Group #33999961 — subgroups

Subgroups — #33999961

Order 18688.

304 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₄	1	4	yes	—
5	#17989453	—	2	2	no	—
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	4	no	—
14	C₂²×#10596831	—	2	1	no	—
15	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
17	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
19	C₄⋊#10596831	—	4	1	no	—
20	C₂²×C₈×C₇₃	—	2	1	no	—
21	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₈×C₇₃	—	4	1	no	—
23	C₂×#10596833	—	2	8	no	—
24	C₂×#10596833	—	4	4	no	—
25	C₂×#10596833	—	4	4	no	—
26	C₂²×Dic₇₃	C₂²⋊C₄	1	2	yes	—
27	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
28	C₂×#10596831	—	4	1	no	—
29	C₂×#10596831	—	4	2	no	—
30	C₄×Dic₇₃	—	2	2	no	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₄×Dic₇₃	—	2	1	no	—
33	C₄×Dic₇₃	—	2	1	no	—
34	C₄×Dic₇₃	—	2	2	no	—
35	C₄×Dic₇₃	—	2	2	no	—
36	C₄×Dic₇₃	—	4	1	no	—
37	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	—	4	1	no	—
41	C₂×C₈×C₇₃	—	4	2	no	—
42	C₄²×C₇₃	—	2	1	no	—
43	C₄²×C₇₃	—	2	1	no	—
44	C₂³×C₇₃	#95	1	1	yes	characteristic
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#101	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#104	1	2	yes	—
52	C₂×C₄×C₇₃	#104	1	2	yes	—
53	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
55	C₂×Dic₇₃	—	2	2	no	—
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	#100	1	2	yes	—
61	C₂×Dic₇₃	#101	1	2	yes	—
62	#10596833	—	4	8	no	—
63	C₈×C₇₃	—	8	1	no	—
64	#10596831	—	8	1	no	—
65	C₂²×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
66	C₂²×C₇₃	#339	1	2	yes	—
67	C₂²×C₇₃	#341	1	1	yes	characteristic
68	C₂²×C₇₃	#343	1	2	yes	—
69	C₂²×C₇₃	#343	1	1	yes	characteristic
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	4	1	no	—
73	C₄×C₇₃	#329	1	1	yes	characteristic
74	C₄×C₇₃	#329	1	1	yes	characteristic
75	C₄×C₇₃	#330	1	2	yes	—
76	Dic₇₃	—	2	2	no	—
77	Dic₇₃	—	2	2	no	—
78	Dic₇₃	—	4	2	no	—
79	#7042	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	#1285	1	2	yes	—
81	C₂×C₇₃	#1290	1	1	yes	characteristic
82	C₂×C₇₃	#1295	1	1	yes	characteristic
83	C₂×C₇₃	#1299	1	2	yes	—
84	C₂×C₇₃	#1309	1	1	yes	characteristic
85	#1831	—	73	1	no	—
86	#1767	—	73	2	no	—
87	C₇₃	#7042	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊C₈	—	73	1	no	—
89	C₂×C₄⋊C₈	—	146	2	no	—
90	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
91	C₂×#97	—	73	4	no	—
92	C₄³	—	73	1	no	—
93	#336	—	146	2	no	—
94	#336	—	146	2	no	—
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	—	146	1	no	—
97	C₂²×C₈	—	146	4	no	—
98	C₂×C₄²	—	73	1	no	—
99	C₂×C₄²	—	73	2	no	—
100	C₂×C₄²	—	73	2	no	—
101	C₂×C₄²	—	73	1	no	—
102	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	—	292	1	no	—
105	C₄⋊C₈	—	146	4	no	—
106	C₄⋊C₈	—	146	4	no	—
107	#97	—	292	4	no	—
108	C₂²×C₄	—	73	2	no	—
109	C₂²×C₄	—	73	2	no	—
110	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	292	4	no	—
116	C₂×C₈	—	292	4	no	—
117	C₂×C₈	—	292	1	no	—
118	C₂×C₈	—	292	2	no	—
119	C₂×C₈	—	146	8	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	292	1	no	—
123	C₄²	—	146	2	no	—
124	C₄²	—	146	2	no	—
125	C₄²	—	146	2	no	—
126	C₄²	—	146	2	no	—
127	C₄²	—	146	1	no	—
128	C₄²	—	146	1	no	—
129	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	73	2	no	—
135	C₂×C₄	—	73	2	no	—
136	C₂×C₄	—	146	2	no	—
137	C₂×C₄	—	146	2	no	—
138	C₂×C₄	—	146	2	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	#14556651	1	1	yes	characteristic
142	C₂×C₄	#14556652	1	1	yes	characteristic
143	C₂×C₄	#14556655	1	2	yes	—
144	C₂×C₄	#14556656	1	2	yes	—
145	C₂×C₄	#14556661	1	1	yes	characteristic
146	C₂×C₄	#14556662	1	1	yes	characteristic
147	C₈	—	8	1	no	—
148	C₈	—	292	8	no	—
149	C₈	—	584	1	no	—
150	C₂²	#17989455	1	1	yes	characteristic
151	C₂²	#17989457	1	2	yes	—
152	C₂²	#17989463	1	1	yes	characteristic
153	C₂²	#17989466	1	2	yes	—
154	C₂²	#17989467	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	4	1	no	—
158	C₄	—	292	2	no	—
159	C₄	—	146	2	no	—
160	C₄	—	146	2	no	—
161	C₄	#17989442	1	1	yes	characteristic
162	C₄	#17989443	1	1	yes	characteristic
163	C₄	#17989444	1	2	yes	—
164	C₂	#22624783	1	2	yes	—
165	C₂	#22624796	1	1	yes	characteristic
166	C₂	#22624807	1	1	yes	characteristic
167	C₂	#22624817	1	2	yes	—
168	C₂	#22624839	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

C₄

yes

—

#17989453

—

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

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

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

#100

yes

—

C₂×Dic₇₃

#101

yes

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

H(3,ℤ/4)

yes

characteristic

C₂²×C₇₃

#339

yes

—

C₂²×C₇₃

#341

yes

characteristic

C₂²×C₇₃

#343

yes

—

C₂²×C₇₃

#343

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

#7042

—

maximal, Sylow

C₂×C₇₃

#1285

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

yes

—

—

C₇₃

#7042

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄⋊C₈

#64408

yes

characteristic

C₂×#97

—

C₄³

—

#336

—

146

—

#336

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×#64408

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

292

—

105

C₄⋊C₈

—

146

—

106

C₄⋊C₈

—

146

—

107

#97

—

292

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×#64408

yes

characteristic

111

C₂²×C₄

C₂²⋊#64408

yes

characteristic

112

C₂²×C₄

C₄⋊#64408

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

292

—

116

C₂×C₈

—

292

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

292

—

119

C₂×C₈

—

146

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

292

—

123

C₄²

—

146

—

124

C₄²

—

146

—

125

C₄²

—

146

—

126

C₄²

—

146

—

127

C₄²

—

146

—

128

C₄²

—

146

—

129

C₂³

(C₂×#64408)⋊C₄

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

#14556651

yes

characteristic

142

C₂×C₄

#14556652

yes

characteristic

143

C₂×C₄

#14556655

yes

—

144

C₂×C₄

#14556656

yes

—

145

C₂×C₄

#14556661

yes

characteristic

146

C₂×C₄

#14556662

yes

characteristic

147

C₈

—

148

C₈

—

292

—

149

C₈

—

584

—

150

C₂²

#17989455

yes

characteristic

151

C₂²

#17989457

yes

—

152

C₂²

#17989463

yes

characteristic

153

C₂²

#17989466

yes

—

154

C₂²

#17989467

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

292

—

159

C₄

—

146

—

160

C₄

—

146

—

161

C₄

#17989442

yes

characteristic

162

C₄

#17989443

yes

characteristic

163

C₄

#17989444

yes

—

164

C₂

#22624783

yes

—

165

C₂

#22624796

yes

characteristic

166

C₂

#22624807

yes

characteristic

167

C₂

#22624817

yes

—

168

C₂

#22624839

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	#7042
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
87. C₇₃	#7042	semidirect

H ≅ G/N

type

87. C₇₃

#7042

semidirect

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

subgroup	index
#22625717	2
#22610047	2
#7042	73

subgroup

index

#22625717

#22610047

#7042

Subgroups — #33999961

Conjugacy classes of subgroups

Complements

Maximal subgroups