Group #32977155 — subgroups

Subgroups — #32977155

Order 18688.

366 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₄×#14556501	C₂	1	1	yes	maximal, characteristic
2	#22608677	C₂	1	2	yes	maximal
3	#22608223	C₂	1	1	yes	maximal, characteristic
4	#22607763	C₂	1	2	yes	maximal
5	C₇₃×#1452	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×#14556501	C₄	1	2	yes	—
8	C₂×#14556501	C₄	1	2	yes	—
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
11	#17988339	C₄	1	4	yes	complemented
12	#17988337	C₄	1	4	yes	complemented
13	#17988309	—	2	2	no	—
14	#431×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#322×C₇₃	—	2	2	no	—
17	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
18	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	—	2	2	no	—
20	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
21	C₄⋊#10596831	C₂×C₄	1	4	yes	—
22	C₂²⋊#10596831	—	2	2	no	—
23	#14556501	—	2	4	no	—
24	#14556501	—	2	4	no	—
25	C₄×#10596831	—	2	2	no	—
26	C₄×#10596831	C₂×C₄	1	2	yes	—
27	C₄×#10596831	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
31	C₂²⋊C₈×C₇₃	—	2	2	no	—
32	C₄×C₈×C₇₃	—	2	2	no	—
33	C₄×C₈×C₇₃	D₄	1	2	yes	—
34	C₄×C₈×C₇₃	Q₈	1	2	yes	—
35	C₂²×Dic₇₃	—	2	2	no	—
36	C₂×#10596831	—	2	2	no	—
37	C₂×#10596831	—	2	4	no	—
38	C₂×#10596831	C₄²	1	1	yes	characteristic
39	C₂×#10596831	C₄²	1	1	yes	characteristic
40	C₂×#10596831	C₄²	1	2	yes	—
41	C₂×#10596831	M₁₆	1	4	yes	—
42	C₄×Dic₇₃	—	2	2	no	—
43	C₄×Dic₇₃	—	4	4	no	—
44	C₄×Dic₇₃	M₁₆	1	4	yes	—
45	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	#41	1	2	yes	—
47	C₂×C₈×C₇₃	—	2	2	no	—
48	C₂×C₈×C₇₃	—	2	2	no	—
49	C₂×C₈×C₇₃	—	2	4	no	—
50	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
51	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
52	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₇₃	#117	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	2	4	no	—
56	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
60	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
61	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
62	C₂×Dic₇₃	—	2	4	no	—
63	C₂×Dic₇₃	—	4	4	no	—
64	C₈×C₇₃	—	2	4	no	—
65	C₈×C₇₃	—	4	4	no	—
66	#10596831	—	2	1	no	—
67	#10596831	—	2	1	no	—
68	#10596831	—	4	4	no	—
69	#10596831	#97	1	4	yes	—
70	C₂²×C₇₃	—	2	2	no	—
71	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₇₃	#432	1	2	yes	—
73	C₄×C₇₃	—	2	2	no	—
74	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
75	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
76	C₄×C₇₃	#423	1	2	yes	—
77	C₄×C₇₃	#444	1	4	yes	—
78	Dic₇₃	—	4	4	no	—
79	#8269	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	2	no	—
81	C₂×C₇₃	#1767	1	1	yes	characteristic
82	C₂×C₇₃	#1773	1	2	yes	—
83	C₄×#97	—	73	1	no	—
84	#1564	—	73	2	no	—
85	#1452	—	73	1	no	—
86	#1452	D₇₃	1	1	yes	characteristic
87	#1276	—	73	2	no	—
88	C₇₃	#8269	1	1	yes	characteristic, complemented, Sylow
89	#431	—	73	2	no	—
90	#431	C₂×D₇₃	1	2	yes	—
91	C₂×#97	—	73	2	no	—
92	C₂×#97	—	73	2	no	—
93	C₂×C₄×C₈	—	73	1	no	—
94	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
95	C₄³	—	73	1	no	—
96	#335	—	73	4	no	—
97	#334	—	73	4	no	—
98	#322	—	2	2	no	—
99	#322	—	146	2	no	—
100	C₂²×C₈	—	73	1	no	—
101	C₂²×C₈	—	73	1	no	—
102	C₂²×C₈	C₄×D₇₃	1	2	yes	—
103	C₂×C₄²	—	73	2	no	—
104	C₂×C₄²	—	146	2	no	—
105	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
106	C₄⋊C₈	—	73	4	no	—
107	C₄⋊C₈	C₄×D₇₃	1	4	yes	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₂²⋊C₈	—	146	2	no	—
110	#97	—	146	4	no	—
111	#97	—	146	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	73	2	no	—
114	C₄×C₈	—	73	2	no	—
115	C₄×C₈	—	146	2	no	—
116	C₄×C₈	Dic₁₄₆	1	2	yes	—
117	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
118	C₂²×C₄	—	146	2	no	—
119	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
120	C₂²×C₄	#11924028	1	2	yes	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	73	2	no	—
125	C₂×C₈	—	73	4	no	—
126	C₂×C₈	—	73	1	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	146	4	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	#11924007	1	4	yes	—
131	C₄²	—	292	4	no	—
132	C₄²	—	73	4	no	—
133	C₄²	—	146	2	no	—
134	C₄²	C₂×C₄×D₇₃	1	2	yes	—
135	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
136	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
137	C₂³	#14556548	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	292	4	no	—
140	C₂×C₄	—	146	4	no	—
141	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
142	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
143	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
144	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
145	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
146	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	4	4	no	—
149	C₈	—	292	4	no	—
150	C₈	—	73	4	no	—
151	C₈	—	146	1	no	—
152	C₈	—	146	1	no	—
153	C₂²	—	2	2	no	—
154	C₂²	C₄×#11924007	1	1	yes	characteristic
155	C₂²	#17988642	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	292	4	no	—
158	C₄	#97×D₇₃	1	1	yes	characteristic
159	C₄	#17988539	1	1	yes	characteristic
160	C₄	#17988610	1	2	yes	—
161	C₄	#17988707	1	4	yes	—
162	C₂	—	2	2	no	—
163	C₂	#22609785	1	1	yes	characteristic
164	C₂	#22609816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556501

C₂

yes

maximal, characteristic

#22608677

C₂

yes

maximal

#22608223

C₂

yes

maximal, characteristic

#22607763

C₂

yes

maximal

C₇₃×#1452

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

C₂×#14556501

C₄

yes

—

C₂×#14556501

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₄²×Dic₇₃

C₂²

yes

characteristic

#17988339

C₄

yes

complemented

#17988337

C₄

yes

complemented

#17988309

—

#431×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#322×C₇₃

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₄⋊#10596831

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₄²

yes

characteristic

C₂×#10596831

C₄²

yes

characteristic

C₂×#10596831

C₄²

yes

—

C₂×#10596831

M₁₆

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₂³×C₇₃

#117

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₄×Q₈

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

—

—

—

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

#432

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

C₄×M₁₆

yes

characteristic

C₄×C₇₃

#423

yes

—

C₄×C₇₃

#444

yes

—

Dic₇₃

—

#8269

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1767

yes

characteristic

C₂×C₇₃

#1773

yes

—

C₄×#97

—

#1564

—

#1452

—

#1452

D₇₃

yes

characteristic

#1276

—

C₇₃

#8269

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₇₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄³

—

—

—

—

—

146

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₄×D₇₃

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

—

146

—

105

C₂×C₄²

C₂²×D₇₃

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

C₄×D₇₃

yes

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

146

—

110

#97

—

146

—

111

#97

—

146

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

146

—

116

C₄×C₈

Dic₁₄₆

yes

—

117

C₄×C₈

C₂²⋊D₇₃

yes

—

118

C₂²×C₄

—

146

—

119

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

120

C₂²×C₄

#11924028

yes

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

146

—

129

C₂×C₈

—

146

—

130

C₂×C₈

#11924007

yes

—

131

C₄²

—

292

—

132

C₄²

—

133

C₄²

—

146

—

134

C₄²

C₂×C₄×D₇₃

yes

—

135

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

136

C₄²

C₂×Dic₁₄₆

yes

characteristic

137

C₂³

#14556548

yes

characteristic

138

C₂×C₄

—

139

C₂×C₄

—

292

—

140

C₂×C₄

—

146

—

141

C₂×C₄

C₄²×D₇₃

yes

characteristic

142

C₂×C₄

C₂×#11924007

yes

characteristic

143

C₂×C₄

C₄×C₂²⋊D₇₃

yes

—

144

C₂×C₄

C₄×Dic₁₄₆

yes

—

145

C₂×C₄

M₁₆×D₇₃

yes

—

146

C₂×C₄

M₁₆×D₇₃

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

292

—

150

C₈

—

151

C₈

—

146

—

152

C₈

—

146

—

153

C₂²

—

154

C₂²

C₄×#11924007

yes

characteristic

155

C₂²

#17988642

yes

—

156

C₄

—

157

C₄

—

292

—

158

C₄

#97×D₇₃

yes

characteristic

159

C₄

#17988539

yes

characteristic

160

C₄

#17988610

yes

—

161

C₄

#17988707

yes

—

162

C₂

—

163

C₂

#22609785

yes

characteristic

164

C₂

#22609816

yes

—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1452	C₂
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1452	D₇₃
hypocenter	C₇₃	#8269
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

subgroup

G/H

center

C₄²

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1452

C₂

socle

C₂²×C₇₃

C₄×C₄⋊C₄

prime radical

C₂³×C₇₃

#117

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

N	H ≅ G/N	type
11. #17988339	C₄	semidirect
12. #17988337	C₄	semidirect
88. C₇₃	#8269	semidirect

H ≅ G/N

type

11. #17988339

C₄

semidirect

12. #17988337

C₄

semidirect

88. C₇₃

#8269

semidirect

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

subgroup	index
C₄×#14556501	2
#22608677	2
#22608223	2
#22607763	2
C₇₃×#1452	2
#8269	73

subgroup

index

C₇₃×#1452

Subgroups — #32977155

Conjugacy classes of subgroups

Complements

Maximal subgroups