Finite Group Databasebeta

← #27437429

Subgroups — #27437429

Order 13568.

1190 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810993C211yesmaximal, characteristic, complemented
2C2×#16810828C211yesmaximal, characteristic, complemented
3#20279668C211yesmaximal, characteristic, complemented
4#20275718C212yesmaximal, complemented
5#20275718C212yesmaximal, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20275027C212yesmaximal, complemented
9#20275022C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#553C211yesmaximal, characteristic
12C22×#11719923C2212yes—
13C2×#13133217C2211yescharacteristic
14C2×#13133196C2212yes—
15C2×#13133196C2212yes—
16#16810993—24no—
17#16810828—24no—
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133140C2212yes—
20C2×#13133140C2212yes—
21C2×#13133140C2212yes—
22C2×#13133138C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×C22⋊#88322C2212yes—
25C2×#13133128C2211yescharacteristic
26#16810559—21no—
27#16810559—21no—
28#16810526—21no—
29#16810492—21no—
30C4×C4⋊Dic53C2211yescharacteristic
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34C53×#553—24no—
35C53×#553—24no—
36C2×#126×C53—21no—
37C2×#125×C53C2211yescharacteristic
38C2×#123×C53—21no—
39C2×#121×C53—21no—
40C2×#121×C53C2211yescharacteristic
41C2×#121×C53C2211yescharacteristic
42C2×C4×D4×C53—21no—
43C2×C4×D4×C53C2211yescharacteristic
44C2×C4×D4×C53C2211yescharacteristic
45C2×#117×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×Dic106C2312yes—
48C2×#11719923—24no—
49C2×#11719923—42no—
50C2×#11719923—42no—
51C22×#88322C2312yes—
52C22×#88322C2312yes—
53#13133217—22no—
54#13133217—22no—
55#13133196—24no—
56#13133196—24no—
57#13133196—42no—
58#13133196—42no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—22no—
66C2×C4×Dic53—21no—
67C4⋊#88322—22no—
68#13133140—24no—
69#13133140—24no—
70#13133140—24no—
71#13133138—24no—
72C22⋊#88322—42no—
73C22⋊#88322—42no—
74#13133128—22no—
75#13133124—21no—
76C22×D4×C53—21no—
77C22×D4×C53—21no—
78C22×D4×C53D411yescharacteristic
79C22×D4×C53C2311yescharacteristic
80C23×C4×C53—21no—
81C23×C4×C53—21no—
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84#126×C53—28no—
85#125×C53—22no—
86#125×C53D414yes—
87#123×C53—28no—
88#121×C53—28no—
89#121×C53—22no—
90#121×C53—22no—
91#121×C53—22no—
92#121×C53—22no—
93C4×D4×C53—28no—
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53—22no—
97C4×D4×C53D414yes—
98#117×C53—22no—
99#117×C53—22no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—22no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C24×C53—21no—
118C24×C53C2×D411yescharacteristic
119C24×C53C2×D411yescharacteristic
120C2×D4×C53—24no—
121C2×D4×C53—22no—
122C2×D4×C53—24no—
123C2×D4×C53—22no—
124C2×D4×C53—21no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—41no—
128C2×D4×C53—42no—
129C2×D4×C53—42no—
130C2×D4×C53—82no—
131C2×D4×C53C2×D412yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—24no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—24no—
145C22×C4×C53—82no—
146C22×C4×C53—82no—
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—22no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53C2×D412yes—
168C4⋊C4×C53C2×D412yes—
169C4⋊C4×C53C2×D412yes—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—24no—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C42×C53—24no—
177C42×C53C2×D412yes—
178C42×C53C2×D412yes—
179C22⋊Dic53—44no—
180C22⋊Dic53—44no—
181#11719923—48no—
182C4⋊Dic53—22no—
183C4⋊Dic53—42no—
184C4⋊Dic53—42no—
185C4⋊Dic53#4114yes—
186#11719919—44no—
187#11719919—44no—
188C4×Dic53—42no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—22no—
195C23×C53—21no—
196C23×C53—22no—
197C23×C53—21no—
198C23×C53—22no—
199C23×C53—21no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—22no—
205C23×C53—42no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—42no—
209C23×C53#13712yes—
210C23×C53C22×D411yescharacteristic
211D4×C53—22no—
212D4×C53—22no—
213D4×C53—44no—
214D4×C53—48no—
215D4×C53—42no—
216D4×C53—84no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—84no—
238C2×C4×C53—84no—
239C2×C4×C53#12012yes—
240C2×C4×C53#12012yes—
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C2×#4112yes—
244C2×C4×C53H(5,𝔽2)12yes—
245C2×Dic53—24no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251Dic106—42no—
252Dic106—42no—
253Dic106—42no—
254Dic106—42no—
255#88322—42no—
256#88322—42no—
257#25582—531nomaximal, Sylow
258C22×C53—22no—
259C22×C53—24no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—42no—
263C22×C53—41no—
264C22×C53—42no—
265C22×C53—41no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—41no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—82no—
272C22×C53—82no—
273C22×C53C2×#12011yescharacteristic
274C22×C53#54612yes—
275C22×C53C2×#13711yescharacteristic
276C22×C53#57511yescharacteristic
277C22×C53#57712yes—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—42no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53#53211yescharacteristic
286C4×C53#53511yescharacteristic
287C4×C53#55012yes—
288Dic53—44no—
289Dic53—82no—
290C2×#553D5311yescharacteristic
291C2×#491—531no—
292C2×#461—531no—
293#2542—531no—
294#2000—532no—
295#2000—532no—
296#1989—531no—
297#1955—531no—
298#1873—532no—
299#1872—532no—
300#1762—531no—
301C2×C53—42no—
302C2×C53—81no—
303C2×C53—82no—
304C2×C53#243111yescharacteristic
305C2×C53#300711yescharacteristic
306C2×C53#300911yescharacteristic
307C2×C53#330712yes—
308C2×C53#332012yes—
309C22×SD16—532no—
310#553—24no—
311#553—24no—
312C2×C4⋊Q8—531no—
313C2×#126—21no—
314C2×#125C2×D5311yescharacteristic
315C2×#123—21no—
316C2×C22⋊Q8—532no—
317C2×C22⋊Q8—532no—
318C2×#121—21no—
319C2×#121C2×D5311yescharacteristic
320C2×#121C2×D5311yescharacteristic
321C2×C4×D4—21no—
322C2×C4×D4C2×D5311yescharacteristic
323C2×C4×D4C2×D5311yescharacteristic
324C2×#117C2×D5311yescharacteristic
325C22×C4⋊C4C2×D5311yescharacteristic
326#491—1064no—
327#461—1064no—
328C2×C4⋊C8—531no—
329C2×#103—532no—
330C2×#103—532no—
331C2×#103—532no—
332C2×#102—532no—
333C2×C22⋊C8—532no—
334C2×C22⋊C8—532no—
335C2×#97—531no—
336#399—1061no—
337#399—1061no—
338#390—1061no—
339#384—1061no—
340C4×C4⋊C4—531no—
341#340—532no—
342#340—532no—
343#340—532no—
344C53#2558211yescharacteristic, complemented, Sylow
345C22×Q8—532no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4C22⋊D5311yescharacteristic
349C22×D4C22×D5311yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4C22×D5311yescharacteristic
353C23×C4C22×D5311yescharacteristic
354C2×SD16—1064no—
355C2×SD16—2122no—
356C2×SD16—2122no—
357C22×C8—532no—
358C22×C8—532no—
359C4⋊Q8—1062no—
360C4⋊Q8—1062no—
361#126—28no—
362#125—22no—
363#125C22⋊D5314yes—
364#123—28no—
365C22⋊Q8—1064no—
366C22⋊Q8—1064no—
367C22⋊Q8—2122no—
368C22⋊Q8—2122no—
369#121—28no—
370#121—22no—
371#121—22no—
372#121—22no—
373#121—22no—
374C4×D4—28no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4C22⋊D5314yes—
379#117—22no—
380#117—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—532no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
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—1062no—
400C2×C22⋊C4—1062no—
401C2×C22⋊C4C22⋊D5311yescharacteristic
402C2×C22⋊C4C22⋊D5311yescharacteristic
403C2×C42—21no—
404C2×C42—1061no—
405C2×C42C22×D5311yescharacteristic
406C4⋊C8—1062no—
407#103—1064no—
408#103—1064no—
409#103—1064no—
410#102—1064no—
411C22⋊C8—2122no—
412C22⋊C8—2122no—
413#97—1062no—
414#95—1061no—
415C24—21no—
416C24C2×C22⋊D5311yescharacteristic
417C24C2×C22⋊D5311yescharacteristic
418C2×Q8—1062no—
419C2×Q8—1062no—
420C2×Q8—1062no—
421C2×Q8—1062no—
422C2×Q8—1062no—
423C2×Q8—1062no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—41no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—82no—
435C2×D4C2×C22⋊D5312yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C23×D5311yescharacteristic
458SD16—2128no—
459C2×C8—1062no—
460C2×C8—1062no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8—1062no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—534no—
473C4⋊C4—1062no—
474C4⋊C4—2124no—
475C4⋊C4—2124no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4C2×C22⋊D5312yes—
479C4⋊C4C2×C22⋊D5312yes—
480C4⋊C4C2×C22⋊D5312yes—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—2124no—
488C22⋊C4—2124no—
489C42—24no—
490C42—2122no—
491C42C2×C22⋊D5312yes—
492C42C2×C22⋊D5312yes—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—22no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23#1313324712yes—
508C23C22×C22⋊D5311yescharacteristic
509Q8—2122no—
510Q8—2122no—
511Q8—2122no—
512Q8—2122no—
513D4—22no—
514D4—22no—
515D4—44no—
516D4—48no—
517D4—42no—
518D4—84no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—1064no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4#1313318612yes—
548C2×C4#1313318612yes—
549C2×C4C22×C22⋊D5311yescharacteristic
550C2×C4C22×C22⋊D5311yescharacteristic
551C2×C4#41×D5312yes—
552C2×C4#1313326312yes—
553C8—2122no—
554C8—2122no—
555C22—22no—
556C22—24no—
557C22—41no—
558C22—41no—
559C22—42no—
560C22—41no—
561C22—42no—
562C22—41no—
563C22—42no—
564C22—42no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—82no—
569C22—82no—
570C22C2×#1313318611yescharacteristic
571C22#1681126112yes—
572C22C2×#1313324711yescharacteristic
573C22#1681142111yescharacteristic
574C22#1681143412yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—4242no—
583C4—2124no—
584C4#1681116911yescharacteristic
585C4#1681118711yescharacteristic
586C4#1681129212yes—
587C2—42no—
588C2—81no—
589C2—82no—
590C2#2027833011yescharacteristic
591C2#2028397511yescharacteristic
592C2#2028399211yescharacteristic
593C2#2028668512yes—
594C2#2028688012yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#553C2
socleC23×C53C22×D4
hypercenterC2×#553D53
hypocenterC53#25582
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. C2×#16810993C2semidirect
2. C2×#16810828C2semidirect
3. #20279668C2semidirect
4. #20275718C2semidirect
5. #20275718C2semidirect
6. #20275673C2semidirect
7. #20275514C2semidirect
8. #20275027C2semidirect
9. #20275022C2semidirect
10. #20274540C2semidirect
344. C53#25582semidirect

Maximal subgroups

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

subgroupindex
C2×#168109932
C2×#168108282
#202796682
#202757182
#202756732
#202755142
#202750272
#202750222
#202745402
C2×C53×#5532
#2558253