Group #28376399 — subgroups

Subgroups — C₂×#20295637 #28376399

Order 13952.

1004 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 485 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	#20295637	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16812694	C₂	1	4	yes	maximal
3	C₂×#16812691	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16812689	C₂	1	4	yes	maximal, complemented
5	C₂×#16812675	C₂	1	2	yes	maximal
6	C₂×Q₈×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
7	C₂×#16812656	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
10	#16812694	C₂²	1	32	yes	—
11	#16812691	C₂²	1	8	yes	complemented
12	#16812689	C₂²	1	32	yes	complemented
13	#16812675	C₂²	1	16	yes	—
14	Q₈×Dic₁₀₉	C₂²	1	16	yes	complemented
15	#16812656	C₂²	1	8	yes	—
16	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
18	C₂×#13182008	C₂²	1	2	yes	—
19	C₂×#13182008	C₂²	1	4	yes	—
20	C₂×#13182008	C₂²	1	4	yes	—
21	C₂×#13182006	C₂²	1	2	yes	—
22	C₂×#13182006	C₂²	1	4	yes	—
23	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₁₀₉	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
28	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
31	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
32	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×D₁₀₉	—	2	4	no	—
34	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
35	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
36	C₄⋊Dic₁₀₉	C₂³	1	16	yes	—
37	#13182008	C₂³	1	8	yes	—
38	#13182008	C₂³	1	16	yes	—
39	#13182008	C₂³	1	16	yes	—
40	#13182006	C₂³	1	8	yes	—
41	#13182006	C₂³	1	16	yes	—
42	C₄×Dic₁₀₉	C₂³	1	4	yes	—
43	C₄×Dic₁₀₉	C₂³	1	16	yes	—
44	C₂×Q₈×C₁₀₉	—	2	8	no	—
45	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
46	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
48	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
50	C₄²×C₁₀₉	C₂³	1	4	yes	—
51	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
52	C₂²×D₁₀₉	—	2	2	no	—
53	C₂²×D₁₀₉	—	2	4	no	—
54	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
55	Q₈×C₁₀₉	—	2	16	no	—
56	C₂×C₄×C₁₀₉	—	2	8	no	—
57	C₂×C₄×C₁₀₉	—	2	4	no	—
58	C₂×C₄×C₁₀₉	#41	1	4	yes	—
59	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
60	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
61	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
62	C₂×Dic₁₀₉	—	2	2	no	—
63	C₂×Dic₁₀₉	—	2	4	no	—
64	C₂×Dic₁₀₉	—	2	8	no	—
65	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
66	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
67	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
68	C₄×D₁₀₉	—	2	8	no	—
69	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
70	C₂²×C₁₀₉	2₋¹⁺⁴	1	4	yes	—
71	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
72	C₂×D₁₀₉	—	2	2	no	—
73	C₂×D₁₀₉	—	2	4	no	—
74	C₂×D₁₀₉	—	4	4	no	—
75	C₄×C₁₀₉	—	2	8	no	—
76	C₄×C₁₀₉	—	2	4	no	—
77	C₄×C₁₀₉	C₂×#41	1	4	yes	—
78	Dic₁₀₉	—	2	2	no	—
79	Dic₁₀₉	—	2	4	no	—
80	Dic₁₀₉	—	2	8	no	—
81	C₂×C₁₀₉	#541	1	4	yes	complemented
82	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
83	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	2	yes	—
84	D₁₀₉	—	4	2	no	—
85	C₂×#541	—	109	1	no	maximal, Sylow
86	C₁₀₉	C₂×#541	1	1	yes	characteristic, complemented, Sylow
87	#541	—	109	16	no	—
88	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
89	C₂×#126	—	109	4	no	—
90	C₂×#125	—	109	4	no	—
91	C₂×#125	—	109	1	no	—
92	C₂×C₂²⋊Q₈	—	109	2	no	—
93	C₂×C₄×Q₈	—	109	2	no	—
94	C₂×#117	—	109	1	no	—
95	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
96	C₂³×C₄	—	109	1	no	—
97	C₄⋊Q₈	C₂×D₁₀₉	1	8	yes	complemented
98	#126	—	109	32	no	—
99	#125	—	109	32	no	—
100	#125	—	109	8	no	—
101	C₂²⋊Q₈	—	109	16	no	—
102	C₄×Q₈	—	109	16	no	—
103	#117	—	109	8	no	—
104	C₂×C₄⋊C₄	—	109	2	no	—
105	C₂×C₄⋊C₄	—	109	2	no	—
106	C₂×C₄⋊C₄	—	109	4	no	—
107	C₂×C₄⋊C₄	—	109	4	no	—
108	C₂×C₄⋊C₄	—	109	4	no	—
109	C₂×C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
110	C₂×C₂²⋊C₄	—	109	2	no	—
111	C₂×C₂²⋊C₄	—	109	4	no	—
112	C₂×C₄²	—	109	1	no	—
113	C₂×C₄²	—	109	4	no	—
114	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
115	C₂⁴	—	109	1	no	—
116	C₂×Q₈	—	2	8	no	—
117	C₂×Q₈	C₂²×D₁₀₉	1	8	yes	—
118	C₂²×C₄	—	109	2	no	—
119	C₂²×C₄	—	109	4	no	—
120	C₂²×C₄	—	109	1	no	—
121	C₂²×C₄	—	109	4	no	—
122	C₂²×C₄	—	218	4	no	—
123	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
124	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
126	C₄⋊C₄	—	109	8	no	—
127	C₄⋊C₄	—	109	16	no	—
128	C₄⋊C₄	—	109	16	no	—
129	C₄⋊C₄	—	109	8	no	—
130	C₄⋊C₄	—	109	16	no	—
131	C₄⋊C₄	C₂²×D₁₀₉	1	16	yes	—
132	C₂²⋊C₄	—	109	8	no	—
133	C₂²⋊C₄	—	109	16	no	—
134	C₄²	—	109	4	no	—
135	C₄²	—	109	16	no	—
136	C₄²	C₂²×D₁₀₉	1	4	yes	—
137	C₂³	—	109	2	no	—
138	C₂³	—	218	2	no	—
139	C₂³	—	218	4	no	—
140	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
141	Q₈	—	2	16	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	109	4	no	—
145	C₂×C₄	—	109	8	no	—
146	C₂×C₄	—	109	2	no	—
147	C₂×C₄	—	218	2	no	—
148	C₂×C₄	—	218	4	no	—
149	C₂×C₄	—	218	8	no	—
150	C₂×C₄	—	218	8	no	—
151	C₂×C₄	#13182028	1	4	yes	—
152	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
153	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
154	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
155	C₂²	—	436	4	no	—
156	C₂²	—	218	2	no	—
157	C₂²	—	218	4	no	—
158	C₂²	C₂×#13182028	1	2	yes	—
159	C₂²	#16812747	1	4	yes	—
160	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
161	C₄	—	2	8	no	—
162	C₄	—	2	4	no	—
163	C₄	—	218	2	no	—
164	C₄	—	218	4	no	—
165	C₄	—	218	8	no	—
166	C₄	C₂×#13182028	1	4	yes	—
167	C₂	—	436	2	no	—
168	C₂	#20295637	1	4	yes	direct product factor
169	C₂	C₂²×#13182028	1	1	yes	characteristic
170	C₂	C₂×#16812747	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295637

C₂

yes

maximal, direct product factor

C₂×#16812694

C₂

yes

maximal

C₂×#16812691

C₂

yes

maximal, characteristic, complemented

C₂×#16812689

C₂

yes

maximal, complemented

C₂×#16812675

C₂

yes

maximal

C₂×Q₈×Dic₁₀₉

C₂

yes

maximal, complemented

C₂×#16812656

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₀₉

C₂²

yes

characteristic

#16812694

C₂²

yes

—

#16812691

C₂²

yes

complemented

#16812689

C₂²

yes

complemented

#16812675

C₂²

yes

—

Q₈×Dic₁₀₉

C₂²

yes

complemented

#16812656

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

complemented

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂³×D₁₀₉

C₂³

yes

characteristic

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

C₂³

yes

—

#13182008

C₂³

yes

—

#13182008

C₂³

yes

—

#13182006

C₂³

yes

—

#13182006

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂⁴

yes

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₄×D₁₀₉

—

C₂²×C₁₀₉

C₂×#41

yes

—

C₂²×C₁₀₉

2₋¹⁺⁴

yes

—

C₂²×C₁₀₉

C₂⁵

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×#41

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#541

yes

complemented

C₂×C₁₀₉

C₂²×#41

yes

characteristic

C₂×C₁₀₉

C₂×2₋¹⁺⁴

yes

—

D₁₀₉

—

C₂×#541

—

109

maximal, Sylow

C₁₀₉

C₂×#541

yes

characteristic, complemented, Sylow

#541

—

109

—

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic, complemented

C₂×#126

—

109

—

C₂×#125

—

109

—

C₂×#125

—

109

—

C₂×C₂²⋊Q₈

—

109

—

C₂×C₄×Q₈

—

109

—

C₂×#117

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₂³×C₄

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

complemented

#126

—

109

—

#125

—

109

—

100

#125

—

109

—

101

C₂²⋊Q₈

—

109

—

102

C₄×Q₈

—

109

—

103

#117

—

109

—

104

C₂×C₄⋊C₄

—

109

—

105

C₂×C₄⋊C₄

—

109

—

106

C₂×C₄⋊C₄

—

109

—

107

C₂×C₄⋊C₄

—

109

—

108

C₂×C₄⋊C₄

—

109

—

109

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

110

C₂×C₂²⋊C₄

—

109

—

111

C₂×C₂²⋊C₄

—

109

—

112

C₂×C₄²

—

109

—

113

C₂×C₄²

—

109

—

114

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

115

C₂⁴

—

109

—

116

C₂×Q₈

—

117

C₂×Q₈

C₂²×D₁₀₉

yes

—

118

C₂²×C₄

—

109

—

119

C₂²×C₄

—

109

—

120

C₂²×C₄

—

109

—

121

C₂²×C₄

—

109

—

122

C₂²×C₄

—

218

—

123

C₂²×C₄

C₂²×D₁₀₉

yes

—

124

C₂²×C₄

C₂²×D₁₀₉

yes

—

125

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

126

C₄⋊C₄

—

109

—

127

C₄⋊C₄

—

109

—

128

C₄⋊C₄

—

109

—

129

C₄⋊C₄

—

109

—

130

C₄⋊C₄

—

109

—

131

C₄⋊C₄

C₂²×D₁₀₉

yes

—

132

C₂²⋊C₄

—

109

—

133

C₂²⋊C₄

—

109

—

134

C₄²

—

109

—

135

C₄²

—

109

—

136

C₄²

C₂²×D₁₀₉

yes

—

137

C₂³

—

109

—

138

C₂³

—

218

—

139

C₂³

—

218

—

140

C₂³

C₂³×D₁₀₉

yes

characteristic

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

109

—

145

C₂×C₄

—

109

—

146

C₂×C₄

—

109

—

147

C₂×C₄

—

218

—

148

C₂×C₄

—

218

—

149

C₂×C₄

—

218

—

150

C₂×C₄

—

218

—

151

C₂×C₄

#13182028

yes

—

152

C₂×C₄

C₂³×D₁₀₉

yes

—

153

C₂×C₄

C₂³×D₁₀₉

yes

—

154

C₂×C₄

C₂³×D₁₀₉

yes

—

155

C₂²

—

436

—

156

C₂²

—

218

—

157

C₂²

—

218

—

158

C₂²

C₂×#13182028

yes

—

159

C₂²

#16812747

yes

—

160

C₂²

C₂⁴×D₁₀₉

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

218

—

164

C₄

—

218

—

165

C₄

—

218

—

166

C₄

C₂×#13182028

yes

—

167

C₂

—

436

—

168

C₂

#20295637

yes

direct product factor

169

C₂

C₂²×#13182028

yes

characteristic

170

C₂

C₂×#16812747

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂³×D₁₀₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#541
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₁₀₉

commutator, cosocle

C₂²×C₁₀₉

C₂⁵

Frattini

C₂²

C₂⁴×D₁₀₉

Fitting

C₂×C₄⋊Q₈×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂⁴

prime radical

C₂³×D₁₀₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #20295637	C₂	direct (N × H)
3. C₂×#16812691	C₂	semidirect
4. C₂×#16812689	C₂	semidirect
6. C₂×Q₈×Dic₁₀₉	C₂	semidirect
8. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
11. #16812691	C₂²	semidirect
12. #16812689	C₂²	semidirect
14. Q₈×Dic₁₀₉	C₂²	semidirect
26. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
81. C₂×C₁₀₉	#541	semidirect
86. C₁₀₉	C₂×#541	semidirect
88. C₂×C₄⋊Q₈	D₁₀₉	semidirect
97. C₄⋊Q₈	C₂×D₁₀₉	semidirect
168. C₂	#20295637	direct (N × H)

H ≅ G/N

type

1. #20295637

C₂

direct (N × H)

3. C₂×#16812691

C₂

semidirect

4. C₂×#16812689

C₂

semidirect

6. C₂×Q₈×Dic₁₀₉

C₂

semidirect

8. C₂×C₄⋊Q₈×C₁₀₉

C₂

semidirect

11. #16812691

C₂²

semidirect

12. #16812689

C₂²

semidirect

14. Q₈×Dic₁₀₉

C₂²

semidirect

26. C₄⋊Q₈×C₁₀₉

C₂²

semidirect

81. C₂×C₁₀₉

#541

semidirect

86. C₁₀₉

C₂×#541

semidirect

88. C₂×C₄⋊Q₈

D₁₀₉

semidirect

97. C₄⋊Q₈

C₂×D₁₀₉

semidirect

168. C₂

#20295637

direct (N × H)

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

subgroup	index
#20295637	2
C₂×#16812694	2
C₂×#16812691	2
C₂×#16812689	2
C₂×#16812675	2
C₂×Q₈×Dic₁₀₉	2
C₂×#16812656	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#541	109

subgroup

index

C₂×Q₈×Dic₁₀₉

C₂×C₄⋊Q₈×C₁₀₉

109

Subgroups — C₂×#20295637 #28376399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295637 #28376399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295637 #28376399