Group #22625810 — subgroups

Subgroups — #22625810

Order 9344.

292 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 82 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×#14556661	C₂	1	2	yes	maximal, complemented
2	#17989558	C₂	1	2	yes	maximal
3	#17989463	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924038	C₂²	1	4	yes	—
6	#14556661	—	2	4	no	—
7	#14556648	—	2	2	no	—
8	C₂²⋊#10596833	—	2	2	no	—
9	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
10	C₂²×#10596831	C₄	1	1	yes	characteristic
11	C₂×C₈×D₇₃	—	2	4	no	—
12	C₂×C₈×D₇₃	C₄	1	1	yes	characteristic
13	C₂×C₈×D₇₃	C₄	1	1	yes	characteristic
14	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	C₄	1	1	yes	characteristic
17	#11924038	—	2	4	no	—
18	#11924038	—	2	4	no	—
19	C₂×#10596833	—	2	4	no	—
20	C₂³×D₇₃	Q₈	1	1	yes	characteristic
21	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
22	C₂×C₄×D₇₃	—	2	2	no	—
23	C₂×C₄×D₇₃	—	2	2	no	—
24	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
26	C₂×C₄×D₇₃	Q₈	1	1	yes	characteristic
27	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
28	C₂×#10596831	—	2	2	no	—
29	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₈×D₇₃	—	2	2	no	—
32	C₈×D₇₃	—	2	2	no	—
33	C₈×D₇₃	—	2	2	no	—
34	C₈×D₇₃	D₄	1	4	yes	—
35	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×D₇₃	—	2	2	no	—
41	C₂²×D₇₃	—	2	2	no	—
42	C₂²×D₇₃	—	2	2	no	—
43	C₂²×D₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×D₇₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₇₃	—	2	2	no	—
50	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
52	C₄×D₇₃	—	2	2	no	—
53	C₄×D₇₃	—	2	1	no	—
54	C₄×D₇₃	—	2	1	no	—
55	C₄×D₇₃	—	2	2	no	—
56	C₄×D₇₃	C₂×D₄	1	2	yes	—
57	C₄×D₇₃	#41	1	2	yes	—
58	#10596833	—	4	4	no	—
59	C₈×C₇₃	—	2	1	no	—
60	C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
61	#10596831	—	2	1	no	—
62	#10596831	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₇₃	—	2	1	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	#108	1	2	yes	complemented
66	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×D₇₃	—	2	8	no	—
68	C₂×D₇₃	—	2	1	no	—
69	C₂×D₇₃	—	2	1	no	—
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	#108	1	2	yes	complemented
73	C₂×D₇₃	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₇₃	#117	1	1	yes	characteristic
77	Dic₇₃	—	2	1	no	—
78	Dic₇₃	#121	1	2	yes	—
79	C₂×C₇₃	—	2	2	no	—
80	C₂×C₇₃	#380	1	1	yes	characteristic
81	C₂×C₇₃	C₂×#108	1	1	yes	characteristic
82	C₂×C₇₃	C₂×#108	1	1	yes	characteristic
83	D₇₃	—	2	2	no	—
84	D₇₃	—	2	2	no	—
85	#1810	—	73	1	no	maximal, Sylow
86	C₇₃	#1810	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₈	—	73	1	no	—
88	C₂×#108	—	73	2	no	—
89	#407	—	73	2	no	—
90	#341	—	73	2	no	—
91	C₂³×C₄	—	73	1	no	—
92	C₂×M₁₆	—	73	4	no	—
93	C₂²×C₈	—	73	1	no	—
94	C₂²×C₈	—	73	1	no	—
95	C₂²×C₈	—	73	1	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	—	73	1	no	—
98	C₂²×C₈	—	146	4	no	—
99	C₂²×C₈	#64408	1	1	yes	characteristic
100	#108	—	146	4	no	—
101	C₂²⋊C₈	—	146	2	no	—
102	C₂²⋊C₈	—	146	2	no	—
103	C₂⁴	—	73	1	no	—
104	C₂²×C₄	—	73	1	no	—
105	C₂²×C₄	—	73	1	no	—
106	C₂²×C₄	—	73	1	no	—
107	C₂²×C₄	—	73	1	no	—
108	C₂²×C₄	—	73	1	no	—
109	C₂²×C₄	—	146	2	no	—
110	C₂²×C₄	—	146	2	no	—
111	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
112	M₁₆	—	146	4	no	—
113	M₁₆	—	146	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	73	1	no	—
116	C₂×C₈	—	73	4	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	146	2	no	—
119	C₂×C₈	—	146	2	no	—
120	C₂×C₈	—	146	4	no	—
121	C₂×C₈	—	146	2	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	C₂×#64408	1	1	yes	characteristic
124	C₂×C₈	C₂×#64408	1	1	yes	characteristic
125	C₂³	—	73	1	no	—
126	C₂³	—	73	1	no	—
127	C₂³	—	146	2	no	—
128	C₂³	—	146	2	no	—
129	C₂³	—	146	2	no	—
130	C₂³	C₄⋊#64408	1	1	yes	characteristic
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	73	2	no	—
135	C₂×C₄	—	73	2	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	—	146	2	no	—
138	C₂×C₄	—	146	2	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	—	146	1	no	—
141	C₂×C₄	—	146	1	no	—
142	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
143	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	292	4	no	—
146	C₈	—	73	2	no	—
147	C₈	—	146	1	no	—
148	C₈	C₂²⋊#64408	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	73	2	no	—
152	C₂²	—	73	2	no	—
153	C₂²	—	146	8	no	—
154	C₂²	—	146	1	no	—
155	C₂²	—	146	1	no	—
156	C₂²	—	146	1	no	—
157	C₂²	—	146	1	no	—
158	C₂²	#14556661	1	2	yes	complemented
159	C₂²	C₂×C₄⋊#64408	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	73	2	no	—
162	C₄	—	146	1	no	—
163	C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
164	C₄	#14556670	1	1	yes	characteristic
165	C₂	—	2	2	no	—
166	C₂	—	146	2	no	—
167	C₂	—	146	2	no	—
168	C₂	#17989529	1	1	yes	characteristic
169	C₂	C₂×#14556661	1	1	yes	characteristic
170	C₂	D₇₃⋊#108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556661

C₂

yes

maximal, complemented

#17989558

C₂

yes

maximal

#17989463

C₂

yes

maximal, complemented

C₂²×C₈×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924038

C₂²

yes

—

#14556661

—

#14556648

—

C₂²⋊#10596833

—

C₂²×C₄×D₇₃

C₂²

yes

characteristic

C₂²×#10596831

C₄

yes

characteristic

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₄

yes

characteristic

C₂×C₈×D₇₃

C₄

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₄

yes

—

—

—

C₂³×D₇₃

Q₈

yes

characteristic

C₂²×Dic₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

Q₈

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₄⋊C₄

yes

characteristic

C₂²×D₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₄

yes

—

C₄×D₇₃

#41

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊C₄

yes

—

#10596831

—

#10596831

C₂²⋊C₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#108

yes

complemented

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#108

yes

complemented

C₂×D₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

#117

yes

characteristic

Dic₇₃

—

Dic₇₃

#121

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#1810

—

maximal, Sylow

C₇₃

#1810

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×#108

—

#407

—

#341

—

C₂³×C₄

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂²×C₈

#64408

yes

characteristic

100

#108

—

146

—

101

C₂²⋊C₈

—

146

—

102

C₂²⋊C₈

—

146

—

103

C₂⁴

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

146

—

110

C₂²×C₄

—

146

—

111

C₂²×C₄

C₂×#64408

yes

characteristic

112

M₁₆

—

146

—

113

M₁₆

—

146

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

146

—

119

C₂×C₈

—

146

—

120

C₂×C₈

—

146

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₂×C₈

C₂×#64408

yes

characteristic

124

C₂×C₈

C₂×#64408

yes

characteristic

125

C₂³

—

126

C₂³

—

127

C₂³

—

146

—

128

C₂³

—

146

—

129

C₂³

—

146

—

130

C₂³

C₄⋊#64408

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

146

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

—

146

—

142

C₂×C₄

C₄⋊#64408

yes

characteristic

143

C₂×C₄

C₂²×#64408

yes

characteristic

144

C₈

—

145

C₈

—

292

—

146

C₈

—

147

C₈

—

146

—

148

C₈

C₂²⋊#64408

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

146

—

154

C₂²

—

146

—

155

C₂²

—

146

—

156

C₂²

—

146

—

157

C₂²

—

146

—

158

C₂²

#14556661

yes

complemented

159

C₂²

C₂×C₄⋊#64408

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

146

—

163

C₄

C₂×C₂²⋊#64408

yes

characteristic

164

C₄

#14556670

yes

characteristic

165

C₂

—

166

C₂

—

146

—

167

C₂

—

146

—

168

C₂

#17989529

yes

characteristic

169

C₂

C₂×#14556661

yes

characteristic

170

C₂

D₇₃⋊#108

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂²×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₄
hypercenter	C₂²×C₈	#64408
hypocenter	C₇₃	#1810
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂²×C₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄⋊C₄

C₂²×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
1. C₂×#14556661	C₂	semidirect
3. #17989463	C₂	semidirect
65. C₂²×C₇₃	#108	semidirect
72. C₂×D₇₃	#108	semidirect
86. C₇₃	#1810	semidirect
158. C₂²	#14556661	semidirect

H ≅ G/N

type

1. C₂×#14556661

C₂

semidirect

3. #17989463

C₂

semidirect

65. C₂²×C₇₃

#108

semidirect

72. C₂×D₇₃

#108

semidirect

86. C₇₃

#1810

semidirect

158. C₂²

#14556661

semidirect

Subgroups — #22625810

Conjugacy classes of subgroups

Complements