Group #32973002 — subgroups

Subgroups — #32973002

Order 18688.

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

normal subgroups only

This group has 200 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	#22611687	C₂	1	1	yes	maximal, characteristic
2	#22608220	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2163	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556531	C₄	1	2	yes	—
5	#17988640	C₄	1	1	yes	characteristic
6	C₄×#11924012	C₄	1	1	yes	characteristic
7	C₄⋊#11924003	—	2	1	no	—
8	C₄⋊#11924003	—	2	1	no	—
9	#17988309	—	2	1	no	—
10	#17988308	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆×C₇₃	—	2	1	no	—
13	C₂²⋊C₁₆×C₇₃	C₄	1	2	yes	—
14	#346×C₇₃	—	2	1	no	—
15	C₂×#11924012	C₈	1	2	yes	—
16	#14556531	—	4	2	no	—
17	C₂×#11924003	—	2	2	no	—
18	C₄⋊#10596831	—	2	1	no	—
19	C₂²⋊#10596831	C₈	1	2	yes	—
20	#14556502	—	2	1	no	—
21	C₄×#10596831	—	2	1	no	—
22	C₄×#10596831	—	2	1	no	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₃	—	2	2	no	—
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
31	#11924012	—	4	2	no	—
32	C₂×#10596831	—	2	2	no	—
33	C₂×#10596831	—	2	2	no	—
34	#11924003	—	4	2	no	—
35	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₇₃	—	2	1	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	—	4	1	no	—
41	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
48	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
49	C₁₆×C₇₃	—	4	2	no	—
50	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	#100	1	1	yes	characteristic
65	C₈×C₇₃	#101	1	1	yes	characteristic
66	#10596831	—	4	2	no	—
67	#10596831	—	4	2	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	—	2	1	no	—
71	C₂²×C₇₃	#336	1	1	yes	characteristic
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	#343	1	1	yes	characteristic
77	C₄×C₇₃	#344	1	1	yes	characteristic
78	C₄×C₇₃	#349	1	2	yes	—
79	#7211	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	4	1	no	—
81	C₂×C₇₃	#1283	1	1	yes	characteristic
82	C₂×C₇₃	#1377	1	1	yes	characteristic
83	C₂×C₇₃	#1378	1	1	yes	characteristic
84	#2163	D₇₃	1	1	yes	characteristic
85	#2150	—	73	1	no	—
86	#1451	—	73	1	no	—
87	C₇₃	#7211	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₃₂	—	73	2	no	—
89	#431	—	73	1	no	—
90	C₄×M₁₆	—	73	1	no	—
91	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
92	C₄⋊C₁₆	—	2	1	no	—
93	C₄⋊C₁₆	—	146	1	no	—
94	C₄⋊C₁₆	—	146	1	no	—
95	C₂²⋊C₁₆	Dic₇₃	1	2	yes	—
96	#346	—	2	1	no	—
97	#322	—	146	1	no	—
98	#322	—	146	1	no	—
99	C₂×M₁₆	—	73	2	no	—
100	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
102	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
103	M₃₂	—	292	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	146	2	no	—
106	C₄⋊C₈	—	146	1	no	—
107	C₂²⋊C₈	—	73	2	no	—
108	#97	—	146	1	no	—
109	C₄×C₈	—	146	1	no	—
110	C₄×C₈	—	146	1	no	—
111	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
112	C₄×C₈	D₂₉₂	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
116	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
117	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
118	M₁₆	—	292	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	—	146	2	no	—
124	C₂×C₈	#11924005	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
126	C₂×C₈	#11924007	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
128	C₄²	#11924005	1	1	yes	characteristic
129	C₄²	#11924007	1	1	yes	characteristic
130	C₄²	#11924011	1	1	yes	characteristic
131	C₄²	#11924011	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	C₁₆	—	292	2	no	—
134	C₂³	C₈×Dic₇₃	1	1	yes	characteristic
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	#14556498	1	1	yes	characteristic
140	C₂×C₄	#14556501	1	1	yes	characteristic
141	C₂×C₄	#14556503	1	1	yes	characteristic
142	C₂×C₄	#14556503	1	1	yes	characteristic
143	C₂×C₄	#14556520	1	1	yes	characteristic
144	C₂×C₄	#14556520	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	292	2	no	—
149	C₈	—	292	2	no	—
150	C₈	#14556507	1	1	yes	characteristic
151	C₈	#14556509	1	1	yes	characteristic
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#17988341	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	#17988355	1	1	yes	characteristic
161	C₄	#17988357	1	1	yes	characteristic
162	C₄	#17988368	1	2	yes	—
163	C₂	—	4	1	no	—
164	C₂	#22607780	1	1	yes	characteristic
165	C₂	#22608014	1	1	yes	characteristic
166	C₂	#22608016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611687

C₂

yes

maximal, characteristic

#22608220

C₂

yes

maximal, characteristic

C₇₃×#2163

C₂

yes

maximal, characteristic

C₂×#14556531

C₄

yes

—

#17988640

C₄

yes

characteristic

C₄×#11924012

C₄

yes

—

—

—

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₂²⋊C₁₆×C₇₃

C₄

yes

—

#346×C₇₃

—

C₂×#11924012

C₈

yes

—

—

—

—

C₈

yes

—

#14556502

—

C₄×#10596831

—

C₄×#10596831

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

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

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#336

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#7211

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

characteristic

#2150

—

#1451

—

C₇₃

#7211

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

Dic₇₃

yes

—

#346

—

#322

—

146

—

#322

—

146

—

C₂×M₁₆

—

100

C₂²×C₈

C₄×D₇₃

yes

characteristic

101

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

102

C₂×C₄²

C₄×D₇₃

yes

characteristic

103

M₃₂

—

292

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

146

—

106

C₄⋊C₈

—

146

—

107

C₂²⋊C₈

—

108

#97

—

146

—

109

C₄×C₈

—

146

—

110

C₄×C₈

—

146

—

111

C₄×C₈

Dic₁₄₆

yes

characteristic

112

C₄×C₈

D₂₉₂

yes

characteristic

113

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

114

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

115

C₂²×C₄

C₄×Dic₇₃

yes

characteristic

116

C₂²×C₄

C₈×D₇₃

yes

characteristic

117

C₂²×C₄

C₈×D₇₃

yes

characteristic

118

M₁₆

—

292

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₂×C₈

#11924005

yes

characteristic

125

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

126

C₂×C₈

#11924007

yes

characteristic

127

C₂×C₈

C₄⋊Dic₇₃

yes

characteristic

128

C₄²

#11924005

yes

characteristic

129

C₄²

#11924007

yes

characteristic

130

C₄²

#11924011

yes

characteristic

131

C₄²

#11924011

yes

characteristic

132

C₁₆

—

133

C₁₆

—

292

—

134

C₂³

C₈×Dic₇₃

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#14556498

yes

characteristic

140

C₂×C₄

#14556501

yes

characteristic

141

C₂×C₄

#14556503

yes

characteristic

142

C₂×C₄

#14556503

yes

characteristic

143

C₂×C₄

#14556520

yes

characteristic

144

C₂×C₄

#14556520

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

292

—

149

C₈

—

292

—

150

C₈

#14556507

yes

characteristic

151

C₈

#14556509

yes

characteristic

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#17988341

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#17988355

yes

characteristic

161

C₄

#17988357

yes

characteristic

162

C₄

#17988368

yes

—

163

C₂

—

164

C₂

#22607780

yes

characteristic

165

C₂

#22608014

yes

characteristic

166

C₂

#22608016

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556498
commutator, prime radical	C₂³×C₇₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₇₃×#2163	C₂
socle	C₂²×C₇₃	#336
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	#2163	D₇₃
hypocenter	C₇₃	#7211
LCS	G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂×C₄

#14556498

commutator, prime radical

C₂³×C₇₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₇₃×#2163

C₂

socle

C₂²×C₇₃

#336

cosocle

C₂×C₄×C₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
87. C₇₃	#7211	semidirect

H ≅ G/N

type

87. C₇₃

#7211

semidirect

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

subgroup	index
#22611687	2
#22608220	2
C₇₃×#2163	2
#7211	73

subgroup

index

#22611687

#22608220

C₇₃×#2163

#7211

Subgroups — #32973002

Conjugacy classes of subgroups

Complements

Maximal subgroups