Finite Group Databasebeta

← #27280574

Subgroups — #27280574

Order 13568.

504 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285476C211yesmaximal, characteristic
2#20273219C211yesmaximal, characteristic, complemented
3C2×#16810358C211yesmaximal, characteristic, complemented
4#20273074C211yesmaximal, characteristic, complemented
5#20272603C211yesmaximal, characteristic, complemented
6#20272557C211yesmaximal, characteristic, complemented
7C53×#2977C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810956C411yescharacteristic, complemented
10#16810947—21no—
11#16810939C411yescharacteristic, complemented
12#16810857—21no—
13#16810854—21no—
14#16810773C411yescharacteristic, complemented
15#16810768—21no—
16#16810756C411yescharacteristic, complemented
17#16810587C2211yescharacteristic
18C2×C22⋊#88322C2211yescharacteristic
19C2×#13133124C2211yescharacteristic
20#16810360—21no—
21#16810358—21no—
22#16810358—21no—
23#16810357—21no—
24#16810346—21no—
25#16810346—21no—
26C53×#525C2211yescharacteristic
27C53×#435—21no—
28C53×#435—21no—
29C8×D4×C53—21no—
30C8×D4×C53—21no—
31C53×#433—21no—
32C53×#408—21no—
33C2×C22⋊C8×C53C2211yescharacteristic
34C2×C22⋊C8×C53C2211yescharacteristic
35C22×D212C2×C411yescharacteristic
36C22×C22⋊D53—21no—
37C23×Dic53—21no—
38C2×#11719924—21no—
39C2×#11719922—21no—
40C2×#11719917—21no—
41C22×#88322—21no—
42#13133191—21no—
43#13133191—21no—
44#13133191—21no—
45#13133191—21no—
46C2×C4⋊Dic53C2×C411yescharacteristic
47C2×#11719921—21no—
48#13133150—21no—
49#13133148—21no—
50#13133141—21no—
51#13133138—21no—
52#13133137—21no—
53#13133137—21no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56C22⋊#88322—21no—
57C22⋊#88322C2×C411yescharacteristic
58C22⋊#88322C2×C411yescharacteristic
59#13133124—21no—
60#13133124—21no—
61C2×#41×C53D411yescharacteristic
62C23×C4×C53C2311yescharacteristic
63C2×M16×C53—21no—
64C22×C8×C53—21no—
65C22×C8×C53—21no—
66C22×C8×C53—21no—
67#123×C53—21no—
68C22⋊Q8×C53C2×C411yescharacteristic
69C22⋊Q8×C53D411yescharacteristic
70#121×C53C2×C411yescharacteristic
71#121×C53D411yescharacteristic
72#120×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75#117×C53D411yescharacteristic
76C4⋊C8×C53—21no—
77C4⋊C8×C53—21no—
78C22⋊C8×C53—21no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53—21no—
81C22⋊C8×C53—21no—
82C22⋊C8×C53D411yescharacteristic
83C22⋊C8×C53D411yescharacteristic
84C22⋊C8×C53D411yescharacteristic
85C22⋊C8×C53D411yescharacteristic
86#97×C53—21no—
87C4×C8×C53—21no—
88C24×C53#4111yescharacteristic
89C23×D53—21no—
90#41×C53—81no—
91C2×Q8×C53C22⋊C411yescharacteristic
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53C22⋊C411yescharacteristic
96C22×C4×C53—21no—
97C22×C4×C53—81no—
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C2×C22⋊D53—41no—
105C2×C22⋊D53—41no—
106C2×C22⋊D53—81no—
107C2×C22⋊D53—81no—
108C22×Dic53—22no—
109C22×Dic53—21no—
110C22×Dic53—41no—
111C22×Dic53—41no—
112C22×Dic53—81no—
113C22×Dic53—81no—
114C2×D212—21no—
115C2×D212—21no—
116C2×D212—81no—
117M16×C53—81no—
118C2×C8×C53—21no—
119C2×C8×C53—21no—
120C2×C8×C53—21no—
121C2×C8×C53—21no—
122C2×C8×C53—81no—
123C2×C8×C53—81no—
124C2×C8×C53—81no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53C22⋊C411yescharacteristic
128C4⋊C4×C53C22⋊C411yescharacteristic
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C22⋊C4×C53—21no—
132C22⋊C4×C53—21no—
133C22⋊C4×C53—21no—
134C42×C53—21no—
135#11719924—81no—
136#11719922—81no—
137#11719921—41no—
138#11719921—41no—
139C4⋊Dic53—21no—
140C4⋊Dic53—21no—
141#11719917—81no—
142C2×#88322—21no—
143C2×#88322—21no—
144C2×#88322—41no—
145C2×#88322—81no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—41no—
152C23×C53C4×D411yescharacteristic
153C23×C53#12111yescharacteristic
154C23×C53#12311yescharacteristic
155C22×D53—41no—
156C22×D53—41no—
157C22×D53—41no—
158C22×D53—81no—
159C22×D53—81no—
160Q8×C53—41no—
161D4×C53—41no—
162D4×C53—41no—
163D4×C53—41no—
164D4×C53—41no—
165D4×C53—41no—
166D4×C53—81no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53C2×C22⋊C411yescharacteristic
179C2×C4×C53C4×D411yescharacteristic
180C2×C4×C53#12011yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C22⋊D53—82no—
183C22⋊D53—81no—
184C22⋊D53—81no—
185C2×Dic53—42no—
186C2×Dic53—41no—
187C2×Dic53—81no—
188C2×Dic53—82no—
189C2×Dic53—81no—
190C2×Dic53—81no—
191C2×Dic53—82no—
192D212—41no—
193D212—41no—
194D212—81no—
195C8×C53—41no—
196C8×C53—41no—
197C8×C53—41no—
198C8×C53—41no—
199#88322—41no—
200#88322—41no—
201#88322—81no—
202#9264—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53#38611yescharacteristic
218C22×C53#44311yescharacteristic
219C22×C53#44411yescharacteristic
220C22×C53#47111yescharacteristic
221C22×C53#47211yescharacteristic
222C2×D53—81no—
223C2×D53—81no—
224C2×D53—81no—
225C2×D53—161no—
226C2×D53—161no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—41no—
234Dic53—82no—
235Dic53—81no—
236#3188—531no—
237#2977D5311yescharacteristic, complemented
238#1504—531no—
239C2×#327—531no—
240#1471—531no—
241#1298—531no—
242#1280—531no—
243C2×C53—21no—
244C2×C53—21no—
245C2×C53—21no—
246C2×C53—21no—
247C2×C53—41no—
248C2×C53—81no—
249C2×C53#187611yescharacteristic
250C2×C53#190711yescharacteristic
251C2×C53#191311yescharacteristic
252D53—161no—
253#525C2×D5311yescharacteristic
254C2×#121—531no—
255#484—531no—
256#482—1061no—
257#481—531no—
258#466—1061no—
259#465—1061no—
260#450—531no—
261#449—1061no—
262#447—531no—
263#435—21no—
264#435—21no—
265C8×D4—21no—
266C8×D4—21no—
267#433—21no—
268#408—21no—
269#407—531no—
270C2×C22⋊C8—531no—
271C2×C22⋊C8C2×D5311yescharacteristic
272C2×C22⋊C8C2×D5311yescharacteristic
273C2×#95—531no—
274#328—1061no—
275#327—1061no—
276#327—1061no—
277#327—1061no—
278#323—1061no—
279#323—1061no—
280C53#926411yescharacteristic, complemented, Sylow
281C2×#41C22⋊D5311yescharacteristic
282C22×D4—531no—
283C22×D4—1061no—
284C23×C4—1061no—
285C23×C4C22×D5311yescharacteristic
286C2×SD16—1061no—
287C2×D8—1061no—
288C2×M16—21no—
289C2×M16—1061no—
290C22×C8—21no—
291C22×C8—21no—
292C22×C8—21no—
293C22×C8—1061no—
294#123—21no—
295C22⋊Q8C4×D5311yescharacteristic
296C22⋊Q8D21211yescharacteristic
297#121—1061no—
298#121—1061no—
299#121—1061no—
300#121—1061no—
301#121C4×D5311yescharacteristic
302#121D21211yescharacteristic
303#120—21no—
304C4×D4—21no—
305C4×D4—21no—
306#117C22⋊D5311yescharacteristic
307C2×C4⋊C4—531no—
308C2×C22⋊C4—1061no—
309#107—1061no—
310#106—1061no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313#103—1061no—
314#102—1061no—
315#102—1061no—
316#102—1061no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321C22⋊C8—531no—
322C22⋊C8—531no—
323C22⋊C8—1061no—
324C22⋊C8—1061no—
325C22⋊C8—1061no—
326C22⋊C8C22⋊D5311yescharacteristic
327C22⋊C8C22⋊D5311yescharacteristic
328C22⋊C8C22⋊D5311yescharacteristic
329C22⋊C8C22⋊D5311yescharacteristic
330#97—21no—
331C4×C8—21no—
332#95—1061no—
333#95—1061no—
334C24—1061no—
335C24#1171994511yescharacteristic
336#41—81no—
337C2×Q8#1171992111yescharacteristic
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—1061no—
342C2×D4—1061no—
343C2×D4—4241no—
344C2×D4—4241no—
345C2×D4—4241no—
346C2×D4—2121no—
347C2×D4—2121no—
348C2×D4#1171992111yescharacteristic
349C22×C4—21no—
350C22×C4—81no—
351C22×C4—1062no—
352C22×C4—1061no—
353C22×C4—4241no—
354C22×C4—4241no—
355C22×C4—2121no—
356C22×C4—2121no—
357C22×C4C2×C4×D5311yescharacteristic
358C22×C4C2×D21211yescharacteristic
359C22×C4#1171994511yescharacteristic
360C22×C4C2×C22⋊D5311yescharacteristic
361C22×C4C2×C22⋊D5311yescharacteristic
362C22×C4C2×C22⋊D5311yescharacteristic
363SD16—4241no—
364D8—4241no—
365M16—81no—
366M16—4241no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—81no—
372C2×C8—81no—
373C2×C8—81no—
374C2×C8—1061no—
375C2×C8—1061no—
376C2×C8—4241no—
377C2×C8—2121no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—1061no—
381C4⋊C4—1061no—
382C4⋊C4#1171992111yescharacteristic
383C4⋊C4#1171992111yescharacteristic
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C42—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—4241no—
399C23—4241no—
400C23—2121no—
401C23—2121no—
402C23—2121no—
403C23C4×C22⋊D5311yescharacteristic
404C23#1313319111yescharacteristic
405C23#1313320011yescharacteristic
406Q8—41no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—41no—
412D4—81no—
413D4—4241no—
414D4—4242no—
415D4—4241no—
416D4—4241no—
417D4—2121no—
418D4—2121no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—4241no—
431C2×C4—4242no—
432C2×C4—4241no—
433C2×C4—4241no—
434C2×C4—4242no—
435C2×C4—2122no—
436C2×C4—2121no—
437C2×C4C2×#1171992111yescharacteristic
438C2×C4C4×C22⋊D5311yescharacteristic
439C2×C4#1313318611yescharacteristic
440C2×C4#1313319111yescharacteristic
441C8—41no—
442C8—41no—
443C8—41no—
444C8—41no—
445C8—4241no—
446C8—2121no—
447C8—2121no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—8481no—
463C22—8481no—
464C22—4241no—
465C22—4241no—
466C22—4241no—
467C22#1681050211yescharacteristic
468C22#1681073811yescharacteristic
469C22#1681074311yescharacteristic
470C22#1681088611yescharacteristic
471C22#1681089311yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—4242no—
480C4—4241no—
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—41no—
486C2—81no—
487C2—8481no—
488C2#2027504211yescharacteristic
489C2#2027522311yescharacteristic
490C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2977C2
socleC22×C53#386
prime radical#20285476C2
cosocleC23×C4×C53C23
hypercenter#2977D53
hypocenterC53#9264
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2977

Complements

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

NH ≅ G/Ntype
2. #20273219C2semidirect
3. C2×#16810358C2semidirect
4. #20273074C2semidirect
5. #20272603C2semidirect
6. #20272557C2semidirect
7. C53×#2977C2semidirect
9. #16810956C4semidirect
11. #16810939C4semidirect
14. #16810773C4semidirect
16. #16810756C4semidirect
237. #2977D53semidirect
280. C53#9264semidirect

Maximal subgroups

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

subgroupindex
#202854762
#202732192
C2×#168103582
#202730742
#202726032
#202725572
C53×#29772
#926453