Group #34000657 — subgroups

Subgroups — #34000657

Order 18688.

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

normal subgroups only

This group has 190 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	#22626295	C₂	1	1	yes	maximal, characteristic
2	#22625492	C₂	1	1	yes	maximal, characteristic
3	#334×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989609	—	2	1	no	—
5	C₄⋊#11924031	—	2	1	no	—
6	#17989482	C₄	1	2	yes	—
7	#17989476	—	2	1	no	—
8	#17989450	—	2	1	no	—
9	#17989448	C₄	1	2	yes	—
10	C₄⋊C₈×D₇₃	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988338	C₄	1	1	yes	characteristic
13	#17988337	—	2	1	no	—
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	#14556646	—	4	1	no	—
19	C₂×C₈×D₇₃	—	4	1	no	—
20	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
22	C₄²×D₇₃	D₄	1	1	yes	characteristic
23	C₄⋊#10596831	—	2	1	no	—
24	#14556520	—	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	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×D₇₃	SD₁₆	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	1	no	—
39	C₈×D₇₃	—	2	1	no	—
40	C₈×D₇₃	—	4	1	no	—
41	C₈×D₇₃	—	8	2	no	—
42	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₇₃	D₈	1	1	yes	characteristic
44	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	4	1	no	—
46	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×D₇₃	#106	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
53	C₂×Dic₇₃	#102	1	1	yes	characteristic
54	C₂×Dic₇₃	#103	1	1	yes	characteristic
55	C₂×Dic₇₃	#107	1	1	yes	characteristic
56	C₄×D₇₃	—	2	2	no	—
57	C₄×D₇₃	—	2	1	no	—
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	—	4	1	no	—
60	#10596833	—	8	2	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	8	1	no	—
63	C₈×C₇₃	#108	1	1	yes	characteristic
64	C₈×C₇₃	#108	1	1	yes	characteristic
65	#10596831	—	2	1	no	—
66	#10596831	—	2	1	no	—
67	#10596831	—	8	1	no	—
68	C₂²×C₇₃	#340	1	1	yes	characteristic
69	C₂×D₇₃	—	2	1	no	—
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	#365	1	2	yes	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	#341	1	1	yes	characteristic
75	C₄×C₇₃	#344	1	1	yes	characteristic
76	Dic₇₃	—	2	1	no	—
77	Dic₇₃	—	2	1	no	—
78	Dic₇₃	—	2	1	no	—
79	#7294	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	#1301	1	1	yes	characteristic
81	C₂×C₇₃	#1383	1	1	yes	characteristic
82	C₂×C₇₃	#1389	1	1	yes	characteristic
83	D₇₃	—	2	2	no	—
84	#2163	—	73	1	no	—
85	#1568	—	73	1	no	—
86	C₂×#334	—	73	1	no	—
87	C₇₃	#7294	1	1	yes	characteristic, complemented, Sylow
88	#432	—	146	1	no	—
89	C₂×C₄⋊C₈	—	146	1	no	—
90	C₂×C₄×C₈	—	73	1	no	—
91	C₄⋊C₁₆	—	146	1	no	—
92	C₂²⋊C₁₆	—	73	2	no	—
93	#346	—	146	1	no	—
94	#335	—	146	1	no	—
95	#334	—	73	2	no	—
96	#334	—	73	1	no	—
97	#334	—	146	1	no	—
98	#334	#64408	1	1	yes	characteristic
99	C₂²×C₈	—	292	1	no	—
100	C₂²×C₈	—	73	1	no	—
101	C₂²×C₈	—	73	1	no	—
102	C₂×C₄²	—	73	1	no	—
103	C₂×C₁₆	—	146	2	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	292	1	no	—
106	C₄⋊C₈	—	146	2	no	—
107	C₄⋊C₈	—	146	1	no	—
108	C₂²⋊C₈	—	292	1	no	—
109	#97	—	292	1	no	—
110	C₄×C₈	—	73	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	C₂×#64408	1	1	yes	characteristic
114	C₂²×C₄	—	73	1	no	—
115	C₂²×C₄	—	73	1	no	—
116	C₂²×C₄	—	73	1	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	292	1	no	—
119	C₂×C₈	—	292	2	no	—
120	C₂×C₈	—	292	1	no	—
121	C₂×C₈	—	584	2	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	146	1	no	—
125	C₂×C₈	—	146	1	no	—
126	C₂×C₈	C₄×#64408	1	1	yes	characteristic
127	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
128	C₄²	—	73	1	no	—
129	C₄²	—	73	1	no	—
130	C₄²	—	73	1	no	—
131	C₄²	C₂²⋊#64408	1	1	yes	characteristic
132	C₁₆	—	292	2	no	—
133	C₂³	—	73	1	no	—
134	C₂×C₄	—	292	1	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	146	2	no	—
139	C₂×C₄	—	146	1	no	—
140	C₂×C₄	—	146	1	no	—
141	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
142	C₂×C₄	#14556653	1	1	yes	characteristic
143	C₂×C₄	#14556654	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	8	1	no	—
146	C₈	—	584	1	no	—
147	C₈	—	584	2	no	—
148	C₈	—	146	1	no	—
149	C₈	—	146	1	no	—
150	C₈	#14556661	1	1	yes	characteristic
151	C₈	#14556662	1	1	yes	characteristic
152	C₂²	—	73	2	no	—
153	C₂²	—	146	1	no	—
154	C₂²	—	146	1	no	—
155	C₂²	#17989460	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	146	1	no	—
159	C₄	—	146	1	no	—
160	C₄	—	146	1	no	—
161	C₄	#17989463	1	1	yes	characteristic
162	C₄	#17989470	1	1	yes	characteristic
163	C₂	—	146	2	no	—
164	C₂	#22624823	1	1	yes	characteristic
165	C₂	#22624995	1	1	yes	characteristic
166	C₂	#22625011	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626295

C₂

yes

maximal, characteristic

#22625492

C₂

yes

maximal, characteristic

#334×D₇₃

C₂

yes

maximal, characteristic

#17989609

—

C₄⋊#11924031

—

#17989482

C₄

yes

—

#17989476

—

#17989450

—

#17989448

C₄

yes

—

C₄⋊C₈×D₇₃

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988338

C₄

yes

characteristic

#17988337

—

#334×C₇₃

C₄

yes

—

—

—

—

C₂×C₈×D₇₃

—

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

SD₁₆

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

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

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

—

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#365

yes

—

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

—

C₂×#334

—

C₇₃

#7294

yes

characteristic, complemented, Sylow

#432

—

146

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

#346

—

146

—

#335

—

146

—

#334

—

#334

—

#334

—

146

—

#334

#64408

yes

characteristic

C₂²×C₈

—

292

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄²

—

103

C₂×C₁₆

—

146

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

292

—

106

C₄⋊C₈

—

146

—

107

C₄⋊C₈

—

146

—

108

C₂²⋊C₈

—

292

—

109

#97

—

292

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂×#64408

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂×C₈

—

118

C₂×C₈

—

292

—

119

C₂×C₈

—

292

—

120

C₂×C₈

—

292

—

121

C₂×C₈

—

584

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

146

—

125

C₂×C₈

—

146

—

126

C₂×C₈

C₄×#64408

yes

characteristic

127

C₂×C₈

C₄⋊#64408

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²⋊#64408

yes

characteristic

132

C₁₆

—

292

—

133

C₂³

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

142

C₂×C₄

#14556653

yes

characteristic

143

C₂×C₄

#14556654

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

584

—

147

C₈

—

584

—

148

C₈

—

146

—

149

C₈

—

146

—

150

C₈

#14556661

yes

characteristic

151

C₈

#14556662

yes

characteristic

152

C₂²

—

153

C₂²

—

146

—

154

C₂²

—

146

—

155

C₂²

#17989460

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

146

—

159

C₄

—

146

—

160

C₄

—

146

—

161

C₄

#17989463

yes

characteristic

162

C₄

#17989470

yes

characteristic

163

C₂

—

146

—

164

C₂

#22624823

yes

characteristic

165

C₂

#22624995

yes

characteristic

166

C₂

#22625011

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	#106
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#334	#64408
hypocenter	C₇₃	#7294
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
87. C₇₃	#7294	semidirect

H ≅ G/N

type

87. C₇₃

#7294

semidirect

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

subgroup	index
#22626295	2
#22625492	2
#334×D₇₃	2
#7294	73

subgroup

index

#22626295

#22625492

#334×D₇₃

#7294

Subgroups — #34000657

Conjugacy classes of subgroups

Complements

Maximal subgroups