Finite Group Databasebeta

← #27437655

Subgroups — #27437655

Order 13568.

1172 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811180C211yesmaximal, characteristic, complemented
2C2×#16811018C211yesmaximal, characteristic, complemented
3C2×#16810982C211yesmaximal, characteristic, complemented
4#20275723C212yesmaximal, complemented
5#20275718C212yesmaximal
6#20275687C211yesmaximal, characteristic, complemented
7#20275534C211yesmaximal, characteristic, complemented
8#20275073C212yesmaximal, complemented
9#20275059C212yesmaximal, complemented
10#20274477C211yesmaximal, characteristic, complemented
11C53×#2850C211yesmaximal, characteristic
12C22×#11719925C2212yes—
13C22×#11719923C2212yes—
14#16811180—24no—
15#16811180—24no—
16C2×#13133217C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133209—21no—
19C2×#13133196—21no—
20C2×#13133196C2212yes—
21C2×C4×Dic106—21no—
22C2×C4×Dic106C2212yes—
23C2×#13133176—21no—
24#16811018—24no—
25#16810982—24no—
26C2×#13133141C2212yes—
27C2×#13133140C2212yes—
28C2×#13133140C2212yes—
29C2×#13133138C2212yes—
30C2×C22⋊#88322C2212yes—
31C2×#13133128C2211yescharacteristic
32C2×#13133128C2211yescharacteristic
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C2×#124×C53C2211yescharacteristic
37C2×C22⋊Q8×C53C2211yescharacteristic
38C2×#121×C53C2211yescharacteristic
39C22×C42×C53C2211yescharacteristic
40C53×#391C2211yescharacteristic
41C53×#390C2211yescharacteristic
42C53×#388—21no—
43C53×#388—21no—
44C53×#386—21no—
45C53×#386—21no—
46C53×#382C2211yescharacteristic
47C22×Dic106—21no—
48C22×Dic106C2312yes—
49C2×#11719925—24no—
50C2×#11719925—24no—
51C2×#11719925—42no—
52C2×#11719923—24no—
53C2×#11719923—24no—
54C2×#11719923—42no—
55C22×#88322C2312yes—
56C22×#88322C2312yes—
57#13133217—22no—
58#13133217—22no—
59#13133217D414yescomplemented
60#13133217D414yescomplemented
61#13133209—28no—
62#13133196—28no—
63#13133196—24no—
64#13133196—24no—
65C4×Dic106—28no—
66C4×Dic106—24no—
67C4×Dic106—24no—
68#13133176—28no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2312yes—
73C2×#11719919—22no—
74C2×#11719919—22no—
75C2×C22⋊Dic53—22no—
76C2×C4×Dic53—22no—
77#13133141—24no—
78#13133140—42no—
79#13133140—42no—
80#13133138—24no—
81C22⋊#88322—24no—
82#13133128—22no—
83#13133128—22no—
84C22×Q8×C53C2311yescharacteristic
85C22×D4×C53—21no—
86C22×D4×C53C2311yescharacteristic
87C23×C4×C53D411yescharacteristic
88C23×C4×C53D411yescharacteristic
89C23×C4×C53C2311yescharacteristic
90#124×C53—22no—
91#124×C53—22no—
92C22⋊Q8×C53—42no—
93#121×C53—42no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53—21no—
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C42×C53—22no—
104C2×C42×C53—22no—
105C2×C42×C53D411yescharacteristic
106C2×C42×C53D411yescharacteristic
107C2×C42×C53C2311yescharacteristic
108C2×C42×C53C2311yescharacteristic
109#95×C53—21no—
110#95×C53—21no—
111#95×C53—21no—
112#95×C53—21no—
113C24×C53—21no—
114C24×C53C2×D411yescharacteristic
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—42no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—22no—
146C22×Dic53—22no—
147C2×Dic106—22no—
148C2×Dic106—22no—
149C2×Dic106—22no—
150C2×Dic106—22no—
151C2×Dic106—24no—
152C2×Dic106—22no—
153C2×Dic106—42no—
154C2×Dic106—42no—
155C2×Dic106C2×D414yes—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—42no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—42no—
162C22⋊C4×C53—42no—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C42×C53—22no—
166C42×C53—22no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53C2×D412yes—
170C42×C53C2×D412yes—
171C22⋊Dic53—28no—
172#11719925—48no—
173#11719923—48no—
174C4⋊Dic53—24no—
175C4⋊Dic53—24no—
176C4⋊Dic53—24no—
177C4⋊Dic53—22no—
178C4⋊Dic53C2×D414yes—
179#11719919—28no—
180#11719919—28no—
181C4×Dic53—28no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322#4114yes—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53#13712yes—
202C23×C53C22×D411yescharacteristic
203Q8×C53—42no—
204Q8×C53—42no—
205D4×C53—42no—
206D4×C53—42no—
207D4×C53—84no—
208C2×C4×C53—28no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—44no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53#12012yes—
234C2×C4×C53#12012yes—
235C2×C4×C53#13712yes—
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C2×#4112yes—
239C2×C4×C53H(5,𝔽2)12yes—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246Dic106—24no—
247Dic106—24no—
248Dic106—48no—
249Dic106—42no—
250Dic106—42no—
251#88322—24no—
252#88322—42no—
253#25603—531nomaximal, Sylow
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53C2×#12011yescharacteristic
269C22×C53#53211yescharacteristic
270C22×C53#53511yescharacteristic
271C22×C53C2×#13711yescharacteristic
272C22×C53C2×#13711yescharacteristic
273C22×C53#57511yescharacteristic
274C22×C53#57511yescharacteristic
275C4×C53—22no—
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53#54612yes—
284C4×C53#55012yes—
285Dic53—44no—
286Dic53—44no—
287C2×#533—531no—
288C2×#497—531no—
289C2×#489—531no—
290#2850D5311yescharacteristic
291#2001—532no—
292#2000—532no—
293#1993—531no—
294#1960—531no—
295#1880—532no—
296#1878—532no—
297#1749—531no—
298C2×C53—42no—
299C2×C53—161no—
300C2×C53#243111yescharacteristic
301C2×C53#300411yescharacteristic
302C2×C53#300511yescharacteristic
303C2×C53#317211yescharacteristic
304C2×C53#318111yescharacteristic
305C2×C53#321511yescharacteristic
306C2×C53#321611yescharacteristic
307C22×Q16—532no—
308C22×SD16—532no—
309#533—1064no—
310#533—1064no—
311C2×C4⋊Q8—531no—
312C2×C4⋊Q8—531no—
313C2×#125—1061no—
314C2×#124C2×D5311yescharacteristic
315C2×C22⋊Q8—532no—
316C2×C22⋊Q8—1061no—
317C2×C22⋊Q8C2×D5311yescharacteristic
318C2×#121C2×D5311yescharacteristic
319C2×C4×Q8—532no—
320C2×C4×Q8—1061no—
321C2×#117—1061no—
322C22×C42C2×D5311yescharacteristic
323#497—1064no—
324#489—1064no—
325C2×#103—532no—
326C2×#103—532no—
327C2×#103—532no—
328C2×#102—532no—
329C2×C22⋊C8—532no—
330C2×#97—531no—
331C2×#97—531no—
332#391C2×D5311yescharacteristic
333#390C2×D5311yescharacteristic
334#388—21no—
335#388—21no—
336#386—21no—
337#386—21no—
338#382C2×D5311yescharacteristic
339#340—532no—
340#340—532no—
341#336—532no—
342C53#2560311yescharacteristic, complemented, Sylow
343C22×Q8—532no—
344C22×Q8—1061no—
345C22×Q8C22×D5311yescharacteristic
346C22×D4—21no—
347C22×D4C22×D5311yescharacteristic
348C23×C4C22⋊D5311yescharacteristic
349C23×C4C22⋊D5311yescharacteristic
350C23×C4C22×D5311yescharacteristic
351C2×Q16—1064no—
352C2×Q16—1064no—
353C2×Q16—2122no—
354C2×SD16—1064no—
355C2×SD16—1064no—
356C2×SD16—2122no—
357C22×C8—532no—
358C22×C8—532no—
359C4⋊Q8—534no—
360C4⋊Q8—534no—
361C4⋊Q8—1062no—
362C4⋊Q8—1062no—
363#125—1068no—
364#124—22no—
365#124—22no—
366C22⋊Q8—42no—
367C22⋊Q8—1064no—
368C22⋊Q8—1064no—
369C22⋊Q8—1068no—
370#121—42no—
371C4×Q8—1064no—
372C4×Q8—1064no—
373C4×Q8—1068no—
374#117—1068no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—532no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1062no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—1062no—
394C2×C42C22⋊D5311yescharacteristic
395C2×C42C22⋊D5311yescharacteristic
396C2×C42C22×D5311yescharacteristic
397C2×C42C22×D5311yescharacteristic
398#103—1064no—
399#103—2122no—
400#103—2122no—
401#102—1064no—
402C22⋊C8—1064no—
403#97—1062no—
404#97—1062no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409C24—21no—
410C24C2×C22⋊D5311yescharacteristic
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—534no—
416C2×Q8—1064no—
417C2×Q8—1062no—
418C2×Q8—1062no—
419C2×Q8—1062no—
420C2×Q8—1062no—
421C2×Q8—1062no—
422C2×Q8—2122no—
423C2×Q8—2122no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C23×D5311yescharacteristic
452Q16—2128no—
453SD16—2128no—
454C2×C8—534no—
455C2×C8—1062no—
456C2×C8—1062no—
457C2×C8—1062no—
458C2×C8—1062no—
459C2×C8—1062no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—534no—
464C4⋊C4—1068no—
465C4⋊C4—1068no—
466C4⋊C4—1064no—
467C4⋊C4—1064no—
468C4⋊C4—1064no—
469C4⋊C4—1062no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—1068no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—21no—
481C42—1068no—
482C42D4×D5312yes—
483C42D4×D5312yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#1313324712yes—
498C23C22×C22⋊D5311yescharacteristic
499Q8—42no—
500Q8—42no—
501Q8—1064no—
502Q8—1064no—
503Q8—2128no—
504Q8—2122no—
505Q8—2122no—
506D4—42no—
507D4—42no—
508D4—84no—
509C2×C4—28no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—1064no—
535C2×C4—1064no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4#1313318512yes—
541C2×C4#1313318512yes—
542C2×C4#1313324712yes—
543C2×C4C2×D4×D5311yescharacteristic
544C2×C4C2×D4×D5311yescharacteristic
545C2×C4#41×D5312yes—
546C2×C4#1313326212yes—
547C8—1064no—
548C8—2122no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22C2×#1313318511yescharacteristic
564C22#1681116911yescharacteristic
565C22#1681119011yescharacteristic
566C22C2×#1313324711yescharacteristic
567C22C2×#1313324711yescharacteristic
568C22#1681142111yescharacteristic
569C22#1681142111yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—41no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—2124no—
579C4—2124no—
580C4#1681127012yes—
581C4#1681129112yes—
582C2—42no—
583C2—161no—
584C2#2027833511yescharacteristic
585C2#2028394811yescharacteristic
586C2#2028395511yescharacteristic
587C2#2028532211yescharacteristic
588C2#2028542511yescharacteristic
589C2#2028571811yescharacteristic
590C2#2028572611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2850C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2850D53
hypocenterC53#25603
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. C2×#16811180C2semidirect
2. C2×#16811018C2semidirect
3. C2×#16810982C2semidirect
4. #20275723C2semidirect
6. #20275687C2semidirect
7. #20275534C2semidirect
8. #20275073C2semidirect
9. #20275059C2semidirect
10. #20274477C2semidirect
59. #13133217D4semidirect
60. #13133217D4semidirect
342. C53#25603semidirect

Maximal subgroups

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

subgroupindex
C2×#168111802
C2×#168110182
C2×#168109822
#202757232
#202757182
#202756872
#202755342
#202750732
#202750592
#202744772
C53×#28502
#2560353