Finite Group Databasebeta

← #27435984

Subgroups — #27435984

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810923C211yesmaximal, characteristic
2C2×Q8×#88322C211yesmaximal, characteristic, complemented
3#20280103C211yesmaximal, characteristic, complemented
4#20276217C211yesmaximal, characteristic, complemented
5#20276211C211yesmaximal, characteristic, complemented
6#20276103C211yesmaximal, characteristic
7#20276089C211yesmaximal, characteristic
8#20275679C211yesmaximal, characteristic, complemented
9#20275481C211yesmaximal, characteristic, complemented
10#20275445C211yesmaximal, characteristic, complemented
11#20275440C211yesmaximal, characteristic, complemented
12#20275401C211yesmaximal, characteristic
13#20275401C211yesmaximal, characteristic
14C4×#13133138C211yesmaximal, characteristic
15C53×#2580C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16810923—24no—
18Q8×#88322—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32C2×C4×#88322C2211yescharacteristic
33#16810554—21no—
34#16810542—21no—
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810518—21no—
40#16810518—21no—
41C4×C4⋊Dic53C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×C4×Q8×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#399—21no—
53C53×#393—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#385—21no—
59C53×#385—21no—
60C53×#382C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133209—22no—
66#13133209—22no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C4×Dic53—21no—
78#13133150—22no—
79#13133150—22no—
80#13133148—22no—
81#13133148—22no—
82C4⋊#88322—22no—
83C4⋊#88322—22no—
84C4⋊#88322D414yes—
85#13133138—22no—
86#13133138—22no—
87#13133138—22no—
88#13133138Q814yescomplemented
89C4×#88322—22no—
90C4×#88322—22no—
91C4×#88322—22no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53D411yescharacteristic
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53—21no—
100C4×Q8×C53—22no—
101C4×Q8×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124C24×C53—21no—
125C2×Q8×C53—22no—
126C2×Q8×C53—22no—
127C2×Q8×C53—22no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53C2×Q812yes—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—42no—
138C22×C4×C53—81no—
139C22×C4×C53—81no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—81no—
161C4⋊C4×C53—81no—
162C22⋊C4×C53—42no—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—81no—
166C22⋊C4×C53—81no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53C2×D412yes—
172C42×C53#4112yes—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53C2×Q812yes—
180C4⋊Dic53#4112yes—
181#11719919—42no—
182#11719919—42no—
183#11719919—42no—
184#11719919—42no—
185#11719919—81no—
186#11719919—81no—
187C4×Dic53—81no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322C2×D412yes—
198C2×#88322C2×D412yes—
199C2×#88322C2×Q812yes—
200C23×C53—22no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53C22×D411yescharacteristic
208Q8×C53—28no—
209D4×C53—22no—
210D4×C53—22no—
211D4×C53—41no—
212D4×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—84no—
237C2×C4×C53#12112yes—
238C2×C4×C53#12112yes—
239C2×C4×C53#13512yes—
240C2×C4×C53#13512yes—
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×Q811yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C532−1+411yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259#88322—22no—
260#88322—22no—
261#88322—22no—
262#88322—41no—
263#25473—531nomaximal, Sylow
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53C2×#12111yescharacteristic
272C22×C53#54811yescharacteristic
273C22×C53D4×Q811yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53C2×#13511yescharacteristic
276C22×C53#57611yescharacteristic
277C22×C53#57711yescharacteristic
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—22no—
282C4×C53—41no—
283C4×C53—41no—
284C4×C53—81no—
285C4×C53—81no—
286C4×C53—81no—
287C4×C53#53311yescharacteristic
288C4×C53#54111yescharacteristic
289C4×C53#54711yescharacteristic
290C4×C53D4×Q811yescharacteristic
291Dic53—42no—
292Dic53—42no—
293Dic53—81no—
294Dic53—81no—
295C2×#478—531no—
296C2×C8×Q8—531no—
297#2584—531no—
298#2580D5311yescharacteristic
299#2088—531no—
300#2087—531no—
301#2067—531no—
302#2065—531no—
303#1991—531no—
304#1949—531no—
305#1942—531no—
306#1941—531no—
307#1934—531no—
308#1934—531no—
309C4×#102—531no—
310C2×C53—82no—
311C2×C53#242911yescharacteristic
312C2×C53#305011yescharacteristic
313C2×C53#306211yescharacteristic
314C2×C53#336111yescharacteristic
315C2×C53#336511yescharacteristic
316C2×C53#338011yescharacteristic
317C2×C53#338111yescharacteristic
318C2×#125—531no—
319C2×C4×Q8C2×D5311yescharacteristic
320C2×C4×D4C2×D5311yescharacteristic
321#478—1064no—
322C8×Q8—1064no—
323C2×#107—531no—
324C2×#107—531no—
325C2×#106—531no—
326C2×#106—531no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8—531no—
330C2×#102—531no—
331C2×#102—531no—
332C2×#102—531no—
333C2×#102—531no—
334C2×C4×C8—531no—
335C2×C4×C8—531no—
336C2×C4×C8—531no—
337#399—21no—
338#398—1061no—
339#395—1061no—
340#393—21no—
341#390C2×D5311yescharacteristic
342#390C2×D5311yescharacteristic
343#389—531no—
344#389—531no—
345#389—531no—
346#389—531no—
347#389—1061no—
348#389—1061no—
349#389C2×D5311yescharacteristic
350#389C2×D5311yescharacteristic
351#385—21no—
352#385—21no—
353#382C2×D5311yescharacteristic
354C4×C4⋊C4—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363C53#2547311yescharacteristic, complemented, Sylow
364C22×Q8C22⋊D5311yescharacteristic
365C22×D4C22×D5311yescharacteristic
366C23×C4—21no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8—531no—
371#125—1062no—
372#125—1062no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×D4—22no—
376C4×D4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4C22⋊D5311yescharacteristic
390C2×C4⋊C4C22⋊D5311yescharacteristic
391C2×C4⋊C4C22⋊D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—1061no—
401C2×C42C22×D5311yescharacteristic
402C2×C42C22×D5311yescharacteristic
403C2×C42C22×D5311yescharacteristic
404#107—1062no—
405#107—1062no—
406#106—1062no—
407#106—1062no—
408C4⋊C8—534no—
409C4⋊C8—1062no—
410C4⋊C8—1062no—
411#102—534no—
412#102—1062no—
413#102—1062no—
414#102—1062no—
415C4×C8—1062no—
416C4×C8—1062no—
417C4×C8—1062no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427C24—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D5312yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C23×D5311yescharacteristic
454C2×C8—532no—
455C2×C8—532no—
456C2×C8—532no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—532no—
478C4⋊C4—532no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—4241no—
486C4⋊C4—4241no—
487C4⋊C4—2122no—
488C4⋊C4—2122no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—4241no—
501C42D4×D5312yes—
502C42#1171994712yes—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C22×C22⋊D5311yescharacteristic
511Q8—28no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4#1313318912yes—
553C2×C4#1313318912yes—
554C2×C4#1313323712yes—
555C2×C4#1313323712yes—
556C2×C4C2×D4×D5311yescharacteristic
557C2×C4C2×Q8×D5311yescharacteristic
558C2×C4#41×D5311yescharacteristic
559C2×C4C2×#1171994711yescharacteristic
560C2×C4#1313326411yescharacteristic
561C2×C4#1313326511yescharacteristic
562C8—1062no—
563C8—1062no—
564C8—1062no—
565C8—2121no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#1313318911yescharacteristic
574C22#1681128011yescharacteristic
575C22Q8×C22⋊D5311yescharacteristic
576C22C2×#1313323711yescharacteristic
577C22C2×#1313323711yescharacteristic
578C22#1681142611yescharacteristic
579C22#1681143411yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—4241no—
590C4—4241no—
591C4—2122no—
592C4—2122no—
593C4#1681117511yescharacteristic
594C4#1681123311yescharacteristic
595C4#1681127511yescharacteristic
596C4#1681128611yescharacteristic
597C2—82no—
598C2#2027831611yescharacteristic
599C2#2028429311yescharacteristic
600C2#2028439111yescharacteristic
601C2#2028737111yescharacteristic
602C2#2028740711yescharacteristic
603C2#2028757811yescharacteristic
604C2#2028759011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2580C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2580D53
hypocenterC53#25473
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#88322C2semidirect
3. #20280103C2semidirect
4. #20276217C2semidirect
5. #20276211C2semidirect
8. #20275679C2semidirect
9. #20275481C2semidirect
10. #20275445C2semidirect
11. #20275440C2semidirect
88. #13133138Q8semidirect
363. C53#25473semidirect

Maximal subgroups

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

subgroupindex
C2×#168109232
C2×Q8×#883222
#202801032
#202762172
#202762112
#202761032
#202760892
#202756792
#202754812
#202754452
#202754402
#202754012
C4×#131331382
C53×#25802
#2547353