Finite Group Databasebeta

← #27411666

Subgroups — #27411666

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×#16810619C211yesmaximal, characteristic, complemented
4C2×#16810591C211yesmaximal, characteristic, complemented
5#20277308C211yesmaximal, characteristic, complemented
6#20275909C214yesmaximal, complemented
7#20275902C214yesmaximal, complemented
8#20274435C211yesmaximal, characteristic, complemented
9C2×C53×#417C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C23×#88322C2211yescharacteristic
12#16811201—24no—
13#16811201—24no—
14C2×#13133196—21no—
15C2×#13133193C2212yes—
16C2×#13133191—21no—
17C2×#13133189—22no—
18C2×#13133188C2212yes—
19C2×#13133176C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C2×#13133148—24no—
22C2×#13133148C2214yes—
23#16810619—24no—
24C2×#13133141C2212yes—
25C2×#13133138C2212yes—
26#16810591—24no—
27C2×#13133129C2214yes—
28#16810528—21no—
29#16810524—21no—
30#16810502—22no—
31#16810480C2214yes—
32#16810386C2214yes—
33C22×#41×C53C2211yescharacteristic
34C22×M16×C53C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C53×#417—24no—
37C2×#103×C53C2212yes—
38C2×#102×C53C2212yes—
39C2×#11719945—22no—
40C2×#11719945—22no—
41C2×#11719945—22no—
42C2×#11719945—22no—
43C2×#11719945—22no—
44C2×#11719945—22no—
45C22×Dic106D411yescharacteristic
46C22×D212D411yescharacteristic
47C22×C22⋊D53—22no—
48C22×C22⋊D53D412yes—
49C23×Dic53—21no—
50C22×C4×D53C2312yes—
51C22×#88322—22no—
52C22×#88322—22no—
53C22×#88322D414yescomplemented
54C22×#88322C2312yes—
55#13133196—28no—
56#13133193—24no—
57#13133193—24no—
58#13133191—28no—
59#13133189—216no—
60#13133188—24no—
61#13133188—24no—
62#13133176—22no—
63#13133176—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53D414yes—
67C2×C4⋊Dic53C2312yes—
68C2×C4⋊Dic53C2312yes—
69C2×#11719919—22no—
70C2×C22⋊Dic53—21no—
71C2×C22⋊Dic53—22no—
72C2×#11719921—22no—
73C2×#11719921—24no—
74C2×C4×Dic53—21no—
75#13133148—28no—
76#13133141—24no—
77#13133138—24no—
78#13133129—28no—
79#13133124—24no—
80C2×#41×C53—22no—
81C2×#41×C53—22no—
82C2×#41×C53—22no—
83C2×#41×C53—22no—
84C2×#41×C53—22no—
85C2×#41×C53D414yes—
86C22×Q8×C53C2311yescharacteristic
87C22×D4×C53—21no—
88C22×D4×C53C2311yescharacteristic
89C23×C4×C53—21no—
90C23×C4×C53—21no—
91C23×C4×C53C2311yescharacteristic
92C2×M16×C53—22no—
93C2×M16×C53—22no—
94C2×M16×C53—22no—
95C22×C8×C53C2312yes—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53C2312yes—
100#103×C53—24no—
101#102×C53—24no—
102C24×C53—21no—
103C24×C53C2×D411yescharacteristic
104C23×D53C2×D412yes—
105#41×C53—28no—
106#41×C53—28no—
107#41×C53—44no—
108#41×C53—44no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53C2×D412yes—
115C2×D4×C53—24no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—42no—
122C2×D4×C53—42no—
123C2×D4×C53C2×D412yes—
124C22×C4×C53—21no—
125C22×C4×C53—24no—
126C22×C4×C53—22no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—82no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53#4112yes—
136C22×C4×C53#4112yes—
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—24no—
140C2×C22⋊D53—24no—
141C2×C22⋊D53—24no—
142C2×C22⋊D53—28no—
143C2×C22⋊D53—84no—
144C22×Dic53—22no—
145C22×Dic53—24no—
146C22×Dic53—22no—
147C22×Dic53—82no—
148C22×Dic53C2×D412yes—
149#11719945—44no—
150#11719945—44no—
151#11719945—44no—
152#11719945—44no—
153C2×D212—22no—
154C2×D212—22no—
155C2×D212—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×Dic106—22no—
163C2×Dic106—22no—
164C2×Dic106—22no—
165M16×C53—44no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—42no—
172C22⋊Dic53—28no—
173C22⋊Dic53—42no—
174#11719921—28no—
175#11719921—84no—
176C4⋊Dic53—24no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53C2×D414yes—
181#11719919—28no—
182C4×Dic53—42no—
183C2×#88322—28no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322C2×D414yes—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53#12012yes—
195C23×C53#12112yes—
196C23×C53#12312yes—
197C23×C53C22×D411yescharacteristic
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—24no—
202C22×D53—24no—
203Q8×C53—22no—
204Q8×C53—22no—
205Q8×C53—41no—
206Q8×C53—41no—
207D4×C53—22no—
208D4×C53—22no—
209D4×C53—48no—
210D4×C53—41no—
211D4×C53—41no—
212C2×C4×C53—22no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—48no—
216C2×C4×C53—42no—
217C2×C4×C53—44no—
218C2×C4×C53—84no—
219C2×C4×C53#12012yes—
220C2×C4×C53#12114yes—
221C2×C4×C53#12114yes—
222C2×C4×C53#12112yes—
223C2×C4×C53#12312yes—
224C2×C4×C53#12314yes—
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—22no—
237C2×Dic53—22no—
238C2×Dic53—44no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—84no—
243D212—44no—
244C4×D53—42no—
245C4×D53—42no—
246C4×D53—42no—
247C4×D53—42no—
248Dic106—44no—
249C8×C53—42no—
250#88322—24no—
251#23489—531nomaximal, Sylow
252C22×C53—24no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—24no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—82no—
260C22×C53—82no—
261C22×C53#39414yes—
262C22×C53C2×#12011yescharacteristic
263C22×C53C2×#12111yescharacteristic
264C22×C53C2×#12311yescharacteristic
265C22×C53#57712yes—
266C22×C53#57712yes—
267C2×D53—44no—
268C2×D53—42no—
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—44no—
272C4×C53—42no—
273C4×C53—82no—
274C4×C53#39414yes—
275C4×C53C2×#12111yescharacteristic
276C4×C53C2×#12111yescharacteristic
277C4×C53C2×#12311yescharacteristic
278C4×C53C2×#12411yescharacteristic
279Dic53—42no—
280Dic53—44no—
281Dic53—82no—
282C2×#537—531no—
283C22×#106—531no—
284C2×#417D5311yescharacteristic, complemented
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×C53C2×#39411yescharacteristic
294C2×C53#300911yescharacteristic
295C2×C53#301011yescharacteristic
296C2×C53#306012yes—
297C2×C53#309912yes—
298D53—82no—
299C22×#41—531no—
300C22×#41C2×D5311yescharacteristic
301C22×M16C2×D5311yescharacteristic
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×#117—531no—
311C22×C4⋊C4—531no—
312C22×C4⋊C4C2×D5311yescharacteristic
313C2×#106—534no—
314C2×#106—1064no—
315#417—24no—
316#416—1064no—
317C2×#103—532no—
318C2×#103C2×D5312yes—
319C2×#102—532no—
320C2×#102C2×D5312yes—
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—1062no—
335C2×#41—1062no—
336C2×#41—1062no—
337C2×#41—1062no—
338C2×#41—1062no—
339C2×#41—1062no—
340C2×#41C22⋊D5314yes—
341C22×Q8—531no—
342C22×Q8C22×D5311yescharacteristic
343C22×D4—21no—
344C22×D4—532no—
345C22×D4—531no—
346C22×D4—1062no—
347C22×D4C22×D5311yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—532no—
351C23×C4—1061no—
352C23×C4C22×D5311yescharacteristic
353C2×M16—22no—
354C2×M16—22no—
355C2×M16—22no—
356C22×C8—534no—
357C22×C8—532no—
358C22×C8—1062no—
359C22×C8—1062no—
360C22×C8C22×D5312yes—
361C22⋊Q8—1064no—
362C22⋊Q8—1068no—
363C22⋊Q8—1064no—
364#121—1064no—
365#121—1068no—
366#121—10616no—
367#121—1064no—
368#117—1062no—
369#117—1062no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—532no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4C22×D5312yes—
380C2×C22⋊C4—1062no—
381C2×C22⋊C4—1062no—
382C2×C22⋊C4—1064no—
383C2×C22⋊C4—1061no—
384C2×C42—1061no—
385#106—1068no—
386#103—24no—
387#103—1064no—
388#102—24no—
389#102—1064no—
390C22⋊C8—1068no—
391#95—1064no—
392C24—21no—
393C24—532no—
394C24D4×D5311yescharacteristic
395#41—28no—
396#41—28no—
397#41—44no—
398#41—44no—
399#41—2124no—
400#41—2124no—
401#41—2124no—
402#41—2124no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—1062no—
409C2×Q8—1062no—
410C2×Q8—1062no—
411C2×Q8C2×C22⋊D5312yes—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—1068no—
421C2×D4—1064no—
422C2×D4—1064no—
423C2×D4—1064no—
424C2×D4—1062no—
425C2×D4—1062no—
426C2×D4—1062no—
427C2×D4—4244no—
428C2×D4C2×C22⋊D5312yes—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—82no—
436C22×C4—532no—
437C22×C4—1064no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—4242no—
447C22×C4#1171994512yes—
448C22×C4#1171994512yes—
449C22×C4D4×D5311yescharacteristic
450C22×C4D4×D5312yes—
451C22×C4D4×D5312yes—
452C22×C4#1171994712yes—
453C22×C4C2×C22⋊D5312yes—
454C22×C4C23×D5311yescharacteristic
455M16—44no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—534no—
460C2×C8—1068no—
461C2×C8—1062no—
462C2×C8—1062no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—534no—
467C4⋊C4—1068no—
468C4⋊C4—1064no—
469C4⋊C4—1062no—
470C4⋊C4—1062no—
471C4⋊C4—1062no—
472C22⋊C4—1068no—
473C22⋊C4—1068no—
474C22⋊C4—4244no—
475C22⋊C4—2122no—
476C42—2122no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—1062no—
485C23—1062no—
486C23—1062no—
487C23—1064no—
488C23—1064no—
489C23#1313318512yes—
490C23#1313319012yes—
491C23#1313319912yes—
492C23C2×D4×D5311yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497Q8—2124no—
498D4—22no—
499D4—22no—
500D4—48no—
501D4—41no—
502D4—41no—
503D4—4248no—
504D4—2128no—
505D4—2124no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—84no—
513C2×C4—1064no—
514C2×C4—1062no—
515C2×C4—1062no—
516C2×C4—1062no—
517C2×C4—4244no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4—2124no—
522C2×C4—2122no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4—2122no—
526C2×C4#1313318512yes—
527C2×C4#1313318914yes—
528C2×C4#1313319012yes—
529C2×C4#1313319014yes—
530C2×C4#1313319912yes—
531C2×C4#1313320014yes—
532C2×C4#1313320314yes—
533C2×C4C2×D4×D5311yescharacteristic
534C2×C4C2×D4×D5311yescharacteristic
535C2×C4C22×C22⋊D5311yescharacteristic
536C2×C4C2×#1171994711yescharacteristic
537C2×C4C2×#1171994511yescharacteristic
538C2×C4C2×#1171994511yescharacteristic
539C8—42no—
540C8—1064no—
541C22—24no—
542C22—22no—
543C22—22no—
544C22—24no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—82no—
549C22—82no—
550C22—2124no—
551C22—2122no—
552C22—2122no—
553C22—2122no—
554C22—2124no—
555C22#1681053814yes—
556C22C2×#1313318511yescharacteristic
557C22C2×#1313319011yescharacteristic
558C22C2×#1313319911yescharacteristic
559C22#1681143212yes—
560C22#1681143312yes—
561C4—42no—
562C4—82no—
563C4—4242no—
564C4—2124no—
565C4—2122no—
566C4#1681053814yes—
567C4C2×#1313318911yescharacteristic
568C4C2×#1313319011yescharacteristic
569C4C2×#1313320011yescharacteristic
570C4C2×#1313320311yescharacteristic
571C2—24no—
572C2—82no—
573C2—4242no—
574C2C2×#1681053811yescharacteristic
575C2#2028399111yescharacteristic
576C2#2028400011yescharacteristic
577C2#2028437612yes—
578C2#2028470312yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#417C2
socleC23×C53C22×D4
prime radicalC2×#16811201C2
hypercenterC2×#417D53
hypocenterC53#23489
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#13133148C2semidirect
3. C2×#16810619C2semidirect
4. C2×#16810591C2semidirect
5. #20277308C2semidirect
6. #20275909C2semidirect
7. #20275902C2semidirect
8. #20274435C2semidirect
9. C2×C53×#417C2semidirect
53. C22×#88322D4semidirect
284. C2×#417D53semidirect
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×#168106192
C2×#168105912
#202773082
#202759092
#202759022
#202744352
C2×C53×#4172
#2348953