Group #21455561 — subgroups

Subgroups — C₂×#16840039 #21455561

Order 7808.

480 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 173 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×C₄×D₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#16840039	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14519710	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14519697	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₂₄₄	C₂²	1	1	yes	characteristic
8	C₂²×C₄×D₆₁	—	2	1	no	—
9	C₂²×#92418	C₂²	1	2	yes	—
10	C₄×D₂₄₄	—	2	2	no	—
11	C₄×D₂₄₄	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂×#11758198	—	2	1	no	—
14	#14519710	C₂²	1	8	yes	complemented
15	C₄⋊#92418	C₂²	1	4	yes	complemented
16	#14519697	C₂²	1	8	yes	—
17	C₂²×Q₈×C₆₁	—	2	1	no	—
18	C₄⋊Q₈×C₆₁	—	2	2	no	—
19	C₄⋊Q₈×C₆₁	C₂²	1	4	yes	complemented
20	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
21	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	C₂³×D₆₁	—	2	1	no	—
24	C₂×Q₈×C₆₁	—	2	4	no	—
25	C₂×Q₈×C₆₁	—	4	4	no	—
26	C₂²×C₄×C₆₁	—	2	2	no	—
27	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²×Dic₆₁	—	2	1	no	—
31	C₂×D₂₄₄	—	2	1	no	—
32	C₂×D₂₄₄	—	2	2	no	—
33	C₂×D₂₄₄	Q₈	1	4	yes	complemented
34	C₂×D₂₄₄	C₂³	1	2	yes	—
35	C₂×C₄×D₆₁	—	2	4	no	—
36	C₂×C₄×D₆₁	—	4	4	no	—
37	C₄⋊C₄×C₆₁	—	2	4	no	—
38	C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
40	C₄²×C₆₁	D₄	1	2	yes	—
41	C₄²×C₆₁	C₂³	1	2	yes	—
42	#11758198	—	2	4	no	—
43	C₄⋊Dic₆₁	—	2	1	no	—
44	C₄⋊Dic₆₁	C₂³	1	2	yes	—
45	C₂×#92418	—	2	4	no	—
46	C₂×#92418	C₂³	1	4	yes	—
47	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×D₆₁	—	2	4	no	—
49	C₂²×D₆₁	—	2	2	no	—
50	C₂²×D₆₁	—	4	2	no	—
51	C₂²×D₆₁	—	4	2	no	—
52	Q₈×C₆₁	—	4	8	no	—
53	C₂×C₄×C₆₁	—	2	2	no	—
54	C₂×C₄×C₆₁	—	2	4	no	—
55	C₂×C₄×C₆₁	—	4	4	no	—
56	C₂×C₄×C₆₁	D₈	1	4	yes	—
57	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₆₁	#41	1	2	yes	—
62	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₆₁	—	2	2	no	—
64	C₂×Dic₆₁	—	4	2	no	—
65	D₂₄₄	—	2	4	no	—
66	D₂₄₄	—	2	2	no	—
67	D₂₄₄	C₂×Q₈	1	4	yes	complemented
68	C₄×D₆₁	—	4	8	no	—
69	#92418	—	2	4	no	—
70	C₂²×C₆₁	C₂²⋊Q₈	1	2	yes	—
71	C₂²×C₆₁	C₂×D₈	1	2	yes	—
72	C₂²×C₆₁	#137	1	2	yes	—
73	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
74	C₂×D₆₁	—	2	4	no	—
75	C₂×D₆₁	—	4	2	no	—
76	C₂×D₆₁	—	4	2	no	—
77	C₂×D₆₁	—	4	8	no	—
78	C₄×C₆₁	—	2	2	no	—
79	C₄×C₆₁	—	4	4	no	—
80	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
81	C₄×C₆₁	C₂×D₈	1	4	yes	—
82	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
84	Dic₆₁	—	4	2	no	—
85	C₂×#474	—	61	1	no	maximal, Sylow
86	C₂×C₆₁	#474	1	4	yes	complemented
87	C₂×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
89	C₂×C₆₁	C₂×#137	1	1	yes	characteristic
90	D₆₁	—	4	4	no	—
91	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic, complemented
92	C₂×C₄×D₄	—	61	1	no	—
93	#474	—	61	8	no	—
94	C₂×#107	—	61	2	no	—
95	C₂×C₄⋊C₈	—	61	1	no	—
96	C₂×#102	—	61	2	no	—
97	C₆₁	C₂×#474	1	1	yes	characteristic, complemented, Sylow
98	C₂²×Q₈	—	2	1	no	—
99	C₂²×D₄	—	61	1	no	—
100	C₂³×C₄	—	122	1	no	—
101	C₂²×C₈	—	61	2	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	C₂×D₆₁	1	4	yes	complemented
104	C₄×D₄	—	61	4	no	—
105	C₄×D₄	—	122	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	61	1	no	—
108	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
109	C₂×C₂²⋊C₄	—	122	1	no	—
110	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
111	#107	—	61	8	no	—
112	C₄⋊C₈	—	61	4	no	—
113	#102	—	61	8	no	—
114	C₂⁴	—	122	1	no	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	4	no	—
117	C₂×D₄	—	61	4	no	—
118	C₂×D₄	—	61	2	no	—
119	C₂×D₄	—	122	2	no	—
120	C₂×D₄	—	122	1	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	122	1	no	—
123	C₂²×C₄	—	122	4	no	—
124	C₂²×C₄	—	244	4	no	—
125	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
128	C₂×C₈	—	61	4	no	—
129	C₂×C₈	—	122	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	61	2	no	—
133	C₄⋊C₄	—	122	1	no	—
134	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
135	C₂²⋊C₄	—	122	4	no	—
136	C₄²	C₂²⋊D₆₁	1	2	yes	—
137	C₄²	C₂²×D₆₁	1	2	yes	—
138	C₂³	—	122	2	no	—
139	C₂³	—	122	4	no	—
140	C₂³	—	244	2	no	—
141	C₂³	—	244	2	no	—
142	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
143	Q₈	—	4	8	no	—
144	D₄	—	61	4	no	—
145	D₄	—	122	2	no	—
146	D₄	—	122	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	122	2	no	—
151	C₂×C₄	—	244	2	no	—
152	C₂×C₄	—	244	8	no	—
153	C₂×C₄	#11758199	1	4	yes	—
154	C₂×C₄	Q₈×D₆₁	1	2	yes	—
155	C₂×C₄	#11758226	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
159	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
160	C₈	—	122	4	no	—
161	C₂²	—	122	4	no	—
162	C₂²	—	244	2	no	—
163	C₂²	—	244	8	no	—
164	C₂²	—	244	2	no	—
165	C₂²	#14519753	1	2	yes	—
166	C₂²	C₂×#11758199	1	2	yes	—
167	C₂²	#14519808	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	244	2	no	—
172	C₄	#14519753	1	2	yes	—
173	C₄	C₂×#11758199	1	4	yes	—
174	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
175	C₄	C₂×#11758226	1	1	yes	characteristic
176	C₂	—	244	4	no	—
177	C₂	#16840039	1	4	yes	direct product factor
178	C₂	C₂×#14519753	1	1	yes	characteristic
179	C₂	C₂²×#11758199	1	1	yes	characteristic
180	C₂	C₂×#14519808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₂₄₄

C₂

yes

maximal, characteristic

#16840039

C₂

yes

maximal, direct product factor

C₂×#14519710

C₂

yes

maximal, complemented

C₂×C₄⋊#92418

C₂

yes

maximal, characteristic, complemented

C₂×#14519697

C₂

yes

maximal

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₄₄

C₂²

yes

characteristic

C₂²×C₄×D₆₁

—

C₂²×#92418

C₂²

yes

—

C₄×D₂₄₄

—

C₄×D₂₄₄

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×#11758198

—

#14519710

C₂²

yes

complemented

C₄⋊#92418

C₂²

yes

complemented

#14519697

C₂²

yes

—

C₂²×Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

complemented

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂³×D₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

Q₈

yes

complemented

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

C₂×#92418

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

C₂×Q₈

yes

complemented

C₄×D₆₁

—

#92418

—

C₂²×C₆₁

C₂²⋊Q₈

yes

—

C₂²×C₆₁

C₂×D₈

yes

—

C₂²×C₆₁

#137

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₆₁

C₂²×D₈

yes

characteristic

C₂×C₆₁

C₂×#137

yes

characteristic

D₆₁

—

C₂×C₄⋊Q₈

D₆₁

yes

characteristic, complemented

C₂×C₄×D₄

—

#474

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×#102

—

C₆₁

C₂×#474

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

100

C₂³×C₄

—

122

—

101

C₂²×C₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

C₂×D₆₁

yes

complemented

104

C₄×D₄

—

105

C₄×D₄

—

122

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

109

C₂×C₂²⋊C₄

—

122

—

110

C₂×C₄²

C₂×D₆₁

yes

characteristic

111

#107

—

112

C₄⋊C₈

—

113

#102

—

114

C₂⁴

—

122

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

122

—

120

C₂×D₄

—

122

—

121

C₂²×C₄

—

122

C₂²×C₄

—

122

—

123

C₂²×C₄

—

122

—

124

C₂²×C₄

—

244

—

125

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

126

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

127

C₂²×C₄

C₂²×D₆₁

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

122

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

122

—

134

C₄⋊C₄

C₂²×D₆₁

yes

—

135

C₂²⋊C₄

—

122

—

136

C₄²

C₂²⋊D₆₁

yes

—

137

C₄²

C₂²×D₆₁

yes

—

138

C₂³

—

122

—

139

C₂³

—

122

—

140

C₂³

—

244

—

141

C₂³

—

244

—

142

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

143

Q₈

—

144

D₄

—

145

D₄

—

122

—

146

D₄

—

122

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

122

—

151

C₂×C₄

—

244

—

152

C₂×C₄

—

244

—

153

C₂×C₄

#11758199

yes

—

154

C₂×C₄

Q₈×D₆₁

yes

—

155

C₂×C₄

#11758226

yes

—

156

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

157

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

159

C₂×C₄

C₂³×D₆₁

yes

characteristic

160

C₈

—

122

—

161

C₂²

—

122

—

162

C₂²

—

244

—

163

C₂²

—

244

—

164

C₂²

—

244

—

165

C₂²

#14519753

yes

—

166

C₂²

C₂×#11758199

yes

—

167

C₂²

#14519808

yes

—

168

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

244

—

172

C₄

#14519753

yes

—

173

C₄

C₂×#11758199

yes

—

174

C₄

C₂×Q₈×D₆₁

yes

characteristic

175

C₄

C₂×#11758226

yes

characteristic

176

C₂

—

244

—

177

C₂

#16840039

yes

direct product factor

178

C₂

C₂×#14519753

yes

characteristic

179

C₂

C₂²×#11758199

yes

characteristic

180

C₂

C₂×#14519808

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	C₂×#474
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₁

hypocenter

C₆₁

C₂×#474

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16840039	C₂	direct (N × H)
3. C₂×#14519710	C₂	semidirect
4. C₂×C₄⋊#92418	C₂	semidirect
6. C₂×C₄⋊Q₈×C₆₁	C₂	semidirect
14. #14519710	C₂²	semidirect
15. C₄⋊#92418	C₂²	semidirect
19. C₄⋊Q₈×C₆₁	C₂²	semidirect
33. C₂×D₂₄₄	Q₈	semidirect
67. D₂₄₄	C₂×Q₈	semidirect
86. C₂×C₆₁	#474	semidirect
91. C₂×C₄⋊Q₈	D₆₁	semidirect
97. C₆₁	C₂×#474	semidirect
103. C₄⋊Q₈	C₂×D₆₁	semidirect
177. C₂	#16840039	direct (N × H)

H ≅ G/N

type

2. #16840039

C₂

direct (N × H)

3. C₂×#14519710

C₂

semidirect

4. C₂×C₄⋊#92418

C₂

semidirect

6. C₂×C₄⋊Q₈×C₆₁

C₂

semidirect

14. #14519710

C₂²

semidirect

15. C₄⋊#92418

C₂²

semidirect

19. C₄⋊Q₈×C₆₁

C₂²

semidirect

33. C₂×D₂₄₄

Q₈

semidirect

67. D₂₄₄

C₂×Q₈

semidirect

86. C₂×C₆₁

#474

semidirect

91. C₂×C₄⋊Q₈

D₆₁

semidirect

97. C₆₁

C₂×#474

semidirect

103. C₄⋊Q₈

C₂×D₆₁

semidirect

177. C₂

#16840039

direct (N × H)

Subgroups — C₂×#16840039 #21455561

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840039 #21455561

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840039 #21455561