Group #34032888 — subgroups

Subgroups — #34032888

Order 18688.

262 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22627117	C₂	1	2	yes	maximal
2	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556686	—	2	2	no	—
4	#17989690	—	2	2	no	—
5	C₄⋊#11924040	C₄	1	4	yes	—
6	C₄×#11924036	C₄	1	1	yes	characteristic
7	C₄×#11924036	C₂²	1	1	yes	characteristic
8	C₈×#10596833	—	2	2	no	—
9	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
10	#14556686	—	4	2	no	—
11	#14556686	—	4	2	no	—
12	C₂×#11924040	—	2	4	no	—
13	C₂×#11924036	C₈	1	2	yes	—
14	C₂×#11924036	D₄	1	1	yes	characteristic
15	C₂×#11924036	Q₈	1	1	yes	characteristic
16	C₄×#10596833	—	2	1	no	—
17	C₄×#10596833	—	2	1	no	—
18	C₄×#10596833	C₂×C₄	1	2	yes	—
19	C₄×#10596833	D₄	1	2	yes	—
20	C₂×C₈×D₇₃	—	2	1	no	—
21	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×#10596831	C₈	1	1	yes	characteristic
23	C₈×Dic₇₃	—	2	1	no	—
24	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
25	#11924040	—	4	4	no	—
26	#11924036	—	2	1	no	—
27	#11924036	—	2	1	no	—
28	#11924036	—	2	2	no	—
29	#11924036	C₁₆	1	4	yes	complemented
30	#11924036	D₈	1	1	yes	characteristic
31	#11924036	SD₁₆	1	2	yes	—
32	#11924036	Q₁₆	1	1	yes	characteristic
33	C₂×#10596833	—	2	2	no	—
34	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
35	C₂×#10596833	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
39	C₂×#10596831	—	2	1	no	—
40	C₈×D₇₃	—	2	4	no	—
41	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
42	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₇₃	—	2	1	no	—
45	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
50	C₂×Dic₇₃	#95	1	1	yes	characteristic
51	C₂×Dic₇₃	#97	1	1	yes	characteristic
52	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₄×D₇₃	—	2	2	no	—
54	C₄×D₇₃	#106	1	1	yes	characteristic
55	C₄×D₇₃	#107	1	1	yes	characteristic
56	C₄×D₇₃	#108	1	1	yes	characteristic
57	C₄×D₇₃	#108	1	1	yes	characteristic
58	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
59	C₄×D₇₃	M₃₂	1	2	yes	—
60	#10596833	—	2	2	no	—
61	#10596833	—	2	4	no	—
62	#10596833	#102	1	2	yes	—
63	#10596833	#103	1	2	yes	—
64	C₈×C₇₃	—	2	2	no	—
65	#10596831	—	2	2	no	—
66	C₂²×C₇₃	#336	1	1	yes	characteristic
67	C₂×D₇₃	#334	1	1	yes	characteristic
68	C₂×D₇₃	#335	1	1	yes	characteristic
69	C₂×D₇₃	C₄⋊C₁₆	1	2	yes	—
70	C₂×D₇₃	#364	1	2	yes	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	#334	1	1	yes	characteristic
73	C₄×C₇₃	#335	1	1	yes	characteristic
74	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
75	C₄×C₇₃	#346	1	1	yes	characteristic
76	Dic₇₃	—	2	1	no	—
77	Dic₇₃	#340	1	1	yes	characteristic
78	Dic₇₃	#341	1	1	yes	characteristic
79	Dic₇₃	C₂²⋊C₁₆	1	2	yes	—
80	#7151	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	#1277	1	1	yes	characteristic
82	C₂×C₇₃	#1374	1	1	yes	characteristic
83	C₂×C₇₃	#1376	1	1	yes	characteristic
84	D₇₃	#1367	1	2	yes	—
85	D₇₃	#1370	1	2	yes	—
86	C₂×C₄⋊C₁₆	—	73	2	no	—
87	C₂×C₈²	—	73	1	no	—
88	C₇₃	#7151	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	146	2	no	—
90	C₂×C₄×C₈	—	73	1	no	—
91	C₂×C₄×C₈	—	73	1	no	—
92	C₂×C₄×C₈	—	73	1	no	—
93	C₄⋊C₁₆	—	73	4	no	—
94	C₄⋊C₁₆	—	146	2	no	—
95	C₈²	—	146	2	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	—	73	2	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂²×C₈	—	146	1	no	—
100	C₂×C₄²	—	73	1	no	—
101	C₂×C₁₆	—	292	2	no	—
102	C₂×C₁₆	—	292	2	no	—
103	C₂×C₁₆	—	146	4	no	—
104	C₄×C₈	—	73	2	no	—
105	C₄×C₈	—	73	2	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	—	146	1	no	—
108	C₄×C₈	—	146	1	no	—
109	C₄×C₈	—	146	1	no	—
110	C₄×C₈	#10596842	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	—	2	1	no	—
115	C₂×C₈	—	73	2	no	—
116	C₂×C₈	—	73	1	no	—
117	C₂×C₈	—	73	2	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	2	no	—
120	C₂×C₈	—	73	4	no	—
121	C₂×C₈	—	146	4	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	—	146	1	no	—
124	C₂×C₈	—	146	1	no	—
125	C₂×C₈	—	146	2	no	—
126	C₂×C₈	—	146	1	no	—
127	C₄²	—	73	1	no	—
128	C₄²	—	73	1	no	—
129	C₄²	—	73	1	no	—
130	C₄²	C₂×#10596842	1	1	yes	characteristic
131	C₁₆	—	292	4	no	—
132	C₂³	—	73	1	no	—
133	C₂×C₄	—	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	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	2	no	—
140	C₂×C₄	—	73	2	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
144	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
145	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	73	2	no	—
148	C₈	—	73	2	no	—
149	C₈	—	146	2	no	—
150	C₈	—	146	2	no	—
151	C₈	—	146	4	no	—
152	C₂²	—	73	1	no	—
153	C₂²	—	73	1	no	—
154	C₂²	—	73	2	no	—
155	C₂²	—	73	2	no	—
156	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	73	1	no	—
159	C₄	—	73	1	no	—
160	C₄	—	73	2	no	—
161	C₄	—	146	1	no	—
162	C₄	#17989676	1	1	yes	characteristic
163	C₄	#17989677	1	1	yes	characteristic
164	C₄	#64408⋊C₁₆	1	1	yes	characteristic
165	C₄	#17989682	1	1	yes	characteristic
166	C₂	—	73	2	no	—
167	C₂	—	73	2	no	—
168	C₂	#22626937	1	1	yes	characteristic
169	C₂	#22627002	1	1	yes	characteristic
170	C₂	#22627005	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627117

C₂

yes

maximal

C₈×#11924036

C₂

yes

maximal, characteristic

C₂×#14556686

—

#17989690

—

C₄⋊#11924040

C₄

yes

—

C₄×#11924036

C₄

yes

characteristic

C₄×#11924036

C₂²

yes

characteristic

C₈×#10596833

—

C₄×C₈×D₇₃

C₄

yes

—

—

—

C₈

yes

—

C₂×#11924036

D₄

yes

characteristic

C₂×#11924036

Q₈

yes

—

—

C₂×C₄

yes

—

C₄×#10596833

D₄

yes

—

C₂×C₈×D₇₃

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

—

C₄×C₈×C₇₃

C₈

yes

—

—

—

—

C₁₆

yes

complemented

#11924036

D₈

yes

characteristic

#11924036

SD₁₆

yes

—

#11924036

Q₁₆

yes

characteristic

C₂×#10596833

—

C₂×#10596833

C₂²⋊C₄

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

M₃₂

yes

—

—

—

yes

—

#10596833

#103

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

C₄⋊C₁₆

yes

—

C₂×D₇₃

#364

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₄×C₁₆

yes

characteristic

C₄×C₇₃

#346

yes

characteristic

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

C₂²⋊C₁₆

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

—

D₇₃

#1370

yes

—

C₂×C₄⋊C₁₆

—

C₂×C₈²

—

C₇₃

#7151

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

146

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

C₈²

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

146

—

100

C₂×C₄²

—

101

C₂×C₁₆

—

292

—

102

C₂×C₁₆

—

292

—

103

C₂×C₁₆

—

146

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

146

—

108

C₄×C₈

—

146

—

109

C₄×C₈

—

146

—

110

C₄×C₈

#10596842

yes

characteristic

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂×C₈

—

115

C₂×C₈

—

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

—

146

—

124

C₂×C₈

—

146

—

125

C₂×C₈

—

146

—

126

C₂×C₈

—

146

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂×#10596842

yes

characteristic

131

C₁₆

—

292

—

132

C₂³

—

133

C₂×C₄

—

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

C₄×#10596842

yes

characteristic

144

C₂×C₄

C₂²⋊#10596842

yes

characteristic

145

C₂×C₄

C₄⋊#10596842

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

146

—

150

C₈

—

146

—

151

C₈

—

146

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

146

—

162

C₄

#17989676

yes

characteristic

163

C₄

#17989677

yes

characteristic

164

C₄

#64408⋊C₁₆

yes

characteristic

165

C₄

#17989682

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#22626937

yes

characteristic

169

C₂

#22627002

yes

characteristic

170

C₂

#22627005

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#10596842
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₄²	C₂×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	C₄⋊C₈
cosocle	C₄×#11924036	C₂²
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#7151
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

C₄⋊#10596842

commutator

C₄×C₇₃

C₄×C₁₆

Frattini

C₄²

C₂×#10596842

Fitting

C₄×C₈×C₇₃

C₈

socle

C₂²×C₇₃

#336

prime radical

C₂²×D₇₃

C₄⋊C₈

C₂²

C₄×C₈

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
29. #11924036	C₁₆	semidirect
88. C₇₃	#7151	semidirect

H ≅ G/N

type

29. #11924036

C₁₆

semidirect

88. C₇₃

#7151

semidirect

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

subgroup	index
#22627117	2
C₈×#11924036	2
#7151	73

subgroup

index

#22627117

C₈×#11924036

#7151

Subgroups — #34032888

Conjugacy classes of subgroups

Complements

Maximal subgroups