C₂²

C₂²×D₄×C₁₀₉

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

#121×C₁₀₉

yes

—

171

C₄

C₂²×D₄×C₁₀₉

yes

characteristic

172

C₄

C₂×#41×C₁₀₉

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

C₂×#121×C₁₀₉

yes

characteristic

176

C₂

#578×C₁₀₉

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂×#41×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×#41×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₀₉

prime radical

C₂×#41×C₁₀₉

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×#41×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. C₂²×Q₁₆×C₁₀₉	C₂	semidirect
4. #470×C₁₀₉	C₂	semidirect
5. #467×C₁₀₉	C₂	semidirect
6. #427×C₁₀₉	C₂	semidirect
7. #419×C₁₀₉	C₂	semidirect
88. #3059	C₁₀₉	direct (N × H)
89. C₁₀₉	#3059	direct (N × H)
90. C₂²×Q₁₆	C₂×C₁₀₉	semidirect
93. #470	C₂×C₁₀₉	semidirect
94. #467	C₂×C₁₀₉	semidirect
95. #427	C₂×C₁₀₉	semidirect
96. #419	C₂×C₁₀₉	semidirect

H ≅ G/N

type

1. C₂²×Q₁₆×C₁₀₉

C₂

semidirect

4. #470×C₁₀₉

C₂

semidirect

5. #467×C₁₀₉

C₂

semidirect

6. #427×C₁₀₉

C₂

semidirect

7. #419×C₁₀₉

C₂

semidirect

88. #3059

C₁₀₉

direct (N × H)

89. C₁₀₉

#3059

direct (N × H)

90. C₂²×Q₁₆

C₂×C₁₀₉

semidirect

93. #470

C₂×C₁₀₉

semidirect

94. #467

C₂×C₁₀₉

semidirect

95. #427

C₂×C₁₀₉

semidirect

96. #419

C₂×C₁₀₉

semidirect

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

subgroup	index
C₂²×Q₁₆×C₁₀₉	2
C₂×#131×C₁₀₉	2
#536×C₁₀₉	2
#470×C₁₀₉	2
#467×C₁₀₉	2
#427×C₁₀₉	2
#419×C₁₀₉	2
#3059	109

subgroup

index

C₂²×Q₁₆×C₁₀₉

C₂×#131×C₁₀₉

#536×C₁₀₉

#470×C₁₀₉

#467×C₁₀₉

#427×C₁₀₉

#419×C₁₀₉

#3059

109

Subgroups — C₁₀₉×#3059 #28360313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#3059 #28360313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3059 #28360313