Group #34032882 — subgroups

Subgroups — #34032882

Order 18688.

302 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 81 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	#22627131	C₂	1	2	yes	maximal
2	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11924040	C₄	1	4	yes	complemented
4	#17989684	—	2	2	no	—
5	C₄×#11924040	—	2	2	no	—
6	C₄×#11924036	C₂²	1	1	yes	characteristic
7	C₂×C₈×#64408	—	2	2	no	—
8	C₄²×#64408	C₄	1	1	yes	characteristic
9	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924040	—	2	4	no	—
11	C₂×#11924036	D₄	1	2	yes	—
12	C₂×C₄×#64408	—	2	1	no	—
13	C₂×C₄×#64408	—	2	1	no	—
14	C₂×C₄×#64408	C₈	1	2	yes	—
15	C₄×#10596833	C₂×C₄	1	2	yes	—
16	C₄×#10596833	D₄	1	1	yes	characteristic
17	C₄×#10596833	Q₈	1	1	yes	characteristic
18	C₈×#64408	—	2	8	no	—
19	C₂×C₈×D₇₃	—	2	1	no	—
20	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
21	C₄×#10596831	C₈	1	1	yes	characteristic
22	C₈×Dic₇₃	—	2	1	no	—
23	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
24	#11924040	—	4	4	no	—
25	C₂²×#64408	—	2	2	no	—
26	#11924036	—	2	4	no	—
27	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
28	C₂×#10596833	C₄⋊C₄	1	2	yes	—
29	C₄×#64408	—	2	4	no	—
30	C₄×#64408	—	2	2	no	—
31	C₄×#64408	—	2	2	no	—
32	C₄×#64408	—	2	2	no	—
33	C₄×#64408	C₁₆	1	4	yes	complemented
34	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
37	C₂×#10596831	—	2	1	no	—
38	C₈×D₇₃	—	2	4	no	—
39	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
40	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×#64408	—	2	4	no	—
46	C₂×#64408	—	2	4	no	—
47	C₂×#64408	—	2	4	no	—
48	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
51	C₂×Dic₇₃	#95	1	1	yes	characteristic
52	C₂×Dic₇₃	#97	1	1	yes	characteristic
53	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
54	C₄×D₇₃	—	2	1	no	—
55	C₄×D₇₃	—	2	1	no	—
56	C₄×D₇₃	—	2	2	no	—
57	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
58	C₄×D₇₃	M₃₂	1	2	yes	—
59	#10596833	—	2	2	no	—
60	#10596833	#104	1	4	yes	—
61	C₈×C₇₃	—	2	2	no	—
62	#10596831	—	2	2	no	—
63	C₂²×C₇₃	#336	1	1	yes	characteristic
64	C₂×D₇₃	—	2	1	no	—
65	C₂×D₇₃	—	2	1	no	—
66	C₂×D₇₃	C₂²⋊C₁₆	1	2	yes	—
67	#64408	—	2	8	no	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
70	C₄×C₇₃	#346	1	1	yes	characteristic
71	C₄×C₇₃	#364	1	1	yes	characteristic
72	C₄×C₇₃	#364	1	1	yes	characteristic
73	Dic₇₃	—	2	1	no	—
74	Dic₇₃	#339	1	2	yes	—
75	Dic₇₃	C₄⋊C₁₆	1	2	yes	—
76	#7148	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	#1276	1	1	yes	characteristic
78	C₂×C₇₃	#1374	1	1	yes	characteristic
79	C₂×C₇₃	#1376	1	1	yes	characteristic
80	D₇₃	—	2	2	no	—
81	#2162	—	73	2	no	—
82	C₄²×C₈	—	73	1	no	—
83	C₇₃	#7148	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₄×C₈	—	73	1	no	—
85	C₂×C₄×C₈	—	73	1	no	—
86	C₂×C₄×C₈	—	146	2	no	—
87	C₄³	—	73	1	no	—
88	C₄⋊C₁₆	—	73	4	no	—
89	C₂²⋊C₁₆	—	146	2	no	—
90	C₄×C₁₆	—	146	2	no	—
91	C₂²×C₈	—	73	2	no	—
92	C₂²×C₈	—	146	1	no	—
93	C₂×C₄²	—	73	1	no	—
94	C₂×C₄²	—	73	2	no	—
95	C₂×C₄²	—	146	1	no	—
96	C₂×C₄²	—	146	1	no	—
97	C₂×C₁₆	—	146	4	no	—
98	C₄×C₈	—	73	1	no	—
99	C₄×C₈	—	73	2	no	—
100	C₄×C₈	—	73	1	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	146	1	no	—
103	C₄×C₈	—	146	8	no	—
104	C₄×C₈	#10596842	1	1	yes	characteristic
105	C₂²×C₄	—	73	1	no	—
106	C₂²×C₄	—	73	1	no	—
107	C₂²×C₄	—	73	1	no	—
108	C₂²×C₄	—	146	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	73	2	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	146	4	no	—
113	C₂×C₈	—	146	4	no	—
114	C₂×C₈	—	146	1	no	—
115	C₄²	—	73	1	no	—
116	C₄²	—	73	1	no	—
117	C₄²	—	73	1	no	—
118	C₄²	—	73	4	no	—
119	C₄²	—	146	2	no	—
120	C₄²	—	146	4	no	—
121	C₄²	—	146	2	no	—
122	C₄²	—	146	2	no	—
123	C₄²	C₂×#10596842	1	1	yes	characteristic
124	C₁₆	—	292	4	no	—
125	C₂³	—	73	1	no	—
126	C₂×C₄	—	73	1	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	73	2	no	—
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	146	2	no	—
132	C₂×C₄	—	146	4	no	—
133	C₂×C₄	—	146	4	no	—
134	C₂×C₄	—	146	4	no	—
135	C₂×C₄	—	146	1	no	—
136	C₂×C₄	—	146	1	no	—
137	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
139	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	73	4	no	—
142	C₈	—	146	2	no	—
143	C₈	—	146	2	no	—
144	C₂²	—	73	2	no	—
145	C₂²	—	146	1	no	—
146	C₂²	—	146	1	no	—
147	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	73	2	no	—
150	C₄	—	73	2	no	—
151	C₄	—	146	8	no	—
152	C₄	—	146	1	no	—
153	C₄	#64408⋊C₁₆	1	1	yes	characteristic
154	C₄	#17989682	1	1	yes	characteristic
155	C₄	#17989691	1	1	yes	characteristic
156	C₄	#17989692	1	1	yes	characteristic
157	C₂	—	146	2	no	—
158	C₂	#22626936	1	1	yes	characteristic
159	C₂	#22627002	1	1	yes	characteristic
160	C₂	#22627006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627131

C₂

yes

maximal

C₄×C₈×#64408

C₂

yes

maximal, characteristic

C₄⋊#11924040

C₄

yes

—

—

C₂²

yes

characteristic

C₂×C₈×#64408

—

C₄²×#64408

C₄

yes

characteristic

C₄×C₈×D₇₃

C₄

yes

characteristic

C₂×#11924040

—

C₂×#11924036

D₄

yes

—

C₂×C₄×#64408

—

C₂×C₄×#64408

—

C₂×C₄×#64408

C₈

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

Q₈

yes

characteristic

C₈×#64408

—

C₂×C₈×D₇₃

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

—

C₄×C₈×C₇₃

C₈

yes

characteristic

#11924040

—

C₂²×#64408

—

#11924036

—

C₂×#10596833

C₂²⋊C₄

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

—

—

—

—

C₁₆

yes

complemented

C₂×C₄×D₇₃

C₂²⋊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

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂²⋊C₈

yes

—

—

—

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

C₄⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

M₃₂

yes

—

#10596833

—

#10596833

#104

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#336

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊C₁₆

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

C₄×C₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

#339

yes

—

Dic₇₃

C₄⋊C₁₆

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2162

—

C₄²×C₈

—

C₇₃

#7148

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

146

—

C₄³

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

C₄×C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

—

146

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

146

—

103

C₄×C₈

—

146

—

104

C₄×C₈

#10596842

yes

characteristic

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

146

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

146

—

113

C₂×C₈

—

146

—

114

C₂×C₈

—

146

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

146

—

120

C₄²

—

146

—

121

C₄²

—

146

—

122

C₄²

—

146

—

123

C₄²

C₂×#10596842

yes

characteristic

124

C₁₆

—

292

—

125

C₂³

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

—

146

—

137

C₂×C₄

C₄×#10596842

yes

characteristic

138

C₂×C₄

C₂²⋊#10596842

yes

characteristic

139

C₂×C₄

C₄⋊#10596842

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

146

—

143

C₈

—

146

—

144

C₂²

—

145

C₂²

—

146

—

146

C₂²

—

146

—

147

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

146

—

152

C₄

—

146

—

153

C₄

#64408⋊C₁₆

yes

characteristic

154

C₄

#17989682

yes

characteristic

155

C₄

#17989691

yes

characteristic

156

C₄

#17989692

yes

characteristic

157

C₂

—

146

—

158

C₂

#22626936

yes

characteristic

159

C₂

#22627002

yes

characteristic

160

C₂

#22627006

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	C₂²⋊C₈
cosocle	C₄×#11924036	C₂²
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#7148
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

C₂²⋊C₈

C₂²

C₄×C₈

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
3. C₄⋊#11924040	C₄	semidirect
33. C₄×#64408	C₁₆	semidirect
83. C₇₃	#7148	semidirect

H ≅ G/N

type

3. C₄⋊#11924040

C₄

semidirect

33. C₄×#64408

C₁₆

semidirect

83. C₇₃

#7148

semidirect

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

subgroup	index
#22627131	2
C₄×C₈×#64408	2
#7148	73

subgroup

index

#22627131

C₄×C₈×#64408

#7148

Subgroups — #34032882

Conjugacy classes of subgroups

Complements

Maximal subgroups