Finite Group Databasebeta

← #27540535

Subgroups — #27540535

Order 13568.

488 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285354C211yesmaximal, characteristic, complemented
2#20276862C211yesmaximal, characteristic, complemented
3#20276859C211yesmaximal, characteristic, complemented
4#20276857C211yesmaximal, characteristic, complemented
5#20276854C211yesmaximal, characteristic, complemented
6#20276821C211yesmaximal, characteristic, complemented
7#20276816C211yesmaximal, characteristic
8#20276813C211yesmaximal, characteristic, complemented
9#20276809C211yesmaximal, characteristic, complemented
10#20276777C211yesmaximal, characteristic, complemented
11#20276774C211yesmaximal, characteristic, complemented
12#20276770C211yesmaximal, characteristic, complemented
13#20276767C211yesmaximal, characteristic, complemented
14#20276474C211yesmaximal, characteristic, complemented
15C53×#3175C211yesmaximal, characteristic, complemented
16#16811244—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133161C2211yescharacteristic
19C2×#13133160C2211yescharacteristic
20C2×#13133158C2211yescharacteristic
21C2×#13133157C2211yescharacteristic
22C22×#11719908C2211yescharacteristic
23#16811028C2211yescharacteristic
24#16811007C2211yescharacteristic
25#16810983—21no—
26#16810869—21no—
27#16810864C2211yescharacteristic
28#16810859C2211yescharacteristic
29#16810725—21no—
30C4×Dic212C2211yescharacteristic
31C4×D424C2211yescharacteristic
32#16810450C2211yescharacteristic
33#16810450C2211yescharacteristic
34#16810450C2211yescharacteristic
35#16810450C2211yescharacteristic
36C4⋊#11719908C2211yescharacteristic
37C4⋊#11719908C2211yescharacteristic
38C4⋊#11719908C2211yescharacteristic
39C4⋊#11719908C2211yescharacteristic
40#16810427C2211yescharacteristic
41#16810427C2211yescharacteristic
42#16810426C2211yescharacteristic
43#16810426C2211yescharacteristic
44#16810424C2211yescharacteristic
45#16810424C2211yescharacteristic
46#16810423C2211yescharacteristic
47#16810423C2211yescharacteristic
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×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—41no—
73#13133161—41no—
74#13133160—41no—
75#13133158—41no—
76#13133157—41no—
77C2×#11719908—41no—
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—
123C2×D212—21no—
124C2×C4×D53—41no—
125C2×Dic106—21no—
126C2×Dic106—41no—
127Q16×C53—21no—
128Q16×C53—21no—
129SD16×C53—42no—
130D8×C53—21no—
131D8×C53—21no—
132M16×C53—41no—
133C2×C8×C53—21no—
134C2×C8×C53—21no—
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2411yescharacteristic
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—41no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53D811yescharacteristic
145C4⋊C4×C53D811yescharacteristic
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53D811yescharacteristic
152C22⋊C4×C53D811yescharacteristic
153C42×C53—41no—
154C42×C53C2×D411yescharacteristic
155C42×C53C2×D411yescharacteristic
156C22⋊Dic53—41no—
157#11719921—41no—
158#11719921—41no—
159#11719921—41no—
160#11719921—41no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—41no—
164#11719919—41no—
165#11719919—41no—
166C4×Dic53—41no—
167Dic212—21no—
168Dic212—21no—
169D424—21no—
170D424—21no—
171#11719913—42no—
172#11719908—21no—
173#11719908—21no—
174#11719908—21no—
175#11719908—21no—
176C23×C53—41no—
177C23×C53C2×D811yescharacteristic
178C22×D53—41no—
179Q8×C53—41no—
180Q8×C53—41no—
181Q8×C53—161no—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—161no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—161no—
191C2×C4×C53C2×D811yescharacteristic
192C2×C4×C53C2×D811yescharacteristic
193C2×C4×C53C2×D811yescharacteristic
194C2×C4×C53C2×D811yescharacteristic
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C22×D411yescharacteristic
197C22⋊D53—161no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201D212—41no—
202D212—41no—
203C4×D53—161no—
204Dic106—41no—
205Dic106—41no—
206Dic106—161no—
207C8×C53—21no—
208C8×C53—21no—
209C8×C53—21no—
210C8×C53H(5,𝔽2)11yescharacteristic
211C8×C532−1+411yescharacteristic
212#33829—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—81no—
217C22×C53—81no—
218C22×C53—81no—
219C22×C53C22×D811yescharacteristic
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—81no—
229C4×C53—81no—
230C4×C53—81no—
231C4×C53#53711yescharacteristic
232C4×C53#57511yescharacteristic
233Dic53—81no—
234Dic53—81no—
235Dic53—81no—
236#3175—531no—
237#3175D5311yescharacteristic, complemented
238#2226—531no—
239#2226—531no—
240#2225—531no—
241#2225—531no—
242#2219—531no—
243#2218—531no—
244#2217—531no—
245#2217—531no—
246#2211—531no—
247#2211—531no—
248#2210—531no—
249#2210—531no—
250#2151—531no—
251C2×C53—41no—
252C2×C53—161no—
253C2×C53#324511yescharacteristic
254C2×C53#341611yescharacteristic
255C2×C53#341611yescharacteristic
256D53—161no—
257#542—21no—
258#542—1061no—
259C2×Q32—531no—
260C2×SD32—531no—
261C2×SD32—531no—
262C2×D16—531no—
263C2×M32—531no—
264C22×C16—531no—
265#500—531no—
266#500C2×D5311yescharacteristic
267#495—531no—
268#495C2×D5311yescharacteristic
269#489—21no—
270#489—1061no—
271#468—21no—
272#468—1061no—
273#467—531no—
274#467C2×D5311yescharacteristic
275#466—531no—
276#466C2×D5311yescharacteristic
277#440—21no—
278#440—1061no—
279C4×Q16—531no—
280C4×Q16C2×D5311yescharacteristic
281C4×D8—531no—
282C4×D8C2×D5311yescharacteristic
283#405C2×D5311yescharacteristic
284#367—531no—
285#367—531no—
286#367—531no—
287#367—531no—
288C4⋊C16—531no—
289C4⋊C16—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—
300C53#3382911yescharacteristic, complemented, Sylow
301C2×Q16—531no—
302C2×Q16C22×D5311yescharacteristic
303C2×SD16—21no—
304C2×SD16—1061no—
305C2×D8—531no—
306C2×D8C22×D5311yescharacteristic
307C2×M16C22⋊D5311yescharacteristic
308C22×C8C22×D5311yescharacteristic
309C4⋊Q8—21no—
310C4⋊Q8—1061no—
311#126—41no—
312#126—2121no—
313#124—21no—
314#124—41no—
315#124—1061no—
316#124—2121no—
317#123—41no—
318#123—2121no—
319C22⋊Q8—21no—
320C22⋊Q8—41no—
321C22⋊Q8—1061no—
322C22⋊Q8—2121no—
323#121—21no—
324#121—1061no—
325C4×Q8—21no—
326C4×Q8—1061no—
327C4×D4—21no—
328C4×D4—1061no—
329#117C22⋊D5311yescharacteristic
330Q32—2121no—
331SD32—2121no—
332SD32—2121no—
333D16—2121no—
334M32—2121no—
335C2×C16—531no—
336C2×C16—531no—
337C2×C16—531no—
338C2×C16—531no—
339C2×C16—2121no—
340#107—21no—
341#107—531no—
342#107—531no—
343#107—1061no—
344#107C22×D5311yescharacteristic
345#107C22×D5311yescharacteristic
346#103—21no—
347#103—21no—
348#103—1061no—
349#103—1061no—
350#102—21no—
351#102—21no—
352#102—1061no—
353#102—1061no—
354#97C22⋊D5311yescharacteristic
355#97C22⋊D5311yescharacteristic
356C4×C8C22×D5311yescharacteristic
357C4×C8C22×D5311yescharacteristic
358C2×Q8—21no—
359C2×Q8—41no—
360C2×Q8—1061no—
361C2×Q8—2121no—
362C2×D4—21no—
363C2×D4—41no—
364C2×D4—1061no—
365C2×D4—2121no—
366C22×C4—41no—
367C22×C4—2121no—
368C22×C4C2×C22⋊D5311yescharacteristic
369Q16—21no—
370Q16—21no—
371Q16—1061no—
372Q16—1061no—
373SD16—42no—
374SD16—2122no—
375D8—21no—
376D8—21no—
377D8—1061no—
378D8—1061no—
379M16—41no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8C2×C22⋊D5311yescharacteristic
383C2×C8C2×C22⋊D5311yescharacteristic
384C2×C8C2×C22⋊D5311yescharacteristic
385C2×C8C23×D5311yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—21no—
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—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—2121no—
404C22⋊C4—2121no—
405C22⋊C4—2121no—
406C22⋊C4—2121no—
407C22⋊C4—2121no—
408C22⋊C4#1171992211yescharacteristic
409C22⋊C4#1171992211yescharacteristic
410C42—41no—
411C42—2121no—
412C42C2×C22⋊D5311yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C16—1061no—
415C16—1061no—
416C16—1061no—
417C16—1061no—
418C23—41no—
419C23—2121no—
420C23C2×#1171992211yescharacteristic
421Q8—41no—
422Q8—41no—
423Q8—161no—
424Q8—8481no—
425Q8—2121no—
426Q8—2121no—
427D4—41no—
428D4—41no—
429D4—161no—
430D4—8481no—
431D4—2121no—
432D4—2121no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—161no—
439C2×C4—8481no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4C2×#1171992211yescharacteristic
444C2×C4C2×#1171992211yescharacteristic
445C2×C4C2×#1171992211yescharacteristic
446C2×C4C2×#1171992211yescharacteristic
447C2×C4C2×#1171992211yescharacteristic
448C2×C4C22×C22⋊D5311yescharacteristic
449C8—21no—
450C8—21no—
451C8—21no—
452C8#1313326311yescharacteristic
453C8#1313326411yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—4241no—
461C22—4241no—
462C22—4241no—
463C22C22×#1171992211yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—4241no—
473C4—4241no—
474C4—4241no—
475C4#1681120111yescharacteristic
476C4#1681142111yescharacteristic
477C2—41no—
478C2—161no—
479C2—8481no—
480C2#2028602211yescharacteristic
481C2#2028785911yescharacteristic
482C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3175C2
socleC22×C53C22×D8
prime radical#20276816C2
hypercenter#3175D53
hypocenterC53#33829
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. #20285354C2semidirect
2. #20276862C2semidirect
3. #20276859C2semidirect
4. #20276857C2semidirect
5. #20276854C2semidirect
6. #20276821C2semidirect
8. #20276813C2semidirect
9. #20276809C2semidirect
10. #20276777C2semidirect
11. #20276774C2semidirect
12. #20276770C2semidirect
13. #20276767C2semidirect
14. #20276474C2semidirect
15. C53×#3175C2semidirect
237. #3175D53semidirect
300. C53#33829semidirect

Maximal subgroups

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

subgroupindex
#202853542
#202768622
#202768592
#202768572
#202768542
#202768212
#202768162
#202768132
#202768092
#202767772
#202767742
#202767702
#202767672
#202764742
C53×#31752
#3382953