Finite Group Databasebeta

← #27274090

Subgroups — #27274090

Order 13568.

1054 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810585C211yesmaximal, characteristic
2C2×#16810485C211yesmaximal, characteristic
3#380×D53C211yesmaximal, characteristic
4#20272595C211yesmaximal, characteristic, complemented
5#20272594C211yesmaximal, characteristic, complemented
6#20272593C211yesmaximal, characteristic, complemented
7C53×#1294C211yesmaximal, characteristic, complemented
8C23×C4×D53C2211yescharacteristic
9C22×#11719912—21no—
10C2×C4⋊C4×D53C412yescomplemented
11C22×#11719921—21no—
12C2×C22⋊C4×D53—21no—
13#16810585—22no—
14#16810585C414yescomplemented
15C2×C22⋊#88322C2211yescharacteristic
16C2×#13133129—21no—
17#16810485—22no—
18#16810485C414yescomplemented
19#16810483C2211yescharacteristic
20#16810482—21no—
21#16810481—21no—
22#16810480C412yescomplemented
23C2×#13133124—21no—
24C2×#13133124C2211yescharacteristic
25#95×D53—21no—
26C2×C22⋊C8×C53C2211yescharacteristic
27C53×#380C2211yescharacteristic
28C2×#95×C53C2211yescharacteristic
29C24×D53D411yescharacteristic
30C23×Dic53—21no—
31C23×Dic53D411yescharacteristic
32C22×C4×D53—22no—
33C22×C4×D53—44no—
34C22×C4×D53C2×C411yescharacteristic
35C22×C4×D53C2×C412yes—
36C22×C4×D53C2×C411yescharacteristic
37C22×C4×D53C2×C412yes—
38C22×C4×D53D411yescharacteristic
39C22×C4×D53D411yescharacteristic
40C22×C4×D53D412yes—
41C22×C4×D53Q812yes—
42C2×#11719912—24no—
43C2×#11719912—44no—
44C22×#88322—21no—
45C2×C4⋊Dic53C2×C412yes—
46C2×#11719919—22no—
47C4⋊C4×D53—44no—
48C2×C22⋊Dic53—21no—
49C2×#11719921—22no—
50C2×#11719921—24no—
51C2×#11719921—44no—
52C22⋊C4×D53—42no—
53C22⋊C4×D53—42no—
54C22⋊#88322—21no—
55C22⋊#88322C2×C412yescomplemented
56#13133129—24no—
57#13133124—21no—
58#13133124—21no—
59#13133124—22no—
60#13133124—24no—
61#13133124C2×C412yescomplemented
62C23×C4×C53—21no—
63C23×C4×C53C2311yescharacteristic
64C22×C8×C53—21no—
65C2×C4⋊C4×C53C2×C412yes—
66C2×C22⋊C4×C53—21no—
67C22⋊C8×C53—21no—
68C22⋊C8×C53C2×C412yescomplemented
69#95×C53—21no—
70#95×C53—21no—
71#95×C53C2×C412yescomplemented
72C24×C53C2×D411yescharacteristic
73C23×D53—21no—
74C23×D53—21no—
75C23×D53—21no—
76C23×D53—21no—
77C23×D53—24no—
78C23×D53—42no—
79C23×D53C22⋊C411yescharacteristic
80C23×D53C22⋊C412yes—
81C23×D53C22⋊C411yescharacteristic
82C23×D53C4⋊C412yes—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C22×C411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×Q811yescharacteristic
96C22×Dic53—22no—
97C22×Dic53—22no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53C22⋊C411yescharacteristic
104C22×Dic53C22⋊C411yescharacteristic
105C22×Dic53C22⋊C412yes—
106C22×Dic53C4⋊C412yes—
107C2×C4×D53—24no—
108C2×C4×D53—24no—
109C2×C4×D53—44no—
110C2×C4×D53—44no—
111C2×C4×D53—44no—
112C2×C4×D53—44no—
113C2×C4×D53—42no—
114C2×C4×D53—42no—
115C2×C4×D53—44no—
116C2×C4×D53—44no—
117C2×C4×D53—44no—
118C2×C4×D53C4212yes—
119C2×C4×D53C4212yes—
120C2×C4×D53C22⋊C412yes—
121C2×C4×D53C22⋊C412yes—
122C2×C4×D53C4⋊C412yes—
123C2×C4×D53C4⋊C412yes—
124C2×C4×D53C4⋊C412yes—
125C2×C4×D53C4⋊C412yes—
126C2×C8×C53—22no—
127C2×C8×C53—42no—
128C4⋊C4×C53—42no—
129C22⋊C4×C53—41no—
130C22⋊C4×C53—41no—
131C22⋊Dic53—41no—
132C22⋊Dic53—41no—
133#11719921—44no—
134#11719921—48no—
135C4⋊Dic53—42no—
136#11719919—44no—
137C2×#88322—22no—
138C2×#88322—42no—
139#11719912—48no—
140C23×C53—22no—
141C23×C53—22no—
142C23×C53C2×C22⋊C411yescharacteristic
143C23×C53C2×C22⋊C411yescharacteristic
144C23×C53C2×C4⋊C411yescharacteristic
145C23×C53AGL(1,ℤ/8)11yescharacteristic
146C23×C53AGL(1,ℤ/8)11yescharacteristic
147C23×C53#13711yescharacteristic
148C23×C53#13711yescharacteristic
149C22×D53—21no—
150C22×D53—21no—
151C22×D53—22no—
152C22×D53—22no—
153C22×D53—22no—
154C22×D53—22no—
155C22×D53—21no—
156C22×D53—21no—
157C22×D53—22no—
158C22×D53—22no—
159C22×D53—22no—
160C22×D53—24no—
161C22×D53—24no—
162C22×D53—42no—
163C22×D53—42no—
164C22×D53—44no—
165C22×D53—44no—
166C22×D53—44no—
167C22×D53—44no—
168C22×D53#9512yes—
169C22×D53#9512yes—
170C22×D53#9914yes—
171C2×C4×C53—22no—
172C2×C4×C53—22no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—82no—
184C2×C4×C53—82no—
185C2×C4×C53C2×C4211yescharacteristic
186C2×C4×C53C2×C22⋊C411yescharacteristic
187C2×C4×C53C2×C4⋊C411yescharacteristic
188C2×C4×C53C2×C4⋊C411yescharacteristic
189C2×Dic53—22no—
190C2×Dic53—22no—
191C2×Dic53—42no—
192C2×Dic53—42no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—82no—
200C2×Dic53—82no—
201C2×Dic53#9512yes—
202C2×Dic53#9512yes—
203C4×D53—24no—
204C4×D53—24no—
205C4×D53—24no—
206C4×D53—24no—
207C4×D53—42no—
208C4×D53—42no—
209C4×D53—42no—
210C4×D53—42no—
211C4×D53—88no—
212C8×C53—42no—
213#88322—42no—
214#8015—531nomaximal, Sylow
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—22no—
220C22×C53—22no—
221C22×C53—22no—
222C22×C53—22no—
223C22×C53C2×#9511yescharacteristic
224C22×C53C2×#9912yes—
225C22×C53#41711yescharacteristic
226C22×C53#41711yescharacteristic
227C22×C53#41811yescharacteristic
228C22×C53#41911yescharacteristic
229C22×C53#42112yes—
230C22×C53#42811yescharacteristic
231C22×C53#42811yescharacteristic
232C2×D53—22no—
233C2×D53—22no—
234C2×D53—24no—
235C2×D53—24no—
236C2×D53—22no—
237C2×D53—22no—
238C2×D53—22no—
239C2×D53—44no—
240C2×D53—44no—
241C2×D53—44no—
242C2×D53—44no—
243C2×D53—42no—
244C2×D53—42no—
245C2×D53#34214yes—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—82no—
251C4×C53—82no—
252Dic53—22no—
253Dic53—22no—
254Dic53—82no—
255Dic53—82no—
256C2×#407—531no—
257C2×#381—531no—
258C2×#380—531no—
259#1294—531no—
260#1294—531no—
261#1294—531no—
262#1294D5311yescharacteristic, complemented
263C2×C53—21no—
264C2×C53—21no—
265C2×C53—21no—
266C2×C53—21no—
267C2×C53#151012yes—
268C2×C53#151112yes—
269C2×C53#173311yescharacteristic
270C2×C53#173611yescharacteristic
271C2×C53C2×#34211yescharacteristic
272D53—24no—
273D53—42no—
274C24×C4—531no—
275C22×M16—1061no—
276C22×C4⋊C4—532no—
277C22×C22⋊C4—1061no—
278C22×C22⋊C4—1061no—
279#407—534no—
280#407—1062no—
281C2×C22⋊C8—531no—
282C2×C22⋊C8—1061no—
283C2×C22⋊C8C2×D5311yescharacteristic
284#381—534no—
285#381—1062no—
286#380—531no—
287#380—532no—
288#380—1061no—
289#380—1061no—
290#380C2×D5311yescharacteristic
291C2×#95—531no—
292C2×#95—1061no—
293C2×#95—1061no—
294C2×#95C2×D5311yescharacteristic
295C53#801511yescharacteristic, complemented, Sylow
296C25—531no—
297C23×C4—21no—
298C23×C4—531no—
299C23×C4—531no—
300C23×C4—531no—
301C23×C4—531no—
302C23×C4—532no—
303C23×C4—532no—
304C23×C4—531no—
305C23×C4—532no—
306C23×C4—532no—
307C23×C4—1062no—
308C23×C4—1061no—
309C23×C4—2124no—
310C23×C4C22×D5311yescharacteristic
311C2×M16—1064no—
312C2×M16—2124no—
313C22×C8—21no—
314C22×C8—1061no—
315C2×C4⋊C4—532no—
316C2×C4⋊C4—1062no—
317C2×C4⋊C4—2124no—
318C2×C4⋊C4C4×D5312yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—1062no—
321C2×C22⋊C4—1064no—
322C2×C22⋊C4—1061no—
323C2×C22⋊C4—2122no—
324C2×C22⋊C4—2122no—
325C2×C22⋊C4—2124no—
326C22⋊C8—21no—
327C22⋊C8—532no—
328C22⋊C8—1064no—
329C22⋊C8—1061no—
330C22⋊C8C4×D5312yescomplemented
331#95—21no—
332#95—21no—
333#95—532no—
334#95—1062no—
335#95—1064no—
336#95—1061no—
337#95—1061no—
338#95C4×D5312yescomplemented
339C24—531no—
340C24—531no—
341C24—532no—
342C24—532no—
343C24—1064no—
344C24—1061no—
345C24—1061no—
346C24—1061no—
347C24—1061no—
348C24—2122no—
349C24D4×D5311yescharacteristic
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—531no—
358C22×C4—531no—
359C22×C4—532no—
360C22×C4—532no—
361C22×C4—532no—
362C22×C4—532no—
363C22×C4—532no—
364C22×C4—532no—
365C22×C4—532no—
366C22×C4—532no—
367C22×C4—532no—
368C22×C4—532no—
369C22×C4—1064no—
370C22×C4—1064no—
371C22×C4—1062no—
372C22×C4—1062no—
373C22×C4—2122no—
374C22×C4—2122no—
375C22×C4—2124no—
376C22×C4—2124no—
377C22×C4—2124no—
378C22×C4—2124no—
379C22×C4—2124no—
380C22×C4—2124no—
381C22×C4—2124no—
382C22×C4—2122no—
383C22×C4—2122no—
384C22×C4—2122no—
385C22×C4—2121no—
386C22×C4—2121no—
387C22×C4C2×C4×D5311yescharacteristic
388C22×C4C2×C4×D5311yescharacteristic
389C22×C4C2×C4×D5311yescharacteristic
390C22×C4D4×D5311yescharacteristic
391C22×C4D4×D5311yescharacteristic
392C22×C4Q8×D5311yescharacteristic
393M16—2128no—
394C2×C8—22no—
395C2×C8—42no—
396C2×C8—1062no—
397C2×C8—2122no—
398C4⋊C4—42no—
399C4⋊C4—2122no—
400C4⋊C4—2124no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—2128no—
404C22⋊C4—2124no—
405C22⋊C4—2121no—
406C22⋊C4—2121no—
407C23—22no—
408C23—22no—
409C23—534no—
410C23—532no—
411C23—532no—
412C23—1064no—
413C23—1064no—
414C23—1061no—
415C23—1061no—
416C23—1062no—
417C23—1062no—
418C23—1062no—
419C23—1062no—
420C23—1061no—
421C23—1061no—
422C23—1062no—
423C23—1062no—
424C23—1062no—
425C23—2122no—
426C23—2122no—
427C23—2124no—
428C23—2124no—
429C23—2124no—
430C23—2124no—
431C23C22⋊C4×D5311yescharacteristic
432C23C22⋊C4×D5311yescharacteristic
433C23C4⋊C4×D5311yescharacteristic
434C23#1313324111yescharacteristic
435C23#1313324411yescharacteristic
436C23#1313324611yescharacteristic
437C23#1313324911yescharacteristic
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—532no—
453C2×C4—532no—
454C2×C4—1064no—
455C2×C4—1064no—
456C2×C4—1064no—
457C2×C4—1064no—
458C2×C4—1062no—
459C2×C4—1062no—
460C2×C4—4248no—
461C2×C4—4242no—
462C2×C4—4242no—
463C2×C4—2122no—
464C2×C4—2122no—
465C2×C4—2122no—
466C2×C4—2122no—
467C2×C4—2122no—
468C2×C4—2122no—
469C2×C4—2122no—
470C2×C4—2122no—
471C2×C4—2122no—
472C2×C4—2122no—
473C2×C4—2122no—
474C2×C4—2122no—
475C2×C4C42×D5311yescharacteristic
476C2×C4C22⋊C4×D5311yescharacteristic
477C2×C4C4⋊C4×D5311yescharacteristic
478C2×C4C4⋊C4×D5311yescharacteristic
479C8—42no—
480C8—2122no—
481C22—21no—
482C22—21no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—21no—
488C22—21no—
489C22—534no—
490C22—1062no—
491C22—1064no—
492C22—1064no—
493C22—1062no—
494C22—1062no—
495C22—1062no—
496C22—1062no—
497C22—2124no—
498C22—2124no—
499C22—2124no—
500C22—2124no—
501C22—2122no—
502C22—2122no—
503C22#95×D5311yescharacteristic
504C22#99×D5312yes—
505C22#1681062111yescharacteristic
506C22#1681062311yescharacteristic
507C22#1681062611yescharacteristic
508C22#1681063011yescharacteristic
509C22#1681063812yes—
510C22#1681066211yescharacteristic
511C22#1681066411yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—82no—
517C4—82no—
518C4—1062no—
519C4—1062no—
520C4—4242no—
521C4—4242no—
522C2—21no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—1064no—
527C2—2122no—
528C2#2027324212yes—
529C2#2027324612yes—
530C2#2027442311yescharacteristic
531C2#2027443311yescharacteristic
532C2#342×D5311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1294C2
socleC23×C53C2×C22⋊C4
prime radicalC24×D53D4
cosocleC23×C4×C53C23
hypercenter#1294D53
hypocenterC53#8015
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
4. #20272595C2semidirect
5. #20272594C2semidirect
6. #20272593C2semidirect
7. C53×#1294C2semidirect
10. C2×C4⋊C4×D53C4semidirect
14. #16810585C4semidirect
18. #16810485C4semidirect
22. #16810480C4semidirect
55. C22⋊#88322C2×C4semidirect
61. #13133124C2×C4semidirect
68. C22⋊C8×C53C2×C4semidirect
71. #95×C53C2×C4semidirect
262. #1294D53semidirect
295. C53#8015semidirect
330. C22⋊C8C4×D53semidirect
338. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105852
C2×#168104852
#380×D532
#202725952
#202725942
#202725932
C53×#12942
#801553