Finite Group Databasebeta

← #27309386

Subgroups — #27309386

Order 13568.

2062 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133182C211yesmaximal, characteristic, complemented
2C2×#16810535C211yesmaximal, characteristic
3C2×#16810495C212yesmaximal
4C2×C22⋊C4×Dic53C211yesmaximal, characteristic
5#20272985C211yesmaximal, characteristic, complemented
6#20272978C211yesmaximal, characteristic, complemented
7#20272952C212yesmaximal
8#20272950C212yesmaximal, complemented
9#20272946C212yesmaximal, complemented
10#20272940C211yesmaximal, characteristic, complemented
11C53×#1439C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C2×#13133182—28no—
15C2×#13133182—24no—
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×C22⋊Dic53C2212yes—
19C22×C22⋊Dic53C2211yescharacteristic
20C22×C4×Dic53C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23#16810535C418yescomplemented
24#16810495—28no—
25C22⋊C4×Dic53—24no—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2212yes—
37C22×C22⋊C4×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C22×Dic106—22no—
41C22×Dic106—22no—
42C22×Dic106—24no—
43C22×Dic106—24no—
44C22×Dic106C2×C414yescomplemented
45C23×Dic53—22no—
46C23×Dic53—22no—
47C23×Dic53—22no—
48C23×Dic53—24no—
49C23×Dic53—42no—
50C23×Dic53C2312yes—
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53C23×Dic53C2312yes—
54C23×Dic53C2311yescharacteristic
55C23×Dic53C2311yescharacteristic
56#13133182—232no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919—42no—
62C2×#11719919D418yes—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—24no—
65C2×C22⋊Dic53—24no—
66C2×C22⋊Dic53—22no—
67C2×C22⋊Dic53C2×C418yes—
68C2×C22⋊Dic53Q814yescomplemented
69C2×C4×Dic53—24no—
70C2×C4×Dic53—24no—
71C2×C4×Dic53—24no—
72C2×C4×Dic53—24no—
73C2×C4×Dic53—22no—
74C2×C4×Dic53—42no—
75C2×C4×Dic53D414yes—
76#13133124—22no—
77#13133124—22no—
78#13133124—24no—
79#13133124—24no—
80#13133124—24no—
81#13133124—24no—
82#13133124—42no—
83#13133124C2×C414yescomplemented
84#13133124C2×C418yescomplemented
85C25×C53C2311yescharacteristic
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2312yes—
89C23×C4×C53C2311yescharacteristic
90C2×C4⋊C4×C53—24no—
91C2×C4⋊C4×C53—22no—
92C2×C22⋊C4×C53—24no—
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53—24no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—42no—
97C2×C22⋊C4×C53C2×C414yes—
98#95×C53—22no—
99#95×C53—42no—
100C24×C53—22no—
101C24×C53—21no—
102C24×C53—21no—
103C24×C53—22no—
104C24×C53—21no—
105C24×C53—21no—
106C24×C53—22no—
107C24×C53—22no—
108C24×C53C22×C412yes—
109C24×C53C2×Q812yes—
110C24×C53#4112yes—
111C24×C53C2411yescharacteristic
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C2×D414yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53#4112yes—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—28no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—21no—
154C22×Dic53—22no—
155C22×Dic53—21no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—48no—
161C22×Dic53—48no—
162C22×Dic53—44no—
163C22×Dic53—44no—
164C22×Dic53C22×C414yes—
165C22×Dic53C22×C414yes—
166C22×Dic53C2×D414yes—
167C22×Dic53C2×D414yes—
168C22×Dic53C2×D412yes—
169C22×Dic53C2×D412yes—
170C22×Dic53C2×Q812yes—
171C22×Dic53C2×Q812yes—
172C22×Dic53#4112yes—
173C22×Dic53#4114yes—
174C22×Dic53#4112yes—
175C2×Dic106—28no—
176C2×Dic106—216no—
177C2×Dic106—28no—
178C2×Dic106—28no—
179C2×Dic106—216no—
180C4⋊C4×C53—28no—
181C22⋊C4×C53—28no—
182C22⋊C4×C53—48no—
183C22⋊Dic53—216no—
184C22⋊Dic53—28no—
185#11719919—216no—
186#11719919—48no—
187C4×Dic53—216no—
188C4×Dic53—28no—
189C4×Dic53—44no—
190C23×C53—24no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—24no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—24no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—24no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—42no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53C4×Q814yescomplemented
213C23×C53C23×C411yescharacteristic
214C23×C53C22×D411yescharacteristic
215C23×C53C22×D412yes—
216C23×C53C22×Q811yescharacteristic
217C23×C53C2×#4112yes—
218C23×C53C2×#4111yescharacteristic
219C23×C53C2×#4111yescharacteristic
220C23×C53C2×#4111yescharacteristic
221C23×C53C2×#4111yescharacteristic
222C23×C53H(5,𝔽2)12yes—
223C23×C532−1+412yes—
224C2×C4×C53—24no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—24no—
228C2×C4×C53—22no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53C4×D414yes—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—22no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—416no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53C4×D418yes—
281C2×Dic53#12014yes—
282C2×Dic53C22⋊Q814yes—
283Dic106—232no—
284#14172—531nomaximal, Sylow
285C22×C53—28no—
286C22×C53—24no—
287C22×C53—42no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—44no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—44no—
294C22×C53—44no—
295C22×C53—42no—
296C22×C53C2×C4×D412yes—
297C22×C53C2×C4×D411yescharacteristic
298C22×C53C2×C4×Q811yescharacteristic
299C22×C53C4×#4111yescharacteristic
300C22×C53#52012yes—
301C22×C53C2×#12011yescharacteristic
302C22×C53C2×C22⋊Q811yescharacteristic
303C22×C53#52511yescharacteristic
304C22×C53#52511yescharacteristic
305C22×C53#52512yes—
306C22×C53#52911yescharacteristic
307C22×C53#53311yescharacteristic
308C22×C53#53511yescharacteristic
309C22×C53#53611yescharacteristic
310C22×C53#54212yes—
311C22×C53#54612yes—
312C22×C53#54612yes—
313C22×C53#54812yes—
314C22×C53D4×Q812yes—
315C22×C53#55212yes—
316C22×C53#55412yes—
317C22×C53#55612yes—
318C22×C53#55912yes—
319C4×C53—24no—
320C4×C53—42no—
321C4×C53—44no—
322C4×C53—82no—
323Dic53—24no—
324Dic53—24no—
325Dic53—28no—
326Dic53—42no—
327Dic53—44no—
328Dic53—42no—
329C22×C4×Q8—531no—
330C2×#393—531no—
331C2×#385—532no—
332C2×C4×C22⋊C4—531no—
333#1446—531no—
334#1444—531no—
335#1439D5311yescharacteristic
336#1437—532no—
337#1436—532no—
338#1436—532no—
339#1434—531no—
340C2×C53—44no—
341C2×C53C4×#12011yescharacteristic
342C2×C53C4×C22⋊Q811yescharacteristic
343C2×C53#232611yescharacteristic
344C2×C53#236012yes—
345C2×C53#236712yes—
346C2×C53#240411yescharacteristic
347C2×C53#240911yescharacteristic
348C2×C53#245812yes—
349C2×C53#246512yes—
350C2×C53#258011yescharacteristic
351C2×C53#259911yescharacteristic
352C23×Q8—531no—
353C24×C4—531no—
354C2×C4×Q8—1064no—
355C2×C4×Q8—1068no—
356C22×C4⋊C4—532no—
357C22×C4⋊C4—532no—
358C22×C4⋊C4C2×D5311yescharacteristic
359C22×C22⋊C4—532no—
360C22×C22⋊C4—531no—
361C22×C22⋊C4C2×D5312yes—
362C22×C22⋊C4C2×D5311yescharacteristic
363C22×C42—531no—
364C22×C42—532no—
365C22×C42—531no—
366#393—538no—
367#385—1068no—
368C4×C22⋊C4—1064no—
369C2×#95—531no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—531no—
374C2×#95—532no—
375C2×#95—532no—
376C2×#95—531no—
377C2×#95—532no—
378C2×#95C2×D5311yescharacteristic
379C2×#95C2×D5312yes—
380C53#1417211yescharacteristic, complemented, Sylow
381C25C22×D5311yescharacteristic
382C22×Q8—534no—
383C22×Q8—1064no—
384C22×Q8—1064no—
385C22×Q8—1062no—
386C22×Q8—1062no—
387C23×C4—532no—
388C23×C4—531no—
389C23×C4—531no—
390C23×C4—532no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4—1062no—
394C23×C4—1062no—
395C23×C4—1064no—
396C23×C4—1062no—
397C23×C4—2122no—
398C23×C4C22×D5312yes—
399C23×C4C22×D5311yescharacteristic
400C23×C4C22×D5312yes—
401C23×C4C22×D5311yescharacteristic
402C4×Q8—10632no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—538no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—2122no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4—538no—
417C2×C22⋊C4—534no—
418C2×C22⋊C4—1062no—
419C2×C22⋊C4—1064no—
420C2×C22⋊C4—1062no—
421C2×C22⋊C4—1064no—
422C2×C22⋊C4C4×D5314yes—
423C2×C42—534no—
424C2×C42—1064no—
425C2×C42—1064no—
426C2×C42—1064no—
427C2×C42—1064no—
428C2×C42—1062no—
429C2×C42—2122no—
430#95—22no—
431#95—42no—
432#95—538no—
433#95—534no—
434#95—1064no—
435#95—1064no—
436#95—1062no—
437#95—1064no—
438#95—1064no—
439#95—1062no—
440#95—2122no—
441C24—22no—
442C24—21no—
443C24—22no—
444C24—21no—
445C24—21no—
446C24—21no—
447C24—22no—
448C24—22no—
449C24C2×C4×D5312yes—
450C24#1171994712yes—
451C24Q8×D5312yes—
452C24C23×D5311yescharacteristic
453C2×Q8—10616no—
454C2×Q8—1068no—
455C2×Q8—1068no—
456C2×Q8—10616no—
457C2×Q8—1068no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—534no—
478C22×C4—532no—
479C22×C4—534no—
480C22×C4—534no—
481C22×C4—532no—
482C22×C4—532no—
483C22×C4—532no—
484C22×C4—534no—
485C22×C4—534no—
486C22×C4—532no—
487C22×C4—532no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1061no—
492C22×C4—1062no—
493C22×C4—1061no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1068no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1061no—
506C22×C4—1062no—
507C22×C4—1061no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—2124no—
513C22×C4—2128no—
514C22×C4—2128no—
515C22×C4—2124no—
516C22×C4C2×C4×D5312yes—
517C22×C4C2×C4×D5312yes—
518C22×C4D4×D5314yes—
519C22×C4D4×D5312yes—
520C22×C4#1171994712yes—
521C4⋊C4—28no—
522C4⋊C4—10616no—
523C4⋊C4—2128no—
524C22⋊C4—28no—
525C22⋊C4—48no—
526C22⋊C4—10616no—
527C22⋊C4—1068no—
528C42—1068no—
529C42—10616no—
530C42—2124no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23#1313318214yescomplemented
554C23C22×C4×D5311yescharacteristic
555C23C2×D4×D5312yes—
556C23C2×D4×D5311yescharacteristic
557C23C2×Q8×D5311yescharacteristic
558C23#41×D5311yescharacteristic
559C23#41×D5312yes—
560C23#41×D5311yescharacteristic
561C23C2×#1171994711yescharacteristic
562C23C2×#1171994711yescharacteristic
563C23#1313326212yes—
564C23#1313326412yes—
565Q8—10632no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—538no—
590C2×C4—534no—
591C2×C4—534no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1062no—
597C2×C4—1064no—
598C2×C4—1064no—
599C2×C4—1062no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—1064no—
607C2×C4—1064no—
608C2×C4—1064no—
609C2×C4—1064no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1064no—
613C2×C4—21216no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2122no—
623C2×C4—2122no—
624C2×C4#1313317714yes—
625C22—28no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—44no—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22C4×D4×D5312yes—
637C22C2×#1313317711yescharacteristic
638C22C2×#1313318211yescharacteristic
639C22#1681109111yescharacteristic
640C22#1681110212yes—
641C22#120×D5311yescharacteristic
642C22C22⋊Q8×D5311yescharacteristic
643C22#1681113111yescharacteristic
644C22#1681113311yescharacteristic
645C22#1681113312yes—
646C22#1681115411yescharacteristic
647C22#1681117511yescharacteristic
648C22#1681118611yescharacteristic
649C22#1681119311yescharacteristic
650C22#1681124012yes—
651C22#1681126012yes—
652C22#1681126212yes—
653C22#1681128112yes—
654C22#1681128612yes—
655C22#1681131012yes—
656C22#1681132712yes—
657C22#1681134012yes—
658C22#1681135712yes—
659C4—24no—
660C4—44no—
661C4—42no—
662C4—82no—
663C4—1064no—
664C4—1064no—
665C4—1068no—
666C4—2122no—
667C4—2122no—
668C4—2124no—
669C2—44no—
670C2#2027722011yescharacteristic
671C2#2027724311yescharacteristic
672C2#2027742511yescharacteristic
673C2#2027771512yes—
674C2#2027778112yes—
675C2#2027802911yescharacteristic
676C2#2027807311yescharacteristic
677C2#2027863812yes—
678C2#2027873712yes—
679C2#2028007011yescharacteristic
680C2#2028029311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1439C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53#14172
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#13133182C2semidirect
5. #20272985C2semidirect
6. #20272978C2semidirect
8. #20272950C2semidirect
9. #20272946C2semidirect
10. #20272940C2semidirect
23. #16810535C4semidirect
44. C22×Dic106C2×C4semidirect
68. C2×C22⋊Dic53Q8semidirect
83. #13133124C2×C4semidirect
84. #13133124C2×C4semidirect
212. C23×C53C4×Q8semidirect
380. C53#14172semidirect
553. C23#13133182semidirect

Maximal subgroups

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

subgroupindex
C22×#131331822
C2×#168105352
C2×#168104952
C2×C22⋊C4×Dic532
#202729852
#202729782
#202729522
#202729502
#202729462
#202729402
C53×#14392
#1417253