Finite Group Databasebeta

← #28317224

Subgroups — #28317224

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291516C211yesmaximal, characteristic, complemented
2#20291457C211yesmaximal, characteristic
3#20290352C211yesmaximal, characteristic, complemented
4#20290350C211yesmaximal, characteristic, complemented
5#20290325C211yesmaximal, characteristic, complemented
6#20290323C211yesmaximal, characteristic, complemented
7#20290319C211yesmaximal, characteristic, complemented
8#20290318C211yesmaximal, characteristic, complemented
9#20290300C211yesmaximal, characteristic, complemented
10#20290298C211yesmaximal, characteristic, complemented
11#20290282C211yesmaximal, characteristic, complemented
12#20290060C211yesmaximal, characteristic
13C2×#16811473C211yesmaximal, characteristic, complemented
14#20289916C211yesmaximal, characteristic, complemented
15#20284743C211yesmaximal, characteristic
16C2×#13133304—21no—
17#16811675—21no—
18#16811675—21no—
19#16811673C2211yescharacteristic
20#16811673C2211yescharacteristic
21#16811672C2211yescharacteristic
22#16811672C2211yescharacteristic
23#16811658C2211yescharacteristic
24#16811657C2211yescharacteristic
25#16811655—21no—
26#16811652—21no—
27#16811651C2211yescharacteristic
28D4×#88324C2211yescharacteristic
29#16811626—21no—
30#16811626C2211yescharacteristic
31C2×C4⋊#88324C2211yescharacteristic
32#16811596—21no—
33C4×#11719940—21no—
34C4×#11719940C2211yescharacteristic
35C2×C4×#88324C2211yescharacteristic
36#16811485C2211yescharacteristic
37#16811485C2211yescharacteristic
38#16811484C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42#16811473C2211yescharacteristic
43#16811473C2211yescharacteristic
44#16811471C2211yescharacteristic
45#16811471C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50C8×#88324C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811168C2211yescharacteristic
53C2×#13133182C2211yescharacteristic
54#16810941C411yescharacteristic
55#16810907C411yescharacteristic
56#16810907C411yescharacteristic
57#16810822C411yescharacteristic
58#16810822C411yescharacteristic
59#16810764C411yescharacteristic
60#16810761C411yescharacteristic
61#16810714C2211yescharacteristic
62#16810714C2211yescharacteristic
63#16810710C2211yescharacteristic
64#16810710C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#481C411yescharacteristic
67#13133304—44no—
68C2×#11719936—21no—
69C2×#11719940—21no—
70C2×#11719940—21no—
71C2×#11719940D411yescharacteristic
72C2×#11719940D411yescharacteristic
73C2×D53⋊C8—21no—
74C22×#88324—21no—
75C22×#88324—21no—
76C22×#88324—21no—
77#13133284—21no—
78#13133284—21no—
79#13133284—21no—
80C4⋊#88324—21no—
81C4⋊#88324—21no—
82C4⋊#88324—21no—
83C4⋊#88324C2311yescharacteristic
84C4⋊#88324C2311yescharacteristic
85C4⋊#88324C2311yescharacteristic
86C4⋊#88324C2311yescharacteristic
87#13133274—21no—
88#13133274—21no—
89C22⋊#88324—21no—
90C22⋊#88324—21no—
91#13133272—21no—
92#13133272—21no—
93#13133272D411yescharacteristic
94#13133272D411yescharacteristic
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324—21no—
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C4×#88324C2311yescharacteristic
102C2×#11719947—21no—
103C22×Dic106C2×C411yescharacteristic
104C2×#11719913C2×C411yescharacteristic
105C2×#11719913C2×C411yescharacteristic
106C2×#11719923C2×C411yescharacteristic
107C2×#11719923C2×C411yescharacteristic
108#13133216C2×C411yescharacteristic
109#13133213C2×C411yescharacteristic
110#13133203—21no—
111#13133191C2×C411yescharacteristic
112#13133189—21no—
113#13133182—21no—
114#13133182—21no—
115#13133182C2×C411yescharacteristic
116#13133182C2×C411yescharacteristic
117#13133182C2311yescharacteristic
118#13133182C2311yescharacteristic
119D4×Dic53C2311yescharacteristic
120#13133179C2311yescharacteristic
121#13133177—21no—
122C2×#11719919—21no—
123C2×C4×Dic53—21no—
124C2×C4×Dic53C2311yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133148C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133146C2×C411yescharacteristic
129#13133142C2×C411yescharacteristic
130#13133142C2×C411yescharacteristic
131#13133140C2×C411yescharacteristic
132#13133140C2×C411yescharacteristic
133#13133137C2×C411yescharacteristic
134#13133137C2×C411yescharacteristic
135C22⋊#88322C2×C411yescharacteristic
136C8×Dic53C2311yescharacteristic
137C8×Dic53C2311yescharacteristic
138#121×C53C2×C411yescharacteristic
139C2×C4⋊C4×C53C2×C411yescharacteristic
140#106×C53C2×C411yescharacteristic
141#106×C53C2×C411yescharacteristic
142#102×C53C2×C411yescharacteristic
143#102×C53C2×C411yescharacteristic
144C22⋊C8×C53C2×C411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53C22×C411yescharacteristic
149C2×C22⋊D53—21no—
150C2×C22⋊D53—21no—
151C2×C22⋊D53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53#4111yescharacteristic
156#11719947—42no—
157#11719947—42no—
158C2×D212C22×C411yescharacteristic
159C2×C4×D53—21no—
160C2×Dic106—22no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—41no—
164C2×Dic106C22×C411yescharacteristic
165C2×Dic106C22×C411yescharacteristic
166#11719940—22no—
167#11719940—42no—
168#11719940—42no—
169#11719940SD1614yescomplemented
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324—21no—
174C2×#88324—21no—
175C2×#88324—21no—
176C2×#88324—42no—
177C2×#88324—42no—
178C2×#88324—42no—
179C2×#88324C2×D411yescharacteristic
180C2×#88324C2×D411yescharacteristic
181C2×#88324C2×D411yescharacteristic
182C2×#88324C2×D411yescharacteristic
183#11719936—42no—
184D53⋊C8—42no—
185C2×C8×C53C22×C411yescharacteristic
186C2×C8×C53C22×C411yescharacteristic
187C4⋊C4×C53—21no—
188C4⋊C4×C53C22×C411yescharacteristic
189C4⋊C4×C53C22×C411yescharacteristic
190C4⋊C4×C53C22×C411yescharacteristic
191C22⋊C4×C53—21no—
192C22⋊Dic53—21no—
193#11719923—22no—
194#11719923—41no—
195#11719921—21no—
196#11719921—21no—
197C4⋊Dic53C22×C411yescharacteristic
198#11719919—21no—
199#11719919—21no—
200#11719919—21no—
201#11719919—21no—
202#11719919—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53C22×C411yescharacteristic
208C4×Dic53C2×D411yescharacteristic
209C4×Dic53#4111yescharacteristic
210C4×Dic53C2411yescharacteristic
211C2×#88322C22×C411yescharacteristic
212C2×#88322C22×C411yescharacteristic
213#11719913—22no—
214#11719913—41no—
215C23×C53—21no—
216C23×C53C4×D411yescharacteristic
217C22×D53—21no—
218D4×C53—21no—
219D4×C53—21no—
220D4×C53—42no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—81no—
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C23×C411yescharacteristic
229C22⋊D53—42no—
230C22⋊D53—42no—
231C22⋊D53—81no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—41no—
236C2×Dic53—42no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53C4×D411yescharacteristic
240C2×Dic53C4×D411yescharacteristic
241C2×Dic53C2×SD1612yes—
242C2×Dic53#13512yes—
243C2×Dic53C22×D411yescharacteristic
244C2×Dic53C2×#4111yescharacteristic
245D212—21no—
246D212—21no—
247C4×D53—41no—
248C4×D53—41no—
249Dic106—21no—
250Dic106—21no—
251Dic106—21no—
252Dic106—42no—
253Dic106—42no—
254Dic106#13112yescomplemented
255#88324—22no—
256#88324—21no—
257#88324—21no—
258#88324—41no—
259#88324—41no—
260#88324—41no—
261#88324—41no—
262#88324—41no—
263#88324C2×SD1612yes—
264#88324C2×SD1612yes—
265C8×C53—21no—
266C8×C53—21no—
267#88322—21no—
268#88322—21no—
269#17997—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53C4×SD1612yes—
276C22×C53C2×C4×D411yescharacteristic
277C2×D53—41no—
278C2×D53—41no—
279C2×D53—41no—
280C4×C53—21no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53—41no—
284C4×C53C2×#13111yescharacteristic
285C4×C53#56811yescharacteristic
286Dic53—22no—
287Dic53—21no—
288Dic53—21no—
289Dic53—41no—
290Dic53—41no—
291Dic53C4×SD1612yes—
292Dic53C22×SD1611yescharacteristic
293Dic53C2×#13511yescharacteristic
294#3103—531no—
295Q8×M16—531no—
296#2935—531no—
297#1590—531no—
298#1590—531no—
299#1583—531no—
300#1583—531no—
301#1582—531no—
302#1582—531no—
303C8×SD16—531no—
304C8×SD16—531no—
305#1570—531no—
306#1492—531no—
307C2×Q8⋊C8—531no—
308C8×M16—531no—
309C2×C53—22no—
310C2×C53—81no—
311C2×C53#293311yescharacteristic
312C2×C53C2×C4×SD1611yescharacteristic
313C2×C53#295611yescharacteristic
314D53—81no—
315C2×#131—1061no—
316#532—531no—
317C2×C4×Q8—531no—
318#481—531no—
319#481#615711yescharacteristic
320#475—531no—
321#475—531no—
322#460—531no—
323#460—531no—
324#448—531no—
325#448—531no—
326#446—531no—
327#446—531no—
328#446—1061no—
329#446—1061no—
330C8×Q8—531no—
331C8×Q8—531no—
332C4×SD16—531no—
333C4×SD16—531no—
334C4×SD16—531no—
335C4×SD16—531no—
336#436—531no—
337#436—531no—
338#435—1061no—
339#435—1061no—
340C8×D4—531no—
341C8×D4—531no—
342#431—531no—
343#423—531no—
344#423—1061no—
345C2×C4⋊C8—531no—
346#408—1061no—
347C4×M16—531no—
348C4×M16—1061no—
349C2×C4×C8—531no—
350#334—531no—
351#334—531no—
352#334—531no—
353#334—531no—
354Q8⋊C8—531no—
355Q8⋊C8—531no—
356Q8⋊C8—531no—
357Q8⋊C8—531no—
358D4⋊C8—531no—
359D4⋊C8—531no—
360D4⋊C8—531no—
361D4⋊C8—531no—
362#322—531no—
363#322—531no—
364C82—531no—
365C82—531no—
366C53#1799711yescharacteristic, complemented, Sylow
367C2×#41—1061no—
368C22×Q8—531no—
369C2×SD16—531no—
370C2×SD16—531no—
371C2×SD16—531no—
372C2×SD16—531no—
373#131—2124no—
374C2×M16—531no—
375C2×M16—531no—
376C2×M16—1061no—
377C2×M16—1061no—
378C2×M16—1061no—
379C22×C8—1061no—
380C22×C8—1061no—
381C22×C8—1061no—
382C22×C8—1061no—
383C4⋊Q8—531no—
384#127—531no—
385#124—1061no—
386#121—531no—
387#121—1061no—
388#121C2×#615711yescharacteristic
389C4×Q8—531no—
390C4×Q8—531no—
391C4×Q8—531no—
392C4×Q8—531no—
393C4×Q8—1061no—
394C4×Q8—1061no—
395C4×D4—531no—
396C4×D4—531no—
397C4×D4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4C2×#615711yescharacteristic
400C2×C42—531no—
401C2×C42—1061no—
402#106—531no—
403#106—531no—
404#106C2×#615711yescharacteristic
405#106C2×#615711yescharacteristic
406C4⋊C8—531no—
407C4⋊C8—531no—
408C4⋊C8—531no—
409C4⋊C8—531no—
410C4⋊C8—531no—
411C4⋊C8—531no—
412C4⋊C8—1061no—
413C4⋊C8—1061no—
414C4⋊C8—1061no—
415C4⋊C8—1061no—
416C4⋊C8—1061no—
417C4⋊C8—1061no—
418#103—531no—
419#103—531no—
420#103—531no—
421#103—531no—
422#102—531no—
423#102—531no—
424#102C2×#615711yescharacteristic
425#102C2×#615711yescharacteristic
426C22⋊C8—531no—
427C22⋊C8—1061no—
428C22⋊C8—1061no—
429C22⋊C8—1061no—
430C22⋊C8—1061no—
431C22⋊C8C2×#615711yescharacteristic
432#97—531no—
433#97—531no—
434#97—1061no—
435#97—1061no—
436C4×C8—531no—
437C4×C8—531no—
438C4×C8—531no—
439C4×C8—531no—
440C4×C8—531no—
441C4×C8—531no—
442C4×C8—1061no—
443C4×C8—1061no—
444C4×C8—1061no—
445#41—2122no—
446#41—2122no—
447C2×Q8—531no—
448C2×Q8—531no—
449C2×Q8—1061no—
450C2×Q8—1062no—
451C2×Q8—1061no—
452C2×Q8—2121no—
453C2×D4—21no—
454C2×D4—531no—
455C2×D4—1061no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4C22×#615711yescharacteristic
459C22×C4—21no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4C22×#615711yescharacteristic
466SD16—1062no—
467SD16—1062no—
468SD16—2121no—
469SD16—2121no—
470M16—534no—
471M16—1062no—
472M16—2122no—
473M16—2122no—
474M16—2122no—
475C2×C8—531no—
476C2×C8—531no—
477C2×C8—531no—
478C2×C8—531no—
479C2×C8—531no—
480C2×C8—531no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—2122no—
488C2×C8—2122no—
489C2×C8—2122no—
490C2×C8—2122no—
491C2×C8C22×#615711yescharacteristic
492C2×C8C22×#615711yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—531no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4C22×#615711yescharacteristic
501C4⋊C4C22×#615711yescharacteristic
502C4⋊C4C22×#615711yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1061no—
505C22⋊C4—1061no—
506C22⋊C4—1061no—
507C42—531no—
508C42—531no—
509C42—531no—
510C42—531no—
511C42—1061no—
512C42—1061no—
513C42—1061no—
514C42—1061no—
515C23—21no—
516C23—1061no—
517C23D4×#615711yescharacteristic
518Q8—532no—
519Q8—1061no—
520Q8—1061no—
521Q8—1061no—
522Q8—2122no—
523Q8—2122no—
524D4—21no—
525D4—21no—
526D4—42no—
527D4—1061no—
528D4—1061no—
529D4—4241no—
530D4—2122no—
531D4—2122no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—81no—
538C2×C4—532no—
539C2×C4—532no—
540C2×C4—531no—
541C2×C4—531no—
542C2×C4—531no—
543C2×C4—531no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2122no—
553C2×C4D4×#615711yescharacteristic
554C2×C4C23×#615711yescharacteristic
555C8—21no—
556C8—21no—
557C8—532no—
558C8—532no—
559C8—1061no—
560C8—1061no—
561C8—1062no—
562C8—1061no—
563C8—1061no—
564C8—2121no—
565C8—2121no—
566C8—2121no—
567C8—2121no—
568C8—2121no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—2121no—
575C22—2121no—
576C22—2121no—
577C22SD16×#615712yes—
578C22C2×D4×#615711yescharacteristic
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—531no—
584C4—531no—
585C4—532no—
586C4—1062no—
587C4—1061no—
588C4—1061no—
589C4—2121no—
590C4—2121no—
591C4D53⋊#13111yescharacteristic
592C4#1681170211yescharacteristic
593C2—22no—
594C2—81no—
595C2—4241no—
596C2#2029144911yescharacteristic
597C2C2×SD16×#615711yescharacteristic
598C2#2029148911yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#481C4
socleC22×C53C2×C4×D4
prime radical#16811168C22
cosocleC4×Dic53C24
hypercenter#481#6157
hypocenterC53#17997
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #20291516C2semidirect
3. #20290352C2semidirect
4. #20290350C2semidirect
5. #20290325C2semidirect
6. #20290323C2semidirect
7. #20290319C2semidirect
8. #20290318C2semidirect
9. #20290300C2semidirect
10. #20290298C2semidirect
11. #20290282C2semidirect
13. C2×#16811473C2semidirect
14. #20289916C2semidirect
169. #11719940SD16semidirect
254. Dic106#131semidirect
366. C53#17997semidirect

Maximal subgroups

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

subgroupindex
#202915162
#202914572
#202903522
#202903502
#202903252
#202903232
#202903192
#202903182
#202903002
#202902982
#202902822
#202900602
C2×#168114732
#202899162
#202847432
#1799753