Finite Group Databasebeta

← #27320370

Subgroups — #27320370

Order 13568.

1378 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810515C211yesmaximal, characteristic
2C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
3#20272971C211yesmaximal, characteristic, complemented
4#20272971C212yesmaximal, complemented
5#20272961C211yesmaximal, characteristic
6#20272957C211yesmaximal, characteristic
7#20272957C212yesmaximal
8#20272951C212yesmaximal
9#20272947C212yesmaximal, complemented
10#20272939C211yesmaximal, characteristic, complemented
11C53×#1444C211yesmaximal, characteristic, complemented
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719921C2211yescharacteristic
15C22×#11719921C2212yes—
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2212yes—
18C22×#11719921C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2212yes—
21#16810515—24no—
22C4⋊C4×Dic53—24no—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2212yes—
34C2×#95×C53C2211yescharacteristic
35C2×#95×C53C2211yescharacteristic
36C2×#95×C53C2212yes—
37C24×D53C2311yescharacteristic
38C23×Dic53C2311yescharacteristic
39C23×Dic53C2312yes—
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C2×C4⋊Dic53—22no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—24no—
46C2×C4⋊Dic53—42no—
47C2×#11719921—22no—
48C2×#11719921—24no—
49C2×#11719921—24no—
50C2×#11719921—22no—
51C2×#11719921—22no—
52C2×#11719921—22no—
53C2×#11719921—22no—
54C2×#11719921—22no—
55C2×#11719921—24no—
56C2×#11719921—42no—
57C2×#11719921—42no—
58C2×#11719921—42no—
59C2×#11719921—42no—
60C2×#11719921Q814yes—
61C2×C4×Dic53—22no—
62C2×C4×Dic53—22no—
63C2×C4×Dic53—24no—
64C2×C4×Dic53—42no—
65C2×C4×Dic53—42no—
66C2×C4×Dic53D414yes—
67#13133124—24no—
68#13133124—24no—
69#13133124—22no—
70#13133124—22no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124—42no—
75#13133124—42no—
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2312yes—
79C23×C4×C53C2312yes—
80C23×C4×C53C2311yescharacteristic
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—24no—
83C2×C4⋊C4×C53—24no—
84C2×C4⋊C4×C53—42no—
85C2×C4⋊C4×C53—42no—
86#95×C53—22no—
87#95×C53—22no—
88#95×C53—42no—
89C24×C53C2411yescharacteristic
90C23×D53—21no—
91C23×D53—21no—
92C23×D53—22no—
93C23×D53—22no—
94C23×D53—21no—
95C23×D53—21no—
96C23×D53—22no—
97C23×D53—21no—
98C23×D53—21no—
99C23×D53—22no—
100C23×D53C2×Q812yes—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
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—21no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×Q812yes—
126C22×C4×C53#4114yes—
127C22×C4×C53#4114yes—
128C22×C4×C53#4112yes—
129C22×C4×C53#4112yes—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53C2×D412yes—
157C22×Dic53C2×D412yes—
158C22×Dic53C2×Q812yes—
159C22×Dic53#4114yes—
160C22×Dic53#4112yes—
161C4⋊C4×C53—28no—
162C4⋊C4×C53—48no—
163#11719921—28no—
164#11719921—48no—
165#11719921—44no—
166#11719921—48no—
167#11719921—44no—
168C4⋊Dic53—28no—
169C4⋊Dic53—44no—
170C4×Dic53—28no—
171C4×Dic53—48no—
172C23×C53C22×D411yescharacteristic
173C23×C53C22×Q811yescharacteristic
174C23×C53C2×#4111yescharacteristic
175C23×C53C2×#4112yes—
176C23×C53C2×#4112yes—
177C23×C53C2×#4111yescharacteristic
178C23×C53C2×#4111yescharacteristic
179C23×C53C2×#4112yes—
180C23×C53H(5,𝔽2)11yescharacteristic
181C23×C53H(5,𝔽2)12yes—
182C23×C53H(5,𝔽2)11yescharacteristic
183C22×D53—22no—
184C22×D53—24no—
185C22×D53—24no—
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—42no—
192C22×D53—42no—
193C22×D53—42no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—41no—
202C22×D53—42no—
203C22×D53—41no—
204C22×D53—42no—
205C22×D53—42no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—22no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—41no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—41no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53#12314yes—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—24no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—41no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53C22⋊Q814yes—
266#15100—531nomaximal, Sylow
267C22×C53C2×C22⋊Q811yescharacteristic
268C22×C53C2×#12311yescharacteristic
269C22×C53#52511yescharacteristic
270C22×C53#52511yescharacteristic
271C22×C53#52912yes—
272C22×C53#52912yes—
273C22×C53#52912yes—
274C22×C53#53211yescharacteristic
275C22×C53#53312yes—
276C22×C53#53511yescharacteristic
277C22×C53#53511yescharacteristic
278C22×C53#53812yes—
279C22×C53#53812yes—
280C22×C53#55112yes—
281C22×C53#55312yes—
282C22×C53#55311yescharacteristic
283C22×C53#55312yes—
284C22×C53#55311yescharacteristic
285C22×C53#55412yes—
286C22×C53#55411yescharacteristic
287C22×C53#55411yescharacteristic
288C22×C53#55512yes—
289C22×C53#55912yes—
290C2×D53—44no—
291C2×D53—42no—
292C2×D53—44no—
293C2×D53—42no—
294C2×D53—42no—
295C2×D53—82no—
296C2×D53—82no—
297C2×D53—81no—
298C2×D53—81no—
299C2×D53—81no—
300C2×D53—81no—
301C4×C53—24no—
302C4×C53—42no—
303C4×C53—44no—
304C4×C53—44no—
305C4×C53—81no—
306Dic53—24no—
307Dic53—44no—
308Dic53—42no—
309Dic53—82no—
310Dic53—81no—
311C2×#388—531no—
312C2×C4×C4⋊C4—531no—
313#1444D5311yescharacteristic, complemented
314#1442—531no—
315#1442—532no—
316#1439—531no—
317#1439—531no—
318#1439—532no—
319#1437—532no—
320#1436—532no—
321#1434—531no—
322C2×C53#239611yescharacteristic
323C2×C53#248711yescharacteristic
324C2×C53#250011yescharacteristic
325C2×C53#257412yes—
326C2×C53#258011yescharacteristic
327C2×C53#258712yes—
328C2×C53#259312yes—
329C2×C53#259512yes—
330C2×C53#260311yescharacteristic
331C2×C53#263111yescharacteristic
332C2×C53#285311yescharacteristic
333D53—82no—
334C22×C4⋊C4—531no—
335C22×C4⋊C4—531no—
336C22×C4⋊C4C2×D5311yescharacteristic
337C22×C4⋊C4C2×D5312yes—
338C22×C22⋊C4—531no—
339C22×C22⋊C4—531no—
340C22×C22⋊C4—532no—
341C22×C22⋊C4—532no—
342C22×C22⋊C4—531no—
343C22×C42—531no—
344C22×C42—532no—
345#388—1064no—
346C4×C4⋊C4—1064no—
347C2×#95—532no—
348C2×#95—532no—
349C2×#95—531no—
350C2×#95—531no—
351C2×#95—532no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95C2×D5311yescharacteristic
357C2×#95C2×D5311yescharacteristic
358C2×#95C2×D5312yes—
359C53#1510011yescharacteristic, complemented, Sylow
360C25—531no—
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—532no—
364C23×C4—531no—
365C23×C4—532no—
366C23×C4C22×D5311yescharacteristic
367C23×C4C22×D5312yes—
368C23×C4C22×D5311yescharacteristic
369C23×C4C22×D5312yes—
370C23×C4C22×D5311yescharacteristic
371C2×C4⋊C4—22no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—42no—
375C2×C4⋊C4—42no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—2122no—
380C2×C22⋊C4—534no—
381C2×C22⋊C4—1062no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4—1064no—
384C2×C22⋊C4—1062no—
385C2×C22⋊C4—1062no—
386C2×C22⋊C4—1062no—
387C2×C22⋊C4—1064no—
388C2×C22⋊C4—1064no—
389C2×C22⋊C4—1062no—
390C2×C22⋊C4—2122no—
391C2×C22⋊C4—2122no—
392C2×C22⋊C4—2122no—
393C2×C22⋊C4—2122no—
394C2×C42—534no—
395C2×C42—1062no—
396C2×C42—1062no—
397C2×C42—1064no—
398C2×C42—2122no—
399C2×C42—2122no—
400#95—22no—
401#95—22no—
402#95—42no—
403#95—1064no—
404#95—1064no—
405#95—1064no—
406#95—1062no—
407#95—1064no—
408#95—1064no—
409#95—1062no—
410#95—2122no—
411#95—2122no—
412C24—532no—
413C24—1061no—
414C24—1062no—
415C24—1061no—
416C24—1062no—
417C24—1061no—
418C24—1061no—
419C24—1062no—
420C24—1062no—
421C24—1061no—
422C24—1061no—
423C24C23×D5311yescharacteristic
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—532no—
450C22×C4—532no—
451C22×C4—534no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1061no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1061no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1061no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1061no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1061no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4#1171994512yes—
479C22×C4D4×D5312yes—
480C22×C4#1171994714yes—
481C22×C4Q8×D5312yes—
482C22×C4#1171994914yes—
483C22×C4#1171994912yes—
484C4⋊C4—28no—
485C4⋊C4—48no—
486C4⋊C4—1068no—
487C4⋊C4—2124no—
488C22⋊C4—1068no—
489C22⋊C4—2128no—
490C22⋊C4—2128no—
491C22⋊C4—2124no—
492C22⋊C4—2124no—
493C42—1068no—
494C42—2128no—
495C23—1064no—
496C23—1062no—
497C23—1062no—
498C23—1062no—
499C23—1064no—
500C23—2122no—
501C23—2122no—
502C23—2121no—
503C23—2122no—
504C23—2121no—
505C23—2122no—
506C23—2122no—
507C23—2121no—
508C23—2121no—
509C23—2121no—
510C23—2122no—
511C23—2122no—
512C23—2122no—
513C23—2121no—
514C23—2121no—
515C23—2121no—
516C23—2122no—
517C23—2122no—
518C23C2×D4×D5311yescharacteristic
519C23C2×Q8×D5311yescharacteristic
520C23#41×D5312yes—
521C23#41×D5311yescharacteristic
522C23C2×#1171994712yes—
523C23C2×#1171994511yescharacteristic
524C23C2×#1171994912yes—
525C23C2×#1171994911yescharacteristic
526C23#1313326211yescharacteristic
527C23#1313326212yes—
528C23#1313326311yescharacteristic
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—534no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1064no—
565C2×C4—1064no—
566C2×C4—1062no—
567C2×C4—1064no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2121no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4#1313320214yes—
589C22—4241no—
590C22—4241no—
591C22—4242no—
592C22—4241no—
593C22—4242no—
594C22—4241no—
595C22—2122no—
596C22—2124no—
597C22—2124no—
598C22—2122no—
599C22—2122no—
600C22C22⋊Q8×D5311yescharacteristic
601C22C2×#1313320211yescharacteristic
602C22#1681112911yescharacteristic
603C22#1681113111yescharacteristic
604C22#1681115312yes—
605C22#1681115912yes—
606C22#1681115912yes—
607C22#1681117411yescharacteristic
608C22#1681117812yes—
609C22#1681118611yescharacteristic
610C22#1681118911yescharacteristic
611C22#1681120812yes—
612C22#1681121012yes—
613C22#1681130012yes—
614C22#1681132112yes—
615C22#1681132311yescharacteristic
616C22#1681132612yes—
617C22#1681132611yescharacteristic
618C22#1681132712yes—
619C22#1681133311yescharacteristic
620C22#1681133411yescharacteristic
621C22#1681133912yes—
622C22#1681135712yes—
623C4—24no—
624C4—42no—
625C4—44no—
626C4—44no—
627C4—81no—
628C4—1064no—
629C4—4241no—
630C4—4242no—
631C4—2124no—
632C4—2122no—
633C2—4242no—
634C2#2027798611yescharacteristic
635C2#2027903011yescharacteristic
636C2#2027917411yescharacteristic
637C2#2027998412yes—
638C2#2028007011yescharacteristic
639C2#2028014112yes—
640C2#2028020512yes—
641C2#2028023312yes—
642C2#2028034311yescharacteristic
643C2#2028056611yescharacteristic
644C2#2028304011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC53×#1444C2
prime radicalC24×D53C23
hypercenter#1444D53
hypocenterC53#15100
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic53C2semidirect
3. #20272971C2semidirect
4. #20272971C2semidirect
9. #20272947C2semidirect
10. #20272939C2semidirect
11. C53×#1444C2semidirect
313. #1444D53semidirect
359. C53#15100semidirect

Maximal subgroups

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

subgroupindex
C2×#168105152
C2×C4⋊C4×Dic532
#202729712
#202729612
#202729572
#202729512
#202729472
#202729392
C53×#14442
#1510053