Group #22610044 — subgroups

Subgroups — #22610044

Order 9344.

236 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 73 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	#17989064	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556509	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556509	C₂	1	1	yes	maximal, characteristic
4	#17988359	C₂	1	1	yes	maximal, characteristic
5	#17988358	C₂	1	1	yes	maximal, characteristic
6	#17988357	C₂	1	1	yes	maximal, characteristic
7	#344×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	C₂×#11924012	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₄₆	—	2	2	no	—
12	#14556556	—	2	2	no	—
13	#14556550	C₂²	1	1	yes	characteristic
14	#14556547	C₄	1	4	yes	—
15	#14556509	—	2	2	no	—
16	#14556509	—	2	2	no	—
17	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	#117×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₄₆	—	2	1	no	—
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×Dic₁₄₆	—	2	1	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×Dic₁₄₆	D₄	1	2	yes	—
25	C₂×Dic₁₄₆	D₄	1	2	yes	—
26	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
27	#11924012	—	2	1	no	—
28	#11924012	—	2	1	no	—
29	#11924012	—	2	1	no	—
30	#11924012	—	2	1	no	—
31	C₂×#10596831	—	2	1	no	—
32	C₂×#10596831	—	2	1	no	—
33	C₄⋊Dic₇₃	—	2	1	no	—
34	#11924007	—	2	4	no	—
35	C₂²⋊Dic₇₃	C₂×C₄	1	2	yes	—
36	C₄×Dic₇₃	—	2	1	no	—
37	C₄×Dic₇₃	—	2	4	no	—
38	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	M₁₆×C₇₃	—	2	1	no	—
40	M₁₆×C₇₃	—	2	1	no	—
41	M₁₆×C₇₃	—	2	1	no	—
42	M₁₆×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	—	2	1	no	—
45	C₄⋊C₄×C₇₃	—	2	1	no	—
46	C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
47	C₄²×C₇₃	—	2	1	no	—
48	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
56	C₂×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	—	2	2	no	—
58	C₂×Dic₇₃	—	2	2	no	—
59	C₂×Dic₇₃	C₂²⋊C₄	1	4	yes	—
60	Dic₁₄₆	—	2	2	no	—
61	Dic₁₄₆	—	2	2	no	—
62	Dic₁₄₆	—	2	2	no	—
63	Dic₁₄₆	—	2	2	no	—
64	C₈×C₇₃	—	4	1	no	—
65	C₈×C₇₃	—	4	1	no	—
66	#10596831	—	4	1	no	—
67	#10596831	—	4	1	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₇₃	#117	1	1	yes	characteristic
71	C₂²×C₇₃	#117	1	1	yes	characteristic
72	C₄×C₇₃	—	4	2	no	—
73	C₄×C₇₃	#123	1	1	yes	characteristic
74	C₄×C₇₃	#123	1	1	yes	characteristic
75	C₄×C₇₃	#123	1	1	yes	characteristic
76	C₄×C₇₃	#123	1	1	yes	characteristic
77	Dic₇₃	—	2	4	no	—
78	Dic₇₃	—	4	2	no	—
79	C₂×C₇₃	—	2	1	no	—
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	2	1	no	—
82	C₂×C₇₃	#381	1	1	yes	characteristic
83	#1830	—	73	1	no	maximal, Sylow
84	C₇₃	#1830	1	1	yes	characteristic, complemented, Sylow
85	#519	—	73	1	no	—
86	C₂×#101	—	73	1	no	—
87	C₂×#101	—	73	1	no	—
88	#344	—	73	1	no	—
89	#344	—	73	1	no	—
90	#344	—	73	1	no	—
91	#344	D₇₃	1	1	yes	characteristic
92	C₂²×Q₈	—	73	1	no	—
93	C₂×M₁₆	—	73	1	no	—
94	C₂×M₁₆	—	73	1	no	—
95	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
96	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
97	C₄×Q₈	—	146	2	no	—
98	C₄×Q₈	—	146	2	no	—
99	#117	—	73	1	no	—
100	#117	—	73	4	no	—
101	#117	C₂×D₇₃	1	1	yes	characteristic
102	#101	—	146	2	no	—
103	#101	—	146	2	no	—
104	C₂×Q₈	—	73	2	no	—
105	C₂×Q₈	—	73	2	no	—
106	C₂×Q₈	—	146	1	no	—
107	C₂×Q₈	—	146	1	no	—
108	C₂×Q₈	—	146	1	no	—
109	C₂×Q₈	—	146	1	no	—
110	C₂²×C₄	—	73	2	no	—
111	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	146	1	no	—
117	M₁₆	—	146	1	no	—
118	M₁₆	—	146	1	no	—
119	M₁₆	—	146	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	146	1	no	—
123	C₂×C₈	—	146	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	146	4	no	—
126	C₄⋊C₄	—	146	1	no	—
127	C₂²⋊C₄	—	73	2	no	—
128	C₂²⋊C₄	C₄×D₇₃	1	2	yes	—
129	C₄²	—	2	1	no	—
130	C₄²	—	146	4	no	—
131	C₄²	—	146	1	no	—
132	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
133	Q₈	—	146	2	no	—
134	Q₈	—	146	2	no	—
135	Q₈	—	146	2	no	—
136	Q₈	—	146	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	73	4	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	—	146	2	no	—
142	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
143	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
144	C₂×C₄	#11924027	1	1	yes	characteristic
145	C₂×C₄	#11924028	1	1	yes	characteristic
146	C₂×C₄	#11924028	1	1	yes	characteristic
147	C₂×C₄	#11924029	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	292	1	no	—
151	C₈	—	292	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
154	C₂²	#14556548	1	1	yes	characteristic
155	C₂²	#14556549	1	1	yes	characteristic
156	C₄	—	4	2	no	—
157	C₄	—	292	2	no	—
158	C₄	—	146	4	no	—
159	C₄	#14556573	1	1	yes	characteristic
160	C₄	#14556573	1	1	yes	characteristic
161	C₄	#14556576	1	1	yes	characteristic
162	C₄	#14556576	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#17988449	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989064

C₂

yes

maximal, characteristic

C₂×#14556509

C₂

yes

maximal, characteristic

C₂×#14556509

C₂

yes

maximal, characteristic

#17988359

C₂

yes

maximal, characteristic

#17988358

C₂

yes

maximal, characteristic

#17988357

C₂

yes

maximal, characteristic

#344×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

C₄×Dic₁₄₆

—

#14556556

—

#14556550

C₂²

yes

characteristic

#14556547

C₄

yes

—

#14556509

—

#14556509

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

—

C₂×Dic₁₄₆

D₄

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₃

yes

characteristic

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

#117

—

100

#117

—

101

#117

C₂×D₇₃

yes

characteristic

102

#101

—

146

—

103

#101

—

146

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

146

—

107

C₂×Q₈

—

146

—

108

C₂×Q₈

—

146

—

109

C₂×Q₈

—

146

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₇₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

146

—

117

M₁₆

—

146

—

118

M₁₆

—

146

—

119

M₁₆

—

146

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

—

146

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

C₄×D₇₃

yes

—

129

C₄²

—

130

C₄²

—

146

—

131

C₄²

—

146

—

132

C₂³

C₂×C₄×D₇₃

yes

characteristic

133

Q₈

—

146

—

134

Q₈

—

146

—

135

Q₈

—

146

—

136

Q₈

—

146

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

—

146

—

142

C₂×C₄

D₄×D₇₃

yes

characteristic

143

C₂×C₄

D₄×D₇₃

yes

characteristic

144

C₂×C₄

#11924027

yes

characteristic

145

C₂×C₄

#11924028

yes

characteristic

146

C₂×C₄

#11924028

yes

characteristic

147

C₂×C₄

#11924029

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

292

—

151

C₈

—

292

—

152

C₂²

—

153

C₂²

C₂²⋊C₄×D₇₃

yes

characteristic

154

C₂²

#14556548

yes

characteristic

155

C₂²

#14556549

yes

characteristic

156

C₄

—

157

C₄

—

292

—

158

C₄

—

146

—

159

C₄

#14556573

yes

characteristic

160

C₄

#14556573

yes

characteristic

161

C₄

#14556576

yes

characteristic

162

C₄

#14556576

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

#17988449

yes

characteristic

	subgroup	G/H
center	C₂	#17988449
commutator	C₂³×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#344×C₇₃	C₂
socle	C₂×C₇₃	#381
hypercenter	#344	D₇₃
hypocenter	C₇₃	#1830
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₂

#17988449

commutator

C₂³×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#344×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
84. C₇₃	#1830	semidirect

H ≅ G/N

type

84. C₇₃

#1830

semidirect

Subgroups — #22610044

Conjugacy classes of subgroups

Complements