Finite Group Databasebeta

← #27539239

Subgroups — #27539239

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#20283584C211yesmaximal, characteristic, complemented
2#20276920C211yesmaximal, characteristic, complemented
3#20276903C211yesmaximal, characteristic
4#20276864C211yesmaximal, characteristic, complemented
5#20276854C211yesmaximal, characteristic, complemented
6#20276774C211yesmaximal, characteristic, complemented
7#20276774C211yesmaximal, characteristic, complemented
8#20276648C211yesmaximal, characteristic, complemented
9#20276631C211yesmaximal, characteristic
10#20276584C211yesmaximal, characteristic, complemented
11#20276579C211yesmaximal, characteristic, complemented
12C4×#13133163C211yesmaximal, characteristic, complemented
13C4×#13133161C211yesmaximal, characteristic, complemented
14#20276522C211yesmaximal, characteristic, complemented
15C53×#3175C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17#16811100—21no—
18C2×#13133163C2211yescharacteristic
19C2×#13133163C2211yescharacteristic
20C2×#13133161C2211yescharacteristic
21C2×#13133161C2211yescharacteristic
22#16810731—21no—
23#16810731—21no—
24C4×Dic212C2211yescharacteristic
25C4×Dic212C2211yescharacteristic
26C4×Dic212C2211yescharacteristic
27C4×Dic212C2211yescharacteristic
28#16810661C2211yescharacteristic
29#16810633—21no—
30#16810450C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35C4⋊#11719908C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44C22⋊#11719908C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#542—21no—
49C53×#500C2211yescharacteristic
50C53×#495C2211yescharacteristic
51C53×#489—21no—
52C53×#468—21no—
53C53×#467C2211yescharacteristic
54C53×#466C2211yescharacteristic
55C53×#440—21no—
56C4×Q16×C53C2211yescharacteristic
57C4×D8×C53C2211yescharacteristic
58C53×#405C2211yescharacteristic
59C22×Dic106—21no—
60C2×Dic212—21no—
61C2×Dic212—22no—
62C2×Dic212—22no—
63C2×Dic212C2311yescharacteristic
64C2×Dic212C2311yescharacteristic
65C4×Dic106—21no—
66C4×Dic106—21no—
67C4×Dic106—21no—
68C4×Dic106—21no—
69#13133182—41no—
70#13133182—41no—
71#13133176—21no—
72#13133173—41no—
73#13133173—41no—
74#13133163—22no—
75#13133163—22no—
76#13133161—22no—
77#13133161—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133151C2311yescharacteristic
84#13133149—21no—
85#13133142—21no—
86#13133142—21no—
87#13133142—21no—
88#13133142—21no—
89C2×Q16×C53C2311yescharacteristic
90C2×SD16×C53—21no—
91C2×D8×C53C2311yescharacteristic
92C2×M16×C53D411yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C4⋊Q8×C53—21no—
95#126×C53—41no—
96#124×C53—21no—
97#124×C53—41no—
98#123×C53—41no—
99C22⋊Q8×C53—21no—
100C22⋊Q8×C53—41no—
101#121×C53—21no—
102C4×Q8×C53—21no—
103C4×D4×C53—21no—
104#117×C53D411yescharacteristic
105#107×C53—21no—
106#107×C53C2311yescharacteristic
107#107×C53C2311yescharacteristic
108#103×C53—21no—
109#103×C53—21no—
110#102×C53—21no—
111#102×C53—21no—
112#97×C53D411yescharacteristic
113#97×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116C2×Q8×C53—21no—
117C2×Q8×C53—41no—
118C2×D4×C53—21no—
119C2×D4×C53—41no—
120C22×C4×C53—41no—
121C22×C4×C53C2×D411yescharacteristic
122C22×Dic53—41no—
123C2×Dic106—24no—
124C2×Dic106—21no—
125C2×Dic106—21no—
126C2×Dic106—42no—
127C2×Dic106—41no—
128Q16×C53—21no—
129Q16×C53—21no—
130SD16×C53—42no—
131D8×C53—21no—
132D8×C53—21no—
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×C53D811yescharacteristic
146C4⋊C4×C53D811yescharacteristic
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53D811yescharacteristic
153C22⋊C4×C53D811yescharacteristic
154C42×C53—41no—
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C22⋊Dic53—41no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160#11719919—41no—
161#11719919—41no—
162#11719919—41no—
163#11719919—41no—
164C4×Dic53—41no—
165C4×Dic53—41no—
166C4×Dic53—41no—
167C4×Dic53—41no—
168C4×Dic53—41no—
169Dic212—24no—
170Dic212—21no—
171Dic212—21no—
172Dic212#4112yes—
173Dic212#4112yes—
174#11719908—21no—
175#11719908—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53—41no—
179C23×C53C2×D811yescharacteristic
180Q8×C53—41no—
181Q8×C53—41no—
182Q8×C53—161no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—161no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—161no—
192C2×C4×C53C2×D811yescharacteristic
193C2×C4×C53C2×D811yescharacteristic
194C2×C4×C53C2×D811yescharacteristic
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×Dic53—44no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203Dic106—22no—
204Dic106—22no—
205Dic106—44no—
206Dic106—41no—
207Dic106—41no—
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×C53—81no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53C22×D811yescharacteristic
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—81no—
227C4×C53—81no—
228C4×C53—81no—
229C4×C53#52511yescharacteristic
230C4×C53#57511yescharacteristic
231Dic53—42no—
232Dic53—42no—
233Dic53—81no—
234Dic53—81no—
235#3175D5311yescharacteristic
236#2950—531no—
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—161no—
252C2×C53#310011yescharacteristic
253C2×C53#341611yescharacteristic
254C2×C53#341711yescharacteristic
255C22×Q16—531no—
256#542—21no—
257#519—1061no—
258C2×Q32—531no—
259C2×Q32—531no—
260C2×SD32—531no—
261C2×SD32—531no—
262#500C2×D5311yescharacteristic
263#495C2×D5311yescharacteristic
264#489—21no—
265#468—21no—
266#467C2×D5311yescharacteristic
267#466C2×D5311yescharacteristic
268#441—1061no—
269#441—1061no—
270#440—21no—
271C4×Q16—531no—
272C4×Q16—531no—
273C4×Q16—531no—
274C4×Q16—531no—
275C4×Q16C2×D5311yescharacteristic
276C4×D8C2×D5311yescharacteristic
277#427—531no—
278#419—1061no—
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—1061no—
300C2×Q16—531no—
301C2×Q16—531no—
302C2×Q16—1062no—
303C2×Q16—1062no—
304C2×Q16—1061no—
305C2×Q16C22×D5311yescharacteristic
306C2×SD16—21no—
307C2×D8C22×D5311yescharacteristic
308C2×M16C22⋊D5311yescharacteristic
309C22×C8C22×D5311yescharacteristic
310C4⋊Q8—21no—
311#126—41no—
312#124—21no—
313#124—41no—
314#123—41no—
315C22⋊Q8—21no—
316C22⋊Q8—41no—
317#121—21no—
318C4×Q8—21no—
319C4×Q8—1061no—
320C4×Q8—1061no—
321C4×Q8—1061no—
322C4×Q8—1061no—
323C4×Q8—2121no—
324C4×Q8—2121no—
325C4×D4—21no—
326#117—1061no—
327#117—2121no—
328#117—2121no—
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—21no—
339#107—531no—
340#107—531no—
341#107C22×D5311yescharacteristic
342#107C22×D5311yescharacteristic
343#106—1061no—
344#103—21no—
345#103—21no—
346#103—1061no—
347#103—1061no—
348#103—1061no—
349#103—1061no—
350#102—21no—
351#102—21no—
352#97C22⋊D5311yescharacteristic
353#97C22⋊D5311yescharacteristic
354C4×C8C22×D5311yescharacteristic
355C4×C8C22×D5311yescharacteristic
356C2×Q8—21no—
357C2×Q8—41no—
358C2×Q8—1064no—
359C2×Q8—1061no—
360C2×Q8—1061no—
361C2×Q8—2122no—
362C2×Q8—2121no—
363C2×D4—21no—
364C2×D4—41no—
365C22×C4—41no—
366C22×C4—2121no—
367C22×C4C2×C22⋊D5311yescharacteristic
368Q16—21no—
369Q16—21no—
370Q16—532no—
371Q16—532no—
372Q16—1064no—
373Q16—1061no—
374Q16—1061no—
375SD16—42no—
376D8—21no—
377D8—21no—
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—1061no—
391C4⋊C4—1061no—
392C4⋊C4—2121no—
393C4⋊C4—2121no—
394C4⋊C4—2121no—
395C4⋊C4—2121no—
396C4⋊C4#1171992211yescharacteristic
397C4⋊C4#1171992211yescharacteristic
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—2121no—
404C22⋊C4#1171992211yescharacteristic
405C22⋊C4#1171992211yescharacteristic
406C42—41no—
407C42—2121no—
408C42—2121no—
409C42—2121no—
410C42—2121no—
411C42—2121no—
412C42C2×C22⋊D5311yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C16—1061no—
415C16—1061no—
416C16—1061no—
417C16—1061no—
418C23—41no—
419C23C2×#1171992211yescharacteristic
420Q8—41no—
421Q8—41no—
422Q8—161no—
423Q8—1062no—
424Q8—1062no—
425Q8—2124no—
426Q8—2121no—
427Q8—2121no—
428D4—41no—
429D4—41no—
430D4—161no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—161no—
437C2×C4—2121no—
438C2×C4—2121no—
439C2×C4—2124no—
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—
451C8#41×D5311yescharacteristic
452C8#41×D5311yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22C22×#1171992211yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—4241no—
469C4—4241no—
470C4—2122no—
471C4—2122no—
472C4#1681113011yescharacteristic
473C4#1681142111yescharacteristic
474C2—41no—
475C2—161no—
476C2#2028471111yescharacteristic
477C2#2028786211yescharacteristic
478C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3175C2
socleC22×C53C22×D8
prime radicalC53×#466C22
hypercenter#3175D53
hypocenterC53#33681
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #20283584C2semidirect
2. #20276920C2semidirect
4. #20276864C2semidirect
5. #20276854C2semidirect
6. #20276774C2semidirect
7. #20276774C2semidirect
8. #20276648C2semidirect
10. #20276584C2semidirect
11. #20276579C2semidirect
12. C4×#13133163C2semidirect
13. C4×#13133161C2semidirect
14. #20276522C2semidirect
298. C53#33681semidirect

Maximal subgroups

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

subgroupindex
#202835842
#202769202
#202769032
#202768642
#202768542
#202767742
#202766482
#202766312
#202765842
#202765792
C4×#131331632
C4×#131331612
#202765222
C53×#31752
#3368153