Finite Group Databasebeta

← #27539272

Subgroups — #27539272

Order 13568.

546 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#20283592C211yesmaximal, characteristic, complemented
2#20276915C211yesmaximal, characteristic, complemented
3#20276893C211yesmaximal, characteristic, complemented
4#20276864C211yesmaximal, characteristic, complemented
5#20276849C211yesmaximal, characteristic, complemented
6#20276774C211yesmaximal, characteristic, complemented
7#20276767C211yesmaximal, characteristic, complemented
8#20276653C211yesmaximal, characteristic, complemented
9#20276636C211yesmaximal, characteristic
10#20276579C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12C4×#13133163C211yesmaximal, characteristic, complemented
13C4×#13133158C211yesmaximal, characteristic, complemented
14#20276522C211yesmaximal, characteristic, complemented
15C53×#3175C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17#16811103—21no—
18C2×#13133163C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20C2×#13133160C2211yescharacteristic
21C2×#13133158C2211yescharacteristic
22#16810725—21no—
23#16810725—21no—
24C4×Dic212C2211yescharacteristic
25C4×Dic212C2211yescharacteristic
26C4×D424C2211yescharacteristic
27C4×D424C2211yescharacteristic
28C2×#13133151C2211yescharacteristic
29#16810625—21no—
30#16810448C2211yescharacteristic
31#16810448C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35C4⋊#11719908C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
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
59C2×#11719945—21no—
60#13133239—22no—
61#13133239—22no—
62C2×Dic212C2311yescharacteristic
63C2×#11719913—21no—
64C2×D424C2311yescharacteristic
65C4×Dic106—21no—
66C4×Dic106—21no—
67C4×D212—21no—
68C4×D212—21no—
69#13133177—41no—
70#13133177—41no—
71#13133175—41no—
72C2×C4⋊Dic53—21no—
73C4⋊C4×D53—41no—
74#13133163—22no—
75#13133161—22no—
76#13133160—22no—
77#13133158—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151—21no—
83#13133151C2311yescharacteristic
84#13133151C2311yescharacteristic
85#13133142—21no—
86#13133142—21no—
87#13133139—21no—
88#13133139—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
122C2×C22⋊D53—41no—
123C22×Dic53—41no—
124#11719945—24no—
125#11719945—42no—
126C2×D212—21no—
127C2×C4×D53—41no—
128C2×C4×D53—41no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131Q16×C53—21no—
132Q16×C53—21no—
133SD16×C53—42no—
134D8×C53—21no—
135D8×C53—21no—
136M16×C53—41no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53C2×D411yescharacteristic
140C2×C8×C53C2×D411yescharacteristic
141C2×C8×C53C2×D411yescharacteristic
142C2×C8×C53C2411yescharacteristic
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53—41no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53D811yescharacteristic
149C4⋊C4×C53D811yescharacteristic
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C22⋊C4×C53—41no—
155C22⋊C4×C53D811yescharacteristic
156C22⋊C4×C53D811yescharacteristic
157C42×C53—41no—
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160#11719921—41no—
161#11719921—41no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—41no—
165#11719919—41no—
166#11719919—41no—
167C4×Dic53—41no—
168C4×Dic53—41no—
169Dic212—21no—
170Dic212#4112yes—
171D424—21no—
172D424#4112yes—
173#11719913—24no—
174#11719908—21no—
175#11719908—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53—41no—
179C23×C53C2×D811yescharacteristic
180C22×D53—41no—
181Q8×C53—41no—
182Q8×C53—41no—
183Q8×C53—161no—
184D4×C53—41no—
185D4×C53—41no—
186D4×C53—161no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—161no—
193C2×C4×C53C2×D811yescharacteristic
194C2×C4×C53C2×D811yescharacteristic
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C22⋊D53—44no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203C2×Dic53—161no—
204D212—22no—
205D212—41no—
206C4×D53—44no—
207C4×D53—161no—
208C4×D53—161no—
209Dic106—22no—
210Dic106—41no—
211C8×C53—21no—
212C8×C53—21no—
213C8×C53—21no—
214C8×C53C2×#4111yescharacteristic
215C8×C53C2×#4111yescharacteristic
216#33684—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53—81no—
223C22×C53C22×D811yescharacteristic
224C2×D53—42no—
225C2×D53—81no—
226C2×D53—81no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—81no—
233C4×C53—81no—
234C4×C53—81no—
235C4×C53#52511yescharacteristic
236C4×C53#57511yescharacteristic
237Dic53—42no—
238Dic53—81no—
239Dic53—81no—
240#3175D5311yescharacteristic, complemented
241#2951—531no—
242#2236—531no—
243#2232—531no—
244#2227—531no—
245#2224—531no—
246#2211—531no—
247#2210—531no—
248#2189—531no—
249#2186—531no—
250#2175—531no—
251#2175—531no—
252C4×Q32—531no—
253C4×D16—531no—
254#2164—531no—
255C2×C53—41no—
256C2×C53—161no—
257C2×C53#310011yescharacteristic
258C2×C53#341511yescharacteristic
259C2×C53#341711yescharacteristic
260D53—82no—
261C2×#135—531no—
262#542—21no—
263#520—1061no—
264C2×Q32—531no—
265C2×SD32—531no—
266C2×SD32—531no—
267C2×D16—531no—
268#500C2×D5311yescharacteristic
269#495C2×D5311yescharacteristic
270#489—21no—
271#468—21no—
272#467C2×D5311yescharacteristic
273#466C2×D5311yescharacteristic
274#440—21no—
275#440—1061no—
276#440—1061no—
277C4×Q16—531no—
278C4×Q16—531no—
279C4×Q16C2×D5311yescharacteristic
280C4×D8—531no—
281C4×D8—531no—
282C4×D8C2×D5311yescharacteristic
283C2×#107—531no—
284#417—1061no—
285#405C2×D5311yescharacteristic
286#366—531no—
287#366—531no—
288#366—531no—
289#366—531no—
290C4⋊C16—531no—
291C4⋊C16—531no—
292#358—531no—
293#358—531no—
294#358—531no—
295#358—531no—
296#357—531no—
297#357—531no—
298#357—531no—
299#357—531no—
300C22⋊C16—531no—
301C22⋊C16—531no—
302#346—531no—
303C4×C16—531no—
304C53#3368411yescharacteristic, complemented, Sylow
305C2×#41—1061no—
306#135—1062no—
307#135—1062no—
308C2×Q16—531no—
309C2×Q16C22×D5311yescharacteristic
310C2×SD16—21no—
311C2×SD16—1061no—
312C2×D8—531no—
313C2×D8C22×D5311yescharacteristic
314C2×M16C22⋊D5311yescharacteristic
315C22×C8C22×D5311yescharacteristic
316C4⋊Q8—21no—
317#126—41no—
318#124—21no—
319#124—41no—
320#123—41no—
321C22⋊Q8—21no—
322C22⋊Q8—41no—
323#121—21no—
324C4×Q8—21no—
325C4×Q8—1061no—
326C4×Q8—1061no—
327C4×D4—21no—
328C4×D4—1061no—
329C4×D4—1061no—
330C4×D4—2121no—
331C4×D4—2121no—
332#117—2121no—
333#117C22⋊D5311yescharacteristic
334C2×C4⋊C4—1061no—
335C2×C4⋊C4—2121no—
336Q32—1062no—
337SD32—1062no—
338SD32—1062no—
339D16—1062no—
340C2×C16—531no—
341C2×C16—531no—
342C2×C16—531no—
343C2×C16—531no—
344#107—21no—
345#107—531no—
346#107—531no—
347#107—1061no—
348#107C22×D5311yescharacteristic
349#107C22×D5311yescharacteristic
350#103—21no—
351#103—21no—
352#103—1061no—
353#103—1061no—
354#102—21no—
355#102—21no—
356#102—1061no—
357#102—1061no—
358#97C22⋊D5311yescharacteristic
359#97C22⋊D5311yescharacteristic
360C4×C8C22×D5311yescharacteristic
361C4×C8C22×D5311yescharacteristic
362#41—1064no—
363#41—2122no—
364C2×Q8—21no—
365C2×Q8—41no—
366C2×Q8—1061no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—1061no—
370C2×D4—2121no—
371C22×C4—41no—
372C22×C4—2121no—
373C22×C4—2121no—
374C22×C4—2121no—
375C22×C4—2121no—
376C22×C4C2×C22⋊D5311yescharacteristic
377Q16—21no—
378Q16—21no—
379Q16—532no—
380Q16—1061no—
381SD16—42no—
382SD16—1064no—
383D8—21no—
384D8—21no—
385D8—532no—
386D8—1061no—
387M16—41no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8C2×C22⋊D5311yescharacteristic
391C2×C8C2×C22⋊D5311yescharacteristic
392C2×C8C2×C22⋊D5311yescharacteristic
393C2×C8C23×D5311yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—1061no—
400C4⋊C4—1061no—
401C4⋊C4—2121no—
402C4⋊C4—2121no—
403C4⋊C4—2121no—
404C4⋊C4#1171992211yescharacteristic
405C4⋊C4#1171992211yescharacteristic
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—2121no—
412C22⋊C4—2121no—
413C22⋊C4#1171992211yescharacteristic
414C22⋊C4#1171992211yescharacteristic
415C42—41no—
416C42—2121no—
417C42—2121no—
418C42C2×C22⋊D5311yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C16—1061no—
421C16—1061no—
422C16—1061no—
423C16—1061no—
424C23—41no—
425C23—2121no—
426C23C2×#1171992211yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—161no—
430Q8—1062no—
431Q8—2121no—
432D4—41no—
433D4—41no—
434D4—161no—
435D4—1062no—
436D4—2124no—
437D4—2121no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—161no—
444C2×C4—8481no—
445C2×C4—8481no—
446C2×C4—8481no—
447C2×C4—2124no—
448C2×C4—2121no—
449C2×C4—2121no—
450C2×C4—2121no—
451C2×C4C2×#1171992211yescharacteristic
452C2×C4C2×#1171992211yescharacteristic
453C2×C4C2×#1171992211yescharacteristic
454C2×C4C2×#1171992211yescharacteristic
455C2×C4C2×#1171992211yescharacteristic
456C2×C4C22×C22⋊D5311yescharacteristic
457C8—21no—
458C8—21no—
459C8—21no—
460C8#41×D5311yescharacteristic
461C8#41×D5311yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—4241no—
469C22—4241no—
470C22—2122no—
471C22C22×#1171992211yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—4241no—
481C4—4241no—
482C4—2122no—
483C4#1681113011yescharacteristic
484C4#1681142111yescharacteristic
485C2—41no—
486C2—161no—
487C2—4242no—
488C2#2028471111yescharacteristic
489C2#2028785411yescharacteristic
490C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3175C2
socleC22×C53C22×D8
prime radical#20276636C2
hypercenter#3175D53
hypocenterC53#33684
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. #20283592C2semidirect
2. #20276915C2semidirect
3. #20276893C2semidirect
4. #20276864C2semidirect
5. #20276849C2semidirect
6. #20276774C2semidirect
7. #20276767C2semidirect
8. #20276653C2semidirect
10. #20276579C2semidirect
11. #20276577C2semidirect
12. C4×#13133163C2semidirect
13. C4×#13133158C2semidirect
14. #20276522C2semidirect
15. C53×#3175C2semidirect
240. #3175D53semidirect
304. C53#33684semidirect

Maximal subgroups

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

subgroupindex
#202835922
#202769152
#202768932
#202768642
#202768492
#202767742
#202767672
#202766532
#202766362
#202765792
#202765772
C4×#131331632
C4×#131331582
#202765222
C53×#31752
#3368453