Finite Group Databasebeta

← #27276491

Subgroups — #27276491

Order 13568.

676 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275689C211yesmaximal, characteristic, complemented
2#20273432C211yesmaximal, characteristic, complemented
3#20273335C211yesmaximal, characteristic, complemented
4#20273335C211yesmaximal, characteristic, complemented
5#20273253C211yesmaximal, characteristic, complemented
6#20273253C211yesmaximal, characteristic, complemented
7C53×#3552C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9#16810680C2211yescharacteristic
10C2×#13133128C2211yescharacteristic
11#16810489C2211yescharacteristic
12#16810385—22no—
13#16810369—22no—
14#16810365—21no—
15#16810365—21no—
16#16810363—21no—
17#16810363—21no—
18#16810360—21no—
19#16810360—21no—
20#16810357—21no—
21#16810357—21no—
22C53×#563—22no—
23C53×#557C411yescharacteristic, complemented
24C53×#556C411yescharacteristic, complemented
25C53×#554—21no—
26C53×#553—21no—
27D4×Q8×C53—21no—
28D4×Q8×C53C411yescharacteristic, complemented
29C53×#548—21no—
30C53×#547C411yescharacteristic, complemented
31C53×#544—21no—
32C53×#542—21no—
33C53×#541—21no—
34C53×#539—22no—
35C53×#536—21no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C53×#529C2211yescharacteristic
38C53×#529C2211yescharacteristic
39C22×#88322—21no—
40C2×C4⋊Dic53—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45#13133128—21no—
46#13133128—21no—
47#13133128—21no—
48#13133128—21no—
49#13133124—21no—
50#13133124—21no—
51C2×#41×C53—21no—
52C22×Q8×C53—21no—
53C22×Q8×C53—21no—
54C4⋊Q8×C53—22no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53C2×C411yescharacteristic
60C4⋊Q8×C53C2×C411yescharacteristic
61C4⋊Q8×C53D412yes—
62#126×C53—22no—
63#126×C53—22no—
64#126×C53—22no—
65#126×C53—21no—
66#126×C53—21no—
67#125×C53—22no—
68#125×C53—22no—
69#125×C53—22no—
70#125×C53—21no—
71#125×C53D411yescharacteristic
72#125×C53D411yescharacteristic
73#124×C53—22no—
74#124×C53—21no—
75#124×C53D411yescharacteristic
76#124×C53D411yescharacteristic
77#123×C53—22no—
78#123×C53—22no—
79#123×C53—22no—
80#123×C53—21no—
81#123×C53—21no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53—22no—
86C22⋊Q8×C53—22no—
87C22⋊Q8×C53—22no—
88C22⋊Q8×C53—21no—
89C22⋊Q8×C53—21no—
90C22⋊Q8×C53—21no—
91C22⋊Q8×C53D411yescharacteristic
92C22⋊Q8×C53D411yescharacteristic
93#121×C53—22no—
94#121×C53—21no—
95#121×C53D411yescharacteristic
96#121×C53D411yescharacteristic
97C4×Q8×C53—21no—
98C4×Q8×C53—21no—
99C4×Q8×C53—21no—
100C4×Q8×C53C2×C411yescharacteristic
101C4×Q8×C53C2×C411yescharacteristic
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53—21no—
105C4×D4×C53—21no—
106C4×D4×C53—21no—
107C4×D4×C53C2×C411yescharacteristic
108C4×D4×C53C2×C411yescharacteristic
109#117×C53—21no—
110#117×C53—21no—
111#117×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53D412yes—
115C2×C42×C53C2311yescharacteristic
116#41×C53—82no—
117C2×Q8×C53—22no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53—22no—
121C2×Q8×C53—82no—
122C2×Q8×C53—82no—
123C2×Q8×C53C22⋊C411yescharacteristic
124C2×Q8×C53C22⋊C411yescharacteristic
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—22no—
128C2×D4×C53C22⋊C411yescharacteristic
129C2×D4×C53C22⋊C411yescharacteristic
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×Dic53—41no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53C22⋊C411yescharacteristic
158C4⋊C4×C53C22⋊C412yes—
159C4⋊C4×C53C22⋊C412yes—
160C4⋊C4×C53C22⋊C411yescharacteristic
161C4⋊C4×C53C22⋊C411yescharacteristic
162C4⋊C4×C53C22⋊C411yescharacteristic
163C22⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—21no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—21no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C42×C53—22no—
178C42×C53—22no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53C22×C411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C4⋊Dic53—42no—
186C2×#88322—22no—
187C2×#88322—41no—
188C2×#88322—41no—
189C2×#88322—41no—
190C2×#88322—41no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53#12011yescharacteristic
194Q8×C53—42no—
195Q8×C53—42no—
196Q8×C53—41no—
197Q8×C53—41no—
198Q8×C53—82no—
199Q8×C53—82no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—41no—
204D4×C53—41no—
205D4×C53—41no—
206D4×C53—82no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—82no—
222C2×C4×C53—82no—
223C2×C4×C53—82no—
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53#12011yescharacteristic
228C2×C4×C53#12011yescharacteristic
229C2×C4×C53#12011yescharacteristic
230C2×Dic53—81no—
231C2×Dic53—81no—
232C2×Dic53—81no—
233#88322—42no—
234#88322—81no—
235#88322—81no—
236#8499—531nomaximal, Sylow
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—41no—
240C22×C53—41no—
241C22×C53—41no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53#37911yescharacteristic
246C22×C53#45411yescharacteristic
247C22×C53#45611yescharacteristic
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—41no—
255C4×C53—41no—
256C4×C53—41no—
257C4×C53—42no—
258C4×C53—42no—
259C4×C53—41no—
260C4×C53—41no—
261C4×C53—41no—
262Dic53—161no—
263#3552D5311yescharacteristic
264#1993—531no—
265#1556—531no—
266#1532—531no—
267#1532—531no—
268#1512—531no—
269#1512—531no—
270C2×C53—21no—
271C2×C53—21no—
272C2×C53—81no—
273C2×C53—81no—
274C2×C53#179311yescharacteristic
275C2×C53#179511yescharacteristic
276C2×C53#179611yescharacteristic
277#563—22no—
278#557Dic5311yescharacteristic, complemented
279#556Dic5311yescharacteristic, complemented
280#554—21no—
281#553—21no—
282D4×Q8—21no—
283D4×Q8Dic5311yescharacteristic, complemented
284#548—21no—
285#547Dic5311yescharacteristic, complemented
286#544—21no—
287#542—21no—
288#541—21no—
289#539—22no—
290#536—21no—
291C2×C4⋊Q8C2×D5311yescharacteristic
292#529C2×D5311yescharacteristic
293#529C2×D5311yescharacteristic
294#432—531no—
295#432—531no—
296C2×#97—531no—
297#382—531no—
298#340—1062no—
299#332—1062no—
300#330—1061no—
301#330—1061no—
302#329—1061no—
303#329—1061no—
304#328—1061no—
305#328—1061no—
306#327—1061no—
307#327—1061no—
308C53#849911yescharacteristic, complemented, Sylow
309C2×#41—21no—
310C22×Q8—21no—
311C22×Q8—21no—
312C22×C8—1061no—
313C4⋊Q8—22no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8C2×Dic5311yescharacteristic
319C4⋊Q8C2×Dic5311yescharacteristic
320C4⋊Q8C22⋊D5312yes—
321#126—22no—
322#126—22no—
323#126—22no—
324#126—21no—
325#126—21no—
326#125—22no—
327#125—22no—
328#125—22no—
329#125—21no—
330#125C22⋊D5311yescharacteristic
331#125C22⋊D5311yescharacteristic
332#124—22no—
333#124—21no—
334#124C22⋊D5311yescharacteristic
335#124C22⋊D5311yescharacteristic
336#123—22no—
337#123—22no—
338#123—21no—
339#123—21no—
340#123—22no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8—21no—
347C22⋊Q8—22no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8C22⋊D5311yescharacteristic
351C22⋊Q8C22⋊D5311yescharacteristic
352#121—22no—
353#121—21no—
354#121C22⋊D5311yescharacteristic
355#121C22⋊D5311yescharacteristic
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8C2×Dic5311yescharacteristic
360C4×Q8C2×Dic5311yescharacteristic
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—21no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4C2×Dic5311yescharacteristic
367C4×D4C2×Dic5311yescharacteristic
368#117—21no—
369#117—21no—
370#117—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4C22⋊D5312yes—
375C2×C42C22×D5311yescharacteristic
376C4⋊C8—1061no—
377C4⋊C8—1061no—
378C22⋊C8—1061no—
379C22⋊C8—1061no—
380#97—1061no—
381#97—1061no—
382#97—1061no—
383#97—1061no—
384#95—1061no—
385#95—1061no—
386#41—82no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×Q8C22⋊Dic5311yescharacteristic
394C2×Q8C22⋊Dic5311yescharacteristic
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4C22⋊Dic5311yescharacteristic
399C2×D4C22⋊Dic5311yescharacteristic
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—2121no—
406C22×C4C2×C22⋊D5311yescharacteristic
407C22×C4C2×C22⋊D5311yescharacteristic
408C22×C4C2×C22⋊D5311yescharacteristic
409C2×C8—1062no—
410C2×C8—2121no—
411C2×C8—2121no—
412C2×C8—2121no—
413C2×C8—2121no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—2122no—
433C4⋊C4C22⋊Dic5311yescharacteristic
434C4⋊C4C22⋊Dic5312yes—
435C4⋊C4C22⋊Dic5312yes—
436C4⋊C4C22⋊Dic5311yescharacteristic
437C4⋊C4C22⋊Dic5311yescharacteristic
438C4⋊C4C22⋊Dic5311yescharacteristic
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—21no—
453C42—22no—
454C42—22no—
455C42—21no—
456C42—21no—
457C42C22×Dic5311yescharacteristic
458C42C2×C22⋊D5311yescharacteristic
459C42C2×C22⋊D5311yescharacteristic
460C42C2×C22⋊D5311yescharacteristic
461C23—21no—
462C23—21no—
463C23#1313318611yescharacteristic
464Q8—41no—
465Q8—41no—
466Q8—42no—
467Q8—42no—
468Q8—82no—
469Q8—82no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—82no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—4241no—
495C2×C4—4241no—
496C2×C4—4241no—
497C2×C4C2×C22⋊Dic5311yescharacteristic
498C2×C4C2×C22⋊Dic5311yescharacteristic
499C2×C4C2×C22⋊Dic5311yescharacteristic
500C2×C4#1313318611yescharacteristic
501C2×C4#1313318611yescharacteristic
502C2×C4#1313318611yescharacteristic
503C8—4241no—
504C8—4241no—
505C8—2122no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22#1681047911yescharacteristic
515C22#1681079611yescharacteristic
516C22#1681080611yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—41no—
531C4—8481no—
532C2—21no—
533C2—21no—
534C2—81no—
535C2—81no—
536C2#2027467411yescharacteristic
537C2#2027468511yescharacteristic
538C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3552C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3552D53
hypocenterC53#8499
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3552

Complements

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

NH ≅ G/Ntype
1. #20275689C2semidirect
2. #20273432C2semidirect
3. #20273335C2semidirect
4. #20273335C2semidirect
5. #20273253C2semidirect
6. #20273253C2semidirect
23. C53×#557C4semidirect
24. C53×#556C4semidirect
28. D4×Q8×C53C4semidirect
30. C53×#547C4semidirect
278. #557Dic53semidirect
279. #556Dic53semidirect
283. D4×Q8Dic53semidirect
285. #547Dic53semidirect
308. C53#8499semidirect

Maximal subgroups

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

subgroupindex
#202756892
#202734322
#202733352
#202732532
C53×#35522
#849953