Finite Group Databasebeta

← #27419210

Subgroups — #27419210

Order 13568.

1558 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810842C212yesmaximal, complemented
2C2×#16810779C211yesmaximal, characteristic, complemented
3C2×#16810774C211yesmaximal, characteristic
4C4×#13133196C211yesmaximal, characteristic, complemented
5#20275951C212yesmaximal, complemented
6#20275809C212yesmaximal, complemented
7#20275648C212yesmaximal, complemented
8#20274695C211yesmaximal, characteristic, complemented
9#20274650C211yesmaximal, characteristic, complemented
10#20274646C211yesmaximal, characteristic
11C22×#124×C53C211yesmaximal, characteristic
12C22×#11719925C2212yescomplemented
13C22×#11719923C2212yes—
14C2×#13133196C2211yescharacteristic
15C2×C4×Dic106C2211yescharacteristic, complemented
16#16810842—28no—
17#16810779—24no—
18#16810774—24no—
19C2×C4⋊#88322C2212yescomplemented
20C2×#13133141C2212yescomplemented
21C2×#13133141C2212yescomplemented
22C2×#13133141C2212yescomplemented
23C2×#13133138C2212yes—
24C2×#13133138C2212yes—
25C2×#13133138C2212yes—
26C2×C22⋊#88322C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810529C2211yescharacteristic, complemented
29#16810521C2211yescharacteristic, complemented
30#16810507—21no—
31#16810496—21no—
32#16810483C2211yescharacteristic
33C4×#11719919—21no—
34C4×C4⋊Dic53C2211yescharacteristic, complemented
35C4×C22⋊Dic53—21no—
36#16810378C2212yescomplemented
37C23×Q8×C53C2211yescharacteristic
38C23×D4×C53C2211yescharacteristic
39C2×#124×C53—28no—
40C2×#124×C53—21no—
41C2×#124×C53—21no—
42C2×#124×C53C2212yes—
43C2×#124×C53C2212yes—
44C22×C22⋊C4×C53—21no—
45C22×C22⋊C4×C53—21no—
46C22×C42×C53C2211yescharacteristic
47C22×Dic106C2311yescharacteristic
48C2×#11719925—24no—
49C2×#11719925—24no—
50C2×#11719925—42no—
51C2×#11719923—24no—
52C2×#11719923—24no—
53C2×#11719923—42no—
54C22×#88322C2312yes—
55C22×#88322C2312yes—
56#13133196—22no—
57#13133196D414yes—
58C4×Dic106—24no—
59C2×C4⋊Dic53C2311yescharacteristic
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C4⋊Dic53C2311yescharacteristic
62C2×#11719919—21no—
63C2×#11719919—21no—
64C2×#11719919—21no—
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53—21no—
69C4⋊#88322—24no—
70#13133141—24no—
71#13133141—24no—
72#13133141—42no—
73#13133138—24no—
74#13133138—24no—
75#13133138—42no—
76C22⋊#88322—22no—
77C22⋊#88322—22no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81#13133124—21no—
82#13133124—21no—
83C25×C53—21no—
84C22×Q8×C53—24no—
85C22×Q8×C53—21no—
86C22×Q8×C53—22no—
87C22×Q8×C53—22no—
88C22×Q8×C53—42no—
89C22×Q8×C53C2312yes—
90C22×D4×C53—24no—
91C22×D4×C53—21no—
92C22×D4×C53—22no—
93C22×D4×C53—22no—
94C22×D4×C53—42no—
95C22×D4×C53C2312yes—
96C23×C4×C53—21no—
97C23×C4×C53D411yescharacteristic
98C23×C4×C53D411yescharacteristic
99C23×C4×C53C2311yescharacteristic
100#124×C53—28no—
101#124×C53—28no—
102#124×C53—24no—
103#124×C53—24no—
104#124×C53—24no—
105#124×C53D418yes—
106C2×C22⋊C4×C53—28no—
107C2×C22⋊C4×C53—28no—
108C2×C22⋊C4×C53—22no—
109C2×C22⋊C4×C53—22no—
110C2×C22⋊C4×C53—22no—
111C2×C22⋊C4×C53—22no—
112C2×C42×C53—24no—
113C2×C42×C53D411yescharacteristic
114C2×C42×C53D411yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C24×C53—24no—
118C24×C53—22no—
119C24×C53—42no—
120C24×C53—42no—
121C24×C53—42no—
122C24×C53—42no—
123C24×C53—42no—
124C24×C53—42no—
125C24×C53C2×D411yescharacteristic
126C2×Q8×C53—24no—
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53—22no—
132C2×Q8×C53—48no—
133C2×Q8×C53—48no—
134C2×Q8×C53—42no—
135C2×Q8×C53—42no—
136C2×Q8×C53C2×D414yes—
137C2×D4×C53—24no—
138C2×D4×C53—22no—
139C2×D4×C53—22no—
140C2×D4×C53—22no—
141C2×D4×C53—22no—
142C2×D4×C53—22no—
143C2×D4×C53—48no—
144C2×D4×C53—48no—
145C2×D4×C53—42no—
146C2×D4×C53—42no—
147C2×D4×C53C2×D414yes—
148C22×C4×C53—22no—
149C22×C4×C53—22no—
150C22×C4×C53—24no—
151C22×C4×C53—22no—
152C22×C4×C53—22no—
153C22×C4×C53—22no—
154C22×C4×C53—22no—
155C22×C4×C53—22no—
156C22×C4×C53—41no—
157C22×C4×C53—41no—
158C22×C4×C53—42no—
159C22×C4×C53—42no—
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D412yes—
166C22×C4×C53#4112yes—
167C22×C4×C53C2411yescharacteristic
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53—21no—
171C22×Dic53—21no—
172C2×Dic106—22no—
173C2×Dic106—22no—
174C2×Dic106—21no—
175C2×Dic106C2×D412yes—
176C22⋊C4×C53—28no—
177C22⋊C4×C53—28no—
178C22⋊C4×C53—28no—
179C22⋊C4×C53—28no—
180C42×C53—22no—
181C42×C53—22no—
182C42×C53—22no—
183C42×C53C2×D414yes—
184C22⋊Dic53—24no—
185C22⋊Dic53—42no—
186#11719925—48no—
187#11719923—48no—
188C4⋊Dic53—21no—
189C4⋊Dic53—22no—
190C4⋊Dic53—22no—
191C4⋊Dic53C2×D412yes—
192#11719919—24no—
193#11719919—42no—
194#11719919—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—24no—
208C23×C53—48no—
209C23×C53—48no—
210C23×C53—48no—
211C23×C53—42no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—84no—
218C23×C53—84no—
219C23×C53—84no—
220C23×C53—84no—
221C23×C53#12112yes—
222C23×C53#13512yes—
223C23×C53#13712yes—
224C23×C53C22×D411yescharacteristic
225Q8×C53—24no—
226Q8×C53—24no—
227Q8×C53—48no—
228Q8×C53—42no—
229Q8×C53—42no—
230D4×C53—24no—
231D4×C53—24no—
232D4×C53—48no—
233D4×C53—42no—
234D4×C53—42no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—44no—
251C2×C4×C53—44no—
252C2×C4×C53—48no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53#12012yes—
256C2×C4×C53#12012yes—
257C2×C4×C53#12112yes—
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×D412yes—
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53H(5,𝔽2)12yes—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274Dic106—24no—
275Dic106—42no—
276#88322—42no—
277#88322—42no—
278#24217—531nomaximal, Sylow
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—44no—
286C22×C53—44no—
287C22×C53—44no—
288C22×C53—816no—
289C22×C53—82no—
290C22×C53—82no—
291C22×C53—82no—
292C22×C53—82no—
293C22×C53#46414yescomplemented
294C22×C53C2×#12011yescharacteristic
295C22×C53C2×#12111yescharacteristic
296C22×C53#53511yescharacteristic
297C22×C53C2×#13511yescharacteristic
298C22×C53C2×#13711yescharacteristic
299C22×C53#57511yescharacteristic
300C22×C53#57511yescharacteristic
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—44no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53D4212yes—
308C4×C53#54612yes—
309Dic53—42no—
310Dic53—42no—
311Dic53—81no—
312Dic53—81no—
313C22×#124D5311yescharacteristic
314C2×#464—532no—
315C2×#451—531no—
316C2×#450—531no—
317C4×C22⋊Q8—531no—
318#2042—532no—
319#2015—532no—
320#1983—532no—
321#1797—531no—
322#1787—531no—
323#1786—531no—
324C2×C53—24no—
325C2×C53—84no—
326C2×C53#240811yescharacteristic
327C2×C53#300211yescharacteristic
328C2×C53#300711yescharacteristic
329C2×C53C2×#46411yescharacteristic
330C2×C53#304011yescharacteristic
331C2×C53#321111yescharacteristic
332C2×C53#321511yescharacteristic
333C23×Q8C2×D5311yescharacteristic
334C23×D4C2×D5311yescharacteristic
335C22×Q16—532no—
336C22×SD16—532no—
337C2×#124—28no—
338C2×#124—21no—
339C2×#124—21no—
340C2×#124C2×D5312yes—
341C2×#124C2×D5312yes—
342C2×C22⋊Q8—531no—
343C2×C4×Q8—531no—
344C22×C22⋊C4—21no—
345C22×C22⋊C4—21no—
346C22×C42C2×D5311yescharacteristic
347#464—1068no—
348#451—1064no—
349#450—1064no—
350C2×C4⋊C8—532no—
351C2×#103—532no—
352C2×#103—532no—
353C2×#103—532no—
354C2×#102—532no—
355C2×#102—532no—
356C2×#102—532no—
357C2×C22⋊C8—531no—
358C2×C22⋊C8—531no—
359#391—531no—
360#389—531no—
361#387—1061no—
362#385—1061no—
363#380—531no—
364C4×C4⋊C4—531no—
365C4×C4⋊C4—1061no—
366C4×C22⋊C4—1061no—
367#336—532no—
368C53#2421711yescharacteristic, complemented, Sylow
369C25—21no—
370C22×Q8—24no—
371C22×Q8—21no—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—42no—
375C22×Q8—531no—
376C22×Q8C22×D5312yes—
377C22×D4—24no—
378C22×D4—22no—
379C22×D4—22no—
380C22×D4—21no—
381C22×D4—42no—
382C22×D4C22×D5312yes—
383C23×C4—21no—
384C23×C4C22⋊D5311yescharacteristic
385C23×C4C22⋊D5311yescharacteristic
386C23×C4C22×D5311yescharacteristic
387C2×Q16—1064no—
388C2×Q16—1064no—
389C2×Q16—2122no—
390C2×SD16—1064no—
391C2×SD16—1064no—
392C2×SD16—2122no—
393C22×C8—532no—
394C22×C8—532no—
395#124—28no—
396#124—28no—
397#124—24no—
398#124—24no—
399#124—24no—
400#124C22⋊D5318yes—
401C22⋊Q8—534no—
402C22⋊Q8—1062no—
403C4×Q8—1064no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C22⋊C4—28no—
411C2×C22⋊C4—28no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C42—24no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42C22⋊D5311yescharacteristic
422C2×C42C22⋊D5311yescharacteristic
423C2×C42C22×D5311yescharacteristic
424C2×C42C22×D5311yescharacteristic
425C4⋊C8—1064no—
426#103—1064no—
427#103—1064no—
428#103—2122no—
429#102—1064no—
430#102—1064no—
431#102—2122no—
432C22⋊C8—1062no—
433C22⋊C8—1062no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439C24—24no—
440C24—22no—
441C24—42no—
442C24—42no—
443C24—42no—
444C24—42no—
445C24—42no—
446C24—42no—
447C24C2×C22⋊D5311yescharacteristic
448C2×Q8—24no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—48no—
455C2×Q8—48no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—532no—
459C2×Q8—1061no—
460C2×Q8—1062no—
461C2×Q8—1062no—
462C2×Q8C2×C22⋊D5314yes—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—48no—
470C2×D4—48no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4C2×C22⋊D5314yes—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4D4×D5312yes—
491C22×C4#1171994712yes—
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C23×D5311yescharacteristic
498Q16—2128no—
499SD16—2128no—
500C2×C8—1062no—
501C2×C8—1062no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—1062no—
506C4⋊C4—532no—
507C4⋊C4—1064no—
508C4⋊C4—1061no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—1064no—
518C22⋊C4—2122no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—2122no—
523C42—2122no—
524C42C2×C22⋊D5314yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—24no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23#1313318812yes—
544C23#1313323712yes—
545C23#1313324712yes—
546C23C22×C22⋊D5311yescharacteristic
547Q8—24no—
548Q8—24no—
549Q8—48no—
550Q8—42no—
551Q8—42no—
552Q8—1064no—
553Q8—2122no—
554D4—24no—
555D4—24no—
556D4—48no—
557D4—42no—
558D4—42no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4#1313318612yes—
592C2×C4#1313318612yes—
593C2×C4#1313318812yes—
594C2×C4C2×D4×D5311yescharacteristic
595C2×C4C22×C22⋊D5312yes—
596C2×C4C2×#1171994711yescharacteristic
597C2×C4#1313326212yes—
598C8—2122no—
599C8—2122no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—816no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22#1681084214yescomplemented
615C22C2×#1313318611yescharacteristic
616C22C2×#1313318811yescharacteristic
617C22#1681119011yescharacteristic
618C22C2×#1313323711yescharacteristic
619C22C2×#1313324711yescharacteristic
620C22#1681142111yescharacteristic
621C22#1681142111yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—44no—
627C4—41no—
628C4—4241no—
629C4—4241no—
630C4—2122no—
631C4—2122no—
632C4D4×C22⋊D5312yes—
633C4#1681126712yes—
634C2—24no—
635C2—84no—
636C2#2027806511yescharacteristic
637C2#2028393311yescharacteristic
638C2#2028397511yescharacteristic
639C2C2×#1681084211yescharacteristic
640C2#2028420511yescharacteristic
641C2#2028568011yescharacteristic
642C2#2028571811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×#124×C53C2
socleC23×C53C22×D4
prime radicalC23×D4×C53C22
hypercenterC22×#124D53
hypocenterC53#24217
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16810842C2semidirect
2. C2×#16810779C2semidirect
4. C4×#13133196C2semidirect
5. #20275951C2semidirect
6. #20275809C2semidirect
7. #20275648C2semidirect
8. #20274695C2semidirect
9. #20274650C2semidirect
12. C22×#11719925C22semidirect
15. C2×C4×Dic106C22semidirect
19. C2×C4⋊#88322C22semidirect
20. C2×#13133141C22semidirect
21. C2×#13133141C22semidirect
22. C2×#13133141C22semidirect
28. #16810529C22semidirect
29. #16810521C22semidirect
34. C4×C4⋊Dic53C22semidirect
36. #16810378C22semidirect
293. C22×C53#464semidirect
368. C53#24217semidirect
614. C22#16810842semidirect

Maximal subgroups

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

subgroupindex
C2×#168108422
C2×#168107792
C2×#168107742
C4×#131331962
#202759512
#202758092
#202756482
#202746952
#202746502
#202746462
C22×#124×C532
#2421753