Finite Group Databasebeta

← #27351935

Subgroups — #27351935

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288567C211yesmaximal, characteristic
2#20283744C211yesmaximal, characteristic, complemented
3#20273269C212yesmaximal, complemented
4#20273265C211yesmaximal, characteristic, complemented
5#20273261C211yesmaximal, characteristic, complemented
6#20273256C212yesmaximal, complemented
7#20273226C211yesmaximal, characteristic, complemented
8#20273222C211yesmaximal, characteristic, complemented
9#20273217C212yesmaximal, complemented
10#20273204C212yesmaximal, complemented
11C53×#2353C211yesmaximal, characteristic, complemented
12#16811356—22no—
13#16811343C412yescomplemented
14#16811338C412yescomplemented
15#16811323—22no—
16#16811294C412yescomplemented
17#16811278C412yescomplemented
18#16811211—21no—
19#16811211—21no—
20#16811190—21no—
21#16811180C2211yescharacteristic
22#16811173C2211yescharacteristic
23#16811134C2212yes—
24#16811134C2212yes—
25#16811104—21no—
26#16810695—22no—
27#16810695—22no—
28#16810680C2211yescharacteristic
29#16810680C2211yescharacteristic
30#16810587C2212yes—
31C2×C22⋊#88322C2212yes—
32#16810359C2212yes—
33#16810359C2212yes—
34#16810359C2212yes—
35#16810359C2212yes—
36#16810356C2212yes—
37#16810356C2212yes—
38#16810356C2212yes—
39#16810356C2212yes—
40C2×C4×D4×C53C2211yescharacteristic
41C53×#388—21no—
42C53×#387—21no—
43C53×#386—21no—
44C53×#383C2211yescharacteristic
45C53×#381C2211yescharacteristic
46C53×#380C2211yescharacteristic
47C4×C22⋊C4×C53—21no—
48C4×C22⋊C4×C53C2211yescharacteristic
49C53×#376C2211yescharacteristic
50C2×#95×C53C2211yescharacteristic
51C2×#11719945—21no—
52C2×#11719945C2×C412yes—
53C2×#11719945C2×C412yescomplemented
54C2×#11719917—22no—
55C22×#88322—22no—
56#13133217C2×C411yescharacteristic, complemented
57#13133217C2×C411yescharacteristic, complemented
58#13133217C2×C411yescharacteristic, complemented
59#13133214C2×C411yescharacteristic, complemented
60#13133213—22no—
61#13133210—22no—
62#13133210—22no—
63#13133209—21no—
64#13133205—21no—
65#13133204—22no—
66#13133203—22no—
67#13133203—22no—
68#13133200—22no—
69#13133200—22no—
70#13133199—22no—
71#13133199—22no—
72#13133196—21no—
73#13133196C2×C412yes—
74#13133196C2×C412yescomplemented
75#13133196C2312yes—
76#13133191—21no—
77#13133191C2×C412yes—
78#13133191C2×C412yescomplemented
79#13133191C2312yes—
80#13133190—22no—
81#13133190—22no—
82#13133188—22no—
83#13133188—22no—
84#13133186—22no—
85#13133186—22no—
86C4×Dic106—21no—
87C4×Dic106C2×C412yescomplemented
88#13133182—22no—
89C4×D212—21no—
90C4×D212C2×C412yescomplemented
91#13133179—22no—
92C4×C22⋊D53—22no—
93C4×C22⋊D53—22no—
94C4×C22⋊D53—22no—
95C4×C22⋊D53—22no—
96#13133176—21no—
97#13133176C2×C412yes—
98#13133176C2×C412yescomplemented
99#13133175—22no—
100C4⋊C4×D53—22no—
101C4⋊#88322—21no—
102C4⋊#88322—21no—
103C22⋊#88322—22no—
104C22⋊#88322—22no—
105C22⋊#88322C2312yes—
106C22⋊#88322C2312yes—
107#13133128—21no—
108#13133128—21no—
109C22×D4×C53D411yescharacteristic
110C23×C4×C53—21no—
111C23×C4×C53C2311yescharacteristic
112C23×C4×C53C2311yescharacteristic
113C4×D4×C53C2×C412yes—
114C4×D4×C53C2×C412yes—
115C4×D4×C53D412yes—
116C4×D4×C53D412yes—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53—21no—
125C2×C42×C53—21no—
126C2×C42×C53C2311yescharacteristic
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132#95×C53C2311yescharacteristic
133#95×C53C2311yescharacteristic
134#95×C53C2311yescharacteristic
135#95×C53C2311yescharacteristic
136C24×C53C2×D411yescharacteristic
137C24×C53C2×D411yescharacteristic
138C2×D4×C53—41no—
139C2×D4×C53—41no—
140C2×D4×C53C22⋊C412yes—
141C2×D4×C53C22⋊C412yes—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53—81no—
149C22×C4×C53—81no—
150C22×C4×C53—81no—
151C22×C4×C53C22×C412yes—
152C22×C4×C53C22×C412yes—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D412yes—
158C22×C4×C53C2×D412yes—
159C22×C4×C53C2411yescharacteristic
160C2×C22⋊D53—22no—
161C2×C22⋊D53—22no—
162C2×C22⋊D53—22no—
163C2×C22⋊D53—22no—
164C2×C22⋊D53—22no—
165#11719945—42no—
166#11719945—42no—
167#11719945—82no—
168#11719945—82no—
169C2×D212—21no—
170C2×D212—21no—
171C2×D212C22×C412yes—
172C2×C4×D53—22no—
173C2×C4×D53—22no—
174C2×C4×D53—22no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106C22×C412yes—
178C4⋊C4×C53—81no—
179C4⋊C4×C53C22⋊C412yes—
180C4⋊C4×C53C22⋊C412yes—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—81no—
184C22⋊C4×C53C22⋊C412yes—
185C22⋊C4×C53C22⋊C412yes—
186C22⋊C4×C53C22⋊C412yes—
187C22⋊C4×C53C22⋊C412yes—
188C42×C53—42no—
189C42×C53—81no—
190C42×C53C22×C411yescharacteristic
191C42×C53C22×C411yescharacteristic
192C42×C53C2×D411yescharacteristic
193C42×C53C2×D411yescharacteristic
194C22⋊Dic53—22no—
195C22⋊Dic53—22no—
196C22⋊Dic53—22no—
197C22⋊Dic53—22no—
198C22⋊Dic53—22no—
199#11719921—22no—
200#11719921—22no—
201#11719921—22no—
202#11719921—22no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53C22×C412yes—
208C4⋊Dic53C22×C412yes—
209#11719919—22no—
210#11719919—22no—
211#11719919—22no—
212#11719919—22no—
213C4×Dic53—22no—
214C4×Dic53—22no—
215C4×Dic53—22no—
216#11719917—82no—
217C2×#88322—22no—
218C2×#88322—22no—
219C2×#88322—82no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53C2×C22⋊C412yes—
229C23×C53C2×C22⋊C412yes—
230C23×C53C22×D411yescharacteristic
231C22×D53—22no—
232D4×C53—42no—
233D4×C53—42no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—84no—
257C2×C4×C53C2×C22⋊C411yescharacteristic
258C2×C4×C53C2×C22⋊C411yescharacteristic
259C2×C4×C53C2×C22⋊C411yescharacteristic
260C2×C4×C53C2×C22⋊C412yes—
261C2×C4×C53C2×C22⋊C412yes—
262C2×C4×C53C2×C22⋊C411yescharacteristic
263C2×C4×C53C23×C411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C22⋊D53—42no—
266C22⋊D53—42no—
267C22⋊D53—82no—
268C22⋊D53—82no—
269C22⋊D53—82no—
270C22⋊D53—82no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274D212—42no—
275D212—42no—
276D212—81no—
277C4×D53—44no—
278C4×D53—82no—
279C4×D53—82no—
280Dic106—42no—
281Dic106—42no—
282Dic106—81no—
283#88322—42no—
284#88322—42no—
285#17629—531nomaximal, Sylow
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53C22×C22⋊C411yescharacteristic
299C22×C53#57511yescharacteristic
300C22×C53#57511yescharacteristic
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C4×C53—21no—
305C4×C53—21no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—21no—
309C4×C53—21no—
310C4×C53—21no—
311C4×C53—21no—
312C4×C53—42no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53—81no—
316Dic53—42no—
317Dic53—42no—
318Dic53—42no—
319#3506—531no—
320#2973—531no—
321#2353D5311yescharacteristic, complemented
322#1516—532no—
323#1515—531no—
324#1514—531no—
325#1513—532no—
326#1506—531no—
327#1505—531no—
328#1504—532no—
329#1501—532no—
330C2×C53—21no—
331C2×C53—21no—
332C2×C53—42no—
333C2×C53—42no—
334C2×C53#288311yescharacteristic
335C2×C53#289611yescharacteristic
336C2×C53#290111yescharacteristic
337D53—82no—
338#559—1062no—
339#556—532no—
340#555—532no—
341#553—1062no—
342#550—532no—
343#547—532no—
344#538—1061no—
345#538—1061no—
346#535—1061no—
347#533—531no—
348#532—531no—
349#525—532no—
350#525—532no—
351#520—1061no—
352C2×C4×D4C2×D5311yescharacteristic
353#435—1062no—
354#435—1062no—
355#432—531no—
356#432—531no—
357#407—532no—
358C2×C22⋊C8—532no—
359#388—21no—
360#387—21no—
361#386—21no—
362#383C2×D5311yescharacteristic
363#381C2×D5311yescharacteristic
364#380C2×D5311yescharacteristic
365C4×C22⋊C4—21no—
366C4×C22⋊C4C2×D5311yescharacteristic
367#376C2×D5311yescharacteristic
368C2×#95C2×D5311yescharacteristic
369#328—532no—
370#328—532no—
371#328—532no—
372#328—532no—
373#327—532no—
374#327—532no—
375#327—532no—
376#327—532no—
377C53#1762911yescharacteristic, complemented, Sylow
378C2×#41—532no—
379C2×#41—532no—
380C2×#41—1061no—
381C22×D4C22⋊D5311yescharacteristic
382C23×C4—21no—
383C23×C4C22×D5311yescharacteristic
384C23×C4C22×D5311yescharacteristic
385C2×M16—1062no—
386C22×C8—1062no—
387C4⋊Q8—531no—
388C4⋊Q8—531no—
389C4⋊Q8—531no—
390#127—531no—
391#127—1062no—
392#126—1062no—
393#126—1062no—
394#125—1061no—
395#124—1062no—
396#124—1062no—
397#124—1062no—
398#124—1061no—
399#123—1062no—
400#123—1062no—
401#123—1062no—
402#123—1062no—
403C22⋊Q8—532no—
404C22⋊Q8—532no—
405C22⋊Q8—532no—
406C22⋊Q8—1061no—
407#121—532no—
408#121—532no—
409#121—532no—
410#121—1062no—
411#121—1062no—
412#121—1061no—
413#121—1062no—
414#121—1062no—
415#120—1062no—
416#120—1062no—
417C4×Q8—532no—
418C4×Q8—1062no—
419C4×Q8—1061no—
420C4×D4—532no—
421C4×D4—1062no—
422C4×D4—1062no—
423C4×D4—1062no—
424C4×D4—1062no—
425C4×D4—1062no—
426C4×D4—1061no—
427C4×D4C4×D5312yes—
428C4×D4C4×D5312yes—
429C4×D4D21212yes—
430C4×D4D21212yes—
431#117—532no—
432#117—532no—
433#117—1061no—
434#117—1062no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—1062no—
437C2×C4⋊C4C22⋊D5311yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4C22⋊D5311yescharacteristic
442C2×C22⋊C4C22⋊D5311yescharacteristic
443C2×C42—21no—
444C2×C42—21no—
445C2×C42C22×D5311yescharacteristic
446C4⋊C8—1061no—
447C4⋊C8—1061no—
448C22⋊C8—532no—
449C22⋊C8—532no—
450C22⋊C8—1062no—
451C22⋊C8—1062no—
452#97—1061no—
453#97—1061no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95C22×D5311yescharacteristic
460#95C22×D5311yescharacteristic
461#95C22×D5311yescharacteristic
462#95C22×D5311yescharacteristic
463C24C2×C22⋊D5311yescharacteristic
464C24C2×C22⋊D5311yescharacteristic
465#41—4242no—
466#41—4242no—
467#41—2122no—
468#41—2122no—
469C2×Q8—532no—
470C2×Q8—1061no—
471C2×Q8—1061no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—532no—
475C2×D4—1062no—
476C2×D4—1062no—
477C2×D4—1062no—
478C2×D4—1062no—
479C2×D4—1062no—
480C2×D4—1061no—
481C2×D4—1061no—
482C2×D4#1171992112yes—
483C2×D4#1171992112yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4C2×C4×D5312yes—
497C22×C4C2×C4×D5312yes—
498C22×C4C2×D21212yes—
499C22×C4C2×D21212yes—
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C23×D5311yescharacteristic
505M16—4242no—
506C2×C8—1062no—
507C2×C8—1062no—
508C2×C8—4242no—
509C4⋊C4—81no—
510C4⋊C4—532no—
511C4⋊C4—532no—
512C4⋊C4—1061no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1062no—
518C4⋊C4—1061no—
519C4⋊C4—1062no—
520C4⋊C4#1171992112yes—
521C4⋊C4#1171992112yes—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—1062no—
526C22⋊C4—1062no—
527C22⋊C4—1062no—
528C22⋊C4—1062no—
529C22⋊C4—1062no—
530C22⋊C4—1062no—
531C22⋊C4—1062no—
532C22⋊C4—1062no—
533C22⋊C4—1062no—
534C22⋊C4#1171992112yes—
535C22⋊C4#1171992112yes—
536C22⋊C4#1171992112yes—
537C22⋊C4#1171992112yes—
538C42—42no—
539C42—81no—
540C42—1062no—
541C42—1062no—
542C42—1062no—
543C42C2×C4×D5311yescharacteristic
544C42C2×C4×D5311yescharacteristic
545C42C2×D21211yescharacteristic
546C42C2×D21211yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—1062no—
556C23C2×#1171992112yes—
557C23C2×#1171992112yes—
558C23C22×C22⋊D5311yescharacteristic
559Q8—4241no—
560Q8—2122no—
561Q8—2122no—
562D4—42no—
563D4—42no—
564D4—4241no—
565D4—4242no—
566D4—4242no—
567D4—4242no—
568D4—4242no—
569D4—2122no—
570D4—2122no—
571D4—2122no—
572D4—2122no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—4242no—
600C2×C4—4242no—
601C2×C4—2124no—
602C2×C4C2×#1171992111yescharacteristic
603C2×C4C2×#1171992112yes—
604C2×C4C2×#1171992112yes—
605C2×C4C2×#1171992111yescharacteristic
606C2×C4C2×#1171992111yescharacteristic
607C2×C4C2×#1171992111yescharacteristic
608C2×C4C22×C4×D5311yescharacteristic
609C2×C4C22×D21211yescharacteristic
610C8—2122no—
611C8—2122no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—2122no—
625C22—2122no—
626C22—2122no—
627C22C22×#1171992111yescharacteristic
628C22#1681142111yescharacteristic
629C22#1681142111yescharacteristic
630C4—21no—
631C4—21no—
632C4—22no—
633C4—22no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—2122no—
643C4—2122no—
644C4—2122no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—4242no—
650C2#2028317511yescharacteristic
651C2#2028323211yescharacteristic
652C2#2028326311yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2353C2
socleC22×C53C22×C22⋊C4
prime radical#20288567C2
cosocleC22×C4×C53C24
hypercenter#2353D53
hypocenterC53#17629
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2353

Complements

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

NH ≅ G/Ntype
2. #20283744C2semidirect
3. #20273269C2semidirect
4. #20273265C2semidirect
5. #20273261C2semidirect
6. #20273256C2semidirect
7. #20273226C2semidirect
8. #20273222C2semidirect
9. #20273217C2semidirect
10. #20273204C2semidirect
11. C53×#2353C2semidirect
13. #16811343C4semidirect
14. #16811338C4semidirect
16. #16811294C4semidirect
17. #16811278C4semidirect
53. C2×#11719945C2×C4semidirect
56. #13133217C2×C4semidirect
57. #13133217C2×C4semidirect
58. #13133217C2×C4semidirect
59. #13133214C2×C4semidirect
74. #13133196C2×C4semidirect
78. #13133191C2×C4semidirect
87. C4×Dic106C2×C4semidirect
90. C4×D212C2×C4semidirect
98. #13133176C2×C4semidirect
321. #2353D53semidirect
377. C53#17629semidirect

Maximal subgroups

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

subgroupindex
#202885672
#202837442
#202732692
#202732652
#202732612
#202732562
#202732262
#202732222
#202732172
#202732042
C53×#23532
#1762953