Finite Group Databasebeta

← #27306845

Subgroups — #27306845

Order 13568.

2150 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810497C212yesmaximal
2C2×#16810494C212yesmaximal
3C2×#16810480C211yesmaximal, characteristic
4C2×#16810470C211yesmaximal, characteristic
5C2×#95×D53C211yesmaximal, characteristic
6#20272980C211yesmaximal, characteristic, complemented
7#20272979C211yesmaximal, characteristic, complemented
8#20272978C211yesmaximal, characteristic, complemented
9#20272944C212yesmaximal, complemented
10#20272943C212yesmaximal, complemented
11C53×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C22×C4⋊Dic53C2212yes—
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×#11719921C2212yes—
20C22×#11719921C2212yes—
21C22×C4×Dic53C2212yes—
22#16810497—28no—
23#16810494—28no—
24#16810480—24no—
25#16810470C418yescomplemented
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34#95×D53—24no—
35C22×C4⋊C4×C53C2212yes—
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C24×D53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2312yes—
46C22×C4×D53—22no—
47C22×C4×D53—22no—
48C22×C4×D53—22no—
49C22×C4×D53—22no—
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—24no—
53C22×C4×D53—24no—
54C22×C4×D53—42no—
55C22×C4×D53—42no—
56C22×C4×D53—42no—
57C22×C4×D53—42no—
58C22×C4×D53C2×C414yes—
59C22×C4×D53D414yes—
60C22×C4×D53Q814yes—
61C2×C4⋊Dic53—24no—
62C2×C4⋊Dic53—22no—
63C2×C4⋊Dic53—24no—
64C2×C4⋊Dic53—42no—
65C2×C4⋊Dic53—42no—
66C2×#11719919—24no—
67C2×#11719919—24no—
68C2×#11719919—24no—
69C2×#11719919—24no—
70C2×#11719919—42no—
71C2×#11719919—42no—
72C2×#11719921—24no—
73C2×#11719921—24no—
74C2×#11719921—24no—
75C2×#11719921C2×C418yes—
76C2×#11719921D418yes—
77C2×#11719921Q818yes—
78C2×C4×Dic53—42no—
79C2×C4×Dic53—42no—
80C2×C4×Dic53C2×C418yescomplemented
81#13133124—22no—
82#13133124—24no—
83#13133124—24no—
84#13133124—22no—
85#13133124—22no—
86#13133124—24no—
87#13133124—24no—
88#13133124C2×C414yescomplemented
89C23×C4×C53C2312yes—
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2312yes—
93C23×C4×C53C2311yescharacteristic
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—24no—
96C2×C4⋊C4×C53—24no—
97C2×C4⋊C4×C53—42no—
98C2×C4⋊C4×C53—42no—
99#95×C53—24no—
100#95×C53—22no—
101#95×C53C2×C414yescomplemented
102C24×C53C2411yescharacteristic
103C23×D53—21no—
104C23×D53—21no—
105C23×D53—21no—
106C23×D53—21no—
107C23×D53—22no—
108C23×D53—22no—
109C23×D53C22×C412yes—
110C23×D53C2×D414yes—
111C23×D53C2×D412yes—
112C23×D53C2×Q814yes—
113C23×D53C2×Q812yes—
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—21no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53C22×C412yes—
136C22×C4×C53C22×C414yes—
137C22×C4×C53C2×D414yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×Q814yes—
140C22×C4×C53C2×Q812yes—
141C22×C4×C53#4112yes—
142C22×C4×C53#4112yes—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53C22×C414yes—
165C22×Dic53C22×C412yes—
166C22×Dic53C2×D412yes—
167C22×Dic53C2×D414yes—
168C22×Dic53C2×Q814yes—
169C22×Dic53C2×Q812yes—
170C22×Dic53#4112yes—
171C22×Dic53#4112yes—
172C2×C4×D53—28no—
173C2×C4×D53—28no—
174C2×C4×D53—48no—
175C2×C4×D53—44no—
176C2×C4×D53—44no—
177C2×C4×D53—48no—
178C2×C4×D53—48no—
179C2×C4×D53—48no—
180C2×C4×D53—48no—
181C2×C4×D53—48no—
182C2×C4×D53—44no—
183C2×C4×D53—44no—
184C2×C4×D53—44no—
185C2×C4×D53—44no—
186C2×C4×D53C4⋊C4116yescomplemented
187C4⋊C4×C53—48no—
188C4⋊C4×C53—44no—
189#11719921—48no—
190#11719921C4⋊C4132yescomplemented
191C4⋊Dic53—48no—
192C4⋊Dic53—44no—
193#11719919—48no—
194#11719919—48no—
195C4×Dic53—48no—
196C23×C53C23×C411yescharacteristic
197C23×C53C22×D411yescharacteristic
198C23×C53C22×D412yes—
199C23×C53C22×Q811yescharacteristic
200C23×C53C22×Q812yes—
201C23×C53C2×#4111yescharacteristic
202C23×C53C2×#4111yescharacteristic
203C23×C53C2×#4111yescharacteristic
204C23×C53C2×#4111yescharacteristic
205C23×C53H(5,𝔽2)12yes—
206C23×C53H(5,𝔽2)12yes—
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—24no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—22no—
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—24no—
217C22×D53—24no—
218C22×D53—24no—
219C22×D53—24no—
220C22×D53—24no—
221C22×D53—24no—
222C22×D53—24no—
223C22×D53—24no—
224C22×D53—24no—
225C22×D53C2×C4⋊C414yes—
226C22×D53C2×C4⋊C418yes—
227C22×D53#12714yes—
228C22×D53C4⋊Q814yes—
229C22×D53C4⋊Q818yes—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53C2×C4⋊C414yes—
258C2×C4×C53C2×C4⋊C418yes—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—24no—
272C2×Dic53—24no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53C2×C4⋊C414yes—
287C2×Dic53C2×C4⋊C418yes—
288C4×D53—416no—
289C4×D53—416no—
290C4×D53—416no—
291#13965—531nomaximal, Sylow
292C22×C53C22×C4⋊C411yescharacteristic
293C22×C53C22×C4⋊C412yes—
294C22×C53C4×#4111yescharacteristic
295C22×C53C4×#4111yescharacteristic
296C22×C53#51812yes—
297C22×C53#52511yescharacteristic
298C22×C53#52511yescharacteristic
299C22×C53C2×#12711yescharacteristic
300C22×C53C2×C4⋊Q811yescharacteristic
301C22×C53C2×C4⋊Q812yes—
302C22×C53#53311yescharacteristic
303C22×C53#53311yescharacteristic
304C22×C53#53411yescharacteristic
305C22×C53#53411yescharacteristic
306C22×C53#54312yes—
307C22×C53#54612yes—
308C22×C53#54612yes—
309C22×C53#54612yes—
310C22×C53#54612yes—
311C22×C53#55412yes—
312C22×C53#55412yes—
313C22×C53#55412yes—
314C22×C53#55412yes—
315C2×D53—24no—
316C2×D53—24no—
317C2×D53—24no—
318C2×D53—28no—
319C2×D53—28no—
320C2×D53—24no—
321C2×D53—24no—
322C2×D53—28no—
323C2×D53—28no—
324C2×D53—24no—
325C2×D53#38418yes—
326C4×C53—24no—
327C4×C53—28no—
328C4×C53—42no—
329C4×C53—42no—
330C4×C53—82no—
331Dic53—24no—
332Dic53—28no—
333Dic53—42no—
334Dic53—42no—
335Dic53—82no—
336C2×#385—532no—
337C2×#385—532no—
338C2×#380—531no—
339C2×C4×C22⋊C4—531no—
340C22×#95—531no—
341#1444—531no—
342#1444—531no—
343#1444—531no—
344#1444D5311yescharacteristic, complemented
345C4⋊#95—532no—
346C4⋊#95—532no—
347C2×C53C2×#38411yescharacteristic
348C2×C53#229611yescharacteristic
349C2×C53#231511yescharacteristic
350C2×C53#234912yes—
351C2×C53#234912yes—
352C2×C53#253911yescharacteristic
353C2×C53#253911yescharacteristic
354C2×C53#254912yes—
355C2×C53#254912yes—
356C2×C53#258311yescharacteristic
357C2×C53#258311yescharacteristic
358D53—28no—
359C24×C4—531no—
360C24×C4—531no—
361C24×C4—531no—
362C22×C4⋊C4—532no—
363C22×C4⋊C4—532no—
364C22×C4⋊C4—532no—
365C22×C4⋊C4—531no—
366C22×C4⋊C4C2×D5312yes—
367C22×C4⋊C4C2×D5311yescharacteristic
368C22×C22⋊C4—532no—
369C22×C22⋊C4—532no—
370C22×C42—532no—
371#385—1068no—
372#385—1068no—
373#380—1064no—
374C4×C22⋊C4—538no—
375C2×#95—531no—
376C2×#95—531no—
377C2×#95—531no—
378C2×#95—532no—
379C2×#95—532no—
380C2×#95—532no—
381C2×#95—532no—
382C2×#95—531no—
383C2×#95—1064no—
384C2×#95C2×D5312yes—
385C2×#95C2×D5311yescharacteristic
386C2×#95C2×D5311yescharacteristic
387C53#1396511yescharacteristic, complemented, Sylow
388C25—531no—
389C23×C4—532no—
390C23×C4—531no—
391C23×C4—534no—
392C23×C4—534no—
393C23×C4—534no—
394C23×C4—531no—
395C23×C4—531no—
396C23×C4—532no—
397C23×C4—1062no—
398C23×C4—1062no—
399C23×C4—1062no—
400C23×C4—1062no—
401C23×C4—1064no—
402C23×C4—1064no—
403C23×C4—1062no—
404C23×C4—1062no—
405C23×C4—2122no—
406C23×C4—2122no—
407C23×C4—2122no—
408C23×C4—2122no—
409C23×C4C22×D5312yes—
410C23×C4C22×D5311yescharacteristic
411C23×C4C22×D5311yescharacteristic
412C23×C4C22×D5312yes—
413C23×C4C22×D5311yescharacteristic
414C2×C4⋊C4—22no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—42no—
418C2×C4⋊C4—42no—
419C2×C4⋊C4—1064no—
420C2×C4⋊C4—1064no—
421C2×C4⋊C4—1064no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4—1062no—
424C2×C4⋊C4—1064no—
425C2×C4⋊C4—1064no—
426C2×C4⋊C4—2122no—
427C2×C4⋊C4—2122no—
428C2×C4⋊C4—2122no—
429C2×C4⋊C4—2122no—
430C2×C22⋊C4—538no—
431C2×C22⋊C4—538no—
432C2×C22⋊C4—538no—
433C2×C22⋊C4—1064no—
434C2×C22⋊C4—1064no—
435C2×C22⋊C4—1064no—
436C2×C42—538no—
437C2×C42—2122no—
438C2×C42—2122no—
439#95—22no—
440#95—24no—
441#95—534no—
442#95—1062no—
443#95—1062no—
444#95—1064no—
445#95—1064no—
446#95—1064no—
447#95—1062no—
448#95—1064no—
449#95C4×D5314yescomplemented
450C24—532no—
451C24—534no—
452C24—532no—
453C24—534no—
454C24—532no—
455C24—1061no—
456C24—1061no—
457C24—1061no—
458C24—1061no—
459C24—1062no—
460C24—1062no—
461C24C23×D5311yescharacteristic
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—534no—
484C22×C4—532no—
485C22×C4—532no—
486C22×C4—534no—
487C22×C4—534no—
488C22×C4—532no—
489C22×C4—5316no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—1068no—
493C22×C4—1068no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—2128no—
516C22×C4—2124no—
517C22×C4—2124no—
518C22×C4—2128no—
519C22×C4—2128no—
520C22×C4—2128no—
521C22×C4—2128no—
522C22×C4—2128no—
523C22×C4—2124no—
524C22×C4—2124no—
525C22×C4—2124no—
526C22×C4—2124no—
527C22×C4C2×C4×D5312yes—
528C22×C4C2×C4×D5314yes—
529C22×C4D4×D5314yes—
530C22×C4D4×D5312yes—
531C22×C4#1171994712yes—
532C22×C4Q8×D5314yes—
533C22×C4Q8×D5312yes—
534C22×C4#1171994912yes—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4—2128no—
538C4⋊C4—2128no—
539C4⋊C4—2124no—
540C4⋊C4—2128no—
541C22⋊C4—5332no—
542C22⋊C4—2128no—
543C42—2128no—
544C23—538no—
545C23—534no—
546C23—534no—
547C23—534no—
548C23—538no—
549C23—1064no—
550C23—1064no—
551C23—1064no—
552C23—1064no—
553C23—1064no—
554C23—1064no—
555C23—1064no—
556C23—1062no—
557C23—1062no—
558C23—1064no—
559C23—1062no—
560C23—1062no—
561C23—1062no—
562C23—1062no—
563C23—1062no—
564C23—1062no—
565C23—1064no—
566C23—1064no—
567C23C22×C4×D5311yescharacteristic
568C23C2×D4×D5312yes—
569C23C2×D4×D5311yescharacteristic
570C23C2×Q8×D5312yes—
571C23C2×Q8×D5311yescharacteristic
572C23#41×D5311yescharacteristic
573C23#41×D5311yescharacteristic
574C23C2×#1171994711yescharacteristic
575C23C2×#1171994911yescharacteristic
576C23#1313326212yes—
577C23#1313326312yes—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—538no—
606C2×C4—534no—
607C2×C4—1064no—
608C2×C4—1064no—
609C2×C4—1064no—
610C2×C4—1064no—
611C2×C4—1064no—
612C2×C4—1064no—
613C2×C4—1064no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1064no—
617C2×C4—1064no—
618C2×C4—1064no—
619C2×C4—1062no—
620C2×C4—1062no—
621C2×C4—21216no—
622C2×C4—21216no—
623C2×C4—21216no—
624C2×C4—2122no—
625C2×C4—2122no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4—2122no—
637C2×C4C4⋊C4×D5318yes—
638C2×C4C4⋊C4×D5314yes—
639C22—538no—
640C22—1064no—
641C22—1064no—
642C22—1064no—
643C22—1064no—
644C22—1064no—
645C22—1068no—
646C22—1068no—
647C22—1068no—
648C22—1068no—
649C22—1064no—
650C22C2×C4⋊C4×D5311yescharacteristic
651C22C2×C4⋊C4×D5312yes—
652C22C4×#1171994711yescharacteristic
653C22C4×#1171994911yescharacteristic
654C22#1681109512yes—
655C22#1681113311yescharacteristic
656C22#1681113611yescharacteristic
657C22#127×D5311yescharacteristic
658C22C4⋊Q8×D5311yescharacteristic
659C22C4⋊Q8×D5312yes—
660C22#1681117811yescharacteristic
661C22#1681117911yescharacteristic
662C22#1681118311yescharacteristic
663C22#1681118411yescharacteristic
664C22#1681124812yes—
665C22#1681126012yes—
666C22#1681126412yes—
667C22#1681126512yes—
668C22#1681127012yes—
669C22#1681132712yes—
670C22#1681132912yes—
671C22#1681133112yes—
672C22#1681133412yes—
673C4—24no—
674C4—28no—
675C4—42no—
676C4—42no—
677C4—82no—
678C4—1064no—
679C4—1068no—
680C4—4242no—
681C4—2122no—
682C4—2122no—
683C2—1068no—
684C2#384×D5311yescharacteristic
685C2#2027719611yescharacteristic
686C2#2027732611yescharacteristic
687C2#2027762112yes—
688C2#2027762412yes—
689C2#2027964011yescharacteristic
690C2#2027964211yescharacteristic
691C2#2027972012yes—
692C2#2027972212yes—
693C2#2028008911yescharacteristic
694C2#2028009111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1444C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1444D53
hypocenterC53#13965
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
6. #20272980C2semidirect
7. #20272979C2semidirect
8. #20272978C2semidirect
9. #20272944C2semidirect
10. #20272943C2semidirect
11. C53×#1444C2semidirect
25. #16810470C4semidirect
80. C2×C4×Dic53C2×C4semidirect
88. #13133124C2×C4semidirect
101. #95×C53C2×C4semidirect
186. C2×C4×D53C4⋊C4semidirect
190. #11719921C4⋊C4semidirect
344. #1444D53semidirect
387. C53#13965semidirect
449. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104972
C2×#168104942
C2×#168104802
C2×#168104702
C2×#95×D532
#202729802
#202729792
#202729782
#202729442
#202729432
C53×#14442
#1396553