Finite Group Databasebeta

← #27411589

Subgroups — #27411589

Order 13568.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810663C211yesmaximal, characteristic, complemented
2C2×#16810632C211yesmaximal, characteristic, complemented
3C2×#16810632C211yesmaximal, characteristic, complemented
4C2×#16810587C211yesmaximal, characteristic
5#20277098C211yesmaximal, characteristic
6#20275915C214yesmaximal, complemented
7#20275908C214yesmaximal, complemented
8#20274434C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8×C53C211yesmaximal, characteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14#16810663—24no—
15C2×#13133150C2212yescomplemented
16C2×#13133148C2212yescomplemented
17#16810632—24no—
18#16810632—24no—
19C2×#13133141C2212yescomplemented
20C2×#13133141C2214yescomplemented
21C2×#13133141C2212yescomplemented
22#16810587—24no—
23C2×C22⋊#88322C2214yes—
24#16810486—21no—
25#16810483C2212yes—
26#16810483C2212yes—
27#16810479—21no—
28C4×C22⋊Dic53—22no—
29#16810385C2214yescomplemented
30C23×Q8×C53C2211yescharacteristic
31C24×C4×C53C2211yescharacteristic
32C2×C22⋊Q8×C53—24no—
33C2×C22⋊Q8×C53—22no—
34C2×C22⋊Q8×C53—24no—
35C2×C22⋊Q8×C53C2214yes—
36C22×C4⋊C4×C53—21no—
37C22×C4⋊C4×C53C2211yescharacteristic
38C22×C22⋊C4×C53—21no—
39C2×#11719917—22no—
40C2×#11719917—22no—
41C2×#11719917—22no—
42C2×#11719917—22no—
43C2×#11719917—22no—
44C2×#11719917D414yes—
45C22×#88322C2312yes—
46C22×#88322C2312yes—
47#13133176—22no—
48#13133176—22no—
49#13133176—22no—
50#13133176—22no—
51C2×C4⋊Dic53C2312yes—
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C22⋊Dic53—21no—
55C2×C22⋊Dic53—21no—
56C2×C22⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C4×Dic53—21no—
59C2×C4×Dic53—21no—
60#13133150—24no—
61#13133148—24no—
62#13133141—28no—
63#13133141—24no—
64#13133141—24no—
65C22⋊#88322—28no—
66#13133124—22no—
67#13133124—22no—
68C25×C53D411yescharacteristic
69C22×Q8×C53—22no—
70C22×Q8×C53—21no—
71C22×Q8×C53—22no—
72C22×Q8×C53—22no—
73C22×Q8×C53—22no—
74C22×Q8×C53—22no—
75C22×Q8×C53D414yes—
76C22×Q8×C53C2312yes—
77C23×C4×C53—21no—
78C23×C4×C53—21no—
79C23×C4×C53—22no—
80C23×C4×C53—22no—
81C23×C4×C53—22no—
82C23×C4×C53—22no—
83C23×C4×C53—22no—
84C23×C4×C53—22no—
85C23×C4×C53D411yescharacteristic
86C23×C4×C53D412yes—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C22⋊Q8×C53—216no—
90C22⋊Q8×C53—28no—
91C22⋊Q8×C53—28no—
92C2×C4⋊C4×C53—28no—
93C2×C4⋊C4×C53—24no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53D414yes—
97C2×C4⋊C4×C53C2312yes—
98C2×C22⋊C4×C53—28no—
99C2×C22⋊C4×C53—24no—
100C24×C53—22no—
101C24×C53—24no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—42no—
105C24×C53C2×D412yes—
106C24×C53C2×D411yescharacteristic
107C2×Q8×C53—24no—
108C2×Q8×C53—22no—
109C2×Q8×C53—28no—
110C2×Q8×C53—28no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53—42no—
116C2×Q8×C53—44no—
117C2×Q8×C53—42no—
118C2×Q8×C53—44no—
119C2×Q8×C53C2×D414yes—
120C22×C4×C53—24no—
121C22×C4×C53—24no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—24no—
128C22×C4×C53—24no—
129C22×C4×C53—24no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—44no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53—44no—
140C22×C4×C53—44no—
141C22×C4×C53—44no—
142C22×C4×C53—44no—
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×D412yes—
148C22×C4×C53#4112yes—
149C22×C4×C53#4112yes—
150C22×C4×C53#4112yes—
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—22no—
155C4⋊C4×C53—216no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53C2×D414yes—
159C22⋊C4×C53—216no—
160C22⋊Dic53—44no—
161C22⋊Dic53—42no—
162C22⋊Dic53—42no—
163C22⋊Dic53—44no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—22no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—22no—
170C4×Dic53—42no—
171C4×Dic53—42no—
172#11719917—28no—
173#11719917—44no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322C2×D414yes—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—28no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—48no—
190C23×C53—48no—
191C23×C53—44no—
192C23×C53—44no—
193C23×C53#12012yes—
194C23×C53#12112yes—
195C23×C53#12312yes—
196C23×C53#13714yes—
197C23×C53C22×D411yescharacteristic
198Q8×C53—24no—
199Q8×C53—24no—
200Q8×C53—42no—
201Q8×C53—42no—
202Q8×C53—48no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—48no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—44no—
212C2×C4×C53—44no—
213C2×C4×C53—48no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—48no—
219C2×C4×C53#12012yes—
220C2×C4×C53#12114yes—
221C2×C4×C53#12114yes—
222C2×C4×C53#12112yes—
223C2×C4×C53#12314yes—
224C2×C4×C53#12312yes—
225C2×C4×C53#12414yes—
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239#88322—24no—
240#88322—42no—
241#23481—531nomaximal, Sylow
242C22×C53—24no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—24no—
246C22×C53—24no—
247C22×C53—24no—
248C22×C53—24no—
249C22×C53—416no—
250C22×C53—42no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53#39414yes—
255C22×C53C2×#12011yescharacteristic
256C22×C53C2×#12111yescharacteristic
257C22×C53C2×#12311yescharacteristic
258C22×C53C2×#13712yes—
259C22×C53#57512yes—
260C4×C53—42no—
261C4×C53—44no—
262C4×C53—44no—
263C4×C53#39414yes—
264C4×C53C2×#12111yescharacteristic
265C4×C53C2×#12111yescharacteristic
266C4×C53C2×#12311yescharacteristic
267C4×C53C2×#12411yescharacteristic
268Dic53—81no—
269Dic53—81no—
270Dic53—82no—
271C22×C22⋊Q8D5311yescharacteristic
272C2×#428—531no—
273C2×#419—531no—
274C2×#419—531no—
275C2×#407—531no—
276#2280—531no—
277#2036—534no—
278#2035—534no—
279#1736—531no—
280C2×C53—24no—
281C2×C53—44no—
282C2×C53C2×#39411yescharacteristic
283C2×C53#300411yescharacteristic
284C2×C53#300511yescharacteristic
285C2×C53#305612yes—
286C2×C53#309512yes—
287C23×Q8C2×D5311yescharacteristic
288C24×C4C2×D5311yescharacteristic
289C22×M16—531no—
290C22×M16—531no—
291C2×C22⋊Q8—24no—
292C2×C22⋊Q8—22no—
293C2×C22⋊Q8—24no—
294C2×C22⋊Q8C2×D5314yes—
295C2×#117—531no—
296C2×#117—531no—
297C22×C4⋊C4—21no—
298C22×C4⋊C4C2×D5311yescharacteristic
299C22×C22⋊C4—21no—
300#428—1064no—
301C2×#107—532no—
302C2×#106—532no—
303#419—1064no—
304#419—1064no—
305C2×#103—532no—
306C2×#103—534no—
307C2×#103—532no—
308#407—1064no—
309C2×C22⋊C8—534no—
310#381—1061no—
311#380—532no—
312#380—532no—
313#379—1061no—
314C4×C22⋊C4—1062no—
315#340—534no—
316C53#2348111yescharacteristic, complemented, Sylow
317C25C22⋊D5311yescharacteristic
318C22×Q8—22no—
319C22×Q8—21no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8C22⋊D5314yes—
325C22×Q8C22×D5312yes—
326C23×C4—21no—
327C23×C4—21no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4C22⋊D5311yescharacteristic
335C23×C4C22⋊D5312yes—
336C23×C4C22×D5312yes—
337C23×C4C22×D5311yescharacteristic
338C2×M16—534no—
339C2×M16—1062no—
340C2×M16—1062no—
341C2×M16—1062no—
342C2×M16—1062no—
343C2×M16—1062no—
344C22×C8—532no—
345C22×C8—532no—
346C22⋊Q8—216no—
347C22⋊Q8—28no—
348C22⋊Q8—28no—
349#117—1062no—
350#117—1062no—
351#117—1062no—
352#117—1062no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—532no—
358C2×C4⋊C4—531no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4C22⋊D5314yes—
361C2×C4⋊C4C22×D5312yes—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—1061no—
365C2×C22⋊C4—1061no—
366C2×C22⋊C4—1062no—
367C2×C22⋊C4—1062no—
368C2×C42—1061no—
369C2×C42—1061no—
370#107—1064no—
371#106—1064no—
372#103—1068no—
373#103—1064no—
374#103—1064no—
375C22⋊C8—1068no—
376#95—1062no—
377#95—1062no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—22no—
382C24—42no—
383C24C2×C22⋊D5312yes—
384C24C2×C22⋊D5311yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×C22⋊D5314yes—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1062no—
424C22×C4D4×D5312yes—
425C22×C4#1171994712yes—
426C22×C4#1171994712yes—
427C22×C4#1171994712yes—
428C22×C4C2×C22⋊D5312yes—
429C22×C4C2×C22⋊D5311yescharacteristic
430C22×C4C2×C22⋊D5312yes—
431C22×C4C2×C22⋊D5312yes—
432C22×C4C23×D5311yescharacteristic
433M16—1068no—
434M16—2124no—
435C2×C8—534no—
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—1062no—
439C2×C8—1062no—
440C2×C8—1062no—
441C4⋊C4—216no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—1062no—
445C4⋊C4—1061no—
446C4⋊C4—1061no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1062no—
450C4⋊C4C2×C22⋊D5314yes—
451C22⋊C4—216no—
452C22⋊C4—2124no—
453C22⋊C4—2122no—
454C22⋊C4—2122no—
455C22⋊C4—2124no—
456C42—2122no—
457C42—2122no—
458C23—22no—
459C23—22no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—48no—
468C23—48no—
469C23—44no—
470C23—44no—
471C23#1313318612yes—
472C23#1313318912yes—
473C23#1313319812yes—
474C23#1313324814yes—
475C23C22×C22⋊D5311yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—48no—
479Q8—42no—
480Q8—42no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—2121no—
498C2×C4—2121no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2122no—
504C2×C4#1313318612yes—
505C2×C4#1313318814yes—
506C2×C4#1313318912yes—
507C2×C4#1313318914yes—
508C2×C4#1313319812yes—
509C2×C4#1313319814yes—
510C2×C4#1313320414yes—
511C2×C4C2×D4×D5311yescharacteristic
512C2×C4C22×C22⋊D5311yescharacteristic
513C2×C4C22×C22⋊D5311yescharacteristic
514C2×C4C2×#1171994711yescharacteristic
515C2×C4C2×#1171994711yescharacteristic
516C2×C4C2×#1171994711yescharacteristic
517C8—1064no—
518C8—2122no—
519C22—22no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—416no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22#1681053714yes—
532C22C2×#1313318611yescharacteristic
533C22C2×#1313318911yescharacteristic
534C22C2×#1313319811yescharacteristic
535C22C2×#1313324812yes—
536C22#1681142112yes—
537C4—44no—
538C4—42no—
539C4—44no—
540C4—4241no—
541C4—4241no—
542C4—4242no—
543C4#1681053714yes—
544C4C2×#1313318811yescharacteristic
545C4C2×#1313318911yescharacteristic
546C4C2×#1313319811yescharacteristic
547C4C2×#1313320411yescharacteristic
548C2—24no—
549C2—44no—
550C2C2×#1681053711yescharacteristic
551C2#2028394911yescharacteristic
552C2#2028395711yescharacteristic
553C2#2028434112yes—
554C2#2028465912yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC25×C53D4
hypercenterC22×C22⋊Q8D53
hypocenterC53#23481
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16810663C2semidirect
2. C2×#16810632C2semidirect
3. C2×#16810632C2semidirect
6. #20275915C2semidirect
7. #20275908C2semidirect
8. #20274434C2semidirect
15. C2×#13133150C22semidirect
16. C2×#13133148C22semidirect
19. C2×#13133141C22semidirect
20. C2×#13133141C22semidirect
21. C2×#13133141C22semidirect
29. #16810385C22semidirect
316. C53#23481semidirect

Maximal subgroups

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

subgroupindex
C2×#168106632
C2×#168106322
C2×#168105872
#202770982
#202759152
#202759082
#202744342
C22×C22⋊Q8×C532
#2348153