Group #22625283 — subgroups

Subgroups — #22625283

Order 9344.

230 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 62 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	#17989609	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11924036	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈⋊#10596833	C₂	1	2	yes	maximal, complemented
4	D₄⋊#10596833	C₂	1	2	yes	maximal, complemented
5	#17989116	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924036	—	2	1	no	—
7	C₄⋊#10596833	C₂²	1	2	yes	—
8	#14556648	—	2	1	no	—
9	#14556646	—	2	1	no	—
10	C₄×#10596833	—	2	1	no	—
11	C₄×#10596833	C₂²	1	2	yes	—
12	#14556584	—	2	1	no	—
13	#14556583	C₄	1	1	yes	characteristic
14	#14556573	—	2	1	no	—
15	#14556567	C₄	1	1	yes	characteristic
16	#14556562	C₄	1	1	yes	characteristic
17	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	D₄×Dic₇₃	C₂²	1	1	yes	characteristic
19	#14556551	—	2	1	no	—
20	#14556547	—	2	1	no	—
21	C₄²×D₇₃	C₂²	1	1	yes	characteristic
22	#124×C₇₃	C₄	1	1	yes	characteristic
23	#11924036	—	2	4	no	—
24	C₂×#10596833	—	2	2	no	—
25	C₂×#10596833	—	2	2	no	—
26	C₂×C₂²⋊D₇₃	—	2	1	no	—
27	C₂²×Dic₇₃	—	2	1	no	—
28	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	#11924007	—	2	1	no	—
35	#11924007	—	2	1	no	—
36	#11924007	—	2	1	no	—
37	C₂²⋊Dic₇₃	—	2	1	no	—
38	C₂²⋊Dic₇₃	—	2	1	no	—
39	#11924005	—	2	1	no	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₄×Dic₇₃	D₄	1	1	yes	characteristic
42	C₄×Dic₇₃	D₄	1	1	yes	characteristic
43	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂²⋊C₄×C₇₃	—	2	1	no	—
47	C₂²⋊C₄×C₇₃	—	2	1	no	—
48	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₇₃	—	2	1	no	—
50	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	Q₈×C₇₃	—	2	2	no	—
52	D₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²⋊D₇₃	—	4	2	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	2	1	no	—
60	C₂×Dic₇₃	—	4	2	no	—
61	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
64	C₄×D₇₃	—	2	2	no	—
65	C₄×D₇₃	—	2	2	no	—
66	C₄×D₇₃	—	2	2	no	—
67	#10596833	—	2	4	no	—
68	#10596833	—	4	2	no	—
69	C₂²×C₇₃	—	4	1	no	—
70	C₂²×C₇₃	—	4	1	no	—
71	C₂²×C₇₃	—	4	1	no	—
72	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×D₇₃	—	2	1	no	—
74	C₂×D₇₃	—	2	1	no	—
75	C₂×D₇₃	#104	1	2	yes	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	4	1	no	—
79	C₄×C₇₃	#131	1	2	yes	—
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	—	2	1	no	—
82	Dic₇₃	—	4	1	no	—
83	Dic₇₃	—	4	1	no	—
84	Dic₇₃	#135	1	2	yes	—
85	C₂×C₇₃	—	8	1	no	—
86	C₂×C₇₃	#408	1	1	yes	characteristic
87	C₂×C₇₃	#416	1	1	yes	characteristic
88	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
89	D₇₃	—	2	2	no	—
90	#1494	—	73	1	no	maximal, Sylow
91	C₇₃	#1494	1	1	yes	characteristic, complemented, Sylow
92	#529	—	73	1	no	—
93	#432	—	73	1	no	—
94	C₂×C₄×C₈	—	73	1	no	—
95	Q₈⋊C₈	—	73	2	no	—
96	D₄⋊C₈	—	73	2	no	—
97	C₂²×C₈	—	146	1	no	—
98	#126	—	146	1	no	—
99	#125	—	73	1	no	—
100	#124	#64408	1	1	yes	characteristic
101	#123	—	146	1	no	—
102	C₂²⋊Q₈	—	73	1	no	—
103	#121	—	73	1	no	—
104	C₄×Q₈	—	73	1	no	—
105	C₄×D₄	—	73	1	no	—
106	C₄×D₄	—	146	1	no	—
107	#117	—	146	1	no	—
108	C₂×C₄²	—	73	1	no	—
109	C₄⋊C₈	—	73	2	no	—
110	C₂²⋊C₈	—	146	1	no	—
111	#97	—	146	1	no	—
112	C₄×C₈	—	73	2	no	—
113	C₄×C₈	—	146	1	no	—
114	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
115	C₂×D₄	—	146	1	no	—
116	C₂×D₄	C₂×#64408	1	1	yes	characteristic
117	C₂²×C₄	—	73	1	no	—
118	C₂²×C₄	—	73	1	no	—
119	C₂²×C₄	—	73	1	no	—
120	C₂²×C₄	—	146	1	no	—
121	C₂×C₈	—	146	2	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	—	146	4	no	—
124	C₄⋊C₄	—	73	1	no	—
125	C₄⋊C₄	—	73	1	no	—
126	C₄⋊C₄	—	146	1	no	—
127	C₄⋊C₄	—	146	1	no	—
128	C₄⋊C₄	—	146	1	no	—
129	C₄⋊C₄	—	146	1	no	—
130	C₂²⋊C₄	—	2	1	no	—
131	C₂²⋊C₄	—	2	1	no	—
132	C₂²⋊C₄	—	146	1	no	—
133	C₂²⋊C₄	—	146	1	no	—
134	C₂²⋊C₄	—	146	1	no	—
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	73	1	no	—
138	C₄²	—	146	1	no	—
139	C₄²	C₂×#64408	1	1	yes	characteristic
140	C₂³	—	2	1	no	—
141	C₂³	—	73	1	no	—
142	Q₈	—	2	2	no	—
143	D₄	—	2	2	no	—
144	D₄	—	292	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	292	2	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	146	2	no	—
151	C₂×C₄	—	146	2	no	—
152	C₂×C₄	—	146	1	no	—
153	C₂×C₄	—	146	1	no	—
154	C₂×C₄	—	146	2	no	—
155	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
156	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
157	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
158	C₈	—	292	2	no	—
159	C₈	—	146	4	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	73	2	no	—
164	C₂²	—	146	1	no	—
165	C₂²	—	146	1	no	—
166	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	292	1	no	—
171	C₄	—	292	1	no	—
172	C₄	—	73	2	no	—
173	C₄	—	146	1	no	—
174	C₄	—	146	1	no	—
175	C₄	#14556678	1	2	yes	—
176	C₂	—	8	1	no	—
177	C₂	—	146	2	no	—
178	C₂	#17989561	1	1	yes	characteristic
179	C₂	#17989577	1	1	yes	characteristic
180	C₂	D₇₃⋊#104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989609

C₂

yes

maximal, characteristic, complemented

C₄×#11924036

C₂

yes

maximal, characteristic, complemented

Q₈⋊#10596833

C₂

yes

maximal, complemented

D₄⋊#10596833

C₂

yes

maximal, complemented

#17989116

C₂

yes

maximal, characteristic

C₂×#11924036

—

C₄⋊#10596833

C₂²

yes

—

—

—

—

C₂²

yes

—

#14556584

—

#14556583

C₄

yes

characteristic

#14556573

—

#14556567

C₄

yes

characteristic

#14556562

C₄

yes

characteristic

Q₈×Dic₇₃

C₂²

yes

characteristic

D₄×Dic₇₃

C₂²

yes

characteristic

#14556551

—

#14556547

—

C₄²×D₇₃

C₂²

yes

characteristic

#124×C₇₃

C₄

yes

—

—

—

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

characteristic

C₄⋊Dic₇₃

C₂×C₄

yes

—

—

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#104

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#131

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#135

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1494

—

maximal, Sylow

C₇₃

#1494

yes

characteristic, complemented, Sylow

#529

—

#432

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₂²×C₈

—

146

—

#126

—

146

—

#125

—

100

yes

101

—

146

—

102

C₂²⋊Q₈

—

103

#121

—

104

C₄×Q₈

—

105

C₄×D₄

—

106

C₄×D₄

—

146

—

107

#117

—

146

—

108

C₂×C₄²

—

109

C₄⋊C₈

—

110

C₂²⋊C₈

—

146

—

111

#97

—

146

—

112

C₄×C₈

—

113

C₄×C₈

—

146

—

114

C₂×Q₈

C₂×#64408

yes

characteristic

115

C₂×D₄

—

146

—

116

C₂×D₄

C₂×#64408

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

146

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

146

—

127

C₄⋊C₄

—

146

—

128

C₄⋊C₄

—

146

—

129

C₄⋊C₄

—

146

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

146

—

133

C₂²⋊C₄

—

146

—

134

C₂²⋊C₄

—

146

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

146

—

139

C₄²

C₂×#64408

yes

characteristic

140

C₂³

—

141

C₂³

—

142

Q₈

—

143

D₄

—

144

D₄

—

292

—

145

C₂×C₄

—

146

C₂×C₄

—

292

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

146

—

151

C₂×C₄

—

146

—

152

C₂×C₄

—

146

—

153

C₂×C₄

—

146

—

154

C₂×C₄

—

146

—

155

C₂×C₄

C₂²⋊#64408

yes

characteristic

156

C₂×C₄

C₂²⋊#64408

yes

characteristic

157

C₂×C₄

C₂²×#64408

yes

characteristic

158

C₈

—

292

—

159

C₈

—

146

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

146

—

165

C₂²

—

146

—

166

C₂²

C₂×C₂²⋊#64408

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

292

—

171

C₄

—

292

—

172

C₄

—

173

C₄

—

146

—

174

C₄

—

146

—

175

C₄

#14556678

yes

—

176

C₂

—

177

C₂

—

146

—

178

C₂

#17989561

yes

characteristic

179

C₂

#17989577

yes

characteristic

180

C₂

D₇₃⋊#104

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	#124×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#124	#64408
hypocenter	C₇₃	#1494
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

#124×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #124

N	H ≅ G/N	type
1. #17989609	C₂	semidirect
2. C₄×#11924036	C₂	semidirect
3. Q₈⋊#10596833	C₂	semidirect
4. D₄⋊#10596833	C₂	semidirect
91. C₇₃	#1494	semidirect

H ≅ G/N

type

1. #17989609

C₂

semidirect

2. C₄×#11924036

C₂

semidirect

3. Q₈⋊#10596833

C₂

semidirect

4. D₄⋊#10596833

C₂

semidirect

91. C₇₃

#1494

semidirect

Subgroups — #22625283

Conjugacy classes of subgroups

Complements