Finite Group Databasebeta

← #37534261

Subgroups — #37534261

Order 23168.

380 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041838C211yesmaximal, characteristic
2#26041754C211yesmaximal, characteristic, complemented
3C2×#19153730C212yesmaximal, complemented
4#26041046C212yesmaximal, complemented
5C2×#99×C181C211yesmaximal, characteristic, complemented
6C22×C22⋊D181C2211yescharacteristic
7#19153799—21no—
8#19153797C412yescomplemented
9#19153788—22no—
10#19153785—21no—
11#19153784C412yescomplemented
12D4×Dic181—24no—
13#19153775—21no—
14C2×C22⋊Dic181—21no—
15C2×C22⋊Dic181C2211yescharacteristic
16C2×C22⋊Dic181C2212yes—
17#19153730—24no—
18C22×D4×C181C2211yescharacteristic
19C2×C22⋊C4×C181C2212yes—
20#99×C181—22no—
21C23×D181C2×C411yescharacteristic, complemented
22C2×C22⋊D181—22no—
23C2×C22⋊D181—22no—
24C2×C22⋊D181—21no—
25C2×C22⋊D181—21no—
26C2×C22⋊D181—22no—
27C22×Dic181—22no—
28C22×Dic181—22no—
29C22×Dic181—21no—
30C22×Dic181C2×C411yescharacteristic, complemented
31C4⋊Dic181—22no—
32#15668428—22no—
33C22⋊Dic181—22no—
34C22⋊Dic181—22no—
35C22⋊Dic181—22no—
36C22⋊Dic181—21no—
37C22⋊Dic181C2×C412yescomplemented
38C22⋊Dic181D414yescomplemented
39#15668426—22no—
40C4×Dic181—22no—
41C24×C181D411yescharacteristic
42C24×C181C2311yescharacteristic
43C2×D4×C181—22no—
44C2×D4×C181—22no—
45C2×D4×C181C2×C412yes—
46C2×D4×C181D412yes—
47C22×C4×C181—22no—
48C22×C4×C181D411yescharacteristic
49C22⋊C4×C181—22no—
50C22⋊C4×C181—22no—
51C22×D181—21no—
52C22×D181—22no—
53C22×D181—81no—
54C22⋊D181—41no—
55C22⋊D181—42no—
56C22⋊D181—41no—
57C22⋊D181—82no—
58C2×Dic181—22no—
59C2×Dic181—21no—
60C2×Dic181—22no—
61C2×Dic181—24no—
62C2×Dic181—82no—
63C2×Dic181—82no—
64C2×Dic181—81no—
65C23×C181—21no—
66C23×C181—21no—
67C23×C181—81no—
68C23×C181—81no—
69C23×C181C22⋊C412yes—
70C23×C181C22×C411yescharacteristic
71C23×C181C2×D412yes—
72C23×C181C2×D411yescharacteristic
73C23×C181C2×D411yescharacteristic
74C23×C181#4112yes—
75D4×C181—24no—
76D4×C181—24no—
77C2×C4×C181—21no—
78C2×C4×C181—21no—
79C2×C4×C181—42no—
80C2×C4×C181—82no—
81C2×C4×C181C22⋊C412yes—
82C22×C181—22no—
83C22×C181—42no—
84C22×C181—41no—
85C22×C181—41no—
86C22×C181—42no—
87C22×C181—41no—
88C22×C181—42no—
89C22×C181—42no—
90C22×C181—81no—
91C22×C181C2×C22⋊C411yescharacteristic
92C22×C181C4×D412yes—
93C22×C181#12011yescharacteristic
94C22×C181#12112yes—
95C22×C181#12311yescharacteristic
96C2×D181—42no—
97C2×D181—41no—
98C2×D181—82no—
99C4×C181—22no—
100C4×C181—82no—
101Dic181—44no—
102Dic181—42no—
103Dic181—41no—
104C2×C181—21no—
105C2×C181—21no—
106C2×C181—22no—
107C2×C181—22no—
108C2×C181—21no—
109C2×C181—42no—
110C2×C181#38611yescharacteristic
111D181—81no—
112C181#191311yescharacteristic, complemented, Sylow
113#1913—1811nomaximal, Sylow
114#534—1811no—
115#518—1811no—
116C2×#99—1812no—
117C2×#99D18111yescharacteristic, complemented
118#342—1812no—
119C22×D4—1811no—
120C22×D4C2×D18111yescharacteristic
121#123—3621no—
122#123—1812no—
123#121—3622no—
124#120—3621no—
125#120—1812no—
126C4×D4—3624no—
127#117—3621no—
128C2×C22⋊C4—3621no—
129C2×C22⋊C4—1812no—
130C2×C22⋊C4—1811no—
131C2×C22⋊C4C2×D18112yes—
132#99—22no—
133#99—3624no—
134C24—1811no—
135C24D72411yescharacteristic
136C24C22×D18111yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—3622no—
140C2×D4—3622no—
141C2×D4—3621no—
142C2×D4—3621no—
143C2×D4—3622no—
144C2×D4C4×D18112yes—
145C2×D4C22⋊D18112yes—
146C22×C4—22no—
147C22×C4—3622no—
148C22×C4—3622no—
149C22×C4—3621no—
150C22×C4—1811no—
151C22×C4D72411yescharacteristic
152C4⋊C4—3622no—
153C4⋊C4—3622no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—3622no—
157C22⋊C4—3622no—
158C22⋊C4—3622no—
159C22⋊C4—3621no—
160C22⋊C4—3622no—
161C22⋊C4—1812no—
162C22⋊C4—1814no—
163C42—3622no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—3622no—
169C23—3621no—
170C23—14481no—
171C23#1566842612yes—
172C23C2×C4×D18111yescharacteristic
173C23D4×D18112yes—
174C23C2×C22⋊D18111yescharacteristic
175C23C2×D72411yescharacteristic
176C23#1566844812yes—
177D4—24no—
178D4—24no—
179D4—14482no—
180D4—7241no—
181D4—7242no—
182D4—7241no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—3622no—
188C2×C4—3621no—
189C2×C4—3622no—
190C2×C4—3624no—
191C2×C4—14482no—
192C2×C4—14482no—
193C2×C4—14481no—
194C2×C4#1566842612yes—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—14482no—
205C22—7242no—
206C22—7241no—
207C22C2×#1566842611yescharacteristic
208C22#1915377612yes—
209C22#1915378611yescharacteristic
210C22#1915378812yes—
211C22#1915380011yescharacteristic
212C4—22no—
213C4—82no—
214C4—7244no—
215C4—7242no—
216C4—7241no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—14481no—
224C2#2604115611yescharacteristic
subgroupG/H
centerC2#26041156
commutatorC23×C181C22×C4
FrattiniC24C22×D181
FittingC2×#99×C181C2
socleC2×C181#386
prime radical#26041838C2
cosocleC24×C181C23
hypercenterC2×#99D181
hypocenterC181#1913
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #26041754C2semidirect
3. C2×#19153730C2semidirect
4. #26041046C2semidirect
5. C2×#99×C181C2semidirect
8. #19153797C4semidirect
11. #19153784C4semidirect
21. C23×D181C2×C4semidirect
30. C22×Dic181C2×C4semidirect
37. C22⋊Dic181C2×C4semidirect
38. C22⋊Dic181D4semidirect
112. C181#1913semidirect
117. C2×#99D181semidirect

Maximal subgroups

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

subgroupindex
#260418382
#260417542
C2×#191537302
#260410462
C2×#99×C1812
#1913181