Group #22611293 — subgroups

Subgroups — #22611293

Order 9344.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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	#17988629	C₂	1	1	yes	maximal, characteristic
2	#17988613	C₂	1	1	yes	maximal, characteristic
3	#17988597	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556516	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556509	C₂	1	1	yes	maximal, characteristic
6	#17988355	C₂	1	1	yes	maximal, characteristic
7	C₂×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	#14556550	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	#14556525	—	2	1	no	—
13	#14556523	—	2	1	no	—
14	#14556520	—	2	2	no	—
15	#14556516	—	2	1	no	—
16	#14556516	—	2	1	no	—
17	#14556516	—	2	2	no	—
18	#14556509	—	2	2	no	—
19	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#108×C₇₃	—	2	2	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×Dic₁₄₆	—	2	1	no	—
27	C₂×Dic₁₄₆	—	2	1	no	—
28	C₂×Dic₁₄₆	Q₈	1	2	yes	—
29	C₂²×Dic₇₃	—	2	1	no	—
30	C₂²×Dic₇₃	—	2	1	no	—
31	#11924012	—	2	1	no	—
32	#11924012	—	2	1	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₄⋊Dic₇₃	—	2	1	no	—
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
37	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
38	C₂²⋊Dic₇₃	—	2	1	no	—
39	C₄×Dic₇₃	—	2	1	no	—
40	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	M₁₆×C₇₃	—	2	1	no	—
42	M₁₆×C₇₃	—	2	1	no	—
43	M₁₆×C₇₃	—	2	1	no	—
44	M₁₆×C₇₃	D₄	1	2	yes	—
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₂×C₈×C₇₃	—	2	1	no	—
48	C₂×C₈×C₇₃	—	2	1	no	—
49	C₂×C₈×C₇₃	D₄	1	2	yes	—
50	C₂³×C₇₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	2	2	no	—
60	C₂×Dic₇₃	—	4	1	no	—
61	C₂×Dic₇₃	—	4	1	no	—
62	C₂×Dic₇₃	—	4	1	no	—
63	C₂×Dic₇₃	—	8	1	no	—
64	Dic₁₄₆	—	2	2	no	—
65	Dic₁₄₆	—	2	2	no	—
66	Dic₁₄₆	—	4	1	no	—
67	Dic₁₄₆	—	4	1	no	—
68	C₈×C₇₃	—	2	2	no	—
69	C₈×C₇₃	—	2	2	no	—
70	C₈×C₇₃	—	4	1	no	—
71	#10596831	—	4	1	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	#121	1	1	yes	characteristic
75	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₇₃	#126	1	1	yes	characteristic
77	C₄×C₇₃	#121	1	1	yes	characteristic
78	C₄×C₇₃	#121	1	1	yes	characteristic
79	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
81	Dic₇₃	—	4	2	no	—
82	Dic₇₃	—	4	1	no	—
83	Dic₇₃	—	4	1	no	—
84	Dic₇₃	—	8	1	no	—
85	C₂×C₇₃	—	2	1	no	—
86	C₂×C₇₃	—	2	1	no	—
87	C₂×C₇₃	#396	1	1	yes	characteristic
88	C₂×C₇₃	#471	1	1	yes	characteristic
89	C₂×C₇₃	#473	1	1	yes	characteristic
90	#2064	—	73	1	no	maximal, Sylow
91	C₇₃	#2064	1	1	yes	characteristic, complemented, Sylow
92	C₂×#108	D₇₃	1	1	yes	characteristic
93	#428	—	73	1	no	—
94	#424	—	73	1	no	—
95	#419	—	73	1	no	—
96	C₂×#103	—	73	1	no	—
97	C₂×#101	—	73	1	no	—
98	#343	—	73	1	no	—
99	C₂²×Q₈	—	73	1	no	—
100	C₂×M₁₆	—	73	1	no	—
101	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
102	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
104	#117	—	73	1	no	—
105	C₂×C₄⋊C₄	—	73	1	no	—
106	#108	—	2	2	no	—
107	#107	—	146	1	no	—
108	#106	—	146	1	no	—
109	C₄⋊C₈	—	146	2	no	—
110	#103	—	146	1	no	—
111	#103	—	146	1	no	—
112	#103	—	146	2	no	—
113	#101	—	146	2	no	—
114	C₂×Q₈	—	73	2	no	—
115	C₂×Q₈	—	146	1	no	—
116	C₂×Q₈	—	146	1	no	—
117	C₂×Q₈	—	146	1	no	—
118	C₂×Q₈	—	146	1	no	—
119	C₂×Q₈	—	146	1	no	—
120	C₂²×C₄	—	146	1	no	—
121	C₂²×C₄	—	146	1	no	—
122	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	1	no	—
125	M₁₆	—	2	1	no	—
126	M₁₆	—	146	1	no	—
127	M₁₆	—	146	1	no	—
128	M₁₆	C₂²⋊D₇₃	1	2	yes	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	146	1	no	—
134	C₂×C₈	D₂₉₂	1	2	yes	—
135	C₄⋊C₄	—	73	1	no	—
136	C₄⋊C₄	—	73	1	no	—
137	C₄⋊C₄	—	146	1	no	—
138	C₄⋊C₄	—	146	1	no	—
139	C₂²⋊C₄	—	146	1	no	—
140	C₄²	—	146	1	no	—
141	C₂³	#11924029	1	1	yes	characteristic
142	Q₈	—	292	1	no	—
143	Q₈	—	292	1	no	—
144	Q₈	—	146	2	no	—
145	Q₈	—	146	2	no	—
146	C₂×C₄	—	292	1	no	—
147	C₂×C₄	—	292	1	no	—
148	C₂×C₄	—	292	1	no	—
149	C₂×C₄	—	584	1	no	—
150	C₂×C₄	—	146	1	no	—
151	C₂×C₄	—	146	1	no	—
152	C₂×C₄	—	146	2	no	—
153	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
155	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
157	C₂×C₄	#11924027	1	1	yes	characteristic
158	C₂×C₄	#11924028	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	—	4	1	no	—
162	C₈	—	292	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#14556566	1	1	yes	characteristic
166	C₂²	#14556567	1	1	yes	characteristic
167	C₂²	#14556586	1	1	yes	characteristic
168	C₄	—	292	2	no	—
169	C₄	—	292	1	no	—
170	C₄	—	292	1	no	—
171	C₄	—	584	1	no	—
172	C₄	#14556563	1	1	yes	characteristic
173	C₄	#14556565	1	1	yes	characteristic
174	C₄	#14556569	1	1	yes	characteristic
175	C₄	#14556571	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#17988509	1	1	yes	characteristic
179	C₂	#17988851	1	1	yes	characteristic
180	C₂	#17988863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988629

C₂

yes

maximal, characteristic

#17988613

C₂

yes

maximal, characteristic

#17988597

C₂

yes

maximal, characteristic

C₂×#14556516

C₂

yes

maximal, characteristic

C₂×#14556509

C₂

yes

maximal, characteristic

#17988355

C₂

yes

maximal, characteristic

C₂×#108×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₂×#11924012

C₂²

yes

characteristic

#14556550

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

—

—

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#108×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

Q₈

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

characteristic

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₂³×C₇₃

#41

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

C₂²×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×#108

D₇₃

yes

—

—

—

—

—

—

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₂×M₁₆

C₂×D₇₃

yes

characteristic

102

C₂×M₁₆

C₂×D₇₃

yes

characteristic

103

C₂²×C₈

C₂×D₇₃

yes

characteristic

104

#117

—

105

C₂×C₄⋊C₄

—

106

#108

—

107

#107

—

146

—

108

#106

—

146

—

109

C₄⋊C₈

—

146

—

110

#103

—

146

—

111

#103

—

146

—

112

#103

—

146

—

113

#101

—

146

—

114

C₂×Q₈

—

115

C₂×Q₈

—

146

—

116

C₂×Q₈

—

146

—

117

C₂×Q₈

—

146

—

118

C₂×Q₈

—

146

—

119

C₂×Q₈

—

146

—

120

C₂²×C₄

—

146

—

121

C₂²×C₄

—

146

—

122

C₂²×C₄

C₂²×D₇₃

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

146

—

127

M₁₆

—

146

—

128

M₁₆

C₂²⋊D₇₃

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

146

—

134

C₂×C₈

D₂₉₂

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

146

—

138

C₄⋊C₄

—

146

—

139

C₂²⋊C₄

—

146

—

140

C₄²

—

146

—

141

C₂³

#11924029

yes

characteristic

142

Q₈

—

292

—

143

Q₈

—

292

—

144

Q₈

—

146

—

145

Q₈

—

146

—

146

C₂×C₄

—

292

—

147

C₂×C₄

—

292

—

148

C₂×C₄

—

292

—

149

C₂×C₄

—

584

—

150

C₂×C₄

—

146

—

151

C₂×C₄

—

146

—

152

C₂×C₄

—

146

—

153

C₂×C₄

D₄×D₇₃

yes

characteristic

154

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

155

C₂×C₄

C₂×D₂₉₂

yes

characteristic

156

C₂×C₄

Q₈×D₇₃

yes

characteristic

157

C₂×C₄

#11924027

yes

characteristic

158

C₂×C₄

#11924028

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

292

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14556566

yes

characteristic

166

C₂²

#14556567

yes

characteristic

167

C₂²

#14556586

yes

characteristic

168

C₄

—

292

—

169

C₄

—

292

—

170

C₄

—

292

—

171

C₄

—

584

—

172

C₄

#14556563

yes

characteristic

173

C₄

#14556565

yes

characteristic

174

C₄

#14556569

yes

characteristic

175

C₄

#14556571

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#17988509

yes

characteristic

179

C₂

#17988851

yes

characteristic

180

C₂

#17988863

yes

characteristic

	subgroup	G/H
center	C₂²	#14556566
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#108×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	C₂×#108	D₇₃
hypocenter	C₇₃	#2064
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14556566

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#108×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
91. C₇₃	#2064	semidirect

H ≅ G/N

type

91. C₇₃

#2064

semidirect

Subgroups — #22611293

Conjugacy classes of subgroups

Complements