Finite Group Databasebeta

← #27320669

Subgroups — #27320669

Order 13568.

1822 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133215C211yesmaximal, characteristic
2C2×#16810544C211yesmaximal, characteristic
3C2×#16810542C211yesmaximal, characteristic
4C2×#16810520C212yesmaximal
5C2×#16810474C211yesmaximal, characteristic
6#20272985C211yesmaximal, characteristic
7#20272979C212yesmaximal
8#20272966C212yesmaximal
9#20272943C211yesmaximal, characteristic
10#20272943C212yesmaximal
11C53×#1446C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C2×#13133215—28no—
14C2×#13133215—24no—
15C22×C4⋊Dic53C2212yes—
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2212yes—
21C22×#11719919C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24C22×C4×Dic53C2211yescharacteristic
25#16810544—24no—
26#16810542—24no—
27#16810520—28no—
28#16810474—24no—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C23×Q8×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C2×#95×C53C2212yes—
41C22×Dic106—22no—
42C22×Dic106—24no—
43C22×Dic106—22no—
44C22×Dic106—22no—
45C22×Dic106—42no—
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2312yes—
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52#13133215—232no—
53C2×C4⋊Dic53—24no—
54C2×C4⋊Dic53—24no—
55C2×C4⋊Dic53—42no—
56C2×#11719919—24no—
57C2×#11719919—22no—
58C2×#11719919—24no—
59C2×#11719919—22no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—22no—
66C2×#11719919—24no—
67C2×#11719919—22no—
68C2×#11719919—24no—
69C2×#11719919—42no—
70C2×#11719919D418yes—
71C2×#11719919D414yes—
72C2×#11719919Q818yescomplemented
73C2×#11719919Q814yescomplemented
74C2×C4×Dic53—24no—
75C2×C4×Dic53—22no—
76C2×C4×Dic53—24no—
77C2×C4×Dic53—22no—
78C2×C4×Dic53—22no—
79C2×C4×Dic53D414yes—
80C2×C4×Dic53Q814yescomplemented
81#13133124—22no—
82#13133124—24no—
83#13133124—24no—
84#13133124—24no—
85#13133124—22no—
86#13133124—24no—
87#13133124—42no—
88C22×Q8×C53—24no—
89C22×Q8×C53—22no—
90C22×Q8×C53—22no—
91C22×Q8×C53—22no—
92C22×Q8×C53—42no—
93C23×C4×C53C2312yes—
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96C23×C4×C53C2312yes—
97C23×C4×C53C2311yescharacteristic
98#95×C53—24no—
99#95×C53—22no—
100#95×C53—22no—
101#95×C53—42no—
102C24×C53C2411yescharacteristic
103C2×Q8×C53—28no—
104C2×Q8×C53—48no—
105C2×Q8×C53—48no—
106C2×Q8×C53—44no—
107C2×Q8×C53—44no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
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—22no—
122C22×C4×C53—21no—
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×C53C2×D414yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×Q812yes—
134C22×C4×C53C2×Q812yes—
135C22×C4×C53C2×Q814yes—
136C22×C4×C53#4112yes—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
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—22no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53C2×D414yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D412yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×D414yes—
165C22×Dic53C2×Q814yes—
166C22×Dic53C2×Q812yes—
167C22×Dic53C2×Q812yes—
168C22×Dic53C2×Q812yes—
169C22×Dic53C2×Q812yes—
170C22×Dic53C2×Q814yes—
171C22×Dic53#4112yes—
172C22×Dic53#4112yes—
173C2×Dic106—28no—
174C2×Dic106—48no—
175C2×Dic106—48no—
176C2×Dic106—44no—
177C2×Dic106—44no—
178C4⋊Dic53—48no—
179#11719919—216no—
180#11719919—28no—
181#11719919—216no—
182#11719919—28no—
183#11719919—216no—
184#11719919—48no—
185C4×Dic53—28no—
186C4×Dic53—28no—
187C4×Dic53—44no—
188C23×C53C22×D411yescharacteristic
189C23×C53C22×D412yes—
190C23×C53C22×D411yescharacteristic
191C23×C53C22×Q811yescharacteristic
192C23×C53C22×Q812yes—
193C23×C53C22×Q811yescharacteristic
194C23×C53C2×#4111yescharacteristic
195C23×C53C2×#4111yescharacteristic
196C23×C53C2×#4111yescharacteristic
197C23×C53H(5,𝔽2)12yes—
198C23×C532−1+412yes—
199Q8×C53—416no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—41no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53#12114yes—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—24no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—22no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—41no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—41no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53#12014yes—
259C2×Dic53C22⋊Q814yes—
260C2×Dic53C22⋊Q814yes—
261C2×Dic53C22⋊Q818yes—
262C2×Dic53C4⋊Q814yes—
263C2×Dic53C4⋊Q818yes—
264C2×Dic53C4⋊Q814yescomplemented
265Dic106—416no—
266#15128—531nomaximal, Sylow
267C22×C53C2×#12011yescharacteristic
268C22×C53C2×#12111yescharacteristic
269C22×C53C2×C22⋊Q811yescharacteristic
270C22×C53C2×C22⋊Q811yescharacteristic
271C22×C53C2×C22⋊Q812yes—
272C22×C53C2×C4⋊Q811yescharacteristic
273C22×C53C2×C4⋊Q811yescharacteristic
274C22×C53C2×C4⋊Q812yes—
275C22×C53#53211yescharacteristic
276C22×C53#53311yescharacteristic
277C22×C53#53311yescharacteristic
278C22×C53#53511yescharacteristic
279C22×C53#53611yescharacteristic
280C22×C53#54212yes—
281C22×C53#54412yes—
282C22×C53D4212yes—
283C22×C53#54612yes—
284C22×C53#54812yes—
285C22×C53D4×Q812yes—
286C22×C53D4×Q812yes—
287C22×C53#55412yes—
288C22×C53#55712yes—
289C22×C53Q8212yes—
290C4×C53—24no—
291C4×C53—42no—
292C4×C53—44no—
293C4×C53—42no—
294C4×C53—44no—
295Dic53—24no—
296Dic53—24no—
297Dic53—44no—
298Dic53—42no—
299Dic53—44no—
300Dic53#39518yescomplemented
301C22×C4⋊Q8—531no—
302C2×#395—531no—
303C2×#395—531no—
304C2×#389—532no—
305C2×C4×C4⋊C4—531no—
306#1446—531no—
307#1446D5311yescharacteristic
308#1444—532no—
309#1441—532no—
310C4⋊#95—532no—
311C4⋊#95—531no—
312C2×C53C2×#39511yescharacteristic
313C2×C53#240811yescharacteristic
314C2×C53#240911yescharacteristic
315C2×C53#245112yes—
316C2×C53#245212yes—
317C2×C53#254611yescharacteristic
318C2×C53#255012yes—
319C2×C53#255112yes—
320C2×C53#256711yescharacteristic
321C2×C53#262811yescharacteristic
322C2×C53#285711yescharacteristic
323C23×Q8—531no—
324C23×Q8C2×D5311yescharacteristic
325C2×C4⋊Q8—1064no—
326C2×C4⋊Q8—1068no—
327C22×C4⋊C4—531no—
328C22×C4⋊C4—532no—
329C22×C4⋊C4—532no—
330C22×C4⋊C4—531no—
331C22×C4⋊C4—532no—
332C22×C4⋊C4—532no—
333C22×C4⋊C4—532no—
334C22×C42—531no—
335C22×C42—531no—
336C22×C42—531no—
337#395—1064no—
338#395—1064no—
339#389—1068no—
340C4×C4⋊C4—1064no—
341C2×#95—532no—
342C2×#95—532no—
343C2×#95—531no—
344C2×#95—532no—
345C2×#95—532no—
346C2×#95—531no—
347C2×#95—532no—
348C2×#95C2×D5311yescharacteristic
349C2×#95C2×D5312yes—
350C2×#95C2×D5311yescharacteristic
351C2×#95C2×D5312yes—
352C53#1512811yescharacteristic, complemented, Sylow
353C22×Q8—22no—
354C22×Q8—24no—
355C22×Q8—22no—
356C22×Q8—22no—
357C22×Q8—42no—
358C22×Q8—1064no—
359C22×Q8—1062no—
360C22×Q8—1062no—
361C22×Q8—1062no—
362C22×Q8—2122no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—531no—
367C23×C4—532no—
368C23×C4—532no—
369C23×C4C22×D5312yes—
370C23×C4C22×D5312yes—
371C23×C4C22×D5311yescharacteristic
372C23×C4C22×D5311yescharacteristic
373C23×C4C22×D5311yescharacteristic
374C4⋊Q8—10632no—
375C2×C4⋊C4—538no—
376C2×C4⋊C4—534no—
377C2×C4⋊C4—538no—
378C2×C4⋊C4—534no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1064no—
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1064no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—1064no—
393C2×C4⋊C4—1064no—
394C2×C4⋊C4—2122no—
395C2×C4⋊C4—2122no—
396C2×C42—534no—
397C2×C42—534no—
398C2×C42—1064no—
399C2×C42—1062no—
400C2×C42—1062no—
401C2×C42—1062no—
402C2×C42—1064no—
403#95—22no—
404#95—24no—
405#95—22no—
406#95—42no—
407#95—1064no—
408#95—1062no—
409#95—1064no—
410#95—1064no—
411#95—1064no—
412#95—1062no—
413#95—2122no—
414C24C23×D5311yescharacteristic
415C2×Q8—28no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—1068no—
421C2×Q8—2128no—
422C2×Q8—2128no—
423C2×Q8—2124no—
424C2×Q8—2124no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—532no—
448C22×C4—534no—
449C22×C4—532no—
450C22×C4—534no—
451C22×C4—532no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—534no—
459C22×C4—532no—
460C22×C4—534no—
461C22×C4—1062no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1061no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4#1171994512yes—
484C22×C4D4×D5314yes—
485C22×C4D4×D5312yes—
486C22×C4D4×D5312yes—
487C22×C4Q8×D5312yes—
488C22×C4Q8×D5312yes—
489C22×C4Q8×D5314yes—
490C4⋊C4—10616no—
491C4⋊C4—10616no—
492C4⋊C4—1068no—
493C4⋊C4—10616no—
494C4⋊C4—1068no—
495C4⋊C4—2128no—
496C4⋊C4—2128no—
497C42—1068no—
498C42—1068no—
499C42—2124no—
500C23C2×D4×D5312yes—
501C23C2×D4×D5311yescharacteristic
502C23C2×D4×D5311yescharacteristic
503C23C2×Q8×D5311yescharacteristic
504C23C2×Q8×D5312yes—
505C23C2×Q8×D5311yescharacteristic
506C23#41×D5311yescharacteristic
507C23#41×D5311yescharacteristic
508C23C2×#1171994511yescharacteristic
509C23#1313326212yes—
510C23#1313326412yes—
511Q8—416no—
512Q8—21216no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—534no—
542C2×C4—534no—
543C2×C4—538no—
544C2×C4—534no—
545C2×C4—538no—
546C2×C4—534no—
547C2×C4—534no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1062no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—1064no—
566C2×C4—2122no—
567C2×C4—2121no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2121no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4#1313319014yes—
578C22#120×D5311yescharacteristic
579C22C2×#1313319011yescharacteristic
580C22C22⋊Q8×D5312yes—
581C22C22⋊Q8×D5311yescharacteristic
582C22C22⋊Q8×D5311yescharacteristic
583C22C4⋊Q8×D5311yescharacteristic
584C22C4⋊Q8×D5312yes—
585C22C4⋊Q8×D5311yescharacteristic
586C22#1681117211yescharacteristic
587C22#1681117711yescharacteristic
588C22#1681117711yescharacteristic
589C22#1681118811yescharacteristic
590C22#1681119311yescharacteristic
591C22#1681123712yes—
592C22#1681125212yes—
593C22#1681125512yes—
594C22#1681126312yes—
595C22#1681128112yes—
596C22#1681128612yes—
597C22#1681128612yes—
598C22#1681132712yes—
599C22#1681135112yes—
600C22#1681135312yes—
601C4—24no—
602C4—44no—
603C4—44no—
604C4—42no—
605C4—42no—
606C4—538no—
607C4—1064no—
608C4—1064no—
609C4—2124no—
610C4—2122no—
611C4—2124no—
612C2#395×D5311yescharacteristic
613C2#2027807011yescharacteristic
614C2#2027807311yescharacteristic
615C2#2027855112yes—
616C2#2027856012yes—
617C2#2027970511yescharacteristic
618C2#2027973312yes—
619C2#2027974012yes—
620C2#2027991911yescharacteristic
621C2#2028053511yescharacteristic
622C2#2028306411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, prime radical, cosocleC24×C53C24
FittingC53×#1446C2
hypercenter#1446D53
hypocenterC53#15128
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
72. C2×#11719919Q8semidirect
73. C2×#11719919Q8semidirect
80. C2×C4×Dic53Q8semidirect
264. C2×Dic53C4⋊Q8semidirect
300. Dic53#395semidirect
352. C53#15128semidirect

Maximal subgroups

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

subgroupindex
C22×#131332152
C2×#168105442
C2×#168105422
C2×#168105202
C2×#168104742
#202729852
#202729792
#202729662
#202729432
C53×#14462
#1512853