Group #34005500 — subgroups

Subgroups — #34005500

Order 18688.

354 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 140 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	#22625870	C₂	1	2	yes	maximal
2	#22625490	C₂	1	2	yes	maximal
3	#22625489	C₂	1	2	yes	maximal
4	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556661	—	2	4	no	—
6	#17989591	C₂²	1	4	yes	—
7	#17989451	—	2	4	no	—
8	#17989450	—	2	4	no	—
9	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	C₈×#10596831	C₄	1	1	yes	characteristic
13	C₈×#10596831	C₄	1	1	yes	characteristic
14	C₈×#10596831	C₄	1	1	yes	characteristic
15	C₈²×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924038	—	2	4	no	—
17	#14556661	—	4	8	no	—
18	C₄⋊#10596833	—	2	4	no	—
19	C₄⋊#10596833	—	2	4	no	—
20	C₂×C₈×D₇₃	D₄	1	2	yes	—
21	C₂×C₈×D₇₃	Q₈	1	2	yes	—
22	C₂×C₈×D₇₃	Q₈	1	2	yes	—
23	C₄²×D₇₃	C₂³	1	1	yes	characteristic
24	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	D₄	1	2	yes	—
28	C₈×Dic₇₃	D₄	1	2	yes	—
29	C₈×Dic₇₃	Q₈	1	2	yes	—
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	#11924038	—	4	8	no	—
34	C₂×#10596833	—	4	4	no	—
35	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×#10596831	C₄⋊C₄	1	2	yes	—
39	C₂×#10596831	C₄⋊C₄	1	2	yes	—
40	C₂×#10596831	C₄⋊C₄	1	2	yes	—
41	C₈×D₇₃	—	2	4	no	—
42	C₈×D₇₃	—	2	4	no	—
43	C₈×D₇₃	D₈	1	4	yes	—
44	C₈×D₇₃	Q₁₆	1	4	yes	—
45	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
46	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
47	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
51	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×Dic₇₃	#127	1	1	yes	characteristic
57	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×D₇₃	—	2	2	no	—
60	C₄×D₇₃	—	2	2	no	—
61	C₄×D₇₃	C₂×D₈	1	2	yes	—
62	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
63	#10596833	—	8	4	no	—
64	C₈×C₇₃	—	2	2	no	—
65	C₈×C₇₃	#107	1	4	yes	—
66	C₈×C₇₃	#108	1	4	yes	complemented
67	#10596831	—	2	2	no	—
68	#10596831	#107	1	4	yes	—
69	#10596831	#108	1	4	yes	complemented
70	C₂²×C₇₃	#384	1	1	yes	characteristic
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	—	2	1	no	—
73	C₂×D₇₃	#500	1	2	yes	—
74	C₄×C₇₃	C₂×#107	1	2	yes	—
75	C₄×C₇₃	C₂×#108	1	2	yes	—
76	C₄×C₇₃	C₂×#108	1	2	yes	—
77	Dic₇₃	—	2	1	no	—
78	Dic₇₃	—	2	1	no	—
79	Dic₇₃	#493	1	1	yes	characteristic
80	Dic₇₃	#494	1	1	yes	characteristic
81	#8823	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	#1845	1	1	yes	characteristic
83	C₂×C₇₃	#1848	1	1	yes	characteristic
84	C₂×C₇₃	#1848	1	1	yes	characteristic
85	D₇₃	—	2	2	no	—
86	#1848	—	73	2	no	—
87	#1567	—	73	2	no	—
88	#1567	—	73	2	no	—
89	C₂×C₈²	—	73	1	no	—
90	C₇₃	#8823	1	1	yes	characteristic, complemented, Sylow
91	C₂×#108	—	146	4	no	—
92	#423	—	73	4	no	—
93	C₂×C₄×C₈	—	73	1	no	—
94	C₂×C₄×C₈	—	73	1	no	—
95	C₂×C₄×C₈	—	73	1	no	—
96	#335	—	146	4	no	—
97	#335	—	146	4	no	—
98	C₈²	—	73	1	no	—
99	C₈²	—	73	1	no	—
100	C₈²	—	73	1	no	—
101	C₈²	#64408	1	1	yes	characteristic
102	C₂×M₁₆	—	146	4	no	—
103	C₂²×C₈	—	73	2	no	—
104	C₂²×C₈	—	73	2	no	—
105	C₂²×C₈	—	73	2	no	—
106	C₂×C₄²	—	73	1	no	—
107	#108	—	292	8	no	—
108	C₄⋊C₈	—	146	4	no	—
109	C₄⋊C₈	—	146	4	no	—
110	C₄×C₈	—	73	2	no	—
111	C₄×C₈	—	73	2	no	—
112	C₄×C₈	—	73	2	no	—
113	C₄×C₈	—	73	1	no	—
114	C₄×C₈	—	73	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	C₂×#64408	1	1	yes	characteristic
117	C₄×C₈	C₂×#64408	1	1	yes	characteristic
118	C₄×C₈	C₂×#64408	1	1	yes	characteristic
119	C₂²×C₄	—	73	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	M₁₆	—	292	8	no	—
123	C₂×C₈	—	292	4	no	—
124	C₂×C₈	—	73	2	no	—
125	C₂×C₈	—	73	2	no	—
126	C₂×C₈	—	73	4	no	—
127	C₂×C₈	—	73	4	no	—
128	C₂×C₈	—	73	2	no	—
129	C₂×C₈	—	146	4	no	—
130	C₂×C₈	—	146	4	no	—
131	C₂×C₈	C₄⋊#64408	1	2	yes	—
132	C₂×C₈	C₄⋊#64408	1	2	yes	—
133	C₂×C₈	C₄⋊#64408	1	2	yes	—
134	C₄²	—	73	1	no	—
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	C₂²×#64408	1	1	yes	characteristic
138	C₂³	—	73	1	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	73	2	no	—
141	C₂×C₄	—	73	2	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	146	2	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	584	4	no	—
151	C₈	—	73	4	no	—
152	C₈	—	73	4	no	—
153	C₈	—	146	2	no	—
154	C₈	#14556660	1	4	yes	—
155	C₈	#14556661	1	4	yes	complemented
156	C₂²	—	73	2	no	—
157	C₂²	—	146	1	no	—
158	C₂²	—	146	1	no	—
159	C₂²	#17989533	1	1	yes	characteristic
160	C₄	—	73	1	no	—
161	C₄	—	73	1	no	—
162	C₄	—	146	1	no	—
163	C₄	—	146	1	no	—
164	C₄	C₂×#14556660	1	2	yes	—
165	C₄	C₂×#14556661	1	2	yes	—
166	C₄	D₇₃⋊#108	1	2	yes	—
167	C₂	—	146	2	no	—
168	C₂	#22625867	1	1	yes	characteristic
169	C₂	#22625870	1	1	yes	characteristic
170	C₂	#22625871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625870

C₂

yes

maximal

#22625490

C₂

yes

maximal

#22625489

C₂

yes

maximal

C₈²×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556661

—

#17989591

C₂²

yes

—

#17989451

—

#17989450

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈²×C₇₃

C₄

yes

—

—

—

—

C₂×C₈×D₇₃

D₄

yes

—

C₂×C₈×D₇₃

Q₈

yes

—

C₂×C₈×D₇₃

Q₈

yes

—

C₄²×D₇₃

C₂³

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924038

—

C₂×#10596833

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×C₄×D₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×D₇₃

C₂×Q₈

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

D₈

yes

—

C₈×D₇₃

Q₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#127

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₈

yes

—

C₄×D₇₃

C₂×Q₁₆

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

#107

yes

—

C₈×C₇₃

yes

—

yes

—

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#500

yes

—

C₄×C₇₃

C₂×#107

yes

—

C₄×C₇₃

C₂×#108

yes

—

C₄×C₇₃

C₂×#108

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1848

—

#1567

—

#1567

—

C₂×C₈²

—

C₇₃

#8823

yes

characteristic, complemented, Sylow

C₂×#108

—

146

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#335

—

146

—

#335

—

146

—

C₈²

—

C₈²

—

100

C₈²

—

101

C₈²

#64408

yes

characteristic

102

C₂×M₁₆

—

146

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

#108

—

292

—

108

C₄⋊C₈

—

146

—

109

C₄⋊C₈

—

146

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

C₂×#64408

yes

characteristic

117

C₄×C₈

C₂×#64408

yes

characteristic

118

C₄×C₈

C₂×#64408

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

M₁₆

—

292

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

146

—

130

C₂×C₈

—

146

—

131

C₂×C₈

C₄⋊#64408

yes

—

132

C₂×C₈

C₄⋊#64408

yes

—

133

C₂×C₈

C₄⋊#64408

yes

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂²×#64408

yes

characteristic

138

C₂³

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

147

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

148

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

149

C₈

—

150

C₈

—

584

—

151

C₈

—

152

C₈

—

153

C₈

—

146

—

154

C₈

#14556660

yes

—

155

C₈

#14556661

yes

complemented

156

C₂²

—

157

C₂²

—

146

—

158

C₂²

—

146

—

159

C₂²

#17989533

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

146

—

163

C₄

—

146

—

164

C₄

C₂×#14556660

yes

—

165

C₄

C₂×#14556661

yes

—

166

C₄

D₇₃⋊#108

yes

—

167

C₂

—

146

—

168

C₂

#22625867

yes

characteristic

169

C₂

#22625870

yes

characteristic

170

C₂

#22625871

yes

characteristic

	subgroup	G/H
center	C₂²	#17989533
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	#384
prime radical	C₂²×D₇₃	C₄⋊Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8823
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#17989533

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

socle

C₂²×C₇₃

#384

prime radical

C₂²×D₇₃

C₄⋊Q₈

cosocle

C₄²×D₇₃

C₂³

C₈²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
66. C₈×C₇₃	#108	semidirect
69. #10596831	#108	semidirect
90. C₇₃	#8823	semidirect
155. C₈	#14556661	semidirect

H ≅ G/N

type

66. C₈×C₇₃

#108

semidirect

69. #10596831

#108

semidirect

90. C₇₃

#8823

semidirect

155. C₈

#14556661

semidirect

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

subgroup	index
#22625870	2
#22625490	2
#22625489	2
C₈²×D₇₃	2
#8823	73

subgroup

index

C₈²×D₇₃

Subgroups — #34005500

Conjugacy classes of subgroups

Complements

Maximal subgroups