Finite Group Databasebeta

← #27280576

Subgroups — #27280576

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#20283771C211yesmaximal, characteristic, complemented
2#20273219C211yesmaximal, characteristic, complemented
3C2×#16810357C211yesmaximal, characteristic, complemented
4#20273075C211yesmaximal, characteristic, complemented
5#20272603C211yesmaximal, characteristic, complemented
6#20272557C211yesmaximal, characteristic, complemented
7C53×#3188C211yesmaximal, characteristic
8#16810695—21no—
9#16810695—21no—
10D4×#88322—21no—
11D4×#88322—21no—
12#16810684—21no—
13#16810590—21no—
14#16810587C2211yescharacteristic
15C2×C22⋊#88322C2211yescharacteristic
16C2×C22⋊#88322C2211yescharacteristic
17C2×#13133124C2211yescharacteristic
18#16810360—21no—
19#16810357—21no—
20#16810357—21no—
21#16810357—21no—
22#16810347—21no—
23#16810347—21no—
24C53×#525C2211yescharacteristic
25C2×#121×C53C2211yescharacteristic
26C53×#484C411yescharacteristic, complemented
27C53×#482—21no—
28C53×#481C411yescharacteristic, complemented
29C53×#466—21no—
30C53×#465—21no—
31C53×#450C411yescharacteristic, complemented
32C53×#449—21no—
33C53×#447C411yescharacteristic, complemented
34C2×C22⋊C8×C53C2211yescharacteristic
35C23×Dic53—21no—
36C2×#11719917—21no—
37C2×#11719917—21no—
38C22×#88322—21no—
39C22×#88322—21no—
40C22×#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322D411yescharacteristic
50C22⋊#88322D411yescharacteristic
51C22⋊#88322D411yescharacteristic
52C22⋊#88322D411yescharacteristic
53#13133128—21no—
54C4×#88322—21no—
55#13133124—21no—
56#13133124—21no—
57C2×#41×C53D411yescharacteristic
58C22×D4×C53—21no—
59C22×D4×C53C2×C411yescharacteristic
60C23×C4×C53C2311yescharacteristic
61C2×SD16×C53—21no—
62C2×D8×C53—21no—
63C22×C8×C53—21no—
64#123×C53—21no—
65C22⋊Q8×C53C2×C411yescharacteristic
66C22⋊Q8×C53D411yescharacteristic
67#121×C53—21no—
68#121×C53—21no—
69#121×C53—21no—
70#121×C53—21no—
71#121×C53C2×C411yescharacteristic
72#121×C53D411yescharacteristic
73#120×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53—21no—
76#117×C53D411yescharacteristic
77C2×C4⋊C4×C53C2×C411yescharacteristic
78C2×C22⋊C4×C53—21no—
79#107×C53—21no—
80#106×C53—21no—
81#103×C53—21no—
82#102×C53—21no—
83#102×C53—21no—
84#102×C53—21no—
85C22⋊C8×C53—21no—
86C22⋊C8×C53C2×C411yescharacteristic
87C22⋊C8×C53C2×C411yescharacteristic
88C24×C53—21no—
89C24×C53#4111yescharacteristic
90#41×C53—81no—
91C2×Q8×C53C22⋊C411yescharacteristic
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—81no—
100C2×D4×C53—81no—
101C2×D4×C53—81no—
102C2×D4×C53C22⋊C411yescharacteristic
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—81no—
106C22×C4×C53C22×C411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×Dic53—22no—
113C22×Dic53—41no—
114C22×Dic53—41no—
115C22×Dic53—81no—
116C22×Dic53—81no—
117SD16×C53—81no—
118D8×C53—81no—
119C2×C8×C53—21no—
120C2×C8×C53—21no—
121C2×C8×C53—81no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53C22⋊C411yescharacteristic
127C4⋊C4×C53C22⋊C411yescharacteristic
128C22⋊C4×C53—21no—
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C22⋊C4×C53—21no—
132C22⋊C4×C53—21no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C42×C53—21no—
136#11719917—81no—
137#11719917—81no—
138C2×#88322—21no—
139C2×#88322—21no—
140C2×#88322—21no—
141C2×#88322—21no—
142C2×#88322—41no—
143C2×#88322—81no—
144C2×#88322—81no—
145C2×#88322—81no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53—41no—
154C23×C53—41no—
155C23×C53—81no—
156C23×C53—81no—
157C23×C53C4×D411yescharacteristic
158C23×C53#12111yescharacteristic
159C23×C53#12311yescharacteristic
160Q8×C53—41no—
161D4×C53—41no—
162D4×C53—41no—
163D4×C53—41no—
164D4×C53—41no—
165D4×C53—41no—
166D4×C53—41no—
167D4×C53—41no—
168D4×C53—81no—
169D4×C53—81no—
170D4×C53—82no—
171D4×C53—81no—
172D4×C53—81no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—81no—
186C2×C4×C53C2×C22⋊C411yescharacteristic
187C2×C4×C53C4×D411yescharacteristic
188C2×C4×C53#12011yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×Dic53—42no—
191C2×Dic53—81no—
192C2×Dic53—82no—
193C2×Dic53—81no—
194C2×Dic53—82no—
195C8×C53—41no—
196C8×C53—41no—
197#88322—41no—
198#88322—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—81no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53—161no—
221C22×C53—161no—
222C22×C53#38611yescharacteristic
223C22×C53#44311yescharacteristic
224C22×C53#44411yescharacteristic
225C22×C53#47111yescharacteristic
226C22×C53#47211yescharacteristic
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—41no—
234C4×C53—81no—
235Dic53—82no—
236#3188D5311yescharacteristic
237#2977—531no—
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—161no—
250C2×C53#187611yescharacteristic
251C2×C53#190711yescharacteristic
252C2×C53#191311yescharacteristic
253#525C2×D5311yescharacteristic
254C2×#121C2×D5311yescharacteristic
255#484Dic5311yescharacteristic, complemented
256#482—21no—
257#481Dic5311yescharacteristic, complemented
258#466—21no—
259#465—21no—
260#450Dic5311yescharacteristic, complemented
261#449—21no—
262#447Dic5311yescharacteristic, complemented
263#435—1061no—
264#435—1061no—
265C8×D4—1061no—
266C8×D4—1061no—
267#433—1061no—
268#408—1061no—
269#407—531no—
270C2×C22⋊C8—531no—
271C2×C22⋊C8—531no—
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—21no—
283C22×D4C2×Dic5311yescharacteristic
284C23×C4—1061no—
285C23×C4C22×D5311yescharacteristic
286C2×SD16—21no—
287C2×D8—21no—
288C2×M16—1061no—
289C2×M16—1061no—
290C22×C8—21no—
291C22×C8—1061no—
292C22×C8—1061no—
293C22×C8—1061no—
294#123—21no—
295C22⋊Q8C2×Dic5311yescharacteristic
296C22⋊Q8C22⋊D5311yescharacteristic
297#121—21no—
298#121—21no—
299#121—21no—
300#121—21no—
301#121C2×Dic5311yescharacteristic
302#121C22⋊D5311yescharacteristic
303#120—21no—
304C4×D4—21no—
305C4×D4—21no—
306#117C22⋊D5311yescharacteristic
307C2×C4⋊C4C2×Dic5311yescharacteristic
308C2×C22⋊C4—21no—
309#107—21no—
310#106—21no—
311C4⋊C8—1061no—
312C4⋊C8—1061no—
313#103—21no—
314#102—21no—
315#102—21no—
316#102—21no—
317C22⋊C8—21no—
318C22⋊C8—531no—
319C22⋊C8—531no—
320C22⋊C8—531no—
321C22⋊C8—531no—
322C22⋊C8—1061no—
323C22⋊C8—1061no—
324C22⋊C8—1061no—
325C22⋊C8—1061no—
326C22⋊C8—1061no—
327C22⋊C8—1061no—
328C22⋊C8C2×Dic5311yescharacteristic
329C22⋊C8C2×Dic5311yescharacteristic
330#97—1061no—
331C4×C8—1061no—
332#95—1061no—
333#95—1061no—
334C24—21no—
335C24#1171994711yescharacteristic
336#41—81no—
337C2×Q8C22⋊Dic5311yescharacteristic
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4C22⋊Dic5311yescharacteristic
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—81no—
352C22×C4—1062no—
353C22×C4—4241no—
354C22×C4—4241no—
355C22×C4—2121no—
356C22×C4—2121no—
357C22×C4D4×D5311yescharacteristic
358C22×C4D4×D5311yescharacteristic
359C22×C4#1171994711yescharacteristic
360C22×C4C22×Dic5311yescharacteristic
361C22×C4C2×C22⋊D5311yescharacteristic
362C22×C4C2×C22⋊D5311yescharacteristic
363SD16—81no—
364D8—81no—
365M16—4241no—
366M16—4241no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—81no—
370C2×C8—1061no—
371C2×C8—1061no—
372C2×C8—1061no—
373C2×C8—1061no—
374C2×C8—4241no—
375C2×C8—4241no—
376C2×C8—4241no—
377C2×C8—2121no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4C22⋊Dic5311yescharacteristic
383C4⋊C4C22⋊Dic5311yescharacteristic
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C42—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—81no—
402C23—81no—
403C23D4×Dic5311yescharacteristic
404C23#1313318911yescharacteristic
405C23#1313319811yescharacteristic
406Q8—41no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—81no—
415D4—81no—
416D4—82no—
417D4—81no—
418D4—81no—
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—41no—
431C2×C4—81no—
432C2×C4—4241no—
433C2×C4—4242no—
434C2×C4—4241no—
435C2×C4—4242no—
436C2×C4—2122no—
437C2×C4C2×C22⋊Dic5311yescharacteristic
438C2×C4D4×Dic5311yescharacteristic
439C2×C4#1313318511yescharacteristic
440C2×C4#1313318911yescharacteristic
441C8—41no—
442C8—41no—
443C8—4241no—
444C8—2121no—
445C8—2121no—
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—81no—
463C22—81no—
464C22—81no—
465C22—161no—
466C22—161no—
467C22#1681050311yescharacteristic
468C22#1681074011yescharacteristic
469C22#1681074511yescharacteristic
470C22#1681088411yescharacteristic
471C22#1681089111yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—81no—
480C4—4242no—
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—41no—
486C2—81no—
487C2—161no—
488C2#2027504411yescharacteristic
489C2#2027522511yescharacteristic
490C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3188C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#3188D53
hypocenterC53#9264
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3188

Complements

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

NH ≅ G/Ntype
1. #20283771C2semidirect
2. #20273219C2semidirect
3. C2×#16810357C2semidirect
4. #20273075C2semidirect
5. #20272603C2semidirect
6. #20272557C2semidirect
26. C53×#484C4semidirect
28. C53×#481C4semidirect
31. C53×#450C4semidirect
33. C53×#447C4semidirect
255. #484Dic53semidirect
257. #481Dic53semidirect
260. #450Dic53semidirect
262. #447Dic53semidirect
280. C53#9264semidirect

Maximal subgroups

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

subgroupindex
#202837712
#202732192
C2×#168103572
#202730752
#202726032
#202725572
C53×#31882
#926453