Finite Group Databasebeta

← #27411668

Subgroups — #27411668

Order 13568.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811201C211yesmaximal, characteristic
2C22×#13133148C211yesmaximal, characteristic, complemented
3C2×#16810622C211yesmaximal, characteristic, complemented
4C2×#16810617C211yesmaximal, characteristic, complemented
5C2×#16810590C211yesmaximal, characteristic, complemented
6#20275911C214yesmaximal, complemented
7#20275904C214yesmaximal, complemented
8#20274436C211yesmaximal, characteristic, complemented
9C53×#2311C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C23×#88322C2211yescharacteristic
13#16811201—24no—
14#16811201—24no—
15C2×#13133196C2212yes—
16C2×#13133193—21no—
17C2×#13133191C2212yes—
18C2×#13133189—22no—
19C2×#13133188—21no—
20C22×C4⋊Dic53C2211yescharacteristic
21C2×#13133148—24no—
22C2×#13133148C2214yes—
23#16810622—24no—
24#16810617—24no—
25C2×#13133140C2212yes—
26C2×#13133140C2212yes—
27C2×#13133137C2212yes—
28C2×#13133137C2212yes—
29#16810590—24no—
30C2×C22⋊#88322C2214yes—
31#16810385C2214yes—
32C22×#41×C53C2211yescharacteristic
33C2×#117×C53C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C53×#391—21no—
36C53×#390—21no—
37C53×#386—22no—
38C53×#380C2214yes—
39C2×#11719945—22no—
40C2×#11719945—22no—
41C2×#11719945—22no—
42C2×#11719945—22no—
43C2×#11719945—22no—
44C2×#11719945D414yes—
45C22×Dic106C2311yescharacteristic
46C22×D212C2311yescharacteristic
47C22×C22⋊D53—21no—
48C22×C22⋊D53—22no—
49C23×Dic53—21no—
50C22×C4×D53—21no—
51C2×#11719917—22no—
52C2×#11719917—22no—
53C2×#11719917—22no—
54C22×#88322—22no—
55C22×#88322—22no—
56C22×#88322D414yescomplemented
57C22×#88322C2312yes—
58C22×#88322C2312yes—
59#13133196—24no—
60#13133196—24no—
61#13133193—28no—
62#13133191—24no—
63#13133191—24no—
64#13133189—216no—
65#13133188—28no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53D414yes—
69C2×C4⋊Dic53C2312yes—
70C2×#11719919—22no—
71C2×C22⋊Dic53—22no—
72C2×#11719921—22no—
73#13133148—28no—
74#13133140—24no—
75#13133140—24no—
76#13133137—24no—
77#13133137—24no—
78C22⋊#88322—28no—
79C2×#41×C53—22no—
80C2×#41×C53—22no—
81C2×#41×C53—22no—
82C2×#41×C53—22no—
83C2×#41×C53—22no—
84C2×#41×C53—22no—
85C22×Q8×C53D411yescharacteristic
86C22×D4×C53D411yescharacteristic
87C22×D4×C53D412yes—
88C23×C4×C53—21no—
89C23×C4×C53C2312yes—
90C23×C4×C53C2311yescharacteristic
91#117×C53—22no—
92#117×C53—22no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53C2312yes—
97C2×C4⋊C4×C53C2312yes—
98C2×C22⋊C4×C53—21no—
99C2×C22⋊C4×C53—24no—
100C2×C42×C53—21no—
101#95×C53—24no—
102C24×C53C2×D412yes—
103C24×C53C2×D411yescharacteristic
104C23×D53—21no—
105#41×C53—44no—
106#41×C53—44no—
107#41×C53—44no—
108#41×C53—44no—
109C2×Q8×C53—22no—
110C2×Q8×C53—22no—
111C2×Q8×C53—22no—
112C2×D4×C53—24no—
113C2×D4×C53—24no—
114C2×D4×C53—24no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—42no—
127C22×C4×C53—82no—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53#4112yes—
134C22×C4×C53#4112yes—
135C22×C4×C53#4112yes—
136C22×C4×C53C2411yescharacteristic
137C2×C22⋊D53—24no—
138C2×C22⋊D53—28no—
139C2×C22⋊D53—42no—
140C2×C22⋊D53—42no—
141C2×C22⋊D53—84no—
142C22×Dic53—22no—
143C22×Dic53—21no—
144C22×Dic53—24no—
145C22×Dic53—82no—
146#11719945—28no—
147#11719945—28no—
148#11719945—44no—
149#11719945—44no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212C2×D412yes—
156C2×C4×D53—24no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106C2×D412yes—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—42no—
170C22⋊C4×C53—42no—
171C22⋊C4×C53—84no—
172C42×C53—42no—
173C22⋊Dic53—28no—
174#11719921—28no—
175C4⋊Dic53—24no—
176C4⋊Dic53—22no—
177C4⋊Dic53C2×D414yes—
178#11719919—28no—
179#11719917—44no—
180C2×#88322—28no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322C2×D414yes—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—24no—
193C23×C53—24no—
194C23×C53#12012yes—
195C23×C53#12112yes—
196C23×C53#12312yes—
197C23×C53C22×D411yescharacteristic
198C22×D53—22no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—42no—
202C22×D53—42no—
203Q8×C53—44no—
204D4×C53—48no—
205D4×C53—44no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—44no—
217C2×C4×C53—42no—
218C2×C4×C53—84no—
219C2×C4×C53#12012yes—
220C2×C4×C53#12112yes—
221C2×C4×C53#12114yes—
222C2×C4×C53#12114yes—
223C2×C4×C53#12314yes—
224C2×C4×C53#12312yes—
225C2×C4×C53#12414yes—
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C22⋊D53—48no—
233C22⋊D53—88no—
234C2×Dic53—22no—
235C2×Dic53—24no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—44no—
239C2×Dic53—84no—
240D212—22no—
241D212—22no—
242D212—41no—
243D212—41no—
244C4×D53—48no—
245Dic106—22no—
246Dic106—22no—
247Dic106—41no—
248Dic106—41no—
249#88322—24no—
250#88322—42no—
251#23489—531nomaximal, Sylow
252C22×C53—24no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—24no—
256C22×C53—44no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—44no—
261C22×C53#39414yes—
262C22×C53C2×#12011yescharacteristic
263C22×C53C2×#12111yescharacteristic
264C22×C53C2×#12311yescharacteristic
265C22×C53#57712yes—
266C22×C53#57712yes—
267C2×D53—42no—
268C2×D53—42no—
269C2×D53—42no—
270C2×D53—82no—
271C2×D53—82no—
272C4×C53—42no—
273C4×C53—82no—
274C4×C53—82no—
275C4×C53#39414yes—
276C4×C53C2×#12111yescharacteristic
277C4×C53C2×#12111yescharacteristic
278C4×C53C2×#12311yescharacteristic
279C4×C53C2×#12411yescharacteristic
280Dic53—42no—
281Dic53—44no—
282C2×#537—531no—
283C22×#106—531no—
284C2×#417—531no—
285C2×#416—531no—
286C2×#408—531no—
287#2311D5311yescharacteristic, complemented
288#2035—534no—
289#2034—534no—
290#1736—531no—
291C2×C53—24no—
292C2×C53—82no—
293C2×C53C2×#39411yescharacteristic
294C2×C53#300911yescharacteristic
295C2×C53#301011yescharacteristic
296C2×C53#306012yes—
297C2×C53#309912yes—
298D53—82no—
299C22×#41—531no—
300C22×#41C2×D5311yescharacteristic
301C22×M16—531no—
302C23×C8—531no—
303#537—1064no—
304#537—1064no—
305C2×C22⋊Q8—532no—
306C2×C22⋊Q8—1061no—
307C2×#121—532no—
308C2×#121—1061no—
309C2×#121—1062no—
310C2×#117C2×D5311yescharacteristic
311C22×C4⋊C4—531no—
312C22×C4⋊C4C2×D5311yescharacteristic
313C2×#106—534no—
314C2×#106—1064no—
315#417—1064no—
316#416—1064no—
317C2×#103—532no—
318C2×#103—532no—
319C2×#102—532no—
320C2×#102—532no—
321#408—1064no—
322C2×C22⋊C8—534no—
323#391—21no—
324#390—21no—
325#386—22no—
326#380C2×D5314yes—
327#340—534no—
328C53#2348911yescharacteristic, complemented, Sylow
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—534no—
336C2×#41—1062no—
337C2×#41—1062no—
338C2×#41—1062no—
339C2×#41—1062no—
340C2×#41—1062no—
341C22×Q8—531no—
342C22×Q8C22⋊D5311yescharacteristic
343C22×D4—531no—
344C22×D4—1062no—
345C22×D4—1061no—
346C22×D4C22⋊D5312yes—
347C22×D4C22⋊D5311yescharacteristic
348C23×C4—21no—
349C23×C4—1061no—
350C23×C4—1061no—
351C23×C4C22×D5312yes—
352C23×C4C22×D5311yescharacteristic
353C2×M16—1062no—
354C2×M16—1062no—
355C2×M16—1062no—
356C22×C8—534no—
357C22×C8—532no—
358C22×C8—532no—
359C22×C8—1062no—
360C22×C8—1062no—
361C22⋊Q8—1064no—
362C22⋊Q8—1064no—
363C22⋊Q8—1068no—
364#121—1064no—
365#121—1064no—
366#121—1068no—
367#121—10616no—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4C22×D5312yes—
379C2×C4⋊C4C22×D5312yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4—1062no—
384C2×C42—21no—
385#106—1068no—
386#103—1064no—
387#103—1064no—
388#102—1064no—
389#102—1064no—
390C22⋊C8—1068no—
391#95—24no—
392C24—1061no—
393C24C2×C22⋊D5312yes—
394C24C2×C22⋊D5311yescharacteristic
395#41—44no—
396#41—44no—
397#41—44no—
398#41—44no—
399#41—1068no—
400#41—1068no—
401#41—2124no—
402#41—2124no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—532no—
407C2×Q8—1061no—
408C2×Q8—1061no—
409C2×Q8—1061no—
410C2×Q8—1061no—
411C2×Q8—1061no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—532no—
419C2×D4—1068no—
420C2×D4—1064no—
421C2×D4—1061no—
422C2×D4—1061no—
423C2×D4—1061no—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4—4244no—
427C2×D4—2122no—
428C2×D4—2122no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—82no—
439C22×C4—1064no—
440C22×C4—1062no—
441C22×C4—1061no—
442C22×C4—1064no—
443C22×C4—4242no—
444C22×C4—2122no—
445C22×C4—2122no—
446C22×C4#1171994512yes—
447C22×C4#1171994512yes—
448C22×C4D4×D5312yes—
449C22×C4D4×D5312yes—
450C22×C4D4×D5312yes—
451C22×C4#1171994712yes—
452C22×C4C2×C22⋊D5312yes—
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C23×D5311yescharacteristic
455M16—2124no—
456C2×C8—534no—
457C2×C8—1068no—
458C2×C8—1062no—
459C2×C8—1062no—
460C2×C8—1062no—
461C2×C8—1062no—
462C2×C8—1062no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—534no—
469C4⋊C4—1068no—
470C4⋊C4—1064no—
471C4⋊C4—1062no—
472C22⋊C4—42no—
473C22⋊C4—84no—
474C22⋊C4—1068no—
475C22⋊C4—1068no—
476C42—42no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—1062no—
485C23—2121no—
486C23—2121no—
487C23—2122no—
488C23—2122no—
489C23#1313318512yes—
490C23#1313318912yes—
491C23#1313320012yes—
492C23C22×C22⋊D5311yescharacteristic
493Q8—44no—
494Q8—1062no—
495Q8—1062no—
496Q8—2121no—
497Q8—2121no—
498D4—48no—
499D4—44no—
500D4—1062no—
501D4—1062no—
502D4—4248no—
503D4—2128no—
504D4—2121no—
505D4—2121no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—84no—
519C2×C4—1064no—
520C2×C4—1062no—
521C2×C4—4244no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2124no—
525C2×C4—2128no—
526C2×C4#1313318512yes—
527C2×C4#1313318912yes—
528C2×C4#1313319014yes—
529C2×C4#1313319014yes—
530C2×C4#1313319914yes—
531C2×C4#1313320012yes—
532C2×C4#1313320314yes—
533C2×C4C2×D4×D5311yescharacteristic
534C2×C4C2×D4×D5311yescharacteristic
535C2×C4C2×D4×D5311yescharacteristic
536C2×C4C2×#1171994711yescharacteristic
537C2×C4C2×#1171994511yescharacteristic
538C2×C4C2×#1171994511yescharacteristic
539C8—1064no—
540C8—2122no—
541C22—24no—
542C22—22no—
543C22—22no—
544C22—24no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—44no—
550C22—4242no—
551C22—4242no—
552C22—2122no—
553C22—2122no—
554C22—2122no—
555C22#1681053814yes—
556C22C2×#1313318511yescharacteristic
557C22C2×#1313318911yescharacteristic
558C22C2×#1313320011yescharacteristic
559C22#1681143112yes—
560C22#1681143412yes—
561C4—42no—
562C4—82no—
563C4—82no—
564C4—2124no—
565C4—2122no—
566C4#1681053814yes—
567C4C2×#1313319011yescharacteristic
568C4C2×#1313319011yescharacteristic
569C4C2×#1313319911yescharacteristic
570C4C2×#1313320311yescharacteristic
571C2—24no—
572C2—82no—
573C2—4242no—
574C2C2×#1681053811yescharacteristic
575C2#2028399011yescharacteristic
576C2#2028399911yescharacteristic
577C2#2028437412yes—
578C2#2028470412yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2311C2
socleC23×C53C22×D4
prime radicalC2×#16811201C2
hypercenter#2311D53
hypocenterC53#23489
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C22×#13133148C2semidirect
3. C2×#16810622C2semidirect
4. C2×#16810617C2semidirect
5. C2×#16810590C2semidirect
6. #20275911C2semidirect
7. #20275904C2semidirect
8. #20274436C2semidirect
9. C53×#2311C2semidirect
56. C22×#88322D4semidirect
287. #2311D53semidirect
328. C53#23489semidirect

Maximal subgroups

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

subgroupindex
C2×#168112012
C22×#131331482
C2×#168106222
C2×#168106172
C2×#168105902
#202759112
#202759042
#202744362
C53×#23112
#2348953