Group #22626097 — subgroups

Subgroups — #22626097

Order 9344.

324 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 88 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	#17989588	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556654	C₂	1	4	yes	maximal
3	C₄×C₄⋊#64408	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924036	C₂²	1	2	yes	—
6	C₂×C₄⋊#64408	C₂²	1	2	yes	—
7	C₂×C₄×#64408	—	2	1	no	—
8	#14556658	—	2	1	no	—
9	C₄⋊#10596833	—	2	1	no	—
10	#14556654	—	2	8	no	—
11	(C₂×#64408)⋊C₄	—	2	1	no	—
12	C₂×Q₈×D₇₃	C₂²	1	2	yes	—
13	#14556591	C₄	1	1	yes	characteristic, complemented
14	#14556590	—	2	1	no	—
15	#14556590	—	2	1	no	—
16	#14556589	C₄	1	2	yes	complemented
17	C₄⋊C₄×D₇₃	—	2	1	no	—
18	C₄⋊C₄×D₇₃	—	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	#11924036	—	2	4	no	—
23	C₂×#10596833	—	2	2	no	—
24	C₄⋊#64408	—	2	2	no	—
25	C₄⋊#64408	D₄	1	4	yes	—
26	C₄×#64408	—	4	2	no	—
27	C₂×Dic₁₄₆	—	2	2	no	—
28	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
29	Q₈×D₇₃	—	2	4	no	—
30	Q₈×D₇₃	—	2	4	no	—
31	C₂×C₄×D₇₃	—	2	2	no	—
32	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
33	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
34	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	—	2	1	no	—
37	#11924007	—	2	2	no	—
38	#11924007	—	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	C₂×D₄	1	1	yes	characteristic
47	C₂×#64408	—	2	4	no	—
48	C₂×#64408	—	4	2	no	—
49	C₂×#64408	—	4	2	no	—
50	Q₈×C₇₃	—	2	4	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₇₃	—	2	2	no	—
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	C₄×D₇₃	—	2	2	no	—
60	C₄×D₇₃	—	4	4	no	—
61	C₄×D₇₃	SD₁₆	1	2	yes	—
62	C₄×D₇₃	Q₁₆	1	2	yes	—
63	C₄×D₇₃	C₂×D₄	1	2	yes	—
64	C₄×D₇₃	#41	1	2	yes	—
65	Dic₁₄₆	—	2	4	no	—
66	Dic₁₄₆	—	4	2	no	—
67	Dic₁₄₆	—	4	2	no	—
68	#10596833	—	4	2	no	—
69	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×D₇₃	#121	1	2	yes	—
71	C₂×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×D₇₃	#137	1	2	yes	—
74	#64408	—	4	4	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	4	2	no	—
77	C₄×C₇₃	#103	1	2	yes	complemented
78	C₄×C₇₃	C₄×D₄	1	2	yes	—
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	—	4	2	no	—
81	Dic₇₃	#103	1	2	yes	complemented
82	Dic₇₃	#124	1	1	yes	characteristic
83	Dic₇₃	#127	1	1	yes	characteristic
84	C₂×C₇₃	#390	1	1	yes	characteristic
85	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
86	C₂×C₇₃	#419	1	1	yes	characteristic
87	D₇₃	#461	1	2	yes	—
88	D₇₃	#462	1	2	yes	—
89	#1981	—	73	1	no	maximal, Sylow
90	C₇₃	#1981	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₄⋊Q₈	—	73	1	no	—
92	C₂×C₄⋊C₈	—	73	1	no	—
93	C₂×#103	—	73	4	no	—
94	C₄×C₄⋊C₄	—	73	1	no	—
95	C₂²×Q₈	—	73	2	no	—
96	C₂²×C₈	—	73	2	no	—
97	C₄⋊Q₈	—	73	2	no	—
98	C₄⋊Q₈	—	73	1	no	—
99	C₄⋊Q₈	—	146	1	no	—
100	C₄⋊Q₈	—	146	1	no	—
101	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
102	C₂×C₄⋊C₄	—	73	2	no	—
103	C₂×C₄⋊C₄	—	146	1	no	—
104	C₂×C₄⋊C₄	—	146	1	no	—
105	C₂×C₄²	—	73	1	no	—
106	C₂×C₄²	—	146	1	no	—
107	C₄⋊C₈	—	146	1	no	—
108	C₄⋊C₈	—	146	1	no	—
109	#103	—	146	8	no	—
110	#95	—	146	1	no	—
111	C₂×Q₈	—	73	2	no	—
112	C₂×Q₈	—	146	2	no	—
113	C₂×Q₈	—	146	4	no	—
114	C₂×Q₈	—	146	4	no	—
115	C₂×Q₈	C₂×#64408	1	2	yes	—
116	C₂²×C₄	—	73	1	no	—
117	C₂²×C₄	—	73	1	no	—
118	C₂²×C₄	—	73	1	no	—
119	C₂²×C₄	—	146	2	no	—
120	C₂²×C₄	—	146	2	no	—
121	C₂×C₈	—	146	2	no	—
122	C₂×C₈	—	146	4	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	73	4	no	—
126	C₄⋊C₄	—	146	2	no	—
127	C₄⋊C₄	—	146	2	no	—
128	C₄⋊C₄	—	146	2	no	—
129	C₄⋊C₄	—	146	1	no	—
130	C₄⋊C₄	—	146	1	no	—
131	C₄²	—	292	2	no	—
132	C₄²	—	73	1	no	—
133	C₄²	—	73	1	no	—
134	C₄²	—	73	1	no	—
135	C₄²	C₂×#64408	1	1	yes	characteristic
136	C₂³	—	73	1	no	—
137	Q₈	—	2	4	no	—
138	Q₈	—	292	2	no	—
139	Q₈	—	292	2	no	—
140	Q₈	—	146	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	292	4	no	—
143	C₂×C₄	—	292	2	no	—
144	C₂×C₄	—	292	2	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	2	no	—
148	C₂×C₄	—	73	2	no	—
149	C₂×C₄	—	73	2	no	—
150	C₂×C₄	—	73	2	no	—
151	C₂×C₄	—	73	1	no	—
152	C₂×C₄	—	146	2	no	—
153	C₂×C₄	—	146	4	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	C₂²	—	73	2	no	—
160	C₂²	—	73	1	no	—
161	C₂²	—	73	1	no	—
162	C₂²	—	73	2	no	—
163	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	292	4	no	—
167	C₄	—	292	2	no	—
168	C₄	—	73	1	no	—
169	C₄	—	73	1	no	—
170	C₄	—	73	2	no	—
171	C₄	—	146	1	no	—
172	C₄	#14556654	1	2	yes	complemented
173	C₄	D₄×#64408	1	2	yes	—
174	C₂	—	73	2	no	—
175	C₂	—	73	2	no	—
176	C₂	#17989539	1	1	yes	characteristic
177	C₂	C₂×#14556654	1	1	yes	characteristic
178	C₂	#17989580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989588

C₂

yes

maximal, characteristic

C₂×#14556654

C₂

yes

maximal

C₄×C₄⋊#64408

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924036

C₂²

yes

—

C₂×C₄⋊#64408

C₂²

yes

—

C₂×C₄×#64408

—

—

—

—

—

C₂×Q₈×D₇₃

C₂²

yes

—

#14556591

C₄

yes

characteristic, complemented

#14556590

—

#14556590

—

#14556589

C₄

yes

complemented

C₄⋊C₄×D₇₃

—

C₄⋊C₄×D₇₃

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

—

—

—

D₄

yes

—

C₄×#64408

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

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

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

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

SD₁₆

yes

—

C₄×D₇₃

Q₁₆

yes

—

C₄×D₇₃

C₂×D₄

yes

—

C₄×D₇₃

#41

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

#121

yes

—

C₂×D₇₃

C₂×SD₁₆

yes

characteristic

C₂×D₇₃

C₂×Q₁₆

yes

characteristic

C₂×D₇₃

#137

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#103

yes

complemented

C₄×C₇₃

C₄×D₄

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

—

D₇₃

#462

yes

—

#1981

—

maximal, Sylow

C₇₃

#1981

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

100

C₄⋊Q₈

—

146

—

101

C₄⋊Q₈

#64408

yes

characteristic, complemented

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

146

—

104

C₂×C₄⋊C₄

—

146

—

105

C₂×C₄²

—

106

C₂×C₄²

—

146

—

107

C₄⋊C₈

—

146

—

108

C₄⋊C₈

—

146

—

109

#103

—

146

—

110

#95

—

146

—

111

C₂×Q₈

—

112

C₂×Q₈

—

146

—

113

C₂×Q₈

—

146

—

114

C₂×Q₈

—

146

—

115

C₂×Q₈

C₂×#64408

yes

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

146

—

120

C₂²×C₄

—

146

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

146

—

127

C₄⋊C₄

—

146

—

128

C₄⋊C₄

—

146

—

129

C₄⋊C₄

—

146

—

130

C₄⋊C₄

—

146

—

131

C₄²

—

292

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₂×#64408

yes

characteristic

136

C₂³

—

137

Q₈

—

138

Q₈

—

292

—

139

Q₈

—

292

—

140

Q₈

—

146

—

141

C₂×C₄

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

292

—

144

C₂×C₄

—

292

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

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

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₂×C₂²⋊#64408

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

292

—

167

C₄

—

292

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

146

—

172

C₄

#14556654

yes

complemented

173

C₄

D₄×#64408

yes

—

174

C₂

—

175

C₂

—

176

C₂

#17989539

yes

characteristic

177

C₂

C₂×#14556654

yes

characteristic

178

C₂

#17989580

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	#1981
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
13. #14556591	C₄	semidirect
16. #14556589	C₄	semidirect
20. C₄⋊Q₈×C₇₃	C₄	semidirect
77. C₄×C₇₃	#103	semidirect
81. Dic₇₃	#103	semidirect
90. C₇₃	#1981	semidirect
101. C₄⋊Q₈	#64408	semidirect
172. C₄	#14556654	semidirect

H ≅ G/N

type

13. #14556591

C₄

semidirect

16. #14556589

C₄

semidirect

20. C₄⋊Q₈×C₇₃

C₄

semidirect

77. C₄×C₇₃

#103

semidirect

81. Dic₇₃

#103

semidirect

90. C₇₃

#1981

semidirect

101. C₄⋊Q₈

#64408

semidirect

172. C₄

#14556654

semidirect

Subgroups — #22626097

Conjugacy classes of subgroups

Complements