Finite Group Databasebeta

← #27312026

Subgroups — #27312026

Order 13568.

1750 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810520C212yesmaximal, complemented
2C2×#16810511C212yesmaximal
3C2×#16810495C211yesmaximal, characteristic
4C2×#16810481C212yesmaximal
5#20272978C212yesmaximal, complemented
6#20272966C212yesmaximal, complemented
7#20272960C211yesmaximal, characteristic
8#20272950C212yesmaximal, complemented
9C53×#1439C211yesmaximal, characteristic
10C24×Dic53C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×#11719919C2212yes—
13C22×#11719919C2212yes—
14C22×#11719919C2212yes—
15C22×#11719919C2212yes—
16C22×C22⋊Dic53C2212yes—
17C22×C22⋊Dic53C2211yescharacteristic
18C22×C4×Dic53C2212yes—
19#16810520—28no—
20#16810511—28no—
21#16810495C418yescomplemented
22#16810481—28no—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C22×C4⋊C4×C53C2211yescharacteristic
31C22×C22⋊C4×C53C2212yes—
32C22×C22⋊C4×C53C2211yescharacteristic
33C2×#95×C53C2212yes—
34C2×#95×C53C2211yescharacteristic
35C23×Dic53—24no—
36C23×Dic53—24no—
37C23×Dic53—41no—
38C23×Dic53—41no—
39C23×Dic53C2×C414yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C2×C4⋊Dic53—24no—
45C2×C4⋊Dic53—41no—
46C2×#11719919—24no—
47C2×#11719919—24no—
48C2×#11719919—24no—
49C2×#11719919—24no—
50C2×#11719919—24no—
51C2×#11719919—24no—
52C2×#11719919—24no—
53C2×#11719919—24no—
54C2×#11719919—42no—
55C2×#11719919—42no—
56C2×#11719919—42no—
57C2×#11719919C2×C418yes—
58C2×C22⋊Dic53—24no—
59C2×C22⋊Dic53—24no—
60C2×C22⋊Dic53—42no—
61C2×C22⋊Dic53C2×C414yes—
62C2×C22⋊Dic53D414yes—
63C2×C22⋊Dic53Q814yes—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—42no—
67#13133124—24no—
68#13133124—24no—
69#13133124—24no—
70#13133124—42no—
71#13133124—41no—
72#13133124C2×C418yes—
73#13133124D418yes—
74C25×C53C2311yescharacteristic
75C23×C4×C53C2312yes—
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2312yes—
79C2×C4⋊C4×C53—41no—
80C2×C4⋊C4×C53—42no—
81C2×C22⋊C4×C53—24no—
82C2×C22⋊C4×C53—24no—
83C2×C22⋊C4×C53—41no—
84C2×C22⋊C4×C53—41no—
85C2×C22⋊C4×C53—42no—
86C2×C22⋊C4×C53C2×C414yes—
87#95×C53—24no—
88#95×C53—41no—
89C24×C53—22no—
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—21no—
93C24×C53—21no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—22no—
97C24×C53C22×C412yes—
98C24×C53C2×D412yes—
99C24×C53C2×Q812yes—
100C24×C53C2411yescharacteristic
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C22×C412yes—
124C22×C4×C53C2×D414yes—
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—216no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—44no—
148C22×Dic53—44no—
149C22×Dic53—48no—
150C22×Dic53—44no—
151C22×Dic53C22×C414yes—
152C22×Dic53C22×C414yes—
153C22×Dic53C2×D414yes—
154C22×Dic53C2×D414yes—
155C22×Dic53C2×D414yes—
156C22×Dic53C2×Q814yes—
157C22×Dic53#4114yes—
158C4⋊C4×C53—44no—
159C22⋊C4×C53—44no—
160C22⋊C4×C53—48no—
161C22⋊Dic53—48no—
162C22⋊Dic53C4⋊C4116yescomplemented
163C4⋊Dic53—44no—
164#11719919—216no—
165#11719919—48no—
166#11719919—48no—
167#11719919—48no—
168C4×Dic53—48no—
169C23×C53—24no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—24no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—24no—
178C23×C53—24no—
179C23×C53—24no—
180C23×C53—24no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—41no—
186C23×C53—42no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—42no—
190C23×C53—41no—
191C23×C53C2×C4⋊C414yes—
192C23×C53C23×C411yescharacteristic
193C23×C53C22×D411yescharacteristic
194C23×C53C22×D412yes—
195C23×C53C22×Q811yescharacteristic
196C23×C53C2×#4112yes—
197C23×C53H(5,𝔽2)11yescharacteristic
198C23×C53H(5,𝔽2)11yescharacteristic
199C23×C53H(5,𝔽2)12yes—
200C23×C532−1+411yescharacteristic
201C23×C532−1+411yescharacteristic
202C23×C532−1+412yes—
203C2×C4×C53—24no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—41no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—24no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—416no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53C2×C4⋊C418yes—
255C2×Dic53C4×D418yes—
256#14402—531nomaximal, Sylow
257C22×C53—24no—
258C22×C53—28no—
259C22×C53—44no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—44no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—44no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53C22×C4⋊C411yescharacteristic
269C22×C53C2×C4×D412yes—
270C22×C53#51811yescharacteristic
271C22×C53#51911yescharacteristic
272C22×C53#52012yes—
273C22×C53#53612yes—
274C22×C53#53711yescharacteristic
275C22×C53#53711yescharacteristic
276C22×C53#53712yes—
277C22×C53#54112yes—
278C22×C53#54212yes—
279C22×C53#54411yescharacteristic
280C22×C53#54411yescharacteristic
281C22×C53D4212yes—
282C22×C53D4211yescharacteristic
283C22×C53#54612yes—
284C22×C53#54712yes—
285C22×C53D4×Q812yes—
286C22×C53#55111yescharacteristic
287C22×C53#55412yes—
288C22×C53#56011yescharacteristic
289C22×C53#56212yes—
290C22×C53#56311yescharacteristic
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—44no—
294C4×C53—82no—
295Dic53—28no—
296Dic53—28no—
297Dic53—44no—
298Dic53—82no—
299C2×#389—532no—
300C2×#388—532no—
301C2×#385—531no—
302C2×#380—532no—
303#1444—532no—
304#1441—532no—
305#1439—531no—
306#1439D5311yescharacteristic
307#1436—532no—
308C2×C53—44no—
309C2×C53#231611yescharacteristic
310C2×C53#232312yes—
311C2×C53D4×C4⋊C412yes—
312C2×C53#235811yescharacteristic
313C2×C53#237611yescharacteristic
314C2×C53#267212yes—
315C2×C53#267511yescharacteristic
316C2×C53#267711yescharacteristic
317C2×C53#270412yes—
318C2×C53#272211yescharacteristic
319C2×C53#274211yescharacteristic
320C24×C4—531no—
321C22×C4⋊C4—532no—
322C22×C4⋊C4—532no—
323C22×C4⋊C4—532no—
324C22×C4⋊C4—532no—
325C22×C4⋊C4—531no—
326C22×C4⋊C4C2×D5311yescharacteristic
327C22×C22⋊C4—531no—
328C22×C22⋊C4—532no—
329C22×C22⋊C4C2×D5312yes—
330C22×C22⋊C4C2×D5311yescharacteristic
331C22×C42—532no—
332#389—1068no—
333#388—1068no—
334#385—538no—
335#380—1068no—
336C2×#95—532no—
337C2×#95—532no—
338C2×#95—532no—
339C2×#95—532no—
340C2×#95—532no—
341C2×#95—532no—
342C2×#95—531no—
343C2×#95C2×D5312yes—
344C2×#95C2×D5311yescharacteristic
345C53#1440211yescharacteristic, complemented, Sylow
346C25C22×D5311yescharacteristic
347C23×C4—532no—
348C23×C4—532no—
349C23×C4—532no—
350C23×C4—534no—
351C23×C4—532no—
352C23×C4—1064no—
353C23×C4—1064no—
354C23×C4—2121no—
355C23×C4—2121no—
356C23×C4C22×D5312yes—
357C23×C4C22×D5311yescharacteristic
358C23×C4C22×D5311yescharacteristic
359C23×C4C22×D5312yes—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—41no—
362C2×C4⋊C4—538no—
363C2×C4⋊C4—1064no—
364C2×C4⋊C4—1064no—
365C2×C4⋊C4—1064no—
366C2×C4⋊C4—1064no—
367C2×C4⋊C4—1064no—
368C2×C4⋊C4—1064no—
369C2×C4⋊C4—1064no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1064no—
372C2×C4⋊C4—2122no—
373C2×C4⋊C4—2122no—
374C2×C4⋊C4—2122no—
375C2×C4⋊C4—2121no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—41no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4—534no—
382C2×C22⋊C4—534no—
383C2×C22⋊C4—534no—
384C2×C22⋊C4—1064no—
385C2×C22⋊C4—1064no—
386C2×C22⋊C4—2122no—
387C2×C22⋊C4C4×D5314yes—
388C2×C42—1064no—
389C2×C42—1064no—
390C2×C42—2122no—
391#95—24no—
392#95—41no—
393#95—538no—
394#95—538no—
395#95—1064no—
396#95—1064no—
397#95—1064no—
398#95—2121no—
399#95—2122no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24C2×C4×D5312yes—
409C24D4×D5312yes—
410C24Q8×D5312yes—
411C24C23×D5311yescharacteristic
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—534no—
434C22×C4—534no—
435C22×C4—534no—
436C22×C4—534no—
437C22×C4—534no—
438C22×C4—534no—
439C22×C4—534no—
440C22×C4—10616no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—2124no—
463C22×C4—2124no—
464C22×C4—2124no—
465C22×C4—2128no—
466C22×C4C2×C4×D5312yes—
467C22×C4C2×C4×D5312yes—
468C22×C4D4×D5314yes—
469C4⋊C4—44no—
470C4⋊C4—10616no—
471C4⋊C4—2128no—
472C4⋊C4—2128no—
473C4⋊C4—2128no—
474C4⋊C4—2124no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—5316no—
478C22⋊C4—2128no—
479C42—2128no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23C4⋊C4×D5314yes—
503C23C22×C4×D5311yescharacteristic
504C23C2×D4×D5311yescharacteristic
505C23C2×D4×D5312yes—
506C23C2×Q8×D5311yescharacteristic
507C23#41×D5312yes—
508C23#1313326211yescharacteristic
509C23#1313326212yes—
510C23#1313326211yescharacteristic
511C23#1313326412yes—
512C23#1313326411yescharacteristic
513C23#1313326411yescharacteristic
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—538no—
539C2×C4—538no—
540C2×C4—1064no—
541C2×C4—1064no—
542C2×C4—1064no—
543C2×C4—1064no—
544C2×C4—1064no—
545C2×C4—1064no—
546C2×C4—1064no—
547C2×C4—1064no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—1064no—
552C2×C4—1064no—
553C2×C4—1064no—
554C2×C4—1064no—
555C2×C4—21216no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C22—24no—
568C22—28no—
569C22—42no—
570C22—44no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22C2×C4⋊C4×D5311yescharacteristic
579C22C4×D4×D5312yes—
580C22#1681109611yescharacteristic
581C22#1681110011yescharacteristic
582C22#1681110212yes—
583C22#1681119512yes—
584C22#1681120011yescharacteristic
585C22#1681120011yescharacteristic
586C22#1681120212yes—
587C22#1681123012yes—
588C22#1681123712yes—
589C22#1681125211yescharacteristic
590C22#1681125211yescharacteristic
591C22#1681125512yes—
592C22#1681125811yescharacteristic
593C22#1681126212yes—
594C22#1681127112yes—
595C22#1681128612yes—
596C22#1681130311yescharacteristic
597C22#1681132712yes—
598C22#1681136211yescharacteristic
599C22#1681136812yes—
600C22#1681137611yescharacteristic
601C4—42no—
602C4—44no—
603C4—42no—
604C4—82no—
605C4—1068no—
606C4—1068no—
607C4—4242no—
608C4—2124no—
609C2—44no—
610C2#2027733211yescharacteristic
611C2#2027739412yes—
612C2#2027761312yes—
613C2#2027769611yescharacteristic
614C2#2027785811yescharacteristic
615C2#2028096812yes—
616C2#2028100711yescharacteristic
617C2#2028103111yescharacteristic
618C2#2028137812yes—
619C2#2028162211yescharacteristic
620C2#2028183811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1439C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53#14402
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C2×#16810520C2semidirect
5. #20272978C2semidirect
6. #20272966C2semidirect
8. #20272950C2semidirect
21. #16810495C4semidirect
162. C22⋊Dic53C4⋊C4semidirect
345. C53#14402semidirect

Maximal subgroups

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

subgroupindex
C2×#168105202
C2×#168105112
C2×#168104952
C2×#168104812
#202729782
#202729662
#202729602
#202729502
C53×#14392
#1440253