Group #32976436 — subgroups

Subgroups — #32976436

Order 18688.

330 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22609619	C₂	1	1	yes	maximal, characteristic
2	#22608224	C₂	1	2	yes	maximal
3	#22607762	C₂	1	1	yes	maximal, characteristic
4	#22607759	C₂	1	2	yes	maximal
5	C₇₃×#1275	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×#14556501	C₄	1	2	yes	—
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
10	#17988433	C₂²	1	1	yes	characteristic
11	#17988309	C₄	1	4	yes	complemented
12	#17988308	C₄	1	4	yes	complemented
13	#431×C₇₃	C₂²	1	2	yes	—
14	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
16	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
18	C₄⋊#10596831	—	2	2	no	—
19	C₂²⋊#10596831	—	2	2	no	—
20	#14556501	—	2	4	no	—
21	C₄×#10596831	C₂×C₄	1	4	yes	—
22	C₄×#10596831	C₂×C₄	1	2	yes	—
23	C₄×#10596831	C₂×C₄	1	2	yes	—
24	C₈×Dic₇₃	—	2	4	no	—
25	#14556498	—	2	1	no	—
26	#14556498	—	2	1	no	—
27	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	2	no	—
30	C₂²⋊C₈×C₇₃	—	2	2	no	—
31	#97×C₇₃	—	2	2	no	—
32	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
33	C₂²×Dic₇₃	—	2	2	no	—
34	C₂×#10596831	—	2	2	no	—
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	—	2	4	no	—
37	C₂×#10596831	C₄²	1	1	yes	characteristic
38	C₂×#10596831	C₄²	1	2	yes	—
39	C₂×#10596831	C₄²	1	1	yes	characteristic
40	C₄×Dic₇₃	—	2	2	no	—
41	C₄×Dic₇₃	M₁₆	1	4	yes	—
42	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	2	2	no	—
46	C₂×C₈×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	—	2	4	no	—
48	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
49	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
50	C₄²×C₇₃	#41	1	2	yes	—
51	C₂³×C₇₃	#117	1	1	yes	characteristic
52	C₂×C₄×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#117	1	2	yes	—
56	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#117	1	2	yes	—
58	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
59	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₇₃	—	2	4	no	—
61	C₂×Dic₇₃	—	4	2	no	—
62	C₂×Dic₇₃	—	4	2	no	—
63	C₈×C₇₃	—	2	4	no	—
64	C₈×C₇₃	—	4	4	no	—
65	#10596831	—	2	1	no	—
66	#10596831	—	2	1	no	—
67	#10596831	—	2	2	no	—
68	#10596831	—	4	4	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	—	2	1	no	—
71	C₂²×C₇₃	#376	1	1	yes	characteristic
72	C₂²×C₇₃	#432	1	2	yes	—
73	C₄×C₇₃	—	2	2	no	—
74	C₄×C₇₃	—	2	2	no	—
75	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
76	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
77	C₄×C₇₃	#432	1	2	yes	—
78	Dic₇₃	—	4	4	no	—
79	#8140	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	2	no	—
81	C₂×C₇₃	#1745	1	1	yes	characteristic
82	C₂×C₇₃	#1747	1	1	yes	characteristic
83	C₂×C₇₃	#1747	1	1	yes	characteristic
84	#1726	—	73	1	no	—
85	#1452	—	73	2	no	—
86	#1275	—	73	1	no	—
87	#1275	D₇₃	1	1	yes	characteristic
88	#1274	—	73	2	no	—
89	C₇₃	#8140	1	1	yes	characteristic, complemented, Sylow
90	#431	—	73	2	no	—
91	#431	C₂×D₇₃	1	2	yes	—
92	C₂×#97	—	73	2	no	—
93	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
94	C₂×C₄×C₈	—	73	1	no	—
95	C₂×C₄×C₈	—	73	2	no	—
96	#376	—	73	1	no	—
97	#322	—	73	4	no	—
98	#322	—	73	4	no	—
99	C₂²×C₈	—	73	1	no	—
100	C₂²×C₈	—	73	1	no	—
101	C₂²×C₈	C₄×D₇₃	1	2	yes	—
102	C₂×C₄²	—	73	2	no	—
103	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	146	2	no	—
106	C₂²⋊C₈	—	2	2	no	—
107	C₂²⋊C₈	—	146	2	no	—
108	#97	—	2	2	no	—
109	#97	—	146	4	no	—
110	C₄×C₈	—	73	2	no	—
111	C₄×C₈	—	73	2	no	—
112	C₄×C₈	—	73	4	no	—
113	C₄×C₈	—	146	4	no	—
114	C₄×C₈	C₄×D₇₃	1	4	yes	—
115	#95	—	146	1	no	—
116	#95	—	146	1	no	—
117	C₂²×C₄	—	146	2	no	—
118	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
119	C₂²×C₄	#11924027	1	1	yes	characteristic
120	C₂²×C₄	#11924029	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	73	2	no	—
127	C₂×C₈	—	146	4	no	—
128	C₂×C₈	—	146	2	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	#11924011	1	4	yes	—
131	C₄²	—	73	4	no	—
132	C₄²	—	146	2	no	—
133	C₄²	C₂×C₄×D₇₃	1	2	yes	—
134	C₄²	#11924027	1	2	yes	—
135	C₂³	#14556549	1	1	yes	characteristic
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	292	2	no	—
139	C₂×C₄	—	292	2	no	—
140	C₂×C₄	—	146	4	no	—
141	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
142	C₂×C₄	#14556547	1	2	yes	—
143	C₂×C₄	#14556547	1	1	yes	characteristic
144	C₂×C₄	#14556549	1	2	yes	—
145	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
146	C₂×C₄	C₂×#11924011	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	4	4	no	—
149	C₈	—	292	4	no	—
150	C₈	—	146	1	no	—
151	C₈	—	146	1	no	—
152	C₈	—	146	2	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#17988432	1	1	yes	characteristic
156	C₂²	#17988643	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	292	4	no	—
160	C₄	C₄×#11924011	1	1	yes	characteristic
161	C₄	#17988539	1	1	yes	characteristic
162	C₄	#17988641	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#22609678	1	1	yes	characteristic
165	C₂	#22609685	1	1	yes	characteristic
166	C₂	#22609687	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22609619

C₂

yes

maximal, characteristic

#22608224

C₂

yes

maximal

#22607762

C₂

yes

maximal, characteristic

#22607759

C₂

yes

maximal

C₇₃×#1275

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

C₂×#14556501

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

—

#17988433

C₂²

yes

characteristic

#17988309

C₄

yes

complemented

#17988308

C₄

yes

complemented

#431×C₇₃

C₂²

yes

—

C₂×#97×C₇₃

C₂²

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₈×Dic₇₃

—

#14556498

—

#14556498

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

—

—

—

—

C₄²

yes

characteristic

C₂×#10596831

C₄²

yes

—

C₂×#10596831

C₄²

yes

characteristic

C₄×Dic₇₃

—

C₄×Dic₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

C₂²×C₄×C₇₃

yes

C₂²×C₄×C₇₃

yes

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₄²×C₇₃

C₂²×C₄

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#117

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

#117

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

—

—

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#376

yes

characteristic

C₂²×C₇₃

#432

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×M₁₆

yes

characteristic

C₄×C₇₃

C₄×M₁₆

yes

characteristic

C₄×C₇₃

#432

yes

—

Dic₇₃

—

#8140

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

D₇₃

yes

characteristic

#1274

—

C₇₃

#8140

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₇₃

yes

—

C₂×#97

—

C₂×#97

C₂×D₇₃

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

#322

—

#322

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

C₄×D₇₃

yes

—

102

C₂×C₄²

—

103

C₂×C₄²

C₂²×D₇₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

146

—

106

C₂²⋊C₈

—

107

C₂²⋊C₈

—

146

—

108

#97

—

109

#97

—

146

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

146

—

114

C₄×C₈

C₄×D₇₃

yes

—

115

#95

—

146

—

116

#95

—

146

—

117

C₂²×C₄

—

146

—

118

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

119

C₂²×C₄

#11924027

yes

characteristic

120

C₂²×C₄

#11924029

yes

characteristic

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

146

—

128

C₂×C₈

—

146

—

129

C₂×C₈

—

146

—

130

C₂×C₈

#11924011

yes

—

131

C₄²

—

132

C₄²

—

146

—

133

C₄²

C₂×C₄×D₇₃

yes

—

134

C₄²

#11924027

yes

—

135

C₂³

#14556549

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

292

—

139

C₂×C₄

—

292

—

140

C₂×C₄

—

146

—

141

C₂×C₄

C₄²×D₇₃

yes

characteristic

142

C₂×C₄

#14556547

yes

—

143

C₂×C₄

#14556547

yes

characteristic

144

C₂×C₄

#14556549

yes

—

145

C₂×C₄

M₁₆×D₇₃

yes

—

146

C₂×C₄

C₂×#11924011

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

292

—

150

C₈

—

146

—

151

C₈

—

146

—

152

C₈

—

146

—

153

C₂²

—

154

C₂²

—

155

C₂²

#17988432

yes

characteristic

156

C₂²

#17988643

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

292

—

160

C₄

C₄×#11924011

yes

characteristic

161

C₄

#17988539

yes

characteristic

162

C₄

#17988641

yes

—

163

C₂

—

164

C₂

#22609678

yes

characteristic

165

C₂

#22609685

yes

characteristic

166

C₂

#22609687

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1275	C₂
socle	C₂²×C₇₃	#376
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1275	D₇₃
hypocenter	C₇₃	#8140
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #1275

subgroup

G/H

center

C₂×C₄

C₄²×D₇₃

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1275

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #1275

N	H ≅ G/N	type
11. #17988309	C₄	semidirect
12. #17988308	C₄	semidirect
89. C₇₃	#8140	semidirect

H ≅ G/N

type

11. #17988309

C₄

semidirect

12. #17988308

C₄

semidirect

89. C₇₃

#8140

semidirect

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

subgroup	index
#22609619	2
#22608224	2
#22607762	2
#22607759	2
C₇₃×#1275	2
#8140	73

subgroup

index

C₇₃×#1275

Subgroups — #32976436

Conjugacy classes of subgroups

Complements

Maximal subgroups