Finite Group Databasebeta

← #27469490

Subgroups — #27469490

Order 13568.

846 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810959C211yesmaximal, characteristic
2C2×#16810965C211yesmaximal, characteristic
3C2×#16810881C211yesmaximal, characteristic
4C2×#16810665C211yesmaximal, characteristic
5#20279200C211yesmaximal, characteristic
6#20279168C211yesmaximal, characteristic
7#20276262C211yesmaximal, characteristic, complemented
8#20276262C211yesmaximal, characteristic, complemented
9#20276262C211yesmaximal, characteristic, complemented
10#20276252C211yesmaximal, characteristic, complemented
11#20275586C211yesmaximal, characteristic, complemented
12#20275576C211yesmaximal, characteristic, complemented
13#20274883C211yesmaximal, characteristic, complemented
14#20274879C211yesmaximal, characteristic, complemented
15C53×#1753C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810965—24no—
22#16810959—24no—
23#16810881—24no—
24#16810665—24no—
25C2×#13133151C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133151C2211yescharacteristic
28C2×#13133151C2211yescharacteristic
29C2×#13133151C2211yescharacteristic
30C2×#13133149C2211yescharacteristic
31C2×#13133149C2211yescharacteristic
32C2×#13133149C2211yescharacteristic
33C2×#13133149C2211yescharacteristic
34C2×#13133142C2211yescharacteristic
35C2×#13133142C2211yescharacteristic
36C2×#13133142C2211yescharacteristic
37C2×#13133142C2211yescharacteristic
38C2×#13133142C2211yescharacteristic
39C2×#13133142C2211yescharacteristic
40#16810563—22no—
41#16810548—22no—
42#16810529C2211yescharacteristic
43#16810529C2211yescharacteristic
44#16810529C2211yescharacteristic
45#16810529C2211yescharacteristic
46#16810508—22no—
47#16810493C2211yescharacteristic
48#16810489C2211yescharacteristic
49#16810489C2211yescharacteristic
50#16810489C2211yescharacteristic
51#16810484—22no—
52C22×M16×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×#97×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×Dic106C2311yescharacteristic
61C23×Dic53—21no—
62C2×Dic212—22no—
63C2×Dic212—22no—
64C2×Dic212—22no—
65#13133196—22no—
66#13133196—22no—
67#13133196—22no—
68#13133196—22no—
69#13133176—22no—
70#13133176—22no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C4×Dic53—21no—
87#13133151—22no—
88#13133151—22no—
89#13133151—22no—
90#13133151—22no—
91#13133151—22no—
92#13133149—22no—
93#13133149—22no—
94#13133149—22no—
95#13133149—22no—
96#13133142—22no—
97#13133142—22no—
98#13133142—22no—
99#13133142—22no—
100#13133142—22no—
101#13133142—22no—
102#13133124—22no—
103#13133124—22no—
104#13133124—22no—
105#13133124—22no—
106#13133124—22no—
107#13133124—22no—
108#13133124—22no—
109C23×C4×C53C2311yescharacteristic
110C2×M16×C53—22no—
111C2×M16×C53—22no—
112C2×M16×C53D414yes—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C22⋊C4×C53D412yes—
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C22⋊C8×C53D414yes—
121#97×C53—22no—
122C4×C8×C53—22no—
123#95×C53D412yes—
124C24×C53C2×D411yescharacteristic
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—44no—
144C22×Dic53—42no—
145C2×Dic106—21no—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157M16×C53—28no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53C2×D412yes—
165C2×C8×C53C2×D412yes—
166C2×C8×C53C2×D412yes—
167C2×C8×C53C2×D412yes—
168C2×C8×C53#4112yes—
169C2×C8×C53#4112yes—
170C22⋊C4×C53—42no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C22⋊Dic53—42no—
176C22⋊Dic53—42no—
177C22⋊Dic53—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—44no—
191C4⋊Dic53—42no—
192#11719919—42no—
193#11719919—42no—
194C4×Dic53—42no—
195Dic212—44no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53#12712yes—
201C23×C53C22×D411yescharacteristic
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53#12712yes—
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C532−1+411yescharacteristic
220C2×C4×C532−1+411yescharacteristic
221C2×Dic53—41no—
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—84no—
240Dic106—41no—
241Dic106—41no—
242Dic106—41no—
243Dic106—41no—
244Dic106—41no—
245Dic106—41no—
246Dic106—41no—
247Dic106—41no—
248C8×C53—22no—
249C8×C53—22no—
250C8×C53—22no—
251C8×C53#12314yes—
252#27943—531nomaximal, Sylow
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53C2×#12711yescharacteristic
259C22×C53#54711yescharacteristic
260C22×C53#54711yescharacteristic
261C22×C53#57512yes—
262C22×C53#57812yes—
263C4×C53—22no—
264C4×C53—42no—
265C4×C53—42no—
266C4×C53C2×#12311yescharacteristic
267C4×C53#53611yescharacteristic
268C4×C53#54711yescharacteristic
269C4×C53#54711yescharacteristic
270Dic53—81no—
271Dic53—81no—
272Dic53—81no—
273Dic53—81no—
274Dic53—81no—
275Dic53—81no—
276Dic53—81no—
277Dic53—81no—
278C2×#484—531no—
279C2×#485—531no—
280C2×#470—531no—
281C2×#428—531no—
282#2503—531no—
283#2499—531no—
284#2097—531no—
285#2097—531no—
286#2097—531no—
287#2095—531no—
288#1971—531no—
289#1969—531no—
290#1847—531no—
291#1846—531no—
292#1753D5311yescharacteristic
293C2×C53—42no—
294C2×C53#250111yescharacteristic
295C2×C53#315311yescharacteristic
296C2×C53#315611yescharacteristic
297C2×C53#334912yes—
298C2×C53#335212yes—
299C22×Q16—531no—
300C22×M16C2×D5311yescharacteristic
301C2×C22⋊Q8—531no—
302C2×C22⋊Q8—531no—
303C2×#117—531no—
304C22×C4⋊C4—531no—
305#485—1064no—
306#484—1064no—
307#470—1064no—
308#428—1064no—
309C2×#107—531no—
310C2×#107—531no—
311C2×#107—531no—
312C2×#107—531no—
313C2×#107—531no—
314C2×#106—531no—
315C2×#106—531no—
316C2×#106—531no—
317C2×#106—531no—
318C2×#103—531no—
319C2×#103—531no—
320C2×#103—531no—
321C2×#103—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×C22⋊C8C2×D5311yescharacteristic
325C2×#97C2×D5311yescharacteristic
326C2×C4×C8C2×D5311yescharacteristic
327#400—1062no—
328#397—1062no—
329#391—531no—
330#391—531no—
331#391—531no—
332#391—531no—
333#387—1062no—
334#384—531no—
335#382—531no—
336#382—531no—
337#382—531no—
338#381—1062no—
339C4×C22⋊C4C2×D5311yescharacteristic
340#336C2×D5311yescharacteristic
341#336C2×D5311yescharacteristic
342C53#2794311yescharacteristic, complemented, Sylow
343C22×Q8—531no—
344C22×Q8—531no—
345C23×C4—1061no—
346C23×C4C22×D5311yescharacteristic
347C2×Q16—1062no—
348C2×Q16—1062no—
349C2×Q16—1062no—
350C2×M16—22no—
351C2×M16—22no—
352C2×M16D21214yes—
353C22×C8C22×D5311yescharacteristic
354C22×C8C22×D5311yescharacteristic
355C22×C8C22×D5311yescharacteristic
356C22×C8C22×D5311yescharacteristic
357C22⋊Q8—1062no—
358C22⋊Q8—1062no—
359C22⋊Q8—1062no—
360C22⋊Q8—1062no—
361#117—1062no—
362#117—1062no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4—531no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1062no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1062no—
375C2×C22⋊C4—1061no—
376C2×C22⋊C4—1061no—
377C2×C22⋊C4—1061no—
378C2×C22⋊C4D21212yes—
379C2×C42—1061no—
380C2×C42C22×D5311yescharacteristic
381C2×C42C22×D5311yescharacteristic
382#107—1062no—
383#107—1062no—
384#107—1062no—
385#107—1062no—
386#107—1062no—
387#106—1062no—
388#106—1062no—
389#106—1062no—
390#106—1062no—
391#103—1062no—
392#103—1062no—
393#103—1062no—
394#103—1062no—
395#103—1062no—
396#103—1062no—
397C22⋊C8D21214yes—
398#97—22no—
399C4×C8—22no—
400#95—1062no—
401#95—1062no—
402#95—1062no—
403#95—1062no—
404#95—1062no—
405#95—1062no—
406#95—1062no—
407#95D21212yes—
408C24C2×D21211yescharacteristic
409C2×Q8—1061no—
410C2×Q8—1061no—
411C2×Q8—1061no—
412C2×Q8—1061no—
413C2×Q8—1061no—
414C2×Q8—1061no—
415C2×Q8—1061no—
416C2×Q8—1061no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—2124no—
434C22×C4—2122no—
435C22×C4C2×D21212yes—
436C22×C4C2×D21212yes—
437C22×C4C2×D21211yescharacteristic
438C22×C4C2×D21212yes—
439C22×C4C2×D21212yes—
440C22×C4C23×D5311yescharacteristic
441Q16—2124no—
442M16—28no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8C2×D21212yes—
450C2×C8C2×D21212yes—
451C2×C8C2×D21212yes—
452C2×C8C2×D21212yes—
453C2×C8#1171994712yes—
454C2×C8#1171994712yes—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—2124no—
468C4⋊C4—2122no—
469C4⋊C4—2122no—
470C4⋊C4—2122no—
471C22⋊C4—42no—
472C22⋊C4—2122no—
473C22⋊C4—2122no—
474C22⋊C4—2122no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—2122no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23#1313321412yes—
485C23C22×D21211yescharacteristic
486Q8—2121no—
487Q8—2121no—
488Q8—2121no—
489Q8—2121no—
490Q8—2121no—
491Q8—2121no—
492Q8—2121no—
493Q8—2121no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—4244no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4#1313321412yes—
526C2×C4C22×D21211yescharacteristic
527C2×C4C22×D21211yescharacteristic
528C2×C4C2×#1171994711yescharacteristic
529C2×C4C2×#1171994711yescharacteristic
530C2×C4#1313326411yescharacteristic
531C2×C4#1313326411yescharacteristic
532C8—22no—
533C8—22no—
534C8—22no—
535C8#1313320114yes—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—42no—
540C22—42no—
541C22C2×#1313321411yescharacteristic
542C22#1681127711yescharacteristic
543C22#1681127711yescharacteristic
544C22#1681142312yes—
545C22#1681144012yes—
546C4—22no—
547C4—42no—
548C4—42no—
549C4—4241no—
550C4—4241no—
551C4—4241no—
552C4—4241no—
553C4—4241no—
554C4—4241no—
555C4—4241no—
556C4—4241no—
557C4C2×#1313320111yescharacteristic
558C4#1681119911yescharacteristic
559C4#1681127711yescharacteristic
560C4#1681127711yescharacteristic
561C2—42no—
562C2#2027918611yescharacteristic
563C2#2028518111yescharacteristic
564C2#2028520911yescharacteristic
565C2#2028724712yes—
566C2#2028728912yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1753C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1753D53
hypocenterC53#27943
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
7. #20276262C2semidirect
8. #20276262C2semidirect
9. #20276262C2semidirect
10. #20276252C2semidirect
11. #20275586C2semidirect
12. #20275576C2semidirect
13. #20274883C2semidirect
14. #20274879C2semidirect
342. C53#27943semidirect

Maximal subgroups

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

subgroupindex
C2×#168109592
C2×#168109652
C2×#168108812
C2×#168106652
#202792002
#202791682
#202762622
#202762522
#202755862
#202755762
#202748832
#202748792
C53×#17532
#2794353