Group #28359757 — subgroups

Subgroups — C₁₀₉×#2503 #28359757

Order 13952.

512 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 216 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	C₂×C₂²⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#400×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	#391×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#382×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#381×C₁₀₉	C₂	1	4	yes	maximal
7	C₄×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
20	#95×C₁₀₉	C₂²	1	2	yes	—
21	#95×C₁₀₉	C₂²	1	8	yes	—
22	#95×C₁₀₉	C₂²	1	8	yes	—
23	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂×Q₈×C₁₀₉	—	2	2	no	—
25	C₂×Q₈×C₁₀₉	—	2	4	no	—
26	C₂²×C₄×C₁₀₉	—	2	4	no	—
27	C₂²×C₄×C₁₀₉	—	2	4	no	—
28	C₂²×C₄×C₁₀₉	—	2	8	no	—
29	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₉	—	2	4	no	—
37	C₄⋊C₄×C₁₀₉	—	2	4	no	—
38	C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₄⋊C₄×C₁₀₉	—	2	4	no	—
40	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
41	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
42	C₄²×C₁₀₉	—	2	4	no	—
43	C₂³×C₁₀₉	—	2	2	no	—
44	C₂³×C₁₀₉	—	2	2	no	—
45	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂³×C₁₀₉	#41	1	4	yes	—
47	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
48	Q₈×C₁₀₉	—	4	4	no	—
49	C₂×C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄×C₁₀₉	—	2	8	no	—
51	C₂×C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄×C₁₀₉	—	2	2	no	—
53	C₂×C₄×C₁₀₉	—	2	4	no	—
54	C₂×C₄×C₁₀₉	—	2	4	no	—
55	C₂×C₄×C₁₀₉	—	2	4	no	—
56	C₂×C₄×C₁₀₉	—	2	8	no	—
57	C₂×C₄×C₁₀₉	—	2	4	no	—
58	C₂×C₄×C₁₀₉	—	2	4	no	—
59	C₂×C₄×C₁₀₉	—	2	4	no	—
60	C₂×C₄×C₁₀₉	—	4	8	no	—
61	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₀₉	#41	1	4	yes	—
64	C₂²×C₁₀₉	—	2	2	no	—
65	C₂²×C₁₀₉	—	2	2	no	—
66	C₂²×C₁₀₉	—	2	8	no	—
67	C₂²×C₁₀₉	#124	1	4	yes	—
68	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
70	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
71	C₂²×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
72	C₄×C₁₀₉	—	2	2	no	—
73	C₄×C₁₀₉	—	4	2	no	—
74	C₄×C₁₀₉	—	4	4	no	—
75	C₄×C₁₀₉	—	4	2	no	—
76	C₄×C₁₀₉	—	4	4	no	—
77	C₄×C₁₀₉	#123	1	4	yes	—
78	C₂×C₁₀₉	—	2	4	no	—
79	C₂×C₁₀₉	C₂×#123	1	1	yes	characteristic
80	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
81	C₂×C₁₀₉	#536	1	1	yes	characteristic
82	C₂×C₁₀₉	#552	1	4	yes	—
83	#2503	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₁₀₉	#2503	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×C₂²⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic
86	C₂²×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
87	#400	C₂×C₁₀₉	1	4	yes	complemented
88	#391	C₂×C₁₀₉	1	2	yes	complemented
89	#382	C₂×C₁₀₉	1	2	yes	complemented
90	#381	C₂×C₁₀₉	1	4	yes	—
91	C₄×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
92	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
93	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
94	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
95	C₂²⋊Q₈	—	2	4	no	—
96	C₂×C₄⋊C₄	—	2	4	no	—
97	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
101	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
102	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
103	C₂×C₄²	C₂²×C₁₀₉	1	2	yes	—
104	#95	C₂²×C₁₀₉	1	2	yes	—
105	#95	C₂²×C₁₀₉	1	8	yes	—
106	#95	C₂²×C₁₀₉	1	8	yes	—
107	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	8	no	—
113	C₂²×C₄	D₄×C₁₀₉	1	4	yes	—
114	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
115	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
116	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
117	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
118	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
119	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	—	2	4	no	—
125	C₂²⋊C₄	—	2	4	no	—
126	C₄²	—	2	4	no	—
127	C₂³	—	2	2	no	—
128	C₂³	—	2	2	no	—
129	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
130	C₂³	#41×C₁₀₉	1	4	yes	—
131	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	4	8	no	—
145	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
147	C₂×C₄	#41×C₁₀₉	1	4	yes	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	8	no	—
151	C₂²	#124×C₁₀₉	1	4	yes	—
152	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
153	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
154	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
155	C₂²	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	4	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	4	2	no	—
160	C₄	—	4	4	no	—
161	C₄	#123×C₁₀₉	1	4	yes	—
162	C₂	—	2	4	no	—
163	C₂	C₂×#123×C₁₀₉	1	1	yes	characteristic
164	C₂	C₂×#124×C₁₀₉	1	1	yes	characteristic
165	C₂	#536×C₁₀₉	1	1	yes	characteristic
166	C₂	#552×C₁₀₉	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

#400×C₁₀₉

C₂

yes

maximal, complemented

#391×C₁₀₉

C₂

yes

maximal, complemented

#382×C₁₀₉

C₂

yes

maximal, complemented

#381×C₁₀₉

C₂

yes

maximal

C₄×C₂²⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂³×C₄×C₁₀₉

C₂²

yes

—

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

—

C₂⁴×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

—

C₂³×C₁₀₉

#41

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

Q₈×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

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#124

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₂²×C₁₀₉

C₂×#41

yes

—

C₂²×C₁₀₉

2₋¹⁺⁴

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#123

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

#2503

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#2503

yes

characteristic, direct product factor, Sylow

C₂×C₂²⋊Q₈

C₂×C₁₀₉

yes

characteristic

C₂²×C₄⋊C₄

C₂×C₁₀₉

yes

characteristic

#400

C₂×C₁₀₉

yes

complemented

#391

C₂×C₁₀₉

yes

complemented

#382

C₂×C₁₀₉

yes

complemented

#381

C₂×C₁₀₉

yes

—

C₄×C₂²⋊C₄

C₂×C₁₀₉

yes

characteristic

C₂²×Q₈

C₂²×C₁₀₉

yes

characteristic

C₂³×C₄

C₂²×C₁₀₉

yes

—

C₂³×C₄

C₂²×C₁₀₉

yes

characteristic

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

characteristic

100

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

101

C₂×C₂²⋊C₄

C₂²×C₁₀₉

yes

—

102

C₂×C₂²⋊C₄

C₂²×C₁₀₉

yes

—

103

C₂×C₄²

C₂²×C₁₀₉

yes

—

104

#95

C₂²×C₁₀₉

yes

—

105

#95

C₂²×C₁₀₉

yes

—

106

#95

C₂²×C₁₀₉

yes

—

107

C₂⁴

C₂³×C₁₀₉

yes

characteristic

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

D₄×C₁₀₉

yes

—

114

C₂²×C₄

C₂³×C₁₀₉

yes

—

115

C₂²×C₄

C₂³×C₁₀₉

yes

—

116

C₂²×C₄

C₂³×C₁₀₉

yes

—

117

C₂²×C₄

C₂³×C₁₀₉

yes

—

118

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

119

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂×D₄×C₁₀₉

yes

—

130

C₂³

#41×C₁₀₉

yes

—

131

C₂³

C₂⁴×C₁₀₉

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

146

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

147

C₂×C₄

#41×C₁₀₉

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#124×C₁₀₉

yes

—

152

C₂²

C₂²×D₄×C₁₀₉

yes

characteristic

153

C₂²

C₂×#41×C₁₀₉

yes

—

154

C₂²

C₂×#41×C₁₀₉

yes

—

155

C₂²

2₋¹⁺⁴×C₁₀₉

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#123×C₁₀₉

yes

—

162

C₂

—

163

C₂

C₂×#123×C₁₀₉

yes

characteristic

164

C₂

C₂×#124×C₁₀₉

yes

characteristic

165

C₂

#536×C₁₀₉

yes

characteristic

166

C₂

#552×C₁₀₉

yes

—

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

subgroup

G/H

center, socle

C₂³×C₁₀₉

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₁₀₉

prime radical

C₂⁴×C₁₀₉

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ G

N	H ≅ G/N	type
3. #400×C₁₀₉	C₂	semidirect
4. #391×C₁₀₉	C₂	semidirect
5. #382×C₁₀₉	C₂	semidirect
83. #2503	C₁₀₉	direct (N × H)
84. C₁₀₉	#2503	direct (N × H)
87. #400	C₂×C₁₀₉	semidirect
88. #391	C₂×C₁₀₉	semidirect
89. #382	C₂×C₁₀₉	semidirect

H ≅ G/N

type

3. #400×C₁₀₉

C₂

semidirect

4. #391×C₁₀₉

C₂

semidirect

5. #382×C₁₀₉

C₂

semidirect

83. #2503

C₁₀₉

direct (N × H)

84. C₁₀₉

#2503

direct (N × H)

87. #400

C₂×C₁₀₉

semidirect

88. #391

C₂×C₁₀₉

semidirect

89. #382

C₂×C₁₀₉

semidirect

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

subgroup	index
C₂×C₂²⋊Q₈×C₁₀₉	2
C₂²×C₄⋊C₄×C₁₀₉	2
#400×C₁₀₉	2
#391×C₁₀₉	2
#382×C₁₀₉	2
#381×C₁₀₉	2
C₄×C₂²⋊C₄×C₁₀₉	2
#2503	109

subgroup

index

C₂×C₂²⋊Q₈×C₁₀₉

C₂²×C₄⋊C₄×C₁₀₉

#400×C₁₀₉

#391×C₁₀₉

#382×C₁₀₉

#381×C₁₀₉

C₄×C₂²⋊C₄×C₁₀₉

#2503

109

Subgroups — C₁₀₉×#2503 #28359757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#2503 #28359757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2503 #28359757