Group #22625253 — subgroups

Subgroups — #22625253

Order 9344.

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

normal subgroups only

This group has 218 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	#17989603	C₂	1	1	yes	maximal, characteristic
2	C₄×#11924037	C₂	1	1	yes	maximal, characteristic
3	#17989437	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#10596833	C₂	1	2	yes	maximal, complemented
5	#17989139	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924037	—	2	1	no	—
7	C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂²⋊#10596833	—	2	1	no	—
9	#14556646	—	2	1	no	—
10	C₄×#10596833	—	2	1	no	—
11	C₄×#10596833	C₂²	1	2	yes	—
12	#14556590	C₄	1	1	yes	characteristic
13	#14556589	C₄	1	1	yes	characteristic
14	#14556581	—	2	1	no	—
15	#14556570	C₄	1	1	yes	characteristic
16	#14556568	—	2	1	no	—
17	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	#14556556	—	2	1	no	—
19	#14556556	C₂²	1	1	yes	characteristic
20	#14556547	—	2	1	no	—
21	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
22	C₂²⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
23	#11924037	—	2	4	no	—
24	C₂×#10596833	—	2	2	no	—
25	C₂×#10596833	—	2	2	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
29	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	#11924007	—	2	1	no	—
33	#11924007	—	2	1	no	—
34	#11924007	—	2	1	no	—
35	#11924007	—	2	1	no	—
36	#11924007	—	2	1	no	—
37	C₂²⋊Dic₇₃	—	2	1	no	—
38	C₄×Dic₇₃	—	2	1	no	—
39	C₄×Dic₇₃	—	2	1	no	—
40	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
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	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₇₃	—	2	1	no	—
47	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₇₃	—	2	1	no	—
49	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	Q₈×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	1	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×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	—	2	2	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	2	1	no	—
60	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
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	Dic₁₄₆	—	2	2	no	—
65	Dic₁₄₆	—	4	2	no	—
66	#10596833	—	2	4	no	—
67	#10596833	—	4	2	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	—	2	1	no	—
71	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	4	1	no	—
74	C₄×C₇₃	—	4	1	no	—
75	C₄×C₇₃	#131	1	2	yes	—
76	Dic₇₃	—	2	1	no	—
77	Dic₇₃	—	2	1	no	—
78	Dic₇₃	—	4	1	no	—
79	Dic₇₃	—	4	1	no	—
80	Dic₇₃	#104	1	2	yes	—
81	Dic₇₃	#137	1	2	yes	—
82	C₂×C₇₃	—	4	1	no	—
83	C₂×C₇₃	#408	1	1	yes	characteristic
84	C₂×C₇₃	#419	1	1	yes	characteristic
85	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
86	#1485	—	73	1	no	maximal, Sylow
87	C₇₃	#1485	1	1	yes	characteristic, complemented, Sylow
88	#533	—	73	1	no	—
89	#431	—	73	1	no	—
90	C₄×M₁₆	—	73	1	no	—
91	Q₈⋊C₈	—	73	2	no	—
92	Q₈⋊C₈	—	73	2	no	—
93	C₂×M₁₆	—	146	1	no	—
94	C₄⋊Q₈	—	73	1	no	—
95	C₄⋊Q₈	—	73	1	no	—
96	#125	—	146	1	no	—
97	C₂²⋊Q₈	—	73	1	no	—
98	C₂²⋊Q₈	—	146	1	no	—
99	C₂²⋊Q₈	#64408	1	1	yes	characteristic
100	C₄×Q₈	—	73	1	no	—
101	C₄×Q₈	—	73	1	no	—
102	C₄×Q₈	—	146	1	no	—
103	#117	—	146	1	no	—
104	C₂×C₄²	—	73	1	no	—
105	C₄⋊C₈	—	73	2	no	—
106	C₂²⋊C₈	—	146	1	no	—
107	#97	—	146	1	no	—
108	C₄×C₈	—	73	2	no	—
109	C₄×C₈	—	146	1	no	—
110	C₂×Q₈	—	73	1	no	—
111	C₂×Q₈	—	146	1	no	—
112	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
113	C₂²×C₄	—	73	1	no	—
114	C₂²×C₄	—	73	1	no	—
115	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
116	M₁₆	—	146	4	no	—
117	C₂×C₈	—	146	2	no	—
118	C₂×C₈	—	146	2	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	73	1	no	—
121	C₄⋊C₄	—	146	1	no	—
122	C₄⋊C₄	—	146	1	no	—
123	C₄⋊C₄	—	146	1	no	—
124	C₄⋊C₄	—	146	1	no	—
125	C₄⋊C₄	—	146	1	no	—
126	C₄⋊C₄	—	146	1	no	—
127	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
128	C₂²⋊C₄	—	2	1	no	—
129	C₂²⋊C₄	—	146	1	no	—
130	C₄²	—	73	1	no	—
131	C₄²	—	73	1	no	—
132	C₄²	—	73	1	no	—
133	C₄²	—	73	1	no	—
134	C₄²	—	146	1	no	—
135	C₄²	—	146	1	no	—
136	C₂³	C₂²⋊#64408	1	1	yes	characteristic
137	Q₈	—	2	2	no	—
138	Q₈	—	292	2	no	—
139	Q₈	—	146	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	146	1	no	—
148	C₂×C₄	—	146	1	no	—
149	C₂×C₄	—	146	2	no	—
150	C₂×C₄	—	146	2	no	—
151	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
152	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
153	C₈	—	292	2	no	—
154	C₈	—	146	4	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	292	1	no	—
163	C₄	—	292	1	no	—
164	C₄	—	73	2	no	—
165	C₄	—	73	2	no	—
166	C₄	—	146	1	no	—
167	C₄	—	146	1	no	—
168	C₄	#14556678	1	2	yes	—
169	C₂	—	4	1	no	—
170	C₂	#17989560	1	1	yes	characteristic
171	C₂	#17989580	1	1	yes	characteristic
172	C₂	D₇₃⋊#104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989603

C₂

yes

maximal, characteristic

C₄×#11924037

C₂

yes

maximal, characteristic

#17989437

C₂

yes

maximal, complemented

Q₈⋊#10596833

C₂

yes

maximal, complemented

#17989139

C₂

yes

maximal, characteristic

C₂×#11924037

—

C₄⋊#10596833

C₂²

yes

—

—

—

—

C₂²

yes

—

#14556590

C₄

yes

characteristic

#14556589

C₄

yes

characteristic

#14556581

—

#14556570

C₄

yes

characteristic

#14556568

—

Q₈×Dic₇₃

C₂²

yes

characteristic

#14556556

—

#14556556

C₂²

yes

characteristic

#14556547

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₃

C₄

yes

—

—

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

—

—

—

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#131

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#104

yes

—

Dic₇₃

#137

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#533

—

#431

—

C₄×M₁₆

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

146

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

146

—

C₂²⋊Q₈

#64408

yes

characteristic

100

C₄×Q₈

—

101

C₄×Q₈

—

102

C₄×Q₈

—

146

—

103

#117

—

146

—

104

C₂×C₄²

—

105

C₄⋊C₈

—

106

C₂²⋊C₈

—

146

—

107

#97

—

146

—

108

C₄×C₈

—

109

C₄×C₈

—

146

—

110

C₂×Q₈

—

111

C₂×Q₈

—

146

—

112

C₂×Q₈

C₂×#64408

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂×#64408

yes

characteristic

116

M₁₆

—

146

—

117

C₂×C₈

—

146

—

118

C₂×C₈

—

146

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

146

—

122

C₄⋊C₄

—

146

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

—

146

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

—

146

—

127

C₄⋊C₄

C₂×#64408

yes

characteristic

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

146

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

146

—

135

C₄²

—

146

—

136

C₂³

C₂²⋊#64408

yes

characteristic

137

Q₈

—

138

Q₈

—

292

—

139

Q₈

—

146

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

146

—

148

C₂×C₄

—

146

—

149

C₂×C₄

—

146

—

150

C₂×C₄

—

146

—

151

C₂×C₄

C₂²⋊#64408

yes

characteristic

152

C₂×C₄

C₂²×#64408

yes

characteristic

153

C₈

—

292

—

154

C₈

—

146

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₂²⋊#64408

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

292

—

163

C₄

—

292

—

164

C₄

—

165

C₄

—

166

C₄

—

146

—

167

C₄

—

146

—

168

C₄

#14556678

yes

—

169

C₂

—

170

C₂

#17989560

yes

characteristic

171

C₂

#17989580

yes

characteristic

172

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	C₂²⋊Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₂²⋊Q₈	#64408
hypocenter	C₇₃	#1485
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂²⋊Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₂²⋊Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

N	H ≅ G/N	type
3. #17989437	C₂	semidirect
4. Q₈⋊#10596833	C₂	semidirect
87. C₇₃	#1485	semidirect

H ≅ G/N

type

3. #17989437

C₂

semidirect

4. Q₈⋊#10596833

C₂

semidirect

87. C₇₃

#1485

semidirect

Subgroups — #22625253

Conjugacy classes of subgroups

Complements