Finite Group Databasebeta

← #27416756

Subgroups — #27416756

Order 13568.

1500 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C212yesmaximal, complemented
2C2×#16810483C211yesmaximal, characteristic
3#20275915C214yesmaximal, complemented
4#20275023C212yesmaximal, complemented
5#20274903C212yesmaximal, complemented
6#20274650C212yesmaximal, complemented
7#20274625C211yesmaximal, characteristic
8C22×C22⋊Q8×C53C211yesmaximal, characteristic
9C22×C4⋊Dic53C2211yescharacteristic
10C22×C4⋊Dic53C2211yescharacteristic
11C22×C22⋊Dic53—21no—
12#16810956—28no—
13C2×#13133150C2214yescomplemented
14C2×#13133141C2214yescomplemented
15C2×#13133141C2214yescomplemented
16C2×C22⋊#88322C2212yes—
17C2×C22⋊#88322C2212yes—
18C2×C22⋊#88322C2212yes—
19#16810483—21no—
20#16810483—21no—
21#16810483C2212yes—
22#16810483C2212yes—
23C2×#13133124—21no—
24#16810385C2214yescomplemented
25C23×Q8×C53C2211yescharacteristic
26C24×C4×C53C2211yescharacteristic
27C2×C22⋊Q8×C53—21no—
28C2×C22⋊Q8×C53—21no—
29C2×C22⋊Q8×C53—28no—
30C2×C22⋊Q8×C53C2212yes—
31C2×C22⋊Q8×C53C2212yes—
32C22×C4⋊C4×C53—21no—
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53—21no—
35C23×Dic53—21no—
36C23×Dic53—21no—
37C22×#88322C2314yes—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—24no—
41C2×C4⋊Dic53—42no—
42C2×C4⋊Dic53C2312yes—
43C2×C4⋊Dic53C2312yes—
44C2×C22⋊Dic53—22no—
45C2×C22⋊Dic53—22no—
46C2×C22⋊Dic53—44no—
47#13133150—28no—
48#13133141—28no—
49#13133141—44no—
50C22⋊#88322—24no—
51C22⋊#88322—24no—
52C22⋊#88322—42no—
53#13133124—22no—
54#13133124—22no—
55C25×C53D411yescharacteristic
56C22×Q8×C53—24no—
57C22×Q8×C53—21no—
58C22×Q8×C53—24no—
59C22×Q8×C53—42no—
60C22×Q8×C53C2312yes—
61C23×C4×C53—21no—
62C23×C4×C53—21no—
63C23×C4×C53—24no—
64C23×C4×C53—24no—
65C23×C4×C53—42no—
66C23×C4×C53D411yescharacteristic
67C23×C4×C53D411yescharacteristic
68C23×C4×C53D411yescharacteristic
69C23×C4×C53C2311yescharacteristic
70C23×C4×C53C2311yescharacteristic
71C23×C4×C53C2311yescharacteristic
72C22⋊Q8×C53—28no—
73C22⋊Q8×C53—28no—
74C22⋊Q8×C53—24no—
75C22⋊Q8×C53—24no—
76C22⋊Q8×C53—24no—
77C22⋊Q8×C53D418yes—
78C2×C4⋊C4×C53—28no—
79C2×C4⋊C4×C53—24no—
80C2×C4⋊C4×C53—22no—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53C2312yes—
84C2×C22⋊C4×C53—28no—
85C2×C22⋊C4×C53—22no—
86C2×C22⋊C4×C53—22no—
87C24×C53—22no—
88C24×C53—22no—
89C24×C53—22no—
90C24×C53—22no—
91C24×C53—41no—
92C24×C53—41no—
93C24×C53Q1614yes—
94C24×C53C2×D411yescharacteristic
95C24×C53C2×D411yescharacteristic
96C24×C53C2×D411yescharacteristic
97C2×Q8×C53—24no—
98C2×Q8×C53—22no—
99C2×Q8×C53—24no—
100C2×Q8×C53—24no—
101C2×Q8×C53—416no—
102C2×Q8×C53—42no—
103C2×Q8×C53—42no—
104C2×Q8×C53C2×D414yes—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—24no—
110C22×C4×C53—24no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—416no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D414yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4114yes—
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53—42no—
141C22×Dic53—42no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—28no—
146C4⋊C4×C53—28no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53C2×D414yes—
149C22⋊C4×C53—28no—
150C22⋊C4×C53—28no—
151C22⋊Dic53—44no—
152C22⋊Dic53—44no—
153C4⋊Dic53—24no—
154C4⋊Dic53—24no—
155C4⋊Dic53—42no—
156C4⋊Dic53—42no—
157C2×#88322—24no—
158C2×#88322—24no—
159C2×#88322—24no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—28no—
175C23×C53—28no—
176C23×C53—44no—
177C23×C53—44no—
178C23×C53—44no—
179C23×C53—44no—
180C23×C53—44no—
181C23×C53#12012yes—
182C23×C53#12312yes—
183C23×C53C2×Q1612yes—
184C23×C53C2×Q1612yes—
185C23×C53C2×Q1612yes—
186C23×C53AGL(1,ℤ/8)12yes—
187C23×C53#13712yes—
188C23×C53C22×D411yescharacteristic
189Q8×C53—28no—
190Q8×C53—48no—
191Q8×C53—44no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—28no—
197C2×C4×C53—28no—
198C2×C4×C53—21no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—44no—
204C2×C4×C53—44no—
205C2×C4×C53—44no—
206C2×C4×C53—48no—
207C2×C4×C53—48no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—44no—
213C2×C4×C53#12012yes—
214C2×C4×C53#12312yes—
215C2×C4×C53C22×D412yes—
216C2×C4×C53C2×#4112yes—
217C2×C4×C53H(5,𝔽2)12yes—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—84no—
225C2×Dic53—84no—
226#88322—44no—
227#24006—531nomaximal, Sylow
228C22×C53—22no—
229C22×C53—22no—
230C22×C53—22no—
231C22×C53—22no—
232C22×C53—22no—
233C22×C53—22no—
234C22×C53—22no—
235C22×C53—22no—
236C22×C53—22no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—416no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53#45114yes—
249C22×C53#48414yescomplemented
250C22×C53C2×#12011yescharacteristic
251C22×C53C2×#12311yescharacteristic
252C22×C53#53411yescharacteristic
253C22×C53C22×Q1611yescharacteristic
254C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C53C2×#13711yescharacteristic
256C22×C53#57511yescharacteristic
257C4×C53—22no—
258C4×C53—22no—
259C4×C53—41no—
260C4×C53—44no—
261C4×C53—44no—
262C4×C53#54614yes—
263Dic53—82no—
264Dic53—82no—
265C22×C22⋊Q8D5311yescharacteristic
266C2×#484—532no—
267C2×#380—531no—
268#2036—534no—
269#1872—532no—
270#1852—532no—
271#1787—532no—
272#1779—531no—
273C2×C53—24no—
274C2×C53—24no—
275C2×C53—42no—
276C2×C53#240511yescharacteristic
277C2×C53C2×#45111yescharacteristic
278C2×C53#300611yescharacteristic
279C2×C53C2×#48411yescharacteristic
280C2×C53#309511yescharacteristic
281C2×C53#318911yescharacteristic
282C2×C53#319311yescharacteristic
283C23×Q8C2×D5311yescharacteristic
284C24×C4C2×D5311yescharacteristic
285C2×C22⋊Q8—21no—
286C2×C22⋊Q8—21no—
287C2×C22⋊Q8—28no—
288C2×C22⋊Q8C2×D5312yes—
289C2×C22⋊Q8C2×D5312yes—
290C22×C4⋊C4—21no—
291C22×C4⋊C4—531no—
292C22×C4⋊C4—531no—
293C22×C4⋊C4C2×D5311yescharacteristic
294C22×C22⋊C4—21no—
295C22×C22⋊C4—1061no—
296#484—1068no—
297C2×#107—534no—
298C2×#103—534no—
299C2×#103—534no—
300C2×C22⋊C8—532no—
301C2×C22⋊C8—532no—
302C2×C22⋊C8—532no—
303#380—532no—
304#380—532no—
305#380—1061no—
306#380—1061no—
307C2×#95—1061no—
308#340—534no—
309C53#2400611yescharacteristic, complemented, Sylow
310C25C22⋊D5311yescharacteristic
311C22×Q8—24no—
312C22×Q8—21no—
313C22×Q8—24no—
314C22×Q8—42no—
315C22×Q8C22×D5312yes—
316C23×C4—21no—
317C23×C4—21no—
318C23×C4—24no—
319C23×C4—24no—
320C23×C4—42no—
321C23×C4—1061no—
322C23×C4—1061no—
323C23×C4C22⋊D5311yescharacteristic
324C23×C4C22⋊D5311yescharacteristic
325C23×C4C22⋊D5311yescharacteristic
326C23×C4C22×D5311yescharacteristic
327C23×C4C22×D5311yescharacteristic
328C23×C4C22×D5311yescharacteristic
329C22×C8—534no—
330C22⋊Q8—28no—
331C22⋊Q8—28no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8C22⋊D5318yes—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—532no—
342C2×C4⋊C4—532no—
343C2×C4⋊C4—1064no—
344C2×C4⋊C4—1061no—
345C2×C4⋊C4—1061no—
346C2×C4⋊C4—2122no—
347C2×C4⋊C4C22×D5312yes—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—1062no—
352C2×C22⋊C4—1062no—
353C2×C22⋊C4—2124no—
354#107—1068no—
355#103—1068no—
356#103—2124no—
357C22⋊C8—1064no—
358C22⋊C8—1064no—
359C22⋊C8—2122no—
360#95—1062no—
361#95—1062no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—41no—
367C24—41no—
368C24#1171992514yes—
369C24C2×C22⋊D5311yescharacteristic
370C24C2×C22⋊D5311yescharacteristic
371C24C2×C22⋊D5311yescharacteristic
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—416no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8C2×C22⋊D5314yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—416no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—1062no—
406C22×C4—1062no—
407C22×C4—2122no—
408C22×C4—2122no—
409C22×C4—2122no—
410C22×C4—2122no—
411C22×C4—2122no—
412C22×C4—2122no—
413C22×C4#1171994714yes—
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C2×C22⋊D5311yescharacteristic
416C22×C4C2×C22⋊D5314yes—
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C23×D5311yescharacteristic
419C2×C8—1064no—
420C2×C8—1064no—
421C2×C8—1064no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—1064no—
427C4⋊C4—1064no—
428C4⋊C4—2122no—
429C4⋊C4—2122no—
430C4⋊C4C2×C22⋊D5314yes—
431C22⋊C4—28no—
432C22⋊C4—28no—
433C22⋊C4—2124no—
434C22⋊C4—2124no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—28no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—28no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23#1313318612yes—
457C23#1313319812yes—
458C23C2×#1171992512yes—
459C23C2×#1171992512yes—
460C23C2×#1171992512yes—
461C23#1313324212yes—
462C23#1313324812yes—
463C23C22×C22⋊D5311yescharacteristic
464Q8—28no—
465Q8—48no—
466Q8—44no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—4244no—
489C2×C4—4244no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2122no—
493C2×C4—2122no—
494C2×C4—2122no—
495C2×C4—2122no—
496C2×C4#1313318612yes—
497C2×C4#1313319812yes—
498C2×C4C22×C22⋊D5312yes—
499C2×C4C2×#1171994712yes—
500C2×C4#1313326212yes—
501C8—2124no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—416no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22#1681077914yes—
523C22#1681095614yescomplemented
524C22C2×#1313318611yescharacteristic
525C22C2×#1313319811yescharacteristic
526C22#1681118211yescharacteristic
527C22C22×#1171992511yescharacteristic
528C22C2×#1313324211yescharacteristic
529C22C2×#1313324811yescharacteristic
530C22#1681142111yescharacteristic
531C4—22no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—44no—
536C4—4242no—
537C4—4242no—
538C4#1681126714yes—
539C2—24no—
540C2—24no—
541C2—42no—
542C2#2027803811yescharacteristic
543C2C2×#1681077911yescharacteristic
544C2#2028396711yescharacteristic
545C2C2×#1681095611yescharacteristic
546C2#2028465911yescharacteristic
547C2#2028548711yescharacteristic
548C2#2028551711yescharacteristic
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#24006
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×#16810956C2semidirect
3. #20275915C2semidirect
4. #20275023C2semidirect
5. #20274903C2semidirect
6. #20274650C2semidirect
13. C2×#13133150C22semidirect
14. C2×#13133141C22semidirect
15. C2×#13133141C22semidirect
24. #16810385C22semidirect
249. C22×C53#484semidirect
309. C53#24006semidirect
523. C22#16810956semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168104832
#202759152
#202750232
#202749032
#202746502
#202746252
C22×C22⋊Q8×C532
#2400653