Finite Group Databasebeta

← #27393194

Subgroups — #27393194

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×#16811194C211yesmaximal, characteristic
2C2×#16810590C211yesmaximal, characteristic
3#20275146C212yesmaximal, complemented
4#20275136C212yesmaximal, complemented
5#20275027C212yesmaximal, complemented
6#20275022C212yesmaximal, complemented
7#20274807C212yesmaximal, complemented
8#20274649C212yesmaximal, complemented
9C53×#2311C211yesmaximal, characteristic
10C23×Dic106C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C23×#88322C2211yescharacteristic
13#16811194—24no—
14#16811194C418yescomplemented
15C2×#13133215—21no—
16C2×#13133204—21no—
17C2×#13133198—21no—
18C2×#13133198—21no—
19C2×#13133196C2212yes—
20C2×#13133196C2212yes—
21C2×#13133176C2211yescharacteristic
22C2×#13133140C2212yes—
23C2×#13133140C2212yes—
24C2×#13133140C2212yes—
25C2×#13133140C2212yes—
26#16810590—24no—
27C2×C22⋊#88322C2212yes—
28C2×C22⋊#88322C2212yes—
29#16810385C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33C22×#41×C53C2211yescharacteristic
34C2×#117×C53C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C53×#391—21no—
37C53×#390—21no—
38C53×#386—21no—
39C53×#386—21no—
40C53×#380C2212yes—
41C53×#380C2212yes—
42C22×Dic106—21no—
43C22×Dic106—22no—
44C22×Dic106—41no—
45C22×Dic106—41no—
46C22×Dic106—41no—
47C22×Dic106—41no—
48C22×Dic106C2×C414yescomplemented
49C22×Dic106C2312yes—
50C23×Dic53—21no—
51C2×#11719917—22no—
52C2×#11719917—41no—
53C2×#11719917—41no—
54C22×#88322—22no—
55C22×#88322—41no—
56C22×#88322—41no—
57C22×#88322C2312yes—
58C22×#88322C2312yes—
59#13133215—28no—
60#13133204—28no—
61#13133198—28no—
62#13133198—28no—
63#13133196—24no—
64#13133196—24no—
65#13133196C2×C418yes—
66#13133196C2×C418yescomplemented
67#13133176—22no—
68#13133176C2×C414yescomplemented
69C2×C4⋊Dic53C2312yes—
70C2×#11719919—22no—
71C2×#11719919—22no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—22no—
74C2×C22⋊Dic53—22no—
75C2×C4×Dic53—21no—
76#13133140—42no—
77#13133140—42no—
78#13133140—42no—
79#13133140—42no—
80C22⋊#88322—24no—
81C22⋊#88322—24no—
82C2×#41×C53—41no—
83C2×#41×C53—41no—
84C2×#41×C53—41no—
85C2×#41×C53—41no—
86C2×#41×C53C2×C414yes—
87C2×#41×C53D414yes—
88C22×Q8×C53D411yescharacteristic
89C22×D4×C53D411yescharacteristic
90C22×D4×C53D411yescharacteristic
91C22×D4×C53D411yescharacteristic
92C23×C4×C53—21no—
93C23×C4×C53C2311yescharacteristic
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96#117×C53—41no—
97#117×C53—41no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—41no—
100C2×C4⋊C4×C53—41no—
101C2×C4⋊C4×C53C2312yes—
102C2×C4⋊C4×C53C2312yes—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—22no—
105C2×C22⋊C4×C53—22no—
106C2×C42×C53—21no—
107#95×C53—22no—
108#95×C53—22no—
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—41no—
117C2×Q8×C53—41no—
118C2×Q8×C53C22⋊C414yes—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53C22⋊C414yes—
126C2×D4×C53C22⋊C414yes—
127C2×D4×C53C22⋊C414yes—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—42no—
143C22×C4×C53—82no—
144C22×C4×C53C22×C412yes—
145C22×C4×C53C22×C412yes—
146C22×C4×C53C22×C412yes—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—24no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C2×Dic106—22no—
160C2×Dic106—22no—
161C2×Dic106—22no—
162C2×Dic106—22no—
163C2×Dic106—24no—
164C2×Dic106—22no—
165C2×Dic106—44no—
166C2×Dic106—44no—
167C2×Dic106—42no—
168C2×Dic106—42no—
169C2×Dic106—44no—
170C2×Dic106—44no—
171C2×Dic106C22×C414yes—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C22⋊C4×C53—82no—
179C22⋊C4×C53—82no—
180C42×C53—42no—
181C22⋊Dic53—24no—
182C22⋊Dic53—28no—
183C22⋊Dic53—28no—
184C4⋊Dic53—22no—
185C4⋊Dic53C22×C414yes—
186#11719919—28no—
187#11719919—28no—
188C4×Dic53—24no—
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—44no—
214D4×C53—44no—
215D4×C53—44no—
216D4×C53—44no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
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—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—84no—
242C2×C4×C53C2×C22⋊C412yes—
243C2×C4×C53C2×C22⋊C412yes—
244C2×C4×C53C2×C22⋊C412yes—
245C2×C4×C53C23×C411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53H(5,𝔽2)11yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×C4×C53H(5,𝔽2)11yescharacteristic
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—48no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258Dic106—24no—
259Dic106—24no—
260Dic106—48no—
261Dic106—42no—
262Dic106—42no—
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—82no—
290C4×C53—82no—
291C4×C53#51811yescharacteristic
292C4×C53#51811yescharacteristic
293C4×C53#53411yescharacteristic
294C4×C53#53411yescharacteristic
295Dic53—44no—
296Dic53—44no—
297C2×#536—531no—
298C2×#408—531no—
299#2311D5311yescharacteristic
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×Q8—531no—
316C22×M16—531no—
317C23×C8—531no—
318#536—538no—
319#536—1064no—
320C2×C4⋊Q8—1061no—
321C2×#124—1061no—
322C2×#123—1061no—
323C2×#123—1061no—
324C2×C22⋊Q8—532no—
325C2×C22⋊Q8—532no—
326C2×#117—531no—
327C2×#117C2×D5311yescharacteristic
328C22×C4⋊C4C2×D5311yescharacteristic
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—21no—
337#390—21no—
338#386—21no—
339#386—21no—
340#380C2×D5312yes—
341#380C2×D5312yes—
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×#41C4×D5314yes—
352C2×#41D21214yes—
353C22×Q8—534no—
354C22×Q8—532no—
355C22×Q8—1062no—
356C22×Q8—1061no—
357C22×Q8—2121no—
358C22×Q8—2121no—
359C22×Q8—2121no—
360C22×Q8—2121no—
361C22×Q8C22⋊D5311yescharacteristic
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—1068no—
379#124—1068no—
380#123—1068no—
381#123—1068no—
382C22⋊Q8—538no—
383C22⋊Q8—538no—
384C22⋊Q8—1064no—
385C22⋊Q8—1064no—
386#117—41no—
387#117—41no—
388#117—534no—
389#117—1062no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—41no—
392C2×C4⋊C4—41no—
393C2×C4⋊C4—532no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4C22×D5312yes—
397C2×C4⋊C4C22×D5312yes—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1062no—
403C2×C22⋊C4—1062no—
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—22no—
413#95—22no—
414C24C2×C22⋊D5311yescharacteristic
415C24C2×C22⋊D5311yescharacteristic
416C24C2×C22⋊D5311yescharacteristic
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—534no—
424C2×Q8—1064no—
425C2×Q8—1062no—
426C2×Q8—1062no—
427C2×Q8—1062no—
428C2×Q8—1062no—
429C2×Q8—1062no—
430C2×Q8—2124no—
431C2×Q8—2124no—
432C2×Q8—2122no—
433C2×Q8—2122no—
434C2×Q8—2124no—
435C2×Q8—2124no—
436C2×Q8#1171992114yes—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4#1171992114yes—
444C2×D4#1171992114yes—
445C2×D4#1171992114yes—
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—22no—
459C22×C4—22no—
460C22×C4—42no—
461C22×C4—82no—
462C22×C4—1064no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4C2×C4×D5312yes—
468C22×C4C2×C4×D5312yes—
469C22×C4C2×C4×D5312yes—
470C22×C4C2×D21212yes—
471C22×C4C2×D21212yes—
472C22×C4C2×D21212yes—
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
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—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—534no—
491C4⋊C4—1068no—
492C4⋊C4—1068no—
493C4⋊C4—1062no—
494C22⋊C4—42no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—1064no—
498C22⋊C4—1068no—
499C22⋊C4—1068no—
500C42—42no—
501C42—1064no—
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×#1171992112yes—
515C23C2×#1171992112yes—
516C23C2×#1171992112yes—
517C23C22×C22⋊D5311yescharacteristic
518Q8—44no—
519Q8—1064no—
520Q8—1064no—
521Q8—2128no—
522Q8—2122no—
523Q8—2122no—
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—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—1064no—
553C2×C4—1064no—
554C2×C4—2128no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4C2×#1171992112yes—
560C2×C4C2×#1171992112yes—
561C2×C4C2×#1171992112yes—
562C2×C4C22×C4×D5311yescharacteristic
563C2×C4C22×D21211yescharacteristic
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×#1171992111yescharacteristic
586C22#1681118211yescharacteristic
587C22#1681118211yescharacteristic
588C22#1681143412yes—
589C22#1681143812yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—82no—
594C4—82no—
595C4—2124no—
596C4—2124no—
597C4#1681109611yescharacteristic
598C4#1681109611yescharacteristic
599C4#1681118511yescharacteristic
600C4#1681118511yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2#2027729011yescharacteristic
605C2#2028324512yes—
606C2#2028558711yescharacteristic
607C2#2028559112yes—
608C2#2028560611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2311C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenter#2311D53
hypocenterC53#21771
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
3. #20275146C2semidirect
4. #20275136C2semidirect
5. #20275027C2semidirect
6. #20275022C2semidirect
7. #20274807C2semidirect
8. #20274649C2semidirect
14. #16811194C4semidirect
48. C22×Dic106C2×C4semidirect
66. #13133196C2×C4semidirect
68. #13133176C2×C4semidirect
346. C53#21771semidirect

Maximal subgroups

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

subgroupindex
C2×#168111942
C2×#168105902
#202751462
#202751362
#202750272
#202750222
#202748072
#202746492
C53×#23112
#2177153