Group #22619682 — subgroups

Subgroups — C₂×#17988877 #22619682

Order 9344.

480 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14556591	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988877	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14556522	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14556512	C₂	1	2	yes	maximal
6	C₂×C₄×D₄×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₄₆	—	2	1	no	—
8	C₂²×#10596831	C₂²	1	2	yes	—
9	#14556591	—	2	2	no	—
10	#14556591	C₂²	1	4	yes	complemented
11	C₂×C₄⋊Dic₇₃	—	2	1	no	—
12	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
13	#14556522	C₂²	1	8	yes	complemented
14	C₄⋊#10596831	C₂²	1	4	yes	complemented
15	#14556512	C₂²	1	8	yes	—
16	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₇₃	—	2	1	no	—
18	C₄×D₄×C₇₃	—	2	2	no	—
19	C₄×D₄×C₇₃	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
22	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	—	2	4	no	—
24	C₂×Dic₁₄₆	—	4	4	no	—
25	C₂²×Dic₇₃	—	2	2	no	—
26	C₂×#10596831	—	2	4	no	—
27	C₂×#10596831	C₂³	1	4	yes	—
28	C₄⋊Dic₇₃	—	2	4	no	—
29	C₄⋊Dic₇₃	—	2	2	no	—
30	C₄⋊Dic₇₃	C₂³	1	4	yes	—
31	C₂⁴×C₇₃	—	2	1	no	—
32	C₂×D₄×C₇₃	—	2	1	no	—
33	C₂×D₄×C₇₃	—	2	2	no	—
34	C₂×D₄×C₇₃	Q₈	1	4	yes	complemented
35	C₂×D₄×C₇₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₇₃	—	2	1	no	—
37	C₂²×C₄×C₇₃	—	2	4	no	—
38	C₂²×C₄×C₇₃	—	4	4	no	—
39	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
44	C₂²⋊C₄×C₇₃	—	2	4	no	—
45	C₄²×C₇₃	D₄	1	2	yes	—
46	C₄²×C₇₃	C₂³	1	2	yes	—
47	C₂³×C₇₃	—	2	4	no	—
48	C₂³×C₇₃	—	2	2	no	—
49	C₂³×C₇₃	—	4	2	no	—
50	C₂³×C₇₃	—	4	2	no	—
51	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
52	D₄×C₇₃	—	2	4	no	—
53	D₄×C₇₃	—	2	2	no	—
54	D₄×C₇₃	C₂×Q₈	1	4	yes	complemented
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	4	2	no	—
58	C₂×C₄×C₇₃	—	4	8	no	—
59	C₂×C₄×C₇₃	SD₁₆	1	4	yes	—
60	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₇₃	#41	1	2	yes	—
65	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₇₃	—	2	4	no	—
67	C₂×Dic₇₃	—	4	4	no	—
68	Dic₁₄₆	—	4	8	no	—
69	#10596831	—	2	4	no	—
70	C₂²×C₇₃	—	2	4	no	—
71	C₂²×C₇₃	—	4	2	no	—
72	C₂²×C₇₃	—	4	2	no	—
73	C₂²×C₇₃	—	4	8	no	—
74	C₂²×C₇₃	C₂²⋊Q₈	1	2	yes	—
75	C₂²×C₇₃	C₂×SD₁₆	1	2	yes	—
76	C₂²×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	4	2	no	—
80	C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
81	C₄×C₇₃	C₂×SD₁₆	1	4	yes	—
82	C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
84	Dic₇₃	—	4	4	no	—
85	C₂×C₇₃	—	4	4	no	—
86	C₂×C₇₃	#476	1	4	yes	complemented
87	C₂×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
89	C₂×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×#476	—	73	1	no	maximal, Sylow
91	C₇₃	C₂×#476	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄⋊Q₈	—	73	1	no	—
93	C₂×C₄×D₄	D₇₃	1	1	yes	characteristic
94	#476	—	73	8	no	—
95	C₂×#106	—	73	2	no	—
96	C₂×C₄⋊C₈	—	73	1	no	—
97	C₂×#102	—	73	2	no	—
98	C₂²×Q₈	—	146	1	no	—
99	C₂²×D₄	C₂×D₇₃	1	1	yes	characteristic
100	C₂³×C₄	—	2	1	no	—
101	C₂²×C₈	—	73	2	no	—
102	C₄⋊Q₈	—	73	4	no	—
103	C₄⋊Q₈	—	146	2	no	—
104	C₄×D₄	—	2	2	no	—
105	C₄×D₄	C₂×D₇₃	1	4	yes	—
106	C₂×C₄⋊C₄	—	73	2	no	—
107	C₂×C₄⋊C₄	—	146	1	no	—
108	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
109	C₂×C₂²⋊C₄	—	2	1	no	—
110	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
111	#106	—	73	8	no	—
112	C₄⋊C₈	—	73	4	no	—
113	#102	—	73	8	no	—
114	C₂⁴	—	2	1	no	—
115	C₂×Q₈	—	292	4	no	—
116	C₂×Q₈	—	146	4	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	Dic₁₄₆	1	4	yes	complemented
120	C₂×D₄	C₂²×D₇₃	1	2	yes	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	4	4	no	—
124	C₂²×C₄	—	146	2	no	—
125	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
128	C₂×C₈	—	73	4	no	—
129	C₂×C₈	—	146	4	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	73	4	no	—
132	C₄⋊C₄	—	146	2	no	—
133	C₄⋊C₄	—	146	4	no	—
134	C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₄²	C₂²⋊D₇₃	1	2	yes	—
137	C₄²	C₂²×D₇₃	1	2	yes	—
138	C₂³	—	2	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	4	2	no	—
141	C₂³	—	4	2	no	—
142	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
143	Q₈	—	292	8	no	—
144	D₄	—	2	4	no	—
145	D₄	—	2	2	no	—
146	D₄	C₂×Dic₁₄₆	1	4	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	8	no	—
151	C₂×C₄	—	292	4	no	—
152	C₂×C₄	—	146	4	no	—
153	C₂×C₄	#11924016	1	4	yes	—
154	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
157	C₂×C₄	C₂×Dic₁₄₆	1	2	yes	—
158	C₂×C₄	#11924028	1	2	yes	—
159	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
160	C₈	—	146	4	no	—
161	C₂²	—	2	4	no	—
162	C₂²	—	4	8	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	2	no	—
165	C₂²	#14556570	1	2	yes	—
166	C₂²	C₂×#11924016	1	2	yes	—
167	C₂²	#14556616	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	292	4	no	—
172	C₄	#14556570	1	2	yes	—
173	C₄	C₂×#11924016	1	4	yes	—
174	C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
175	C₄	C₂×#11924028	1	1	yes	characteristic
176	C₂	—	4	4	no	—
177	C₂	#17988877	1	4	yes	direct product factor
178	C₂	C₂×#14556570	1	1	yes	characteristic
179	C₂	C₂²×#11924016	1	1	yes	characteristic
180	C₂	C₂×#14556616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal, characteristic, complemented

#17988877

C₂

yes

maximal, direct product factor

C₂×#14556522

C₂

yes

maximal, complemented

C₂×C₄⋊#10596831

C₂

yes

maximal, characteristic, complemented

C₂×#14556512

C₂

yes

maximal

C₂×C₄×D₄×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

—

C₂²×#10596831

C₂²

yes

—

#14556591

—

#14556591

C₂²

yes

complemented

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

#14556522

C₂²

yes

complemented

C₄⋊#10596831

C₂²

yes

complemented

#14556512

C₂²

yes

—

C₂²×D₄×C₇₃

C₂²

yes

characteristic

C₂³×C₄×C₇₃

—

C₄×D₄×C₇₃

—

C₄×D₄×C₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

C₂⁴×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

Q₈

yes

complemented

C₂×D₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

C₂×Q₈

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊Q₈

yes

—

C₂²×C₇₃

C₂×SD₁₆

yes

—

C₂²×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

C₂×SD₁₆

yes

—

C₄×C₇₃

C₂²×Q₈

yes

characteristic

C₄×C₇₃

C₂×#41

yes

characteristic

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#476

yes

complemented

C₂×C₇₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₇₃

C₂²×SD₁₆

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄×D₄

D₇₃

yes

characteristic

#476

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×#102

—

C₂²×Q₈

—

146

—

C₂²×D₄

C₂×D₇₃

yes

characteristic

100

C₂³×C₄

—

101

C₂²×C₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

146

—

104

C₄×D₄

—

105

C₄×D₄

C₂×D₇₃

yes

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

146

—

108

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

109

C₂×C₂²⋊C₄

—

110

C₂×C₄²

C₂×D₇₃

yes

characteristic

111

#106

—

112

C₄⋊C₈

—

113

#102

—

114

C₂⁴

—

115

C₂×Q₈

—

292

—

116

C₂×Q₈

—

146

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

Dic₁₄₆

yes

complemented

120

C₂×D₄

C₂²×D₇₃

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

146

—

125

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

126

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

127

C₂²×C₄

C₂²×D₇₃

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

146

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

—

146

—

134

C₄⋊C₄

C₂²×D₇₃

yes

—

135

C₂²⋊C₄

—

136

C₄²

C₂²⋊D₇₃

yes

—

137

C₄²

C₂²×D₇₃

yes

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

143

Q₈

—

292

—

144

D₄

—

145

D₄

—

146

D₄

C₂×Dic₁₄₆

yes

complemented

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

146

—

153

C₂×C₄

#11924016

yes

—

154

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

155

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

156

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

157

C₂×C₄

C₂×Dic₁₄₆

yes

—

158

C₂×C₄

#11924028

yes

—

159

C₂×C₄

C₂³×D₇₃

yes

characteristic

160

C₈

—

146

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14556570

yes

—

166

C₂²

C₂×#11924016

yes

—

167

C₂²

#14556616

yes

—

168

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

292

—

172

C₄

#14556570

yes

—

173

C₄

C₂×#11924016

yes

—

174

C₄

C₂²×Dic₁₄₆

yes

characteristic

175

C₄

C₂×#11924028

yes

characteristic

176

C₂

—

177

C₂

#17988877

yes

direct product factor

178

C₂

C₂×#14556570

yes

characteristic

179

C₂

C₂²×#11924016

yes

characteristic

180

C₂

C₂×#14556616

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	C₂×C₄×D₄×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄×D₄	D₇₃
hypocenter	C₇₃	C₂×#476
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

C₂×C₄×D₄×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄×D₄

D₇₃

hypocenter

C₇₃

C₂×#476

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×#14556591	C₂	semidirect
2. #17988877	C₂	direct (N × H)
3. C₂×#14556522	C₂	semidirect
4. C₂×C₄⋊#10596831	C₂	semidirect
10. #14556591	C₂²	semidirect
13. #14556522	C₂²	semidirect
14. C₄⋊#10596831	C₂²	semidirect
34. C₂×D₄×C₇₃	Q₈	semidirect
54. D₄×C₇₃	C₂×Q₈	semidirect
86. C₂×C₇₃	#476	semidirect
91. C₇₃	C₂×#476	semidirect
119. C₂×D₄	Dic₁₄₆	semidirect
146. D₄	C₂×Dic₁₄₆	semidirect
177. C₂	#17988877	direct (N × H)

H ≅ G/N

type

1. C₂×#14556591

C₂

semidirect

2. #17988877

C₂

direct (N × H)

3. C₂×#14556522

C₂

semidirect

4. C₂×C₄⋊#10596831

C₂

semidirect

10. #14556591

C₂²

semidirect

13. #14556522

C₂²

semidirect

14. C₄⋊#10596831

C₂²

semidirect

34. C₂×D₄×C₇₃

Q₈

semidirect

54. D₄×C₇₃

C₂×Q₈

semidirect

86. C₂×C₇₃

#476

semidirect

91. C₇₃

C₂×#476

semidirect

119. C₂×D₄

Dic₁₄₆

semidirect

146. D₄

C₂×Dic₁₄₆

semidirect

177. C₂

#17988877

direct (N × H)

Subgroups — C₂×#17988877 #22619682

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17988877 #22619682

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17988877 #22619682