Finite Group Databasebeta

← #27539245

Subgroups — #27539245

Order 13568.

532 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285354C211yesmaximal, characteristic
2#20276923C211yesmaximal, characteristic, complemented
3#20276906C211yesmaximal, characteristic
4#20276867C211yesmaximal, characteristic, complemented
5#20276857C211yesmaximal, characteristic, complemented
6#20276777C211yesmaximal, characteristic, complemented
7#20276777C211yesmaximal, characteristic, complemented
8#20276649C211yesmaximal, characteristic, complemented
9#20276632C211yesmaximal, characteristic
10#20276584C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12C4×#13133163C211yesmaximal, characteristic, complemented
13C4×#13133160C211yesmaximal, characteristic, complemented
14#20276522C211yesmaximal, characteristic, complemented
15C53×#2950C211yesmaximal, characteristic, complemented
16#16811244—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133163C2211yescharacteristic
19C2×#13133160C2211yescharacteristic
20C2×#13133160C2211yescharacteristic
21#16811028C2211yescharacteristic
22#16811007C2211yescharacteristic
23#16810983—21no—
24#16810869—21no—
25#16810864C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810725—21no—
28C4×Dic212C2211yescharacteristic
29C4×D424C2211yescharacteristic
30#16810450C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35C4⋊#11719908C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810427C2211yescharacteristic
39#16810427C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44C22⋊#11719908C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C22×Q16×C53C2211yescharacteristic
49C53×#519—21no—
50C53×#441—21no—
51C53×#441—21no—
52C4×Q16×C53C2211yescharacteristic
53C4×Q16×C53C2211yescharacteristic
54C4×Q16×C53C2211yescharacteristic
55C4×Q16×C53C2211yescharacteristic
56C53×#427C2211yescharacteristic
57C53×#419—21no—
58C53×#405C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60C2×#11719913—21no—
61C2×D424C2311yescharacteristic
62#13133217—21no—
63#13133212—41no—
64#13133205—21no—
65#13133203—41no—
66#13133199—41no—
67#13133196—21no—
68#13133195—41no—
69#13133191—21no—
70C4×Dic106—21no—
71C4×D212—21no—
72#13133163—22no—
73#13133163—22no—
74#13133160—22no—
75#13133160—22no—
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80#13133151—21no—
81#13133151C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133142—21no—
84#13133142—21no—
85#13133139—21no—
86#13133139—21no—
87C22×Q8×C53—21no—
88C2×Q16×C53—22no—
89C2×Q16×C53—22no—
90C2×Q16×C53—21no—
91C2×Q16×C53C2311yescharacteristic
92C2×Q16×C53C2311yescharacteristic
93C2×M16×C53D411yescharacteristic
94C22×C8×C53C2311yescharacteristic
95C4×Q8×C53—21no—
96C4×Q8×C53—21no—
97C4×Q8×C53—21no—
98C4×Q8×C53—21no—
99C4×Q8×C53—41no—
100C4×Q8×C53—41no—
101#117×C53—21no—
102#117×C53—41no—
103#117×C53—41no—
104#117×C53D411yescharacteristic
105#107×C53C2311yescharacteristic
106#107×C53C2311yescharacteristic
107#106×C53—21no—
108#103×C53—21no—
109#103×C53—21no—
110#103×C53—21no—
111#103×C53—21no—
112#97×C53D411yescharacteristic
113#97×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116C2×Q8×C53—24no—
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—42no—
120C2×Q8×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53C2×D411yescharacteristic
123C2×C22⋊D53—41no—
124C2×D212—21no—
125C2×C4×D53—41no—
126C2×Dic106—21no—
127C2×Dic106—41no—
128Q16×C53—24no—
129Q16×C53—21no—
130Q16×C53—21no—
131Q16×C53#4112yes—
132Q16×C53#4112yes—
133M16×C53—41no—
134C2×C8×C53—21no—
135C2×C8×C53—21no—
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2411yescharacteristic
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53D811yescharacteristic
147C4⋊C4×C53D811yescharacteristic
148C22⋊C4×C53—41no—
149C22⋊C4×C53D811yescharacteristic
150C22⋊C4×C53D811yescharacteristic
151C42×C53—41no—
152C42×C53—41no—
153C42×C53—41no—
154C42×C53—41no—
155C42×C53—41no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C22⋊Dic53—41no—
159#11719921—41no—
160#11719921—41no—
161#11719921—41no—
162#11719921—41no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—41no—
166#11719919—41no—
167#11719919—41no—
168C4×Dic53—41no—
169Dic212—21no—
170Dic212—21no—
171D424—21no—
172D424—21no—
173#11719913—42no—
174#11719908—21no—
175#11719908—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53C2×D811yescharacteristic
179C22×D53—41no—
180Q8×C53—22no—
181Q8×C53—22no—
182Q8×C53—44no—
183Q8×C53—41no—
184Q8×C53—41no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—44no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53C2×D811yescharacteristic
193C2×C4×C53C2×D811yescharacteristic
194C2×C4×C53C2×D811yescharacteristic
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C22⋊D53—161no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202D212—41no—
203D212—41no—
204C4×D53—161no—
205Dic106—41no—
206Dic106—41no—
207Dic106—161no—
208C8×C53—21no—
209C8×C53—21no—
210C8×C53—21no—
211C8×C53C2×#4111yescharacteristic
212C8×C53C2×#4111yescharacteristic
213#33681—531nomaximal, Sylow
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53C22×D811yescharacteristic
218C2×D53—81no—
219C2×D53—81no—
220C2×D53—81no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—42no—
227C4×C53—42no—
228C4×C53—81no—
229C4×C53—81no—
230C4×C53#52511yescharacteristic
231C4×C53#57511yescharacteristic
232Dic53—81no—
233Dic53—81no—
234Dic53—81no—
235#3175—531no—
236#2950D5311yescharacteristic, complemented
237#2237—531no—
238#2234—531no—
239#2227—531no—
240#2225—531no—
241#2211—531no—
242#2211—531no—
243#2188—531no—
244#2185—531no—
245#2176—531no—
246#2175—531no—
247C4×Q32—531no—
248C4×SD32—531no—
249#2164—531no—
250C2×C53—41no—
251C2×C53#310011yescharacteristic
252C2×C53#341611yescharacteristic
253C2×C53#341711yescharacteristic
254D53—161no—
255C22×Q16C2×D5311yescharacteristic
256#542—1061no—
257#519—21no—
258C2×Q32—531no—
259C2×Q32—531no—
260C2×SD32—531no—
261C2×SD32—531no—
262#500—531no—
263#495—531no—
264#489—1061no—
265#468—1061no—
266#467—531no—
267#466—531no—
268#441—21no—
269#441—21no—
270#440—1061no—
271C4×Q16—531no—
272C4×Q16C2×D5311yescharacteristic
273C4×Q16C2×D5311yescharacteristic
274C4×Q16C2×D5311yescharacteristic
275C4×Q16C2×D5311yescharacteristic
276C4×D8—531no—
277#427C2×D5311yescharacteristic
278#419—21no—
279#405C2×D5311yescharacteristic
280#367—531no—
281#367—531no—
282#366—531no—
283#366—531no—
284C4⋊C16—531no—
285C4⋊C16—531no—
286#358—531no—
287#358—531no—
288#358—531no—
289#358—531no—
290#358—531no—
291#358—531no—
292#357—531no—
293#357—531no—
294C22⋊C16—531no—
295C22⋊C16—531no—
296#346—531no—
297C4×C16—531no—
298C53#3368111yescharacteristic, complemented, Sylow
299C22×Q8—21no—
300C2×Q16—22no—
301C2×Q16—22no—
302C2×Q16—21no—
303C2×Q16—531no—
304C2×Q16C22×D5311yescharacteristic
305C2×Q16C22×D5311yescharacteristic
306C2×SD16—1061no—
307C2×D8—531no—
308C2×M16C22⋊D5311yescharacteristic
309C22×C8C22×D5311yescharacteristic
310C4⋊Q8—1061no—
311#126—2121no—
312#124—1061no—
313#124—2121no—
314#123—2121no—
315C22⋊Q8—1061no—
316C22⋊Q8—2121no—
317#121—1061no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—41no—
323C4×Q8—41no—
324C4×Q8—1061no—
325C4×D4—1061no—
326#117—21no—
327#117—41no—
328#117—41no—
329#117C22⋊D5311yescharacteristic
330Q32—1062no—
331Q32—1062no—
332SD32—1062no—
333SD32—1062no—
334C2×C16—531no—
335C2×C16—531no—
336C2×C16—531no—
337C2×C16—531no—
338#107—531no—
339#107—531no—
340#107—1061no—
341#107C22×D5311yescharacteristic
342#107C22×D5311yescharacteristic
343#106—21no—
344#103—21no—
345#103—21no—
346#103—21no—
347#103—21no—
348#103—1061no—
349#103—1061no—
350#102—1061no—
351#102—1061no—
352#97C22⋊D5311yescharacteristic
353#97C22⋊D5311yescharacteristic
354C4×C8C22×D5311yescharacteristic
355C4×C8C22×D5311yescharacteristic
356C2×Q8—24no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—42no—
360C2×Q8—41no—
361C2×Q8—1061no—
362C2×Q8—2121no—
363C2×D4—1061no—
364C2×D4—2121no—
365C22×C4—41no—
366C22×C4—2121no—
367C22×C4C2×C22⋊D5311yescharacteristic
368Q16—24no—
369Q16—21no—
370Q16—21no—
371Q16—1061no—
372Q16—1061no—
373Q16#1171994512yes—
374Q16#1171994512yes—
375SD16—2122no—
376D8—1061no—
377D8—1061no—
378M16—41no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8C2×C22⋊D5311yescharacteristic
382C2×C8C2×C22⋊D5311yescharacteristic
383C2×C8C2×C22⋊D5311yescharacteristic
384C2×C8C23×D5311yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—1061no—
392C4⋊C4—1061no—
393C4⋊C4—2121no—
394C4⋊C4—2121no—
395C4⋊C4—2121no—
396C4⋊C4#1171992211yescharacteristic
397C4⋊C4#1171992211yescharacteristic
398C22⋊C4—41no—
399C22⋊C4—2121no—
400C22⋊C4—2121no—
401C22⋊C4—2121no—
402C22⋊C4—2121no—
403C22⋊C4—2121no—
404C22⋊C4#1171992211yescharacteristic
405C22⋊C4#1171992211yescharacteristic
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—2121no—
412C42C2×C22⋊D5311yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C16—1061no—
415C16—1061no—
416C16—1061no—
417C16—1061no—
418C23—2121no—
419C23C2×#1171992211yescharacteristic
420Q8—22no—
421Q8—22no—
422Q8—44no—
423Q8—41no—
424Q8—41no—
425Q8—8481no—
426Q8—2121no—
427Q8—2121no—
428D4—8481no—
429D4—2121no—
430D4—2121no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—8481no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4C2×#1171992211yescharacteristic
443C2×C4C2×#1171992211yescharacteristic
444C2×C4C2×#1171992211yescharacteristic
445C2×C4C2×#1171992211yescharacteristic
446C2×C4C2×#1171992211yescharacteristic
447C2×C4C22×C22⋊D5311yescharacteristic
448C8—21no—
449C8—21no—
450C8—21no—
451C8C2×#1171994511yescharacteristic
452C8C2×#1171994511yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—4241no—
457C22—4241no—
458C22—4241no—
459C22C22×#1171992211yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—4241no—
470C4—4241no—
471C4—4241no—
472C4#1681113411yescharacteristic
473C4#1681142111yescharacteristic
474C2—41no—
475C2—8481no—
476C2#2028471511yescharacteristic
477C2#2028785911yescharacteristic
478C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2950C2
socleC22×C53C22×D8
prime radical#16810859C22
hypercenter#2950D53
hypocenterC53#33681
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2950

Complements

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

NH ≅ G/Ntype
2. #20276923C2semidirect
4. #20276867C2semidirect
5. #20276857C2semidirect
6. #20276777C2semidirect
7. #20276777C2semidirect
8. #20276649C2semidirect
10. #20276584C2semidirect
11. #20276577C2semidirect
12. C4×#13133163C2semidirect
13. C4×#13133160C2semidirect
14. #20276522C2semidirect
15. C53×#2950C2semidirect
236. #2950D53semidirect
298. C53#33681semidirect

Maximal subgroups

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

subgroupindex
#202853542
#202769232
#202769062
#202768672
#202768572
#202767772
#202766492
#202766322
#202765842
#202765772
C4×#131331632
C4×#131331602
#202765222
C53×#29502
#3368153