Group #34032884 — subgroups

Subgroups — #34032884

Order 18688.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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	#22627135	C₂	1	2	yes	maximal
2	C₄×#14556645	C₂	1	1	yes	maximal, characteristic
3	#17989690	C₄	1	4	yes	complemented
4	#17989684	—	2	2	no	—
5	#17989681	—	2	2	no	—
6	C₂×#14556645	—	2	2	no	—
7	C₄×#11924036	C₂²	1	1	yes	characteristic
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	#14556645	—	2	8	no	—
16	C₄×#10596833	C₂×C₄	1	2	yes	—
17	C₄×#10596833	D₄	1	1	yes	characteristic
18	C₄×#10596833	Q₈	1	1	yes	characteristic
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	—	4	2	no	—
31	C₄×#64408	—	4	2	no	—
32	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
35	C₂×#10596831	—	2	1	no	—
36	C₈×D₇₃	—	2	4	no	—
37	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
38	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×#64408	—	2	4	no	—
44	C₂×#64408	—	4	2	no	—
45	C₂×#64408	—	4	2	no	—
46	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×Dic₇₃	#95	1	1	yes	characteristic
50	C₂×Dic₇₃	#97	1	1	yes	characteristic
51	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
52	C₄×D₇₃	—	2	2	no	—
53	C₄×D₇₃	—	2	2	no	—
54	C₄×D₇₃	—	2	2	no	—
55	#10596833	—	2	2	no	—
56	#10596833	#104	1	4	yes	—
57	C₈×C₇₃	—	2	2	no	—
58	#10596831	—	2	2	no	—
59	C₂²×C₇₃	#336	1	1	yes	characteristic
60	C₂×D₇₃	—	2	1	no	—
61	C₂×D₇₃	—	2	1	no	—
62	C₂×D₇₃	#349	1	2	yes	—
63	#64408	—	4	4	no	—
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	2	1	no	—
67	Dic₇₃	—	2	1	no	—
68	Dic₇₃	#339	1	2	yes	—
69	Dic₇₃	#347	1	2	yes	—
70	#7149	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	#1276	1	1	yes	characteristic
72	C₂×C₇₃	#1375	1	1	yes	characteristic
73	C₂×C₇₃	#1377	1	1	yes	characteristic
74	D₇₃	—	2	2	no	—
75	#2163	—	73	2	no	—
76	C₄×#97	—	73	1	no	—
77	C₇₃	#7149	1	1	yes	characteristic, complemented, Sylow
78	C₂×#97	—	146	2	no	—
79	C₂×C₄×C₈	—	73	1	no	—
80	C₂×C₄×C₈	—	73	1	no	—
81	C₄³	—	73	1	no	—
82	C₄⋊C₁₆	—	73	4	no	—
83	C₂²⋊C₁₆	—	146	2	no	—
84	#346	—	146	2	no	—
85	C₂²×C₈	—	73	2	no	—
86	C₂²×C₈	—	146	1	no	—
87	C₂×C₄²	—	73	2	no	—
88	C₂×C₄²	—	73	1	no	—
89	C₂×C₄²	—	146	1	no	—
90	C₂×C₄²	—	146	1	no	—
91	C₂×C₁₆	—	146	4	no	—
92	#97	—	146	8	no	—
93	C₄×C₈	—	73	1	no	—
94	C₄×C₈	—	73	2	no	—
95	C₄×C₈	—	73	1	no	—
96	C₄×C₈	—	73	1	no	—
97	C₄×C₈	—	146	1	no	—
98	C₄×C₈	#10596842	1	1	yes	characteristic
99	C₂²×C₄	—	73	1	no	—
100	C₂²×C₄	—	73	1	no	—
101	C₂²×C₄	—	73	1	no	—
102	C₂²×C₄	—	146	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	73	2	no	—
105	C₂×C₈	—	73	2	no	—
106	C₂×C₈	—	146	4	no	—
107	C₂×C₈	—	146	4	no	—
108	C₂×C₈	—	146	1	no	—
109	C₄²	—	292	2	no	—
110	C₄²	—	292	2	no	—
111	C₄²	—	73	1	no	—
112	C₄²	—	73	1	no	—
113	C₄²	—	73	1	no	—
114	C₄²	—	146	4	no	—
115	C₄²	C₂×#10596842	1	1	yes	characteristic
116	C₁₆	—	292	4	no	—
117	C₂³	—	73	1	no	—
118	C₂×C₄	—	292	2	no	—
119	C₂×C₄	—	292	2	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	146	2	no	—
124	C₂×C₄	—	146	2	no	—
125	C₂×C₄	—	146	2	no	—
126	C₂×C₄	—	146	4	no	—
127	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
129	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	—	73	4	no	—
132	C₈	—	146	2	no	—
133	C₈	—	146	2	no	—
134	C₂²	—	73	2	no	—
135	C₂²	—	146	1	no	—
136	C₂²	—	146	1	no	—
137	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	292	4	no	—
142	C₄	—	73	2	no	—
143	C₄	—	73	2	no	—
144	C₄	—	146	1	no	—
145	C₂	—	146	2	no	—
146	C₂	#22626936	1	1	yes	characteristic
147	C₂	#22627004	1	1	yes	characteristic
148	C₂	#22627008	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627135

C₂

yes

maximal

C₄×#14556645

C₂

yes

maximal, characteristic

#17989690

C₄

yes

—

—

—

C₂²

yes

characteristic

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

—

#14556645

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

Q₈

yes

characteristic

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

—

C₄×#64408

—

C₄×#64408

—

C₄×#64408

—

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

—

#10596833

—

#10596833

#104

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#336

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#349

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

#339

yes

—

Dic₇₃

#347

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2163

—

C₄×#97

—

C₇₃

#7149

yes

characteristic, complemented, Sylow

C₂×#97

—

146

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

#346

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

—

146

—

C₂×C₁₆

—

146

—

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₄×C₈

#10596842

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

146

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

146

—

107

C₂×C₈

—

146

—

108

C₂×C₈

—

146

—

109

C₄²

—

292

—

110

C₄²

—

292

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

146

—

115

C₄²

C₂×#10596842

yes

characteristic

116

C₁₆

—

292

—

117

C₂³

—

118

C₂×C₄

—

292

—

119

C₂×C₄

—

292

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

146

—

124

C₂×C₄

—

146

—

125

C₂×C₄

—

146

—

126

C₂×C₄

—

146

—

127

C₂×C₄

C₄×#10596842

yes

characteristic

128

C₂×C₄

C₂²⋊#10596842

yes

characteristic

129

C₂×C₄

C₄⋊#10596842

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

146

—

133

C₈

—

146

—

134

C₂²

—

135

C₂²

—

146

—

136

C₂²

—

146

—

137

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

292

—

142

C₄

—

143

C₄

—

144

C₄

—

146

—

145

C₂

—

146

—

146

C₂

#22626936

yes

characteristic

147

C₂

#22627004

yes

characteristic

148

C₂

#22627008

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#10596842)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
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	#7149
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

(C₂×#10596842)⋊C₄

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₄²

C₂×#10596842

Fitting

C₄×C₈×C₇₃

C₈

socle

C₂²×C₇₃

#336

prime radical

C₂²×D₇₃

C₂²⋊C₈

C₂²

C₄×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
3. #17989690	C₄	semidirect
77. C₇₃	#7149	semidirect

H ≅ G/N

type

3. #17989690

C₄

semidirect

77. C₇₃

#7149

semidirect

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

subgroup	index
#22627135	2
C₄×#14556645	2
#7149	73

subgroup

index

#22627135

C₄×#14556645

#7149

Subgroups — #34032884

Conjugacy classes of subgroups

Complements

Maximal subgroups