Group #34000464 — subgroups

Subgroups — #34000464

Order 18688.

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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	#22625811	C₂	1	2	yes	maximal
2	C₂×#17988366	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556662	—	2	2	no	—
4	#17989558	—	2	2	no	—
5	#17989463	C₄	1	4	yes	—
6	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
7	C₂²×#11924003	C₄	1	1	yes	characteristic
8	#17988366	—	2	2	no	—
9	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924038	—	2	4	no	—
11	#14556662	—	4	2	no	—
12	#14556662	—	4	2	no	—
13	#14556648	—	4	2	no	—
14	C₂²⋊#10596833	—	4	2	no	—
15	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
16	C₂²×#10596831	Q₈	1	1	yes	characteristic
17	C₂×C₈×D₇₃	—	4	1	no	—
18	C₂×C₈×D₇₃	—	4	1	no	—
19	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×D₇₃	D₄	1	2	yes	—
21	C₂×#11924003	—	2	1	no	—
22	C₂×#11924003	—	4	1	no	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₃	—	2	1	no	—
25	C₂×C₁₆×C₇₃	—	4	1	no	—
26	#11924038	—	4	4	no	—
27	#11924038	—	4	4	no	—
28	C₂×#10596833	—	4	4	no	—
29	C₂³×D₇₃	Q₁₆	1	1	yes	characteristic
30	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
31	C₂×C₄×D₇₃	—	4	1	no	—
32	C₂×C₄×D₇₃	—	4	1	no	—
33	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×D₇₃	SD₁₆	1	2	yes	—
35	C₂×#10596831	—	4	1	no	—
36	C₂×#10596831	C₄⋊C₄	1	2	yes	—
37	C₈×D₇₃	—	2	2	no	—
38	C₈×D₇₃	—	2	2	no	—
39	C₈×D₇₃	—	4	2	no	—
40	#11924003	—	4	1	no	—
41	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₇₃	—	4	1	no	—
47	C₂³×C₇₃	#107	1	1	yes	characteristic
48	C₂²×D₇₃	—	2	1	no	—
49	C₂²×D₇₃	—	2	1	no	—
50	C₂²×D₇₃	—	4	1	no	—
51	C₂²×D₇₃	—	4	1	no	—
52	C₂²×D₇₃	#103	1	2	yes	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
57	C₂×Dic₇₃	—	4	1	no	—
58	C₂×Dic₇₃	#102	1	2	yes	—
59	C₄×D₇₃	—	2	2	no	—
60	C₄×D₇₃	—	2	2	no	—
61	C₄×D₇₃	—	4	2	no	—
62	#10596833	—	8	4	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	#99	1	1	yes	characteristic
65	C₈×C₇₃	#99	1	1	yes	characteristic
66	#10596831	—	2	2	no	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	#340	1	1	yes	characteristic
70	C₂²×C₇₃	#368	1	1	yes	characteristic
71	C₂²×C₇₃	#368	1	1	yes	characteristic
72	C₂×D₇₃	—	2	2	no	—
73	C₂×D₇₃	—	2	2	no	—
74	C₂×D₇₃	—	2	2	no	—
75	C₂×D₇₃	—	4	2	no	—
76	C₂×D₇₃	—	4	2	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
79	C₄×C₇₃	#342	1	1	yes	characteristic
80	Dic₇₃	—	2	2	no	—
81	#7231	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	—	2	1	no	—
83	C₂×C₇₃	—	2	1	no	—
84	C₂×C₇₃	#1289	1	1	yes	characteristic
85	C₂×C₇₃	#1381	1	1	yes	characteristic
86	C₂×C₇₃	#1385	1	1	yes	characteristic
87	D₇₃	—	4	2	no	—
88	C₂×C₂²⋊C₁₆	—	73	1	no	—
89	#1810	—	73	2	no	—
90	C₇₃	#7231	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₈	—	73	1	no	—
92	C₂²×C₁₆	—	73	1	no	—
93	C₂²×C₁₆	#64408	1	1	yes	characteristic
94	C₂×#108	—	146	2	no	—
95	#407	—	146	2	no	—
96	C₂²⋊C₁₆	—	146	2	no	—
97	#341	—	73	4	no	—
98	C₂³×C₄	—	73	1	no	—
99	C₂×M₁₆	—	146	4	no	—
100	C₂²×C₈	—	292	1	no	—
101	C₂²×C₈	—	292	1	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂²×C₈	—	73	2	no	—
104	C₂²×C₈	—	73	2	no	—
105	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	4	1	no	—
108	C₂×C₁₆	—	292	1	no	—
109	C₂×C₁₆	—	146	1	no	—
110	#108	—	292	2	no	—
111	#108	—	292	2	no	—
112	C₂²⋊C₈	—	292	2	no	—
113	C₂²⋊C₈	—	292	2	no	—
114	C₂⁴	—	73	1	no	—
115	C₂²×C₄	—	292	1	no	—
116	C₂²×C₄	—	292	1	no	—
117	C₂²×C₄	—	73	1	no	—
118	C₂²×C₄	—	73	2	no	—
119	C₂²×C₄	—	73	2	no	—
120	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
121	M₁₆	—	292	4	no	—
122	M₁₆	—	292	4	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	292	2	no	—
126	C₂×C₈	—	292	4	no	—
127	C₂×C₈	—	292	1	no	—
128	C₂×C₈	—	73	2	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	—	146	2	no	—
131	C₂×C₈	C₄×#64408	1	1	yes	characteristic
132	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
133	C₁₆	—	4	1	no	—
134	C₁₆	—	292	1	no	—
135	C₂³	—	292	1	no	—
136	C₂³	—	292	1	no	—
137	C₂³	—	73	2	no	—
138	C₂³	—	146	1	no	—
139	C₂³	—	146	1	no	—
140	C₂³	#14556660	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	292	2	no	—
144	C₂×C₄	—	292	1	no	—
145	C₂×C₄	—	73	2	no	—
146	C₂×C₄	—	146	2	no	—
147	C₂×C₄	—	146	2	no	—
148	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#14556659	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	584	4	no	—
152	C₈	—	146	2	no	—
153	C₈	#14556649	1	1	yes	characteristic
154	C₈	#14556650	1	1	yes	characteristic
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	292	2	no	—
158	C₂²	—	292	2	no	—
159	C₂²	—	146	2	no	—
160	C₂²	—	146	2	no	—
161	C₂²	—	146	2	no	—
162	C₂²	#17989461	1	1	yes	characteristic
163	C₂²	#17989515	1	1	yes	characteristic
164	C₂²	#17989516	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	146	2	no	—
167	C₄	#17989455	1	1	yes	characteristic
168	C₄	#17989465	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	292	2	no	—
172	C₂	#22624794	1	1	yes	characteristic
173	C₂	#22624991	1	1	yes	characteristic
174	C₂	#22625001	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625811

C₂

yes

maximal

C₂×#17988366

C₂

yes

maximal, characteristic

C₂×#14556662

—

#17989558

—

#17989463

C₄

yes

—

C₂²×C₈×D₇₃

C₂²

yes

characteristic

C₂²×#11924003

C₄

yes

characteristic

#17988366

—

C₂²×C₁₆×C₇₃

C₄

yes

—

—

—

—

—

C₂²×C₄×D₇₃

D₄

yes

characteristic

C₂²×#10596831

Q₈

yes

characteristic

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₂×C₈×D₇₃

D₄

yes

—

C₂×#11924003

—

C₂×#11924003

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

#11924038

—

#11924038

—

C₂×#10596833

—

C₂³×D₇₃

Q₁₆

yes

characteristic

C₂²×Dic₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

—

C₂×C₄×D₇₃

SD₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

#11924003

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#107

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

#103

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

#102

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#7231

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

C₂×C₂²⋊C₁₆

—

#1810

—

C₇₃

#7231

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

#64408

yes

characteristic

C₂×#108

—

146

—

#407

—

146

—

C₂²⋊C₁₆

—

146

—

#341

—

C₂³×C₄

—

C₂×M₁₆

—

146

—

100

C₂²×C₈

—

292

—

101

C₂²×C₈

—

292

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂×#64408

yes

characteristic

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

292

—

109

C₂×C₁₆

—

146

—

110

#108

—

292

—

111

#108

—

292

—

112

C₂²⋊C₈

—

292

—

113

C₂²⋊C₈

—

292

—

114

C₂⁴

—

115

C₂²×C₄

—

292

—

116

C₂²×C₄

—

292

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₄⋊#64408

yes

characteristic

121

M₁₆

—

292

—

122

M₁₆

—

292

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

292

—

126

C₂×C₈

—

292

—

127

C₂×C₈

—

292

—

128

C₂×C₈

—

129

C₂×C₈

—

146

—

130

C₂×C₈

—

146

—

131

C₂×C₈

C₄×#64408

yes

characteristic

132

C₂×C₈

C₂²⋊#64408

yes

characteristic

133

C₁₆

—

134

C₁₆

—

292

—

135

C₂³

—

292

—

136

C₂³

—

292

—

137

C₂³

—

138

C₂³

—

146

—

139

C₂³

—

146

—

140

C₂³

#14556660

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

292

—

144

C₂×C₄

—

292

—

145

C₂×C₄

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

149

C₂×C₄

#14556659

yes

characteristic

150

C₈

—

151

C₈

—

584

—

152

C₈

—

146

—

153

C₈

#14556649

yes

characteristic

154

C₈

#14556650

yes

characteristic

155

C₂²

—

156

C₂²

—

157

C₂²

—

292

—

158

C₂²

—

292

—

159

C₂²

—

146

—

160

C₂²

—

146

—

161

C₂²

—

146

—

162

C₂²

#17989461

yes

characteristic

163

C₂²

#17989515

yes

characteristic

164

C₂²

#17989516

yes

characteristic

165

C₄

—

166

C₄

—

146

—

167

C₄

#17989455

yes

characteristic

168

C₄

#17989465

yes

characteristic

169

C₂

—

170

C₂

—

171

C₂

—

292

—

172

C₂

#22624794

yes

characteristic

173

C₂

#22624991

yes

characteristic

174

C₂

#22625001

yes

characteristic

	subgroup	G/H
center	C₂²	#17989461
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂³×D₇₃	Q₁₆
cosocle	C₂²×C₈×D₇₃	C₂²
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#7231
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

#17989461

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂²×C₁₆×C₇₃

C₄

socle

C₂²×C₇₃

#340

prime radical

C₂³×D₇₃

Q₁₆

cosocle

C₂²×C₈×D₇₃

C₂²

C₂²×C₁₆

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
90. C₇₃	#7231	semidirect

H ≅ G/N

type

90. C₇₃

#7231

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22625811	2
C₂×#17988366	2
#7231	73

subgroup

index

#22625811

C₂×#17988366

#7231

Subgroups — #34000464

Conjugacy classes of subgroups

Complements

Maximal subgroups