Finite Group Databasebeta

← #27385681

Subgroups — #27385681

Order 13568.

1694 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811194C211yesmaximal, characteristic
2C2×#16811100C211yesmaximal, characteristic, complemented
3#20283234C211yesmaximal, characteristic
4C22×#13133135C211yesmaximal, characteristic, complemented
5#20275115C214yesmaximal, complemented
6#20275105C214yesmaximal, complemented
7#20274426C211yesmaximal, characteristic, complemented
8#20274424C211yesmaximal, characteristic, complemented
9C53×#2896C211yesmaximal, characteristic
10C23×Dic106C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12#16811194—24no—
13#16811194C418yescomplemented
14C2×#13133215C2212yes—
15C2×#13133204C2212yes—
16C2×#13133198—22no—
17C2×#13133196—22no—
18#16811100—22no—
19#16811100—22no—
20#16811100—22no—
21#16811100C2214yes—
22C2×C4×Dic106—22no—
23C2×#13133182—22no—
24C2×#13133176C2211yescharacteristic
25C2×#13133176C2211yescharacteristic
26C2×#13133173—22no—
27#16810632C2212yes—
28#16810628C2212yes—
29#16810624—22no—
30#16810619—22no—
31C2×#13133135—24no—
32C2×#13133135C2214yes—
33#16810384C2214yes—
34#16810382C2214yes—
35C22×#41×C53C2211yescharacteristic
36C22×M16×C53C2211yescharacteristic
37C2×#117×C53C2211yescharacteristic
38C53×#419C2212yes—
39C53×#418C2212yes—
40C53×#417—22no—
41C53×#416—22no—
42C22×Dic106—21no—
43C22×Dic106—21no—
44C22×Dic106—21no—
45C22×Dic106—21no—
46C22×Dic106—22no—
47C22×Dic106—22no—
48C22×Dic106C2×C414yes—
49C22×Dic106D412yes—
50C22×Dic106D414yes—
51C22×Dic106C2312yes—
52C23×Dic53D412yes—
53C2×#11719917—21no—
54C2×#11719917—22no—
55C2×#11719917—22no—
56C2×#11719917C2312yes—
57C22×#88322—21no—
58#13133215—24no—
59#13133215C2×C418yescomplemented
60#13133204—24no—
61#13133204C2×C418yescomplemented
62#13133198—216no—
63#13133196—216no—
64C4×Dic106—28no—
65C4×Dic106—28no—
66#13133182—28no—
67#13133182—28no—
68#13133176—21no—
69#13133176—21no—
70#13133176—22no—
71#13133176C2×C414yes—
72#13133176D412yes—
73#13133176C2312yes—
74#13133173—28no—
75#13133173—28no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53D412yes—
78C2×#11719919—24no—
79C2×#11719919—24no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—24no—
83C2×C4×Dic53—24no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2312yes—
86#13133141—24no—
87#13133138—24no—
88#13133135—28no—
89C2×#41×C53—22no—
90C2×#41×C53—21no—
91C2×#41×C53—21no—
92C2×#41×C53—21no—
93C2×#41×C53—21no—
94C2×#41×C53—22no—
95C2×#41×C53—22no—
96C2×#41×C53C2×C414yes—
97C22×Q8×C53C2311yescharacteristic
98C22×D4×C53—21no—
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53—21no—
101C23×C4×C53C2311yescharacteristic
102C2×M16×C53—21no—
103C2×M16×C53—22no—
104C2×M16×C53—22no—
105C2×M16×C53C2312yes—
106C22×C8×C53—21no—
107#117×C53—21no—
108#117×C53—21no—
109#117×C53D412yes—
110#117×C53C2312yes—
111C2×C4⋊C4×C53D412yes—
112C2×C22⋊C4×C53—21no—
113C2×C42×C53—21no—
114#103×C53—24no—
115#102×C53—24no—
116C24×C53—21no—
117C24×C53C2×D411yescharacteristic
118#41×C53—24no—
119#41×C53—24no—
120#41×C53—24no—
121#41×C53—24no—
122#41×C53—42no—
123#41×C53—42no—
124#41×C53—42no—
125#41×C53—42no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C22×C412yes—
132C2×D4×C53—24no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—41no—
139C2×D4×C53—41no—
140C2×D4×C53—42no—
141C2×D4×C53C22×C412yes—
142C22×C4×C53—22no—
143C22×C4×C53—21no—
144C22×C4×C53—24no—
145C22×C4×C53—41no—
146C22×C4×C53—41no—
147C22×C4×C53—42no—
148C22×C4×C53C22×C412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53#4112yes—
156C22×C4×C53#4112yes—
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—24no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53C22⋊C418yes—
164C22×Dic53C2×D414yes—
165C2×Dic106—24no—
166C2×Dic106—24no—
167C2×Dic106—24no—
168C2×Dic106—24no—
169C2×Dic106—22no—
170C2×Dic106—22no—
171C2×Dic106—24no—
172C2×Dic106—24no—
173C2×Dic106—24no—
174C2×Dic106—24no—
175C2×Dic106—24no—
176C2×Dic106—22no—
177C2×Dic106—22no—
178C2×Dic106—22no—
179C2×Dic106—22no—
180C2×Dic106C22⋊C418yes—
181C2×Dic106C22×C414yes—
182C2×Dic106C2×D414yes—
183M16×C53—22no—
184M16×C53—22no—
185M16×C53—42no—
186C2×C8×C53—22no—
187C2×C8×C53—42no—
188C4⋊C4×C53—22no—
189C4⋊C4×C53C2×D414yes—
190C22⋊C4×C53—22no—
191C22⋊C4×C53—22no—
192C42×C53—22no—
193C42×C53—22no—
194C22⋊Dic53—24no—
195C22⋊Dic53—22no—
196C22⋊Dic53—22no—
197C22⋊Dic53—216no—
198C4⋊Dic53—24no—
199C4⋊Dic53—22no—
200C4⋊Dic53C2×D414yes—
201#11719919—216no—
202#11719919—216no—
203C4×Dic53—216no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53C22×C414yes—
208#11719917—22no—
209#11719917—22no—
210#11719917—42no—
211C2×#88322—22no—
212C2×#88322—42no—
213C23×C53—22no—
214C23×C53—42no—
215C23×C53—44no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53C2×C22⋊C412yes—
219C23×C53#12011yescharacteristic
220C23×C53#12011yescharacteristic
221C23×C53#12312yes—
222C23×C53C22×D411yescharacteristic
223Q8×C53—22no—
224Q8×C53—22no—
225Q8×C53—41no—
226Q8×C53—41no—
227D4×C53—22no—
228D4×C53—22no—
229D4×C53—44no—
230D4×C53—44no—
231D4×C53—41no—
232D4×C53—41no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—44no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—44no—
239C2×C4×C53—44no—
240C2×C4×C53C2×C22⋊C412yes—
241C2×C4×C53C4×D412yes—
242C2×C4×C53C4×D412yes—
243C2×C4×C53C4×D412yes—
244C2×C4×C53C4×D412yes—
245C2×C4×C53#12011yescharacteristic
246C2×C4×C53#12011yescharacteristic
247C2×C4×C53#12112yes—
248C2×C4×C53#12112yes—
249C2×C4×C53#12112yes—
250C2×C4×C53#12112yes—
251C2×C4×C53#12312yes—
252C2×C4×C53C23×C411yescharacteristic
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—28no—
261C2×Dic53—28no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—44no—
265C2×Dic53—44no—
266C2×Dic53C2×C22⋊C418yes—
267Dic106—24no—
268Dic106—24no—
269Dic106—24no—
270Dic106—24no—
271Dic106—28no—
272Dic106—28no—
273C8×C53—42no—
274#88322—42no—
275#20954—531nomaximal, Sylow
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—44no—
282C22×C53—42no—
283C22×C53—82no—
284C22×C53—82no—
285C22×C53#38612yes—
286C22×C53#38612yes—
287C22×C53C22×C22⋊C411yescharacteristic
288C22×C53C2×#12011yescharacteristic
289C22×C53C2×#12311yescharacteristic
290C4×C53—44no—
291C4×C53—42no—
292C4×C53#38612yes—
293C4×C53#38612yes—
294C4×C53C2×C4×D411yescharacteristic
295C4×C53C2×C4×D411yescharacteristic
296C4×C53C2×#12111yescharacteristic
297C4×C53C2×#12111yescharacteristic
298Dic53—28no—
299Dic53—44no—
300Dic53—44no—
301C2×#536—531no—
302C2×#519—531no—
303#2896—531no—
304#2896D5311yescharacteristic
305C22×#101—531no—
306#1888—534no—
307#1886—534no—
308#1733—531no—
309#1733—531no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53—22no—
313C2×C53—22no—
314C2×C53—82no—
315C2×C53C2×#38611yescharacteristic
316C2×C53#302012yes—
317C22×#41C2×D5311yescharacteristic
318C23×Q8—531no—
319C22×M16—531no—
320C22×M16C2×D5311yescharacteristic
321#536—538no—
322#536—1064no—
323C2×C4⋊Q8—532no—
324C2×#124—532no—
325C2×#123—1062no—
326C2×C22⋊Q8—1062no—
327#519—534no—
328#519—1062no—
329#519—1062no—
330#519—1062no—
331C2×C4×Q8—1062no—
332C2×C4×Q8—1062no—
333C2×#117—531no—
334C2×#117—531no—
335C2×#117—1062no—
336C2×#117C2×D5311yescharacteristic
337#419—532no—
338#419C2×D5312yes—
339#418—532no—
340#418C2×D5312yes—
341#417—22no—
342#417—1062no—
343#416—22no—
344#416—1062no—
345C2×#101—534no—
346C2×#101—1064no—
347#339—534no—
348#339—534no—
349C53#2095411yescharacteristic, complemented, Sylow
350C2×#41—22no—
351C2×#41—22no—
352C2×#41—21no—
353C2×#41—21no—
354C2×#41—22no—
355C2×#41—21no—
356C2×#41—21no—
357C2×#41C4×D5314yes—
358C22×Q8—534no—
359C22×Q8—532no—
360C22×Q8—534no—
361C22×Q8—532no—
362C22×Q8—1062no—
363C22×Q8—1061no—
364C22×Q8—1061no—
365C22×Q8—1062no—
366C22×Q8—1061no—
367C22×Q8—1061no—
368C22×Q8C22×D5311yescharacteristic
369C22×D4—21no—
370C22×D4C22×D5311yescharacteristic
371C23×C4—21no—
372C23×C4—532no—
373C23×C4C22×D5311yescharacteristic
374C2×M16—21no—
375C2×M16—22no—
376C2×M16—22no—
377C2×M16—532no—
378C2×M16—1061no—
379C2×M16—1062no—
380C2×M16—1062no—
381C2×M16C22×D5312yes—
382C22×C8—21no—
383C22×C8—1061no—
384C4⋊Q8—538no—
385C4⋊Q8—1064no—
386#124—538no—
387#124—1064no—
388#123—10616no—
389C22⋊Q8—10616no—
390C4×Q8—1068no—
391C4×Q8—1068no—
392C4×Q8—1068no—
393C4×Q8—1068no—
394#117—21no—
395#117—21no—
396#117—534no—
397#117—532no—
398#117—532no—
399#117—1068no—
400#117—1068no—
401#117—1061no—
402#117—1062no—
403#117—1061no—
404#117C22⋊D5312yes—
405#117C22×D5312yes—
406C2×C4⋊C4—532no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4C22⋊D5312yes—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1064no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C42—21no—
416C2×C42—532no—
417C2×C42—1064no—
418C2×C42—1061no—
419#103—24no—
420#103—1064no—
421#102—24no—
422#102—1064no—
423#101—1068no—
424C24—21no—
425C24D4×D5311yescharacteristic
426#41—24no—
427#41—24no—
428#41—24no—
429#41—24no—
430#41—42no—
431#41—42no—
432#41—42no—
433#41—42no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—538no—
440C2×Q8—534no—
441C2×Q8—534no—
442C2×Q8—1064no—
443C2×Q8—1064no—
444C2×Q8—1064no—
445C2×Q8—1064no—
446C2×Q8—1062no—
447C2×Q8—1062no—
448C2×Q8—1064no—
449C2×Q8—1064no—
450C2×Q8—1064no—
451C2×Q8—1064no—
452C2×Q8—1064no—
453C2×Q8—1062no—
454C2×Q8—1062no—
455C2×Q8—1062no—
456C2×Q8—1062no—
457C2×Q8C2×C4×D5312yes—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4—42no—
467C2×D4C2×C4×D5312yes—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—538no—
475C22×C4—534no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1064no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4C2×C4×D5312yes—
482C22×C4#1171994512yes—
483C22×C4D4×D5311yescharacteristic
484C22×C4D4×D5312yes—
485C22×C4D4×D5311yescharacteristic
486C22×C4D4×D5311yescharacteristic
487C22×C4#1171994712yes—
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C23×D5311yescharacteristic
491M16—22no—
492M16—22no—
493M16—42no—
494M16—1062no—
495M16—1062no—
496M16—2122no—
497C2×C8—22no—
498C2×C8—42no—
499C2×C8—1062no—
500C2×C8—2122no—
501C4⋊C4—22no—
502C4⋊C4—534no—
503C4⋊C4—10616no—
504C4⋊C4—10616no—
505C4⋊C4—1064no—
506C4⋊C4—1062no—
507C4⋊C4C2×C22⋊D5314yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—10616no—
511C22⋊C4—1064no—
512C22⋊C4—1062no—
513C22⋊C4—1062no—
514C42—22no—
515C42—22no—
516C42—534no—
517C42—10616no—
518C42—1062no—
519C42—1062no—
520C42—1062no—
521C23—22no—
522C23—42no—
523C23—44no—
524C23—41no—
525C23—41no—
526C23C22⋊C4×D5312yes—
527C23#1313318511yescharacteristic
528C23#1313318511yescharacteristic
529C23#1313319912yes—
530C23C2×D4×D5311yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535Q8—1068no—
536Q8—1068no—
537Q8—1064no—
538Q8—1064no—
539Q8—1064no—
540Q8—1064no—
541D4—22no—
542D4—22no—
543D4—44no—
544D4—44no—
545D4—41no—
546D4—41no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—538no—
555C2×C4—1068no—
556C2×C4—1068no—
557C2×C4—1064no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—2124no—
562C2×C4—2124no—
563C2×C4C22⋊C4×D5312yes—
564C2×C4#1313317712yes—
565C2×C4#1313317712yes—
566C2×C4C4×C22⋊D5312yes—
567C2×C4C4×C22⋊D5312yes—
568C2×C4#1313318511yescharacteristic
569C2×C4#1313318511yescharacteristic
570C2×C4#1313318812yes—
571C2×C4#1313318812yes—
572C2×C4#1313319012yes—
573C2×C4#1313319012yes—
574C2×C4#1313319912yes—
575C2×C4C22×C4×D5311yescharacteristic
576C2×C4C2×D4×D5311yescharacteristic
577C2×C4C2×D4×D5311yescharacteristic
578C2×C4C22×C22⋊D5311yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4C2×#1171994511yescharacteristic
581C8—42no—
582C8—2122no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22—82no—
591C22—82no—
592C22#1681049912yes—
593C22#1681049912yes—
594C22C2×C22⋊C4×D5311yescharacteristic
595C22C2×#1313318511yescharacteristic
596C22C2×#1313319911yescharacteristic
597C4—42no—
598C4—44no—
599C4—1068no—
600C4—2124no—
601C4—2124no—
602C4#1681049912yes—
603C4#1681049912yes—
604C4C2×#1313317711yescharacteristic
605C4C2×C4×C22⋊D5311yescharacteristic
606C4C2×#1313318811yescharacteristic
607C4C2×#1313319011yescharacteristic
608C2—21no—
609C2—21no—
610C2—22no—
611C2—22no—
612C2—82no—
613C2C2×#1681049911yescharacteristic
614C2#2028407712yes—
subgroupG/H
centerC22C2×#13133185
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2896C2
socleC22×C53C2×#120
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenter#2896D53
hypocenterC53#20954
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. C2×#16811100C2semidirect
4. C22×#13133135C2semidirect
5. #20275115C2semidirect
6. #20275105C2semidirect
7. #20274426C2semidirect
8. #20274424C2semidirect
13. #16811194C4semidirect
59. #13133215C2×C4semidirect
61. #13133204C2×C4semidirect
349. C53#20954semidirect

Maximal subgroups

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

subgroupindex
C2×#168111942
C2×#168111002
#202832342
C22×#131331352
#202751152
#202751052
#202744262
#202744242
C53×#28962
#2095453