Group #34000086 — subgroups

Subgroups — #34000086

Order 18688.

268 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 56 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	#22624759	C₂	1	2	yes	maximal, complemented
2	#22610758	C₂	1	1	yes	maximal, characteristic
3	#17989603	—	2	2	no	—
4	#17989137	C₄	1	1	yes	characteristic
5	C₂×#14556517	—	2	2	no	—
6	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
7	#423×C₇₃	C₄	1	1	yes	characteristic
8	C₄⋊#10596833	—	2	4	no	—
9	C₂²⋊#10596833	—	4	2	no	—
10	C₄×#10596833	—	4	2	no	—
11	C₂×#11924028	—	2	1	no	—
12	#14556591	C₈	1	1	yes	characteristic, complemented
13	#14556588	C₈	1	1	yes	characteristic, complemented
14	#14556579	—	2	1	no	—
15	#14556565	—	4	1	no	—
16	C₄×D₂₉₂	—	4	1	no	—
17	C₂×C₄×Dic₇₃	—	4	1	no	—
18	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
19	#14556517	—	4	2	no	—
20	#14556517	—	4	2	no	—
21	C₂×M₁₆×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	#11924028	—	4	2	no	—
26	#11924028	—	4	2	no	—
27	C₂×Dic₁₄₆	—	2	1	no	—
28	C₂×D₂₉₂	—	2	1	no	—
29	C₂×D₂₉₂	—	4	1	no	—
30	C₂×C₂²⋊D₇₃	—	4	1	no	—
31	C₂²×Dic₇₃	—	2	2	no	—
32	C₂×C₄×D₇₃	—	4	1	no	—
33	C₄⋊Dic₇₃	—	4	1	no	—
34	#11924005	—	4	2	no	—
35	C₄×Dic₇₃	—	2	2	no	—
36	C₄×Dic₇₃	—	4	4	no	—
37	C₄×Dic₇₃	D₈	1	2	yes	—
38	C₄×Dic₇₃	SD₁₆	1	2	yes	—
39	C₂²×C₄×C₇₃	—	2	1	no	—
40	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	M₁₆×C₇₃	—	4	2	no	—
42	C₂×C₈×C₇₃	—	4	1	no	—
43	C₄²×C₇₃	—	2	1	no	—
44	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
46	C₂³×C₇₃	#99	1	1	yes	characteristic
47	C₂²×D₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	4	2	no	—
51	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#102	1	2	yes	—
54	C₂×C₄×C₇₃	#104	1	2	yes	—
55	C₂²⋊D₇₃	—	8	2	no	—
56	C₂×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	—	2	2	no	—
58	C₂×Dic₇₃	—	4	4	no	—
59	C₂×Dic₇₃	—	4	1	no	—
60	D₂₉₂	—	4	2	no	—
61	D₂₉₂	—	8	2	no	—
62	C₄×D₇₃	—	8	1	no	—
63	C₄×D₇₃	—	8	1	no	—
64	Dic₁₄₆	—	4	2	no	—
65	#10596833	—	8	4	no	—
66	C₈×C₇₃	—	8	1	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	#323	1	1	yes	characteristic
69	C₂²×C₇₃	#327	1	2	yes	—
70	C₂×D₇₃	—	8	2	no	—
71	C₂×D₇₃	—	8	1	no	—
72	C₄×C₇₃	—	2	2	no	—
73	C₄×C₇₃	—	4	1	no	—
74	C₄×C₇₃	D₄⋊C₈	1	2	yes	—
75	C₄×C₇₃	#329	1	1	yes	characteristic
76	C₄×C₇₃	#331	1	1	yes	characteristic
77	Dic₇₃	—	2	4	no	—
78	Dic₇₃	—	4	2	no	—
79	Dic₇₃	—	8	1	no	—
80	#7106	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	—	2	2	no	—
82	C₂×C₇₃	—	2	1	no	—
83	C₂×C₇₃	#1270	1	1	yes	characteristic
84	D₇₃	—	16	1	no	—
85	#1962	—	73	1	no	—
86	#1276	—	73	2	no	—
87	C₇₃	#7106	1	1	yes	characteristic, complemented, Sylow
88	#532	—	73	1	no	—
89	#431	—	146	2	no	—
90	#423	#64408	1	1	yes	characteristic
91	C₂×#104	—	146	2	no	—
92	C₄³	—	73	1	no	—
93	C₂×#41	—	146	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₄⋊Q₈	—	73	1	no	—
96	#127	—	73	1	no	—
97	#124	—	146	1	no	—
98	#121	—	292	1	no	—
99	C₄×D₄	—	292	1	no	—
100	C₂×C₄²	—	292	1	no	—
101	C₂×C₄²	—	73	2	no	—
102	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	—	146	4	no	—
105	#104	—	292	2	no	—
106	#104	—	292	2	no	—
107	C₂²⋊C₈	—	292	2	no	—
108	C₄×C₈	—	292	2	no	—
109	#41	—	292	2	no	—
110	#41	—	292	2	no	—
111	C₂×Q₈	—	146	1	no	—
112	C₂×D₄	—	292	1	no	—
113	C₂×D₄	—	292	1	no	—
114	C₂×D₄	—	146	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	292	1	no	—
117	C₂²×C₄	—	146	2	no	—
118	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	292	4	no	—
122	C₄⋊C₄	—	292	1	no	—
123	C₂²⋊C₄	—	292	2	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	292	4	no	—
126	C₄²	—	73	2	no	—
127	C₄²	—	73	2	no	—
128	C₄²	—	146	2	no	—
129	C₄²	#11924036	1	1	yes	characteristic
130	C₄²	#11924038	1	1	yes	characteristic
131	C₂³	—	292	1	no	—
132	C₂³	#14556650	1	1	yes	characteristic
133	Q₈	—	292	2	no	—
134	D₄	—	292	2	no	—
135	D₄	—	584	2	no	—
136	D₄	—	584	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	292	4	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	—	584	1	no	—
143	C₂×C₄	—	584	1	no	—
144	C₂×C₄	—	146	2	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	#14556648	1	1	yes	characteristic
147	C₂×C₄	#14556651	1	1	yes	characteristic
148	C₂×C₄	#14556653	1	2	yes	—
149	C₂×C₄	#14556656	1	2	yes	—
150	C₈	—	8	1	no	—
151	C₈	—	584	4	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	584	2	no	—
154	C₂²	—	584	1	no	—
155	C₂²	#17989431	1	1	yes	characteristic
156	C₂²	#17989439	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	4	1	no	—
159	C₄	—	292	2	no	—
160	C₄	—	584	1	no	—
161	C₄	—	146	4	no	—
162	C₄	#17989435	1	2	yes	—
163	C₄	#17989443	1	1	yes	characteristic
164	C₄	#17989445	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	2	no	—
167	C₂	—	1168	1	no	—
168	C₂	#22624748	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22624759

C₂

yes

maximal, complemented

#22610758

C₂

yes

maximal, characteristic

#17989603

—

#17989137

C₄

yes

characteristic

C₂×#14556517

—

C₄²×Dic₇₃

C₂²

yes

characteristic

#423×C₇₃

C₄

yes

—

—

—

—

C₈

yes

characteristic, complemented

#14556588

C₈

yes

characteristic, complemented

#14556579

—

#14556565

—

C₄×D₂₉₂

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

D₄

yes

—

#14556517

—

#14556517

—

C₂×M₁₆×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂×#10596833

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₈

yes

—

C₄×Dic₇₃

SD₁₆

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

#99

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#104

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#323

yes

characteristic

C₂²×C₇₃

#327

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

D₄⋊C₈

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

#7106

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1270

yes

characteristic

D₇₃

—

#1962

—

#1276

—

C₇₃

#7106

yes

characteristic, complemented, Sylow

#532

—

#431

—

146

—

yes

—

146

—

C₄³

—

C₂×#41

—

146

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

146

—

#121

—

292

—

C₄×D₄

—

292

—

100

C₂×C₄²

—

292

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×#64408

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

146

—

105

#104

—

292

—

106

#104

—

292

—

107

C₂²⋊C₈

—

292

—

108

C₄×C₈

—

292

—

109

#41

—

292

—

110

#41

—

292

—

111

C₂×Q₈

—

146

—

112

C₂×D₄

—

292

—

113

C₂×D₄

—

292

—

114

C₂×D₄

—

146

—

115

C₂²×C₄

—

116

C₂²×C₄

—

292

—

117

C₂²×C₄

—

146

—

118

C₂²×C₄

C₂²⋊#64408

yes

characteristic

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

292

—

122

C₄⋊C₄

—

292

—

123

C₂²⋊C₄

—

292

—

124

C₄²

—

125

C₄²

—

292

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

146

—

129

C₄²

#11924036

yes

characteristic

130

C₄²

#11924038

yes

characteristic

131

C₂³

—

292

—

132

C₂³

#14556650

yes

characteristic

133

Q₈

—

292

—

134

D₄

—

292

—

135

D₄

—

584

—

136

D₄

—

584

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

292

—

142

C₂×C₄

—

584

—

143

C₂×C₄

—

584

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

#14556648

yes

characteristic

147

C₂×C₄

#14556651

yes

characteristic

148

C₂×C₄

#14556653

yes

—

149

C₂×C₄

#14556656

yes

—

150

C₈

—

151

C₈

—

584

—

152

C₂²

—

153

C₂²

—

584

—

154

C₂²

—

584

—

155

C₂²

#17989431

yes

characteristic

156

C₂²

#17989439

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

292

—

160

C₄

—

584

—

161

C₄

—

146

—

162

C₄

#17989435

yes

—

163

C₄

#17989443

yes

characteristic

164

C₄

#17989445

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

1168

—

168

C₂

#22624748

yes

characteristic

	subgroup	G/H
center	C₄	#17989445
commutator	C₄²×C₇₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	#423×C₇₃	C₄
socle	C₂×C₇₃	#1270
prime radical	#17989137	C₄
cosocle	C₄²×Dic₇₃	C₂²
hypercenter	#423	#64408
hypocenter	C₇₃	#7106
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₄

#17989445

commutator

C₄²×C₇₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#64408

Fitting

#423×C₇₃

C₄

C₂×C₇₃

C₄

C₄²×Dic₇₃

C₂²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #22624759	C₂	semidirect
12. #14556591	C₈	semidirect
13. #14556588	C₈	semidirect
87. C₇₃	#7106	semidirect

H ≅ G/N

type

1. #22624759

C₂

semidirect

12. #14556591

C₈

semidirect

13. #14556588

C₈

semidirect

87. C₇₃

#7106

semidirect

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

subgroup	index
#22624759	2
#22610758	2
#7106	73

subgroup

index

#22624759

#22610758

#7106

Subgroups — #34000086

Conjugacy classes of subgroups

Complements

Maximal subgroups