Finite Group Databasebeta

← #36362419

Subgroups — #36362419

Order 22144.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D173C211yesmaximal, characteristic
2#26011967C211yesmaximal, characteristic, complemented
3#26011963C211yesmaximal, characteristic
4#26011593C211yesmaximal, characteristic, complemented
5#26011592C211yesmaximal, characteristic, complemented
6#26011590C211yesmaximal, characteristic, complemented
7#26011582C211yesmaximal, characteristic, complemented
8C4×#14583813C211yesmaximal, characteristic, complemented
9C4×#14583814C211yesmaximal, characteristic, complemented
10#26011576C211yesmaximal, characteristic, complemented
11#26011528C211yesmaximal, characteristic, complemented
12#26011498C211yesmaximal, characteristic, complemented
13#26011494C211yesmaximal, characteristic
14#26011448C211yesmaximal, characteristic, complemented
15#440×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21SD16×D173C418yescomplemented
22C2×#14583809C2211yescharacteristic
23C2×C8×D173C2211yescharacteristic
24Q8×Dic173C2211yescharacteristic, complemented
25C4×Dic346—21no—
26C4×Dic346C2211yescharacteristic, complemented
27#19149644—21no—
28#19149644C2211yescharacteristic, complemented
29C4×D692C2211yescharacteristic
30D4×Dic173C2211yescharacteristic
31#19149641C2211yescharacteristic
32C4×C22⋊D173—21no—
33#19149639—21no—
34#19149637—21no—
35#19149637C2211yescharacteristic
36#19149637C2211yescharacteristic
37#19149636—21no—
38#19149636C2211yescharacteristic
39C2×#14583803—21no—
40C22⋊C4×D173—21no—
41#19149613C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149610C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149602C2211yescharacteristic, complemented
47#19149601C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C4×#13063436C2211yescharacteristic, complemented
52C8×Dic173C2211yescharacteristic, complemented
53C2×SD16×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic, complemented
55C4×D4×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic
59#97×C173C2211yescharacteristic, complemented
60C23×D173—21no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173C2×C414yescomplemented
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67D4×D173—22no—
68D4×D173C2×C414yes—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74#14583815C2×C414yescomplemented
75#14583814C2×C414yescomplemented
76#14583813C2×C414yescomplemented
77#14583809—22no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173C2×C414yescomplemented
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87#14583805D411yescharacteristic
88C22⋊Dic173—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99C4×Dic173—21no—
100C4×Dic173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C22×C4×C173—21no—
104SD16×C173C2×C414yescomplemented
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C42×C173—21no—
112C42×C173C2311yescharacteristic
113C22×D173—21no—
114C22×D173—24no—
115C22×D173—81no—
116C22×D173—81no—
117C22×D173C2×D411yescharacteristic
118Dic346—21no—
119Dic346—24no—
120Dic346C22×C412yes—
121D692—21no—
122D692C22×C412yes—
123C22⋊D173—24no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—81no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C4×D173—24no—
132C4×D173—81no—
133C4×D173—81no—
134C4×D173C22×C412yes—
135C4×D173#4112yes—
136#13063436—21no—
137#13063436C22×C412yes—
138C23×C173—21no—
139Q8×C173—21no—
140Q8×C173C22×C412yes—
141D4×C173—21no—
142D4×C173C22×C412yes—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—81no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173C22×C412yes—
152C22×C173—22no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173C22×D411yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—22no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—84no—
162C2×D173C4×D412yes—
163C4×C173—22no—
164C4×C173—21no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173C23×C411yescharacteristic
169C4×C173C2×#4111yescharacteristic
170Dic173—22no—
171Dic173—21no—
172Dic173—21no—
173Dic173—41no—
174Dic173—41no—
175Dic173C4×D412yes—
176C2×C173—42no—
177C2×C173C2×C4×D411yescharacteristic
178C2×C173#57711yescharacteristic
179C2×C173#57711yescharacteristic
180D173—22no—
181D173—42no—
182C173#294911yescharacteristic, complemented, Sylow
183#2949—1731nomaximal, Sylow
184C22×SD16—1731no—
185#519—1731no—
186#518—1731no—
187#440—1731no—
188#440—1731no—
189#440—1731no—
190#440D17311yescharacteristic, complemented
191C4×SD16—1731no—
192C4×SD16—1731no—
193C4×SD16—1731no—
194C4×SD16—1731no—
195#427—1731no—
196#419—1731no—
197#418—1731no—
198#405—1731no—
199C22×Q8—1731no—
200C22×D4—1731no—
201C2×SD16—1731no—
202C2×SD16—1731no—
203C2×SD16—1738no—
204C2×SD16—1731no—
205C2×SD16C2×D17311yescharacteristic
206C2×M16—1731no—
207C22×C8—1731no—
208C4×Q8—3461no—
209C4×Q8—3461no—
210C4×Q8—1731no—
211C4×Q8—1731no—
212C4×Q8—1731no—
213C4×Q8C2×D17311yescharacteristic, complemented
214C4×D4—3461no—
215C4×D4—3461no—
216C4×D4—1731no—
217C4×D4—1731no—
218C4×D4—1731no—
219C4×D4C2×D17311yescharacteristic
220#117—3461no—
221#117—3461no—
222#117—1731no—
223#117—1731no—
224#117—1731no—
225C2×C22⋊C4—3461no—
226C2×C22⋊C4—3461no—
227#107—1731no—
228#107C2×D17311yescharacteristic, complemented
229#106—1731no—
230#106—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103C2×D17311yescharacteristic, complemented
235#102—1731no—
236#102—1731no—
237#102—1731no—
238#102C2×D17311yescharacteristic
239#97—1731no—
240#97C2×D17311yescharacteristic, complemented
241C4×C8—1731no—
242C4×C8—1731no—
243C24—3461no—
244C2×Q8—3461no—
245C2×Q8—3462no—
246C2×Q8—1734no—
247C2×Q8—1731no—
248C2×Q8C22×D17311yescharacteristic
249C2×D4—3462no—
250C2×D4—3461no—
251C2×D4—1734no—
252C2×D4—1731no—
253C2×D4C22×D17311yescharacteristic
254C22×C4—21no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—1731no—
260SD16—1734no—
261SD16—1734no—
262SD16—1734no—
263SD16C4×D17314yescomplemented
264M16—3462no—
265C2×C8—1731no—
266C2×C8—1731no—
267C2×C8—1734no—
268C2×C8C22×D17311yescharacteristic
269C2×C8C22×D17311yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—1731no—
275C4⋊C4—1731no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4C22×D17311yescharacteristic
279C4⋊C4C22×D17311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—1731no—
287C22⋊C4—1731no—
288C42—21no—
289C42—3461no—
290C42—3461no—
291C42—3461no—
292C42—3461no—
293C42—3461no—
294C42—1731no—
295C42C22×D17311yescharacteristic
296C23—21no—
297C23—3464no—
298C23—3461no—
299C23—13841no—
300C23—13841no—
301C23—1731no—
302Q8—21no—
303Q8—3464no—
304Q8—3461no—
305Q8—1732no—
306Q8C2×C4×D17312yes—
307D4—21no—
308D4—3464no—
309D4—3461no—
310D4—1732no—
311D4C2×C4×D17312yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—3464no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—13841no—
321C2×C4—13841no—
322C2×C4—13841no—
323C2×C4—1731no—
324C2×C4—1731no—
325C2×C4—1731no—
326C2×C4—1732no—
327C2×C4—1732no—
328C2×C4D4×D17311yescharacteristic
329C2×C4D4×D17311yescharacteristic
330C2×C4C23×D17311yescharacteristic
331C8—21no—
332C8—3461no—
333C8—1732no—
334C8C2×C4×D17312yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—3461no—
339C22—3461no—
340C22—3462no—
341C22—13844no—
342C22—1732no—
343C22—6921no—
344C22—6921no—
345C22C2×D4×D17311yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—3462no—
352C4—3461no—
353C4—3461no—
354C4—1732no—
355C4—6921no—
356C4—6921no—
357C4C22×C4×D17311yescharacteristic
358C4#41×D17311yescharacteristic
359C2—42no—
360C2—3462no—
361C2—6922no—
362C2C4×D4×D17311yescharacteristic
363C2#2601230011yescharacteristic
364C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutatorC4×C173C23×C4
FrattiniC2×C4C23×D173
Fitting#440×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×D173C22
cosocleC2×C4×C173C24
hypercenter#440D173
hypocenterC173#2949
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #26011967C2semidirect
4. #26011593C2semidirect
5. #26011592C2semidirect
6. #26011590C2semidirect
7. #26011582C2semidirect
8. C4×#14583813C2semidirect
9. C4×#14583814C2semidirect
10. #26011576C2semidirect
11. #26011528C2semidirect
12. #26011498C2semidirect
14. #26011448C2semidirect
15. #440×C173C2semidirect
21. SD16×D173C4semidirect
24. Q8×Dic173C22semidirect
26. C4×Dic346C22semidirect
28. #19149644C22semidirect
41. #19149613C22semidirect
42. #19149610C22semidirect
43. #19149610C22semidirect
44. #19149604C22semidirect
45. #19149603C22semidirect
46. #19149602C22semidirect
50. #19149589C22semidirect
51. C4×#13063436C22semidirect
52. C8×Dic173C22semidirect
54. C4×Q8×C173C22semidirect
56. #107×C173C22semidirect
57. #103×C173C22semidirect
59. #97×C173C22semidirect
64. Q8×D173C2×C4semidirect
74. #14583815C2×C4semidirect
75. #14583814C2×C4semidirect
76. #14583813C2×C4semidirect
80. C8×D173C2×C4semidirect
104. SD16×C173C2×C4semidirect
182. C173#2949semidirect
190. #440D173semidirect
213. C4×Q8C2×D173semidirect
228. #107C2×D173semidirect
234. #103C2×D173semidirect
240. #97C2×D173semidirect
263. SD16C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1732
#260119672
#260119632
#260115932
#260115922
#260115902
#260115822
C4×#145838132
C4×#145838142
#260115762
#260115282
#260114982
#260114942
#260114482
#440×C1732
#2949173