Finite Group Databasebeta

← #27393199

Subgroups — #27393199

Order 13568.

1326 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810590C211yesmaximal, characteristic
2#20277307C211yesmaximal, characteristic
3#20275149C212yesmaximal, complemented
4#20275139C212yesmaximal, complemented
5#20275028C212yesmaximal, complemented
6#20275023C212yesmaximal, complemented
7#20274810C212yesmaximal, complemented
8#20274650C212yesmaximal, complemented
9C2×C53×#536C211yesmaximal, characteristic
10C22×#11719917C2211yescharacteristic
11C23×#88322C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C2×#13133141C2212yes—
15C2×#13133141C2212yes—
16C2×#13133141C2212yes—
17C2×#13133141C2212yes—
18#16810590—24no—
19C2×C22⋊#88322C2212yes—
20C2×C22⋊#88322C2212yes—
21#16810530—21no—
22#16810526—21no—
23#16810503—21no—
24#16810503—21no—
25#16810483C2212yes—
26#16810483C2212yes—
27#16810385C2212yes—
28#16810385C2212yes—
29#16810385C2212yes—
30#16810385C2212yes—
31C22×#41×C53C2211yescharacteristic
32C23×Q8×C53C2211yescharacteristic
33C53×#536—24no—
34C53×#536C418yescomplemented
35C2×C4⋊Q8×C53—21no—
36C2×#124×C53—21no—
37C2×#123×C53—21no—
38C2×#123×C53—21no—
39C2×C22⋊Q8×C53C2212yes—
40C2×C22⋊Q8×C53C2212yes—
41C2×#117×C53C2211yescharacteristic
42C23×Dic53—21no—
43C2×#11719917—22no—
44C2×#11719917—41no—
45C2×#11719917—41no—
46C22×#88322—22no—
47C22×#88322—41no—
48C22×#88322—41no—
49C22×#88322C2312yes—
50C22×#88322C2312yes—
51#13133176—41no—
52#13133176—41no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—41no—
55C2×C4⋊Dic53—41no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2312yes—
58C2×C22⋊Dic53—21no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—22no—
61C2×C4×Dic53—21no—
62#13133141—42no—
63#13133141—42no—
64#13133141—42no—
65#13133141—42no—
66C22⋊#88322—24no—
67C22⋊#88322—24no—
68#13133124—22no—
69#13133124—22no—
70C2×#41×C53—41no—
71C2×#41×C53—41no—
72C2×#41×C53—41no—
73C2×#41×C53—41no—
74C2×#41×C53C2×C414yes—
75C2×#41×C53D414yes—
76C22×Q8×C53—21no—
77C22×Q8×C53—22no—
78C22×Q8×C53—41no—
79C22×Q8×C53—41no—
80C22×Q8×C53—41no—
81C22×Q8×C53—41no—
82C22×Q8×C53C2×C414yescomplemented
83C22×Q8×C53D411yescharacteristic
84C22×Q8×C53C2312yes—
85C22×D4×C53D411yescharacteristic
86C22×D4×C53D411yescharacteristic
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C4⋊Q8×C53—28no—
93#124×C53—28no—
94#123×C53—28no—
95#123×C53—28no—
96C22⋊Q8×C53—24no—
97C22⋊Q8×C53—24no—
98C22⋊Q8×C53C2×C418yes—
99C22⋊Q8×C53C2×C418yescomplemented
100#117×C53—22no—
101#117×C53C2×C414yescomplemented
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53C2312yes—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—22no—
107C2×C22⋊C4×C53—22no—
108C2×C42×C53—21no—
109C24×C53C2×D411yescharacteristic
110C24×C53C2×D411yescharacteristic
111C24×C53C2×D411yescharacteristic
112#41×C53—44no—
113#41×C53—44no—
114#41×C53—44no—
115#41×C53—44no—
116C2×Q8×C53—24no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53—22no—
120C2×Q8×C53—22no—
121C2×Q8×C53—22no—
122C2×Q8×C53—41no—
123C2×Q8×C53—41no—
124C2×Q8×C53—44no—
125C2×Q8×C53—44no—
126C2×Q8×C53—42no—
127C2×Q8×C53—42no—
128C2×Q8×C53—44no—
129C2×Q8×C53—44no—
130C2×Q8×C53C22⋊C414yes—
131C2×Q8×C53C22×C414yes—
132C2×D4×C53—41no—
133C2×D4×C53—41no—
134C2×D4×C53—41no—
135C2×D4×C53—41no—
136C2×D4×C53—41no—
137C2×D4×C53—41no—
138C2×D4×C53C22⋊C414yes—
139C2×D4×C53C22⋊C414yes—
140C2×D4×C53C22⋊C414yes—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—24no—
154C22×C4×C53—22no—
155C22×C4×C53—22no—
156C22×C4×C53—42no—
157C22×C4×C53—42no—
158C22×C4×C53C22×C412yes—
159C22×C4×C53C22×C412yes—
160C22×C4×C53C22×C412yes—
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D412yes—
165C22×C4×C53C2×D412yes—
166C22×C4×C53C2×D412yes—
167C22×C4×C53C2411yescharacteristic
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—42no—
171C22×Dic53—82no—
172C4⋊C4×C53—28no—
173C4⋊C4×C53—28no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53C22×C414yes—
176C22⋊C4×C53—24no—
177C22⋊C4×C53—28no—
178C22⋊C4×C53—28no—
179C42×C53—24no—
180C22⋊Dic53—42no—
181C22⋊Dic53—82no—
182C22⋊Dic53—82no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187C4⋊Dic53—42no—
188C4×Dic53—42no—
189#11719917—44no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—44no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—22no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53C2×C22⋊C412yes—
210C23×C53C2×C22⋊C412yes—
211C23×C53C2×C22⋊C412yes—
212C23×C53C22×D411yescharacteristic
213Q8×C53—24no—
214Q8×C53—24no—
215Q8×C53—44no—
216Q8×C53—48no—
217Q8×C53—42no—
218Q8×C53—42no—
219D4×C53—44no—
220D4×C53—44no—
221D4×C53—44no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—48no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53C2×C22⋊C412yes—
248C2×C4×C53C2×C22⋊C412yes—
249C2×C4×C53C2×C22⋊C412yes—
250C2×C4×C53C23×C411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×C4×C53H(5,𝔽2)11yescharacteristic
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—84no—
263#88322—42no—
264#88322—42no—
265#21771—531nomaximal, Sylow
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—24no—
270C22×C53—24no—
271C22×C53—24no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53C22×C22⋊C411yescharacteristic
282C22×C53#53411yescharacteristic
283C22×C53#53411yescharacteristic
284C22×C53#57712yes—
285C22×C53#57812yes—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—44no—
290C4×C53—44no—
291C4×C53#51811yescharacteristic
292C4×C53#51811yescharacteristic
293C4×C53#53411yescharacteristic
294C4×C53#53411yescharacteristic
295Dic53—82no—
296Dic53—82no—
297C2×#536D5311yescharacteristic
298C2×#408—531no—
299#2311—531no—
300#1894—532no—
301#1892—532no—
302#1873—532no—
303#1872—532no—
304#1825—532no—
305#1787—532no—
306C2×C53—42no—
307C2×C53—42no—
308C2×C53—42no—
309C2×C53#230811yescharacteristic
310C2×C53#289812yes—
311C2×C53#320011yescharacteristic
312C2×C53#320112yes—
313C2×C53#320211yescharacteristic
314C22×#41C2×D5311yescharacteristic
315C23×Q8C2×D5311yescharacteristic
316C22×M16—531no—
317C23×C8—531no—
318#536—24no—
319#536Dic5318yescomplemented
320C2×C4⋊Q8—21no—
321C2×#124—21no—
322C2×#123—21no—
323C2×#123—21no—
324C2×C22⋊Q8C2×D5312yes—
325C2×C22⋊Q8C2×D5312yes—
326C2×#117—531no—
327C2×#117C2×D5311yescharacteristic
328C22×C4⋊C4—531no—
329C2×#103—532no—
330C2×#103—532no—
331C2×#103—532no—
332C2×#103—532no—
333#408—1064no—
334C2×C22⋊C8—532no—
335C2×C22⋊C8—532no—
336#391—1061no—
337#390—1061no—
338#386—1061no—
339#386—1061no—
340#380—532no—
341#380—532no—
342#340—532no—
343#340—532no—
344#340—532no—
345#340—532no—
346C53#2177111yescharacteristic, complemented, Sylow
347C2×#41—41no—
348C2×#41—41no—
349C2×#41—41no—
350C2×#41—41no—
351C2×#41C2×Dic5314yes—
352C2×#41C22⋊D5314yes—
353C22×Q8—21no—
354C22×Q8—22no—
355C22×Q8—41no—
356C22×Q8—41no—
357C22×Q8—41no—
358C22×Q8—41no—
359C22×Q8C2×Dic5314yescomplemented
360C22×Q8C22⋊D5311yescharacteristic
361C22×Q8C22×D5312yes—
362C22×D4C22⋊D5311yescharacteristic
363C22×D4C22⋊D5311yescharacteristic
364C22×D4C22⋊D5311yescharacteristic
365C23×C4—21no—
366C23×C4—1061no—
367C23×C4C22×D5311yescharacteristic
368C23×C4C22×D5311yescharacteristic
369C23×C4C22×D5311yescharacteristic
370C2×M16—1062no—
371C2×M16—2121no—
372C2×M16—2121no—
373C22×C8—532no—
374C22×C8—532no—
375C22×C8—1062no—
376C22×C8—2121no—
377C22×C8—2121no—
378C4⋊Q8—28no—
379#124—28no—
380#123—28no—
381#123—28no—
382C22⋊Q8—24no—
383C22⋊Q8—24no—
384C22⋊Q8C2×Dic5318yes—
385C22⋊Q8C2×Dic5318yescomplemented
386#117—22no—
387#117—2121no—
388#117—2121no—
389#117C2×Dic5314yescomplemented
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—532no—
393C2×C4⋊C4—532no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—2121no—
396C2×C4⋊C4—2121no—
397C2×C4⋊C4C22×D5312yes—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—1062no—
402C2×C22⋊C4—1062no—
403C2×C22⋊C4—1061no—
404C2×C42—21no—
405C2×C42—1061no—
406#103—2122no—
407#103—2122no—
408#103—2122no—
409#103—2122no—
410C22⋊C8—1064no—
411C22⋊C8—1064no—
412#95—1062no—
413#95—1062no—
414C24C2×C22⋊D5311yescharacteristic
415C24C2×C22⋊D5311yescharacteristic
416C24C2×C22⋊D5311yescharacteristic
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—41no—
428C2×Q8—41no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8C22⋊Dic5314yes—
436C2×Q8C22×Dic5314yes—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4C22⋊Dic5314yes—
444C2×D4C22⋊Dic5314yes—
445C2×D4C22⋊Dic5314yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—4242no—
466C22×C4—2122no—
467C22×C4C22×Dic5312yes—
468C22×C4C22×Dic5312yes—
469C22×C4C22×Dic5312yes—
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5312yes—
474C22×C4C2×C22⋊D5312yes—
475C22×C4C2×C22⋊D5312yes—
476C22×C4C23×D5311yescharacteristic
477M16—2124no—
478C2×C8—1062no—
479C2×C8—1062no—
480C2×C8—1062no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—2124no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—22no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—2122no—
493C4⋊C4C22×Dic5314yes—
494C22⋊C4—24no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—4242no—
498C22⋊C4—4242no—
499C22⋊C4—2122no—
500C42—24no—
501C42—2122no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23C2×C22⋊Dic5312yes—
515C23C2×C22⋊Dic5312yes—
516C23C2×C22⋊Dic5312yes—
517C23C22×C22⋊D5311yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—44no—
521Q8—48no—
522Q8—42no—
523Q8—42no—
524D4—44no—
525D4—44no—
526D4—44no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—4244no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4C2×C22⋊Dic5312yes—
560C2×C4C2×C22⋊Dic5312yes—
561C2×C4C2×C22⋊Dic5312yes—
562C2×C4C23×Dic5311yescharacteristic
563C2×C4C22×C22⋊D5311yescharacteristic
564C2×C4#1313326211yescharacteristic
565C2×C4#1313326211yescharacteristic
566C2×C4#1313326211yescharacteristic
567C2×C4#1313326211yescharacteristic
568C8—2122no—
569C8—2122no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C22×C22⋊Dic5311yescharacteristic
586C22#1681118211yescharacteristic
587C22#1681118211yescharacteristic
588C22#1681143112yes—
589C22#1681143812yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—44no—
594C4—44no—
595C4—4242no—
596C4—4242no—
597C4#1681109811yescharacteristic
598C4#1681109811yescharacteristic
599C4#1681118211yescharacteristic
600C4#1681118211yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2#2027729311yescharacteristic
605C2#2028324812yes—
606C2#2028558211yescharacteristic
607C2#2028559712yes—
608C2#2028560611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#536C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#536D53
hypocenterC53#21771
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
3. #20275149C2semidirect
4. #20275139C2semidirect
5. #20275028C2semidirect
6. #20275023C2semidirect
7. #20274810C2semidirect
8. #20274650C2semidirect
34. C53×#536C4semidirect
82. C22×Q8×C53C2×C4semidirect
99. C22⋊Q8×C53C2×C4semidirect
101. #117×C53C2×C4semidirect
319. #536Dic53semidirect
346. C53#21771semidirect
359. C22×Q8C2×Dic53semidirect
385. C22⋊Q8C2×Dic53semidirect
389. #117C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105902
#202773072
#202751492
#202751392
#202750282
#202750232
#202748102
#202746502
C2×C53×#5362
#2177153