Finite Group Databasebeta

← #27411665

Subgroups — #27411665

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
1C22×#13133148C211yesmaximal, characteristic, complemented
2C2×#16810624C211yesmaximal, characteristic, complemented
3C2×#16810619C211yesmaximal, characteristic, complemented
4C2×#16810590C211yesmaximal, characteristic, complemented
5#20277307C211yesmaximal, characteristic, complemented
6#20275908C214yesmaximal, complemented
7#20275901C214yesmaximal, complemented
8#20274434C211yesmaximal, characteristic, complemented
9C2×C53×#537C211yesmaximal, characteristic
10C22×#11719917C2211yescharacteristic
11C23×#88322C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C2×#13133148—24no—
15C2×#13133148C2214yes—
16#16810624—24no—
17#16810619—24no—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22#16810590—24no—
23C2×C22⋊#88322C2214yes—
24#16810530—21no—
25#16810526—21no—
26#16810503—22no—
27#16810483C2214yes—
28#16810385C2214yes—
29C22×#41×C53C2211yescharacteristic
30C22×#41×C53C2211yescharacteristic
31C53×#537—24no—
32C53×#537—24no—
33C2×C22⋊Q8×C53—21no—
34C2×C22⋊Q8×C53C2212yes—
35C2×#121×C53—21no—
36C2×#121×C53—22no—
37C2×#121×C53C2212yes—
38C22×C4⋊C4×C53C2211yescharacteristic
39C23×Dic53—21no—
40C2×#11719917—22no—
41C2×#11719917—22no—
42C2×#11719917—22no—
43C22×#88322—22no—
44C22×#88322—22no—
45C22×#88322D414yescomplemented
46C22×#88322C2312yes—
47C22×#88322C2312yes—
48#13133176—22no—
49#13133176—22no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2312yes—
55C2×C22⋊Dic53—21no—
56C2×C22⋊Dic53—24no—
57C2×C4×Dic53—21no—
58#13133148—28no—
59#13133141—24no—
60#13133141—24no—
61#13133138—24no—
62#13133138—24no—
63C22⋊#88322—28no—
64#13133124—24no—
65C2×#41×C53—22no—
66C2×#41×C53—22no—
67C2×#41×C53—22no—
68C2×#41×C53—22no—
69C2×#41×C53—22no—
70C2×#41×C53—22no—
71C2×#41×C53—22no—
72C2×#41×C53—22no—
73C2×#41×C53—22no—
74C2×#41×C53—22no—
75C2×#41×C53—22no—
76C2×#41×C53D414yes—
77C22×Q8×C53D411yescharacteristic
78C22×Q8×C53C2311yescharacteristic
79C22×D4×C53—21no—
80C22×D4×C53—22no—
81C22×D4×C53D412yes—
82C22×D4×C53D411yescharacteristic
83C22×D4×C53C2311yescharacteristic
84C23×C4×C53—21no—
85C23×C4×C53—21no—
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C22⋊Q8×C53—28no—
89C22⋊Q8×C53—24no—
90C22⋊Q8×C53—24no—
91#121×C53—216no—
92#121×C53—28no—
93#121×C53—24no—
94#121×C53—24no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53D414yes—
99C2×C4⋊C4×C53C2312yes—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—22no—
102C24×C53—21no—
103C24×C53C2×D412yes—
104C24×C53C2×D411yescharacteristic
105#41×C53—28no—
106#41×C53—28no—
107#41×C53—44no—
108#41×C53—44no—
109#41×C53—44no—
110#41×C53—44no—
111#41×C53—44no—
112#41×C53—44no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53—21no—
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53C2×D412yes—
122C2×D4×C53—24no—
123C2×D4×C53—24no—
124C2×D4×C53—24no—
125C2×D4×C53—24no—
126C2×D4×C53—28no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—22no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—42no—
136C2×D4×C53—42no—
137C2×D4×C53—84no—
138C2×D4×C53C2×D412yes—
139C22×C4×C53—22no—
140C22×C4×C53—21no—
141C22×C4×C53—24no—
142C22×C4×C53—24no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—42no—
150C22×C4×C53—42no—
151C22×C4×C53—82no—
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2×D412yes—
157C22×C4×C53#4112yes—
158C22×C4×C53#4112yes—
159C22×C4×C53#4112yes—
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—42no—
164C22×Dic53—82no—
165C4⋊C4×C53—28no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53C2×D414yes—
169C22⋊C4×C53—28no—
170C22⋊C4×C53—28no—
171C22⋊Dic53—42no—
172C22⋊Dic53—84no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177C4⋊Dic53—42no—
178C4×Dic53—42no—
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—22no—
193C23×C53—24no—
194C23×C53—24no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53#12012yes—
200C23×C53#12112yes—
201C23×C53#12312yes—
202C23×C53C22×D411yescharacteristic
203Q8×C53—22no—
204Q8×C53—22no—
205Q8×C53—41no—
206Q8×C53—41no—
207Q8×C53—44no—
208D4×C53—22no—
209D4×C53—22no—
210D4×C53—48no—
211D4×C53—48no—
212D4×C53—41no—
213D4×C53—41no—
214D4×C53—44no—
215D4×C53—88no—
216C2×C4×C53—22no—
217C2×C4×C53—24no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—44no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—48no—
229C2×C4×C53—84no—
230C2×C4×C53#12012yes—
231C2×C4×C53#12114yes—
232C2×C4×C53#12114yes—
233C2×C4×C53#12112yes—
234C2×C4×C53#12314yes—
235C2×C4×C53#12312yes—
236C2×C4×C53#12414yes—
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—44no—
247C2×Dic53—42no—
248C2×Dic53—84no—
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—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—44no—
264C22×C53—82no—
265C22×C53—82no—
266C22×C53#39414yes—
267C22×C53C2×#12011yescharacteristic
268C22×C53C2×#12111yescharacteristic
269C22×C53C2×#12311yescharacteristic
270C22×C53#57712yes—
271C22×C53#57712yes—
272C4×C53—42no—
273C4×C53—42no—
274C4×C53—44no—
275C4×C53#39414yes—
276C4×C53C2×#12111yescharacteristic
277C4×C53C2×#12111yescharacteristic
278C4×C53C2×#12311yescharacteristic
279C4×C53C2×#12411yescharacteristic
280Dic53—82no—
281Dic53—82no—
282C2×#537D5311yescharacteristic
283C22×#106—531no—
284C2×#417—531no—
285C2×#416—531no—
286C2×#408—531no—
287#2311—531no—
288#2035—534no—
289#2034—534no—
290#1736—531no—
291C2×C53—24no—
292C2×C53—82no—
293C2×C53—82no—
294C2×C53C2×#39411yescharacteristic
295C2×C53#300911yescharacteristic
296C2×C53#301011yescharacteristic
297C2×C53#306012yes—
298C2×C53#309912yes—
299C22×#41C2×D5311yescharacteristic
300C22×#41C2×D5311yescharacteristic
301C22×M16—531no—
302C23×C8—531no—
303#537—24no—
304#537—24no—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8C2×D5312yes—
307C2×#121—21no—
308C2×#121—22no—
309C2×#121C2×D5312yes—
310C2×#117—531no—
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—1061no—
324#390—1061no—
325#386—1062no—
326#380—534no—
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—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41C22⋊D5314yes—
341C22×Q8C22⋊D5311yescharacteristic
342C22×Q8C22×D5311yescharacteristic
343C22×D4—21no—
344C22×D4—22no—
345C22×D4C22⋊D5312yes—
346C22×D4C22⋊D5311yescharacteristic
347C22×D4C22×D5311yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
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—28no—
362C22⋊Q8—24no—
363C22⋊Q8—24no—
364#121—216no—
365#121—28no—
366#121—24no—
367#121—24no—
368#117—1062no—
369#117—1062no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4C22⋊D5314yes—
379C2×C4⋊C4C22×D5312yes—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1064no—
383C2×C22⋊C4—1061no—
384C2×C42—1061no—
385#106—1068no—
386#103—1064no—
387#103—1064no—
388#102—1064no—
389#102—1064no—
390C22⋊C8—1068no—
391#95—1064no—
392C24—21no—
393C24C2×C22⋊D5312yes—
394C24C2×C22⋊D5311yescharacteristic
395#41—28no—
396#41—28no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401#41—44no—
402#41—44no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8C2×C22⋊D5312yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—84no—
428C2×D4C2×C22⋊D5312yes—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—4242no—
445C22×C4—2122no—
446C22×C4D4×D5312yes—
447C22×C4#1171994712yes—
448C22×C4#1171994712yes—
449C22×C4#1171994712yes—
450C22×C4C2×C22⋊D5312yes—
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5312yes—
453C22×C4C2×C22⋊D5312yes—
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—28no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—1062no—
467C4⋊C4—1062no—
468C4⋊C4—1062no—
469C4⋊C4—1062no—
470C4⋊C4—2122no—
471C4⋊C4C2×C22⋊D5314yes—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—4244no—
475C22⋊C4—2122no—
476C42—2122no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23#1313318612yes—
490C23#1313318912yes—
491C23#1313319812yes—
492C23C22×C22⋊D5311yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—44no—
496Q8—41no—
497Q8—41no—
498D4—22no—
499D4—22no—
500D4—48no—
501D4—48no—
502D4—44no—
503D4—41no—
504D4—41no—
505D4—88no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—4244no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2122no—
524C2×C4—2124no—
525C2×C4—2122no—
526C2×C4#1313318612yes—
527C2×C4#1313318814yes—
528C2×C4#1313318912yes—
529C2×C4#1313318914yes—
530C2×C4#1313319812yes—
531C2×C4#1313319814yes—
532C2×C4#1313320414yes—
533C2×C4C2×D4×D5311yescharacteristic
534C2×C4C22×C22⋊D5311yescharacteristic
535C2×C4C22×C22⋊D5311yescharacteristic
536C2×C4C2×#1171994711yescharacteristic
537C2×C4C2×#1171994711yescharacteristic
538C2×C4C2×#1171994711yescharacteristic
539C8—1064no—
540C8—2122no—
541C22—24no—
542C22—22no—
543C22—22no—
544C22—24no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—82no—
554C22—82no—
555C22#1681053714yes—
556C22C2×#1313318611yescharacteristic
557C22C2×#1313318911yescharacteristic
558C22C2×#1313319811yescharacteristic
559C22#1681143112yes—
560C22#1681143412yes—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—4242no—
565C4—4242no—
566C4#1681053714yes—
567C4C2×#1313318811yescharacteristic
568C4C2×#1313318911yescharacteristic
569C4C2×#1313319811yescharacteristic
570C4C2×#1313320411yescharacteristic
571C2—24no—
572C2—82no—
573C2—82no—
574C2C2×#1681053711yescharacteristic
575C2#2028399211yescharacteristic
576C2#2028400111yescharacteristic
577C2#2028437412yes—
578C2#2028470012yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#537C2
socleC23×C53C22×D4
hypercenterC2×#537D53
hypocenterC53#23489
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. C22×#13133148C2semidirect
2. C2×#16810624C2semidirect
3. C2×#16810619C2semidirect
4. C2×#16810590C2semidirect
5. #20277307C2semidirect
6. #20275908C2semidirect
7. #20275901C2semidirect
8. #20274434C2semidirect
45. C22×#88322D4semidirect
328. C53#23489semidirect

Maximal subgroups

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

subgroupindex
C22×#131331482
C2×#168106242
C2×#168106192
C2×#168105902
#202773072
#202759082
#202759012
#202744342
C2×C53×#5372
#2348953