Group #32989104 — subgroups

Subgroups — #32989104

Order 18688.

364 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22610908	C₂	1	1	yes	maximal, characteristic
2	#22608494	C₂	1	2	yes	maximal
3	#22607789	C₂	1	2	yes	maximal
4	#22607764	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#2864	C₂	1	1	yes	maximal, characteristic
6	#17989144	C₂²	1	1	yes	characteristic
7	#17988640	C₂²	1	2	yes	—
8	#17988611	C₂²	1	1	yes	characteristic
9	#17988347	—	2	2	no	—
10	#17988323	—	2	4	no	—
11	#389×C₇₃	—	2	2	no	—
12	#387×C₇₃	—	2	4	no	—
13	#382×C₇₃	C₂²	1	2	yes	—
14	C₄³×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#11924012	—	2	1	no	—
16	#14556591	Q₈	1	2	yes	—
17	#14556583	—	2	1	no	—
18	#14556570	—	2	2	no	—
19	C₄×Dic₁₄₆	—	2	2	no	—
20	#14556550	—	2	1	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₂²⋊#10596831	—	2	2	no	—
24	C₄×#10596831	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
27	C₂×C₄²×C₇₃	—	2	2	no	—
28	C₂×C₄²×C₇₃	D₄	1	2	yes	—
29	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
30	#95×C₇₃	—	2	4	no	—
31	#95×C₇₃	—	2	2	no	—
32	#95×C₇₃	D₄	1	4	yes	—
33	C₂×Dic₁₄₆	—	2	2	no	—
34	#11924012	—	4	2	no	—
35	C₂×#10596831	—	4	2	no	—
36	C₂×#10596831	—	4	1	no	—
37	C₄⋊Dic₇₃	—	2	2	no	—
38	C₄⋊Dic₇₃	—	2	2	no	—
39	#11924007	—	4	2	no	—
40	C₂²⋊Dic₇₃	—	4	1	no	—
41	C₄×Dic₇₃	—	4	1	no	—
42	C₂²×C₄×C₇₃	—	2	2	no	—
43	C₂²×C₄×C₇₃	—	2	4	no	—
44	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₇₃	—	4	8	no	—
47	C₄⋊C₄×C₇₃	—	4	2	no	—
48	C₄⋊C₄×C₇₃	—	4	2	no	—
49	C₄²×C₇₃	—	2	2	no	—
50	C₄²×C₇₃	—	2	2	no	—
51	C₄²×C₇₃	—	4	4	no	—
52	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₇₃	#41	1	1	yes	characteristic
54	C₄²×C₇₃	#41	1	2	yes	—
55	C₂³×C₇₃	#120	1	1	yes	characteristic
56	C₂×C₄×C₇₃	—	2	4	no	—
57	C₂×C₄×C₇₃	—	4	2	no	—
58	C₂×C₄×C₇₃	—	4	2	no	—
59	C₂×C₄×C₇₃	—	4	2	no	—
60	C₂×C₄×C₇₃	—	4	8	no	—
61	C₂×C₄×C₇₃	—	4	4	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	2	no	—
69	Dic₁₄₆	—	8	2	no	—
70	#10596831	—	8	2	no	—
71	#10596831	—	8	1	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	4	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	2	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	D₇₃	1	1	yes	characteristic
88	#1994	—	73	1	no	—
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	—	73	1	no	—
94	#431	—	73	2	no	—
95	#423	—	73	1	no	—
96	#389	—	2	2	no	—
97	#387	—	2	4	no	—
98	#382	C₂×D₇₃	1	2	yes	—
99	C₄³	C₂×D₇₃	1	1	yes	characteristic
100	#339	—	146	2	no	—
101	#328	—	146	4	no	—
102	C₂×M₁₆	—	146	1	no	—
103	C₄⋊Q₈	—	73	2	no	—
104	#125	—	146	1	no	—
105	C₂²⋊Q₈	—	146	2	no	—
106	C₄×Q₈	—	146	2	no	—
107	#117	—	146	1	no	—
108	C₂×C₄⋊C₄	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	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	—	146	2	no	—
114	C₄⋊C₈	—	146	2	no	—
115	C₂²⋊C₈	—	146	2	no	—
116	C₄×C₈	—	146	2	no	—
117	#95	—	2	4	no	—
118	#95	—	2	2	no	—
119	#95	C₂²⋊D₇₃	1	4	yes	—
120	C₂×Q₈	—	146	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
125	M₁₆	—	292	2	no	—
126	C₂×C₈	—	292	2	no	—
127	C₂×C₈	—	292	1	no	—
128	C₄⋊C₄	—	4	8	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	292	2	no	—
132	C₄⋊C₄	—	146	2	no	—
133	C₄⋊C₄	—	146	2	no	—
134	C₂²⋊C₄	—	292	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	292	1	no	—
139	C₄²	Q₈×D₇₃	1	1	yes	characteristic
140	C₄²	#11924029	1	1	yes	characteristic
141	C₄²	#11924029	1	2	yes	—
142	C₂³	#14556560	1	1	yes	characteristic
143	Q₈	—	584	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	8	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	292	2	no	—
151	C₂×C₄	#14556567	1	2	yes	—
152	C₂×C₄	#14556567	1	1	yes	characteristic
153	C₂×C₄	#14556580	1	2	yes	—
154	C₂×C₄	#14556580	1	1	yes	characteristic
155	C₂×C₄	#14556611	1	2	yes	—
156	C₂×C₄	#14556621	1	2	yes	—
157	C₈	—	584	2	no	—
158	C₈	—	584	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#17988498	1	1	yes	characteristic
161	C₂²	#17988749	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	4	no	—
166	C₄	—	584	2	no	—
167	C₄	#17988891	1	2	yes	—
168	C₄	#17988944	1	1	yes	characteristic
169	C₄	#17988957	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#22611079	1	1	yes	characteristic
172	C₂	#22611100	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610908

C₂

yes

maximal, characteristic

#22608494

C₂

yes

maximal

#22607789

C₂

yes

maximal

#22607764

C₂

yes

maximal, characteristic

C₇₃×#2864

C₂

yes

maximal, characteristic

#17989144

C₂²

yes

characteristic

#17988640

C₂²

yes

—

#17988611

C₂²

yes

characteristic

#17988347

—

#17988323

—

#389×C₇₃

—

#387×C₇₃

—

#382×C₇₃

C₂²

yes

—

C₄³×C₇₃

C₂²

yes

characteristic

C₂×#11924012

—

#14556591

Q₈

yes

—

#14556583

—

#14556570

—

C₄×Dic₁₄₆

—

—

—

—

—

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#95×C₇₃

—

#95×C₇₃

—

#95×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

#11924012

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×Q₈

yes

C₄²×C₇₃

yes

C₄²×C₇₃

yes

—

C₂³×C₇₃

#120

yes

characteristic

C₂×C₄×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

—

Dic₁₄₆

—

#10596831

—

#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

D₇₃

yes

—

—

—

—

C₇₃

#10369

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×D₇₃

yes

—

C₄³

C₂×D₇₃

yes

characteristic

100

#339

—

146

—

101

#328

—

146

—

102

C₂×M₁₆

—

146

—

103

C₄⋊Q₈

—

104

#125

—

146

—

105

C₂²⋊Q₈

—

146

—

106

C₄×Q₈

—

146

—

107

#117

—

146

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂²⋊D₇₃

yes

—

112

C₂×C₄²

C₂²×D₇₃

yes

characteristic

113

C₄⋊C₈

—

146

—

114

C₄⋊C₈

—

146

—

115

C₂²⋊C₈

—

146

—

116

C₄×C₈

—

146

—

117

#95

—

118

#95

—

119

#95

C₂²⋊D₇₃

yes

—

120

C₂×Q₈

—

146

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

125

M₁₆

—

292

—

126

C₂×C₈

—

292

—

127

C₂×C₈

—

292

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

292

—

132

C₄⋊C₄

—

146

—

133

C₄⋊C₄

—

146

—

134

C₂²⋊C₄

—

292

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

292

—

139

C₄²

Q₈×D₇₃

yes

characteristic

140

C₄²

#11924029

yes

characteristic

141

C₄²

#11924029

yes

—

142

C₂³

#14556560

yes

characteristic

143

Q₈

—

584

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

292

—

151

C₂×C₄

#14556567

yes

—

152

C₂×C₄

#14556567

yes

characteristic

153

C₂×C₄

#14556580

yes

—

154

C₂×C₄

#14556580

yes

characteristic

155

C₂×C₄

#14556611

yes

—

156

C₂×C₄

#14556621

yes

—

157

C₈

—

584

—

158

C₈

—

584

—

159

C₂²

—

160

C₂²

#17988498

yes

characteristic

161

C₂²

#17988749

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

584

—

167

C₄

#17988891

yes

—

168

C₄

#17988944

yes

characteristic

169

C₄

#17988957

yes

characteristic

170

C₂

—

171

C₂

#22611079

yes

characteristic

172

C₂

#22611100

yes

—

	subgroup	G/H
center	C₂²	#17988498
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#2864	C₂
socle	C₂²×C₇₃	#393
prime radical	C₂³×C₇₃	#120
hypercenter	#2864	D₇₃
hypocenter	C₇₃	#10369
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

subgroup

G/H

center

C₂²

#17988498

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#2864

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

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
#22610908	2
#22608494	2
#22607789	2
#22607764	2
C₇₃×#2864	2
#10369	73

subgroup

index

C₇₃×#2864

Subgroups — #32989104

Conjugacy classes of subgroups

Complements

Maximal subgroups