Group #22625816 — subgroups

Subgroups — #22625816

Order 9344.

252 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 72 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	D₇₃⋊#108	C₂	1	2	yes	maximal, complemented
2	#17989573	C₂	1	2	yes	maximal
3	#17989456	C₂	1	2	yes	maximal, complemented
4	C₂×#14556599	C₂	1	1	yes	maximal, characteristic
5	D₇₃⋊M₁₆	C₂²	1	4	yes	—
6	#14556662	—	2	2	no	—
7	#14556661	—	2	2	no	—
8	#14556652	—	2	2	no	—
9	#14556651	—	2	2	no	—
10	C₂×#11924028	C₂²	1	1	yes	characteristic
11	#14556599	—	2	2	no	—
12	#14556599	—	2	2	no	—
13	C₂×#11924011	C₄	1	1	yes	characteristic
14	C₂×#11924011	C₂²	1	1	yes	characteristic
15	C₂²×#10596831	C₄	1	1	yes	characteristic
16	C₂×C₈×D₇₃	C₄	1	1	yes	characteristic
17	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₇₃	C₄	1	1	yes	characteristic
19	#11924038	—	2	4	no	—
20	#11924037	—	2	4	no	—
21	#11924036	—	2	4	no	—
22	#11924028	—	2	2	no	—
23	#11924028	—	2	2	no	—
24	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
25	C₂×D₂₉₂	Q₈	1	1	yes	characteristic
26	C₂×C₂²⋊D₇₃	D₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₇₃	Q₈	1	1	yes	characteristic
28	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
30	#11924011	—	2	2	no	—
31	#11924011	—	2	2	no	—
32	C₂×#10596831	—	2	1	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
36	C₈×D₇₃	—	2	1	no	—
37	C₈×D₇₃	—	2	1	no	—
38	C₈×D₇₃	D₄	1	2	yes	—
39	C₈×D₇₃	D₄	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
41	M₁₆×C₇₃	—	2	2	no	—
42	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×D₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×D₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²⋊D₇₃	—	2	2	no	—
52	C₂²⋊D₇₃	—	2	2	no	—
53	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
55	D₂₉₂	—	2	2	no	—
56	C₄×D₇₃	—	2	1	no	—
57	C₄×D₇₃	—	2	1	no	—
58	C₄×D₇₃	C₂×D₄	1	2	yes	—
59	C₄×D₇₃	#41	1	2	yes	—
60	Dic₁₄₆	—	2	2	no	—
61	#10596833	—	4	4	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
64	#10596831	—	2	1	no	—
65	#10596831	C₂²⋊C₄	1	1	yes	characteristic
66	#10596831	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	—	4	1	no	—
72	C₂×D₇₃	—	4	1	no	—
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	1	no	—
80	C₂×C₇₃	—	4	1	no	—
81	C₂×C₇₃	#380	1	1	yes	characteristic
82	D₇₃	—	2	2	no	—
83	D₇₃	—	4	1	no	—
84	#1813	—	73	1	no	maximal, Sylow
85	C₇₃	#1813	1	1	yes	characteristic, complemented, Sylow
86	C₂×#131	—	73	1	no	—
87	C₂×#108	—	73	2	no	—
88	#413	—	73	2	no	—
89	#338	—	73	2	no	—
90	C₂×#41	—	73	1	no	—
91	#131	—	146	2	no	—
92	#131	—	146	2	no	—
93	C₂×M₁₆	—	73	1	no	—
94	C₂×M₁₆	—	73	1	no	—
95	C₂×M₁₆	—	73	4	no	—
96	C₂×M₁₆	#64408	1	1	yes	characteristic
97	C₂²×C₈	—	73	1	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂²×C₈	—	73	1	no	—
100	#108	—	146	2	no	—
101	#108	—	146	2	no	—
102	#101	—	146	2	no	—
103	#100	—	146	2	no	—
104	#41	—	146	2	no	—
105	#41	—	146	2	no	—
106	C₂×Q₈	—	73	1	no	—
107	C₂×D₄	—	73	1	no	—
108	C₂×D₄	—	73	1	no	—
109	C₂×D₄	—	73	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂²×C₄	—	73	1	no	—
112	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	146	2	no	—
115	M₁₆	—	146	4	no	—
116	M₁₆	—	146	4	no	—
117	M₁₆	—	146	2	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	2	no	—
120	C₂×C₈	—	73	2	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	146	1	no	—
123	C₂×C₈	—	146	1	no	—
124	C₂×C₈	—	146	4	no	—
125	C₂×C₈	—	146	1	no	—
126	C₂×C₈	—	146	1	no	—
127	C₂×C₈	C₂×#64408	1	1	yes	characteristic
128	C₂×C₈	C₂×#64408	1	1	yes	characteristic
129	C₂³	—	73	1	no	—
130	C₂³	—	73	1	no	—
131	C₂³	C₄⋊#64408	1	1	yes	characteristic
132	Q₈	—	146	2	no	—
133	D₄	—	146	2	no	—
134	D₄	—	146	2	no	—
135	D₄	—	146	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	2	no	—
140	C₂×C₄	—	73	2	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	146	1	no	—
143	C₂×C₄	—	146	1	no	—
144	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
145	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	292	4	no	—
148	C₈	—	73	1	no	—
149	C₈	—	73	1	no	—
150	C₈	—	146	1	no	—
151	C₈	C₂²⋊#64408	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	292	1	no	—
155	C₂²	—	292	1	no	—
156	C₂²	—	73	2	no	—
157	C₂²	—	146	1	no	—
158	C₂²	C₂×C₄⋊#64408	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	73	2	no	—
161	C₄	—	146	1	no	—
162	C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
163	C₄	#14556670	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	4	1	no	—
166	C₂	—	292	1	no	—
167	C₂	—	146	2	no	—
168	C₂	#17989529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₇₃⋊#108

C₂

yes

maximal, complemented

#17989573

C₂

yes

maximal

#17989456

C₂

yes

maximal, complemented

C₂×#14556599

C₂

yes

maximal, characteristic

D₇₃⋊M₁₆

C₂²

yes

—

—

—

—

—

C₂²

yes

—

—

C₄

yes

characteristic

C₂×#11924011

C₂²

yes

characteristic

C₂²×#10596831

C₄

yes

characteristic

C₂×C₈×D₇₃

C₄

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₄

yes

—

—

—

—

—

C₂×Dic₁₄₆

D₄

yes

characteristic

C₂×D₂₉₂

Q₈

yes

characteristic

C₂×C₂²⋊D₇₃

D₄

yes

characteristic

C₂×C₂²⋊D₇₃

Q₈

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

D₄

yes

—

C₈×D₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

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

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₄

yes

—

C₄×D₇₃

#41

yes

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊C₄

yes

—

#10596831

—

#10596831

C₂²⋊C₄

yes

characteristic

#10596831

C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

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

—

C₂×C₇₃

#380

yes

characteristic

D₇₃

—

D₇₃

—

#1813

—

maximal, Sylow

C₇₃

#1813

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×#41

—

#131

—

146

—

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

#64408

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

100

#108

—

146

—

101

#108

—

146

—

102

#101

—

146

—

103

#100

—

146

—

104

#41

—

146

—

105

#41

—

146

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×#64408

yes

characteristic

113

M₁₆

—

114

M₁₆

—

146

—

115

M₁₆

—

146

—

116

M₁₆

—

146

—

117

M₁₆

—

146

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₂×C₈

—

146

—

125

C₂×C₈

—

146

—

126

C₂×C₈

—

146

—

127

C₂×C₈

C₂×#64408

yes

characteristic

128

C₂×C₈

C₂×#64408

yes

characteristic

129

C₂³

—

130

C₂³

—

131

C₂³

C₄⋊#64408

yes

characteristic

132

Q₈

—

146

—

133

D₄

—

146

—

134

D₄

—

146

—

135

D₄

—

146

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

C₄⋊#64408

yes

characteristic

145

C₂×C₄

C₂²×#64408

yes

characteristic

146

C₈

—

147

C₈

—

292

—

148

C₈

—

149

C₈

—

150

C₈

—

146

—

151

C₈

C₂²⋊#64408

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

292

—

155

C₂²

—

292

—

156

C₂²

—

157

C₂²

—

146

—

158

C₂²

C₂×C₄⋊#64408

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

146

—

162

C₄

C₂×C₂²⋊#64408

yes

characteristic

163

C₄

#14556670

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

292

—

167

C₂

—

146

—

168

C₂

#17989529

yes

characteristic

	subgroup	G/H
center	C₄	#14556670
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂×M₁₆×C₇₃	C₄
socle	C₂×C₇₃	#380
hypercenter	C₂×M₁₆	#64408
hypocenter	C₇₃	#1813
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

#14556670

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂×M₁₆×C₇₃

C₄

C₂×C₇₃

C₂×M₁₆

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. D₇₃⋊#108	C₂	semidirect
3. #17989456	C₂	semidirect
85. C₇₃	#1813	semidirect

H ≅ G/N

type

1. D₇₃⋊#108

C₂

semidirect

3. #17989456

C₂

semidirect

85. C₇₃

#1813

semidirect

Subgroups — #22625816

Conjugacy classes of subgroups

Complements