Group #34013269 — subgroups

Subgroups — #34013269

Order 18688.

366 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 164 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	C₂×C₄⋊#11924031	C₂	1	2	yes	maximal
2	#22624970	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	C₂²	1	4	yes	—
5	C₄⋊#11924031	—	2	4	no	—
6	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
9	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924031	—	2	4	no	—
12	C₂×#11924031	—	2	8	no	—
13	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
14	C₂²×#10596831	C₂³	1	1	yes	characteristic
15	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
16	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
19	C₄×#10596831	D₄	1	2	yes	—
20	C₄×#10596831	Q₈	1	2	yes	—
21	C₈×Dic₇₃	—	2	4	no	—
22	C₈×Dic₇₃	—	2	4	no	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₇₃	—	2	2	no	—
26	#11924031	—	4	4	no	—
27	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
28	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
29	C₂×#10596831	—	2	2	no	—
30	C₂×#10596831	C₄⋊C₄	1	2	yes	—
31	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
32	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
33	C₂×#10596831	#41	1	4	yes	—
34	C₄×Dic₇₃	—	2	2	no	—
35	C₄×Dic₇₃	—	2	2	no	—
36	C₄×Dic₇₃	—	2	2	no	—
37	C₄×Dic₇₃	—	2	4	no	—
38	C₄×Dic₇₃	C₁₆	1	4	yes	complemented
39	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	2	no	—
43	C₂×C₈×C₇₃	—	2	2	no	—
44	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
45	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
46	C₄²×C₇₃	M₁₆	1	2	yes	—
47	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
50	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
52	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#117	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
57	C₂×Dic₇₃	—	2	2	no	—
58	C₂×Dic₇₃	—	2	2	no	—
59	C₂×Dic₇₃	—	2	2	no	—
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	C₂×C₁₆	1	4	yes	—
62	C₂×Dic₇₃	M₃₂	1	4	yes	—
63	C₈×C₇₃	—	2	4	no	—
64	#10596831	—	2	2	no	—
65	#10596831	#124	1	2	yes	—
66	#10596831	#125	1	2	yes	—
67	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₇₃	#431	1	2	yes	—
69	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
70	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
71	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
72	C₄×C₇₃	—	2	2	no	—
73	C₄×C₇₃	C₄⋊C₁₆	1	4	yes	—
74	C₄×C₇₃	#382	1	1	yes	characteristic
75	C₄×C₇₃	#423	1	1	yes	characteristic
76	C₄×C₇₃	#432	1	2	yes	—
77	Dic₇₃	—	2	4	no	—
78	Dic₇₃	—	2	4	no	—
79	#12879	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	#1831	1	1	yes	characteristic
81	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
82	C₂×C₇₃	#2150	1	1	yes	characteristic
83	C₂×C₇₃	#2162	1	2	yes	—
84	C₂×C₇₃	#2163	1	2	yes	—
85	C₂×C₄⋊C₁₆	—	73	2	no	—
86	C₄²×C₈	—	73	1	no	—
87	#1374	—	73	4	no	—
88	C₇₃	#12879	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	73	4	no	—
90	C₂×C₄×C₈	—	73	1	no	—
91	C₂×C₄×C₈	—	73	2	no	—
92	C₂×C₄×C₈	—	73	2	no	—
93	C₂×C₄×C₈	#64408	1	1	yes	characteristic
94	C₄³	—	73	1	no	—
95	C₄⋊C₁₆	—	146	4	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	—	73	1	no	—
98	C₂²×C₈	C₂×#64408	1	2	yes	—
99	C₂×C₄²	—	73	1	no	—
100	C₂×C₄²	—	73	2	no	—
101	C₂×C₄²	—	73	1	no	—
102	C₂×C₄²	—	73	2	no	—
103	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
104	C₂×C₁₆	—	146	4	no	—
105	C₂×C₁₆	—	146	8	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	73	2	no	—
108	C₄×C₈	—	73	2	no	—
109	C₄×C₈	—	146	4	no	—
110	C₄×C₈	—	146	4	no	—
111	C₂²×C₄	—	73	2	no	—
112	C₂²×C₄	—	73	2	no	—
113	C₂²×C₄	#11924036	1	1	yes	characteristic
114	C₂²×C₄	#11924036	1	1	yes	characteristic
115	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	73	4	no	—
121	C₂×C₈	—	73	2	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	#11924038	1	4	yes	—
124	C₄²	—	73	4	no	—
125	C₄²	—	146	4	no	—
126	C₄²	—	146	2	no	—
127	C₄²	—	146	2	no	—
128	C₄²	—	146	2	no	—
129	C₄²	C₄⋊#64408	1	2	yes	—
130	C₄²	#11924037	1	2	yes	—
131	C₁₆	—	292	4	no	—
132	C₂³	C₂×#11924036	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	73	4	no	—
135	C₂×C₄	—	73	4	no	—
136	C₂×C₄	—	146	2	no	—
137	C₂×C₄	—	146	2	no	—
138	C₂×C₄	—	146	2	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	#14556658	1	2	yes	—
141	C₂×C₄	#14556658	1	2	yes	—
142	C₂×C₄	#14556664	1	2	yes	—
143	C₂×C₄	#14556666	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
145	C₂×C₄	#14556670	1	2	yes	—
146	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
147	C₂×C₄	C₂×#11924038	1	2	yes	—
148	C₈	—	2	4	no	—
149	C₈	—	73	2	no	—
150	C₈	—	73	2	no	—
151	C₈	—	146	2	no	—
152	C₂²	C₂×#14556658	1	1	yes	characteristic
153	C₂²	#17989606	1	2	yes	—
154	C₂²	C₂×#14556664	1	1	yes	characteristic
155	C₂²	#17989643	1	2	yes	—
156	C₂²	C₂×#14556666	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	146	4	no	—
159	C₄	—	146	4	no	—
160	C₄	#17989508	1	4	yes	—
161	C₄	#17989531	1	1	yes	characteristic
162	C₄	#17989591	1	1	yes	characteristic
163	C₄	#17989608	1	2	yes	—
164	C₂	#22625848	1	1	yes	characteristic
165	C₂	C₂×#17989508	1	1	yes	characteristic
166	C₂	#22626265	1	1	yes	characteristic
167	C₂	#22626291	1	2	yes	—
168	C₂	#22626295	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11924031

C₂

yes

maximal

#22624970

C₂

yes

maximal

C₄×C₈×Dic₇₃

C₂

yes

maximal, characteristic

C₂²×#11924031

C₂²

yes

—

C₄⋊#11924031

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₄²×Dic₇₃

C₄

yes

characteristic

C₂×C₄×C₈×C₇₃

C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

C₈

yes

—

C₂×C₄×Dic₇₃

C₈

yes

characteristic

C₂×C₄×Dic₇₃

C₈

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

Q₈

yes

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

—

#11924031

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂×#10596831

#41

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₁₆

yes

complemented

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₂×Dic₇₃

M₃₂

yes

—

C₈×C₇₃

—

#10596831

—

#10596831

#124

yes

—

#10596831

#125

yes

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂²×C₇₃

C₂×M₃₂

yes

—

C₂²×C₇₃

C₂×M₃₂

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄⋊C₁₆

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×C₄⋊C₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2163

yes

—

C₂×C₄⋊C₁₆

—

C₄²×C₈

—

#1374

—

C₇₃

#12879

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#64408

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₂×C₄²

C₂×#64408

yes

characteristic

104

C₂×C₁₆

—

146

—

105

C₂×C₁₆

—

146

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

146

—

110

C₄×C₈

—

146

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

#11924036

yes

characteristic

114

C₂²×C₄

#11924036

yes

characteristic

115

C₂²×C₄

C₂²×#64408

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₂×C₈

#11924038

yes

—

124

C₄²

—

125

C₄²

—

146

—

126

C₄²

—

146

—

127

C₄²

—

146

—

128

C₄²

—

146

—

129

C₄²

C₄⋊#64408

yes

—

130

C₄²

#11924037

yes

—

131

C₁₆

—

292

—

132

C₂³

C₂×#11924036

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

#14556658

yes

—

141

C₂×C₄

#14556658

yes

—

142

C₂×C₄

#14556664

yes

—

143

C₂×C₄

#14556666

yes

—

144

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

145

C₂×C₄

#14556670

yes

—

146

C₂×C₄

C₂×#11924037

yes

characteristic

147

C₂×C₄

C₂×#11924038

yes

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

146

—

152

C₂²

C₂×#14556658

yes

characteristic

153

C₂²

#17989606

yes

—

154

C₂²

C₂×#14556664

yes

characteristic

155

C₂²

#17989643

yes

—

156

C₂²

C₂×#14556666

yes

characteristic

157

C₄

—

158

C₄

—

146

—

159

C₄

—

146

—

160

C₄

#17989508

yes

—

161

C₄

#17989531

yes

characteristic

162

C₄

#17989591

yes

characteristic

163

C₄

#17989608

yes

—

164

C₂

#22625848

yes

characteristic

165

C₂

C₂×#17989508

yes

characteristic

166

C₂

#22626265

yes

characteristic

167

C₂

#22626291

yes

—

168

C₂

#22626295

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12879
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#64408

commutator

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₂×C₄×C₈×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₂²×C₈

C₂³

C₂×C₄×C₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
38. C₄×Dic₇₃	C₁₆	semidirect
88. C₇₃	#12879	semidirect

H ≅ G/N

type

38. C₄×Dic₇₃

C₁₆

semidirect

88. C₇₃

#12879

semidirect

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

subgroup	index
C₂×C₄⋊#11924031	2
#22624970	2
C₄×C₈×Dic₇₃	2
#12879	73

subgroup

index

C₂×C₄⋊#11924031

#22624970

C₄×C₈×Dic₇₃

#12879

Subgroups — #34013269

Conjugacy classes of subgroups

Complements

Maximal subgroups