Finite Group Databasebeta

← #27411580

Subgroups — #27411580

Order 13568.

1902 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810663C211yesmaximal, characteristic, complemented
2C2×#16810628C211yesmaximal, characteristic, complemented
3C2×#16810628C211yesmaximal, characteristic, complemented
4C2×#16810587C211yesmaximal, characteristic, complemented
5#20277098C211yesmaximal, characteristic, complemented
6#20275901C214yesmaximal, complemented
7#20275894C214yesmaximal, complemented
8#20274434C211yesmaximal, characteristic, complemented
9C22×#121×C53C211yesmaximal, characteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14#16810663—24no—
15C2×#13133150C2212yescomplemented
16C2×#13133148C2212yescomplemented
17#16810628—24no—
18#16810628—24no—
19C2×#13133138C2212yescomplemented
20C2×#13133138C2214yescomplemented
21C2×#13133138C2212yescomplemented
22#16810587—24no—
23C2×C22⋊#88322C2214yescomplemented
24#16810486—21no—
25#16810483C2212yescomplemented
26#16810483C2212yescomplemented
27#16810479—21no—
28C4×C22⋊Dic53—22no—
29#16810385C2214yescomplemented
30C23×D4×C53—21no—
31C23×D4×C53C2211yescharacteristic
32C24×C4×C53C2211yescharacteristic
33C2×#121×C53—24no—
34C2×#121×C53—22no—
35C2×#121×C53—24no—
36C2×#121×C53C2214yes—
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×#11719917D414yescomplemented
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#13133138—28no—
63#13133138—24no—
64#13133138—24no—
65C22⋊#88322—28no—
66#13133124—22no—
67#13133124—22no—
68C25×C53—21no—
69C25×C53D411yescharacteristic
70C22×D4×C53—28no—
71C22×D4×C53—24no—
72C22×D4×C53—22no—
73C22×D4×C53—21no—
74C22×D4×C53—22no—
75C22×D4×C53—22no—
76C22×D4×C53—22no—
77C22×D4×C53—22no—
78C22×D4×C53—48no—
79C22×D4×C53D414yes—
80C22×D4×C53C2312yes—
81C23×C4×C53—21no—
82C23×C4×C53—22no—
83C23×C4×C53—22no—
84C23×C4×C53—22no—
85C23×C4×C53—22no—
86C23×C4×C53—22no—
87C23×C4×C53—22no—
88C23×C4×C53D411yescharacteristic
89C23×C4×C53D412yes—
90C23×C4×C53C2312yes—
91C23×C4×C53C2311yescharacteristic
92#121×C53—216no—
93#121×C53—28no—
94#121×C53—28no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D414yes—
98C2×C4⋊C4×C53C2312yes—
99C2×C22⋊C4×C53—28no—
100C2×C22⋊C4×C53—24no—
101C24×C53—22no—
102C24×C53—24no—
103C24×C53—24no—
104C24×C53—22no—
105C24×C53—22no—
106C24×C53—22no—
107C24×C53—42no—
108C24×C53—44no—
109C24×C53—44no—
110C24×C53—42no—
111C24×C53—42no—
112C24×C53C2×D412yes—
113C24×C53C2×D411yescharacteristic
114C2×D4×C53—216no—
115C2×D4×C53—24no—
116C2×D4×C53—28no—
117C2×D4×C53—28no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—432no—
124C2×D4×C53—48no—
125C2×D4×C53—48no—
126C2×D4×C53—44no—
127C2×D4×C53—42no—
128C2×D4×C53—42no—
129C2×D4×C53—44no—
130C2×D4×C53C2×D414yes—
131C22×C4×C53—22no—
132C22×C4×C53—24no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—24no—
137C22×C4×C53—24no—
138C22×C4×C53—24no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—44no—
146C22×C4×C53—44no—
147C22×C4×C53—44no—
148C22×C4×C53—44no—
149C22×C4×C53—44no—
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D412yes—
155C22×C4×C53#4112yes—
156C22×C4×C53#4112yes—
157C22×C4×C53#4112yes—
158C22×C4×C53C2411yescharacteristic
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—22no—
162C4⋊C4×C53—24no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53C2×D414yes—
165C22⋊C4×C53—216no—
166C22⋊Dic53—44no—
167C22⋊Dic53—42no—
168C22⋊Dic53—42no—
169C22⋊Dic53—44no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—22no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—22no—
176C4×Dic53—42no—
177C4×Dic53—42no—
178#11719917—28no—
179#11719917—44no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322C2×D414yescomplemented
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—28no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—24no—
196C23×C53—48no—
197C23×C53—48no—
198C23×C53—44no—
199C23×C53—48no—
200C23×C53—48no—
201C23×C53—44no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—44no—
205C23×C53—44no—
206C23×C53—48no—
207C23×C53—88no—
208C23×C53—88no—
209C23×C53#12012yes—
210C23×C53#12112yes—
211C23×C53#12312yes—
212C23×C53AGL(1,ℤ/8)14yes—
213C23×C53C22×D411yescharacteristic
214D4×C53—24no—
215D4×C53—24no—
216D4×C53—432no—
217D4×C53—42no—
218D4×C53—42no—
219D4×C53—48no—
220C2×C4×C53—22no—
221C2×C4×C53—24no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—48no—
225C2×C4×C53—44no—
226C2×C4×C53—44no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—48no—
232C2×C4×C53#12012yes—
233C2×C4×C53#12114yes—
234C2×C4×C53#12114yes—
235C2×C4×C53#12112yes—
236C2×C4×C53#12314yes—
237C2×C4×C53#12312yes—
238C2×C4×C53#12414yes—
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252#88322—24no—
253#88322—42no—
254#23480—531nomaximal, Sylow
255C22×C53—24no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—24no—
259C22×C53—24no—
260C22×C53—24no—
261C22×C53—24no—
262C22×C53—416no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—44no—
268C22×C53—44no—
269C22×C53—44no—
270C22×C53—816no—
271C22×C53—84no—
272C22×C53—84no—
273C22×C53#39414yes—
274C22×C53C2×#12011yescharacteristic
275C22×C53C2×#12111yescharacteristic
276C22×C53C2×#12311yescharacteristic
277C22×C53C2×AGL(1,ℤ/8)12yes—
278C22×C53#57512yes—
279C4×C53—42no—
280C4×C53—44no—
281C4×C53#39414yes—
282C4×C53C2×#12111yescharacteristic
283C4×C53C2×#12111yescharacteristic
284C4×C53C2×#12311yescharacteristic
285C4×C53C2×#12411yescharacteristic
286Dic53—81no—
287Dic53—81no—
288Dic53—82no—
289C22×#121D5311yescharacteristic
290C2×#428—531no—
291C2×#418—531no—
292C2×#418—531no—
293C2×#407—531no—
294#2280—531no—
295#2034—534no—
296#2033—534no—
297#1736—531no—
298C2×C53—24no—
299C2×C53—44no—
300C2×C53—84no—
301C2×C53C2×#39411yescharacteristic
302C2×C53#300311yescharacteristic
303C2×C53#300511yescharacteristic
304C2×C53#305512yes—
305C2×C53#309412yes—
306C23×D4—21no—
307C23×D4C2×D5311yescharacteristic
308C24×C4C2×D5311yescharacteristic
309C22×M16—531no—
310C22×M16—531no—
311C2×#121—24no—
312C2×#121—22no—
313C2×#121—24no—
314C2×#121C2×D5314yes—
315C2×#117—531no—
316C2×#117—531no—
317C22×C4⋊C4C2×D5311yescharacteristic
318C22×C22⋊C4—21no—
319#428—1064no—
320C2×#107—532no—
321C2×#106—532no—
322#418—1064no—
323#418—1064no—
324C2×#102—532no—
325C2×#102—534no—
326C2×#102—532no—
327#407—1064no—
328C2×C22⋊C8—534no—
329#381—1061no—
330#380—532no—
331#380—532no—
332#379—1061no—
333C4×C22⋊C4—1062no—
334#340—534no—
335C53#2348011yescharacteristic, complemented, Sylow
336C25—21no—
337C25C22⋊D5311yescharacteristic
338C22×D4—28no—
339C22×D4—24no—
340C22×D4—22no—
341C22×D4—22no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—21no—
345C22×D4—22no—
346C22×D4—48no—
347C22×D4C22⋊D5314yes—
348C22×D4C22×D5312yes—
349C23×C4—21no—
350C23×C4—22no—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—22no—
354C23×C4—22no—
355C23×C4—22no—
356C23×C4C22⋊D5311yescharacteristic
357C23×C4C22⋊D5312yes—
358C23×C4C22×D5312yes—
359C23×C4C22×D5311yescharacteristic
360C2×M16—534no—
361C2×M16—1062no—
362C2×M16—1062no—
363C2×M16—1062no—
364C2×M16—1062no—
365C2×M16—1062no—
366C22×C8—532no—
367C22×C8—532no—
368#121—216no—
369#121—28no—
370#121—28no—
371#117—1062no—
372#117—1062no—
373#117—1062no—
374#117—1062no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—532no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4C22⋊D5314yes—
381C2×C4⋊C4C22×D5312yes—
382C2×C22⋊C4—28no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1062no—
387C2×C22⋊C4—1062no—
388C2×C42—1061no—
389C2×C42—1061no—
390#107—1064no—
391#106—1064no—
392#102—1068no—
393#102—1064no—
394#102—1064no—
395C22⋊C8—1068no—
396#95—1062no—
397#95—1062no—
398C24—22no—
399C24—24no—
400C24—24no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—42no—
405C24—44no—
406C24—44no—
407C24—42no—
408C24—42no—
409C24C2×C22⋊D5312yes—
410C24C2×C22⋊D5311yescharacteristic
411C2×D4—216no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—432no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4C2×C22⋊D5314yes—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1062no—
450C22×C4D4×D5312yes—
451C22×C4#1171994712yes—
452C22×C4#1171994712yes—
453C22×C4#1171994712yes—
454C22×C4C2×C22⋊D5312yes—
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5312yes—
457C22×C4C2×C22⋊D5312yes—
458C22×C4C23×D5311yescharacteristic
459M16—1068no—
460M16—2124no—
461C2×C8—534no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8—1062no—
465C2×C8—1062no—
466C2×C8—1062no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—1062no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1062no—
475C4⋊C4C2×C22⋊D5314yes—
476C22⋊C4—216no—
477C22⋊C4—2124no—
478C22⋊C4—2122no—
479C22⋊C4—2122no—
480C22⋊C4—2124no—
481C42—2122no—
482C42—2122no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—48no—
494C23—48no—
495C23—44no—
496C23—48no—
497C23—48no—
498C23—44no—
499C23—48no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—88no—
505C23—88no—
506C23#1313318612yes—
507C23#1313318912yes—
508C23#1313319812yes—
509C23#1313324314yes—
510C23C22×C22⋊D5311yescharacteristic
511D4—24no—
512D4—24no—
513D4—432no—
514D4—48no—
515D4—42no—
516D4—42no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4#1313318612yes—
537C2×C4#1313318814yes—
538C2×C4#1313318912yes—
539C2×C4#1313318914yes—
540C2×C4#1313319812yes—
541C2×C4#1313319814yes—
542C2×C4#1313320414yes—
543C2×C4C2×D4×D5311yescharacteristic
544C2×C4C22×C22⋊D5311yescharacteristic
545C2×C4C22×C22⋊D5311yescharacteristic
546C2×C4C2×#1171994711yescharacteristic
547C2×C4C2×#1171994711yescharacteristic
548C2×C4C2×#1171994711yescharacteristic
549C8—1064no—
550C8—2122no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—816no—
567C22—84no—
568C22—84no—
569C22#1681053714yes—
570C22C2×#1313318611yescharacteristic
571C22C2×#1313318911yescharacteristic
572C22C2×#1313319811yescharacteristic
573C22C2×#1313324312yes—
574C22#1681142112yes—
575C4—44no—
576C4—42no—
577C4—4241no—
578C4—4241no—
579C4—4242no—
580C4#1681053714yes—
581C4C2×#1313318811yescharacteristic
582C4C2×#1313318911yescharacteristic
583C4C2×#1313319811yescharacteristic
584C4C2×#1313320411yescharacteristic
585C2—24no—
586C2—44no—
587C2—84no—
588C2C2×#1681053711yescharacteristic
589C2#2028394211yescharacteristic
590C2#2028395711yescharacteristic
591C2#2028433012yes—
592C2#2028465112yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC22×#121×C53C2
socleC23×C53C22×D4
hypercenterC22×#121D53
hypocenterC53#23480
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16810663C2semidirect
2. C2×#16810628C2semidirect
3. C2×#16810628C2semidirect
4. C2×#16810587C2semidirect
5. #20277098C2semidirect
6. #20275901C2semidirect
7. #20275894C2semidirect
8. #20274434C2semidirect
15. C2×#13133150C22semidirect
16. C2×#13133148C22semidirect
19. C2×#13133138C22semidirect
20. C2×#13133138C22semidirect
21. C2×#13133138C22semidirect
23. C2×C22⋊#88322C22semidirect
25. #16810483C22semidirect
26. #16810483C22semidirect
29. #16810385C22semidirect
44. C2×#11719917D4semidirect
185. C2×#88322C2×D4semidirect
335. C53#23480semidirect

Maximal subgroups

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

subgroupindex
C2×#168106632
C2×#168106282
C2×#168105872
#202770982
#202759012
#202758942
#202744342
C22×#121×C532
#2348053