Finite Group Databasebeta

← #27539125

Subgroups — #27539125

Order 13568.

554 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#20285285C211yesmaximal, characteristic, complemented
2#20276923C211yesmaximal, characteristic, complemented
3#20276906C211yesmaximal, characteristic, complemented
4#20276879C211yesmaximal, characteristic, complemented
5#20276872C211yesmaximal, characteristic, complemented
6#20276777C211yesmaximal, characteristic, complemented
7#20276770C211yesmaximal, characteristic, complemented
8#20276655C211yesmaximal, characteristic, complemented
9#20276638C211yesmaximal, characteristic
10#20276589C211yesmaximal, characteristic, complemented
11#20276584C211yesmaximal, characteristic, complemented
12#20276579C211yesmaximal, characteristic, complemented
13#20276577C211yesmaximal, characteristic, complemented
14#20276518C211yesmaximal, characteristic, complemented
15C4×#135×C53C211yesmaximal, characteristic, complemented
16#16811157—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133161C2211yescharacteristic
19C2×#13133160C2211yescharacteristic
20C2×#13133158C2211yescharacteristic
21#16811025C2211yescharacteristic
22#16811007C2211yescharacteristic
23#16810978—21no—
24#16810864C2211yescharacteristic
25#16810859C2211yescharacteristic
26#16810852—21no—
27C4×Dic212C2211yescharacteristic
28C4×#11719913—21no—
29C4×D424C2211yescharacteristic
30#16810450C2211yescharacteristic
31#16810450C2211yescharacteristic
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
47#16810399C2211yescharacteristic
48C2×#135×C53C2211yescharacteristic
49C4×#41×C53—21no—
50C4×Q16×C53C2211yescharacteristic
51C4×Q16×C53C2211yescharacteristic
52C4×SD16×C53—21no—
53C4×SD16×C53—21no—
54C4×D8×C53C2211yescharacteristic
55C4×D8×C53C2211yescharacteristic
56C53×#427C2211yescharacteristic
57C53×#416—21no—
58C2×C4×C8×C53C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60C2×#11719913—21no—
61C2×D424C2311yescharacteristic
62#13133211—41no—
63#13133209—21no—
64#13133205—21no—
65#13133200—41no—
66#13133196—21no—
67#13133191—21no—
68C4×Dic106—21no—
69C4×D212—21no—
70C4×C22⋊D53—41no—
71#13133174—41no—
72#13133163—22no—
73#13133161—22no—
74#13133160—22no—
75#13133158—22no—
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80#13133151C2311yescharacteristic
81#13133151C2311yescharacteristic
82#13133149—21no—
83#13133142—21no—
84#13133142—21no—
85#13133139—21no—
86#13133139—21no—
87C2×#41×C53—21no—
88#135×C53—22no—
89#135×C53—22no—
90C2×Q16×C53C2311yescharacteristic
91C2×SD16×C53—21no—
92C2×D8×C53C2311yescharacteristic
93C22×C8×C53D411yescharacteristic
94C22×C8×C53C2311yescharacteristic
95C4×Q8×C53—21no—
96C4×Q8×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—41no—
100C4×D4×C53—41no—
101#117×C53—21no—
102#117×C53—41no—
103C2×C42×C53—41no—
104C2×C42×C53D411yescharacteristic
105#107×C53C2311yescharacteristic
106#107×C53C2311yescharacteristic
107#106×C53—21no—
108#103×C53—21no—
109#103×C53—21no—
110#102×C53—21no—
111#102×C53—21no—
112C4×C8×C53D411yescharacteristic
113C4×C8×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116#41×C53—24no—
117#41×C53—42no—
118C2×Q8×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53D811yescharacteristic
125C22×C4×C53D811yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C2×C22⋊D53—41no—
128C2×D212—21no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131Q16×C53—21no—
132Q16×C53#4112yes—
133SD16×C53—24no—
134D8×C53—21no—
135D8×C53#4112yes—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53—41no—
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—
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53—41no—
151C42×C53—41no—
152C42×C53—41no—
153C42×C53D811yescharacteristic
154C42×C53D811yescharacteristic
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C22⋊Dic53—41no—
158#11719921—41no—
159#11719921—41no—
160#11719921—41no—
161#11719921—41no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164#11719919—41no—
165#11719919—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×C53—41no—
179C23×C53C2×D811yescharacteristic
180C22×D53—41no—
181Q8×C53—22no—
182Q8×C53—41no—
183D4×C53—22no—
184D4×C53—44no—
185D4×C53—41no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—44no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—82no—
195C2×C4×C53—82no—
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C22⋊D53—82no—
203C2×Dic53—41no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206D212—41no—
207D212—41no—
208C4×D53—82no—
209Dic106—41no—
210Dic106—41no—
211C8×C53—21no—
212C8×C53—21no—
213C8×C53—21no—
214C8×C53C2×#4111yescharacteristic
215C8×C53C2×#4111yescharacteristic
216#33671—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—42no—
221C22×C53—81no—
222C22×C53—81no—
223C22×C53C22×D811yescharacteristic
224C2×D53—81no—
225C2×D53—81no—
226C2×D53—81no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—42no—
233C4×C53—81no—
234C4×C53—81no—
235C4×C53#52511yescharacteristic
236C4×C53#57511yescharacteristic
237Dic53—81no—
238Dic53—81no—
239Dic53—81no—
240#3168—531no—
241C4×#135D5311yescharacteristic, complemented
242#2237—531no—
243#2234—531no—
244#2229—531no—
245#2228—531no—
246#2211—531no—
247#2210—531no—
248#2189—531no—
249#2186—531no—
250#2177—531no—
251#2176—531no—
252#2175—531no—
253#2175—531no—
254#2163—531no—
255C2×C53—41no—
256C2×C53—82no—
257C2×C53#310011yescharacteristic
258C2×C53#341411yescharacteristic
259C2×C53#341411yescharacteristic
260D53—161no—
261C2×#135C2×D5311yescharacteristic
262#529—1061no—
263C4×#41—21no—
264C2×Q32—531no—
265C2×SD32—531no—
266C2×SD32—531no—
267C2×D16—531no—
268#499—531no—
269#495—531no—
270#488—1061no—
271#467—531no—
272#466—531no—
273#465—1061no—
274C4×Q16—531no—
275C4×Q16C2×D5311yescharacteristic
276C4×Q16C2×D5311yescharacteristic
277C4×SD16—21no—
278C4×SD16—21no—
279C4×SD16—1061no—
280C4×D8—531no—
281C4×D8C2×D5311yescharacteristic
282C4×D8C2×D5311yescharacteristic
283#427C2×D5311yescharacteristic
284#416—21no—
285C2×C4×C8C2×D5311yescharacteristic
286#367—531no—
287#367—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—
303#346—531no—
304C53#3367111yescharacteristic, complemented, Sylow
305C2×#41—21no—
306#135—22no—
307#135—22no—
308C2×Q16—531no—
309C2×Q16C22×D5311yescharacteristic
310C2×SD16—21no—
311C2×SD16—1061no—
312C2×D8—531no—
313C2×D8C22×D5311yescharacteristic
314C22×C8C22⋊D5311yescharacteristic
315C22×C8C22×D5311yescharacteristic
316#126—2121no—
317#125—1061no—
318#124—1061no—
319#123—2121no—
320C22⋊Q8—1061no—
321#121—1061no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—1061no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—41no—
328C4×D4—41no—
329C4×D4—1061no—
330C4×D4—2121no—
331#117—21no—
332#117—41no—
333#117—2121no—
334C2×C42—41no—
335C2×C42C22⋊D5311yescharacteristic
336Q32—1062no—
337SD32—1062no—
338SD32—1062no—
339D16—1062no—
340C2×C16—531no—
341C2×C16—531no—
342C2×C16—531no—
343C2×C16—531no—
344#107—531no—
345#107—531no—
346#107C22×D5311yescharacteristic
347#107C22×D5311yescharacteristic
348#106—21no—
349#106—1061no—
350#103—21no—
351#103—21no—
352#103—1061no—
353#103—1061no—
354#102—21no—
355#102—21no—
356#102—1061no—
357#102—1061no—
358C4×C8C22⋊D5311yescharacteristic
359C4×C8C22⋊D5311yescharacteristic
360C4×C8C22×D5311yescharacteristic
361C4×C8C22×D5311yescharacteristic
362#41—24no—
363#41—42no—
364C2×Q8—21no—
365C2×Q8—1061no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—1061no—
369C2×D4—2121no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—2121no—
374C22×C4#1171992211yescharacteristic
375C22×C4#1171992211yescharacteristic
376C22×C4C2×C22⋊D5311yescharacteristic
377Q16—21no—
378Q16—1061no—
379Q16—1061no—
380Q16#1171994512yes—
381SD16—24no—
382SD16—2122no—
383D8—21no—
384D8—1061no—
385D8—1061no—
386D8#1171994512yes—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—41no—
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—1061no—
399C4⋊C4—1061no—
400C4⋊C4—2121no—
401C4⋊C4—2121no—
402C4⋊C4—2121no—
403C4⋊C4—2121no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—2121no—
408C22⋊C4—2121no—
409C22⋊C4—2121no—
410C22⋊C4—2121no—
411C22⋊C4—2121no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—2121no—
416C42#1171992211yescharacteristic
417C42#1171992211yescharacteristic
418C42C2×C22⋊D5311yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C16—1061no—
421C16—1061no—
422C16—1061no—
423C16—1061no—
424C23—41no—
425C23—2121no—
426C23C2×#1171992211yescharacteristic
427Q8—22no—
428Q8—41no—
429Q8—2121no—
430Q8—2121no—
431D4—22no—
432D4—44no—
433D4—41no—
434D4—4242no—
435D4—2121no—
436D4—2121no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—4242no—
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—
460C8C2×#1171994511yescharacteristic
461C8C2×#1171994511yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—42no—
466C22—81no—
467C22—81no—
468C22—4241no—
469C22—4241no—
470C22—4241no—
471C22C22×#1171992211yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—81no—
479C4—81no—
480C4—4241no—
481C4—4241no—
482C4—4241no—
483C4#1681113411yescharacteristic
484C4#1681142111yescharacteristic
485C2—41no—
486C2—82no—
487C2—8481no—
488C2#2028471511yescharacteristic
489C2#2028784911yescharacteristic
490C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC4×#135×C53C2
socleC22×C53C22×D8
prime radical#20276638C2
hypercenterC4×#135D53
hypocenterC53#33671
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #20285285C2semidirect
2. #20276923C2semidirect
3. #20276906C2semidirect
4. #20276879C2semidirect
5. #20276872C2semidirect
6. #20276777C2semidirect
7. #20276770C2semidirect
8. #20276655C2semidirect
10. #20276589C2semidirect
11. #20276584C2semidirect
12. #20276579C2semidirect
13. #20276577C2semidirect
14. #20276518C2semidirect
15. C4×#135×C53C2semidirect
241. C4×#135D53semidirect
304. C53#33671semidirect

Maximal subgroups

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

subgroupindex
#202852852
#202769232
#202769062
#202768792
#202768722
#202767772
#202767702
#202766552
#202766382
#202765892
#202765842
#202765792
#202765772
#202765182
C4×#135×C532
#3367153