Finite Group Databasebeta

← #27417851

Subgroups — #27417851

Order 13568.

1146 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×#16810939C211yesmaximal, characteristic, complemented
2C2×#16810666C211yesmaximal, characteristic, complemented
3#20279023C211yesmaximal, characteristic, complemented
4#20276045C212yesmaximal, complemented
5#20275901C212yesmaximal, complemented
6#20275154C211yesmaximal, characteristic, complemented
7#20275013C211yesmaximal, characteristic, complemented
8#20274962C212yesmaximal, complemented
9#20274914C212yesmaximal, complemented
10#20274810C211yesmaximal, characteristic, complemented
11C2×C53×#555C211yesmaximal, characteristic
12C22×#11719917C2211yescharacteristic
13C2×#13133198—21no—
14C22×C4⋊Dic53C2211yescharacteristic
15#16810939—24no—
16#16810666—24no—
17C2×#13133150C2212yes—
18C2×#13133148C2212yes—
19C2×#13133148C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2212yes—
24#16810553—21no—
25#16810548—21no—
26#16810521C2212yes—
27#16810484—21no—
28#16810483C2211yescharacteristic
29#16810483C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#555—24no—
35C53×#555—24no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#124×C53—21no—
38C2×#124×C53—21no—
39C2×#121×C53—21no—
40C2×#121×C53C2211yescharacteristic
41C2×#121×C53C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×#117×C53—21no—
45C2×#117×C53C2211yescharacteristic
46C2×#117×C53C2211yescharacteristic
47C23×Dic53—21no—
48C2×#11719917—24no—
49C2×#11719917—22no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52#13133198—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2312yes—
58C2×#11719919—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—21no—
61#13133150—24no—
62#13133148—24no—
63#13133148—24no—
64#13133138—24no—
65#13133138—24no—
66C22⋊#88322—24no—
67C22⋊#88322—22no—
68#13133124—22no—
69#13133124—21no—
70#13133124—22no—
71C22×Q8×C53—21no—
72C22×D4×C53—21no—
73C22×D4×C53—21no—
74C22×D4×C53D411yescharacteristic
75C22×D4×C53C2311yescharacteristic
76C23×C4×C53—21no—
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2311yescharacteristic
79C4⋊Q8×C53—22no—
80C4⋊Q8×C53—22no—
81#124×C53—28no—
82#124×C53—28no—
83#121×C53—28no—
84#121×C53—22no—
85#121×C53—22no—
86#121×C53—22no—
87#121×C53D414yes—
88C4×D4×C53—22no—
89C4×D4×C53—22no—
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92#117×C53—28no—
93#117×C53—22no—
94#117×C53—22no—
95#117×C53—22no—
96#117×C53D414yes—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C42×C53—21no—
110C2×C42×C53—21no—
111C2×C42×C53C2311yescharacteristic
112C24×C53—21no—
113C24×C53C2×D411yescharacteristic
114C24×C53C2×D411yescharacteristic
115C2×Q8×C53—24no—
116C2×Q8×C53—82no—
117C2×D4×C53—22no—
118C2×D4×C53—24no—
119C2×D4×C53—24no—
120C2×D4×C53—22no—
121C2×D4×C53—21no—
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—41no—
125C2×D4×C53—82no—
126C2×D4×C53—82no—
127C2×D4×C53C2×D412yes—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—24no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—42no—
139C22×C4×C53—42no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53C2×D412yes—
158C4⋊C4×C53C2×D412yes—
159C4⋊C4×C53C2×D412yes—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—24no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—22no—
167C42×C53—24no—
168C42×C53—24no—
169C42×C53—21no—
170C42×C53—21no—
171C22⋊Dic53—44no—
172C22⋊Dic53—42no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177C4⋊Dic53—42no—
178#11719919—44no—
179#11719917—28no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322#4114yes—
186C23×C53—22no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—22no—
190C23×C53—21no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—22no—
196C23×C53—42no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—42no—
200C23×C53#12012yes—
201C23×C53C22×D411yescharacteristic
202Q8×C53—84no—
203D4×C53—24no—
204D4×C53—44no—
205D4×C53—41no—
206D4×C53—41no—
207D4×C53—84no—
208D4×C53—84no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—24no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—48no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53#12012yes—
231C2×C4×C53AGL(1,ℤ/8)12yes—
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4112yes—
235C2×C4×C532−1+412yes—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—84no—
243#88322—24no—
244#88322—42no—
245#24104—531nomaximal, Sylow
246C22×C53—22no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—42no—
250C22×C53—42no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—41no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—42no—
259C22×C53—82no—
260C22×C53—82no—
261C22×C53C2×#12011yescharacteristic
262C22×C53#54712yes—
263C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C53#57511yescharacteristic
265C22×C53#57712yes—
266C4×C53—22no—
267C4×C53—22no—
268C4×C53—41no—
269C4×C53—42no—
270C4×C53—41no—
271C4×C53—42no—
272C4×C53—42no—
273C4×C53#52511yescharacteristic
274C4×C53#53611yescharacteristic
275C4×C53#54712yes—
276Dic53—82no—
277Dic53—82no—
278C2×#555D5311yescharacteristic
279C2×#481—531no—
280C2×#428—531no—
281#2486—531no—
282#2058—532no—
283#2034—532no—
284#1895—531no—
285#1870—531no—
286#1862—532no—
287#1854—532no—
288#1825—531no—
289C2×C53—42no—
290C2×C53—81no—
291C2×C53—82no—
292C2×C53#240611yescharacteristic
293C2×C53#300611yescharacteristic
294C2×C53#300911yescharacteristic
295C2×C53#334812yes—
296C2×C53#335412yes—
297C22×M16—531no—
298#555—24no—
299#555—24no—
300C2×C4⋊Q8C2×D5311yescharacteristic
301C2×#124—21no—
302C2×#124—21no—
303C2×#123—1061no—
304C2×#121—21no—
305C2×#121C2×D5311yescharacteristic
306C2×#121C2×D5311yescharacteristic
307C2×C4×D4C2×D5311yescharacteristic
308C2×C4×D4C2×D5311yescharacteristic
309C2×#117—21no—
310C2×#117C2×D5311yescharacteristic
311C2×#117C2×D5311yescharacteristic
312C22×C4⋊C4—531no—
313#481—1064no—
314#428—1064no—
315C2×#107—532no—
316C2×#106—532no—
317C2×#106—532no—
318C2×#102—532no—
319C2×#102—532no—
320C2×C22⋊C8—531no—
321C2×C22⋊C8—532no—
322#398—1061no—
323#397—1061no—
324#389—532no—
325#381—1061no—
326#380—531no—
327#380—532no—
328#340—532no—
329#340—532no—
330#340—532no—
331#336—532no—
332C53#2410411yescharacteristic, complemented, Sylow
333C22×Q8—21no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C22⋊D5311yescharacteristic
337C22×D4C22×D5311yescharacteristic
338C23×C4—21no—
339C23×C4—1061no—
340C23×C4C22×D5311yescharacteristic
341C23×C4C22×D5311yescharacteristic
342C2×M16—1064no—
343C2×M16—1062no—
344C22×C8—532no—
345C22×C8—532no—
346C4⋊Q8—22no—
347C4⋊Q8—22no—
348#124—28no—
349#124—28no—
350#123—2124no—
351#121—28no—
352#121—22no—
353#121—22no—
354#121—22no—
355#121C22⋊D5314yes—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360#117—28no—
361#117—22no—
362#117—22no—
363#117—22no—
364#117C22⋊D5314yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—532no—
367C2×C4⋊C4—532no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4C22⋊D5311yescharacteristic
373C2×C4⋊C4C22×D5311yescharacteristic
374C2×C4⋊C4C22×D5311yescharacteristic
375C2×C4⋊C4C22×D5311yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4C22⋊D5311yescharacteristic
384C2×C22⋊C4C22⋊D5311yescharacteristic
385C2×C42—21no—
386C2×C42—21no—
387C2×C42C22×D5311yescharacteristic
388#107—1064no—
389#106—1064no—
390#106—1064no—
391#102—1064no—
392#102—1064no—
393C22⋊C8—1064no—
394C22⋊C8—1062no—
395#95—1062no—
396#95—1061no—
397#95—1062no—
398C24—21no—
399C24C2×C22⋊D5311yescharacteristic
400C24C2×C22⋊D5311yescharacteristic
401C2×Q8—24no—
402C2×Q8—82no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—41no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4C2×C22⋊D5312yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—2122no—
429C22×C4—2122no—
430C22×C4—2122no—
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5312yes—
435C22×C4C2×C22⋊D5312yes—
436C22×C4C2×C22⋊D5311yescharacteristic
437C22×C4C23×D5311yescharacteristic
438M16—1068no—
439C2×C8—534no—
440C2×C8—1062no—
441C2×C8—1062no—
442C2×C8—1062no—
443C2×C8—1062no—
444C2×C8—1062no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—1062no—
451C4⋊C4—1062no—
452C4⋊C4—1062no—
453C4⋊C4—1062no—
454C4⋊C4—2122no—
455C4⋊C4—2124no—
456C4⋊C4C2×C22⋊D5312yes—
457C4⋊C4C2×C22⋊D5312yes—
458C4⋊C4C2×C22⋊D5312yes—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—22no—
464C22⋊C4—24no—
465C22⋊C4—22no—
466C22⋊C4—2122no—
467C22⋊C4—2124no—
468C42—24no—
469C42—24no—
470C42—21no—
471C42—21no—
472C23—22no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—22no—
477C23—21no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23#1313318612yes—
487C23C22×C22⋊D5311yescharacteristic
488Q8—84no—
489D4—24no—
490D4—44no—
491D4—41no—
492D4—41no—
493D4—84no—
494D4—84no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—48no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—4244no—
517C2×C4—2122no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4#1313318612yes—
524C2×C4#1313324212yes—
525C2×C4C22×C22⋊D5311yescharacteristic
526C2×C4C22×C22⋊D5311yescharacteristic
527C2×C4#41×D5312yes—
528C2×C4#1313326412yes—
529C8—1064no—
530C8—2122no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—42no—
538C22—42no—
539C22—41no—
540C22—42no—
541C22—41no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22C2×#1313318611yescharacteristic
547C22#1681127212yes—
548C22C2×#1313324211yescharacteristic
549C22#1681142111yescharacteristic
550C22#1681143412yes—
551C4—22no—
552C4—22no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—4242no—
559C4—4242no—
560C4#1681113011yescharacteristic
561C4#1681119411yescharacteristic
562C4#1681127212yes—
563C2—42no—
564C2—81no—
565C2—82no—
566C2#2027804611yescharacteristic
567C2#2028396711yescharacteristic
568C2#2028399211yescharacteristic
569C2#2028723012yes—
570C2#2028730412yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#555C2
socleC23×C53C22×D4
hypercenterC2×#555D53
hypocenterC53#24104
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#16810939C2semidirect
2. C2×#16810666C2semidirect
3. #20279023C2semidirect
4. #20276045C2semidirect
5. #20275901C2semidirect
6. #20275154C2semidirect
7. #20275013C2semidirect
8. #20274962C2semidirect
9. #20274914C2semidirect
10. #20274810C2semidirect
332. C53#24104semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168106662
#202790232
#202760452
#202759012
#202751542
#202750132
#202749622
#202749142
#202748102
C2×C53×#5552
#2410453