Group #34000652 — subgroups

Subgroups — #34000652

Order 18688.

180 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 60 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	#22625481	C₂	1	1	yes	maximal, characteristic
3	#22608691	C₂	1	1	yes	maximal, characteristic
4	#17989606	—	2	1	no	—
5	#17989508	—	2	1	no	—
6	#17989482	C₄	1	2	yes	—
7	#17989475	—	2	1	no	—
8	#17989450	—	2	1	no	—
9	#17989449	C₄	1	2	yes	—
10	#17988643	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988339	—	2	1	no	—
13	#17988338	C₄	1	1	yes	characteristic
14	#334×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924031	—	2	2	no	—
16	C₄⋊#10596833	—	2	2	no	—
17	#14556648	—	4	1	no	—
18	C₄×#10596833	—	4	1	no	—
19	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
21	C₄²×D₇₃	D₄	1	1	yes	characteristic
22	C₄⋊#10596831	—	2	1	no	—
23	#14556503	—	4	1	no	—
24	#14556501	—	4	1	no	—
25	C₄×#10596831	D₄	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	—	2	1	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#10596833	—	4	2	no	—
31	#11924031	—	4	2	no	—
32	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
34	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
35	C₂×#10596831	—	4	1	no	—
36	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
38	C₈×D₇₃	—	2	2	no	—
39	C₈×D₇₃	—	4	1	no	—
40	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
42	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₇₃	—	4	1	no	—
44	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×D₇₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
49	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
51	C₂×Dic₇₃	#102	1	1	yes	characteristic
52	C₂×Dic₇₃	#103	1	1	yes	characteristic
53	C₂×Dic₇₃	#106	1	1	yes	characteristic
54	C₄×D₇₃	—	2	1	no	—
55	C₄×D₇₃	—	2	1	no	—
56	C₄×D₇₃	—	4	1	no	—
57	C₄×D₇₃	—	4	1	no	—
58	#10596833	—	8	2	no	—
59	C₈×C₇₃	—	2	1	no	—
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	—	8	1	no	—
62	#10596831	—	2	1	no	—
63	#10596831	—	8	1	no	—
64	#10596831	#108	1	2	yes	—
65	C₂²×C₇₃	#340	1	1	yes	characteristic
66	C₂×D₇₃	—	2	1	no	—
67	C₂×D₇₃	—	2	1	no	—
68	C₂×D₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	#341	1	1	yes	characteristic
72	C₄×C₇₃	#344	1	1	yes	characteristic
73	Dic₇₃	—	2	1	no	—
74	Dic₇₃	—	2	1	no	—
75	Dic₇₃	—	2	1	no	—
76	#7293	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	#1301	1	1	yes	characteristic
78	C₂×C₇₃	#1384	1	1	yes	characteristic
79	C₂×C₇₃	#1388	1	1	yes	characteristic
80	D₇₃	—	4	1	no	—
81	#2163	—	73	1	no	—
82	#1568	—	73	1	no	—
83	#1564	—	73	1	no	—
84	C₇₃	#7293	1	1	yes	characteristic, complemented, Sylow
85	#432	—	146	1	no	—
86	#431	—	146	1	no	—
87	C₂×C₄×C₈	—	73	1	no	—
88	C₄⋊C₁₆	—	146	1	no	—
89	C₂²⋊C₁₆	—	73	2	no	—
90	#346	—	146	1	no	—
91	#335	—	146	1	no	—
92	#335	—	146	1	no	—
93	#334	—	73	2	no	—
94	#334	—	73	1	no	—
95	#334	#64408	1	1	yes	characteristic
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	—	73	1	no	—
98	C₂×C₄²	—	73	1	no	—
99	C₂×C₁₆	—	146	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	146	2	no	—
102	C₄⋊C₈	—	146	1	no	—
103	C₂²⋊C₈	—	292	1	no	—
104	C₂²⋊C₈	—	292	1	no	—
105	#97	—	292	1	no	—
106	C₄×C₈	—	292	1	no	—
107	C₄×C₈	—	73	1	no	—
108	C₄×C₈	—	73	1	no	—
109	C₄×C₈	—	73	1	no	—
110	C₄×C₈	C₂×#64408	1	1	yes	characteristic
111	C₂²×C₄	—	73	1	no	—
112	C₂²×C₄	—	73	1	no	—
113	C₂²×C₄	—	73	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	292	1	no	—
116	C₂×C₈	—	292	2	no	—
117	C₂×C₈	—	292	1	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	146	2	no	—
121	C₂×C₈	C₄×#64408	1	1	yes	characteristic
122	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
123	C₄²	—	73	1	no	—
124	C₄²	—	73	1	no	—
125	C₄²	—	73	1	no	—
126	C₄²	C₂²⋊#64408	1	1	yes	characteristic
127	C₁₆	—	292	2	no	—
128	C₂³	—	73	1	no	—
129	C₂×C₄	—	292	1	no	—
130	C₂×C₄	—	292	1	no	—
131	C₂×C₄	—	73	1	no	—
132	C₂×C₄	—	73	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	146	1	no	—
135	C₂×C₄	—	146	1	no	—
136	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	#14556653	1	1	yes	characteristic
138	C₂×C₄	#14556654	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	8	1	no	—
142	C₈	—	584	1	no	—
143	C₈	—	584	2	no	—
144	C₈	—	73	2	no	—
145	C₈	—	146	1	no	—
146	C₂²	—	146	1	no	—
147	C₂²	—	146	1	no	—
148	C₂²	—	146	1	no	—
149	C₂²	#17989460	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	146	1	no	—
153	C₄	—	146	1	no	—
154	C₄	—	146	1	no	—
155	C₄	#17989462	1	1	yes	characteristic
156	C₄	#17989469	1	1	yes	characteristic
157	C₂	—	292	1	no	—
158	C₂	#22624821	1	1	yes	characteristic
159	C₂	#22624998	1	1	yes	characteristic
160	C₂	#22625010	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626294

C₂

yes

maximal, characteristic

#22625481

C₂

yes

maximal, characteristic

#22608691

C₂

yes

maximal, characteristic

#17989606

—

#17989508

—

#17989482

C₄

yes

—

#17989475

—

#17989450

—

#17989449

C₄

yes

—

#17988643

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988339

—

#17988338

C₄

yes

characteristic

#334×C₇₃

C₄

yes

—

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

—

D₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×#10596833

—

#11924031

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

Q₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

SD₁₆

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

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

—

—

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2163

—

#1568

—

#1564

—

C₇₃

#7293

yes

characteristic, complemented, Sylow

#432

—

146

—

#431

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

#346

—

146

—

#335

—

146

—

#335

—

146

—

—

—

yes

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

146

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

146

—

102

C₄⋊C₈

—

146

—

103

C₂²⋊C₈

—

292

—

104

C₂²⋊C₈

—

292

—

105

#97

—

292

—

106

C₄×C₈

—

292

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₂×#64408

yes

characteristic

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂×C₈

—

115

C₂×C₈

—

292

—

116

C₂×C₈

—

292

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

146

—

121

C₂×C₈

C₄×#64408

yes

characteristic

122

C₂×C₈

C₄⋊#64408

yes

characteristic

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₂²⋊#64408

yes

characteristic

127

C₁₆

—

292

—

128

C₂³

—

129

C₂×C₄

—

292

—

130

C₂×C₄

—

292

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

137

C₂×C₄

#14556653

yes

characteristic

138

C₂×C₄

#14556654

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

584

—

143

C₈

—

584

—

144

C₈

—

145

C₈

—

146

—

146

C₂²

—

146

—

147

C₂²

—

146

—

148

C₂²

—

146

—

149

C₂²

#17989460

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

146

—

153

C₄

—

146

—

154

C₄

—

146

—

155

C₄

#17989462

yes

characteristic

156

C₄

#17989469

yes

characteristic

157

C₂

—

292

—

158

C₂

#22624821

yes

characteristic

159

C₂

#22624998

yes

characteristic

160

C₂

#22625010

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	#107
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#334	#64408
hypocenter	C₇₃	#7293
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
84. C₇₃	#7293	semidirect

H ≅ G/N

type

84. C₇₃

#7293

semidirect

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

subgroup	index
#22626294	2
#22625481	2
#22608691	2
#7293	73

subgroup

index

Subgroups — #34000652

Conjugacy classes of subgroups

Complements

Maximal subgroups