Group #34034073 — subgroups

Subgroups — #34034073

Order 18688.

414 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 169 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	C₄×#14556686	C₂	1	2	yes	maximal
2	#22627002	C₂	1	4	yes	maximal
3	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556686	C₂²	1	4	yes	—
5	C₄×#11924040	C₄	1	4	yes	complemented
6	C₄×#11924040	C₄	1	4	yes	complemented
7	C₄×#11924036	C₂²	1	1	yes	characteristic
8	C₂×C₈×#64408	C₄	1	2	yes	—
9	C₂×C₈×#64408	C₂²	1	2	yes	—
10	C₄²×#64408	C₄	1	1	yes	characteristic
11	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
12	#14556686	—	2	8	no	—
13	C₂×#11924040	C₂×C₄	1	8	yes	—
14	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
15	C₂×#11924036	C₂³	1	1	yes	characteristic
16	C₂×C₄×#64408	C₈	1	2	yes	—
17	C₂×C₄×#64408	C₈	1	2	yes	—
18	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
19	C₄×#10596833	C₂×C₄	1	2	yes	—
20	C₄×#10596833	C₂×C₄	1	2	yes	—
21	C₈×#64408	—	2	4	no	—
22	C₈×#64408	—	2	4	no	—
23	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
24	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	C₈	1	1	yes	characteristic
26	C₈×Dic₇₃	C₈	1	2	yes	—
27	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
28	#11924040	—	2	8	no	—
29	C₂²×#64408	C₂×C₈	1	4	yes	—
30	#11924036	M₁₆	1	4	yes	—
31	#11924036	#41	1	4	yes	—
32	C₂×#10596833	C₄²	1	2	yes	—
33	C₂×#10596833	C₂²×C₄	1	2	yes	—
34	C₄×#64408	—	2	4	no	—
35	C₄×#64408	—	2	2	no	—
36	C₄×#64408	—	2	2	no	—
37	C₄×#64408	C₁₆	1	8	yes	complemented
38	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×#10596831	C₂×C₈	1	2	yes	—
42	C₈×D₇₃	—	2	2	no	—
43	C₈×D₇₃	—	2	2	no	—
44	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
45	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
46	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
48	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×#64408	—	2	4	no	—
51	C₂×#64408	—	2	4	no	—
52	C₂×#64408	C₂×C₁₆	1	8	yes	—
53	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
57	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
58	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
59	C₄×D₇₃	C₂×C₁₆	1	4	yes	—
60	C₄×D₇₃	M₃₂	1	4	yes	—
61	C₄×D₇₃	#117	1	2	yes	—
62	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
63	#10596833	#97	1	4	yes	—
64	#10596833	#117	1	4	yes	—
65	C₈×C₇₃	—	2	2	no	—
66	#10596831	—	2	2	no	—
67	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
68	C₂×D₇₃	#431	1	2	yes	—
69	C₂×D₇₃	C₂²×C₁₆	1	2	yes	—
70	C₂×D₇₃	C₂×M₃₂	1	2	yes	—
71	#64408	—	2	8	no	—
72	C₄×C₇₃	C₄×C₁₆	1	2	yes	—
73	C₄×C₇₃	#346	1	2	yes	—
74	C₄×C₇₃	#431	1	2	yes	—
75	Dic₇₃	C₄×C₁₆	1	2	yes	—
76	Dic₇₃	#346	1	2	yes	—
77	Dic₇₃	#376	1	1	yes	characteristic
78	Dic₇₃	C₂×#97	1	1	yes	characteristic
79	#12590	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	#1744	1	1	yes	characteristic
81	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
82	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
83	D₇₃	#2162	1	4	yes	—
84	C₂×C₄×C₁₆	—	73	2	no	—
85	C₄²×C₈	—	73	1	no	—
86	#1374	—	73	4	no	—
87	C₇₃	#12590	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	73	4	no	—
89	C₂×C₄×C₈	—	73	1	no	—
90	C₂×C₄×C₈	—	73	1	no	—
91	C₂×C₄×C₈	—	73	2	no	—
92	C₂×C₄×C₈	—	73	2	no	—
93	C₄³	—	73	1	no	—
94	C₄×C₁₆	—	73	4	no	—
95	C₄×C₁₆	—	73	4	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	—	73	1	no	—
98	C₂²×C₈	—	73	2	no	—
99	C₂×C₄²	—	73	2	no	—
100	C₂×C₄²	—	73	1	no	—
101	C₂×C₄²	—	73	2	no	—
102	C₂×C₄²	—	73	2	no	—
103	C₂×C₁₆	—	73	8	no	—
104	C₂×C₁₆	—	146	8	no	—
105	C₄×C₈	—	73	2	no	—
106	C₄×C₈	—	73	2	no	—
107	C₄×C₈	—	73	2	no	—
108	C₄×C₈	—	73	1	no	—
109	C₄×C₈	—	146	4	no	—
110	C₄×C₈	—	146	4	no	—
111	C₄×C₈	#10596842	1	1	yes	characteristic
112	C₂²×C₄	—	73	1	no	—
113	C₂²×C₄	—	73	1	no	—
114	C₂²×C₄	—	73	1	no	—
115	C₂²×C₄	—	73	4	no	—
116	C₂×C₈	—	73	2	no	—
117	C₂×C₈	—	73	2	no	—
118	C₂×C₈	—	73	4	no	—
119	C₂×C₈	—	73	4	no	—
120	C₂×C₈	—	73	2	no	—
121	C₂×C₈	—	146	2	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	C₂×#10596842	1	2	yes	—
124	C₄²	—	73	8	no	—
125	C₄²	—	73	1	no	—
126	C₄²	—	73	1	no	—
127	C₄²	—	73	1	no	—
128	C₄²	—	146	2	no	—
129	C₄²	—	146	2	no	—
130	C₄²	—	146	4	no	—
131	C₄²	C₂×#10596842	1	1	yes	characteristic
132	C₁₆	—	146	8	no	—
133	C₂³	—	73	1	no	—
134	C₂×C₄	—	73	1	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	—	73	4	no	—
138	C₂×C₄	—	73	4	no	—
139	C₂×C₄	—	73	2	no	—
140	C₂×C₄	—	73	2	no	—
141	C₂×C₄	—	73	8	no	—
142	C₂×C₄	—	146	4	no	—
143	C₂×C₄	—	146	4	no	—
144	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
145	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
146	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	73	4	no	—
149	C₈	—	73	4	no	—
150	C₈	—	146	2	no	—
151	C₂²	—	73	2	no	—
152	C₂²	—	73	2	no	—
153	C₂²	—	73	2	no	—
154	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
155	C₄	—	73	2	no	—
156	C₄	—	73	2	no	—
157	C₄	—	73	1	no	—
158	C₄	—	73	1	no	—
159	C₄	—	146	8	no	—
160	C₄	#64408⋊C₁₆	1	2	yes	—
161	C₄	#17989682	1	2	yes	—
162	C₄	#17989700	1	2	yes	—
163	C₂	—	73	4	no	—
164	C₂	#22627065	1	1	yes	characteristic
165	C₂	C₂×#64408⋊C₁₆	1	1	yes	characteristic
166	C₂	C₂×#17989682	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556686

C₂

yes

maximal

#22627002

C₂

yes

maximal

C₄×C₈×#64408

C₂

yes

maximal, characteristic

C₂×#14556686

C₂²

yes

—

C₄×#11924040

C₄

yes

complemented

C₄×#11924040

C₄

yes

complemented

C₄×#11924036

C₂²

yes

characteristic

C₂×C₈×#64408

C₄

yes

—

C₂×C₈×#64408

C₂²

yes

—

C₄²×#64408

C₄

yes

characteristic

C₄×C₈×D₇₃

C₄

yes

characteristic

#14556686

—

C₂×#11924040

C₂×C₄

yes

—

C₂×#11924036

C₂×C₄

yes

characteristic

C₂×#11924036

C₂³

yes

characteristic

C₂×C₄×#64408

C₈

yes

—

C₂×C₄×#64408

C₈

yes

—

C₂×C₄×#64408

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₈×#64408

—

C₈×#64408

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₈

yes

characteristic

#11924040

—

C₂²×#64408

C₂×C₈

yes

—

#11924036

M₁₆

yes

—

#11924036

#41

yes

—

C₂×#10596833

C₄²

yes

—

C₂×#10596833

C₂²×C₄

yes

—

—

—

—

C₁₆

yes

complemented

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂²×C₈

yes

—

—

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₄²

yes

characteristic

C₄×D₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

M₃₂

yes

—

C₄×D₇₃

#117

yes

—

C₄×D₇₃

C₂×M₁₆

yes

—

#10596833

#97

yes

—

#10596833

#117

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

#431

yes

—

C₂×D₇₃

C₂²×C₁₆

yes

—

C₂×D₇₃

C₂×M₃₂

yes

—

#64408

—

C₄×C₇₃

C₄×C₁₆

yes

—

C₄×C₇₃

#346

yes

—

C₄×C₇₃

#431

yes

—

Dic₇₃

C₄×C₁₆

yes

—

Dic₇₃

#346

yes

—

Dic₇₃

#376

yes

characteristic

Dic₇₃

C₂×#97

yes

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×C₄×C₁₆

yes

C₂×C₇₃

yes

D₇₃

yes

—

C₂×C₄×C₁₆

—

C₄²×C₈

—

#1374

—

C₇₃

#12590

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

146

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

146

—

110

C₄×C₈

—

146

—

111

C₄×C₈

#10596842

yes

characteristic

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₂×C₈

C₂×#10596842

yes

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

146

—

129

C₄²

—

146

—

130

C₄²

—

146

—

131

C₄²

C₂×#10596842

yes

characteristic

132

C₁₆

—

146

—

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

C₄×#10596842

yes

characteristic

145

C₂×C₄

C₄×#10596842

yes

characteristic

146

C₂×C₄

C₂²×#10596842

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

146

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

C₂×C₄×#10596842

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

146

—

160

C₄

#64408⋊C₁₆

yes

—

161

C₄

#17989682

yes

—

162

C₄

#17989700

yes

—

163

C₂

—

164

C₂

#22627065

yes

characteristic

165

C₂

C₂×#64408⋊C₁₆

yes

characteristic

166

C₂

C₂×#17989682

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×#10596842
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#11924036	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#12590
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₄²

C₂×#10596842

commutator

C₂×C₇₃

C₂×C₄×C₁₆

Frattini

C₂×C₄

C₂²×#10596842

Fitting

C₄×C₈×C₇₃

C₈

socle

C₂²×C₇₃

C₂×C₄×C₈

prime radical

C₂²×D₇₃

C₂²×C₈

C₂³

C₄×C₈

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
5. C₄×#11924040	C₄	semidirect
6. C₄×#11924040	C₄	semidirect
37. C₄×#64408	C₁₆	semidirect
87. C₇₃	#12590	semidirect

H ≅ G/N

type

5. C₄×#11924040

C₄

semidirect

6. C₄×#11924040

C₄

semidirect

37. C₄×#64408

C₁₆

semidirect

87. C₇₃

#12590

semidirect

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

subgroup	index
C₄×#14556686	2
#22627002	2
C₄×C₈×#64408	2
#12590	73

subgroup

index

C₄×#14556686

#22627002

C₄×C₈×#64408

#12590

Subgroups — #34034073

Conjugacy classes of subgroups

Complements

Maximal subgroups