Group #22619749 — subgroups

Subgroups — #22619749

Order 9344.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₄×#11924028	C₂	1	1	yes	maximal, characteristic
2	#17988891	C₂	1	4	yes	maximal, complemented
3	#17988884	C₂	1	4	yes	maximal, complemented
4	#17988627	C₂	1	2	yes	maximal, complemented
5	#17988611	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988586	C₂	1	2	yes	maximal
7	C₂×#125×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924028	C₂²	1	1	yes	characteristic
9	C₂×#11924012	C₂²	1	2	yes	—
10	C₄×Dic₁₄₆	C₂²	1	2	yes	—
11	C₄×D₂₉₂	C₂²	1	2	yes	—
12	C₄×C₂²⋊D₇₃	—	2	2	no	—
13	#14556550	C₂²	1	1	yes	characteristic
14	#14556548	—	2	1	no	—
15	C₄²×D₇₃	—	2	1	no	—
16	#14556524	C₂²	1	4	yes	—
17	#14556522	C₂²	1	4	yes	—
18	C₄⋊#10596831	C₂²	1	4	yes	—
19	#14556514	C₂²	1	4	yes	—
20	#14556511	C₂²	1	4	yes	—
21	#125×C₇₃	—	2	1	no	—
22	#125×C₇₃	—	2	1	no	—
23	#125×C₇₃	C₂²	1	4	yes	—
24	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
27	#11924028	—	2	2	no	—
28	#11924028	Q₈	1	4	yes	—
29	C₂×Dic₁₄₆	C₂³	1	1	yes	characteristic
30	C₂×D₂₉₂	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₇₃	—	2	1	no	—
32	C₂×C₄×D₇₃	—	2	1	no	—
33	C₂×C₄×D₇₃	—	2	2	no	—
34	#11924012	—	2	4	no	—
35	C₂×#10596831	C₂³	1	4	yes	—
36	C₄⋊Dic₇₃	C₂³	1	2	yes	—
37	#11924007	—	2	2	no	—
38	C₂²⋊Dic₇₃	—	2	1	no	—
39	#11924005	—	2	2	no	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₄×Dic₇₃	—	2	2	no	—
42	C₂²×C₄×C₇₃	—	2	2	no	—
43	C₂²×C₄×C₇₃	D₄	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₇₃	—	2	4	no	—
46	C₄⋊C₄×C₇₃	—	2	4	no	—
47	C₄⋊C₄×C₇₃	—	2	2	no	—
48	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
49	C₄²×C₇₃	D₄	1	1	yes	characteristic
50	C₄²×C₇₃	D₄	1	1	yes	characteristic
51	C₄²×C₇₃	C₂³	1	2	yes	—
52	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×D₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	4	no	—
55	C₂×C₄×C₇₃	—	4	2	no	—
56	C₂×C₄×C₇₃	—	4	4	no	—
57	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₇₃	#41	1	2	yes	—
62	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
63	C₂²⋊D₇₃	—	2	4	no	—
64	C₂×Dic₇₃	—	2	1	no	—
65	C₂×Dic₇₃	—	2	2	no	—
66	D₂₉₂	—	2	1	no	—
67	D₂₉₂	C₂×Q₈	1	2	yes	—
68	C₄×D₇₃	—	2	4	no	—
69	C₄×D₇₃	—	4	4	no	—
70	Dic₁₄₆	—	2	1	no	—
71	Dic₁₄₆	C₂×Q₈	1	2	yes	—
72	#10596831	—	2	4	no	—
73	C₂²×C₇₃	—	2	2	no	—
74	C₂²×C₇₃	C₂²⋊Q₈	1	2	yes	—
75	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×D₇₃	—	2	2	no	—
77	C₂×D₇₃	—	4	2	no	—
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	2	2	no	—
80	C₄×C₇₃	—	4	4	no	—
81	C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
82	C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
84	Dic₇₃	—	2	2	no	—
85	Dic₇₃	—	4	2	no	—
86	C₂×C₇₃	—	2	2	no	—
87	C₂×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₇₃	#575	1	2	yes	—
89	D₇₃	—	4	2	no	—
90	#3080	—	73	1	no	maximal, Sylow
91	C₇₃	#3080	1	1	yes	characteristic, complemented, Sylow
92	C₂×#125	D₇₃	1	1	yes	characteristic, complemented
93	C₄×#41	—	73	1	no	—
94	#479	—	73	4	no	—
95	#478	—	73	4	no	—
96	#428	—	73	2	no	—
97	#423	—	73	1	no	—
98	#417	—	73	2	no	—
99	C₂×#41	—	73	1	no	—
100	C₂×M₁₆	—	73	2	no	—
101	#125	—	2	1	no	—
102	#125	—	2	1	no	—
103	#125	C₂×D₇₃	1	4	yes	—
104	C₄×Q₈	—	73	2	no	—
105	C₄×D₄	—	73	2	no	—
106	C₄×D₄	—	146	2	no	—
107	#117	—	73	1	no	—
108	#117	—	146	1	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
111	C₂×C₄²	—	146	1	no	—
112	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
113	#107	—	73	4	no	—
114	#106	—	73	4	no	—
115	C₄⋊C₈	—	73	4	no	—
116	#103	—	73	4	no	—
117	#102	—	73	4	no	—
118	#41	—	73	4	no	—
119	#41	—	146	2	no	—
120	C₂×Q₈	—	73	1	no	—
121	C₂×D₄	—	73	1	no	—
122	C₂×D₄	—	146	1	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	146	2	no	—
125	C₂²×C₄	—	146	1	no	—
126	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
127	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
128	M₁₆	—	146	4	no	—
129	C₂×C₈	—	73	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	73	2	no	—
134	C₄⋊C₄	—	146	2	no	—
135	C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
136	C₂²⋊C₄	—	146	1	no	—
137	C₂²⋊C₄	—	146	2	no	—
138	C₄²	—	146	1	no	—
139	C₄²	—	146	2	no	—
140	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
141	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
142	C₄²	C₂²×D₇₃	1	2	yes	—
143	C₂³	—	146	1	no	—
144	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
145	Q₈	—	73	2	no	—
146	Q₈	—	146	1	no	—
147	D₄	—	73	2	no	—
148	D₄	—	146	4	no	—
149	D₄	—	146	1	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	292	4	no	—
154	C₂×C₄	—	146	2	no	—
155	C₂×C₄	—	146	1	no	—
156	C₂×C₄	—	146	4	no	—
157	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
158	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
159	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
160	C₂×C₄	Q₈×D₇₃	1	2	yes	—
161	C₂×C₄	#11924029	1	2	yes	—
162	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
163	C₈	—	146	4	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	292	2	no	—
166	C₂²	—	146	2	no	—
167	C₂²	#14556567	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	—	292	2	no	—
173	C₄	—	146	2	no	—
174	C₄	#14556567	1	2	yes	—
175	C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
176	C₄	C₂×#11924029	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	—	292	2	no	—
179	C₂	C₂×#14556567	1	1	yes	characteristic
180	C₂	#17989385	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11924028

C₂

yes

maximal, characteristic

#17988891

C₂

yes

maximal, complemented

#17988884

C₂

yes

maximal, complemented

#17988627

C₂

yes

maximal, complemented

#17988611

C₂

yes

maximal, characteristic, complemented

#17988586

C₂

yes

maximal

C₂×#125×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

—

C₄×Dic₁₄₆

C₂²

yes

—

C₄×D₂₉₂

C₂²

yes

—

C₄×C₂²⋊D₇₃

—

#14556550

C₂²

yes

characteristic

#14556548

—

C₄²×D₇₃

—

#14556524

C₂²

yes

—

#14556522

C₂²

yes

—

C₄⋊#10596831

C₂²

yes

—

#14556514

C₂²

yes

—

#14556511

C₂²

yes

—

#125×C₇₃

—

#125×C₇₃

—

#125×C₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#11924028

—

#11924028

Q₈

yes

—

C₂×Dic₁₄₆

C₂³

yes

characteristic

C₂×D₂₉₂

C₂³

yes

characteristic

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

#11924012

—

C₂×#10596831

C₂³

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

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

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

C₂×Q₈

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

C₂×Q₈

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊Q₈

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

C₂²×Q₈

yes

characteristic

C₄×C₇₃

C₂×#41

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₇₃

#575

yes

—

D₇₃

—

#3080

—

maximal, Sylow

C₇₃

#3080

yes

characteristic, complemented, Sylow

C₂×#125

D₇₃

yes

characteristic, complemented

C₄×#41

—

—

—

—

—

—

C₂×#41

—

100

C₂×M₁₆

—

101

#125

—

102

#125

—

103

#125

C₂×D₇₃

yes

—

104

C₄×Q₈

—

105

C₄×D₄

—

106

C₄×D₄

—

146

—

107

#117

—

108

#117

—

146

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

111

C₂×C₄²

—

146

—

112

C₂×C₄²

C₂×D₇₃

yes

characteristic

113

#107

—

114

#106

—

115

C₄⋊C₈

—

116

#103

—

117

#102

—

118

#41

—

119

#41

—

146

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

146

—

123

C₂²×C₄

—

124

C₂²×C₄

—

146

—

125

C₂²×C₄

—

146

—

126

C₂²×C₄

C₂²⋊D₇₃

yes

—

127

C₂²×C₄

C₂²×D₇₃

yes

characteristic

128

M₁₆

—

146

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

146

—

135

C₄⋊C₄

C₂²×D₇₃

yes

—

136

C₂²⋊C₄

—

146

—

137

C₂²⋊C₄

—

146

—

138

C₄²

—

146

—

139

C₄²

—

146

—

140

C₄²

C₂²⋊D₇₃

yes

characteristic

141

C₄²

C₂²⋊D₇₃

yes

characteristic

142

C₄²

C₂²×D₇₃

yes

—

143

C₂³

—

146

—

144

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

145

Q₈

—

146

Q₈

—

146

—

147

D₄

—

148

D₄

—

146

—

149

D₄

—

146

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

292

—

154

C₂×C₄

—

146

—

155

C₂×C₄

—

146

—

156

C₂×C₄

—

146

—

157

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

158

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

159

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

160

C₂×C₄

Q₈×D₇₃

yes

—

161

C₂×C₄

#11924029

yes

—

162

C₂×C₄

C₂³×D₇₃

yes

characteristic

163

C₈

—

146

—

164

C₂²

—

165

C₂²

—

292

—

166

C₂²

—

146

—

167

C₂²

#14556567

yes

—

168

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

292

—

173

C₄

—

146

—

174

C₄

#14556567

yes

—

175

C₄

C₂×Q₈×D₇₃

yes

characteristic

176

C₄

C₂×#11924029

yes

characteristic

177

C₂

—

178

C₂

—

292

—

179

C₂

C₂×#14556567

yes

characteristic

180

C₂

#17989385

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	C₂×#125×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	C₂×#125	D₇₃
hypocenter	C₇₃	#3080
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

C₂×#125×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #17988891	C₂	semidirect
3. #17988884	C₂	semidirect
4. #17988627	C₂	semidirect
5. #17988611	C₂	semidirect
7. C₂×#125×C₇₃	C₂	semidirect
91. C₇₃	#3080	semidirect
92. C₂×#125	D₇₃	semidirect

H ≅ G/N

type

2. #17988891

C₂

semidirect

3. #17988884

C₂

semidirect

4. #17988627

C₂

semidirect

5. #17988611

C₂

semidirect

7. C₂×#125×C₇₃

C₂

semidirect

91. C₇₃

#3080

semidirect

92. C₂×#125

D₇₃

semidirect

Subgroups — #22619749

Conjugacy classes of subgroups

Complements