Finite Group Databasebeta

← #27438169

Subgroups — #27438169

Order 13568.

1018 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
1C2×#16810977C211yesmaximal, characteristic
2C2×#16810930C211yesmaximal, characteristic, complemented
3#20280103C211yesmaximal, characteristic, complemented
4#20275958C211yesmaximal, characteristic, complemented
5#20275958C211yesmaximal, characteristic, complemented
6#20275951C211yesmaximal, characteristic
7#20275951C211yesmaximal, characteristic
8#20275652C211yesmaximal, characteristic, complemented
9#20275648C211yesmaximal, characteristic
10#20274983C211yesmaximal, characteristic, complemented
11#20274976C211yesmaximal, characteristic
12#20274969C211yesmaximal, characteristic, complemented
13#20274962C211yesmaximal, characteristic
14C4×#13133141C211yesmaximal, characteristic, complemented
15C2×C53×#556C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16810977—24no—
18#16810930—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35#16810554—21no—
36#16810542—21no—
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810521C2211yescharacteristic
41#16810518—21no—
42#16810518—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C53×#556—24no—
51C53×#556—24no—
52C2×C4⋊Q8×C53C2211yescharacteristic
53C2×#126×C53—21no—
54C2×#126×C53—21no—
55C2×#124×C53C2211yescharacteristic
56C2×#124×C53C2211yescharacteristic
57C2×C22⋊Q8×C53—21no—
58C2×C4×Q8×C53C2211yescharacteristic
59C2×C4×Q8×C53C2211yescharacteristic
60C2×C4×Q8×C53C2211yescharacteristic
61C2×C4×D4×C53C2211yescharacteristic
62C2×#117×C53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133209—22no—
68#13133209—22no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C4×Dic53—21no—
80#13133150—22no—
81#13133148—22no—
82C4⋊#88322—22no—
83#13133141—22no—
84#13133141—22no—
85#13133141—22no—
86#13133141—22no—
87#13133141—22no—
88#13133141—22no—
89#13133141—22no—
90#13133141—22no—
91#13133138—22no—
92#13133138—22no—
93#13133138—22no—
94#13133138—22no—
95C4×#88322—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C22×Q8×C53D411yescharacteristic
102C22×Q8×C53C2311yescharacteristic
103C22×Q8×C53C2311yescharacteristic
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C4⋊Q8×C53—22no—
107C4⋊Q8×C53—22no—
108#126×C53—28no—
109#126×C53—28no—
110#124×C53—22no—
111#124×C53—22no—
112#124×C53—22no—
113#124×C53D414yes—
114C22⋊Q8×C53—28no—
115C4×Q8×C53—22no—
116C4×Q8×C53—22no—
117C4×Q8×C53—22no—
118C4×Q8×C53—22no—
119C4×Q8×C53—22no—
120C4×Q8×C53D414yes—
121C4×D4×C53—22no—
122C4×D4×C53—22no—
123#117×C53—28no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—21no—
128C2×C4⋊C4×C53D411yescharacteristic
129C2×C4⋊C4×C53D411yescharacteristic
130C2×C4⋊C4×C53D411yescharacteristic
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C22⋊C4×C53—21no—
133C2×C22⋊C4×C53—21no—
134C2×C22⋊C4×C53—21no—
135C2×C22⋊C4×C53—21no—
136C2×C22⋊C4×C53—21no—
137C2×C42×C53—21no—
138C2×C42×C53—21no—
139C2×C42×C53C2311yescharacteristic
140C2×C42×C53C2311yescharacteristic
141C2×C42×C53C2311yescharacteristic
142C24×C53—21no—
143C2×Q8×C53—22no—
144C2×Q8×C53—22no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53—21no—
149C2×Q8×C53—21no—
150C2×Q8×C53—21no—
151C2×Q8×C53—21no—
152C2×Q8×C53—21no—
153C2×Q8×C53—21no—
154C2×Q8×C53—21no—
155C2×Q8×C53—41no—
156C2×Q8×C53C2×D412yes—
157C2×Q8×C53C2×D412yes—
158C2×D4×C53—21no—
159C2×D4×C53—21no—
160C2×D4×C53—21no—
161C2×D4×C53—21no—
162C2×D4×C53—21no—
163C2×D4×C53C2×D412yes—
164C22×C4×C53—24no—
165C22×C4×C53—21no—
166C22×C4×C53—21no—
167C22×C4×C53—21no—
168C22×C4×C53—42no—
169C22×C4×C53—42no—
170C22×C4×C53C2×D411yescharacteristic
171C22×C4×C53C2×D411yescharacteristic
172C22×C4×C53C2×D411yescharacteristic
173C22×C4×C53C2×D411yescharacteristic
174C22×C4×C53C2×D411yescharacteristic
175C22×C4×C53C2×D411yescharacteristic
176C22×C4×C53C2411yescharacteristic
177C22×Dic53—21no—
178C22×Dic53—21no—
179C22×Dic53—21no—
180C22×Dic53—21no—
181C4⋊C4×C53—24no—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—24no—
184C4⋊C4×C53—24no—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53—22no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53C2×D412yes—
190C22⋊C4×C53—24no—
191C22⋊C4×C53—24no—
192C22⋊C4×C53—24no—
193C22⋊C4×C53—24no—
194C22⋊C4×C53—24no—
195C42×C53—24no—
196C42×C53—24no—
197C42×C53—21no—
198C42×C53—21no—
199C42×C53—21no—
200C42×C53—21no—
201C42×C53C2×D412yes—
202C42×C53C2×D412yes—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53#4112yes—
210C4⋊Dic53#4112yes—
211#11719919—42no—
212#11719919—42no—
213#11719919—42no—
214#11719919—42no—
215#11719919—81no—
216#11719919—81no—
217C4×Dic53—81no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—21no—
227C2×#88322—21no—
228C2×#88322—21no—
229C2×#88322—21no—
230C23×C53—22no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53—41no—
234C23×C53—41no—
235C23×C53—41no—
236C23×C53—41no—
237C23×C53C22×D411yescharacteristic
238Q8×C53—22no—
239Q8×C53—22no—
240Q8×C53—22no—
241Q8×C53—22no—
242Q8×C53—44no—
243Q8×C53—41no—
244Q8×C53—41no—
245Q8×C53—41no—
246Q8×C53—41no—
247D4×C53—22no—
248D4×C53—22no—
249D4×C53—41no—
250D4×C53—41no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—22no—
258C2×C4×C53—22no—
259C2×C4×C53—22no—
260C2×C4×C53—21no—
261C2×C4×C53—21no—
262C2×C4×C53—21no—
263C2×C4×C53—21no—
264C2×C4×C53—21no—
265C2×C4×C53—21no—
266C2×C4×C53—21no—
267C2×C4×C53—21no—
268C2×C4×C53—21no—
269C2×C4×C53—48no—
270C2×C4×C53—41no—
271C2×C4×C53—41no—
272C2×C4×C53—41no—
273C2×C4×C53—41no—
274C2×C4×C53—41no—
275C2×C4×C53—41no—
276C2×C4×C53#12012yes—
277C2×C4×C53#12012yes—
278C2×C4×C53#13512yes—
279C2×C4×C53C22×D411yescharacteristic
280C2×C4×C53C22×D411yescharacteristic
281C2×C4×C53C2×#4111yescharacteristic
282C2×C4×C53C2×#4111yescharacteristic
283C2×C4×C532−1+411yescharacteristic
284C2×C4×C532−1+411yescharacteristic
285C2×Dic53—22no—
286C2×Dic53—22no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293C2×Dic53—41no—
294C2×Dic53—41no—
295C2×Dic53—41no—
296C2×Dic53—41no—
297#88322—41no—
298#88322—41no—
299#88322—41no—
300#88322—41no—
301#25654—531nomaximal, Sylow
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53C2×#12011yescharacteristic
310C22×C53#54811yescharacteristic
311C22×C53#54811yescharacteristic
312C22×C53C2×#13511yescharacteristic
313C22×C53#57511yescharacteristic
314C22×C53#57711yescharacteristic
315C22×C53#57811yescharacteristic
316C4×C53—22no—
317C4×C53—22no—
318C4×C53—22no—
319C4×C53—42no—
320C4×C53—42no—
321C4×C53—41no—
322C4×C53—42no—
323C4×C53—41no—
324C4×C53—41no—
325C4×C53#53211yescharacteristic
326C4×C53#53611yescharacteristic
327C4×C53#54711yescharacteristic
328C4×C53#54711yescharacteristic
329Dic53—42no—
330Dic53—42no—
331Dic53—81no—
332Dic53—81no—
333C2×#556D5311yescharacteristic
334C2×#488—531no—
335C2×#479—531no—
336#2584—531no—
337#2043—531no—
338#2043—531no—
339#2042—531no—
340#2042—531no—
341#1984—531no—
342#1983—531no—
343#1865—531no—
344#1864—531no—
345#1863—531no—
346#1862—531no—
347C4×#103—531no—
348C2×C53—82no—
349C2×C53#243211yescharacteristic
350C2×C53#300211yescharacteristic
351C2×C53#301311yescharacteristic
352C2×C53#336011yescharacteristic
353C2×C53#336411yescharacteristic
354C2×C53#337611yescharacteristic
355C2×C53#337711yescharacteristic
356#556—24no—
357#556—24no—
358C2×C4⋊Q8C2×D5311yescharacteristic
359C2×#126—21no—
360C2×#126—21no—
361C2×#125—531no—
362C2×#124C2×D5311yescharacteristic
363C2×#124C2×D5311yescharacteristic
364C2×C22⋊Q8—21no—
365C2×C4×Q8C2×D5311yescharacteristic
366C2×C4×Q8C2×D5311yescharacteristic
367C2×C4×Q8C2×D5311yescharacteristic
368C2×C4×D4C2×D5311yescharacteristic
369C2×#117—21no—
370#488—1064no—
371#479—1064no—
372C2×#107—531no—
373C2×#106—531no—
374C2×C4⋊C8—531no—
375C2×#103—531no—
376C2×#103—531no—
377C2×#103—531no—
378C2×#103—531no—
379C2×#103—531no—
380C2×#103—531no—
381C2×#103—531no—
382C2×#103—531no—
383C2×#102—531no—
384C2×#102—531no—
385C2×#102—531no—
386C2×#102—531no—
387C2×C4×C8—531no—
388#398—1061no—
389#395—1061no—
390#389—531no—
391#389—531no—
392#389—531no—
393#389—531no—
394#389—1061no—
395#389—1061no—
396C4×C4⋊C4—531no—
397#340—531no—
398#340—531no—
399#336—531no—
400#336—531no—
401#336—531no—
402#336—531no—
403C53#2565411yescharacteristic, complemented, Sylow
404C22×Q8C22⋊D5311yescharacteristic
405C22×Q8C22×D5311yescharacteristic
406C22×Q8C22×D5311yescharacteristic
407C22×D4C22×D5311yescharacteristic
408C23×C4—21no—
409C22×C8—531no—
410C22×C8—531no—
411C22×C8—531no—
412C22×C8—531no—
413C4⋊Q8—22no—
414C4⋊Q8—22no—
415#126—28no—
416#126—28no—
417#125—1062no—
418#125—1062no—
419#124—22no—
420#124—22no—
421#124—22no—
422#124C22⋊D5314yes—
423C22⋊Q8—28no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×Q8—22no—
429C4×Q8C22⋊D5314yes—
430C4×D4—22no—
431C4×D4—22no—
432#117—28no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4C22⋊D5311yescharacteristic
448C2×C4⋊C4C22⋊D5311yescharacteristic
449C2×C4⋊C4C22⋊D5311yescharacteristic
450C2×C4⋊C4C22×D5311yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C42—21no—
457C2×C42—21no—
458C2×C42—1061no—
459C2×C42C22×D5311yescharacteristic
460C2×C42C22×D5311yescharacteristic
461C2×C42C22×D5311yescharacteristic
462#107—1062no—
463#106—1062no—
464C4⋊C8—1062no—
465#103—1062no—
466#103—1062no—
467#103—1062no—
468#103—1062no—
469#103—1062no—
470#103—1062no—
471#103—1062no—
472#103—1062no—
473#102—1062no—
474#102—1062no—
475#102—1062no—
476#102—1062no—
477C4×C8—1062no—
478#95—1061no—
479#95—1061no—
480#95—1061no—
481#95—1061no—
482#95—1061no—
483C24—21no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—41no—
497C2×Q8C2×C22⋊D5312yes—
498C2×Q8C2×C22⋊D5312yes—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4C2×C22⋊D5312yes—
505C22×C4—24no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—1061no—
514C22×C4—1061no—
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C2×C22⋊D5311yescharacteristic
517C22×C4C2×C22⋊D5311yescharacteristic
518C22×C4C2×C22⋊D5311yescharacteristic
519C22×C4C2×C22⋊D5311yescharacteristic
520C22×C4C2×C22⋊D5311yescharacteristic
521C22×C4C23×D5311yescharacteristic
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8—1061no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—532no—
543C4⋊C4—532no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—4241no—
551C4⋊C4—4241no—
552C4⋊C4—2122no—
553C4⋊C4—2122no—
554C4⋊C4—2122no—
555C4⋊C4—2122no—
556C4⋊C4C2×C22⋊D5312yes—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C42—24no—
563C42—24no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—4241no—
569C42C2×C22⋊D5312yes—
570C42C2×C22⋊D5312yes—
571C23—22no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23C22×C22⋊D5311yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—44no—
584Q8—41no—
585Q8—41no—
586Q8—41no—
587Q8—41no—
588D4—22no—
589D4—22no—
590D4—41no—
591D4—41no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—1062no—
618C2×C4—1062no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4#1313318612yes—
630C2×C4#1313318612yes—
631C2×C4#1313323712yes—
632C2×C4C22×C22⋊D5311yescharacteristic
633C2×C4C22×C22⋊D5311yescharacteristic
634C2×C4#41×D5311yescharacteristic
635C2×C4#41×D5311yescharacteristic
636C2×C4#1313326411yescharacteristic
637C2×C4#1313326411yescharacteristic
638C8—2121no—
639C8—2121no—
640C8—2121no—
641C8—2121no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22C2×#1313318611yescharacteristic
650C22#1681128011yescharacteristic
651C22#1681128011yescharacteristic
652C22C2×#1313323711yescharacteristic
653C22#1681142111yescharacteristic
654C22#1681143111yescharacteristic
655C22#1681143811yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—41no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—4241no—
666C4—4241no—
667C4—2122no—
668C4—2122no—
669C4#1681116911yescharacteristic
670C4#1681119411yescharacteristic
671C4#1681127211yescharacteristic
672C4#1681127211yescharacteristic
673C2—82no—
674C2#2027833911yescharacteristic
675C2#2028393311yescharacteristic
676C2#2028403011yescharacteristic
677C2#2028736011yescharacteristic
678C2#2028739611yescharacteristic
679C2#2028753611yescharacteristic
680C2#2028755111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#556C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenterC2×#556D53
hypocenterC53#25654
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#16810930C2semidirect
3. #20280103C2semidirect
4. #20275958C2semidirect
5. #20275958C2semidirect
8. #20275652C2semidirect
10. #20274983C2semidirect
12. #20274969C2semidirect
14. C4×#13133141C2semidirect
403. C53#25654semidirect

Maximal subgroups

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

subgroupindex
C2×#168109772
C2×#168109302
#202801032
#202759582
#202759512
#202756522
#202756482
#202749832
#202749762
#202749692
#202749622
C4×#131331412
C2×C53×#5562
#2565453