Finite Group Databasebeta

← #27396352

Subgroups — #27396352

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×#16810904C211yesmaximal, characteristic
2#20277461C211yesmaximal, characteristic, complemented
3#20277353C211yesmaximal, characteristic
4#20275663C211yesmaximal, characteristic, complemented
5#20275451C211yesmaximal, characteristic, complemented
6#20275446C211yesmaximal, characteristic, complemented
7#20275410C211yesmaximal, characteristic
8#20275396C211yesmaximal, characteristic
9#20275341C211yesmaximal, characteristic
10#20275334C211yesmaximal, characteristic
11#20275306C211yesmaximal, characteristic, complemented
12#20275285C211yesmaximal, characteristic, complemented
13#20274722C211yesmaximal, characteristic
14#20274574C211yesmaximal, characteristic, complemented
15C53×#1949C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18#16810904C418yescomplemented
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133139C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133137C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133127C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810527—21no—
29#16810527—21no—
30#16810524C2211yescharacteristic
31#16810524C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810514—21no—
37#16810514—21no—
38#16810508—21no—
39#16810508—21no—
40#16810485—21no—
41#16810482—21no—
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44#16810469C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C53×#382C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106—21no—
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133217—22no—
66#13133217C2×C414yes—
67C4×D212—22no—
68C4×D212C2×C414yes—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133151C2×C414yes—
87#13133151C2×C414yes—
88C4⋊#88322—22no—
89#13133139C2×C414yes—
90#13133139C2×C414yes—
91#13133137—41no—
92#13133137—41no—
93#13133127—41no—
94C8×Dic53—41no—
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×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C42×C53—21no—
112C2×C42×C53C2311yescharacteristic
113#107×C53—41no—
114#106×C53—41no—
115C4⋊C8×C53C2×C414yes—
116#95×C53—21no—
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×C53C22×C412yes—
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—21no—
159C4⋊C4×C53—21no—
160C42×C53—81no—
161C42×C53C22×C412yes—
162C42×C53#4112yes—
163#11719921—24no—
164#11719921—81no—
165#11719921—81no—
166#11719921—81no—
167#11719921—81no—
168C4⋊Dic53—24no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53C22×C412yes—
173C4⋊Dic53C22×C412yes—
174C4⋊Dic53C22×C412yes—
175#11719919—81no—
176#11719919—81no—
177#11719919—81no—
178#11719919—81no—
179C4×Dic53—21no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—21no—
183C4×Dic53—81no—
184C4×Dic53—81no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
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—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
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×C53C4×D412yes—
216C2×C4×C53C4×D412yes—
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—22no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244D212—22no—
245D212—22no—
246D212—41no—
247D212—41no—
248C4×D53—84no—
249Dic106—84no—
250C8×C53—22no—
251C8×C53—22no—
252#88322—41no—
253#88322—41no—
254#22127—531nomaximal, Sylow
255C22×C53C2×C4×D411yescharacteristic
256C22×C53#53611yescharacteristic
257C22×C53#53711yescharacteristic
258C22×C53#57611yescharacteristic
259C22×C53#57611yescharacteristic
260C22×C53#57711yescharacteristic
261C22×C53#57811yescharacteristic
262C2×D53—42no—
263C2×D53—42no—
264C2×D53—42no—
265C2×D53—81no—
266C2×D53—81no—
267C2×D53—81no—
268C2×D53—81no—
269C4×C53—22no—
270C4×C53—22no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53#51911yescharacteristic
276C4×C53#52011yescharacteristic
277C4×C53#54111yescharacteristic
278C4×C53#54211yescharacteristic
279Dic53—42no—
280Dic53—42no—
281Dic53—42no—
282Dic53—41no—
283Dic53—41no—
284Dic53—41no—
285Dic53—41no—
286C2×#474—531no—
287#2331—531no—
288#2319—531no—
289#1987—531no—
290#1949D5311yescharacteristic, complemented
291#1943—531no—
292#1942—531no—
293#1936—531no—
294#1934—531no—
295#1926—531no—
296#1925—531no—
297#1921—531no—
298#1918—531no—
299#1804—531no—
300#1768—531no—
301C2×C53#232311yescharacteristic
302C2×C53#294711yescharacteristic
303C2×C53#294811yescharacteristic
304C2×C53#324111yescharacteristic
305C2×C53#324311yescharacteristic
306C2×C53#325711yescharacteristic
307C2×C53#326211yescharacteristic
308D53—82no—
309C2×C4⋊Q8—531no—
310C2×C4×D4—531no—
311#474—538no—
312C2×#107—531no—
313C2×#107—531no—
314C2×#107C2×D5311yescharacteristic
315C2×#106C2×D5311yescharacteristic
316C2×C4⋊C8—531no—
317C2×C4⋊C8C2×D5311yescharacteristic
318C2×#102—531no—
319C2×#102—531no—
320C2×#102—531no—
321C2×#102—531no—
322C2×#97—531no—
323C2×C4×C8—531no—
324#391—1061no—
325#391—1061no—
326#390—531no—
327#390—531no—
328#389—531no—
329#389—531no—
330#389—531no—
331#389—531no—
332#388—1061no—
333#388—1061no—
334#387—1061no—
335#387—1061no—
336#382C2×D5311yescharacteristic
337#381—1061no—
338#380—1061no—
339C4×C4⋊C4—531no—
340C4×C4⋊C4—531no—
341C4×C4⋊C4C2×D5311yescharacteristic
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#2212711yescharacteristic, 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—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4—1061no—
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—21no—
383C2×C42—531no—
384C2×C42—531no—
385C2×C42—1061no—
386C2×C42—1061no—
387C2×C42C22×D5311yescharacteristic
388#107—41no—
389#107—534no—
390#107—534no—
391#106—41no—
392C4⋊C8—1062no—
393C4⋊C8C4×D5314yes—
394#102—534no—
395#102—534no—
396#102—2121no—
397#102—2121no—
398#97—2121no—
399C4×C8—2121no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—531no—
404#95—531no—
405#95—531no—
406#95—531no—
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—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1064no—
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—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8C2×C4×D5312yes—
451C2×C8C2×C4×D5312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—532no—
461C4⋊C4—532no—
462C4⋊C4—532no—
463C4⋊C4—1064no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—4241no—
468C4⋊C4—4241no—
469C4⋊C4—4241no—
470C4⋊C4—4241no—
471C22⋊C4—1064no—
472C22⋊C4—4241no—
473C22⋊C4—4241no—
474C22⋊C4—4241no—
475C22⋊C4—4241no—
476C42—81no—
477C42—1061no—
478C42—1061no—
479C42—1061no—
480C42—1061no—
481C42—4241no—
482C42—4241no—
483C42C2×C4×D5312yes—
484C42#1171994912yes—
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—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
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—1062no—
515C2×C4—1062no—
516C2×C4—1062no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
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#1313317912yes—
537C2×C4#1313317912yes—
538C2×C4C22×C4×D5311yescharacteristic
539C2×C4C2×#1171994911yescharacteristic
540C2×C4#1313326211yescharacteristic
541C2×C4#1313326411yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326511yescharacteristic
544C8—22no—
545C8—22no—
546C8—2121no—
547C8—2121no—
548C22—4241no—
549C22—4241no—
550C22—4241no—
551C22—4241no—
552C22—2122no—
553C22—2122no—
554C22—2122no—
555C22C2×#1313317911yescharacteristic
556C22#1681119611yescharacteristic
557C22#1681120211yescharacteristic
558C22#1681142511yescharacteristic
559C22#1681142511yescharacteristic
560C22#1681143311yescharacteristic
561C22#1681143911yescharacteristic
562C4—22no—
563C4—22no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—2121no—
569C4—2121no—
570C4—2121no—
571C4—2122no—
572C4—2122no—
573C4—2122no—
574C4—2121no—
575C4#1681109911yescharacteristic
576C4#1681110211yescharacteristic
577C4#1681123611yescharacteristic
578C4#1681124711yescharacteristic
579C2—4242no—
580C2#2027739511yescharacteristic
581C2#2028355611yescharacteristic
582C2#2028356611yescharacteristic
583C2#2028596511yescharacteristic
584C2#2028600511yescharacteristic
585C2#2028612311yescharacteristic
586C2#2028617411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1949C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1949D53
hypocenterC53#22127
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. #20277461C2semidirect
4. #20275663C2semidirect
5. #20275451C2semidirect
6. #20275446C2semidirect
11. #20275306C2semidirect
12. #20275285C2semidirect
14. #20274574C2semidirect
15. C53×#1949C2semidirect
18. #16810904C4semidirect
290. #1949D53semidirect
353. C53#22127semidirect

Maximal subgroups

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

subgroupindex
C2×#168109042
#202774612
#202773532
#202756632
#202754512
#202754462
#202754102
#202753962
#202753412
#202753342
#202753062
#202752852
#202747222
#202745742
C53×#19492
#2212753