Finite Group Databasebeta

← #27310508

Subgroups — #27310508

Order 13568.

2142 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810494C212yesmaximal
3C2×#16810485C211yesmaximal, characteristic
4C2×#16810485C211yesmaximal, characteristic
5#20272979C211yesmaximal, characteristic, complemented
6#20272978C212yesmaximal, complemented
7#20272974C211yesmaximal, characteristic, complemented
8#20272972C211yesmaximal, characteristic, complemented
9#20272970C212yesmaximal, complemented
10#20272936C212yesmaximal
11C53×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C23×C4×D53C2212yes—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919C2212yes—
16C22×#11719919C2212yes—
17C2×C4⋊C4×D53—28no—
18C2×C4⋊C4×D53—24no—
19C22×#11719921C2211yescharacteristic
20C22×#11719921C2211yescharacteristic
21C22×#11719921C2212yes—
22#16810494—28no—
23#16810485—24no—
24#16810485C418yescomplemented
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2212yes—
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C24×D53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2312yes—
45C23×Dic53C2311yescharacteristic
46C22×C4×D53—24no—
47C22×C4×D53—24no—
48C22×C4×D53—24no—
49C22×C4×D53—24no—
50C22×C4×D53—42no—
51C22×C4×D53—42no—
52C22×C4×D53—42no—
53C22×C4×D53—42no—
54C22×C4×D53C2×C414yes—
55C22×C4×D53D414yes—
56C22×C4×D53D418yes—
57C22×C4×D53Q814yes—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53—24no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—42no—
65C2×#11719919—42no—
66C4⋊C4×D53—232no—
67C2×#11719921—24no—
68C2×#11719921—22no—
69C2×#11719921—22no—
70C2×#11719921—24no—
71C2×#11719921—24no—
72C2×#11719921—24no—
73C2×#11719921C2×C418yes—
74#13133124—22no—
75#13133124—24no—
76#13133124—22no—
77#13133124—22no—
78#13133124—24no—
79#13133124—42no—
80#13133124—42no—
81#13133124—42no—
82#13133124C2×C414yescomplemented
83#13133124C2×C418yescomplemented
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2312yes—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53—24no—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53—24no—
92C2×C4⋊C4×C53—42no—
93C2×C4⋊C4×C53—42no—
94#95×C53—22no—
95#95×C53—42no—
96#95×C53C2×C414yescomplemented
97C24×C53C2411yescharacteristic
98C23×D53—21no—
99C23×D53—22no—
100C23×D53—21no—
101C23×D53—21no—
102C23×D53—22no—
103C23×D53—21no—
104C23×D53C22×C412yes—
105C23×D53C2×D412yes—
106C23×D53C2×D414yes—
107C23×D53C2×Q812yes—
108C23×D53#4114yes—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
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×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C22×C414yes—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D414yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×Q812yes—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53C22×C414yes—
163C22×Dic53C22×C412yes—
164C22×Dic53C2×D414yes—
165C22×Dic53C2×D412yes—
166C22×Dic53C2×Q812yes—
167C2×C4×D53—216no—
168C2×C4×D53—28no—
169C2×C4×D53—216no—
170C2×C4×D53—28no—
171C2×C4×D53—216no—
172C2×C4×D53—48no—
173C2×C4×D53—48no—
174C2×C4×D53—48no—
175C2×C4×D53—48no—
176C2×C4×D53—48no—
177C2×C4×D53—48no—
178C2×C4×D53C4⋊C4116yescomplemented
179C4⋊C4×C53—28no—
180C4⋊C4×C53—48no—
181#11719921—216no—
182#11719921—44no—
183#11719921—44no—
184C4⋊Dic53—28no—
185#11719919—216no—
186#11719919—48no—
187C23×C53C23×C411yescharacteristic
188C23×C53C22×D411yescharacteristic
189C23×C53C22×D412yes—
190C23×C53C22×Q811yescharacteristic
191C23×C53C2×#4112yes—
192C23×C53H(5,𝔽2)11yescharacteristic
193C23×C53H(5,𝔽2)12yes—
194C23×C53H(5,𝔽2)11yescharacteristic
195C23×C532−1+411yescharacteristic
196C23×C532−1+412yes—
197C23×C532−1+411yescharacteristic
198C22×D53—22no—
199C22×D53—24no—
200C22×D53—22no—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—24no—
207C22×D53—22no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—24no—
213C22×D53—24no—
214C22×D53—22no—
215C22×D53—22no—
216C22×D53C2×C4⋊C414yes—
217C22×D53C4×D418yes—
218C22×D53#12014yes—
219C22×D53C22⋊Q818yes—
220C22×D53#12314yes—
221C2×C4×C53—24no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53C2×C4⋊C414yes—
250C2×Dic53—24no—
251C2×Dic53—22no—
252C2×Dic53—24no—
253C2×Dic53—22no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—42no—
273C2×Dic53—41no—
274C2×Dic53—42no—
275C2×Dic53—41no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53C2×C4⋊C414yes—
279C4×D53—232no—
280C4×D53—432no—
281#14263—531nomaximal, Sylow
282C22×C53C22×C4⋊C411yescharacteristic
283C22×C53C2×C4×D412yes—
284C22×C53#52011yescharacteristic
285C22×C53#52012yes—
286C22×C53#52011yescharacteristic
287C22×C53C2×#12011yescharacteristic
288C22×C53C2×C22⋊Q812yes—
289C22×C53C2×#12311yescharacteristic
290C22×C53#53412yes—
291C22×C53#53412yes—
292C22×C53#53511yescharacteristic
293C22×C53#53511yescharacteristic
294C22×C53#53611yescharacteristic
295C22×C53#53612yes—
296C22×C53#53611yescharacteristic
297C22×C53#53612yes—
298C22×C53#53912yes—
299C22×C53#53912yes—
300C22×C53#53912yes—
301C22×C53#53912yes—
302C22×C53#54412yes—
303C22×C53#54411yescharacteristic
304C22×C53#54411yescharacteristic
305C2×D53—24no—
306C2×D53—24no—
307C2×D53—28no—
308C2×D53—28no—
309C2×D53—24no—
310C2×D53—24no—
311C2×D53—24no—
312C2×D53—24no—
313C2×D53—28no—
314C2×D53—28no—
315C2×D53#38518yes—
316C4×C53—24no—
317C4×C53—44no—
318C4×C53—44no—
319C4×C53—81no—
320C4×C53—81no—
321Dic53—24no—
322Dic53—44no—
323Dic53—44no—
324Dic53—81no—
325Dic53—81no—
326C23×C4⋊C4—531no—
327C2×#385—532no—
328C2×#381—531no—
329C2×#381—531no—
330#1444—531no—
331#1444—532no—
332#1444D5311yescharacteristic, complemented
333#1442—532no—
334#1442—531no—
335#1442—531no—
336C22⋊#95—532no—
337C2×C53C2×#38511yescharacteristic
338C2×C53#231711yescharacteristic
339C2×C53#231711yescharacteristic
340C2×C53#232212yes—
341C2×C53#232212yes—
342C2×C53#261311yescharacteristic
343C2×C53#261411yescharacteristic
344C2×C53#262311yescharacteristic
345C2×C53#262311yescharacteristic
346C2×C53#262712yes—
347C2×C53#262712yes—
348D53—28no—
349C24×C4—531no—
350C24×C4—532no—
351C22×C4⋊C4—532no—
352C22×C4⋊C4—531no—
353C22×C4⋊C4—532no—
354C22×C4⋊C4—1064no—
355C22×C4⋊C4—1068no—
356C22×C4⋊C4C2×D5311yescharacteristic
357C22×C4⋊C4C2×D5312yes—
358C22×C22⋊C4—532no—
359C22×C22⋊C4—531no—
360C22×C22⋊C4—531no—
361#385—1068no—
362#381—538no—
363#381—1064no—
364C2×#95—532no—
365C2×#95—531no—
366C2×#95—531no—
367C2×#95—531no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—531no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—532no—
374C2×#95C2×D5311yescharacteristic
375C2×#95C2×D5312yes—
376C2×#95C2×D5311yescharacteristic
377C53#1426311yescharacteristic, complemented, Sylow
378C25—531no—
379C23×C4—531no—
380C23×C4—534no—
381C23×C4—534no—
382C23×C4—534no—
383C23×C4—532no—
384C23×C4—532no—
385C23×C4—538no—
386C23×C4—531no—
387C23×C4—531no—
388C23×C4—1064no—
389C23×C4—1064no—
390C23×C4—1064no—
391C23×C4—1064no—
392C23×C4—2122no—
393C23×C4—2122no—
394C23×C4—2122no—
395C23×C4—2122no—
396C23×C4C22×D5311yescharacteristic
397C23×C4C22×D5311yescharacteristic
398C23×C4C22×D5312yes—
399C23×C4C22×D5312yes—
400C23×C4C22×D5311yescharacteristic
401C2×C4⋊C4—24no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—42no—
405C2×C4⋊C4—42no—
406C2×C4⋊C4—10632no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—1064no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1064no—
413C2×C4⋊C4—2122no—
414C2×C4⋊C4—2122no—
415C2×C22⋊C4—538no—
416C2×C22⋊C4—1064no—
417C2×C22⋊C4—1064no—
418C2×C22⋊C4—1064no—
419C2×C22⋊C4—1062no—
420C2×C22⋊C4—1062no—
421C2×C22⋊C4—1064no—
422#95—22no—
423#95—42no—
424#95—538no—
425#95—534no—
426#95—1062no—
427#95—1062no—
428#95—1064no—
429#95—1062no—
430#95—1064no—
431#95—2122no—
432#95—2122no—
433#95—2122no—
434#95C4×D5314yescomplemented
435C24—532no—
436C24—532no—
437C24—534no—
438C24—532no—
439C24—534no—
440C24—1061no—
441C24—1062no—
442C24—1061no—
443C24—1061no—
444C24—1062no—
445C24—1061no—
446C24C23×D5311yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—532no—
472C22×C4—532no—
473C22×C4—532no—
474C22×C4—5316no—
475C22×C4—534no—
476C22×C4—534no—
477C22×C4—1068no—
478C22×C4—10616no—
479C22×C4—10616no—
480C22×C4—1068no—
481C22×C4—10616no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1061no—
497C22×C4—1062no—
498C22×C4—1061no—
499C22×C4—1062no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—1062no—
504C22×C4—1061no—
505C22×C4—1062no—
506C22×C4—2128no—
507C22×C4—2128no—
508C22×C4—2128no—
509C22×C4—2128no—
510C22×C4—2128no—
511C22×C4—2128no—
512C22×C4C2×C4×D5314yes—
513C22×C4C2×C4×D5312yes—
514C22×C4D4×D5314yes—
515C22×C4D4×D5312yes—
516C22×C4Q8×D5312yes—
517C4⋊C4—28no—
518C4⋊C4—48no—
519C4⋊C4—10616no—
520C4⋊C4—1068no—
521C4⋊C4—2128no—
522C22⋊C4—10616no—
523C22⋊C4—2124no—
524C22⋊C4—2124no—
525C23—534no—
526C23—538no—
527C23—534no—
528C23—538no—
529C23—534no—
530C23—1062no—
531C23—1064no—
532C23—1062no—
533C23—1064no—
534C23—1064no—
535C23—1062no—
536C23—1062no—
537C23—1062no—
538C23—1064no—
539C23—1062no—
540C23—1064no—
541C23—1064no—
542C23—1064no—
543C23—1064no—
544C23—1064no—
545C23—1064no—
546C23—1062no—
547C23—1062no—
548C23C22×C4×D5311yescharacteristic
549C23C2×D4×D5311yescharacteristic
550C23C2×D4×D5312yes—
551C23C2×Q8×D5311yescharacteristic
552C23#41×D5312yes—
553C23#1313326211yescharacteristic
554C23#1313326212yes—
555C23#1313326311yescharacteristic
556C23#1313326411yescharacteristic
557C23#1313326512yes—
558C23#1313326511yescharacteristic
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—534no—
588C2×C4—10632no—
589C2×C4—1064no—
590C2×C4—1062no—
591C2×C4—1064no—
592C2×C4—1062no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1064no—
597C2×C4—1064no—
598C2×C4—1064no—
599C2×C4—21232no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2122no—
605C2×C4—2121no—
606C2×C4—2122no—
607C2×C4—2121no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4C4⋊C4×D5314yes—
619C22—538no—
620C22—1064no—
621C22—1064no—
622C22—1064no—
623C22—1064no—
624C22—1068no—
625C22—1068no—
626C22—1064no—
627C22—1064no—
628C22—1068no—
629C22—1068no—
630C22C2×C4⋊C4×D5311yescharacteristic
631C22C4×D4×D5312yes—
632C22#1681110211yescharacteristic
633C22#1681110412yes—
634C22#1681110511yescharacteristic
635C22#120×D5311yescharacteristic
636C22C22⋊Q8×D5312yes—
637C22#123×D5311yescharacteristic
638C22#1681118112yes—
639C22#1681118312yes—
640C22#1681118611yescharacteristic
641C22#1681118911yescharacteristic
642C22#1681119611yescharacteristic
643C22#1681119612yes—
644C22#1681119811yescharacteristic
645C22#1681119812yes—
646C22#1681121612yes—
647C22#1681121912yes—
648C22#1681121912yes—
649C22#1681122112yes—
650C22#1681125012yes—
651C22#1681125011yescharacteristic
652C22#1681125311yescharacteristic
653C4—24no—
654C4—44no—
655C4—44no—
656C4—81no—
657C4—81no—
658C4—1064no—
659C4—4241no—
660C4—4241no—
661C4—2124no—
662C4—2124no—
663C2—1068no—
664C2#385×D5311yescharacteristic
665C2#2027733611yescharacteristic
666C2#2027733711yescharacteristic
667C2#2027737912yes—
668C2#2027738212yes—
669C2#2028042611yescharacteristic
670C2#2028043511yescharacteristic
671C2#2028049611yescharacteristic
672C2#2028050011yescharacteristic
673C2#2028052712yes—
674C2#2028053312yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1444C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1444D53
hypocenterC53#14263
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
5. #20272979C2semidirect
6. #20272978C2semidirect
7. #20272974C2semidirect
8. #20272972C2semidirect
9. #20272970C2semidirect
11. C53×#1444C2semidirect
24. #16810485C4semidirect
82. #13133124C2×C4semidirect
83. #13133124C2×C4semidirect
96. #95×C53C2×C4semidirect
178. C2×C4×D53C4⋊C4semidirect
332. #1444D53semidirect
377. C53#14263semidirect
434. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D532
C2×#168104942
C2×#168104852
#202729792
#202729782
#202729742
#202729722
#202729702
#202729362
C53×#14442
#1426353