Finite Group Databasebeta

← #27306764

Subgroups — #27306764

Order 13568.

1630 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810520C212yesmaximal
2C2×#16810518C212yesmaximal
3C2×#16810487C211yesmaximal, characteristic
4C2×C4⋊C4×Dic53C211yesmaximal, characteristic
5C2×#16810474C211yesmaximal, characteristic
6#20272980C211yesmaximal, characteristic
7#20272979C211yesmaximal, characteristic
8#20272978C211yesmaximal, characteristic
9#20272944C212yesmaximal
10#20272943C212yesmaximal
11C53×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2212yes—
14C22×#11719919C2211yescharacteristic
15C22×#11719919C2211yescharacteristic
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×C4×Dic53C2212yes—
20C22×C4×Dic53C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22#16810520—28no—
23#16810518—28no—
24#16810487—24no—
25C4⋊C4×Dic53—24no—
26#16810474C418yescomplemented
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2212yes—
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2211yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2312yes—
45C23×Dic53C2311yescharacteristic
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—24no—
48C2×C4⋊Dic53—24no—
49C2×C4⋊Dic53—24no—
50C2×C4⋊Dic53—42no—
51C2×#11719919—24no—
52C2×#11719919—24no—
53C2×#11719919—24no—
54C2×#11719919—24no—
55C2×#11719919—24no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—24no—
59C2×#11719919—42no—
60C2×#11719919—42no—
61C2×#11719919C2×C418yes—
62C2×#11719919D418yes—
63C2×#11719919Q818yes—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—42no—
68C2×C4×Dic53—42no—
69C2×C4×Dic53C2×C418yes—
70C2×C4×Dic53C2×C414yes—
71C2×C4×Dic53D414yes—
72C2×C4×Dic53Q814yes—
73#13133124—24no—
74#13133124—22no—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—22no—
79#13133124—22no—
80#13133124C2×C414yes—
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2312yes—
84C23×C4×C53C2312yes—
85C23×C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53—22no—
87C2×C4⋊C4×C53—24no—
88C2×C4⋊C4×C53—24no—
89C2×C4⋊C4×C53—24no—
90C2×C4⋊C4×C53—42no—
91#95×C53—24no—
92#95×C53—22no—
93#95×C53C2×C414yes—
94C24×C53C2411yescharacteristic
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53C22×C412yes—
118C22×C4×C53C22×C414yes—
119C22×C4×C53C2×D412yes—
120C22×C4×C53C2×D414yes—
121C22×C4×C53C2×Q812yes—
122C22×C4×C53C2×Q814yes—
123C22×C4×C53#4114yes—
124C22×Dic53—22no—
125C22×Dic53—22no—
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53C22×C414yes—
149C22×Dic53C22×C412yes—
150C22×Dic53C22×C412yes—
151C22×Dic53C2×D412yes—
152C22×Dic53C2×D414yes—
153C22×Dic53C2×D412yes—
154C22×Dic53C2×D414yes—
155C22×Dic53C2×Q812yes—
156C22×Dic53C2×Q814yes—
157C22×Dic53C2×Q812yes—
158C22×Dic53C2×Q814yes—
159C22×Dic53#4114yes—
160C4⋊C4×C53—28no—
161C4⋊C4×C53—48no—
162C4⋊Dic53—28no—
163C4⋊Dic53—48no—
164#11719919—48no—
165#11719919—48no—
166#11719919—44no—
167#11719919—44no—
168#11719919C4⋊C4132yescomplemented
169C4×Dic53—48no—
170C4×Dic53—48no—
171C4×Dic53C4⋊C4116yescomplemented
172C23×C53C23×C411yescharacteristic
173C23×C53C22×D411yescharacteristic
174C23×C53C22×D412yes—
175C23×C53C22×Q811yescharacteristic
176C23×C53C22×Q812yes—
177C23×C53C2×#4112yes—
178C23×C53C2×#4112yes—
179C23×C53H(5,𝔽2)11yescharacteristic
180C23×C53H(5,𝔽2)11yescharacteristic
181C23×C532−1+411yescharacteristic
182C23×C532−1+411yescharacteristic
183C2×C4×C53—24no—
184C2×C4×C53—24no—
185C2×C4×C53—24no—
186C2×C4×C53—22no—
187C2×C4×C53—22no—
188C2×C4×C53—24no—
189C2×C4×C53—24no—
190C2×C4×C53—24no—
191C2×C4×C53—24no—
192C2×C4×C53—24no—
193C2×C4×C53—24no—
194C2×C4×C53—24no—
195C2×C4×C53—24no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53C2×C4⋊C414yes—
211C2×C4×C53C2×C4⋊C418yes—
212C2×Dic53—24no—
213C2×Dic53—24no—
214C2×Dic53—22no—
215C2×Dic53—22no—
216C2×Dic53—22no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—24no—
221C2×Dic53—24no—
222C2×Dic53—24no—
223C2×Dic53—24no—
224C2×Dic53—24no—
225C2×Dic53—24no—
226C2×Dic53—24no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53C2×C4⋊C414yes—
242C2×Dic53C2×C4⋊C418yes—
243C2×Dic53C2×C4⋊C414yes—
244C2×Dic53C2×C4⋊C418yes—
245C2×Dic53#12714yes—
246C2×Dic53C4⋊Q814yes—
247C2×Dic53C4⋊Q818yes—
248#13954—531nomaximal, Sylow
249C22×C53C22×C4⋊C411yescharacteristic
250C22×C53C22×C4⋊C412yes—
251C22×C53C4×#4112yes—
252C22×C53#51811yescharacteristic
253C22×C53#51911yescharacteristic
254C22×C53#52511yescharacteristic
255C22×C53#52511yescharacteristic
256C22×C53C2×#12711yescharacteristic
257C22×C53C2×C4⋊Q811yescharacteristic
258C22×C53C2×C4⋊Q812yes—
259C22×C53#53312yes—
260C22×C53#53711yescharacteristic
261C22×C53#53711yescharacteristic
262C22×C53#54411yescharacteristic
263C22×C53#54411yescharacteristic
264C22×C53#54612yes—
265C22×C53#54612yes—
266C22×C53#54812yes—
267C22×C53#54812yes—
268C22×C53#55412yes—
269C22×C53#55412yes—
270C22×C53#55712yes—
271C22×C53#55712yes—
272C4×C53—28no—
273C4×C53—24no—
274C4×C53—44no—
275C4×C53—81no—
276C4×C53—81no—
277Dic53—24no—
278Dic53—28no—
279Dic53—44no—
280Dic53—81no—
281Dic53—81no—
282Dic53#38418yes—
283C2×#389—532no—
284C2×#389—532no—
285C2×#382—531no—
286C2×C4×C4⋊C4—531no—
287C2×C4×C4⋊C4—531no—
288#1444—531no—
289#1444—531no—
290#1444—531no—
291#1444D5311yescharacteristic
292C4⋊#95—532no—
293C4⋊#95—532no—
294C2×C53C2×#38411yescharacteristic
295C2×C53#229611yescharacteristic
296C2×C53#231611yescharacteristic
297C2×C53#234912yes—
298C2×C53#235112yes—
299C2×C53#254011yescharacteristic
300C2×C53#254011yescharacteristic
301C2×C53#255012yes—
302C2×C53#255012yes—
303C2×C53#258411yescharacteristic
304C2×C53#258411yescharacteristic
305C22×C4⋊C4—532no—
306C22×C4⋊C4—531no—
307C22×C4⋊C4—531no—
308C22×C4⋊C4—531no—
309C22×C4⋊C4—532no—
310C22×C4⋊C4—532no—
311C22×C4⋊C4—532no—
312C22×C4⋊C4C2×D5311yescharacteristic
313C22×C4⋊C4C2×D5312yes—
314C22×C42—531no—
315C22×C42—532no—
316C22×C42—532no—
317#389—1068no—
318#389—1068no—
319#382—1064no—
320C4×C4⋊C4—538no—
321C4×C4⋊C4—1064no—
322C2×#95—531no—
323C2×#95—532no—
324C2×#95—531no—
325C2×#95—531no—
326C2×#95—532no—
327C2×#95—532no—
328C2×#95—532no—
329C2×#95—531no—
330C2×#95C2×D5312yes—
331C2×#95C2×D5311yescharacteristic
332C2×#95C2×D5311yescharacteristic
333C53#1395411yescharacteristic, complemented, Sylow
334C23×C4—531no—
335C23×C4—532no—
336C23×C4—531no—
337C23×C4—532no—
338C23×C4—531no—
339C23×C4—531no—
340C23×C4C22×D5311yescharacteristic
341C23×C4C22×D5311yescharacteristic
342C23×C4C22×D5312yes—
343C23×C4C22×D5312yes—
344C23×C4C22×D5311yescharacteristic
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—538no—
351C2×C4⋊C4—538no—
352C2×C4⋊C4—538no—
353C2×C4⋊C4—1064no—
354C2×C4⋊C4—1062no—
355C2×C4⋊C4—1064no—
356C2×C4⋊C4—1062no—
357C2×C4⋊C4—1062no—
358C2×C4⋊C4—1064no—
359C2×C4⋊C4—1064no—
360C2×C4⋊C4—1064no—
361C2×C4⋊C4—1064no—
362C2×C4⋊C4—1064no—
363C2×C4⋊C4—1064no—
364C2×C4⋊C4—1064no—
365C2×C4⋊C4—2122no—
366C2×C4⋊C4—2122no—
367C2×C4⋊C4—2122no—
368C2×C42—534no—
369C2×C42—534no—
370C2×C42—534no—
371C2×C42—538no—
372C2×C42—1064no—
373C2×C42—1064no—
374C2×C42—1064no—
375C2×C42—2122no—
376C2×C42—2122no—
377#95—22no—
378#95—24no—
379#95—534no—
380#95—1064no—
381#95—1062no—
382#95—1062no—
383#95—1064no—
384#95—1064no—
385#95—1062no—
386#95—1064no—
387#95C4×D5314yes—
388C24C23×D5311yescharacteristic
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—532no—
412C22×C4—532no—
413C22×C4—534no—
414C22×C4—532no—
415C22×C4—534no—
416C22×C4—534no—
417C22×C4—532no—
418C22×C4—532no—
419C22×C4—534no—
420C22×C4—532no—
421C22×C4—534no—
422C22×C4—534no—
423C22×C4—1062no—
424C22×C4—1062no—
425C22×C4—1061no—
426C22×C4—1062no—
427C22×C4—1061no—
428C22×C4—1062no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4C2×C4×D5312yes—
448C22×C4C2×C4×D5314yes—
449C22×C4D4×D5312yes—
450C22×C4D4×D5314yes—
451C22×C4#1171994714yes—
452C22×C4Q8×D5312yes—
453C22×C4Q8×D5314yes—
454C4⋊C4—28no—
455C4⋊C4—48no—
456C4⋊C4—5332no—
457C4⋊C4—1068no—
458C4⋊C4—2128no—
459C4⋊C4—2128no—
460C4⋊C4—2124no—
461C4⋊C4—2124no—
462C4⋊C4—2128no—
463C42—5316no—
464C42—2128no—
465C42—2128no—
466C23C22×C4×D5311yescharacteristic
467C23C2×D4×D5312yes—
468C23C2×D4×D5311yescharacteristic
469C23C2×Q8×D5312yes—
470C23C2×Q8×D5311yescharacteristic
471C23#41×D5312yes—
472C23C2×#1171994712yes—
473C23#1313326211yescharacteristic
474C23#1313326311yescharacteristic
475C23#1313326411yescharacteristic
476C23#1313326511yescharacteristic
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—534no—
505C2×C4—538no—
506C2×C4—534no—
507C2×C4—534no—
508C2×C4—538no—
509C2×C4—534no—
510C2×C4—538no—
511C2×C4—1064no—
512C2×C4—1064no—
513C2×C4—1064no—
514C2×C4—1064no—
515C2×C4—1062no—
516C2×C4—1064no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1062no—
521C2×C4—1062no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—2122no—
527C2×C4—2122no—
528C2×C4—2122no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4C4⋊C4×D5314yes—
541C2×C4C4⋊C4×D5318yes—
542C22C2×C4⋊C4×D5311yescharacteristic
543C22C2×C4⋊C4×D5312yes—
544C22C4×#1171994712yes—
545C22#1681109511yescharacteristic
546C22#1681109911yescharacteristic
547C22#1681112911yescharacteristic
548C22#1681113111yescharacteristic
549C22#127×D5311yescharacteristic
550C22C4⋊Q8×D5311yescharacteristic
551C22C4⋊Q8×D5312yes—
552C22#1681117512yes—
553C22#1681120211yescharacteristic
554C22#1681120311yescharacteristic
555C22#1681125011yescharacteristic
556C22#1681125311yescharacteristic
557C22#1681126012yes—
558C22#1681126512yes—
559C22#1681127912yes—
560C22#1681128312yes—
561C22#1681132712yes—
562C22#1681133112yes—
563C22#1681134812yes—
564C22#1681135012yes—
565C4—28no—
566C4—24no—
567C4—44no—
568C4—81no—
569C4—81no—
570C4—538no—
571C4—1064no—
572C4—1068no—
573C4—4241no—
574C4—4241no—
575C4—2124no—
576C2#384×D5311yescharacteristic
577C2#2027719411yescharacteristic
578C2#2027733011yescharacteristic
579C2#2027762112yes—
580C2#2027763412yes—
581C2#2027964811yescharacteristic
582C2#2027965011yescharacteristic
583C2#2027972912yes—
584C2#2027973112yes—
585C2#2028009711yescharacteristic
586C2#2028009911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1444C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1444D53
hypocenterC53#13954
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
26. #16810474C4semidirect
168. #11719919C4⋊C4semidirect
171. C4×Dic53C4⋊C4semidirect
333. C53#13954semidirect

Maximal subgroups

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

subgroupindex
C2×#168105202
C2×#168105182
C2×#168104872
C2×C4⋊C4×Dic532
C2×#168104742
#202729802
#202729792
#202729782
#202729442
#202729432
C53×#14442
#1395453