Group #33999975 — subgroups

Subgroups — #33999975

Order 18688.

304 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 94 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	#22625717	C₂	1	2	yes	maximal
2	#22610047	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#10596833	—	2	2	no	—
4	C₂×#14556646	—	2	2	no	—
5	#17989453	C₄	1	4	yes	—
6	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
7	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
8	#17988341	—	2	2	no	—
9	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
10	C₂²×#10596833	—	2	4	no	—
11	C₄⋊#10596833	—	2	4	no	—
12	C₄⋊#10596833	—	2	4	no	—
13	#14556646	—	4	2	no	—
14	#14556646	—	4	2	no	—
15	C₂²×#10596831	—	2	1	no	—
16	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
18	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
20	C₄⋊#10596831	—	4	1	no	—
21	C₂²×C₈×C₇₃	—	2	1	no	—
22	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₈×C₇₃	—	4	1	no	—
24	C₂×#10596833	—	2	8	no	—
25	C₂×#10596833	—	4	4	no	—
26	C₂×#10596833	—	4	4	no	—
27	C₂²×Dic₇₃	C₂²⋊C₄	1	2	yes	—
28	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
29	C₂×#10596831	—	4	1	no	—
30	C₂×#10596831	—	4	2	no	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₄×Dic₇₃	—	2	2	no	—
33	C₄×Dic₇₃	—	4	2	no	—
34	C₄×Dic₇₃	—	4	1	no	—
35	C₄×Dic₇₃	D₈	1	1	yes	characteristic
36	C₄×Dic₇₃	SD₁₆	1	2	yes	—
37	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	—	4	1	no	—
42	C₂×C₈×C₇₃	—	4	2	no	—
43	C₄²×C₇₃	—	2	1	no	—
44	C₄²×C₇₃	—	2	1	no	—
45	C₂³×C₇₃	#95	1	1	yes	characteristic
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#101	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#104	1	2	yes	—
53	C₂×C₄×C₇₃	#104	1	2	yes	—
54	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#107	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	—	2	2	no	—
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	#102	1	2	yes	—
62	C₂×Dic₇₃	#103	1	2	yes	—
63	#10596833	—	4	8	no	—
64	C₈×C₇₃	—	8	1	no	—
65	#10596831	—	8	1	no	—
66	C₂²×C₇₃	#338	1	1	yes	characteristic
67	C₂²×C₇₃	#339	1	2	yes	—
68	C₂²×C₇₃	#340	1	1	yes	characteristic
69	C₂²×C₇₃	#343	1	1	yes	characteristic
70	C₂²×C₇₃	#344	1	2	yes	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	4	1	no	—
74	C₄×C₇₃	#329	1	1	yes	characteristic
75	C₄×C₇₃	#329	1	1	yes	characteristic
76	C₄×C₇₃	#330	1	2	yes	—
77	Dic₇₃	—	2	2	no	—
78	Dic₇₃	—	2	2	no	—
79	Dic₇₃	—	4	2	no	—
80	#7049	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	#1291	1	2	yes	—
82	C₂×C₇₃	#1292	1	1	yes	characteristic
83	C₂×C₇₃	#1295	1	1	yes	characteristic
84	C₂×C₇₃	#1303	1	2	yes	—
85	C₂×C₇₃	#1307	1	1	yes	characteristic
86	#1831	—	73	1	no	—
87	#1767	—	73	2	no	—
88	C₇₃	#7049	1	1	yes	characteristic, complemented, Sylow
89	C₂×C₄⋊C₈	—	73	1	no	—
90	C₂×C₄⋊C₈	—	146	2	no	—
91	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
92	C₂×#97	—	146	2	no	—
93	C₄³	—	73	1	no	—
94	#336	—	73	4	no	—
95	#336	—	146	2	no	—
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	—	146	1	no	—
98	C₂²×C₈	—	146	4	no	—
99	C₂×C₄²	—	73	2	no	—
100	C₂×C₄²	—	73	1	no	—
101	C₂×C₄²	—	73	1	no	—
102	C₂×C₄²	—	73	2	no	—
103	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
104	C₄⋊C₈	—	4	1	no	—
105	C₄⋊C₈	—	292	1	no	—
106	C₄⋊C₈	—	146	4	no	—
107	C₄⋊C₈	—	146	4	no	—
108	#97	—	292	2	no	—
109	#97	—	292	2	no	—
110	C₂²×C₄	—	73	2	no	—
111	C₂²×C₄	—	73	2	no	—
112	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
113	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	292	4	no	—
118	C₂×C₈	—	292	4	no	—
119	C₂×C₈	—	292	1	no	—
120	C₂×C₈	—	292	2	no	—
121	C₂×C₈	—	146	8	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	292	1	no	—
125	C₄²	—	292	2	no	—
126	C₄²	—	73	2	no	—
127	C₄²	—	73	1	no	—
128	C₄²	—	73	1	no	—
129	C₄²	—	146	2	no	—
130	C₄²	—	146	2	no	—
131	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	73	2	no	—
137	C₂×C₄	—	73	2	no	—
138	C₂×C₄	—	146	2	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	—	146	2	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	#14556651	1	1	yes	characteristic
144	C₂×C₄	#14556652	1	1	yes	characteristic
145	C₂×C₄	#14556655	1	2	yes	—
146	C₂×C₄	#14556656	1	2	yes	—
147	C₂×C₄	#14556659	1	1	yes	characteristic
148	C₂×C₄	#14556660	1	1	yes	characteristic
149	C₈	—	8	1	no	—
150	C₈	—	292	8	no	—
151	C₈	—	584	1	no	—
152	C₂²	#17989456	1	1	yes	characteristic
153	C₂²	#17989457	1	2	yes	—
154	C₂²	#17989461	1	1	yes	characteristic
155	C₂²	#17989467	1	1	yes	characteristic
156	C₂²	#17989470	1	2	yes	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	1	no	—
160	C₄	—	292	2	no	—
161	C₄	—	146	2	no	—
162	C₄	—	146	2	no	—
163	C₄	#17989442	1	1	yes	characteristic
164	C₄	#17989443	1	1	yes	characteristic
165	C₄	#17989444	1	2	yes	—
166	C₂	#22624797	1	2	yes	—
167	C₂	#22624801	1	1	yes	characteristic
168	C₂	#22624807	1	1	yes	characteristic
169	C₂	#22624828	1	2	yes	—
170	C₂	#22624837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625717

C₂

yes

maximal

#22610047

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596833

—

C₂×#14556646

—

#17989453

C₄

yes

—

C₂×C₄⋊#10596831

C₄

yes

characteristic

C₄²×Dic₇₃

C₂²

yes

characteristic

#17988341

—

C₂×C₄⋊C₈×C₇₃

C₄

yes

—

—

—

—

—

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

D₄

yes

—

C₂×C₄×Dic₇₃

D₄

yes

characteristic

C₂×C₄×Dic₇₃

Q₈

yes

characteristic

C₄⋊#10596831

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂×#10596833

—

C₂×#10596833

—

C₂×#10596833

—

C₂²×Dic₇₃

C₂²⋊C₄

yes

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

—

C₄×Dic₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#104

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#102

yes

—

C₂×Dic₇₃

#103

yes

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#338

yes

characteristic

C₂²×C₇₃

#339

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

#7049

—

maximal, Sylow

C₂×C₇₃

#1291

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

yes

—

—

C₇₃

#7049

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄⋊C₈

#64408

yes

characteristic

C₂×#97

—

146

—

C₄³

—

#336

—

#336

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂²×C₈

—

146

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₂×C₄²

C₂×#64408

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

292

—

106

C₄⋊C₈

—

146

—

107

C₄⋊C₈

—

146

—

108

#97

—

292

—

109

#97

—

292

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₄×#64408

yes

characteristic

113

C₂²×C₄

C₂²⋊#64408

yes

characteristic

114

C₂²×C₄

C₄⋊#64408

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

292

—

119

C₂×C₈

—

292

—

120

C₂×C₈

—

292

—

121

C₂×C₈

—

146

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

292

—

125

C₄²

—

292

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

146

—

130

C₄²

—

146

—

131

C₂³

(C₂×#64408)⋊C₄

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

#14556651

yes

characteristic

144

C₂×C₄

#14556652

yes

characteristic

145

C₂×C₄

#14556655

yes

—

146

C₂×C₄

#14556656

yes

—

147

C₂×C₄

#14556659

yes

characteristic

148

C₂×C₄

#14556660

yes

characteristic

149

C₈

—

150

C₈

—

292

—

151

C₈

—

584

—

152

C₂²

#17989456

yes

characteristic

153

C₂²

#17989457

yes

—

154

C₂²

#17989461

yes

characteristic

155

C₂²

#17989467

yes

characteristic

156

C₂²

#17989470

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

292

—

161

C₄

—

146

—

162

C₄

—

146

—

163

C₄

#17989442

yes

characteristic

164

C₄

#17989443

yes

characteristic

165

C₄

#17989444

yes

—

166

C₂

#22624797

yes

—

167

C₂

#22624801

yes

characteristic

168

C₂

#22624807

yes

characteristic

169

C₂

#22624828

yes

—

170

C₂

#22624837

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×#64408)⋊C₄
commutator	C₂²×C₄×C₇₃	C₄²
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	#95
cosocle	C₄²×Dic₇₃	C₂²
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	#7049
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

(C₂×#64408)⋊C₄

commutator

C₂²×C₄×C₇₃

C₄²

Frattini

C₂×C₄²

C₂×#64408

Fitting

C₂×C₄⋊C₈×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

#95

cosocle

C₄²×Dic₇₃

C₂²

C₂×C₄⋊C₈

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
88. C₇₃	#7049	semidirect

H ≅ G/N

type

88. C₇₃

#7049

semidirect

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

subgroup	index
#22625717	2
#22610047	2
#7049	73

subgroup

index

#22625717

#22610047

#7049

Subgroups — #33999975

Conjugacy classes of subgroups

Complements

Maximal subgroups