Group #32973220 — subgroups

Subgroups — #32973220

Order 18688.

198 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22608687	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2163	C₂	1	1	yes	maximal, characteristic
4	#17988611	—	2	1	no	—
5	C₂×C₄⋊#10596831	—	2	1	no	—
6	#17988340	—	2	1	no	—
7	#17988340	C₄	1	2	yes	—
8	#17988338	—	2	1	no	—
9	#17988338	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆×C₇₃	—	2	1	no	—
12	C₂²⋊C₁₆×C₇₃	C₄	1	2	yes	—
13	#346×C₇₃	—	2	1	no	—
14	C₂×#11924012	—	4	1	no	—
15	C₂²×#10596831	—	4	1	no	—
16	C₄⋊#10596831	—	2	2	no	—
17	C₄⋊#10596831	—	2	2	no	—
18	C₄⋊#10596831	—	4	1	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	C₂×C₁₆×C₇₃	—	2	2	no	—
24	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	#11924012	—	8	2	no	—
29	C₂×#10596831	—	4	2	no	—
30	C₂×#10596831	—	4	2	no	—
31	C₂×#10596831	—	8	2	no	—
32	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	—	4	1	no	—
37	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₃	D₈	1	1	yes	characteristic
44	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
45	C₁₆×C₇₃	—	4	2	no	—
46	C₂³×C₇₃	#106	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	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	#107	1	1	yes	characteristic
58	C₈×C₇₃	—	2	1	no	—
59	C₈×C₇₃	—	2	1	no	—
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	#108	1	2	yes	—
62	#10596831	—	8	2	no	—
63	#10596831	—	8	2	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	#340	1	1	yes	characteristic
67	C₂²×C₇₃	#365	1	2	yes	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	#341	1	1	yes	characteristic
73	C₄×C₇₃	#344	1	1	yes	characteristic
74	C₄×C₇₃	#361	1	1	yes	characteristic
75	C₄×C₇₃	#362	1	1	yes	characteristic
76	#7292	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	2	2	no	—
78	C₂×C₇₃	#1301	1	1	yes	characteristic
79	C₂×C₇₃	#1383	1	1	yes	characteristic
80	C₂×C₇₃	#1387	1	1	yes	characteristic
81	#2163	D₇₃	1	1	yes	characteristic
82	#1566	—	73	1	no	—
83	C₂×#335	—	73	1	no	—
84	C₇₃	#7292	1	1	yes	characteristic, complemented, Sylow
85	#423	—	146	1	no	—
86	C₂×C₄⋊C₈	—	146	1	no	—
87	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
88	C₄⋊C₁₆	—	2	1	no	—
89	C₂²⋊C₁₆	Dic₇₃	1	2	yes	—
90	#346	—	2	1	no	—
91	#335	—	73	2	no	—
92	#335	—	146	1	no	—
93	#334	—	73	2	no	—
94	#334	—	146	1	no	—
95	C₂×M₁₆	—	292	1	no	—
96	C₂²×C₈	—	292	1	no	—
97	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
98	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
99	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
100	C₂×C₁₆	—	2	2	no	—
101	C₄⋊C₈	—	292	1	no	—
102	C₄⋊C₈	—	292	1	no	—
103	C₄⋊C₈	—	146	2	no	—
104	C₄⋊C₈	—	146	2	no	—
105	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
106	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
109	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
110	C₂²×C₄	#11924017	1	1	yes	characteristic
111	C₂²×C₄	#11924017	1	1	yes	characteristic
112	M₁₆	—	584	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	292	2	no	—
116	C₂×C₈	—	292	2	no	—
117	C₂×C₈	—	584	2	no	—
118	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
120	C₂×C₈	#11924007	1	1	yes	characteristic
121	C₂×C₈	#11924007	1	1	yes	characteristic
122	C₄²	#11924005	1	1	yes	characteristic
123	C₄²	#11924005	1	1	yes	characteristic
124	C₄²	D₅₈₄	1	1	yes	characteristic
125	C₄²	Dic₂₉₂	1	1	yes	characteristic
126	C₁₆	—	4	2	no	—
127	C₂³	#14556523	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	#14556498	1	1	yes	characteristic
134	C₂×C₄	#14556513	1	1	yes	characteristic
135	C₂×C₄	#14556513	1	1	yes	characteristic
136	C₂×C₄	#14556516	1	1	yes	characteristic
137	C₂×C₄	#14556516	1	1	yes	characteristic
138	C₂×C₄	#14556525	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	584	2	no	—
143	C₈	—	584	2	no	—
144	C₈	#14556526	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#17988350	1	1	yes	characteristic
148	C₂²	#17988410	1	2	yes	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#17988351	1	1	yes	characteristic
154	C₄	#17988357	1	1	yes	characteristic
155	C₄	#17988401	1	1	yes	characteristic
156	C₄	#17988404	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	#22607827	1	1	yes	characteristic
159	C₂	#22608030	1	1	yes	characteristic
160	C₂	#22608040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22608687

C₂

yes

maximal, characteristic

C₂×#17988340

C₂

yes

maximal, characteristic

C₇₃×#2163

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#17988338

—

#17988338

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₂²⋊C₁₆×C₇₃

C₄

yes

—

#346×C₇₃

—

—

—

—

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

Q₈

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×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

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₄²×C₇₃

Q₁₆

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

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

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#108

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₂²×C₇₃

#365

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

characteristic

#1566

—

C₂×#335

—

C₇₃

#7292

yes

characteristic, complemented, Sylow

#423

—

146

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

Dic₇₃

yes

—

#346

—

#335

—

#335

—

146

—

#334

—

#334

—

146

—

C₂×M₁₆

—

292

—

C₂²×C₈

—

292

—

C₂²×C₈

Dic₁₄₆

yes

characteristic

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

C₂×C₄²

D₂₉₂

yes

characteristic

100

C₂×C₁₆

—

101

C₄⋊C₈

—

292

—

102

C₄⋊C₈

—

292

—

103

C₄⋊C₈

—

146

—

104

C₄⋊C₈

—

146

—

105

C₄×C₈

C₄×D₇₃

yes

characteristic

106

C₄×C₈

C₄×D₇₃

yes

characteristic

107

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

108

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

109

C₂²×C₄

C₄⋊Dic₇₃

yes

characteristic

110

C₂²×C₄

#11924017

yes

characteristic

111

C₂²×C₄

#11924017

yes

characteristic

112

M₁₆

—

584

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

292

—

116

C₂×C₈

—

292

—

117

C₂×C₈

—

584

—

118

C₂×C₈

C₄×Dic₇₃

yes

characteristic

119

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

120

C₂×C₈

#11924007

yes

characteristic

121

C₂×C₈

#11924007

yes

characteristic

122

C₄²

#11924005

yes

characteristic

123

C₄²

#11924005

yes

characteristic

124

C₄²

D₅₈₄

yes

characteristic

125

C₄²

Dic₂₉₂

yes

characteristic

126

C₁₆

—

127

C₂³

#14556523

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#14556498

yes

characteristic

134

C₂×C₄

#14556513

yes

characteristic

135

C₂×C₄

#14556513

yes

characteristic

136

C₂×C₄

#14556516

yes

characteristic

137

C₂×C₄

#14556516

yes

characteristic

138

C₂×C₄

#14556525

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

584

—

143

C₈

—

584

—

144

C₈

#14556526

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#17988350

yes

characteristic

148

C₂²

#17988410

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#17988351

yes

characteristic

154

C₄

#17988357

yes

characteristic

155

C₄

#17988401

yes

characteristic

156

C₄

#17988404

yes

characteristic

157

C₂

—

158

C₂

#22607827

yes

characteristic

159

C₂

#22608030

yes

characteristic

160

C₂

#22608040

yes

characteristic

	subgroup	G/H
center	C₂²	#17988350
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₇₃×#2163	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#106
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	#2163	D₇₃
hypocenter	C₇₃	#7292
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂²

#17988350

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₇₃×#2163

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×C₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
84. C₇₃	#7292	semidirect

H ≅ G/N

type

84. C₇₃

#7292

semidirect

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

subgroup	index
#22608687	2
C₂×#17988340	2
C₇₃×#2163	2
#7292	73

subgroup

index

#22608687

C₂×#17988340

C₇₃×#2163

#7292

Subgroups — #32973220

Conjugacy classes of subgroups

Complements

Maximal subgroups