Finite Group Databasebeta

← #27437659

Subgroups — #27437659

Order 13568.

1178 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811018C211yesmaximal, characteristic, complemented
2C2×#16810993C211yesmaximal, characteristic, complemented
3#20283067C211yesmaximal, characteristic, complemented
4#20275723C212yesmaximal, complemented
5#20275718C212yesmaximal
6#20275685C211yesmaximal, characteristic, complemented
7#20275532C211yesmaximal, characteristic, complemented
8#20275056C212yesmaximal, complemented
9#20275049C212yesmaximal, complemented
10#20274477C211yesmaximal, characteristic, complemented
11C2×C53×#529C211yesmaximal, characteristic
12C22×#11719925C2212yes—
13C22×#11719923C2212yes—
14C2×#13133217C2211yescharacteristic
15C2×#13133196C2212yes—
16#16811018—24no—
17#16810993—24no—
18C2×#13133141C2212yes—
19C2×#13133140C2212yes—
20C2×#13133140C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2212yes—
23C2×#13133128C2211yescharacteristic
24C2×#13133128C2211yescharacteristic
25#16810529C2212yes—
26#16810521—21no—
27#16810521—21no—
28#16810493C2211yescharacteristic
29#16810483—21no—
30#16810483—21no—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#529—24no—
35C53×#529—24no—
36C2×#126×C53—21no—
37C2×#125×C53C2211yescharacteristic
38C2×#124×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×#121×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53—21no—
44C2×C4×D4×C53C2211yescharacteristic
45C2×#117×C53—21no—
46C22×C42×C53C2211yescharacteristic
47C22×Dic106C2312yes—
48C2×#11719925—24no—
49C2×#11719925—24no—
50C2×#11719925—42no—
51C2×#11719923—24no—
52C2×#11719923—24no—
53C2×#11719923—42no—
54C22×#88322C2312yes—
55C22×#88322C2312yes—
56#13133217—22no—
57#13133217—22no—
58#13133196—42no—
59#13133196—42no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53C2312yes—
65C2×#11719919—22no—
66C2×C22⋊Dic53—22no—
67#13133141—42no—
68#13133140—24no—
69#13133140—24no—
70#13133138—42no—
71C22⋊#88322—24no—
72#13133128—22no—
73#13133128—22no—
74#13133124—22no—
75#13133124—22no—
76C22×Q8×C53C2311yescharacteristic
77C22×D4×C53—21no—
78C22×D4×C53C2311yescharacteristic
79C23×C4×C53—21no—
80C23×C4×C53D411yescharacteristic
81C23×C4×C53D411yescharacteristic
82C23×C4×C53C2311yescharacteristic
83#126×C53—28no—
84#125×C53—22no—
85#125×C53D414yes—
86#124×C53—22no—
87#124×C53D414yes—
88#123×C53—28no—
89C22⋊Q8×C53—22no—
90C22⋊Q8×C53—22no—
91#121×C53—22no—
92#121×C53—22no—
93C4×Q8×C53—22no—
94C4×Q8×C53—22no—
95C4×D4×C53—28no—
96C4×D4×C53—22no—
97C4×D4×C53—22no—
98#117×C53—28no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C42×C53—22no—
111C2×C42×C53—22no—
112C2×C42×C53—21no—
113C2×C42×C53D411yescharacteristic
114C2×C42×C53D411yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C24×C53—21no—
118C24×C53C2×D411yescharacteristic
119C2×Q8×C53—21no—
120C2×Q8×C53—22no—
121C2×Q8×C53—22no—
122C2×Q8×C53C2×D412yes—
123C2×D4×C53—24no—
124C2×D4×C53—21no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—42no—
128C2×D4×C53—42no—
129C2×D4×C53C2×D412yes—
130C22×C4×C53—24no—
131C22×C4×C53—22no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—41no—
143C22×C4×C53—41no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—22no—
153C22×Dic53—22no—
154C2×Dic106—22no—
155C2×Dic106—22no—
156C2×Dic106—22no—
157C2×Dic106—22no—
158C2×Dic106—22no—
159C2×Dic106—22no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—24no—
162C4⋊C4×C53—24no—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53C2×D412yes—
167C4⋊C4×C53C2×D412yes—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—24no—
172C22⋊C4×C53—24no—
173C42×C53—24no—
174C42×C53—22no—
175C42×C53—22no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53C2×D412yes—
179C42×C53C2×D412yes—
180C22⋊Dic53—44no—
181#11719925—48no—
182#11719923—48no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53—42no—
188C4⋊Dic53—42no—
189#11719919—44no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322#4114yes—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—22no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53#13712yes—
210C23×C53C22×D411yescharacteristic
211Q8×C53—24no—
212Q8×C53—42no—
213D4×C53—24no—
214D4×C53—48no—
215D4×C53—42no—
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—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—44no—
232C2×C4×C53—44no—
233C2×C4×C53—48no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53#12012yes—
243C2×C4×C53#12012yes—
244C2×C4×C53#13712yes—
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C2×#4112yes—
248C2×C4×C53H(5,𝔽2)12yes—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255Dic106—42no—
256Dic106—42no—
257Dic106—42no—
258Dic106—42no—
259#88322—24no—
260#88322—42no—
261#25604—531nomaximal, Sylow
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53C2×#12011yescharacteristic
277C22×C53#53211yescharacteristic
278C22×C53#53511yescharacteristic
279C22×C53C2×#13711yescharacteristic
280C22×C53C2×#13711yescharacteristic
281C22×C53#57511yescharacteristic
282C22×C53#57511yescharacteristic
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—22no—
286C4×C53—41no—
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—42no—
290C4×C53—42no—
291C4×C53#54612yes—
292C4×C53#55012yes—
293Dic53—82no—
294Dic53—82no—
295C2×#529D5311yescharacteristic
296C2×#497—531no—
297C2×#491—531no—
298#2857—531no—
299#2001—532no—
300#2000—532no—
301#1992—531no—
302#1959—531no—
303#1878—532no—
304#1877—532no—
305#1749—531no—
306C2×C53—42no—
307C2×C53—82no—
308C2×C53#243111yescharacteristic
309C2×C53#300711yescharacteristic
310C2×C53#300711yescharacteristic
311C2×C53#317211yescharacteristic
312C2×C53#318111yescharacteristic
313C2×C53#321811yescharacteristic
314C2×C53#321911yescharacteristic
315C22×Q16—532no—
316C22×SD16—532no—
317C2×C4⋊Q8—531no—
318#529—24no—
319#529—24no—
320C2×#126—21no—
321C2×#125C2×D5311yescharacteristic
322C2×#124C2×D5311yescharacteristic
323C2×#123—21no—
324C2×C22⋊Q8—532no—
325C2×C22⋊Q8C2×D5311yescharacteristic
326C2×#121C2×D5311yescharacteristic
327C2×C4×Q8C2×D5311yescharacteristic
328C2×C4×D4—21no—
329C2×C4×D4C2×D5311yescharacteristic
330C2×#117—21no—
331C22×C42C2×D5311yescharacteristic
332#497—1064no—
333#491—1064no—
334C2×#103—532no—
335C2×#103—532no—
336C2×#103—532no—
337C2×#102—532no—
338C2×C22⋊C8—532no—
339C2×#97—531no—
340C2×#97—531no—
341#391—532no—
342#389—1061no—
343#389—1061no—
344#384—531no—
345#380—1061no—
346#380—1061no—
347#340—532no—
348#340—532no—
349#336—532no—
350C53#2560411yescharacteristic, complemented, Sylow
351C22×Q8—532no—
352C22×Q8C22×D5311yescharacteristic
353C22×D4—21no—
354C22×D4C22×D5311yescharacteristic
355C23×C4—21no—
356C23×C4C22⋊D5311yescharacteristic
357C23×C4C22⋊D5311yescharacteristic
358C23×C4C22×D5311yescharacteristic
359C2×Q16—1064no—
360C2×Q16—1064no—
361C2×Q16—2122no—
362C2×SD16—1064no—
363C2×SD16—1064no—
364C2×SD16—2122no—
365C22×C8—532no—
366C22×C8—532no—
367C4⋊Q8—1062no—
368C4⋊Q8—1062no—
369#126—28no—
370#125—22no—
371#125C22⋊D5314yes—
372#124—22no—
373#124C22⋊D5314yes—
374#123—28no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—2122no—
378C22⋊Q8—2122no—
379#121—22no—
380#121—22no—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×D4—28no—
384C4×D4—22no—
385C4×D4—22no—
386#117—28no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—532no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1062no—
405C2×C42—21no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42C22⋊D5311yescharacteristic
409C2×C42C22⋊D5311yescharacteristic
410C2×C42C22×D5311yescharacteristic
411C2×C42C22×D5311yescharacteristic
412#103—1064no—
413#103—1064no—
414#103—2122no—
415#102—2122no—
416C22⋊C8—1064no—
417#97—1062no—
418#97—1062no—
419#95—1062no—
420#95—1062no—
421C24—21no—
422C24C2×C22⋊D5311yescharacteristic
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—1062no—
427C2×Q8—1062no—
428C2×Q8—1062no—
429C2×Q8—1062no—
430C2×Q8—1062no—
431C2×Q8—1062no—
432C2×Q8C2×C22⋊D5312yes—
433C2×D4—24no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C2×C22⋊D5312yes—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C23×D5311yescharacteristic
464Q16—2128no—
465SD16—2128no—
466C2×C8—534no—
467C2×C8—1062no—
468C2×C8—1062no—
469C2×C8—1062no—
470C2×C8—1062no—
471C2×C8—1062no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—1062no—
479C4⋊C4—1062no—
480C4⋊C4—2124no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C4⋊C4—2122no—
484C4⋊C4—2122no—
485C4⋊C4C2×C22⋊D5312yes—
486C4⋊C4C2×C22⋊D5312yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—2124no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42C2×C22⋊D5312yes—
499C42C2×C22⋊D5312yes—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23#1313324712yes—
514C23C22×C22⋊D5311yescharacteristic
515Q8—24no—
516Q8—42no—
517Q8—2122no—
518Q8—2122no—
519Q8—2122no—
520Q8—2122no—
521D4—24no—
522D4—48no—
523D4—42no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4#1313318612yes—
557C2×C4#1313318612yes—
558C2×C4#1313324712yes—
559C2×C4C22×C22⋊D5311yescharacteristic
560C2×C4C22×C22⋊D5311yescharacteristic
561C2×C4#41×D5312yes—
562C2×C4#1313326312yes—
563C8—1064no—
564C8—2122no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22C2×#1313318611yescharacteristic
580C22#1681116911yescharacteristic
581C22#1681118711yescharacteristic
582C22C2×#1313324711yescharacteristic
583C22C2×#1313324711yescharacteristic
584C22#1681142111yescharacteristic
585C22#1681142111yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—4242no—
595C4—4242no—
596C4#1681126112yes—
597C4#1681129212yes—
598C2—42no—
599C2—82no—
600C2#2027833011yescharacteristic
601C2#2028397511yescharacteristic
602C2#2028397511yescharacteristic
603C2#2028532211yescharacteristic
604C2#2028542511yescharacteristic
605C2#2028573911yescharacteristic
606C2#2028574811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#529C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#529D53
hypocenterC53#25604
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#16811018C2semidirect
2. C2×#16810993C2semidirect
3. #20283067C2semidirect
4. #20275723C2semidirect
6. #20275685C2semidirect
7. #20275532C2semidirect
8. #20275056C2semidirect
9. #20275049C2semidirect
10. #20274477C2semidirect
350. C53#25604semidirect

Maximal subgroups

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

subgroupindex
C2×#168110182
C2×#168109932
#202830672
#202757232
#202757182
#202756852
#202755322
#202750562
#202750492
#202744772
C2×C53×#5292
#2560453