Group #22615320 — subgroups

Subgroups — #22615320

Order 9344.

468 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 169 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×C₄×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	#17988517	C₂	1	2	yes	maximal
3	#17988507	C₂	1	2	yes	maximal
4	#17988483	C₂	1	2	yes	maximal
5	#17988483	C₂	1	2	yes	maximal
6	#17988482	C₂	1	4	yes	maximal
7	C₄×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₄₆	—	2	4	no	—
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
13	C₂×#11924007	C₂²	1	2	yes	—
14	C₂×#11924007	C₂²	1	4	yes	—
15	C₂×#11924007	C₂²	1	4	yes	—
16	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
18	#14556498	C₂²	1	2	yes	—
19	#14556498	C₂²	1	4	yes	—
20	#14556498	C₂²	1	4	yes	—
21	#125×C₇₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×Dic₁₄₆	—	2	2	no	—
28	C₂×Dic₁₄₆	—	2	4	no	—
29	C₂²×Dic₇₃	C₂³	1	2	yes	—
30	C₂²×Dic₇₃	C₂³	1	4	yes	—
31	C₂²×Dic₇₃	C₂³	1	2	yes	—
32	C₄⋊Dic₇₃	—	2	2	no	—
33	C₄⋊Dic₇₃	Q₈	1	8	yes	complemented
34	#11924007	—	2	4	no	—
35	#11924007	—	2	8	no	—
36	#11924007	—	4	4	no	—
37	C₄×Dic₇₃	—	2	4	no	—
38	C₄×Dic₇₃	—	4	2	no	—
39	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₇₃	—	2	4	no	—
45	C₄⋊C₄×C₇₃	—	2	4	no	—
46	C₄⋊C₄×C₇₃	—	2	2	no	—
47	C₄⋊C₄×C₇₃	—	2	2	no	—
48	C₄²×C₇₃	D₄	1	4	yes	—
49	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	4	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	2	no	—
58	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₇₃	C₂×Q₈	1	4	yes	—
62	C₂×Dic₇₃	—	2	2	no	—
63	C₂×Dic₇₃	—	2	4	no	—
64	C₂×Dic₇₃	—	2	2	no	—
65	C₂×Dic₇₃	—	2	4	no	—
66	C₂×Dic₇₃	—	2	2	no	—
67	C₂×Dic₇₃	—	2	2	no	—
68	C₂×Dic₇₃	—	2	2	no	—
69	C₂×Dic₇₃	C₂×Q₈	1	8	yes	—
70	C₂×Dic₇₃	#41	1	4	yes	—
71	Dic₁₄₆	—	2	8	no	—
72	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₇₃	C₂²×Q₈	1	2	yes	—
74	C₂²×C₇₃	C₂×#41	1	2	yes	—
75	C₂²×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂²×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
77	C₄×C₇₃	—	2	2	no	—
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	4	2	no	—
80	C₄×C₇₃	—	4	2	no	—
81	C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
82	Dic₇₃	—	2	4	no	—
83	Dic₇₃	—	2	8	no	—
84	Dic₇₃	—	4	2	no	—
85	C₂×C₇₃	#525	1	1	yes	characteristic
86	C₂×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₇₃	#537	1	1	yes	characteristic
88	C₂×C₇₃	#554	1	2	yes	—
89	C₂×C₇₃	#557	1	2	yes	—
90	#2584	—	73	1	no	maximal, Sylow
91	C₇₃	#2584	1	1	yes	characteristic, complemented, Sylow
92	C₂×#125	D₇₃	1	1	yes	characteristic
93	C₂×C₄×Q₈	—	73	1	no	—
94	#398	—	73	2	no	—
95	#395	—	73	2	no	—
96	#389	—	73	4	no	—
97	#389	—	73	2	no	—
98	#389	—	73	2	no	—
99	C₄×C₄⋊C₄	—	73	1	no	—
100	C₂²×Q₈	—	73	1	no	—
101	#125	—	2	4	no	—
102	C₄×Q₈	—	146	4	no	—
103	C₂×C₄⋊C₄	—	73	4	no	—
104	C₂×C₄⋊C₄	—	73	2	no	—
105	C₂×C₄⋊C₄	—	73	2	no	—
106	C₂×C₄⋊C₄	—	73	1	no	—
107	C₂×C₄⋊C₄	—	73	4	no	—
108	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
109	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
112	C₂×C₄²	—	73	2	no	—
113	C₂×C₄²	—	73	2	no	—
114	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
115	#95	—	73	2	no	—
116	#95	—	73	4	no	—
117	#95	—	73	4	no	—
118	C₂×Q₈	—	146	4	no	—
119	C₂×Q₈	—	146	2	no	—
120	C₂²×C₄	—	73	4	no	—
121	C₂²×C₄	—	73	2	no	—
122	C₂²×C₄	—	73	2	no	—
123	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
124	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
125	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	292	4	no	—
133	C₄⋊C₄	—	73	8	no	—
134	C₄⋊C₄	—	146	8	no	—
135	C₄⋊C₄	—	146	4	no	—
136	C₄⋊C₄	—	146	2	no	—
137	C₄²	—	292	2	no	—
138	C₄²	—	146	4	no	—
139	C₄²	C₂²⋊D₇₃	1	4	yes	—
140	C₂³	C₂³×D₇₃	1	1	yes	characteristic
141	Q₈	—	146	8	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	73	8	no	—
151	C₂×C₄	—	73	4	no	—
152	C₂×C₄	—	146	2	no	—
153	C₂×C₄	—	146	2	no	—
154	C₂×C₄	—	146	2	no	—
155	C₂×C₄	—	146	2	no	—
156	C₂×C₄	—	146	4	no	—
157	C₂×C₄	—	146	2	no	—
158	C₂×C₄	—	146	4	no	—
159	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
162	C₂×C₄	Q₈×D₇₃	1	4	yes	—
163	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
164	C₂²	C₂×Q₈×D₇₃	1	2	yes	—
165	C₂²	#41×D₇₃	1	2	yes	—
166	C₂²	#14556637	1	1	yes	characteristic
167	C₂²	#14556638	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	2	no	—
172	C₄	—	292	2	no	—
173	C₄	—	146	8	no	—
174	C₄	—	146	4	no	—
175	C₄	#14556589	1	4	yes	—
176	C₂	#17989094	1	1	yes	characteristic
177	C₂	C₂×#14556589	1	1	yes	characteristic
178	C₂	#17989165	1	1	yes	characteristic
179	C₂	#17989293	1	2	yes	—
180	C₂	#17989314	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₄₆

C₂

yes

maximal, characteristic

#17988517

C₂

yes

maximal

#17988507

C₂

yes

maximal

#17988483

C₂

yes

maximal

#17988483

C₂

yes

maximal

#17988482

C₂

yes

maximal

C₄×C₄⋊Dic₇₃

C₂

yes

maximal, characteristic

C₂×#125×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924007

C₂²

yes

—

C₂×#11924007

C₂²

yes

—

C₂×#11924007

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

#125×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

Q₈

yes

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂⁴

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×Q₈

yes

—

C₂×Dic₇₃

#41

yes

—

Dic₁₄₆

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂²×C₇₃

C₂²×Q₈

yes

—

C₂²×C₇₃

C₂×#41

yes

—

C₂²×C₇₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₇₃

2₋¹⁺⁴

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄⋊Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#525

yes

characteristic

C₂×C₇₃

C₂×C₄⋊Q₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#557

yes

—

#2584

—

maximal, Sylow

C₇₃

#2584

yes

characteristic, complemented, Sylow

C₂×#125

D₇₃

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

—

C₄×C₄⋊C₄

—

100

C₂²×Q₈

—

101

#125

—

102

C₄×Q₈

—

146

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

109

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

110

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

111

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂×D₇₃

yes

characteristic

115

#95

—

116

#95

—

117

#95

—

118

C₂×Q₈

—

146

—

119

C₂×Q₈

—

146

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²×D₇₃

yes

—

124

C₂²×C₄

C₂²×D₇₃

yes

—

125

C₂²×C₄

C₂²×D₇₃

yes

characteristic

126

C₂²×C₄

C₂²×D₇₃

yes

characteristic

127

C₂²×C₄

C₂²×D₇₃

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

292

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

146

—

135

C₄⋊C₄

—

146

—

136

C₄⋊C₄

—

146

—

137

C₄²

—

292

—

138

C₄²

—

146

—

139

C₄²

C₂²⋊D₇₃

yes

—

140

C₂³

C₂³×D₇₃

yes

characteristic

141

Q₈

—

146

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

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

—

146

—

156

C₂×C₄

—

146

—

157

C₂×C₄

—

146

—

158

C₂×C₄

—

146

—

159

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

160

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

161

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

162

C₂×C₄

Q₈×D₇₃

yes

—

163

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

164

C₂²

C₂×Q₈×D₇₃

yes

—

165

C₂²

#41×D₇₃

yes

—

166

C₂²

#14556637

yes

characteristic

167

C₂²

#14556638

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

292

—

173

C₄

—

146

—

174

C₄

—

146

—

175

C₄

#14556589

yes

—

176

C₂

#17989094

yes

characteristic

177

C₂

C₂×#14556589

yes

characteristic

178

C₂

#17989165

yes

characteristic

179

C₂

#17989293

yes

—

180

C₂

#17989314

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	C₂×#125×C₇₃	C₂
hypercenter	C₂×#125	D₇₃
hypocenter	C₇₃	#2584
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

C₂×#125×C₇₃

C₂

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
33. C₄⋊Dic₇₃	Q₈	semidirect
91. C₇₃	#2584	semidirect

H ≅ G/N

type

33. C₄⋊Dic₇₃

Q₈

semidirect

91. C₇₃

#2584

semidirect

Subgroups — #22615320

Conjugacy classes of subgroups

Complements