Group #32985448 — subgroups

Subgroups — #32985448

Order 18688.

478 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 159 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	#22610103	C₂	1	1	yes	maximal, characteristic
2	#22608603	C₂	1	4	yes	maximal
3	#22608214	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₇₃	C₂	1	1	yes	maximal, characteristic
5	#17988640	C₂²	1	2	yes	—
6	C₂×#14556527	—	2	2	no	—
7	#17988611	C₂²	1	2	yes	—
8	#17988333	—	2	8	no	—
9	#17988308	—	2	1	no	—
10	C₈×#10596831	—	2	1	no	—
11	C₂²×Q₁₆×C₇₃	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
13	#494×C₇₃	—	2	1	no	—
14	#494×C₇₃	—	2	1	no	—
15	#494×C₇₃	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924012	—	2	2	no	—
18	#14556527	—	4	4	no	—
19	C₄⋊#10596831	—	2	4	no	—
20	C₄⋊#10596831	D₄	1	4	yes	—
21	C₂²⋊#10596831	—	2	2	no	—
22	C₄×#10596831	—	2	2	no	—
23	C₂²×Q₈×C₇₃	—	2	2	no	—
24	C₂×Q₁₆×C₇₃	—	2	4	no	—
25	C₂×Q₁₆×C₇₃	—	2	4	no	—
26	C₂×Q₁₆×C₇₃	—	4	4	no	—
27	C₂×Q₁₆×C₇₃	—	4	4	no	—
28	C₂²×C₈×C₇₃	D₄	1	2	yes	—
29	C₄⋊Q₈×C₇₃	—	2	4	no	—
30	C₄⋊Q₈×C₇₃	—	2	2	no	—
31	C₄⋊Q₈×C₇₃	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
33	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
37	#11924012	—	4	4	no	—
38	C₂×#10596831	—	2	4	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₂×Q₈×C₇₃	—	2	4	no	—
41	C₂×Q₈×C₇₃	—	4	2	no	—
42	C₂×Q₈×C₇₃	—	4	4	no	—
43	C₂×Q₈×C₇₃	—	4	4	no	—
44	C₂²×C₄×C₇₃	—	4	2	no	—
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	#41	1	2	yes	—
47	Q₁₆×C₇₃	—	4	8	no	—
48	Q₁₆×C₇₃	—	4	8	no	—
49	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₃	D₈	1	4	yes	—
52	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₇₃	—	2	8	no	—
54	C₄⋊C₄×C₇₃	—	4	4	no	—
55	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₇₃	C₂×D₄	1	2	yes	—
57	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₇₃	#124	1	1	yes	characteristic
59	Q₈×C₇₃	—	4	8	no	—
60	Q₈×C₇₃	—	8	2	no	—
61	Q₈×C₇₃	—	8	2	no	—
62	C₂×C₄×C₇₃	—	2	4	no	—
63	C₂×C₄×C₇₃	—	4	4	no	—
64	C₂×C₄×C₇₃	—	8	4	no	—
65	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₇₃	#121	1	2	yes	—
68	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
71	C₈×C₇₃	#102	1	4	yes	—
72	C₈×C₇₃	#102	1	4	yes	—
73	#10596831	—	4	4	no	—
74	#10596831	—	8	2	no	—
75	C₂²×C₇₃	—	2	2	no	—
76	C₂²×C₇₃	#390	1	1	yes	characteristic
77	C₂²×C₇₃	#486	1	2	yes	—
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	8	4	no	—
80	C₄×C₇₃	C₂×#102	1	2	yes	—
81	C₄×C₇₃	#473	1	4	yes	—
82	C₄×C₇₃	#492	1	1	yes	characteristic
83	C₄×C₇₃	#493	1	1	yes	characteristic
84	#9772	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	—	2	2	no	—
86	C₂×C₇₃	#1957	1	1	yes	characteristic
87	C₂×C₇₃	#1979	1	2	yes	—
88	C₂×#494	D₇₃	1	1	yes	characteristic
89	#1848	—	73	1	no	—
90	#1546	—	73	4	no	—
91	#1450	—	73	1	no	—
92	C₇₃	#9772	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₁₆	—	2	2	no	—
94	C₂×C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
95	#494	—	2	1	no	—
96	#494	—	2	1	no	—
97	#494	Dic₇₃	1	4	yes	—
98	#431	—	73	2	no	—
99	C₂×#108	—	146	2	no	—
100	#423	—	73	2	no	—
101	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
102	#332	—	146	8	no	—
103	#322	—	146	1	no	—
104	C₈²	—	146	1	no	—
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂×Q₁₆	—	4	4	no	—
110	C₂×M₁₆	—	146	2	no	—
111	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂×Dic₇₃	1	4	yes	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
117	#108	—	292	4	no	—
118	C₄⋊C₈	—	73	4	no	—
119	C₄⋊C₈	—	146	4	no	—
120	C₂²⋊C₈	—	146	2	no	—
121	C₄×C₈	—	146	2	no	—
122	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
123	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
131	C₂²×C₄	#11924027	1	2	yes	—
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	8	no	—
134	M₁₆	—	292	4	no	—
135	C₂×C₈	—	292	2	no	—
136	C₂×C₈	—	146	4	no	—
137	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
138	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
139	C₂×C₈	#11924013	1	4	yes	—
140	C₂×C₈	#11924016	1	4	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
144	C₄²	D₄×D₇₃	1	2	yes	—
145	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
146	C₂³	#14556578	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
154	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
155	C₂×C₄	#14556562	1	2	yes	—
156	C₂×C₄	#14556587	1	1	yes	characteristic
157	C₂×C₄	C₂×#11924013	1	2	yes	—
158	C₂×C₄	C₂×#11924016	1	2	yes	—
159	C₈	—	292	4	no	—
160	C₈	—	584	2	no	—
161	C₈	#14556512	1	4	yes	—
162	C₈	#14556512	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	#17988490	1	1	yes	characteristic
165	C₂²	#17988933	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	C₂×#14556512	1	2	yes	—
169	C₄	#17988859	1	4	yes	—
170	C₄	#17988959	1	1	yes	characteristic
171	C₄	#17988963	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	#22610738	1	1	yes	characteristic
174	C₂	#22610847	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22610103

C₂

yes

maximal, characteristic

#22608603

C₂

yes

maximal

#22608214

C₂

yes

maximal, characteristic

C₂×#494×C₇₃

C₂

yes

maximal, characteristic

#17988640

C₂²

yes

—

C₂×#14556527

—

#17988611

C₂²

yes

—

#17988333

—

#17988308

—

C₈×#10596831

—

C₂²×Q₁₆×C₇₃

—

C₂×C₄⋊Q₈×C₇₃

C₂²

yes

—

#494×C₇₃

—

#494×C₇₃

—

#494×C₇₃

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

—

—

D₄

yes

—

C₂²⋊#10596831

—

C₄×#10596831

—

C₂²×Q₈×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

—

—

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

#124

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

#102

yes

—

C₈×C₇₃

#102

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#390

yes

characteristic

C₂²×C₇₃

#486

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#102

yes

—

C₄×C₇₃

#473

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1957

yes

characteristic

C₂×C₇₃

#1979

yes

—

C₂×#494

D₇₃

yes

—

—

—

C₇₃

#9772

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₇₃

yes

—

#494

—

#494

—

#494

Dic₇₃

yes

—

#431

—

C₂×#108

—

146

—

100

#423

—

101

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

102

#332

—

146

—

103

#322

—

146

—

104

C₈²

—

146

—

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×M₁₆

—

146

—

111

C₂²×C₈

C₂²⋊D₇₃

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×Dic₇₃

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂²×D₇₃

yes

characteristic

117

#108

—

292

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

146

—

120

C₂²⋊C₈

—

146

—

121

C₄×C₈

—

146

—

122

C₄×C₈

C₂×Dic₇₃

yes

—

123

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

124

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

131

C₂²×C₄

#11924027

yes

—

132

Q₁₆

—

133

Q₁₆

—

134

M₁₆

—

292

—

135

C₂×C₈

—

292

—

136

C₂×C₈

—

146

—

137

C₂×C₈

C₂²⋊Dic₇₃

yes

—

138

C₂×C₈

C₂²⋊Dic₇₃

yes

—

139

C₂×C₈

#11924013

yes

—

140

C₂×C₈

#11924016

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

C₂²×Dic₇₃

yes

characteristic

144

C₄²

D₄×D₇₃

yes

—

145

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

146

C₂³

#14556578

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₂²⋊Dic₇₃

yes

characteristic

154

C₂×C₄

D₄×Dic₇₃

yes

—

155

C₂×C₄

#14556562

yes

—

156

C₂×C₄

#14556587

yes

characteristic

157

C₂×C₄

C₂×#11924013

yes

—

158

C₂×C₄

C₂×#11924016

yes

—

159

C₈

—

292

—

160

C₈

—

584

—

161

C₈

#14556512

yes

—

162

C₈

#14556512

yes

—

163

C₂²

—

164

C₂²

#17988490

yes

characteristic

165

C₂²

#17988933

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₂×#14556512

yes

—

169

C₄

#17988859

yes

—

170

C₄

#17988959

yes

characteristic

171

C₄

#17988963

yes

characteristic

172

C₂

—

173

C₂

#22610738

yes

characteristic

174

C₂

#22610847

yes

—

	subgroup	G/H
center	C₂²	#17988490
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#494×C₇₃	C₂
socle	C₂²×C₇₃	#390
prime radical	C₂³×C₇₃	#124
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#494	D₇₃
hypocenter	C₇₃	#9772
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#17988490

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₂×#494×C₇₃

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
92. C₇₃	#9772	semidirect

H ≅ G/N

type

92. C₇₃

#9772

semidirect

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

subgroup	index
#22610103	2
#22608603	2
#22608214	2
C₂×#494×C₇₃	2
#9772	73

subgroup

index

C₂×#494×C₇₃

Subgroups — #32985448

Conjugacy classes of subgroups

Complements

Maximal subgroups