Finite Group Databasebeta

← #36363500

Subgroups — #36363500

Order 22144.

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

normal subgroups only

This group has 462 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#121×D173C211yesmaximal, characteristic
2C2×#19149637C211yesmaximal, characteristic, complemented
3#26011817C212yesmaximal, complemented
4#26011813C212yesmaximal, complemented
5#26011809C211yesmaximal, characteristic, complemented
6#26011805C211yesmaximal, characteristic, complemented
7#26011802C211yesmaximal, characteristic, complemented
8#26011530C212yesmaximal, complemented
9#26011494C212yesmaximal, complemented
10#26011453C211yesmaximal, characteristic, complemented
11#481×C173C211yesmaximal, characteristic, complemented
12C2×D4×D173—21no—
13C2×D4×D173C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15C2×#14583809C2212yes—
16#19149654C2211yescharacteristic
17#19149653C2211yescharacteristic
18#19149652—21no—
19#19149651—21no—
20#19149650C2211yescharacteristic
21#19149637—21no—
22#19149637—21no—
23#19149637C2212yes—
24#19149637C2212yescomplemented
25C2×C4⋊Dic173C2211yescharacteristic, complemented
26C4⋊C4×D173C2211yescharacteristic
27C2×#14583803—21no—
28C22⋊C4×D173—21no—
29C2×C4×Dic173—21no—
30#19149613C2212yescomplemented
31#19149612C2212yescomplemented
32#19149610C2212yescomplemented
33#19149601C2212yescomplemented
34#19149600C2212yescomplemented
35#19149599C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic, complemented
37#19149591C2212yescomplemented
38#121×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic, complemented
40#106×C173C2212yescomplemented
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic, complemented
43C23×D173—21no—
44C23×D173D411yescharacteristic
45C2×D692—21no—
46C2×D692C2311yescharacteristic
47C2×C22⋊D173—21no—
48C2×C22⋊D173—21no—
49C2×C22⋊D173—21no—
50D4×D173—24no—
51D4×D173—44no—
52C22×Dic173—21no—
53C22×Dic173D411yescharacteristic
54C2×C4×D173—24no—
55C2×C4×D173—21no—
56C2×C4×D173D411yescharacteristic
57C2×C4×D173D411yescharacteristic
58C2×C4×D173C2311yescharacteristic
59C2×C4×D173C2311yescharacteristic
60#14583809—24no—
61C2×#13063436C2312yes—
62C4⋊Dic173—21no—
63C4⋊Dic173C2312yes—
64C4⋊Dic173C2311yescharacteristic
65#14583805—21no—
66C22⋊Dic173—21no—
67#14583803—22no—
68#14583803—22no—
69#14583803—21no—
70#14583803—21no—
71C4×Dic173—22no—
72C4×Dic173—22no—
73C2×D4×C173—21no—
74C2×D4×C173C2311yescharacteristic
75C22×C4×C173—21no—
76C22×C4×C173C2311yescharacteristic
77C2×C8×C173C2312yes—
78C4⋊C4×C173—21no—
79C4⋊C4×C173C2312yes—
80C4⋊C4×C173C2311yescharacteristic
81C22⋊C4×C173—21no—
82C22×D173—22no—
83C22×D173—22no—
84C22×D173—21no—
85C22×D173—41no—
86C22×D173—42no—
87C22×D173—42no—
88C22×D173—42no—
89C22×D173C2×D411yescharacteristic
90C22×D173C2×D411yescharacteristic
91D692—22no—
92D692—42no—
93C22⋊D173—42no—
94C22⋊D173—42no—
95C22⋊D173—42no—
96C2×Dic173—22no—
97C2×Dic173—21no—
98C2×Dic173—22no—
99C2×Dic173—42no—
100C2×Dic173C2×D411yescharacteristic
101C2×Dic173C2×D411yescharacteristic
102C4×D173—22no—
103C4×D173—22no—
104C4×D173—22no—
105C4×D173—42no—
106C4×D173#4114yes—
107#13063436—22no—
108C23×C173—21no—
109C23×C173C2×D411yescharacteristic
110D4×C173—22no—
111D4×C173—42no—
112C2×C4×C173—22no—
113C2×C4×C173—21no—
114C2×C4×C173—22no—
115C2×C4×C173—42no—
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2411yescharacteristic
118C8×C173—22no—
119C22×C173—21no—
120C22×C173—21no—
121C22×C173—41no—
122C22×C173—41no—
123C22×C173—41no—
124C22×C173AGL(1,ℤ/8)12yes—
125C22×C173C22×D411yescharacteristic
126C2×D173—21no—
127C2×D173—21no—
128C2×D173—21no—
129C2×D173—21no—
130C2×D173—21no—
131C2×D173—44no—
132C2×D173—41no—
133C2×D173—41no—
134C2×D173—41no—
135C2×D173—84no—
136C2×D173#12312yes—
137C4×C173—21no—
138C4×C173—41no—
139C4×C173—42no—
140C4×C173C2×#4112yes—
141Dic173—21no—
142Dic173—41no—
143Dic173—42no—
144Dic173#12312yes—
145C2×C173—22no—
146C2×C173—81no—
147C2×C173C2×#12311yescharacteristic
148C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C173#57511yescharacteristic
150D173—22no—
151D173—41no—
152D173—81no—
153C173#309411yescharacteristic, complemented, Sylow
154#3094—1731nomaximal, Sylow
155C2×#121—1731no—
156C2×#117—1731no—
157#482—1732no—
158#482—1732no—
159#481—1731no—
160#481D17311yescharacteristic, complemented
161#480—1731no—
162#480—1731no—
163#428—1732no—
164#418—1732no—
165#407—1731no—
166C22×D4—3461no—
167C22×D4—1731no—
168C23×C4—1731no—
169C2×M16—1732no—
170#121—3461no—
171#121—3461no—
172#121—1731no—
173#121—1731no—
174#121—1731no—
175#121C2×D17311yescharacteristic
176#117—3461no—
177#117—3461no—
178#117—1732no—
179#117—1732no—
180C2×C4⋊C4—1731no—
181C2×C4⋊C4—1731no—
182C2×C4⋊C4C2×D17311yescharacteristic, complemented
183C2×C22⋊C4—3461no—
184C2×C22⋊C4—3461no—
185C2×C42—3461no—
186#107—1732no—
187#107—1732no—
188#106—1732no—
189#106C2×D17312yescomplemented
190#102—1732no—
191#102—1732no—
192#102—1732no—
193#102C2×D17312yescomplemented
194C22⋊C8—1731no—
195C22⋊C8—1732no—
196C22⋊C8C2×D17311yescharacteristic, complemented
197C24—3461no—
198C24—1731no—
199C2×D4—21no—
200C2×D4—3461no—
201C2×D4—3461no—
202C2×D4—3461no—
203C2×D4—3464no—
204C2×D4—3461no—
205C2×D4—1731no—
206C2×D4—6924no—
207C2×D4C22×D17311yescharacteristic
208C22×C4—21no—
209C22×C4—3464no—
210C22×C4—3461no—
211C22×C4—3461no—
212C22×C4—1731no—
213C22×C4—1731no—
214C22×C4—1731no—
215C22×C4—1731no—
216C22×C4—1731no—
217C22×C4C22×D17311yescharacteristic
218M16—3464no—
219C2×C8—1732no—
220C2×C8C22×D17312yes—
221C4⋊C4—21no—
222C4⋊C4—3461no—
223C4⋊C4—3461no—
224C4⋊C4—1732no—
225C4⋊C4—1731no—
226C4⋊C4C22×D17312yes—
227C4⋊C4C22×D17311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—3462no—
230C22⋊C4—3462no—
231C22⋊C4—3461no—
232C22⋊C4—3461no—
233C22⋊C4—3461no—
234C42—3462no—
235C42—3462no—
236C23—21no—
237C23—3462no—
238C23—3462no—
239C23—3461no—
240C23—1731no—
241C23—1731no—
242C23—6921no—
243C23—6922no—
244C23—6922no—
245C23—6922no—
246C23D4×D17311yescharacteristic
247D4—22no—
248D4—42no—
249D4—3462no—
250D4—6922no—
251D4—6922no—
252D4—6922no—
253D4—6922no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—3462no—
259C2×C4—3462no—
260C2×C4—3462no—
261C2×C4—3462no—
262C2×C4—3462no—
263C2×C4—3461no—
264C2×C4—1731no—
265C2×C4—1731no—
266C2×C4—1734no—
267C2×C4—6922no—
268C2×C4—6922no—
269C2×C4D4×D17311yescharacteristic
270C2×C4C23×D17311yescharacteristic
271C8—22no—
272C8—3462no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—3461no—
279C22—3461no—
280C22—3461no—
281C22—3461no—
282C22—3461no—
283C22—13844no—
284C22—1732no—
285C22—6924no—
286C22—6921no—
287C22—6921no—
288C22—6921no—
289C22#1914970612yes—
290C22C2×D4×D17311yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—3461no—
295C4—1732no—
296C4—6921no—
297C4—6922no—
298C4#41×D17312yes—
299C2—22no—
300C2—81no—
301C2—3462no—
302C2—13841no—
303C2—6921no—
304C2#123×D17311yescharacteristic
305C2C2×#1914970611yescharacteristic
306C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#481×C173C2
socleC22×C173C22×D4
prime radical#121×D173C2
hypercenter#481D173
hypocenterC173#3094
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#19149637C2semidirect
3. #26011817C2semidirect
4. #26011813C2semidirect
5. #26011809C2semidirect
6. #26011805C2semidirect
7. #26011802C2semidirect
8. #26011530C2semidirect
9. #26011494C2semidirect
10. #26011453C2semidirect
11. #481×C173C2semidirect
24. #19149637C22semidirect
25. C2×C4⋊Dic173C22semidirect
30. #19149613C22semidirect
31. #19149612C22semidirect
32. #19149610C22semidirect
33. #19149601C22semidirect
34. #19149600C22semidirect
35. #19149599C22semidirect
36. C22⋊#13063436C22semidirect
37. #19149591C22semidirect
39. C2×C4⋊C4×C173C22semidirect
40. #106×C173C22semidirect
41. #102×C173C22semidirect
42. C22⋊C8×C173C22semidirect
153. C173#3094semidirect
160. #481D173semidirect
182. C2×C4⋊C4C2×D173semidirect
189. #106C2×D173semidirect
193. #102C2×D173semidirect
196. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#121×D1732
C2×#191496372
#260118172
#260118132
#260118092
#260118052
#260118022
#260115302
#260114942
#260114532
#481×C1732
#3094173