Finite Group Databasebeta

← #27396289

Subgroups — #27396289

Order 13568.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810912C211yesmaximal, characteristic
2#20277462C211yesmaximal, characteristic, complemented
3#20277354C211yesmaximal, characteristic
4#20275665C211yesmaximal, characteristic, complemented
5#20275480C211yesmaximal, characteristic, complemented
6#20275440C211yesmaximal, characteristic, complemented
7#20275412C211yesmaximal, characteristic
8#20275397C211yesmaximal, characteristic
9#20275340C211yesmaximal, characteristic
10#20275334C211yesmaximal, characteristic
11#20275302C211yesmaximal, characteristic, complemented
12#20275296C211yesmaximal, characteristic, complemented
13#20274724C211yesmaximal, characteristic
14#20274575C211yesmaximal, characteristic, complemented
15C53×#1944C211yesmaximal, characteristic, complemented
16C2×#13133216C2211yescharacteristic
17C2×#13133179C2211yescharacteristic
18#16810912C418yescomplemented
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810528—21no—
29#16810527—21no—
30#16810525C2211yescharacteristic
31#16810524C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810515—21no—
35#16810513—21no—
36#16810505—21no—
37#16810504—21no—
38#16810488C2211yescharacteristic
39#16810485—21no—
40#16810480—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44#16810469C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106—21no—
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133216—22no—
66#13133216C2×C414yes—
67#13133179—22no—
68#13133179C2×C414yes—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85#13133148—41no—
86#13133148C2×C414yes—
87#13133146—22no—
88#13133146C2×C414yes—
89#13133139—41no—
90#13133139C2×C414yes—
91#13133137—41no—
92#13133137C2×C414yes—
93C4×#88322—41no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124D411yescharacteristic
102#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114#107×C53—41no—
115#106×C53C2×C414yes—
116#97×C53—41no—
117#95×C53—21no—
118#95×C53—21no—
119C23×D53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53C2×D411yescharacteristic
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212C22×C412yes—
140C2×C4×D53—24no—
141C2×C4×D53—81no—
142C2×C4×D53—81no—
143C2×Dic106—24no—
144C2×Dic106—81no—
145C2×Dic106—81no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53C22×C412yes—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—81no—
159C4⋊C4×C53—81no—
160C4⋊C4×C53C22×C412yes—
161C4⋊C4×C53C22×C412yes—
162C42×C53—21no—
163C42×C53—21no—
164#11719921—24no—
165#11719921—81no—
166#11719921—81no—
167#11719921—81no—
168#11719921—81no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—81no—
175C4⋊Dic53C22×C412yes—
176#11719919—24no—
177#11719919—81no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—81no—
181C4×Dic53—81no—
182C4×Dic53—81no—
183C4×Dic53C22×C412yes—
184C4×Dic53#4112yes—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322C22×C412yes—
191C23×C53C22×D411yescharacteristic
192C22×D53—22no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53C23×C411yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C53H(5,𝔽2)11yescharacteristic
220C2×C4×C532−1+411yescharacteristic
221C2×C4×C532−1+411yescharacteristic
222C2×C4×C532−1+411yescharacteristic
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—22no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53C4×D412yes—
243C2×Dic53C4×D412yes—
244D212—22no—
245D212—22no—
246D212—41no—
247D212—41no—
248C4×D53—84no—
249Dic106—84no—
250C8×C53—22no—
251C8×C53—41no—
252#88322—22no—
253#88322—41no—
254#22123—531nomaximal, Sylow
255C22×C53C2×C4×D411yescharacteristic
256C22×C53#53611yescharacteristic
257C22×C53#53711yescharacteristic
258C22×C53#57611yescharacteristic
259C22×C53#57711yescharacteristic
260C22×C53#57711yescharacteristic
261C22×C53#57711yescharacteristic
262C2×D53—42no—
263C2×D53—42no—
264C2×D53—42no—
265C2×D53—81no—
266C2×D53—81no—
267C2×D53—81no—
268C2×D53—81no—
269C4×C53—42no—
270C4×C53—42no—
271C4×C53—41no—
272C4×C53—41no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53#51911yescharacteristic
276C4×C53#52011yescharacteristic
277C4×C53#54111yescharacteristic
278C4×C53#54211yescharacteristic
279Dic53—22no—
280Dic53—22no—
281Dic53—42no—
282Dic53—41no—
283Dic53—41no—
284Dic53—81no—
285Dic53—81no—
286C2×#476—531no—
287#2331—531no—
288#2319—531no—
289#1988—531no—
290#1949—531no—
291#1944D5311yescharacteristic, complemented
292#1941—531no—
293#1936—531no—
294#1934—531no—
295#1926—531no—
296#1925—531no—
297#1920—531no—
298#1919—531no—
299#1804—531no—
300#1768—531no—
301C2×C53#232311yescharacteristic
302C2×C53#294611yescharacteristic
303C2×C53#294911yescharacteristic
304C2×C53#324111yescharacteristic
305C2×C53#324411yescharacteristic
306C2×C53#325711yescharacteristic
307C2×C53#326311yescharacteristic
308D53—82no—
309C2×C4⋊Q8—531no—
310C2×C4×D4—531no—
311#476—538no—
312C2×#107C2×D5311yescharacteristic
313C2×#106—531no—
314C2×#106—531no—
315C2×#106C2×D5311yescharacteristic
316C2×C4⋊C8—531no—
317C2×C4⋊C8—531no—
318C2×#102—531no—
319C2×#102—531no—
320C2×#102—531no—
321C2×#102—531no—
322C2×#97C2×D5311yescharacteristic
323C2×C4×C8—531no—
324#391—1061no—
325#391—1061no—
326#390—531no—
327#390—531no—
328#389—531no—
329#389—531no—
330#389C2×D5311yescharacteristic
331#389C2×D5311yescharacteristic
332#388—1061no—
333#388—1061no—
334#387—1061no—
335#387—1061no—
336#382—531no—
337#381—1061no—
338#380—1061no—
339C4×C4⋊C4—531no—
340C4×C4⋊C4—531no—
341C4×C4⋊C4—531no—
342#376—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340C2×D5311yescharacteristic
350#340C2×D5311yescharacteristic
351#336—531no—
352#336—531no—
353C53#2212311yescharacteristic, complemented, Sylow
354C22×Q8—1061no—
355C22×D4—531no—
356C23×C4—1061no—
357C22×C8—531no—
358C22×C8—531no—
359C22×C8C22×D5311yescharacteristic
360C22×C8C22×D5311yescharacteristic
361C4⋊Q8—534no—
362C4⋊Q8—1062no—
363C4×D4—534no—
364C4×D4—1062no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4C22×D5311yescharacteristic
374C2×C4⋊C4C22×D5311yescharacteristic
375C2×C4⋊C4C22×D5311yescharacteristic
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C22⋊C4—1061no—
378C2×C22⋊C4—1061no—
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—1061no—
381C2×C22⋊C4—1061no—
382C2×C42—531no—
383C2×C42—531no—
384C2×C42—1061no—
385C2×C42—1061no—
386C2×C42—1061no—
387C2×C42C22×D5311yescharacteristic
388#107—41no—
389#106—534no—
390#106—2121no—
391#106C4×D5314yes—
392C4⋊C8—534no—
393C4⋊C8—1062no—
394#102—534no—
395#102—534no—
396#102—2121no—
397#102—2121no—
398#97—41no—
399C4×C8—2121no—
400#95—21no—
401#95—21no—
402#95—531no—
403#95—531no—
404#95—531no—
405#95—531no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413C24—1061no—
414C2×Q8—1064no—
415C2×Q8—4241no—
416C2×Q8—4241no—
417C2×D4—532no—
418C2×D4—1061no—
419C2×D4—1061no—
420C2×D4—1061no—
421C2×D4—1061no—
422C2×D4—1061no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—531no—
431C22×C4—1064no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—4241no—
436C22×C4—4241no—
437C22×C4D4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4C23×D5311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—532no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8C2×C4×D5312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—532no—
461C4⋊C4—1064no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—4241no—
468C4⋊C4—4241no—
469C4⋊C4C2×C4×D5312yes—
470C4⋊C4C2×C4×D5312yes—
471C22⋊C4—1064no—
472C22⋊C4—4241no—
473C22⋊C4—4241no—
474C22⋊C4—4241no—
475C22⋊C4—4241no—
476C42—21no—
477C42—21no—
478C42—532no—
479C42—532no—
480C42—1061no—
481C42—1061no—
482C42—4241no—
483C42—4241no—
484C42—4241no—
485C23—1062no—
486C23—2121no—
487C23—2121no—
488C23—2121no—
489C23—2121no—
490C23—2121no—
491C23—2121no—
492C23C2×D4×D5311yescharacteristic
493Q8—4244no—
494D4—1062no—
495D4—1062no—
496D4—2121no—
497D4—2121no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—532no—
517C2×C4—532no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1062no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—4244no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4C22×C4×D5311yescharacteristic
539C2×C4#41×D5311yescharacteristic
540C2×C4#1313326211yescharacteristic
541C2×C4#1313326411yescharacteristic
542C2×C4#1313326511yescharacteristic
543C2×C4#1313326511yescharacteristic
544C8—22no—
545C8—41no—
546C8—1062no—
547C8—2121no—
548C22—4241no—
549C22—4241no—
550C22—4241no—
551C22—4241no—
552C22—2122no—
553C22—2122no—
554C22—2122no—
555C22C4×D4×D5311yescharacteristic
556C22#1681119811yescharacteristic
557C22#1681120011yescharacteristic
558C22#1681142711yescharacteristic
559C22#1681143211yescharacteristic
560C22#1681143211yescharacteristic
561C22#1681143311yescharacteristic
562C4—42no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—1062no—
569C4—1062no—
570C4—4241no—
571C4—4241no—
572C4—2121no—
573C4—2121no—
574C4—2122no—
575C4#1681109911yescharacteristic
576C4#1681110411yescharacteristic
577C4#1681123211yescharacteristic
578C4#1681124311yescharacteristic
579C2—4242no—
580C2#2027739211yescharacteristic
581C2#2028354611yescharacteristic
582C2#2028357111yescharacteristic
583C2#2028596911yescharacteristic
584C2#2028601411yescharacteristic
585C2#2028612711yescharacteristic
586C2#2028618111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1944C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1944D53
hypocenterC53#22123
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
2. #20277462C2semidirect
4. #20275665C2semidirect
5. #20275480C2semidirect
6. #20275440C2semidirect
11. #20275302C2semidirect
12. #20275296C2semidirect
14. #20274575C2semidirect
15. C53×#1944C2semidirect
18. #16810912C4semidirect
291. #1944D53semidirect
353. C53#22123semidirect

Maximal subgroups

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

subgroupindex
C2×#168109122
#202774622
#202773542
#202756652
#202754802
#202754402
#202754122
#202753972
#202753402
#202753342
#202753022
#202752962
#202747242
#202745752
C53×#19442
#2212353