Group #22626110 — subgroups

Subgroups — #22626110

Order 9344.

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

normal subgroups only

This group has 292 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	C₄×C₄⋊#64408	C₂	1	1	yes	maximal, characteristic
3	#17989460	C₂	1	4	yes	maximal
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	(C₂×#64408)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₇₃	—	2	1	no	—
12	#14556591	C₄	1	1	yes	characteristic, complemented
13	#14556590	—	2	1	no	—
14	#14556590	C₄	1	2	yes	complemented
15	#14556589	—	2	1	no	—
16	C₄⋊C₄×D₇₃	—	2	1	no	—
17	C₄⋊C₄×D₇₃	C₂²	1	2	yes	—
18	C₄²×D₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#64408	—	2	2	no	—
21	#11924036	—	2	4	no	—
22	C₂×#10596833	—	2	2	no	—
23	C₄⋊#64408	—	2	2	no	—
24	C₄⋊#64408	Q₈	1	4	yes	complemented
25	C₄×#64408	—	4	2	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	C₂×Dic₁₄₆	—	2	2	no	—
28	Q₈×D₇₃	—	4	2	no	—
29	Q₈×D₇₃	—	4	2	no	—
30	C₂×C₄×D₇₃	—	2	2	no	—
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
33	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₇₃	—	2	1	no	—
35	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
36	#11924007	—	2	2	no	—
37	#11924007	—	2	2	no	—
38	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₇₃	D₄	1	1	yes	characteristic
40	C₄×Dic₇₃	D₄	1	1	yes	characteristic
41	C₂×Q₈×C₇₃	—	2	1	no	—
42	C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
44	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×#64408	—	2	4	no	—
47	C₂×#64408	—	4	2	no	—
48	C₂×#64408	—	4	2	no	—
49	Q₈×C₇₃	—	4	2	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
58	C₄×D₇₃	—	2	2	no	—
59	C₄×D₇₃	—	4	2	no	—
60	C₄×D₇₃	—	4	2	no	—
61	C₄×D₇₃	D₈	1	2	yes	—
62	C₄×D₇₃	SD₁₆	1	2	yes	—
63	C₄×D₇₃	C₂×Q₈	1	2	yes	—
64	C₄×D₇₃	#41	1	2	yes	—
65	Dic₁₄₆	—	4	2	no	—
66	Dic₁₄₆	—	4	4	no	—
67	#10596833	—	4	2	no	—
68	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×D₇₃	C₂²⋊Q₈	1	2	yes	—
70	C₂×D₇₃	C₂×D₈	1	1	yes	characteristic
71	C₂×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×D₇₃	#137	1	2	yes	—
73	#64408	—	4	4	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	4	2	no	—
76	C₄×C₇₃	#102	1	2	yes	—
77	C₄×C₇₃	C₄×Q₈	1	2	yes	—
78	Dic₇₃	—	2	1	no	—
79	Dic₇₃	—	4	2	no	—
80	Dic₇₃	#102	1	2	yes	—
81	Dic₇₃	#124	1	1	yes	characteristic
82	Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₇₃	#391	1	1	yes	characteristic
84	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
85	C₂×C₇₃	#419	1	1	yes	characteristic
86	D₇₃	#474	1	2	yes	—
87	D₇₃	#475	1	2	yes	—
88	#1988	—	73	1	no	maximal, Sylow
89	C₇₃	#1988	1	1	yes	characteristic, complemented, Sylow
90	C₂×C₄⋊Q₈	—	73	1	no	—
91	C₂×C₄⋊C₈	—	73	1	no	—
92	C₄×C₄⋊C₄	—	73	1	no	—
93	#340	—	73	4	no	—
94	C₂²×Q₈	—	146	1	no	—
95	C₂²×C₈	—	73	2	no	—
96	C₄⋊Q₈	—	73	2	no	—
97	C₄⋊Q₈	—	73	1	no	—
98	C₄⋊Q₈	—	146	1	no	—
99	C₄⋊Q₈	—	146	1	no	—
100	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
101	C₂×C₄⋊C₄	—	73	2	no	—
102	C₂×C₄⋊C₄	—	73	2	no	—
103	C₂×C₄⋊C₄	—	146	1	no	—
104	C₂×C₄²	—	73	1	no	—
105	C₂×C₄²	—	146	1	no	—
106	C₄⋊C₈	—	146	1	no	—
107	C₄⋊C₈	—	146	1	no	—
108	#95	—	146	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	292	2	no	—
111	C₂×Q₈	—	292	2	no	—
112	C₂×Q₈	—	146	2	no	—
113	C₂×Q₈	—	146	1	no	—
114	C₂²×C₄	—	73	1	no	—
115	C₂²×C₄	—	73	1	no	—
116	C₂²×C₄	—	73	1	no	—
117	C₂²×C₄	—	146	2	no	—
118	C₂²×C₄	—	146	2	no	—
119	C₂×C₈	—	146	2	no	—
120	C₂×C₈	—	146	4	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	73	4	no	—
123	C₄⋊C₄	—	73	2	no	—
124	C₄⋊C₄	—	146	2	no	—
125	C₄⋊C₄	—	146	2	no	—
126	C₄⋊C₄	—	146	2	no	—
127	C₄⋊C₄	—	146	1	no	—
128	C₄⋊C₄	C₂×#64408	1	2	yes	—
129	C₄²	—	292	2	no	—
130	C₄²	—	73	1	no	—
131	C₄²	—	73	1	no	—
132	C₄²	—	73	1	no	—
133	C₄²	C₂×#64408	1	1	yes	characteristic
134	C₂³	—	73	1	no	—
135	Q₈	—	4	2	no	—
136	Q₈	—	292	4	no	—
137	Q₈	—	292	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	292	2	no	—
140	C₂×C₄	—	292	2	no	—
141	C₂×C₄	—	292	2	no	—
142	C₂×C₄	—	292	2	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	2	no	—
146	C₂×C₄	—	73	2	no	—
147	C₂×C₄	—	73	2	no	—
148	C₂×C₄	—	73	2	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	146	2	no	—
151	C₂×C₄	—	146	4	no	—
152	C₂×C₄	—	146	2	no	—
153	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
154	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
155	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
156	C₈	—	292	2	no	—
157	C₂²	—	73	2	no	—
158	C₂²	—	73	1	no	—
159	C₂²	—	73	1	no	—
160	C₂²	—	73	2	no	—
161	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	4	2	no	—
164	C₄	—	292	4	no	—
165	C₄	—	292	2	no	—
166	C₄	—	73	1	no	—
167	C₄	—	73	1	no	—
168	C₄	—	73	2	no	—
169	C₄	—	146	1	no	—
170	C₄	#14556653	1	2	yes	—
171	C₄	Q₈×#64408	1	2	yes	—
172	C₂	—	73	2	no	—
173	C₂	—	73	2	no	—
174	C₂	(C₂×#64408)⋊Q₈	1	1	yes	characteristic
175	C₂	C₂×#14556653	1	1	yes	characteristic
176	C₂	#17989580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989588

C₂

yes

maximal, characteristic

C₄×C₄⋊#64408

C₂

yes

maximal, characteristic

#17989460

C₂

yes

maximal

C₄⋊Q₈×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924036

C₂²

yes

—

C₂×C₄⋊#64408

C₂²

yes

—

C₂×C₄×#64408

—

#14556658

—

C₄⋊#10596833

—

(C₂×#64408)⋊C₄

—

C₂×Q₈×D₇₃

—

#14556591

C₄

yes

characteristic, complemented

#14556590

—

#14556590

C₄

yes

complemented

#14556589

—

C₄⋊C₄×D₇₃

—

C₄⋊C₄×D₇₃

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

—

—

—

Q₈

yes

complemented

C₄×#64408

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

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

C₂×C₄

yes

—

#11924007

—

#11924007

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

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

—

C₄×D₇₃

D₈

yes

—

C₄×D₇₃

SD₁₆

yes

—

C₄×D₇₃

C₂×Q₈

yes

—

C₄×D₇₃

#41

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

C₂²⋊Q₈

yes

—

C₂×D₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

C₂×SD₁₆

yes

characteristic

C₂×D₇₃

#137

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#102

yes

—

C₄×C₇₃

C₄×Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#102

yes

—

Dic₇₃

#124

yes

characteristic

Dic₇₃

C₄⋊Q₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

—

D₇₃

#475

yes

—

#1988

—

maximal, Sylow

C₇₃

#1988

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₂²×Q₈

—

146

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

100

C₄⋊Q₈

#64408

yes

characteristic, complemented

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

146

—

104

C₂×C₄²

—

105

C₂×C₄²

—

146

—

106

C₄⋊C₈

—

146

—

107

C₄⋊C₈

—

146

—

108

#95

—

146

—

109

C₂×Q₈

—

110

C₂×Q₈

—

292

—

111

C₂×Q₈

—

292

—

112

C₂×Q₈

—

146

—

113

C₂×Q₈

—

146

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

146

—

118

C₂²×C₄

—

146

—

119

C₂×C₈

—

146

—

120

C₂×C₈

—

146

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

146

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

—

146

—

127

C₄⋊C₄

—

146

—

128

C₄⋊C₄

C₂×#64408

yes

—

129

C₄²

—

292

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂×#64408

yes

characteristic

134

C₂³

—

135

Q₈

—

136

Q₈

—

292

—

137

Q₈

—

292

—

138

C₂×C₄

—

139

C₂×C₄

—

292

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

292

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

146

—

151

C₂×C₄

—

146

—

152

C₂×C₄

—

146

—

153

C₂×C₄

C₂²⋊#64408

yes

characteristic

154

C₂×C₄

C₂²⋊#64408

yes

characteristic

155

C₂×C₄

C₂²×#64408

yes

characteristic

156

C₈

—

292

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

C₂×C₂²⋊#64408

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

292

—

165

C₄

—

292

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

146

—

170

C₄

#14556653

yes

—

171

C₄

Q₈×#64408

yes

—

172

C₂

—

173

C₂

—

174

C₂

(C₂×#64408)⋊Q₈

yes

characteristic

175

C₂

C₂×#14556653

yes

characteristic

176

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	#1988
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
12. #14556591	C₄	semidirect
14. #14556590	C₄	semidirect
19. C₄⋊Q₈×C₇₃	C₄	semidirect
24. C₄⋊#64408	Q₈	semidirect
89. C₇₃	#1988	semidirect
100. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

12. #14556591

C₄

semidirect

14. #14556590

C₄

semidirect

19. C₄⋊Q₈×C₇₃

C₄

semidirect

24. C₄⋊#64408

Q₈

semidirect

89. C₇₃

#1988

semidirect

100. C₄⋊Q₈

#64408

semidirect

Subgroups — #22626110

Conjugacy classes of subgroups

Complements