Finite Group Databasebeta

← #27274106

Subgroups — #27274106

Order 13568.

1162 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810585C211yesmaximal, characteristic
2#380×D53C211yesmaximal, characteristic
3C2×#16810470C211yesmaximal, 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—
13C22×C4×Dic53—21no—
14#16810585—22no—
15#16810585C414yescomplemented
16C2×C22⋊#88322C2211yescharacteristic
17C2×#13133129—21no—
18#16810483C2211yescharacteristic
19#16810482—21no—
20#16810481—21no—
21#16810480C412yescomplemented
22#16810470—22no—
23#16810470C414yescomplemented
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×C412yes—
37C22×C4×D53C2×C411yescharacteristic
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—28no—
50C2×#11719921—24no—
51C2×#11719921—22no—
52C22⋊C4×D53—42no—
53C22⋊C4×D53—42no—
54C2×C4×Dic53—24no—
55C2×C4×Dic53—44no—
56C22⋊#88322—21no—
57C22⋊#88322C2×C412yescomplemented
58#13133129—24no—
59#13133124—21no—
60#13133124—21no—
61#13133124—22no—
62#13133124C2×C412yescomplemented
63C23×C4×C53—21no—
64C23×C4×C53C2311yescharacteristic
65C22×C8×C53—21no—
66C2×C4⋊C4×C53C2×C412yes—
67C2×C22⋊C4×C53—21no—
68C22⋊C8×C53—21no—
69C22⋊C8×C53C2×C412yescomplemented
70#95×C53—21no—
71#95×C53—21no—
72#95×C53C2×C412yescomplemented
73C24×C53C2×D411yescharacteristic
74C23×D53—21no—
75C23×D53—21no—
76C23×D53—21no—
77C23×D53—21no—
78C23×D53—24no—
79C23×D53—42no—
80C23×D53C22⋊C411yescharacteristic
81C23×D53C22⋊C412yes—
82C23×D53C22⋊C411yescharacteristic
83C23×D53C4⋊C412yes—
84C22×C4×C53—24no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C22×C411yescharacteristic
93C22×C4×C53C22×C411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×Q811yescharacteristic
97C22×Dic53—24no—
98C22×Dic53—24no—
99C22×Dic53—22no—
100C22×Dic53—22no—
101C22×Dic53—42no—
102C22×Dic53—41no—
103C22×Dic53—41no—
104C22×Dic53C22⋊C411yescharacteristic
105C22×Dic53C22⋊C412yes—
106C22×Dic53C22⋊C411yescharacteristic
107C22×Dic53C4⋊C412yes—
108C2×C4×D53—24no—
109C2×C4×D53—24no—
110C2×C4×D53—44no—
111C2×C4×D53—44no—
112C2×C4×D53—44no—
113C2×C4×D53—44no—
114C2×C4×D53—42no—
115C2×C4×D53—42no—
116C2×C4×D53—44no—
117C2×C4×D53—44no—
118C2×C4×D53—44no—
119C2×C4×D53C4212yes—
120C2×C4×D53C4212yes—
121C2×C4×D53C22⋊C412yes—
122C2×C4×D53C22⋊C412yes—
123C2×C4×D53C4⋊C412yes—
124C2×C4×D53C4⋊C412yes—
125C2×C4×D53C4⋊C412yes—
126C2×C4×D53C4⋊C412yes—
127C2×C8×C53—22no—
128C2×C8×C53—42no—
129C4⋊C4×C53—42no—
130C22⋊C4×C53—41no—
131C22⋊C4×C53—41no—
132C22⋊Dic53—41no—
133C22⋊Dic53—41no—
134#11719921—216no—
135#11719921—44no—
136C4⋊Dic53—42no—
137#11719919—44no—
138C4×Dic53—48no—
139C2×#88322—22no—
140C2×#88322—42no—
141#11719912—48no—
142C23×C53—22no—
143C23×C53—22no—
144C23×C53C2×C22⋊C411yescharacteristic
145C23×C53C2×C22⋊C411yescharacteristic
146C23×C53C2×C4⋊C411yescharacteristic
147C23×C53AGL(1,ℤ/8)11yescharacteristic
148C23×C53AGL(1,ℤ/8)11yescharacteristic
149C23×C53#13711yescharacteristic
150C23×C53#13711yescharacteristic
151C22×D53—22no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—22no—
155C22×D53—22no—
156C22×D53—22no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53—22no—
160C22×D53—22no—
161C22×D53—22no—
162C22×D53—24no—
163C22×D53—24no—
164C22×D53—42no—
165C22×D53—42no—
166C22×D53—44no—
167C22×D53—44no—
168C22×D53—44no—
169C22×D53—44no—
170C22×D53#9512yes—
171C22×D53#9512yes—
172C22×D53#10414yes—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—24no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—44no—
181C2×C4×C53—44no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53C2×C4211yescharacteristic
188C2×C4×C53C2×C22⋊C411yescharacteristic
189C2×C4×C53C2×C4⋊C411yescharacteristic
190C2×C4×C53C2×C4⋊C411yescharacteristic
191C2×Dic53—22no—
192C2×Dic53—24no—
193C2×Dic53—24no—
194C2×Dic53—24no—
195C2×Dic53—44no—
196C2×Dic53—44no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—42no—
200C2×Dic53—42no—
201C2×Dic53—42no—
202C2×Dic53#9512yes—
203C2×Dic53#9512yes—
204C2×Dic53#9914yes—
205C4×D53—24no—
206C4×D53—24no—
207C4×D53—24no—
208C4×D53—24no—
209C4×D53—42no—
210C4×D53—42no—
211C4×D53—42no—
212C4×D53—42no—
213C4×D53—88no—
214C8×C53—42no—
215#88322—42no—
216#8019—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—21no—
221C22×C53—22no—
222C22×C53—22no—
223C22×C53—22no—
224C22×C53—22no—
225C22×C53C2×#9511yescharacteristic
226C22×C53C2×#9912yes—
227C22×C53#41711yescharacteristic
228C22×C53#41711yescharacteristic
229C22×C53#41811yescharacteristic
230C22×C53#41911yescharacteristic
231C22×C53C2×#10412yes—
232C22×C53#42811yescharacteristic
233C22×C53#42811yescharacteristic
234C2×D53—22no—
235C2×D53—22no—
236C2×D53—24no—
237C2×D53—24no—
238C2×D53—22no—
239C2×D53—22no—
240C2×D53—22no—
241C2×D53—44no—
242C2×D53—44no—
243C2×D53—44no—
244C2×D53—44no—
245C2×D53—42no—
246C2×D53—42no—
247C2×D53#33914yes—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—44no—
253C4×C53—82no—
254Dic53—22no—
255Dic53—44no—
256Dic53—82no—
257Dic53#34214yes—
258C2×#407—531no—
259C2×#380—531no—
260C2×C4×C22⋊C4—531no—
261#1294—531no—
262#1294—531no—
263#1294—531no—
264#1294D5311yescharacteristic, complemented
265C2×C53—21no—
266C2×C53—21no—
267C2×C53—21no—
268C2×C53—21no—
269C2×C53#150712yes—
270C2×C53#150812yes—
271C2×C53C2×#33911yescharacteristic
272C2×C53#173611yescharacteristic
273C2×C53C2×#34211yescharacteristic
274D53—24no—
275D53—42no—
276C24×C4—531no—
277C22×M16—1061no—
278C22×C4⋊C4—532no—
279C22×C22⋊C4—1061no—
280C22×C22⋊C4—1061no—
281C22×C42—1061no—
282#407—534no—
283#407—1062no—
284C2×C22⋊C8—531no—
285C2×C22⋊C8—1061no—
286C2×C22⋊C8C2×D5311yescharacteristic
287#380—531no—
288#380—532no—
289#380—1061no—
290#380—1061no—
291#380C2×D5311yescharacteristic
292C4×C22⋊C4—534no—
293C4×C22⋊C4—1062no—
294C2×#95—531no—
295C2×#95—1061no—
296C2×#95C2×D5311yescharacteristic
297C53#801911yescharacteristic, complemented, Sylow
298C25—531no—
299C23×C4—21no—
300C23×C4—531no—
301C23×C4—531no—
302C23×C4—531no—
303C23×C4—532no—
304C23×C4—531no—
305C23×C4—532no—
306C23×C4—531no—
307C23×C4—532no—
308C23×C4—532no—
309C23×C4—1062no—
310C23×C4—1061no—
311C23×C4—2124no—
312C23×C4C22×D5311yescharacteristic
313C2×M16—1064no—
314C2×M16—2124no—
315C22×C8—21no—
316C22×C8—1061no—
317C2×C4⋊C4—532no—
318C2×C4⋊C4—1062no—
319C2×C4⋊C4—2124no—
320C2×C4⋊C4C4×D5312yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—1064no—
323C2×C22⋊C4—1068no—
324C2×C22⋊C4—1062no—
325C2×C22⋊C4—1061no—
326C2×C22⋊C4—2122no—
327C2×C22⋊C4—2122no—
328C2×C42—1064no—
329C2×C42—2124no—
330C22⋊C8—21no—
331C22⋊C8—532no—
332C22⋊C8—1064no—
333C22⋊C8—1061no—
334C22⋊C8C4×D5312yescomplemented
335#95—21no—
336#95—21no—
337#95—532no—
338#95—1062no—
339#95—1061no—
340#95—1061no—
341#95C4×D5312yescomplemented
342C24—531no—
343C24—531no—
344C24—532no—
345C24—532no—
346C24—1064no—
347C24—1061no—
348C24—1061no—
349C24—1061no—
350C24—1061no—
351C24—2122no—
352C24D4×D5311yescharacteristic
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—531no—
361C22×C4—532no—
362C22×C4—531no—
363C22×C4—532no—
364C22×C4—532no—
365C22×C4—532no—
366C22×C4—532no—
367C22×C4—532no—
368C22×C4—532no—
369C22×C4—532no—
370C22×C4—532no—
371C22×C4—532no—
372C22×C4—1064no—
373C22×C4—1064no—
374C22×C4—1064no—
375C22×C4—1064no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—2122no—
379C22×C4—2122no—
380C22×C4—2124no—
381C22×C4—2124no—
382C22×C4—2124no—
383C22×C4—2124no—
384C22×C4—2124no—
385C22×C4—2124no—
386C22×C4—2124no—
387C22×C4—2122no—
388C22×C4—2121no—
389C22×C4—2121no—
390C22×C4C2×C4×D5311yescharacteristic
391C22×C4C2×C4×D5311yescharacteristic
392C22×C4C2×C4×D5311yescharacteristic
393C22×C4D4×D5311yescharacteristic
394C22×C4D4×D5311yescharacteristic
395C22×C4Q8×D5311yescharacteristic
396M16—2128no—
397C2×C8—22no—
398C2×C8—42no—
399C2×C8—1062no—
400C2×C8—2122no—
401C4⋊C4—42no—
402C4⋊C4—2122no—
403C4⋊C4—2124no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—10616no—
407C22⋊C4—2124no—
408C22⋊C4—2121no—
409C22⋊C4—2121no—
410C42—2128no—
411C23—22no—
412C23—22no—
413C23—534no—
414C23—532no—
415C23—532no—
416C23—1064no—
417C23—1064no—
418C23—1062no—
419C23—1061no—
420C23—1061no—
421C23—1062no—
422C23—1062no—
423C23—1062no—
424C23—1061no—
425C23—1061no—
426C23—1062no—
427C23—1062no—
428C23—1062no—
429C23—2122no—
430C23—2122no—
431C23—2124no—
432C23—2124no—
433C23—2124no—
434C23—2124no—
435C23C22⋊C4×D5311yescharacteristic
436C23C22⋊C4×D5311yescharacteristic
437C23C4⋊C4×D5311yescharacteristic
438C23#1313324111yescharacteristic
439C23#1313324411yescharacteristic
440C23#1313324611yescharacteristic
441C23#1313324911yescharacteristic
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—534no—
457C2×C4—532no—
458C2×C4—532no—
459C2×C4—1064no—
460C2×C4—1064no—
461C2×C4—1064no—
462C2×C4—1064no—
463C2×C4—1064no—
464C2×C4—1064no—
465C2×C4—1062no—
466C2×C4—1064no—
467C2×C4—4248no—
468C2×C4—2122no—
469C2×C4—2122no—
470C2×C4—2122no—
471C2×C4—2122no—
472C2×C4—2122no—
473C2×C4—2122no—
474C2×C4—2122no—
475C2×C4—2122no—
476C2×C4—2124no—
477C2×C4—2124no—
478C2×C4—2122no—
479C2×C4C42×D5311yescharacteristic
480C2×C4C22⋊C4×D5311yescharacteristic
481C2×C4C4⋊C4×D5311yescharacteristic
482C2×C4C4⋊C4×D5311yescharacteristic
483C8—42no—
484C8—2122no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—21no—
492C22—21no—
493C22—534no—
494C22—1062no—
495C22—1062no—
496C22—1064no—
497C22—1064no—
498C22—1062no—
499C22—1062no—
500C22—1062no—
501C22—2124no—
502C22—2124no—
503C22—2124no—
504C22—2124no—
505C22—2122no—
506C22—2122no—
507C22#95×D5311yescharacteristic
508C22#99×D5312yes—
509C22#1681062111yescharacteristic
510C22#1681062311yescharacteristic
511C22#1681062611yescharacteristic
512C22#1681063011yescharacteristic
513C22#104×D5312yes—
514C22#1681066211yescharacteristic
515C22#1681066411yescharacteristic
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—44no—
521C4—82no—
522C4—534no—
523C4—1062no—
524C4—4242no—
525C4—2124no—
526C2—21no—
527C2—21no—
528C2—21no—
529C2—21no—
530C2—1064no—
531C2—2122no—
532C2#2027322912yes—
533C2#2027323312yes—
534C2#339×D5311yescharacteristic
535C2#2027443311yescharacteristic
536C2#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#8019
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
15. #16810585C4semidirect
21. #16810480C4semidirect
23. #16810470C4semidirect
57. C22⋊#88322C2×C4semidirect
62. #13133124C2×C4semidirect
69. C22⋊C8×C53C2×C4semidirect
72. #95×C53C2×C4semidirect
264. #1294D53semidirect
297. C53#8019semidirect
334. C22⋊C8C4×D53semidirect
341. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105852
#380×D532
C2×#168104702
#202725952
#202725942
#202725932
C53×#12942
#801953