Group #34010029 — subgroups

Subgroups — #34010029

Order 18688.

330 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 110 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	#22626292	C₂	1	2	yes	maximal
2	#22624867	C₂	1	4	yes	maximal
3	C₂×Q₈×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924031	—	2	2	no	—
5	C₂²⋊#11924031	C₂²	1	4	yes	—
6	#17989475	—	2	2	no	—
7	Q₈×#10596831	—	2	2	no	—
8	Q₈×#10596831	—	2	2	no	—
9	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
10	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
11	C₂×C₄×#10596831	C₄	1	2	yes	—
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
14	C₂×#11924031	—	2	4	no	—
15	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
16	C₂²×#10596831	C₂×C₄	1	2	yes	—
17	C₂²×#10596831	C₂³	1	1	yes	characteristic
18	C₄⋊#10596831	—	2	4	no	—
19	C₄⋊#10596831	—	2	2	no	—
20	C₄×#10596831	—	2	2	no	—
21	C₄×#10596831	—	2	2	no	—
22	C₄×#10596831	D₄	1	2	yes	—
23	C₄×#10596831	D₄	1	2	yes	—
24	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic
25	C₄×Q₈×C₇₃	—	2	2	no	—
26	C₄×Q₈×C₇₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₇₃	C₈	1	2	yes	—
29	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
31	#11924031	—	4	4	no	—
32	C₂×#10596831	—	2	2	no	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	—	4	1	no	—
35	C₂×#10596831	—	4	2	no	—
36	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×#10596831	C₂×D₄	1	2	yes	—
39	C₂×Q₈×C₇₃	—	2	2	no	—
40	C₂×Q₈×C₇₃	—	2	4	no	—
41	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₇₃	—	2	4	no	—
47	C₄⋊C₄×C₇₃	—	2	2	no	—
48	C₄²×C₇₃	—	2	2	no	—
49	C₄²×C₇₃	—	2	2	no	—
50	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₇₃	M₁₆	1	2	yes	—
52	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
53	Q₈×C₇₃	—	2	8	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	2	2	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	2	no	—
60	C₂×C₄×C₇₃	—	4	1	no	—
61	C₂×C₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
65	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
67	#10596831	—	2	2	no	—
68	#10596831	—	2	1	no	—
69	#10596831	—	4	2	no	—
70	#10596831	#101	1	2	yes	—
71	C₂²×C₇₃	—	2	1	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
75	C₄×C₇₃	—	2	2	no	—
76	C₄×C₇₃	—	2	4	no	—
77	C₄×C₇₃	—	2	2	no	—
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	—	4	2	no	—
80	C₄×C₇₃	#410	1	1	yes	characteristic
81	C₄×C₇₃	C₂×#101	1	1	yes	characteristic
82	#11494	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	2	no	—
84	C₂×C₇₃	#1461	1	1	yes	characteristic
85	C₂×C₇₃	#2112	1	1	yes	characteristic
86	C₂×C₇₃	#2115	1	1	yes	characteristic
87	C₂×C₈×Q₈	—	73	1	no	—
88	#2163	—	73	2	no	—
89	#1322	—	73	4	no	—
90	C₇₃	#11494	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₄×Q₈	#64408	1	1	yes	characteristic
92	C₈×Q₈	—	146	2	no	—
93	C₈×Q₈	—	146	2	no	—
94	C₂×C₄⋊C₈	—	73	2	no	—
95	C₂×C₄⋊C₈	—	73	1	no	—
96	C₂×C₄×C₈	—	73	1	no	—
97	C₂×C₄×C₈	—	73	2	no	—
98	C₄⋊C₁₆	—	146	2	no	—
99	C₂²⋊C₁₆	—	73	4	no	—
100	#346	—	146	2	no	—
101	C₂²×Q₈	#10596833	1	1	yes	characteristic
102	C₂²×C₈	—	73	1	no	—
103	C₂²×C₈	—	73	2	no	—
104	C₂²×C₈	—	73	1	no	—
105	C₄×Q₈	—	2	2	no	—
106	C₄×Q₈	—	2	2	no	—
107	C₂×C₄⋊C₄	#10596833	1	1	yes	characteristic
108	C₂×C₄⋊C₄	#10596833	1	2	yes	—
109	C₂×C₄²	C₂×#64408	1	2	yes	—
110	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
111	C₂×C₁₆	—	146	4	no	—
112	C₄⋊C₈	—	146	4	no	—
113	C₄⋊C₈	—	146	2	no	—
114	C₄×C₈	—	73	2	no	—
115	C₄×C₈	—	73	2	no	—
116	C₄×C₈	—	146	2	no	—
117	C₄×C₈	—	146	2	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
121	C₂²×C₄	C₂×#10596833	1	2	yes	—
122	C₂²×C₄	C₂×#10596833	1	2	yes	—
123	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
124	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
125	C₂×C₈	—	292	2	no	—
126	C₂×C₈	—	292	1	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	2	no	—
130	C₂×C₈	—	146	2	no	—
131	C₂×C₈	—	146	2	no	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂²⋊#64408	1	2	yes	—
137	C₄²	#11924037	1	2	yes	—
138	C₁₆	—	292	4	no	—
139	C₂³	C₂²×#10596833	1	1	yes	characteristic
140	Q₈	—	2	8	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
150	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
151	C₂×C₄	#14556665	1	4	yes	—
152	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
153	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
154	C₈	—	292	2	no	—
155	C₈	—	73	2	no	—
156	C₈	—	146	2	no	—
157	C₈	—	146	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂×C₂²⋊#10596833	1	1	yes	characteristic
161	C₂²	C₂×#14556665	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	4	no	—
164	C₄	—	2	2	no	—
165	C₄	—	2	1	no	—
166	C₄	—	4	2	no	—
167	C₄	#17989568	1	1	yes	characteristic
168	C₄	D₇₃⋊#101	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#22625174	1	1	yes	characteristic
171	C₂	#22626158	1	1	yes	characteristic
172	C₂	#22626167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626292

C₂

yes

maximal

#22624867

C₂

yes

maximal

C₂×Q₈×#10596831

C₂

yes

maximal, characteristic

C₄⋊#11924031

—

C₂²⋊#11924031

C₂²

yes

—

—

—

—

C₄

yes

characteristic

C₂×C₄⋊#10596831

C₂²

yes

—

C₂×C₄×#10596831

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₄×Q₈×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₂²×Q₈×C₇₃

C₈

yes

characteristic

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₈

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₈

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

—

C₄²×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

—

—

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

C₂×M₃₂

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₈×Q₈

—

#2163

—

#1322

—

C₇₃

#11494

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

#64408

yes

characteristic

C₈×Q₈

—

146

—

C₈×Q₈

—

146

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

100

#346

—

146

—

101

C₂²×Q₈

#10596833

yes

characteristic

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₄×Q₈

—

106

C₄×Q₈

—

107

C₂×C₄⋊C₄

#10596833

yes

characteristic

108

C₂×C₄⋊C₄

#10596833

yes

—

109

C₂×C₄²

C₂×#64408

yes

—

110

C₂×C₄²

C₂×#64408

yes

characteristic

111

C₂×C₁₆

—

146

—

112

C₄⋊C₈

—

146

—

113

C₄⋊C₈

—

146

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

146

—

117

C₄×C₈

—

146

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

C₂×#10596833

yes

characteristic

121

C₂²×C₄

C₂×#10596833

yes

—

122

C₂²×C₄

C₂×#10596833

yes

—

123

C₂²×C₄

C₂×#10596833

yes

characteristic

124

C₂²×C₄

C₂²×#64408

yes

characteristic

125

C₂×C₈

—

292

—

126

C₂×C₈

—

292

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

146

—

131

C₂×C₈

—

146

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²⋊#64408

yes

—

137

C₄²

#11924037

yes

—

138

C₁₆

—

292

—

139

C₂³

C₂²×#10596833

yes

characteristic

140

Q₈

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

C₂²⋊#10596833

yes

—

150

C₂×C₄

C₂²⋊#10596833

yes

—

151

C₂×C₄

#14556665

yes

—

152

C₂×C₄

C₂×C₂²⋊#64408

yes

characteristic

153

C₂×C₄

C₂×#11924037

yes

characteristic

154

C₈

—

292

—

155

C₈

—

156

C₈

—

146

—

157

C₈

—

146

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂×C₂²⋊#10596833

yes

characteristic

161

C₂²

C₂×#14556665

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#17989568

yes

characteristic

168

C₄

D₇₃⋊#101

yes

characteristic

169

C₂

—

170

C₂

#22625174

yes

characteristic

171

C₂

#22626158

yes

characteristic

172

C₂

#22626167

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#64408
commutator, prime radical	C₂³×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×Q₈	#64408
hypocenter	C₇₃	#11494
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#64408

commutator, prime radical

C₂³×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₂×C₄×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₈

C₂³

C₂×C₄×Q₈

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
90. C₇₃	#11494	semidirect

H ≅ G/N

type

90. C₇₃

#11494

semidirect

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

subgroup	index
#22626292	2
#22624867	2
C₂×Q₈×#10596831	2
#11494	73

subgroup

index

Subgroups — #34010029

Conjugacy classes of subgroups

Complements

Maximal subgroups