Group #34000679 — subgroups

Subgroups — #34000679

Order 18688.

238 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22626294	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556659	C₂	1	1	yes	maximal, characteristic
3	#22608685	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556659	—	2	1	no	—
5	C₄×C₄⋊#64408	—	2	1	no	—
6	#17989508	C₄	1	2	yes	complemented
7	#17989482	—	2	1	no	—
8	#17989475	—	2	1	no	—
9	#17989461	C₄	1	2	yes	—
10	#17988609	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988338	C₄	1	1	yes	characteristic, complemented
13	#17988337	—	2	1	no	—
14	#334×C₇₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#64408	—	2	2	no	—
16	C₂×C₄×#64408	—	4	1	no	—
17	C₂×#11924031	—	2	2	no	—
18	#14556659	—	2	4	no	—
19	(C₂×#64408)⋊C₄	—	4	1	no	—
20	C₂×#11924011	—	4	1	no	—
21	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
23	C₄²×D₇₃	D₄	1	1	yes	characteristic
24	C₄⋊#10596831	—	2	1	no	—
25	#14556520	—	4	1	no	—
26	C₄×#10596831	D₄	1	1	yes	characteristic
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	1	no	—
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×#64408	—	4	2	no	—
32	C₄⋊#64408	—	2	4	no	—
33	C₄⋊#64408	—	4	2	no	—
34	C₄×#64408	—	8	2	no	—
35	#11924031	—	4	2	no	—
36	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
38	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
39	#11924011	—	8	2	no	—
40	C₂×#10596831	—	4	1	no	—
41	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
43	C₈×D₇₃	—	2	2	no	—
44	C₈×D₇₃	—	4	1	no	—
45	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
47	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
48	C₂×C₈×C₇₃	—	4	1	no	—
49	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×D₇₃	#102	1	1	yes	characteristic
53	C₂×#64408	—	4	4	no	—
54	C₂×#64408	—	8	2	no	—
55	C₂×#64408	—	8	2	no	—
56	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
59	C₂×Dic₇₃	#103	1	1	yes	characteristic
60	C₂×Dic₇₃	#106	1	1	yes	characteristic
61	C₂×Dic₇₃	#107	1	1	yes	characteristic
62	C₄×D₇₃	—	2	2	no	—
63	C₄×D₇₃	—	2	1	no	—
64	C₄×D₇₃	—	2	1	no	—
65	C₄×D₇₃	—	2	2	no	—
66	C₈×C₇₃	—	2	1	no	—
67	C₈×C₇₃	—	2	1	no	—
68	C₈×C₇₃	—	8	1	no	—
69	#10596831	—	2	1	no	—
70	#10596831	—	8	1	no	—
71	#10596831	#104	1	2	yes	—
72	C₂²×C₇₃	#340	1	1	yes	characteristic
73	C₂×D₇₃	—	2	1	no	—
74	C₂×D₇₃	—	2	1	no	—
75	C₂×D₇₃	#360	1	2	yes	—
76	#64408	—	8	4	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	#339	1	1	yes	characteristic
80	C₄×C₇₃	#344	1	1	yes	characteristic
81	Dic₇₃	—	2	1	no	—
82	Dic₇₃	—	2	1	no	—
83	Dic₇₃	#365	1	2	yes	—
84	#7302	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	#1303	1	1	yes	characteristic
86	C₂×C₇₃	#1383	1	1	yes	characteristic
87	C₂×C₇₃	#1386	1	1	yes	characteristic
88	D₇₃	—	2	2	no	—
89	#2163	—	73	1	no	—
90	C₄×#106	—	73	1	no	—
91	#1566	—	73	1	no	—
92	C₇₃	#7302	1	1	yes	characteristic, complemented, Sylow
93	C₂×#106	—	146	1	no	—
94	#423	—	146	1	no	—
95	C₂×C₄×C₈	—	73	1	no	—
96	C₄×C₄⋊C₄	—	146	1	no	—
97	C₄⋊C₁₆	—	73	2	no	—
98	C₂²⋊C₁₆	—	146	1	no	—
99	#346	—	146	1	no	—
100	#340	—	73	2	no	—
101	#334	—	73	1	no	—
102	#334	—	146	1	no	—
103	#334	#64408	1	1	yes	characteristic, complemented
104	C₂×M₁₆	—	292	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂×C₄⋊C₄	—	146	2	no	—
108	C₂×C₄²	—	292	1	no	—
109	C₂×C₄²	—	73	1	no	—
110	C₂×C₁₆	—	146	2	no	—
111	#106	—	146	4	no	—
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	—	292	1	no	—
114	C₄⋊C₈	—	146	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	C₂×#64408	1	1	yes	characteristic
119	#95	—	292	1	no	—
120	C₂²×C₄	—	292	2	no	—
121	C₂²×C₄	—	73	1	no	—
122	C₂²×C₄	—	73	1	no	—
123	C₂²×C₄	—	73	1	no	—
124	M₁₆	—	584	2	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	292	1	no	—
127	C₂×C₈	—	292	1	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	146	2	no	—
131	C₂×C₈	C₄×#64408	1	1	yes	characteristic
132	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
133	C₄⋊C₄	—	292	2	no	—
134	C₄⋊C₄	—	146	4	no	—
135	C₄²	—	584	2	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	73	1	no	—
138	C₄²	—	73	1	no	—
139	C₄²	C₂²⋊#64408	1	1	yes	characteristic
140	C₁₆	—	292	2	no	—
141	C₂³	—	73	1	no	—
142	C₂×C₄	—	292	4	no	—
143	C₂×C₄	—	584	2	no	—
144	C₂×C₄	—	584	2	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	146	2	no	—
149	C₂×C₄	—	146	2	no	—
150	C₂×C₄	—	146	1	no	—
151	C₂×C₄	—	146	1	no	—
152	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
153	C₂×C₄	#14556653	1	1	yes	characteristic
154	C₂×C₄	#14556654	1	1	yes	characteristic
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	8	1	no	—
158	C₈	—	584	1	no	—
159	C₈	—	73	2	no	—
160	C₈	—	146	1	no	—
161	C₂²	—	73	2	no	—
162	C₂²	—	146	1	no	—
163	C₂²	—	146	1	no	—
164	C₂²	#17989460	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	584	4	no	—
168	C₄	—	73	2	no	—
169	C₄	—	146	1	no	—
170	C₄	—	146	1	no	—
171	C₄	#17989459	1	1	yes	characteristic
172	C₄	#17989469	1	1	yes	characteristic
173	C₂	—	146	2	no	—
174	C₂	#22624827	1	1	yes	characteristic
175	C₂	#22624997	1	1	yes	characteristic
176	C₂	#22625004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626294

C₂

yes

maximal, characteristic

C₄×#14556659

C₂

yes

maximal, characteristic

#22608685

C₂

yes

maximal, characteristic

C₂×#14556659

—

C₄×C₄⋊#64408

—

#17989508

C₄

yes

—

—

C₄

yes

—

#17988609

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988338

C₄

yes

characteristic, complemented

#17988337

—

#334×C₇₃

C₄

yes

characteristic, complemented

C₂×C₄⋊#64408

—

C₂×C₄×#64408

—

—

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

—

—

—

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

C₂²×D₇₃

yes

—

—

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

—

—

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#360

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#365

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2163

—

C₄×#106

—

#1566

—

C₇₃

#7302

yes

characteristic, complemented, Sylow

C₂×#106

—

146

—

#423

—

146

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

146

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

#346

—

146

—

100

#340

—

101

#334

—

102

#334

—

146

—

103

#334

#64408

yes

characteristic, complemented

104

C₂×M₁₆

—

292

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄⋊C₄

—

146

—

108

C₂×C₄²

—

292

—

109

C₂×C₄²

—

110

C₂×C₁₆

—

146

—

111

#106

—

146

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

292

—

114

C₄⋊C₈

—

146

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

C₂×#64408

yes

characteristic

119

#95

—

292

—

120

C₂²×C₄

—

292

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

M₁₆

—

584

—

125

C₂×C₈

—

126

C₂×C₈

—

292

—

127

C₂×C₈

—

292

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

146

—

131

C₂×C₈

C₄×#64408

yes

characteristic

132

C₂×C₈

C₄⋊#64408

yes

characteristic

133

C₄⋊C₄

—

292

—

134

C₄⋊C₄

—

146

—

135

C₄²

—

584

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂²⋊#64408

yes

characteristic

140

C₁₆

—

292

—

141

C₂³

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

584

—

144

C₂×C₄

—

584

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

146

—

149

C₂×C₄

—

146

—

150

C₂×C₄

—

146

—

151

C₂×C₄

—

146

—

152

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

153

C₂×C₄

#14556653

yes

characteristic

154

C₂×C₄

#14556654

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

584

—

159

C₈

—

160

C₈

—

146

—

161

C₂²

—

162

C₂²

—

146

—

163

C₂²

—

146

—

164

C₂²

#17989460

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

584

—

168

C₄

—

169

C₄

—

146

—

170

C₄

—

146

—

171

C₄

#17989459

yes

characteristic

172

C₄

#17989469

yes

characteristic

173

C₂

—

146

—

174

C₂

#22624827

yes

characteristic

175

C₂

#22624997

yes

characteristic

176

C₂

#22625004

yes

characteristic

	subgroup	G/H
center	C₂²	#17989460
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₄×C₈	C₂×#64408
Fitting	#334×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#102
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#334	#64408
hypocenter	C₇₃	#7302
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#17989460

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₄×C₈

C₂×#64408

Fitting

#334×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×D₇₃

C₂²

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
6. #17989508	C₄	semidirect
12. #17988338	C₄	semidirect
14. #334×C₇₃	C₄	semidirect
92. C₇₃	#7302	semidirect
103. #334	#64408	semidirect

H ≅ G/N

type

6. #17989508

C₄

semidirect

12. #17988338

C₄

semidirect

14. #334×C₇₃

C₄

semidirect

92. C₇₃

#7302

semidirect

103. #334

#64408

semidirect

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

subgroup	index
#22626294	2
C₄×#14556659	2
#22608685	2
#7302	73

subgroup

index

Subgroups — #34000679

Conjugacy classes of subgroups

Complements

Maximal subgroups