Finite Group Databasebeta

← #28319260

Subgroups — #28319260

Order 13568.

1490 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291191C212yesmaximal, complemented
2#20290611C212yesmaximal, complemented
3#20290577C212yesmaximal, complemented
4#20290575C212yesmaximal, complemented
5#20290541C212yesmaximal
6#20290462C212yesmaximal, complemented
7C4×#13133275C212yesmaximal, complemented
8C2×#16811096C211yesmaximal, characteristic
9C2×#13133275C2214yes—
10C2×#13133275C2214yes—
11#16811573—28no—
12#16811565C2214yescomplemented
13#16811564C2214yes—
14C4×C22⋊#6157C2214yescomplemented
15#16811501C2214yescomplemented
16#16811501C2214yescomplemented
17C23×C22⋊D53C2211yescharacteristic
18#16811096—28no—
19C2×C4×C22⋊D53C411yescharacteristic, complemented
20C2×C4×C22⋊D53C411yescharacteristic, complemented
21C2×C4×C22⋊D53C2211yescharacteristic
22C2×C4×C22⋊D53C2211yescharacteristic
23C2×#13133177C411yescharacteristic, complemented
24C2×#13133177C411yescharacteristic, complemented
25C2×#13133177C2211yescharacteristic
26C2×#13133177C2211yescharacteristic
27C2×#13133173C411yescharacteristic, complemented
28C2×#13133173C411yescharacteristic, complemented
29C22×C22⋊Dic53C411yescharacteristic, complemented
30C2×C22⋊C4×D53C2211yescharacteristic
31C2×C22⋊C4×D53C2211yescharacteristic
32C22×C22⋊C4×C53C411yescharacteristic, complemented
33C2×C4⋊#6157—24no—
34C2×C22⋊#6157—24no—
35C2×C22⋊#6157—24no—
36C2×C22⋊#6157C2318yes—
37C2×C4×#6157—24no—
38#13133275—28no—
39#13133275—28no—
40(C2×#6157)⋊C4—24no—
41(C2×#6157)⋊C4—24no—
42C24×D53C2311yescharacteristic
43C22×C22⋊D53—24no—
44C22×C22⋊D53—24no—
45C22×C22⋊D53—24no—
46C22×C22⋊D53C2×C411yescharacteristic
47C22×C22⋊D53C2×C411yescharacteristic
48C22×C22⋊D53C2311yescharacteristic
49C22×C22⋊D53C2311yescharacteristic
50C23×Dic53C2×C411yescharacteristic
51C23×Dic53C2×C411yescharacteristic
52C23×Dic53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2×C411yescharacteristic
55C22×C4×D53C2311yescharacteristic
56C22×C4×D53C2311yescharacteristic
57C4×C22⋊D53—24no—
58C4×C22⋊D53—24no—
59C4×C22⋊D53—24no—
60C4×C22⋊D53—24no—
61#13133177—24no—
62#13133177—24no—
63#13133177—24no—
64#13133177—24no—
65#13133173—24no—
66#13133173—24no—
67C2×#11719919C2×C411yescharacteristic
68C2×#11719919C2×C411yescharacteristic
69C2×#11719919C2×C411yescharacteristic
70C2×#11719919C2×C411yescharacteristic
71C2×C22⋊Dic53—24no—
72C2×C22⋊Dic53C2×C411yescharacteristic
73C2×C22⋊Dic53C2×C411yescharacteristic
74C2×C22⋊Dic53C2×C411yescharacteristic
75C2×C22⋊Dic53C2×C411yescharacteristic
76C2×#11719921C2×C411yescharacteristic
77C2×#11719921C2×C411yescharacteristic
78C2×#11719921C2311yescharacteristic
79C2×#11719921C2311yescharacteristic
80C22⋊C4×D53—24no—
81C22⋊C4×D53D418yes—
82C2×C4×Dic53C2×C411yescharacteristic
83C2×C4×Dic53C2×C411yescharacteristic
84C2×C4×Dic53C2×C411yescharacteristic
85C2×C4×Dic53C2×C411yescharacteristic
86C25×C53C2×C411yescharacteristic
87C23×C4×C53C2×C411yescharacteristic
88C23×C4×C53C2×C411yescharacteristic
89C2×C22⋊C4×C53—24no—
90C2×C22⋊C4×C53C2×C411yescharacteristic
91C2×C22⋊C4×C53C2×C411yescharacteristic
92C2×C22⋊C4×C53C2×C411yescharacteristic
93C2×C22⋊C4×C53C2×C411yescharacteristic
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—81no—
98C24×C53—81no—
99C24×C53C22×C411yescharacteristic
100C24×C53C22×C411yescharacteristic
101C24×C53C22×C411yescharacteristic
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—81no—
105C23×D53—81no—
106C23×D53C22×C411yescharacteristic
107C23×D53C2×D414yes—
108C23×D53C2411yescharacteristic
109C22×#6157—28no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—81no—
113C22×C4×C53—81no—
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C22×C411yescharacteristic
117C22×C4×C53C22×C411yescharacteristic
118C2×C22⋊D53—232no—
119C2×C22⋊D53—22no—
120C2×C22⋊D53—22no—
121C2×C22⋊D53—22no—
122C2×C22⋊D53—22no—
123C2×C22⋊D53—22no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—22no—
128C2×C22⋊D53—22no—
129C2×C22⋊D53#4114yes—
130C2×C22⋊D53#4114yes—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—81no—
137C22×Dic53—81no—
138C22×Dic53C22×C411yescharacteristic
139C22×Dic53C22×C411yescharacteristic
140C22×Dic53C22×C411yescharacteristic
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C22×C411yescharacteristic
143C22×Dic53C22×C411yescharacteristic
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—81no—
147C2×C4×D53—81no—
148C2×C4×D53—81no—
149C2×C4×D53—81no—
150C2×C4×D53C2×D414yes—
151C2×C4×D53C2×D414yes—
152C22⋊#6157—28no—
153C22⋊#6157—28no—
154C22⋊#6157—84no—
155C22⋊#6157—84no—
156C4⋊#6157—84no—
157C4×#6157—84no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—22no—
161C22⋊C4×C53C22⋊C414yes—
162C22⋊Dic53—22no—
163C22⋊Dic53—22no—
164C22⋊Dic53—22no—
165C22⋊Dic53C22⋊C414yes—
166#11719921—22no—
167#11719921—22no—
168#11719921C22⋊C414yes—
169#11719921C22⋊C414yes—
170#11719919—22no—
171#11719919—22no—
172#11719919—22no—
173#11719919—22no—
174C4×Dic53—22no—
175C4×Dic53—22no—
176C4×Dic53—22no—
177C4×Dic53—22no—
178C23×C53—28no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—82no—
192C23×C53—82no—
193C23×C53—84no—
194C23×C53—84no—
195C23×C53C2×C22⋊C412yes—
196C23×C53#11712yes—
197C23×C53#11712yes—
198C23×C53C23×C411yescharacteristic
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—28no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—44no—
207C22×D53—44no—
208C22×D53—84no—
209C22×D53—84no—
210C22×D53C2×C22⋊C412yes—
211C22×D53C22×D412yes—
212C22×D53C2×#4112yes—
213C22×D53C2×#4112yes—
214C2×#6157—48no—
215C2×#6157—48no—
216C2×#6157—48no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—82no—
224C2×C4×C53—82no—
225C2×C4×C53C2×C22⋊C412yes—
226C2×C4×C53C2×C22⋊C412yes—
227C22⋊D53—28no—
228C22⋊D53—28no—
229C22⋊D53—28no—
230C22⋊D53—28no—
231C2×Dic53—28no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—82no—
243C2×Dic53—82no—
244C2×Dic53C2×C22⋊C412yes—
245C2×Dic53C2×C22⋊C412yes—
246C2×Dic53#11712yes—
247C2×Dic53#11712yes—
248C4×D53—44no—
249C4×D53—44no—
250C4×D53—82no—
251C4×D53—82no—
252#20724—531nomaximal, Sylow
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—88no—
271C22×C53C22×C22⋊C411yescharacteristic
272C22×C53C2×#11711yescharacteristic
273C22×C53C2×#11711yescharacteristic
274C2×D53—22no—
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—22no—
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—42no—
282C2×D53—42no—
283C2×D53—88no—
284C2×D53#52514yes—
285#6157—88no—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—41no—
289C4×C53—41no—
290Dic53—22no—
291Dic53—22no—
292Dic53—22no—
293Dic53—22no—
294Dic53—41no—
295Dic53—41no—
296C2×#518—531no—
297#2312—532no—
298#1819—532no—
299#1799—532no—
300#1799—532no—
301#1781—532no—
302C4×#99—532no—
303C4×#99—532no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53—42no—
307C2×C53—42no—
308C2×C53—42no—
309C2×C53#228011yescharacteristic
310C2×C53#288311yescharacteristic
311C2×C53#288311yescharacteristic
312D53—24no—
313D53—42no—
314C23×D4—531no—
315#518—1068no—
316C2×C4×D4—531no—
317C2×C4×D4—531no—
318C2×C4×D4—531no—
319C2×C4×D4—531no—
320C2×C4×D4—531no—
321C2×C4×D4—531no—
322C2×C4×D4—531no—
323C2×C4×D4—531no—
324C2×#117—531no—
325C2×#117—531no—
326C22×C22⋊C4—531no—
327C22×C22⋊C4—531no—
328C22×C22⋊C4—531no—
329C22×C22⋊C4#615711yescharacteristic, complemented
330C2×#99—534no—
331C2×#99—534no—
332#388—1068no—
333#380—534no—
334#379—534no—
335C4×C22⋊C4—534no—
336#342—534no—
337#342—534no—
338C53#2072411yescharacteristic, complemented, Sylow
339C25—531no—
340C25C2×#615711yescharacteristic
341C22×D4—531no—
342C22×D4—531no—
343C22×D4—531no—
344C22×D4—531no—
345C22×D4—1064no—
346C22×D4—1064no—
347C22×D4—1064no—
348C23×C4—531no—
349C23×C4—531no—
350C23×C4—531no—
351C23×C4—531no—
352C23×C4—531no—
353C23×C4—531no—
354C23×C4—531no—
355C23×C4C2×#615711yescharacteristic
356C23×C4C2×#615711yescharacteristic
357C4×D4—1064no—
358C4×D4—1064no—
359C4×D4—1064no—
360C4×D4—1064no—
361C4×D4—1064no—
362C4×D4—1064no—
363C4×D4—1064no—
364C4×D4—1064no—
365#117—1064no—
366#117—1064no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—1064no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—531no—
374C2×C22⋊C4—531no—
375C2×C22⋊C4—531no—
376C2×C22⋊C4—538no—
377C2×C22⋊C4—531no—
378C2×C22⋊C4—531no—
379C2×C22⋊C4—531no—
380C2×C22⋊C4—531no—
381C2×C22⋊C4—531no—
382C2×C22⋊C4—538no—
383C2×C22⋊C4—1064no—
384C2×C22⋊C4—1064no—
385C2×C22⋊C4—1064no—
386C2×C22⋊C4—1064no—
387C2×C22⋊C4C2×#615711yescharacteristic
388C2×C22⋊C4C2×#615711yescharacteristic
389C2×C22⋊C4C2×#615711yescharacteristic
390C2×C22⋊C4C2×#615711yescharacteristic
391C2×C42—531no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—1064no—
396#99—1068no—
397#99—1068no—
398#95—1064no—
399#95—1064no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—81no—
404C24—81no—
405C24—531no—
406C24—534no—
407C24—531no—
408C24—1062no—
409C24—1062no—
410C24—4241no—
411C24—4241no—
412C24C22×#615711yescharacteristic
413C24C22×#615711yescharacteristic
414C24C22×#615711yescharacteristic
415C2×D4—534no—
416C2×D4—534no—
417C2×D4—1062no—
418C2×D4—1062no—
419C2×D4—1062no—
420C2×D4—1062no—
421C2×D4—1062no—
422C2×D4—1062no—
423C2×D4—1062no—
424C2×D4—1062no—
425C2×D4—10632no—
426C2×D4—1062no—
427C2×D4—1062no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—531no—
433C22×C4—531no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—534no—
437C22×C4—534no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1068no—
448C22×C4—4241no—
449C22×C4—4241no—
450C22×C4—4241no—
451C22×C4—4241no—
452C22×C4—4241no—
453C22×C4—4241no—
454C22×C4C22×#615711yescharacteristic
455C22×C4C22×#615711yescharacteristic
456C22×C4C22×#615711yescharacteristic
457C22×C4C22×#615711yescharacteristic
458C4⋊C4—1062no—
459C4⋊C4—1062no—
460C4⋊C4—1062no—
461C4⋊C4—1062no—
462C4⋊C4—4244no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—534no—
467C22⋊C4—534no—
468C22⋊C4—534no—
469C22⋊C4—1062no—
470C22⋊C4—1062no—
471C22⋊C4—1062no—
472C22⋊C4—1062no—
473C22⋊C4—1062no—
474C22⋊C4—1068no—
475C22⋊C4—1068no—
476C22⋊C4—4244no—
477C22⋊C4—4244no—
478C22⋊C4C22⋊#615714yes—
479C42—1062no—
480C42—1062no—
481C42—1062no—
482C42—1062no—
483C42—4244no—
484C23—28no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—82no—
498C23—82no—
499C23—84no—
500C23—84no—
501C23—532no—
502C23—532no—
503C23—532no—
504C23—532no—
505C23—1068no—
506C23—1061no—
507C23—1061no—
508C23—4244no—
509C23—4244no—
510C23—2124no—
511C23—2124no—
512C23—2121no—
513C23—2121no—
514C23—2121no—
515C23—2121no—
516C23C2×C22⋊#615712yes—
517C23#1313329612yes—
518C23#1313329612yes—
519C23C23×#615711yescharacteristic
520D4—1068no—
521D4—1068no—
522D4—1068no—
523D4—1068no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—532no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—1068no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—4242no—
544C2×C4—4242no—
545C2×C4—4242no—
546C2×C4—4242no—
547C2×C4—2124no—
548C2×C4—2124no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2128no—
552C2×C4—2128no—
553C2×C4—2128no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4C2×C22⋊#615712yes—
557C2×C4C2×C22⋊#615712yes—
558C22—21no—
559C22—21no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—88no—
576C22—534no—
577C22—1062no—
578C22—1062no—
579C22—1062no—
580C22—1062no—
581C22—1062no—
582C22—4248no—
583C22—2122no—
584C22—2122no—
585C22—2122no—
586C22—2122no—
587C22C22×C22⋊#615711yescharacteristic
588C22C2×#1313329611yescharacteristic
589C22C2×#1313329611yescharacteristic
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—1062no—
595C4—1062no—
596C4—1062no—
597C4—1062no—
598C4—4248no—
599C4—2121no—
600C4—2121no—
601C2—21no—
602C2—21no—
603C2—42no—
604C2—42no—
605C2—42no—
606C2—1064no—
607C2—2122no—
608C2#2029115911yescharacteristic
609C2#2029131411yescharacteristic
610C2#2029131411yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C22⋊C4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC23×C22⋊D53C22
cosocleC23×D53C24
hypercenterC22×C22⋊C4#6157
hypocenterC53#20724
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20291191C2semidirect
2. #20290611C2semidirect
3. #20290577C2semidirect
4. #20290575C2semidirect
6. #20290462C2semidirect
7. C4×#13133275C2semidirect
12. #16811565C22semidirect
14. C4×C22⋊#6157C22semidirect
15. #16811501C22semidirect
16. #16811501C22semidirect
19. C2×C4×C22⋊D53C4semidirect
20. C2×C4×C22⋊D53C4semidirect
23. C2×#13133177C4semidirect
24. C2×#13133177C4semidirect
27. C2×#13133173C4semidirect
28. C2×#13133173C4semidirect
29. C22×C22⋊Dic53C4semidirect
32. C22×C22⋊C4×C53C4semidirect
329. C22×C22⋊C4#6157semidirect
338. C53#20724semidirect

Maximal subgroups

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

subgroupindex
#202911912
#202906112
#202905772
#202905752
#202905412
#202904622
C4×#131332752
C2×#168110962
#2072453