Finite Group Databasebeta

← #37544686

Subgroups — #37544686

Order 23168.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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#26041813C211yesmaximal, characteristic, complemented
2#26041661C212yesmaximal, complemented
3#26041633C212yesmaximal, complemented
4#26041498C212yesmaximal, complemented
5#26041491C212yesmaximal, complemented
6#26041435C211yesmaximal, characteristic, complemented
7#26041430C211yesmaximal, characteristic, complemented
8#26041418C211yesmaximal, characteristic, complemented
9#26041413C211yesmaximal, characteristic
10#26041332C211yesmaximal, characteristic, complemented
11C2×#124×C181C211yesmaximal, characteristic, complemented
12C2×#15668439C2212yescomplemented
13C2×#15668437C2212yes—
14C2×#15668436C2212yescomplemented
15C2×#15668434C2212yes—
16#19153810—21no—
17#19153808C2211yescharacteristic, complemented
18#19153804C2211yescharacteristic, complemented
19#19153799—21no—
20#19153795C2211yescharacteristic
21#19153790C2211yescharacteristic
22C4×Dic362C2211yescharacteristic, complemented
23C4×D724C2211yescharacteristic, complemented
24C4×C22⋊D181—21no—
25#19153773—21no—
26C4⋊#13089678C2212yescomplemented
27#19153740C2212yescomplemented
28#19153740C2212yescomplemented
29#19153737C2212yes—
30#19153737C2212yes—
31C22⋊#13089678C2211yescharacteristic
32C22⋊#13089678C2211yescharacteristic
33C4×#13089678C2212yescomplemented
34C22×Q8×C181C2211yescharacteristic
35C22×D4×C181C2211yescharacteristic
36#124×C181—21no—
37#124×C181—21no—
38#124×C181C2212yes—
39#124×C181C2212yes—
40C2×C22⋊C4×C181—21no—
41C2×C22⋊C4×C181—21no—
42C2×C42×C181C2211yescharacteristic
43C2×Dic362C2311yescharacteristic
44C2×D724C2311yescharacteristic
45C2×C22⋊D181—21no—
46C2×C4×D181—21no—
47#15668439—42no—
48#15668437—42no—
49#15668436—42no—
50#15668434—42no—
51C2×#13089678C2312yes—
52C2×#13089678C2312yes—
53C4⋊Dic181C2311yescharacteristic
54C4⋊Dic181C2311yescharacteristic
55#15668428—21no—
56#15668428—21no—
57#15668428—21no—
58#15668428—21no—
59C22⋊Dic181—21no—
60#15668426—21no—
61#15668426—21no—
62#15668426—21no—
63#15668426—21no—
64C4×Dic181—21no—
65C24×C181—21no—
66C2×Q8×C181—21no—
67C2×Q8×C181—42no—
68C2×Q8×C181C2312yes—
69C2×D4×C181—21no—
70C2×D4×C181—42no—
71C2×D4×C181C2312yes—
72C22×C4×C181—21no—
73C22×C4×C181D411yescharacteristic
74C22×C4×C181D411yescharacteristic
75C22×C4×C181C2311yescharacteristic
76C22⋊C4×C181—22no—
77C22⋊C4×C181—22no—
78C22⋊C4×C181—22no—
79C22⋊C4×C181—22no—
80C42×C181D411yescharacteristic
81C42×C181D411yescharacteristic
82C42×C181C2311yescharacteristic
83C42×C181C2311yescharacteristic
84C22×D181—21no—
85Dic362—22no—
86D724—22no—
87C22⋊D181—42no—
88C2×Dic181—21no—
89C2×Dic181—21no—
90C2×Dic181—21no—
91C4×D181—42no—
92#13089678—22no—
93#13089678—22no—
94C23×C181—22no—
95C23×C181—42no—
96C23×C181—42no—
97C23×C181—42no—
98C23×C181C2×D411yescharacteristic
99Q8×C181—22no—
100Q8×C181—22no—
101Q8×C181—42no—
102D4×C181—22no—
103D4×C181—22no—
104D4×C181—42no—
105C2×C4×C181—22no—
106C2×C4×C181—22no—
107C2×C4×C181—22no—
108C2×C4×C181—41no—
109C2×C4×C181—42no—
110C2×C4×C181C2×D411yescharacteristic
111C2×C4×C181C2×D411yescharacteristic
112C2×C4×C181C2×D411yescharacteristic
113C2×C4×C181C2×D411yescharacteristic
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2411yescharacteristic
116C22×C181—21no—
117C22×C181—21no—
118C22×C181—42no—
119C22×C181—42no—
120C22×C181—42no—
121C22×C181—84no—
122C22×C181#13512yes—
123C22×C181C22×D411yescharacteristic
124C2×D181—41no—
125C2×D181—41no—
126C2×D181—41no—
127C4×C181—21no—
128C4×C181—21no—
129C4×C181—21no—
130C4×C181—21no—
131C4×C181—21no—
132C4×C181—42no—
133C4×C181H(5,𝔽2)12yes—
134Dic181—41no—
135Dic181—41no—
136Dic181—41no—
137C2×C181—22no—
138C2×C181—82no—
139C2×C181#53511yescharacteristic
140C2×C181C2×#13511yescharacteristic
141C2×C181#57511yescharacteristic
142D181—81no—
143C181#321111yescharacteristic, complemented, Sylow
144#3211—1811nomaximal, Sylow
145#529—1811no—
146C2×#124D18111yescharacteristic, complemented
147#495—1812no—
148#488—1812no—
149#464—1812no—
150#463—1812no—
151#451—1811no—
152#450—1811no—
153#448—1811no—
154#447—1811no—
155#431—1811no—
156C22×Q8C2×D18111yescharacteristic
157C22×D4C2×D18111yescharacteristic
158C2×Q16—1812no—
159C2×SD16—1812no—
160C2×SD16—1812no—
161C2×D8—1812no—
162#126—3621no—
163#125—1811no—
164#124—21no—
165#124—21no—
166#124—1811no—
167#124C2×D18112yes—
168#124C2×D18112yes—
169#123—3621no—
170C22⋊Q8—1811no—
171#121—1811no—
172C4×Q8—1811no—
173C4×D4—3621no—
174C4×D4—1811no—
175#117—3621no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42C2×D18111yescharacteristic
179C4⋊C8—1812no—
180#103—1812no—
181#103—1812no—
182#102—1812no—
183#102—1812no—
184C22⋊C8—1811no—
185C22⋊C8—1811no—
186C4×C8—1812no—
187C24—21no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8—1811no—
191C2×Q8C22×D18112yes—
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—3621no—
195C2×D4—1811no—
196C2×D4C22×D18112yes—
197C22×C4—21no—
198C22×C4—3621no—
199C22×C4C22⋊D18111yescharacteristic
200C22×C4C22⋊D18111yescharacteristic
201C22×C4C22×D18111yescharacteristic
202Q16—7242no—
203SD16—7242no—
204SD16—7242no—
205D8—7242no—
206C2×C8—1812no—
207C2×C8—1812no—
208C4⋊C4—3621no—
209C4⋊C4—3621no—
210C4⋊C4—3621no—
211C4⋊C4—3621no—
212C4⋊C4—1811no—
213C4⋊C4—1811no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—3621no—
219C22⋊C4—3621no—
220C22⋊C4—3621no—
221C22⋊C4—3621no—
222C22⋊C4—3621no—
223C42—3621no—
224C42C22⋊D18111yescharacteristic
225C42C22⋊D18111yescharacteristic
226C42C22×D18111yescharacteristic
227C42C22×D18111yescharacteristic
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—3621no—
233C23C2×C22⋊D18111yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—42no—
237Q8—3622no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—3622no—
242D4—7242no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—3621no—
249C2×C4—3621no—
250C2×C4—3621no—
251C2×C4—7242no—
252C2×C4C2×C22⋊D18111yescharacteristic
253C2×C4C2×C22⋊D18111yescharacteristic
254C2×C4C2×C22⋊D18111yescharacteristic
255C2×C4C2×C22⋊D18111yescharacteristic
256C2×C4C2×C22⋊D18111yescharacteristic
257C2×C4C23×D18111yescharacteristic
258C8—3622no—
259C8—3622no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—7241no—
267C22—7241no—
268C22—7241no—
269C22#1915383612yes—
270C22C22×C22⋊D18111yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—42no—
277C4—7241no—
278C4—7241no—
279C4—7241no—
280C4#1915386112yes—
281C2—22no—
282C2—82no—
283C2—14481no—
284C2#2604184611yescharacteristic
285C2C2×#1915383611yescharacteristic
286C2#2604207711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#124×C181C2
socleC22×C181C22×D4
prime radical#26041413C2
hypercenterC2×#124D181
hypocenterC181#3211
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #26041813C2semidirect
2. #26041661C2semidirect
3. #26041633C2semidirect
4. #26041498C2semidirect
5. #26041491C2semidirect
6. #26041435C2semidirect
7. #26041430C2semidirect
8. #26041418C2semidirect
10. #26041332C2semidirect
11. C2×#124×C181C2semidirect
12. C2×#15668439C22semidirect
14. C2×#15668436C22semidirect
17. #19153808C22semidirect
18. #19153804C22semidirect
22. C4×Dic362C22semidirect
23. C4×D724C22semidirect
26. C4⋊#13089678C22semidirect
27. #19153740C22semidirect
28. #19153740C22semidirect
33. C4×#13089678C22semidirect
143. C181#3211semidirect
146. C2×#124D181semidirect

Maximal subgroups

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

subgroupindex
#260418132
#260416612
#260416332
#260414982
#260414912
#260414352
#260414302
#260414182
#260414132
#260413322
C2×#124×C1812
#3211181