Finite Group Databasebeta

← #27418834

Subgroups — #27418834

Order 13568.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810825C211yesmaximal, characteristic, complemented
2C2×C4×#11719922C211yesmaximal, characteristic
3#20279633C211yesmaximal, characteristic, complemented
4#20275951C212yesmaximal, complemented
5#20275792C212yesmaximal
6#20275652C211yesmaximal, characteristic, complemented
7#20275518C211yesmaximal, characteristic, complemented
8#20275070C212yesmaximal, complemented
9#20274969C212yesmaximal, complemented
10#20274715C211yesmaximal, characteristic, complemented
11C2×C53×#556C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C22×#11719922C2211yescharacteristic
14C2×#13133193—21no—
15C2×C4×D212C2211yescharacteristic
16#16810825—24no—
17C4×#11719922—24no—
18C2×#13133150C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133141C2212yes—
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2212yes—
25C2×#13133137C2211yescharacteristic
26C2×C4×#88322C2211yescharacteristic
27#16810545—22no—
28#16810525C2212yes—
29#16810521C2212yes—
30#16810493C2211yescharacteristic
31#16810480—21no—
32#16810385C2212yes—
33#16810378C2212yes—
34#16810378C2212yes—
35C53×#556—24no—
36C53×#556—24no—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#126×C53—22no—
39C2×#124×C53C2212yes—
40C2×C22⋊Q8×C53—21no—
41C2×C4×Q8×C53C2212yes—
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×#117×C53—21no—
45C22×D212C2311yescharacteristic
46C22×C4×D53—21no—
47C2×#11719924—24no—
48C2×#11719924—24no—
49C2×#11719924—42no—
50C2×#11719922—22no—
51C2×#11719922—22no—
52C2×#11719922—22no—
53C22×#88322C2311yescharacteristic
54C22×#88322C2312yes—
55C22×#88322C2311yescharacteristic
56#13133193—44no—
57C4×D212—22no—
58C4×D212—22no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C4⋊Dic53C2312yes—
63C2×#11719919—22no—
64C2×#11719921—22no—
65C2×#11719921—21no—
66C2×#11719921—22no—
67#13133150—22no—
68C4⋊#88322—22no—
69#13133141—24no—
70#13133141—24no—
71#13133141—42no—
72#13133138—22no—
73#13133138—42no—
74#13133137—22no—
75C4×#88322—22no—
76#13133124—22no—
77C22×Q8×C53D411yescharacteristic
78C22×Q8×C53C2312yes—
79C22×D4×C53C2311yescharacteristic
80C23×C4×C53—21no—
81C4⋊Q8×C53—22no—
82C4⋊Q8×C53D414yes—
83#126×C53—216no—
84#124×C53—24no—
85#124×C53—24no—
86C22⋊Q8×C53—28no—
87C4×Q8×C53—24no—
88C4×Q8×C53—22no—
89C4×Q8×C53—22no—
90C4×Q8×C53—24no—
91C4×D4×C53—22no—
92C4×D4×C53D414yes—
93#117×C53—28no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53D412yes—
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—22no—
103C2×C42×C53—22no—
104C2×C42×C53C2312yes—
105C2×C42×C53C2311yescharacteristic
106C24×C53—21no—
107C23×D53—21no—
108C2×Q8×C53—22no—
109C2×Q8×C53—22no—
110C2×Q8×C53—22no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53C2×D414yes—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53C2×D412yes—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—24no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—22no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212#4112yes—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C4⋊C4×C53—28no—
145C4⋊C4×C53—24no—
146C4⋊C4×C53—24no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—24no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53C2×D412yes—
151C22⋊C4×C53—28no—
152C22⋊C4×C53—28no—
153C22⋊C4×C53—24no—
154C42×C53—28no—
155C42×C53—22no—
156C42×C53—22no—
157C42×C53C2×D412yes—
158C42×C53C2×D412yes—
159#11719924—48no—
160#11719922—28no—
161#11719921—44no—
162#11719921—42no—
163#11719921—82no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—22no—
167C4⋊Dic53—22no—
168C4⋊Dic53—42no—
169C4⋊Dic53—81no—
170#11719919—44no—
171C2×#88322—21no—
172C2×#88322—21no—
173C2×#88322—21no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—21no—
178C2×#88322—21no—
179C2×#88322#4112yes—
180C23×C53—22no—
181C23×C53—41no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53C22×D411yescharacteristic
188C22×D53—22no—
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—41no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—41no—
195Q8×C53—24no—
196Q8×C53—24no—
197Q8×C53—42no—
198Q8×C53—42no—
199Q8×C53—44no—
200D4×C53—22no—
201D4×C53—22no—
202D4×C53—41no—
203D4×C53—41no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—24no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—48no—
219C2×C4×C53#12012yes—
220C2×C4×C53#12012yes—
221C2×C4×C53#13512yes—
222C2×C4×C53AGL(1,ℤ/8)12yes—
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53H(5,𝔽2)11yescharacteristic
228C2×C4×C532−1+411yescharacteristic
229C2×Dic53—41no—
230C2×Dic53—42no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235D212—22no—
236D212—22no—
237D212—41no—
238D212—41no—
239C4×D53—84no—
240#88322—22no—
241#88322—41no—
242#88322—42no—
243#24185—531nomaximal, Sylow
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—81no—
248C22×C53—81no—
249C22×C53—81no—
250C22×C53—81no—
251C22×C53C2×#12011yescharacteristic
252C22×C53#54811yescharacteristic
253C22×C53#55011yescharacteristic
254C22×C53C2×#13511yescharacteristic
255C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C53#57711yescharacteristic
257C22×C53#57811yescharacteristic
258C2×D53—42no—
259C2×D53—42no—
260C2×D53—42no—
261C2×D53—81no—
262C2×D53—81no—
263C2×D53—81no—
264C2×D53—81no—
265C4×C53—22no—
266C4×C53—44no—
267C4×C53—42no—
268C4×C53—42no—
269C4×C53—42no—
270C4×C53#44914yes—
271C4×C53#52511yescharacteristic
272C4×C53#53711yescharacteristic
273C4×C53#54611yescharacteristic
274C4×C53#54711yescharacteristic
275Dic53—82no—
276Dic53—81no—
277C2×#556D5311yescharacteristic, complemented
278C2×#461—531no—
279C2×C4×D8—531no—
280#2538—531no—
281#2042—532no—
282#2012—532no—
283#1984—531no—
284#1956—531no—
285#1880—532no—
286#1863—532no—
287#1802—531no—
288C2×C53—82no—
289C2×C53#240711yescharacteristic
290C2×C53C2×#44911yescharacteristic
291C2×C53#301311yescharacteristic
292C2×C53#336011yescharacteristic
293C2×C53#336711yescharacteristic
294C2×C53#339311yescharacteristic
295C2×C53#339711yescharacteristic
296D53—82no—
297C22×SD16—532no—
298C22×D8—531no—
299#556—24no—
300#556—24no—
301C2×C4⋊Q8C2×D5311yescharacteristic
302C2×#126—22no—
303C2×#124C2×D5312yes—
304C2×C22⋊Q8—21no—
305C2×C22⋊Q8—1061no—
306C2×C4×Q8C2×D5312yes—
307C2×C4×Q8C2×D5311yescharacteristic
308C2×C4×D4—531no—
309C2×C4×D4C2×D5311yescharacteristic
310C2×#117—21no—
311#461—1064no—
312C4×D8—1064no—
313C2×#107—531no—
314C2×C4⋊C8—531no—
315C2×#103—532no—
316C2×#103—532no—
317C2×#103—532no—
318C2×#102—531no—
319C2×#102—531no—
320C2×#102—532no—
321C2×C4×C8—531no—
322#396—1062no—
323#390—532no—
324#389—532no—
325#384—531no—
326#380—1061no—
327#340—532no—
328#336—532no—
329#336—532no—
330C53#2418511yescharacteristic, complemented, Sylow
331C22×Q8C22⋊D5311yescharacteristic
332C22×Q8C22×D5312yes—
333C22×D4—531no—
334C22×D4C22×D5311yescharacteristic
335C23×C4—21no—
336C23×C4—1061no—
337C2×SD16—1064no—
338C2×SD16—1064no—
339C2×SD16—2122no—
340C2×D8—1062no—
341C2×D8—1062no—
342C2×D8—1062no—
343C22×C8—531no—
344C22×C8—532no—
345C22×C8—531no—
346C4⋊Q8—22no—
347C4⋊Q8C22⋊D5314yes—
348#126—216no—
349#124—24no—
350#124—24no—
351C22⋊Q8—28no—
352C22⋊Q8—2124no—
353C4×Q8—24no—
354C4×Q8—24no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×D4—22no—
358C4×D4—1062no—
359C4×D4—1062no—
360C4×D4C22⋊D5314yes—
361#117—28no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—532no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4C22⋊D5311yescharacteristic
371C2×C4⋊C4C22⋊D5312yes—
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—1062no—
377C2×C22⋊C4—1061no—
378C2×C22⋊C4—1062no—
379C2×C42—22no—
380C2×C42C22×D5312yes—
381C2×C42C22×D5311yescharacteristic
382#107—1062no—
383C4⋊C8—1062no—
384#103—1064no—
385#103—1064no—
386#103—2122no—
387#102—1062no—
388#102—1062no—
389#102—2122no—
390C4×C8—1062no—
391#95—1062no—
392C24—21no—
393C24—1061no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8C2×C22⋊D5314yes—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—532no—
409C2×D4—1061no—
410C2×D4—1061no—
411C2×D4—1061no—
412C2×D4—1061no—
413C2×D4—1061no—
414C2×D4C2×C22⋊D5312yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—1061no—
421C22×C4—1062no—
422C22×C4—2122no—
423C22×C4—2122no—
424C22×C4—2122no—
425C22×C4C2×C22⋊D5312yes—
426C22×C4C2×C22⋊D5312yes—
427C22×C4C2×C22⋊D5311yescharacteristic
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C23×D5311yescharacteristic
430SD16—2128no—
431D8—1068no—
432C2×C8—532no—
433C2×C8—1061no—
434C2×C8—1061no—
435C2×C8—1061no—
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—1062no—
439C2×C8—1061no—
440C2×C8—1061no—
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1062no—
450C4⋊C4—1062no—
451C4⋊C4—4241no—
452C4⋊C4—2124no—
453C4⋊C4—2122no—
454C4⋊C4C2×C22⋊D5312yes—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—4242no—
459C22⋊C4—2124no—
460C22⋊C4—2122no—
461C42—28no—
462C42—22no—
463C42—22no—
464C42C2×C22⋊D5312yes—
465C42C2×C22⋊D5312yes—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—1062no—
474C23—2121no—
475C23—2121no—
476C23—2121no—
477C23—2121no—
478C23—2121no—
479C23—2121no—
480C23C22×C22⋊D5311yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—44no—
484Q8—42no—
485Q8—42no—
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490D4—1062no—
491D4—1062no—
492D4—2121no—
493D4—2121no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—48no—
509C2×C4—4244no—
510C2×C4—2122no—
511C2×C4—2121no—
512C2×C4—2121no—
513C2×C4—2122no—
514C2×C4—2121no—
515C2×C4—2122no—
516C2×C4#1313318612yes—
517C2×C4#1313318612yes—
518C2×C4#1313323712yes—
519C2×C4#1313324212yes—
520C2×C4C22×C22⋊D5311yescharacteristic
521C2×C4C22×C22⋊D5311yescharacteristic
522C2×C4#41×D5311yescharacteristic
523C2×C4#41×D5311yescharacteristic
524C2×C4#1313326311yescharacteristic
525C2×C4#1313326411yescharacteristic
526C8—1062no—
527C8—2121no—
528C8—2122no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—4241no—
537C22—4241no—
538C22—4241no—
539C22—4241no—
540C22—2122no—
541C22—2122no—
542C22—2122no—
543C22C2×#1313318611yescharacteristic
544C22#1681128011yescharacteristic
545C22#1681129211yescharacteristic
546C22C2×#1313323711yescharacteristic
547C22C2×#1313324211yescharacteristic
548C22#1681143111yescharacteristic
549C22#1681143811yescharacteristic
550C4—22no—
551C4—42no—
552C4—42no—
553C4—44no—
554C4—42no—
555C4—4241no—
556C4—4242no—
557C4#1681076814yes—
558C4#1681113011yescharacteristic
559C4#1681120111yescharacteristic
560C4#1681126111yescharacteristic
561C4#1681127211yescharacteristic
562C2—82no—
563C2—4242no—
564C2#2027805411yescharacteristic
565C2C2×#1681076811yescharacteristic
566C2#2028403011yescharacteristic
567C2#2028736011yescharacteristic
568C2#2028742911yescharacteristic
569C2#2028769911yescharacteristic
570C2#2028773511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#556C2
socleC23×C53C22×D4
prime radicalC22×#11719922C22
hypercenterC2×#556D53
hypocenterC53#24185
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
1. C2×#16810825C2semidirect
3. #20279633C2semidirect
4. #20275951C2semidirect
6. #20275652C2semidirect
7. #20275518C2semidirect
8. #20275070C2semidirect
9. #20274969C2semidirect
10. #20274715C2semidirect
11. C2×C53×#556C2semidirect
277. C2×#556D53semidirect
330. C53#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#168108252
C2×C4×#117199222
#202796332
#202759512
#202757922
#202756522
#202755182
#202750702
#202749692
#202747152
C2×C53×#5562
#2418553