Finite Group Databasebeta

← #27541702

Subgroups — #27541702

Order 13568.

488 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287118C211yesmaximal, characteristic, complemented
2#20276991C211yesmaximal, characteristic, complemented
3#20276906C211yesmaximal, characteristic, complemented
4#20276903C211yesmaximal, characteristic, complemented
5#20276895C211yesmaximal, characteristic, complemented
6#20276893C211yesmaximal, characteristic, complemented
7#20276872C211yesmaximal, characteristic, complemented
8#20276869C211yesmaximal, characteristic, complemented
9#20276816C211yesmaximal, characteristic, complemented
10#20276816C211yesmaximal, characteristic, complemented
11C4×#13133158C211yesmaximal, characteristic, complemented
12#20276547C211yesmaximal, characteristic, complemented
13#20276504C211yesmaximal, characteristic, complemented
14#20276501C211yesmaximal, characteristic, complemented
15C53×#3339C211yesmaximal, characteristic, complemented
16#16811319—21no—
17C2×#13133158C2211yescharacteristic
18C2×#13133157C2211yescharacteristic
19C2×#13133157C2211yescharacteristic
20#16811025C2211yescharacteristic
21#16810948—21no—
22#16810942—21no—
23#16810924—21no—
24#16810859C2211yescharacteristic
25#16810859C2211yescharacteristic
26C4×D424C2211yescharacteristic
27#16810661C2211yescharacteristic
28C2×#13133151C2211yescharacteristic
29#16810450C2211yescharacteristic
30#16810450C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810450C2211yescharacteristic
33#16810448C2211yescharacteristic
34#16810448C2211yescharacteristic
35#16810448C2211yescharacteristic
36#16810448C2211yescharacteristic
37#16810448C2211yescharacteristic
38C4⋊#11719908C2211yescharacteristic
39#16810424C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#553—21no—
49C53×#499C2211yescharacteristic
50C53×#482—21no—
51C53×#481—21no—
52C53×#478—21no—
53C53×#466C2211yescharacteristic
54C53×#466C2211yescharacteristic
55C4×D8×C53C2211yescharacteristic
56C8×D4×C53C2211yescharacteristic
57C53×#427C2211yescharacteristic
58C2×#107×C53C2211yescharacteristic
59C2×D424C2311yescharacteristic
60#13133212—41no—
61#13133209—21no—
62#13133199—41no—
63#13133191—21no—
64#13133191—21no—
65#13133189—41no—
66C4×D212—21no—
67#13133177—41no—
68#13133176—21no—
69C2×C4⋊Dic53—21no—
70#13133158—22no—
71#13133157—22no—
72#13133157—22no—
73C2×#11719908C2311yescharacteristic
74C2×#11719908C2311yescharacteristic
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77#13133151—21no—
78#13133151C2311yescharacteristic
79#13133151C2311yescharacteristic
80#13133151C2311yescharacteristic
81#13133149—21no—
82#13133149—21no—
83#13133139—21no—
84#13133139—21no—
85#13133139—21no—
86C2×D8×C53C2311yescharacteristic
87C22×C8×C53C2311yescharacteristic
88C22×C8×C53C2311yescharacteristic
89#126×C53—41no—
90#125×C53—21no—
91#123×C53—41no—
92#121×C53—21no—
93#121×C53—21no—
94#121×C53—41no—
95C4×D4×C53—21no—
96C4×D4×C53—41no—
97C4×D4×C53D411yescharacteristic
98#117×C53—21no—
99C2×C4⋊C4×C53—21no—
100#107×C53—21no—
101#107×C53C2311yescharacteristic
102#107×C53C2311yescharacteristic
103#107×C53C2311yescharacteristic
104#106×C53—21no—
105#106×C53—21no—
106C4⋊C8×C53D411yescharacteristic
107#102×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110C22⋊C8×C53D411yescharacteristic
111C22⋊C8×C53D411yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C2×D4×C53—21no—
114C2×D4×C53—41no—
115C2×D4×C53—41no—
116C2×D4×C53D811yescharacteristic
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C2×C22⋊D53—41no—
122C2×C22⋊D53—41no—
123C22×Dic53—41no—
124C2×D212—21no—
125C2×C4×D53—41no—
126D8×C53—21no—
127D8×C53—21no—
128C2×C8×C53—21no—
129C2×C8×C53—21no—
130C2×C8×C53—21no—
131C2×C8×C53—21no—
132C2×C8×C53C2×D411yescharacteristic
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2411yescharacteristic
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53—41no—
142C4⋊C4×C53D811yescharacteristic
143C22⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53D811yescharacteristic
148C22⋊C4×C53D811yescharacteristic
149C42×C53—41no—
150C42×C53C2×D411yescharacteristic
151C22⋊Dic53—41no—
152#11719921—41no—
153#11719921—41no—
154#11719921—41no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—41no—
159#11719919—41no—
160#11719919—41no—
161C4×Dic53—41no—
162D424—21no—
163D424—21no—
164#11719908—21no—
165#11719908—21no—
166#11719908—21no—
167#11719908#4112yes—
168C23×C53—41no—
169C23×C53C2×D811yescharacteristic
170C23×C53C2×D811yescharacteristic
171C22×D53—41no—
172D4×C53—41no—
173D4×C53—41no—
174D4×C53—41no—
175D4×C53—82no—
176D4×C53—161no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—161no—
185C2×C4×C53—161no—
186C2×C4×C53C2×D811yescharacteristic
187C2×C4×C53C2×D811yescharacteristic
188C2×C4×C53C2×D811yescharacteristic
189C2×C4×C53C2×D811yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C22⋊D53—82no—
192C22⋊D53—161no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—161no—
197D212—41no—
198D212—41no—
199C4×D53—161no—
200C8×C53—21no—
201C8×C53—21no—
202C8×C53—21no—
203C8×C53C2×#4111yescharacteristic
204C8×C532−1+411yescharacteristic
205#33922—531nomaximal, Sylow
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—81no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53C22×D811yescharacteristic
216C2×D53—81no—
217C2×D53—81no—
218C2×D53—81no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—81no—
224C4×C53—81no—
225C4×C53—81no—
226C4×C53#54711yescharacteristic
227C4×C53#57811yescharacteristic
228Dic53—81no—
229Dic53—81no—
230Dic53—81no—
231#3339—531no—
232#3339D5311yescharacteristic, complemented
233#2253—531no—
234#2234—531no—
235#2234—531no—
236#2232—531no—
237#2232—531no—
238#2228—531no—
239#2228—531no—
240#2218—531no—
241#2218—531no—
242C4×D16—531no—
243#2169—531no—
244#2159—531no—
245#2159—531no—
246C2×C53—41no—
247C2×C53—41no—
248C2×C53—161no—
249C2×C53#333411yescharacteristic
250C2×C53#341411yescharacteristic
251C2×C53#341511yescharacteristic
252D53—161no—
253#553—21no—
254#553—1061no—
255C2×D16—531no—
256C2×M32—531no—
257C2×M32—531no—
258#499—531no—
259#499C2×D5311yescharacteristic
260#482—21no—
261#482—1061no—
262#481—21no—
263#481—1061no—
264#478—21no—
265#478—1061no—
266#466—531no—
267#466—531no—
268#466C2×D5311yescharacteristic
269#466C2×D5311yescharacteristic
270C4×D8—531no—
271C4×D8C2×D5311yescharacteristic
272C8×D4C2×D5311yescharacteristic
273#427—531no—
274#427C2×D5311yescharacteristic
275C2×#107—531no—
276C2×#107C2×D5311yescharacteristic
277#367—531no—
278#367—531no—
279#367—531no—
280#367—531no—
281#366—531no—
282#366—531no—
283#366—531no—
284#366—531no—
285#366—531no—
286C4⋊C16—531no—
287#357—531no—
288#357—531no—
289#357—531no—
290#357—531no—
291#357—531no—
292#357—531no—
293C22⋊C16—531no—
294C22⋊C16—531no—
295C4×C16—531no—
296C53#3392211yescharacteristic, complemented, Sylow
297C2×D8—531no—
298C2×D8C22×D5311yescharacteristic
299C22×C8C22×D5311yescharacteristic
300C22×C8C22×D5311yescharacteristic
301#126—41no—
302#126—2121no—
303#125—21no—
304#125—1061no—
305#123—41no—
306#123—2121no—
307#121—21no—
308#121—21no—
309#121—41no—
310#121—1061no—
311#121—1061no—
312#121—2121no—
313C4×D4—21no—
314C4×D4—41no—
315C4×D4—1061no—
316C4×D4—2121no—
317C4×D4C22⋊D5311yescharacteristic
318#117—21no—
319#117—1061no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—1061no—
322D16—1062no—
323M32—1062no—
324M32—1062no—
325C2×C16—531no—
326C2×C16—531no—
327C2×C16—531no—
328C2×C16—531no—
329#107—21no—
330#107—531no—
331#107—531no—
332#107—531no—
333#107—1061no—
334#107C22×D5311yescharacteristic
335#107C22×D5311yescharacteristic
336#107C22×D5311yescharacteristic
337#106—21no—
338#106—21no—
339#106—1061no—
340#106—1061no—
341C4⋊C8C22⋊D5311yescharacteristic
342#102—21no—
343#102—21no—
344#102—21no—
345#102—1061no—
346#102—1061no—
347#102—1061no—
348C22⋊C8C22⋊D5311yescharacteristic
349C22⋊C8C22⋊D5311yescharacteristic
350C4×C8C22×D5311yescharacteristic
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—1061no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4#1171992211yescharacteristic
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—2121no—
361C22×C4—2121no—
362C22×C4C2×C22⋊D5311yescharacteristic
363C22×C4C2×C22⋊D5311yescharacteristic
364D8—21no—
365D8—21no—
366D8—1061no—
367D8—1061no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8C2×C22⋊D5311yescharacteristic
373C2×C8C2×C22⋊D5311yescharacteristic
374C2×C8C2×C22⋊D5311yescharacteristic
375C2×C8C23×D5311yescharacteristic
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—1061no—
383C4⋊C4—1061no—
384C4⋊C4—1061no—
385C4⋊C4—2121no—
386C4⋊C4—2121no—
387C4⋊C4—2121no—
388C4⋊C4#1171992211yescharacteristic
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C22⋊C4—2121no—
396C22⋊C4—2121no—
397C22⋊C4#1171992211yescharacteristic
398C22⋊C4#1171992211yescharacteristic
399C42—41no—
400C42—2121no—
401C42C2×C22⋊D5311yescharacteristic
402C16—532no—
403C16—1061no—
404C16—1061no—
405C16—1061no—
406C23—41no—
407C23—2121no—
408C23C2×#1171992211yescharacteristic
409C23C2×#1171992211yescharacteristic
410D4—41no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—161no—
415D4—8481no—
416D4—4242no—
417D4—2121no—
418D4—2121no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—161no—
428C2×C4—8481no—
429C2×C4—8481no—
430C2×C4—2121no—
431C2×C4—2121no—
432C2×C4—2121no—
433C2×C4C2×#1171992211yescharacteristic
434C2×C4C2×#1171992211yescharacteristic
435C2×C4C2×#1171992211yescharacteristic
436C2×C4C2×#1171992211yescharacteristic
437C2×C4C22×C22⋊D5311yescharacteristic
438C8—21no—
439C8—21no—
440C8—21no—
441C8#41×D5311yescharacteristic
442C8#1313326411yescharacteristic
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—4241no—
453C22—4241no—
454C22—4241no—
455C22C22×#1171992211yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—4241no—
464C4—4241no—
465C4—4241no—
466C4#1681127211yescharacteristic
467C4#1681143811yescharacteristic
468C2—41no—
469C2—41no—
470C2—161no—
471C2—8481no—
472C2#2028705811yescharacteristic
473C2#2028784911yescharacteristic
474C2#2028785411yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3339C2
socleC22×C53C22×D8
hypercenter#3339D53
hypocenterC53#33922
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. #20287118C2semidirect
2. #20276991C2semidirect
3. #20276906C2semidirect
4. #20276903C2semidirect
5. #20276895C2semidirect
6. #20276893C2semidirect
7. #20276872C2semidirect
8. #20276869C2semidirect
9. #20276816C2semidirect
10. #20276816C2semidirect
11. C4×#13133158C2semidirect
12. #20276547C2semidirect
13. #20276504C2semidirect
14. #20276501C2semidirect
15. C53×#3339C2semidirect
232. #3339D53semidirect
296. C53#33922semidirect

Maximal subgroups

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

subgroupindex
#202871182
#202769912
#202769062
#202769032
#202768952
#202768932
#202768722
#202768692
#202768162
C4×#131331582
#202765472
#202765042
#202765012
C53×#33392
#3392253