Group #32972866 — subgroups

Subgroups — #32972866

Order 18688.

222 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14556531	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556531	C₄	1	2	yes	—
5	C₂×C₄⋊#10596831	—	2	1	no	—
6	#17988363	—	2	1	no	—
7	C₄×#11924003	—	2	1	no	—
8	#17988340	—	2	1	no	—
9	#17988340	C₄	1	2	yes	—
10	#431×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	#335×C₇₃	—	2	1	no	—
13	#334×C₇₃	C₄	1	2	yes	—
14	C₂²×#10596831	—	4	1	no	—
15	#14556531	—	2	2	no	—
16	#14556531	C₈	1	4	yes	complemented
17	C₂×#11924003	—	2	2	no	—
18	C₄⋊#10596831	—	2	2	no	—
19	C₄⋊#10596831	—	4	1	no	—
20	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
23	C₄⋊C₈×C₇₃	—	2	2	no	—
24	C₂²⋊C₈×C₇₃	—	4	1	no	—
25	C₄×C₈×C₇₃	—	4	1	no	—
26	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂×#10596831	—	4	2	no	—
31	C₂×#10596831	—	8	2	no	—
32	#11924003	—	2	4	no	—
33	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	—	4	2	no	—
38	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
44	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₃	D₈	1	1	yes	characteristic
47	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
48	C₂³×C₇₃	#106	1	1	yes	characteristic
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#108	1	2	yes	—
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	8	2	no	—
63	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
64	C₈×C₇₃	#97	1	1	yes	characteristic
65	C₈×C₇₃	C₂²⋊C₈	1	2	yes	—
66	#10596831	—	8	2	no	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	#334	1	2	yes	—
70	C₂²×C₇₃	#340	1	1	yes	characteristic
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	#335	1	2	yes	—
75	C₄×C₇₃	#336	1	1	yes	characteristic
76	C₄×C₇₃	#341	1	1	yes	characteristic
77	C₄×C₇₃	#361	1	1	yes	characteristic
78	C₄×C₇₃	#362	1	1	yes	characteristic
79	#7159	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	2	no	—
81	C₂×C₇₃	#1277	1	1	yes	characteristic
82	C₂×C₇₃	#1387	1	1	yes	characteristic
83	C₂×C₇₃	#1389	1	1	yes	characteristic
84	C₄×M₃₂	—	73	1	no	—
85	#1564	D₇₃	1	1	yes	characteristic
86	C₂×#335	—	73	1	no	—
87	C₇₃	#7159	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₃₂	—	73	2	no	—
89	#431	—	2	1	no	—
90	C₂×C₄⋊C₈	—	146	1	no	—
91	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
92	#346	—	146	1	no	—
93	C₄×C₁₆	—	146	1	no	—
94	#335	—	2	1	no	—
95	#335	—	73	2	no	—
96	#335	—	146	1	no	—
97	#334	Dic₇₃	1	2	yes	—
98	C₂²×C₈	—	292	1	no	—
99	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
100	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
102	M₃₂	—	73	4	no	—
103	M₃₂	—	146	2	no	—
104	C₂×C₁₆	—	146	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	292	1	no	—
107	C₄⋊C₈	—	146	2	no	—
108	C₂²⋊C₈	—	4	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
111	C₄×C₈	D₂₉₂	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
114	C₂²×C₄	#11924007	1	1	yes	characteristic
115	C₂²×C₄	#11924015	1	1	yes	characteristic
116	C₂²×C₄	#11924016	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	292	2	no	—
120	C₂×C₈	—	584	2	no	—
121	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
122	C₂×C₈	#11924005	1	1	yes	characteristic
123	C₂×C₈	#11924007	1	1	yes	characteristic
124	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
125	C₂×C₈	C₈×D₇₃	1	2	yes	—
126	C₂×C₈	#11924011	1	2	yes	—
127	C₄²	#11924005	1	1	yes	characteristic
128	C₄²	C₂²⋊Dic₇₃	1	1	yes	characteristic
129	C₄²	#11924013	1	1	yes	characteristic
130	C₄²	#11924018	1	1	yes	characteristic
131	C₁₆	—	146	4	no	—
132	C₂³	#14556522	1	1	yes	characteristic
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	#14556498	1	1	yes	characteristic
137	C₂×C₄	#14556511	1	1	yes	characteristic
138	C₂×C₄	#14556512	1	1	yes	characteristic
139	C₂×C₄	#14556514	1	1	yes	characteristic
140	C₂×C₄	#14556515	1	1	yes	characteristic
141	C₂×C₄	#14556520	1	2	yes	—
142	C₂×C₄	#14556524	1	1	yes	characteristic
143	C₂×C₄	#14556526	1	2	yes	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	584	2	no	—
148	C₈	C₈×Dic₇₃	1	1	yes	characteristic
149	C₈	#14556501	1	1	yes	characteristic
150	C₈	#14556503	1	2	yes	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	#17988337	1	2	yes	—
154	C₂²	#17988349	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#17988339	1	2	yes	—
159	C₄	#17988341	1	1	yes	characteristic
160	C₄	#17988351	1	1	yes	characteristic
161	C₄	#17988399	1	1	yes	characteristic
162	C₄	#17988402	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	#22607765	1	1	yes	characteristic
165	C₂	#22608041	1	1	yes	characteristic
166	C₂	#22608047	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556531

C₂

yes

maximal, characteristic

C₂×#17988340

C₂

yes

maximal, characteristic

C₇₃×#1564

C₂

yes

maximal, characteristic

C₂×#14556531

C₄

yes

—

—

—

—

—

C₄

yes

—

#431×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#335×C₇₃

—

#334×C₇₃

C₄

yes

—

C₂²×#10596831

—

#14556531

—

#14556531

C₈

yes

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

Q₈

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

—

—

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

Q₁₆

yes

characteristic

C₂³×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

#107

yes

characteristic

C₂×C₄×C₇₃

#108

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

characteristic

C₈×C₇₃

#97

yes

characteristic

C₈×C₇₃

C₂²⋊C₈

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#334

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#335

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×M₃₂

—

#1564

D₇₃

yes

characteristic

C₂×#335

—

C₇₃

#7159

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#431

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#346

—

146

—

C₄×C₁₆

—

146

—

#335

—

#335

—

#335

—

146

—

#334

Dic₇₃

yes

—

C₂²×C₈

—

292

—

C₂²×C₈

Dic₁₄₆

yes

characteristic

100

C₂²×C₈

C₄×D₇₃

yes

characteristic

101

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

102

M₃₂

—

103

M₃₂

—

146

—

104

C₂×C₁₆

—

146

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

292

—

107

C₄⋊C₈

—

146

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₇₃

yes

characteristic

111

C₄×C₈

D₂₉₂

yes

characteristic

112

C₄×C₈

C₂×Dic₇₃

yes

characteristic

113

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

114

C₂²×C₄

#11924007

yes

characteristic

115

C₂²×C₄

#11924015

yes

characteristic

116

C₂²×C₄

#11924016

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

292

—

120

C₂×C₈

—

584

—

121

C₂×C₈

C₄×Dic₇₃

yes

characteristic

122

C₂×C₈

#11924005

yes

characteristic

123

C₂×C₈

#11924007

yes

characteristic

124

C₂×C₈

C₄⋊Dic₇₃

yes

characteristic

125

C₂×C₈

C₈×D₇₃

yes

—

126

C₂×C₈

#11924011

yes

—

127

C₄²

#11924005

yes

characteristic

128

C₄²

C₂²⋊Dic₇₃

yes

characteristic

129

C₄²

#11924013

yes

characteristic

130

C₄²

#11924018

yes

characteristic

131

C₁₆

—

146

—

132

C₂³

#14556522

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#14556498

yes

characteristic

137

C₂×C₄

#14556511

yes

characteristic

138

C₂×C₄

#14556512

yes

characteristic

139

C₂×C₄

#14556514

yes

characteristic

140

C₂×C₄

#14556515

yes

characteristic

141

C₂×C₄

#14556520

yes

—

142

C₂×C₄

#14556524

yes

characteristic

143

C₂×C₄

#14556526

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

584

—

148

C₈

C₈×Dic₇₃

yes

characteristic

149

C₈

#14556501

yes

characteristic

150

C₈

#14556503

yes

—

151

C₂²

—

152

C₂²

—

153

C₂²

#17988337

yes

—

154

C₂²

#17988349

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#17988339

yes

—

159

C₄

#17988341

yes

characteristic

160

C₄

#17988351

yes

characteristic

161

C₄

#17988399

yes

characteristic

162

C₄

#17988402

yes

characteristic

163

C₂

—

164

C₂

#22607765

yes

characteristic

165

C₂

#22608041

yes

characteristic

166

C₂

#22608047

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556511
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₇₃×#1564	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#106
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	#1564	D₇₃
hypocenter	C₇₃	#7159
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

subgroup

G/H

center

C₂×C₄

#14556511

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₇₃×#1564

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×C₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

N	H ≅ G/N	type
16. #14556531	C₈	semidirect
87. C₇₃	#7159	semidirect

H ≅ G/N

type

16. #14556531

C₈

semidirect

87. C₇₃

#7159

semidirect

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

subgroup	index
C₄×#14556531	2
C₂×#17988340	2
C₇₃×#1564	2
#7159	73

subgroup

index

C₄×#14556531

C₂×#17988340

C₇₃×#1564

#7159

Subgroups — #32972866

Conjugacy classes of subgroups

Complements

Maximal subgroups