Group #22626094 — subgroups

Subgroups — #22626094

Order 9344.

336 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 74 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×#14556656	C₂	1	4	yes	maximal, complemented
2	C₄×#11924038	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊#64408	C₂	1	1	yes	maximal, characteristic, complemented
4	#17989138	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924038	C₂²	1	2	yes	—
6	C₂×C₄⋊#64408	—	2	1	no	—
7	C₂×C₄×#64408	C₂²	1	2	yes	—
8	#14556656	—	2	8	no	—
9	#14556646	—	2	1	no	—
10	C₄×#10596833	—	2	1	no	—
11	(C₂×#64408)⋊C₄	—	2	1	no	—
12	C₂×#11924029	C₂²	1	2	yes	—
13	#14556588	C₄	1	1	yes	characteristic, complemented
14	#14556580	C₄	1	2	yes	complemented
15	#14556566	—	2	1	no	—
16	#14556566	—	2	1	no	—
17	#14556553	—	2	1	no	—
18	#14556553	—	2	1	no	—
19	C₄²×D₇₃	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
21	C₂²×#64408	—	2	2	no	—
22	#11924038	—	2	2	no	—
23	#11924038	D₄	1	4	yes	complemented
24	C₂×#10596833	—	2	2	no	—
25	C₄⋊#64408	—	2	4	no	—
26	C₄×#64408	—	2	4	no	—
27	#11924029	—	2	4	no	—
28	#11924029	—	2	4	no	—
29	C₂×D₂₉₂	—	2	2	no	—
30	C₂×D₂₉₂	—	2	1	no	—
31	C₂×D₂₉₂	—	2	1	no	—
32	C₂×D₂₉₂	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₇₃	—	2	2	no	—
34	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
35	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
37	#11924005	—	2	2	no	—
38	#11924005	—	2	2	no	—
39	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₇₃	D₄	1	1	yes	characteristic
41	C₄×Dic₇₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₇₃	—	2	1	no	—
44	C₄⋊C₄×C₇₃	—	2	1	no	—
45	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×D₇₃	—	2	2	no	—
47	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×#64408	—	2	4	no	—
49	C₂×#64408	—	4	2	no	—
50	C₂×#64408	—	4	2	no	—
51	Q₈×C₇₃	—	2	4	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
59	D₂₉₂	—	2	4	no	—
60	D₂₉₂	—	4	2	no	—
61	D₂₉₂	—	4	2	no	—
62	D₂₉₂	—	4	2	no	—
63	D₂₉₂	—	4	2	no	—
64	C₄×D₇₃	—	2	2	no	—
65	C₄×D₇₃	—	4	1	no	—
66	C₄×D₇₃	—	4	4	no	—
67	C₄×D₇₃	C₂×D₄	1	2	yes	—
68	C₄×D₇₃	#41	1	2	yes	—
69	#10596833	—	2	4	no	—
70	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	—	2	1	no	—
73	C₂×D₇₃	—	4	2	no	—
74	C₂×D₇₃	—	4	2	no	—
75	C₂×D₇₃	—	4	2	no	—
76	C₂×D₇₃	#124	1	1	yes	characteristic
77	C₂×D₇₃	#127	1	1	yes	characteristic
78	#64408	—	4	4	no	—
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	—	4	2	no	—
81	C₄×C₇₃	#104	1	2	yes	complemented
82	C₄×C₇₃	C₄×D₄	1	2	yes	—
83	Dic₇₃	—	2	1	no	—
84	Dic₇₃	—	2	1	no	—
85	Dic₇₃	#121	1	2	yes	—
86	C₂×C₇₃	#390	1	1	yes	characteristic
87	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
88	C₂×C₇₃	#421	1	1	yes	characteristic
89	D₇₃	—	2	1	no	—
90	D₇₃	—	2	1	no	—
91	D₇₃	—	8	2	no	—
92	#1980	—	73	1	no	maximal, Sylow
93	C₇₃	#1980	1	1	yes	characteristic, complemented, Sylow
94	#532	—	73	1	no	—
95	C₂×#104	—	73	4	no	—
96	C₄×M₁₆	—	73	1	no	—
97	C₄×C₄⋊C₄	—	73	1	no	—
98	C₂×#41	—	73	2	no	—
99	C₂×M₁₆	—	73	2	no	—
100	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
101	#127	—	73	1	no	—
102	#124	—	73	2	no	—
103	#121	—	146	1	no	—
104	#121	—	146	1	no	—
105	C₄×D₄	—	146	1	no	—
106	C₄×D₄	—	146	1	no	—
107	C₂×C₄⋊C₄	—	146	1	no	—
108	C₂×C₄²	—	73	1	no	—
109	C₂×C₄²	—	73	2	no	—
110	#104	—	146	8	no	—
111	#97	—	146	1	no	—
112	C₄×C₈	—	146	1	no	—
113	#95	—	146	1	no	—
114	#41	—	146	4	no	—
115	#41	—	146	4	no	—
116	C₂×Q₈	C₂×#64408	1	2	yes	—
117	C₂×D₄	—	73	2	no	—
118	C₂×D₄	—	146	2	no	—
119	C₂×D₄	—	146	1	no	—
120	C₂×D₄	—	146	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	C₂²×C₄	—	73	1	no	—
123	C₂²×C₄	—	73	1	no	—
124	C₂²×C₄	—	146	2	no	—
125	C₂²×C₄	—	146	2	no	—
126	M₁₆	—	73	4	no	—
127	M₁₆	—	146	2	no	—
128	C₂×C₈	—	146	2	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	146	4	no	—
132	C₂²⋊C₄	—	146	2	no	—
133	C₂²⋊C₄	—	146	2	no	—
134	C₄²	—	73	1	no	—
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	146	4	no	—
138	C₄²	C₂×#64408	1	1	yes	characteristic
139	C₂³	—	73	1	no	—
140	C₂³	—	146	2	no	—
141	Q₈	—	2	4	no	—
142	D₄	—	292	2	no	—
143	D₄	—	292	2	no	—
144	D₄	—	292	2	no	—
145	D₄	—	292	2	no	—
146	D₄	—	146	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	292	1	no	—
149	C₂×C₄	—	292	4	no	—
150	C₂×C₄	—	292	2	no	—
151	C₂×C₄	—	292	2	no	—
152	C₂×C₄	—	73	1	no	—
153	C₂×C₄	—	73	2	no	—
154	C₂×C₄	—	73	2	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	73	1	no	—
157	C₂×C₄	—	146	4	no	—
158	C₂×C₄	—	146	2	no	—
159	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
160	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
161	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
162	C₈	—	146	4	no	—
163	C₂²	—	292	2	no	—
164	C₂²	—	292	2	no	—
165	C₂²	—	292	2	no	—
166	C₂²	—	73	1	no	—
167	C₂²	—	73	1	no	—
168	C₂²	—	146	1	no	—
169	C₂²	—	146	1	no	—
170	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	—	292	4	no	—
174	C₄	—	73	2	no	—
175	C₄	—	146	1	no	—
176	C₄	—	146	1	no	—
177	C₄	#14556656	1	2	yes	complemented
178	C₄	D₄×#64408	1	2	yes	—
179	C₂	—	584	2	no	—
180	C₂	—	146	1	no	—
181	C₂	—	146	1	no	—
182	C₂	#17989539	1	1	yes	characteristic
183	C₂	C₂×#14556656	1	1	yes	characteristic
184	C₂	#17989586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556656

C₂

yes

maximal, complemented

C₄×#11924038

C₂

yes

maximal, characteristic, complemented

C₄×C₄⋊#64408

C₂

yes

maximal, characteristic, complemented

#17989138

C₂

yes

maximal, characteristic

C₂×#11924038

C₂²

yes

—

C₂×C₄⋊#64408

—

C₂×C₄×#64408

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#14556588

C₄

yes

characteristic, complemented

#14556580

C₄

yes

—

—

—

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

#11924038

—

#11924038

D₄

yes

—

—

—

—

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

#11924005

—

#11924005

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

C₂×D₄

yes

—

—

—

Q₈×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

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₄

yes

—

C₄×D₇₃

#41

yes

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#104

yes

complemented

C₄×C₇₃

C₄×D₄

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#121

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

D₇₃

—

#1980

—

maximal, Sylow

C₇₃

#1980

yes

characteristic, complemented, Sylow

#532

—

C₂×#104

—

C₄×M₁₆

—

C₄×C₄⋊C₄

—

C₂×#41

—

C₂×M₁₆

—

100

C₄⋊Q₈

#64408

yes

characteristic, complemented

101

#127

—

102

#124

—

103

#121

—

146

—

104

#121

—

146

—

105

C₄×D₄

—

146

—

106

C₄×D₄

—

146

—

107

C₂×C₄⋊C₄

—

146

—

108

C₂×C₄²

—

109

C₂×C₄²

—

110

#104

—

146

—

111

#97

—

146

—

112

C₄×C₈

—

146

—

113

#95

—

146

—

114

#41

—

146

—

115

#41

—

146

—

116

C₂×Q₈

C₂×#64408

yes

—

117

C₂×D₄

—

118

C₂×D₄

—

146

—

119

C₂×D₄

—

146

—

120

C₂×D₄

—

146

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

146

—

125

C₂²×C₄

—

146

—

126

M₁₆

—

127

M₁₆

—

146

—

128

C₂×C₈

—

146

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

146

—

132

C₂²⋊C₄

—

146

—

133

C₂²⋊C₄

—

146

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

146

—

138

C₄²

C₂×#64408

yes

characteristic

139

C₂³

—

140

C₂³

—

146

—

141

Q₈

—

142

D₄

—

292

—

143

D₄

—

292

—

144

D₄

—

292

—

145

D₄

—

292

—

146

D₄

—

146

—

147

C₂×C₄

—

148

C₂×C₄

—

292

—

149

C₂×C₄

—

292

—

150

C₂×C₄

—

292

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

146

—

158

C₂×C₄

—

146

—

159

C₂×C₄

C₂²⋊#64408

yes

characteristic

160

C₂×C₄

C₂²⋊#64408

yes

characteristic

161

C₂×C₄

C₂²×#64408

yes

characteristic

162

C₈

—

146

—

163

C₂²

—

292

—

164

C₂²

—

292

—

165

C₂²

—

292

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

146

—

169

C₂²

—

146

—

170

C₂²

C₂×C₂²⋊#64408

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

292

—

174

C₄

—

175

C₄

—

146

—

176

C₄

—

146

—

177

C₄

#14556656

yes

complemented

178

C₄

D₄×#64408

yes

—

179

C₂

—

584

—

180

C₂

—

146

—

181

C₂

—

146

—

182

C₂

#17989539

yes

characteristic

183

C₂

C₂×#14556656

yes

characteristic

184

C₂

#17989586

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄⋊Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#64408
hypocenter	C₇₃	#1980
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄⋊Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×#14556656	C₂	semidirect
2. C₄×#11924038	C₂	semidirect
3. C₄×C₄⋊#64408	C₂	semidirect
13. #14556588	C₄	semidirect
14. #14556580	C₄	semidirect
20. C₄⋊Q₈×C₇₃	C₄	semidirect
23. #11924038	D₄	semidirect
81. C₄×C₇₃	#104	semidirect
93. C₇₃	#1980	semidirect
100. C₄⋊Q₈	#64408	semidirect
177. C₄	#14556656	semidirect

H ≅ G/N

type

1. C₂×#14556656

C₂

semidirect

2. C₄×#11924038

C₂

semidirect

3. C₄×C₄⋊#64408

C₂

semidirect

13. #14556588

C₄

semidirect

14. #14556580

C₄

semidirect

20. C₄⋊Q₈×C₇₃

C₄

semidirect

23. #11924038

D₄

semidirect

81. C₄×C₇₃

#104

semidirect

93. C₇₃

#1980

semidirect

100. C₄⋊Q₈

#64408

semidirect

177. C₄

#14556656

semidirect

Subgroups — #22626094

Conjugacy classes of subgroups

Complements