Group #32989107 — subgroups

Subgroups — #32989107

Order 18688.

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

normal subgroups only

This group has 364 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	#22618318	C₂	1	1	yes	maximal, characteristic
2	#22608495	C₂	1	2	yes	maximal
3	#22607791	C₂	1	2	yes	maximal
4	#22607764	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1994	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	#17988485	—	2	2	no	—
8	#17988471	—	2	4	no	—
9	#17988453	C₂²	1	2	yes	—
10	#17988324	—	2	4	no	—
11	#533×C₇₃	C₂²	1	1	yes	characteristic
12	#423×C₇₃	C₂²	1	1	yes	characteristic
13	C₄³×C₇₃	C₂²	1	1	yes	characteristic
14	#339×C₇₃	—	2	2	no	—
15	C₂×C₄⋊Dic₇₃	—	2	2	no	—
16	C₂×#11924007	—	2	4	no	—
17	C₄⋊#10596831	—	2	2	no	—
18	C₂²⋊#10596831	—	2	2	no	—
19	C₄×#10596831	—	2	2	no	—
20	#14556498	—	2	4	no	—
21	#14556498	—	2	2	no	—
22	#14556498	D₄	1	4	yes	—
23	C₂×M₁₆×C₇₃	—	2	1	no	—
24	C₄⋊Q₈×C₇₃	Q₈	1	2	yes	—
25	#125×C₇₃	—	2	1	no	—
26	C₂²⋊Q₈×C₇₃	—	2	2	no	—
27	C₄×Q₈×C₇₃	—	2	2	no	—
28	#117×C₇₃	—	2	1	no	—
29	C₂×C₄²×C₇₃	—	2	2	no	—
30	C₂×C₄²×C₇₃	D₄	1	2	yes	—
31	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₃	—	2	2	no	—
33	C₂²×Dic₇₃	—	2	4	no	—
34	C₂×#10596831	—	4	2	no	—
35	C₄⋊Dic₇₃	—	4	2	no	—
36	C₄⋊Dic₇₃	—	4	2	no	—
37	#11924007	—	4	8	no	—
38	C₂×Q₈×C₇₃	—	2	2	no	—
39	C₂²×C₄×C₇₃	—	2	2	no	—
40	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	M₁₆×C₇₃	—	4	2	no	—
43	C₂×C₈×C₇₃	—	4	1	no	—
44	C₄⋊C₄×C₇₃	—	2	2	no	—
45	C₄⋊C₄×C₇₃	—	2	2	no	—
46	C₄⋊C₄×C₇₃	—	4	2	no	—
47	C₂²⋊C₄×C₇₃	—	4	1	no	—
48	C₄²×C₇₃	—	2	2	no	—
49	C₄²×C₇₃	—	2	2	no	—
50	C₄²×C₇₃	—	4	4	no	—
51	C₄²×C₇₃	—	4	1	no	—
52	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₇₃	#41	1	2	yes	—
54	C₄²×C₇₃	#41	1	1	yes	characteristic
55	C₂³×C₇₃	#120	1	1	yes	characteristic
56	Q₈×C₇₃	—	8	2	no	—
57	C₂×C₄×C₇₃	—	2	4	no	—
58	C₂×C₄×C₇₃	—	4	2	no	—
59	C₂×C₄×C₇₃	—	4	2	no	—
60	C₂×C₄×C₇₃	—	4	2	no	—
61	C₂×C₄×C₇₃	—	4	2	no	—
62	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#124	1	2	yes	—
65	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#135	1	2	yes	—
67	C₂×C₄×C₇₃	#137	1	2	yes	—
68	C₂×Dic₇₃	—	4	8	no	—
69	C₂×Dic₇₃	—	4	4	no	—
70	C₈×C₇₃	—	8	1	no	—
71	#10596831	—	8	2	no	—
72	C₂²×C₇₃	—	2	2	no	—
73	C₂²×C₇₃	#393	1	1	yes	characteristic
74	C₂²×C₇₃	#452	1	2	yes	—
75	C₄×C₇₃	—	2	2	no	—
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	—	4	4	no	—
78	C₄×C₇₃	—	8	2	no	—
79	C₄×C₇₃	#479	1	2	yes	—
80	C₄×C₇₃	#488	1	1	yes	characteristic
81	C₄×C₇₃	#491	1	1	yes	characteristic
82	Dic₇₃	—	8	4	no	—
83	#10369	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	—	2	2	no	—
85	C₂×C₇₃	#2026	1	1	yes	characteristic
86	C₂×C₇₃	#2031	1	2	yes	—
87	#2864	—	73	1	no	—
88	#1994	D₇₃	1	1	yes	characteristic
89	#1518	—	73	2	no	—
90	#1286	—	73	2	no	—
91	#1276	—	73	1	no	—
92	C₇₃	#10369	1	1	yes	characteristic, complemented, Sylow
93	#533	C₂×D₇₃	1	1	yes	characteristic
94	#431	—	73	2	no	—
95	#423	C₂×D₇₃	1	1	yes	characteristic
96	#389	—	146	2	no	—
97	#387	—	146	4	no	—
98	#382	—	73	2	no	—
99	C₄³	C₂×D₇₃	1	1	yes	characteristic
100	#339	—	2	2	no	—
101	#328	—	146	4	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₄⋊Q₈	Dic₁₄₆	1	2	yes	—
104	#125	—	2	1	no	—
105	C₂²⋊Q₈	—	2	2	no	—
106	C₄×Q₈	—	2	2	no	—
107	#117	—	2	1	no	—
108	C₂×C₄⋊C₄	—	146	4	no	—
109	C₂×C₄⋊C₄	—	146	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
112	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	146	2	no	—
115	C₂²⋊C₈	—	146	2	no	—
116	C₄×C₈	—	146	2	no	—
117	#95	—	73	4	no	—
118	#95	—	146	4	no	—
119	#95	—	146	2	no	—
120	C₂×Q₈	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	146	4	no	—
123	C₂²×C₄	D₄×D₇₃	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	292	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	292	8	no	—
132	C₄⋊C₄	—	292	2	no	—
133	C₄⋊C₄	—	292	2	no	—
134	C₂²⋊C₄	—	4	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	4	1	no	—
139	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
140	C₄²	#11924027	1	2	yes	—
141	C₄²	#11924028	1	1	yes	characteristic
142	C₂³	#14556559	1	1	yes	characteristic
143	Q₈	—	8	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	292	8	no	—
150	C₂×C₄	—	292	4	no	—
151	C₂×C₄	#14556568	1	2	yes	—
152	C₂×C₄	#14556570	1	1	yes	characteristic
153	C₂×C₄	#14556577	1	2	yes	—
154	C₂×C₄	#14556578	1	1	yes	characteristic
155	C₂×C₄	#14556611	1	2	yes	—
156	C₂×C₄	#14556622	1	2	yes	—
157	C₈	—	8	1	no	—
158	C₈	—	584	2	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#17988499	1	1	yes	characteristic
161	C₂²	#17988747	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	2	no	—
166	C₄	—	584	4	no	—
167	C₄	#17988894	1	2	yes	—
168	C₄	#17988941	1	1	yes	characteristic
169	C₄	#17988955	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#22611082	1	1	yes	characteristic
172	C₂	#22611103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22618318

C₂

yes

maximal, characteristic

#22608495

C₂

yes

maximal

#22607791

C₂

yes

maximal

#22607764

C₂

yes

maximal, characteristic

C₇₃×#1994

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

#17988485

—

#17988471

—

#17988453

C₂²

yes

—

#17988324

—

#533×C₇₃

C₂²

yes

characteristic

#423×C₇₃

C₂²

yes

characteristic

C₄³×C₇₃

C₂²

yes

characteristic

#339×C₇₃

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

Q₈

yes

—

#125×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₄²×C₇₃

yes

C₂³×C₇₃

yes

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

#124

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#393

yes

characteristic

C₂²×C₇₃

#452

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#479

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#10369

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2026

yes

characteristic

C₂×C₇₃

#2031

yes

—

#2864

—

#1994

D₇₃

yes

—

—

—

C₇₃

#10369

yes

characteristic, complemented, Sylow

#533

C₂×D₇₃

yes

characteristic

#431

—

#423

C₂×D₇₃

yes

characteristic

#389

—

146

—

#387

—

146

—

#382

—

C₄³

C₂×D₇₃

yes

characteristic

100

#339

—

101

#328

—

146

—

102

C₂×M₁₆

—

103

C₄⋊Q₈

Dic₁₄₆

yes

—

104

#125

—

105

C₂²⋊Q₈

—

106

C₄×Q₈

—

107

#117

—

108

C₂×C₄⋊C₄

—

146

—

109

C₂×C₄⋊C₄

—

146

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂²⋊D₇₃

yes

—

112

C₂×C₄²

C₂²×D₇₃

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

146

—

115

C₂²⋊C₈

—

146

—

116

C₄×C₈

—

146

—

117

#95

—

118

#95

—

146

—

119

#95

—

146

—

120

C₂×Q₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

146

—

123

C₂²×C₄

D₄×D₇₃

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

292

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

292

—

132

C₄⋊C₄

—

292

—

133

C₄⋊C₄

—

292

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂×Dic₁₄₆

yes

characteristic

140

C₄²

#11924027

yes

—

141

C₄²

#11924028

yes

characteristic

142

C₂³

#14556559

yes

characteristic

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

292

—

150

C₂×C₄

—

292

—

151

C₂×C₄

#14556568

yes

—

152

C₂×C₄

#14556570

yes

characteristic

153

C₂×C₄

#14556577

yes

—

154

C₂×C₄

#14556578

yes

characteristic

155

C₂×C₄

#14556611

yes

—

156

C₂×C₄

#14556622

yes

—

157

C₈

—

158

C₈

—

584

—

159

C₂²

—

160

C₂²

#17988499

yes

characteristic

161

C₂²

#17988747

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

584

—

167

C₄

#17988894

yes

—

168

C₄

#17988941

yes

characteristic

169

C₄

#17988955

yes

characteristic

170

C₂

—

171

C₂

#22611082

yes

characteristic

172

C₂

#22611103

yes

—

	subgroup	G/H
center	C₂²	#17988499
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1994	C₂
socle	C₂²×C₇₃	#393
prime radical	C₂³×C₇₃	#120
hypercenter	#1994	D₇₃
hypocenter	C₇₃	#10369
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#17988499

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1994

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

N	H ≅ G/N	type
92. C₇₃	#10369	semidirect

H ≅ G/N

type

92. C₇₃

#10369

semidirect

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

subgroup	index
#22618318	2
#22608495	2
#22607791	2
#22607764	2
C₇₃×#1994	2
#10369	73

subgroup

index

C₇₃×#1994

Subgroups — #32989107

Conjugacy classes of subgroups

Complements

Maximal subgroups