Finite Group Databasebeta

← #27418448

Subgroups — #27418448

Order 13568.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810915C211yesmaximal, characteristic, complemented
2C2×#16810830C211yesmaximal, characteristic, complemented
3C4×#13133196C211yesmaximal, characteristic
4#20275915C212yesmaximal
5#20275723C212yesmaximal
6#20275664C211yesmaximal, characteristic, complemented
7#20275639C211yesmaximal, characteristic, complemented
8#20275056C212yesmaximal, complemented
9#20274969C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic
11C2×C53×#533C211yesmaximal, characteristic
12C22×#11719925C2212yes—
13C2×#13133196C2211yescharacteristic
14C2×C4×Dic106C2211yescharacteristic
15#16810915—24no—
16#16810830—24no—
17C2×#13133150C2212yes—
18C2×C4⋊#88322C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133141C2212yes—
23C2×#13133140C2212yes—
24C2×C22⋊#88322C2212yes—
25#16810529C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810507—21no—
28#16810496—21no—
29#16810483C2211yescharacteristic
30C4×#11719919—21no—
31C4×C4⋊Dic53C2211yescharacteristic
32C4×C22⋊Dic53—21no—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C53×#533—24no—
37C53×#533—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×C4⋊Q8×C53C2211yescharacteristic
40C2×#125×C53—21no—
41C2×C22⋊Q8×C53—21no—
42C2×C22⋊Q8×C53C2212yes—
43C2×C4×Q8×C53—21no—
44C2×C4×Q8×C53C2212yes—
45C2×#117×C53—21no—
46C22×C42×C53C2211yescharacteristic
47C22×Dic106C2311yescharacteristic
48C2×#11719925—24no—
49C2×#11719925—24no—
50C2×#11719925—42no—
51C22×#88322C2312yes—
52C22×#88322C2312yes—
53#13133196—22no—
54#13133196—22no—
55C4×Dic106—22no—
56C4×Dic106—22no—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2311yescharacteristic
60C2×#11719919—21no—
61C2×#11719919—21no—
62C2×#11719919—21no—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×C4×Dic53—21no—
66C2×C4×Dic53—21no—
67#13133150—24no—
68C4⋊#88322—22no—
69C4⋊#88322—22no—
70#13133141—24no—
71#13133141—24no—
72#13133141—42no—
73#13133140—24no—
74C22⋊#88322—42no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78#13133124—21no—
79#13133124—21no—
80C22×Q8×C53—21no—
81C22×Q8×C53C2312yes—
82C23×C4×C53D411yescharacteristic
83C23×C4×C53D411yescharacteristic
84C23×C4×C53C2311yescharacteristic
85C4⋊Q8×C53—22no—
86C4⋊Q8×C53—22no—
87C4⋊Q8×C53D414yes—
88C4⋊Q8×C53D414yes—
89#125×C53—28no—
90C22⋊Q8×C53—28no—
91C22⋊Q8×C53—24no—
92C22⋊Q8×C53—24no—
93C4×Q8×C53—28no—
94C4×Q8×C53—24no—
95C4×Q8×C53—24no—
96#117×C53—28no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—22no—
102C2×C4⋊C4×C53C2312yes—
103C2×C22⋊C4×C53—22no—
104C2×C42×C53—22no—
105C2×C42×C53—22no—
106C2×C42×C53—22no—
107C2×C42×C53D411yescharacteristic
108C2×C42×C53D411yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C24×C53C2×D411yescharacteristic
112C2×Q8×C53—24no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—42no—
119C2×Q8×C53—42no—
120C2×Q8×C53C2×D414yes—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—41no—
133C22×C4×C53—41no—
134C22×C4×C53Q1614yescomplemented
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C2×Dic106—21no—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106#4112yes—
151C4⋊C4×C53—28no—
152C4⋊C4×C53—28no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—24no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53C2×D414yes—
158C22⋊C4×C53—28no—
159C42×C53—28no—
160C42×C53—22no—
161C42×C53—22no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53C2×D412yes—
165C42×C53C2×D412yes—
166C22⋊Dic53—24no—
167C22⋊Dic53—42no—
168#11719925—48no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53#4112yes—
175#11719919—24no—
176#11719919—42no—
177#11719919—42no—
178C4×Dic53—42no—
179C4×Dic53—42no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53C2×Q1612yes—
193C23×C53C22×D411yescharacteristic
194Q8×C53—24no—
195Q8×C53—24no—
196Q8×C53—48no—
197Q8×C53—42no—
198Q8×C53—42no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—28no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—44no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53#12012yes—
222C2×C4×C53#12012yes—
223C2×C4×C53C2×Q1612yes—
224C2×C4×C53C2×Q1612yes—
225C2×C4×C53#13712yes—
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53H(5,𝔽2)11yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244Dic106—22no—
245Dic106—22no—
246Dic106—41no—
247Dic106—41no—
248#88322—42no—
249#88322—42no—
250#24155—531nomaximal, Sylow
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53C2×#12011yescharacteristic
259C22×C53#52511yescharacteristic
260C22×C53#53711yescharacteristic
261C22×C53C22×Q1611yescharacteristic
262C22×C53C2×#13711yescharacteristic
263C22×C53#57511yescharacteristic
264C22×C53#57511yescharacteristic
265C4×C53—22no—
266C4×C53—22no—
267C4×C53—41no—
268C4×C53—44no—
269C4×C53—41no—
270C4×C53—44no—
271C4×C53#45114yes—
272C4×C53#54611yescharacteristic
273C4×C53#54711yescharacteristic
274C4×C53#54811yescharacteristic
275C4×C53#55011yescharacteristic
276Dic53—42no—
277Dic53—42no—
278Dic53—81no—
279Dic53—81no—
280C2×#533D5311yescharacteristic
281C2×#477—531no—
282C2×#462—531no—
283C4×C22⋊Q8—531no—
284#2036—532no—
285#2001—532no—
286#1987—531no—
287#1981—531no—
288#1878—532no—
289#1863—532no—
290#1797—531no—
291C2×C53—42no—
292C2×C53#240711yescharacteristic
293C2×C53C2×#45111yescharacteristic
294C2×C53#300511yescharacteristic
295C2×C53#310411yescharacteristic
296C2×C53#311611yescharacteristic
297C2×C53#324911yescharacteristic
298C2×C53#325211yescharacteristic
299C22×Q16—532no—
300#533—24no—
301#533—24no—
302C2×C4⋊Q8C2×D5311yescharacteristic
303C2×C4⋊Q8C2×D5311yescharacteristic
304C2×#125—21no—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8—531no—
307C2×C22⋊Q8C2×D5312yes—
308C2×C4×Q8—21no—
309C2×C4×Q8—531no—
310C2×C4×Q8C2×D5312yes—
311C2×#117—21no—
312C22×C42C2×D5311yescharacteristic
313#477—1064no—
314#462—1064no—
315C2×#107—532no—
316C2×C4⋊C8—531no—
317C2×C4⋊C8—531no—
318C2×#103—532no—
319C2×#103—532no—
320C2×#103—532no—
321C2×#103—532no—
322C2×C22⋊C8—532no—
323#391—531no—
324#389—531no—
325#387—1061no—
326#385—1061no—
327#380—531no—
328C4×C4⋊C4—531no—
329C4×C4⋊C4—1061no—
330C4×C22⋊C4—1061no—
331#340—532no—
332#340—532no—
333#336—532no—
334C53#2415511yescharacteristic, complemented, Sylow
335C22×Q8—21no—
336C22×Q8—531no—
337C22×Q8C22×D5312yes—
338C23×C4C22⋊D5311yescharacteristic
339C23×C4C22⋊D5311yescharacteristic
340C23×C4C22×D5311yescharacteristic
341C2×Q16—1064no—
342C2×Q16—1064no—
343C2×Q16—2122no—
344C22×C8—532no—
345C22×C8—532no—
346C4⋊Q8—22no—
347C4⋊Q8—22no—
348C4⋊Q8C22⋊D5314yes—
349C4⋊Q8C22⋊D5314yes—
350#125—28no—
351C22⋊Q8—28no—
352C22⋊Q8—24no—
353C22⋊Q8—24no—
354C22⋊Q8—1062no—
355C22⋊Q8—1062no—
356C4×Q8—28no—
357C4×Q8—24no—
358C4×Q8—24no—
359C4×Q8—1062no—
360C4×Q8—1062no—
361#117—28no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4C22×D5312yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—1061no—
376C2×C22⋊C4—1061no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—1061no—
381C2×C42—1061no—
382C2×C42C22⋊D5311yescharacteristic
383C2×C42C22⋊D5311yescharacteristic
384C2×C42C22×D5311yescharacteristic
385C2×C42C22×D5311yescharacteristic
386#107—1064no—
387C4⋊C8—1062no—
388C4⋊C8—1062no—
389#103—1064no—
390#103—1064no—
391#103—1064no—
392#103—2122no—
393C22⋊C8—2122no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397#95—1061no—
398#95—1061no—
399C24C2×C22⋊D5311yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—532no—
409C2×Q8—1061no—
410C2×Q8—1061no—
411C2×Q8—1061no—
412C2×Q8—1061no—
413C2×Q8—1061no—
414C2×Q8C2×C22⋊D5314yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4#1171992514yescomplemented
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C2×C22⋊D5311yescharacteristic
436C22×C4C2×C22⋊D5311yescharacteristic
437C22×C4C2×C22⋊D5311yescharacteristic
438C22×C4C23×D5311yescharacteristic
439Q16—2128no—
440C2×C8—1062no—
441C2×C8—1062no—
442C2×C8—1062no—
443C2×C8—1062no—
444C2×C8—1062no—
445C2×C8—1062no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—532no—
453C4⋊C4—1064no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—2122no—
460C4⋊C4—2122no—
461C4⋊C4C2×C22⋊D5314yes—
462C22⋊C4—28no—
463C22⋊C4—1064no—
464C22⋊C4—2122no—
465C42—28no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—2122no—
471C42—2122no—
472C42C2×C22⋊D5312yes—
473C42C2×C22⋊D5312yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23C2×#1171992512yes—
481C23C22×C22⋊D5311yescharacteristic
482Q8—24no—
483Q8—24no—
484Q8—48no—
485Q8—42no—
486Q8—42no—
487Q8—1062no—
488Q8—1062no—
489Q8—2121no—
490Q8—2121no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—28no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—1062no—
514C2×C4—1062no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4#1313318612yes—
526C2×C4#1313318612yes—
527C2×C4C2×#1171992512yes—
528C2×C4C2×#1171992512yes—
529C2×C4#1313324812yes—
530C2×C4C22×C22⋊D5311yescharacteristic
531C2×C4C22×C22⋊D5311yescharacteristic
532C2×C4#41×D5311yescharacteristic
533C2×C4#41×D5311yescharacteristic
534C2×C4#1313326311yescharacteristic
535C2×C4#1313326411yescharacteristic
536C8—2122no—
537C8—2122no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×#1313318611yescharacteristic
546C22#1681113011yescharacteristic
547C22#1681120111yescharacteristic
548C22C22×#1171992511yescharacteristic
549C22C2×#1313324811yescharacteristic
550C22#1681142111yescharacteristic
551C22#1681142111yescharacteristic
552C4—22no—
553C4—22no—
554C4—41no—
555C4—44no—
556C4—41no—
557C4—44no—
558C4—4241no—
559C4—4241no—
560C4—2122no—
561C4—2122no—
562C4#1681077914yes—
563C4#1681126111yescharacteristic
564C4#1681127211yescharacteristic
565C4#1681128011yescharacteristic
566C4#1681129211yescharacteristic
567C2—42no—
568C2#2027805411yescharacteristic
569C2C2×#1681077911yescharacteristic
570C2#2028395711yescharacteristic
571C2#2028475011yescharacteristic
572C2#2028487111yescharacteristic
573C2#2028605711yescharacteristic
574C2#2028608211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#533C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#533D53
hypocenterC53#24155
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#16810915C2semidirect
2. C2×#16810830C2semidirect
6. #20275664C2semidirect
7. #20275639C2semidirect
8. #20275056C2semidirect
9. #20274969C2semidirect
134. C22×C4×C53Q16semidirect
334. C53#24155semidirect
432. C22×C4#11719925semidirect

Maximal subgroups

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

subgroupindex
C2×#168109152
C2×#168108302
C4×#131331962
#202759152
#202757232
#202756642
#202756392
#202750562
#202749692
#202746952
C2×C53×#5332
#2415553