Group #32973023 — subgroups

Subgroups — #32973023

Order 18688.

202 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22611730	C₂	1	1	yes	maximal, characteristic
2	#22608694	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1835	C₂	1	1	yes	maximal, characteristic
4	#17988644	—	2	1	no	—
5	C₄⋊#11924003	—	2	1	no	—
6	C₂²⋊#11924003	—	2	1	no	—
7	C₄×#11924003	C₄	1	2	yes	complemented
8	#17988340	—	2	1	no	—
9	#17988338	C₄	1	2	yes	complemented
10	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
11	#382×C₇₃	—	2	1	no	—
12	#340×C₇₃	—	2	1	no	—
13	#340×C₇₃	C₄	1	2	yes	—
14	C₂×#11924003	—	2	2	no	—
15	C₄⋊#10596831	—	2	2	no	—
16	C₂²⋊#10596831	—	4	1	no	—
17	#14556502	—	4	1	no	—
18	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
21	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
26	#95×C₇₃	—	4	1	no	—
27	#95×C₇₃	—	4	1	no	—
28	C₂×#10596831	—	4	2	no	—
29	#11924003	—	4	2	no	—
30	C₂²×C₄×C₇₃	—	4	2	no	—
31	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
34	C₂×C₈×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	—	4	1	no	—
36	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	—	4	2	no	—
41	C₄⋊C₄×C₇₃	—	4	2	no	—
42	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
45	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
46	C₂³×C₇₃	#102	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	—	4	1	no	—
51	C₂×C₄×C₇₃	—	8	2	no	—
52	C₂×C₄×C₇₃	—	8	2	no	—
53	C₂×C₄×C₇₃	—	8	2	no	—
54	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	2	1	no	—
64	#10596831	—	8	2	no	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	—	2	1	no	—
67	C₂²×C₇₃	#340	1	1	yes	characteristic
68	C₂²×C₇₃	#359	1	2	yes	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	16	2	no	—
75	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
76	C₄×C₇₃	#344	1	1	yes	characteristic
77	#7218	—	73	1	no	maximal, Sylow
78	C₂×C₇₃	—	2	2	no	—
79	C₂×C₇₃	#1286	1	1	yes	characteristic
80	C₂×C₇₃	#1382	1	1	yes	characteristic
81	C₂×C₇₃	#1384	1	1	yes	characteristic
82	#2162	—	73	1	no	—
83	#1835	D₇₃	1	1	yes	characteristic
84	#1568	—	73	1	no	—
85	C₇₃	#7218	1	1	yes	characteristic, complemented, Sylow
86	#432	—	146	1	no	—
87	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
88	#382	—	2	1	no	—
89	C₄⋊C₁₆	—	146	1	no	—
90	C₂²⋊C₁₆	—	146	1	no	—
91	C₄×C₁₆	—	73	2	no	—
92	#340	—	2	1	no	—
93	#340	Dic₇₃	1	2	yes	—
94	#335	—	146	1	no	—
95	#334	—	73	2	no	—
96	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
97	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
100	C₂×C₁₆	—	146	2	no	—
101	C₄⋊C₈	—	146	2	no	—
102	C₂²⋊C₈	—	292	1	no	—
103	#97	—	292	1	no	—
104	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
105	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
106	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
107	C₄×C₈	D₂₉₂	1	1	yes	characteristic
108	#95	—	4	1	no	—
109	#95	—	4	1	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
112	C₂²×C₄	#11924013	1	1	yes	characteristic
113	C₂²×C₄	#11924016	1	1	yes	characteristic
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	292	2	no	—
117	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
118	C₂×C₈	#11924005	1	1	yes	characteristic
119	C₂×C₈	#11924007	1	1	yes	characteristic
120	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄²	#11924005	1	1	yes	characteristic
124	C₄²	#11924007	1	1	yes	characteristic
125	C₄²	#11924015	1	1	yes	characteristic
126	C₄²	#11924018	1	1	yes	characteristic
127	C₁₆	—	292	2	no	—
128	C₂³	#14556512	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	#14556498	1	1	yes	characteristic
137	C₂×C₄	#14556511	1	1	yes	characteristic
138	C₂×C₄	#14556514	1	1	yes	characteristic
139	C₂×C₄	#14556515	1	1	yes	characteristic
140	C₂×C₄	#14556522	1	1	yes	characteristic
141	C₂×C₄	#14556524	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	584	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#17988349	1	1	yes	characteristic
150	C₂²	#17988394	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	16	2	no	—
157	C₄	#17988343	1	1	yes	characteristic
158	C₄	#17988357	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	#22607789	1	1	yes	characteristic
161	C₂	#22608026	1	1	yes	characteristic
162	C₂	#22608032	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611730

C₂

yes

maximal, characteristic

#22608694

C₂

yes

maximal, characteristic

C₇₃×#1835

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

#17988340

—

#17988338

C₄

yes

complemented

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#382×C₇₃

—

#340×C₇₃

—

#340×C₇₃

C₄

yes

—

—

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

characteristic

#95×C₇₃

—

#95×C₇₃

—

C₂×#10596831

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

D₈

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

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

characteristic

C₄²×C₇₃

Q₁₆

yes

characteristic

C₂³×C₇₃

#102

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

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

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₂²×C₇₃

#359

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

characteristic

#1568

—

C₇₃

#7218

yes

characteristic, complemented, Sylow

#432

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#382

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

146

—

C₄×C₁₆

—

#340

—

#340

Dic₇₃

yes

—

#335

—

146

—

#334

—

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

100

C₂×C₁₆

—

146

—

101

C₄⋊C₈

—

146

—

102

C₂²⋊C₈

—

292

—

103

#97

—

292

—

104

C₄×C₈

Dic₁₄₆

yes

characteristic

105

C₄×C₈

C₄×D₇₃

yes

characteristic

106

C₄×C₈

C₄×D₇₃

yes

characteristic

107

C₄×C₈

D₂₉₂

yes

characteristic

108

#95

—

109

#95

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

112

C₂²×C₄

#11924013

yes

characteristic

113

C₂²×C₄

#11924016

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

292

—

117

C₂×C₈

C₄×Dic₇₃

yes

characteristic

118

C₂×C₈

#11924005

yes

characteristic

119

C₂×C₈

#11924007

yes

characteristic

120

C₂×C₈

C₄⋊Dic₇₃

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

#11924005

yes

characteristic

124

C₄²

#11924007

yes

characteristic

125

C₄²

#11924015

yes

characteristic

126

C₄²

#11924018

yes

characteristic

127

C₁₆

—

292

—

128

C₂³

#14556512

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#14556498

yes

characteristic

137

C₂×C₄

#14556511

yes

characteristic

138

C₂×C₄

#14556514

yes

characteristic

139

C₂×C₄

#14556515

yes

characteristic

140

C₂×C₄

#14556522

yes

characteristic

141

C₂×C₄

#14556524

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

584

—

147

C₂²

—

148

C₂²

—

149

C₂²

#17988349

yes

characteristic

150

C₂²

#17988394

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#17988343

yes

characteristic

158

C₄

#17988357

yes

characteristic

159

C₂

—

160

C₂

#22607789

yes

characteristic

161

C₂

#22608026

yes

characteristic

162

C₂

#22608032

yes

characteristic

	subgroup	G/H
center	C₂²	#17988349
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₇₃×#1835	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#102
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	#1835	D₇₃
hypocenter	C₇₃	#7218
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1835

subgroup

G/H

center

C₂²

#17988349

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₇₃×#1835

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×C₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1835

N	H ≅ G/N	type
7. C₄×#11924003	C₄	semidirect
9. #17988338	C₄	semidirect
85. C₇₃	#7218	semidirect

H ≅ G/N

type

7. C₄×#11924003

C₄

semidirect

9. #17988338

C₄

semidirect

85. C₇₃

#7218

semidirect

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

subgroup	index
#22611730	2
#22608694	2
C₇₃×#1835	2
#7218	73

subgroup

index

#22611730

#22608694

C₇₃×#1835

#7218

Subgroups — #32973023

Conjugacy classes of subgroups

Complements

Maximal subgroups