Group #33999956 — subgroups

Subgroups — #33999956

Order 18688.

292 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 102 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	#22625848	C₂	1	2	yes	maximal
2	#22610082	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#10596833	—	2	2	no	—
4	#17989453	—	2	2	no	—
5	#17989453	C₄	1	4	yes	—
6	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
7	C₂×#14556520	—	2	2	no	—
8	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
10	C₂²×#10596833	—	2	4	no	—
11	C₄⋊#10596833	—	4	2	no	—
12	C₄⋊#10596833	—	4	2	no	—
13	C₂²×#10596831	—	2	1	no	—
14	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
16	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
18	C₄⋊#10596831	—	4	1	no	—
19	#14556520	—	4	2	no	—
20	#14556520	—	4	2	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	—	4	4	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	1	no	—
32	C₄×Dic₇₃	—	2	1	no	—
33	C₄×Dic₇₃	—	4	2	no	—
34	C₄×Dic₇₃	—	4	1	no	—
35	C₄×Dic₇₃	D₈	1	2	yes	—
36	C₄×Dic₇₃	SD₁₆	1	2	yes	—
37	C₄×Dic₇₃	SD₁₆	1	2	yes	—
38	C₄×Dic₇₃	Q₁₆	1	2	yes	—
39	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	4	1	no	—
43	C₂×C₈×C₇₃	—	4	2	no	—
44	C₄²×C₇₃	—	2	1	no	—
45	C₄²×C₇₃	—	2	1	no	—
46	C₂³×C₇₃	#95	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#101	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#102	1	2	yes	—
54	C₂×C₄×C₇₃	#103	1	2	yes	—
55	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
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	C₂×Dic₇₃	#106	1	2	yes	—
64	C₂×Dic₇₃	#107	1	2	yes	—
65	#10596833	—	8	4	no	—
66	C₈×C₇₃	—	8	1	no	—
67	#10596831	—	8	1	no	—
68	C₂²×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
69	C₂²×C₇₃	#340	1	2	yes	—
70	C₂²×C₇₃	#341	1	1	yes	characteristic
71	C₂²×C₇₃	#343	1	1	yes	characteristic
72	C₂²×C₇₃	#344	1	2	yes	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	4	1	no	—
76	C₄×C₇₃	#331	1	1	yes	characteristic
77	C₄×C₇₃	#332	1	2	yes	—
78	C₄×C₇₃	#333	1	1	yes	characteristic
79	Dic₇₃	—	2	2	no	—
80	Dic₇₃	—	2	2	no	—
81	Dic₇₃	—	2	2	no	—
82	Dic₇₃	—	2	2	no	—
83	#7040	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	#1286	1	2	yes	—
85	C₂×C₇₃	#1290	1	1	yes	characteristic
86	C₂×C₇₃	#1296	1	1	yes	characteristic
87	C₂×C₇₃	#1301	1	2	yes	—
88	C₂×C₇₃	#1307	1	1	yes	characteristic
89	#1842	—	73	1	no	—
90	#1831	—	73	2	no	—
91	C₇₃	#7040	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄⋊C₈	—	73	1	no	—
93	C₂×C₄⋊C₈	—	146	2	no	—
94	C₂×C₄⋊C₈	—	146	2	no	—
95	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
96	C₄³	—	73	1	no	—
97	#336	—	73	4	no	—
98	#336	—	146	2	no	—
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	146	4	no	—
101	C₂²×C₈	—	146	1	no	—
102	C₂×C₄²	—	73	2	no	—
103	C₂×C₄²	—	73	1	no	—
104	C₂×C₄²	—	73	2	no	—
105	C₂×C₄²	—	73	1	no	—
106	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
107	C₄⋊C₈	—	4	1	no	—
108	C₄⋊C₈	—	292	1	no	—
109	C₄⋊C₈	—	292	2	no	—
110	C₄⋊C₈	—	292	2	no	—
111	C₄⋊C₈	—	292	2	no	—
112	C₄⋊C₈	—	292	2	no	—
113	C₂²×C₄	—	73	2	no	—
114	C₂²×C₄	—	73	2	no	—
115	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
117	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	292	4	no	—
121	C₂×C₈	—	292	4	no	—
122	C₂×C₈	—	292	4	no	—
123	C₂×C₈	—	292	1	no	—
124	C₂×C₈	—	292	2	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	292	1	no	—
128	C₄²	—	292	2	no	—
129	C₄²	—	73	2	no	—
130	C₄²	—	73	2	no	—
131	C₄²	—	73	2	no	—
132	C₄²	—	73	2	no	—
133	C₄²	—	146	1	no	—
134	C₄²	—	146	1	no	—
135	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	73	2	no	—
141	C₂×C₄	—	73	2	no	—
142	C₂×C₄	—	73	2	no	—
143	C₂×C₄	—	73	2	no	—
144	C₂×C₄	—	146	2	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	—	146	2	no	—
147	C₂×C₄	—	146	2	no	—
148	C₂×C₄	#14556651	1	1	yes	characteristic
149	C₂×C₄	#14556652	1	1	yes	characteristic
150	C₂×C₄	#14556653	1	2	yes	—
151	C₂×C₄	#14556654	1	2	yes	—
152	C₂×C₄	#14556661	1	1	yes	characteristic
153	C₂×C₄	#14556662	1	1	yes	characteristic
154	C₈	—	8	1	no	—
155	C₈	—	584	4	no	—
156	C₈	—	584	1	no	—
157	C₂²	#17989455	1	1	yes	characteristic
158	C₂²	#17989460	1	2	yes	—
159	C₂²	#17989463	1	1	yes	characteristic
160	C₂²	#17989467	1	1	yes	characteristic
161	C₂²	#17989470	1	2	yes	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	146	2	no	—
166	C₄	—	146	2	no	—
167	C₄	—	146	2	no	—
168	C₄	—	146	2	no	—
169	C₄	#17989445	1	1	yes	characteristic
170	C₄	#17989446	1	2	yes	—
171	C₄	#17989447	1	1	yes	characteristic
172	C₂	#22624786	1	2	yes	—
173	C₂	#22624796	1	1	yes	characteristic
174	C₂	#22624809	1	1	yes	characteristic
175	C₂	#22624823	1	2	yes	—
176	C₂	#22624837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625848

C₂

yes

maximal

#22610082

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596833

—

#17989453

—

#17989453

C₄

yes

—

C₂×C₄⋊#10596831

C₄

yes

characteristic

C₂×#14556520

—

C₄²×Dic₇₃

C₂²

yes

characteristic

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

—

—

—

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

—

C₄×Dic₇₃

SD₁₆

yes

—

C₄×Dic₇₃

SD₁₆

yes

—

C₄×Dic₇₃

Q₁₆

yes

—

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

#103

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#102

yes

—

C₂×Dic₇₃

#103

yes

—

C₂×Dic₇₃

#106

yes

—

C₂×Dic₇₃

#107

yes

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

H(3,ℤ/4)

yes

characteristic

C₂²×C₇₃

#340

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#331

yes

characteristic

C₄×C₇₃

#332

yes

—

C₄×C₇₃

#333

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

#7040

—

maximal, Sylow

C₂×C₇₃

#1286

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

yes

—

—

C₇₃

#7040

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄⋊C₈

#64408

yes

characteristic

C₄³

—

#336

—

#336

—

146

—

C₂²×C₈

—

100

C₂²×C₈

—

146

—

101

C₂²×C₈

—

146

—

102

C₂×C₄²

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×#64408

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

292

—

109

C₄⋊C₈

—

292

—

110

C₄⋊C₈

—

292

—

111

C₄⋊C₈

—

292

—

112

C₄⋊C₈

—

292

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₄×#64408

yes

characteristic

116

C₂²×C₄

C₂²⋊#64408

yes

characteristic

117

C₂²×C₄

C₄⋊#64408

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

292

—

121

C₂×C₈

—

292

—

122

C₂×C₈

—

292

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

292

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

292

—

128

C₄²

—

292

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

146

—

134

C₄²

—

146

—

135

C₂³

(C₂×#64408)⋊C₄

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

#14556651

yes

characteristic

149

C₂×C₄

#14556652

yes

characteristic

150

C₂×C₄

#14556653

yes

—

151

C₂×C₄

#14556654

yes

—

152

C₂×C₄

#14556661

yes

characteristic

153

C₂×C₄

#14556662

yes

characteristic

154

C₈

—

155

C₈

—

584

—

156

C₈

—

584

—

157

C₂²

#17989455

yes

characteristic

158

C₂²

#17989460

yes

—

159

C₂²

#17989463

yes

characteristic

160

C₂²

#17989467

yes

characteristic

161

C₂²

#17989470

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

146

—

166

C₄

—

146

—

167

C₄

—

146

—

168

C₄

—

146

—

169

C₄

#17989445

yes

characteristic

170

C₄

#17989446

yes

—

171

C₄

#17989447

yes

characteristic

172

C₂

#22624786

yes

—

173

C₂

#22624796

yes

characteristic

174

C₂

#22624809

yes

characteristic

175

C₂

#22624823

yes

—

176

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	#7040
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
91. C₇₃	#7040	semidirect

H ≅ G/N

type

91. C₇₃

#7040

semidirect

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

subgroup	index
#22625848	2
#22610082	2
#7040	73

subgroup

index

#22625848

#22610082

#7040

Subgroups — #33999956

Conjugacy classes of subgroups

Complements

Maximal subgroups