Finite Group Databasebeta

← #27275967

Subgroups — #27275967

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274525C211yesmaximal, characteristic, complemented
2#20273322C211yesmaximal, characteristic, complemented
3#20273227C211yesmaximal, characteristic, complemented
4#20273201C211yesmaximal, characteristic, complemented
5#20273149C211yesmaximal, characteristic, complemented
6#20273136C211yesmaximal, characteristic, complemented
7C53×#3515C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9#16810676C2211yescharacteristic
10C2×#13133144—21no—
11C4×#11719917C2211yescharacteristic
12#16810469C2211yescharacteristic
13#16810382—21no—
14#16810365—21no—
15#16810363—21no—
16#16810360—21no—
17#16810360—22no—
18#16810357—21no—
19Q8⋊#88322—22no—
20Q8⋊#88322—21no—
21D4⋊#88322—21no—
22C53×#557C411yescharacteristic, complemented
23C53×#554C411yescharacteristic, complemented
24C53×#554C411yescharacteristic, complemented
25C53×#553—21no—
26C53×#552—21no—
27C53×#552—22no—
28C53×#552—21no—
29C53×#547—21no—
30C53×#547C411yescharacteristic, complemented
31C53×#541—21no—
32C53×#539—22no—
33C53×#533C2211yescharacteristic
34C53×#529—21no—
35C53×#529C2211yescharacteristic
36C2×#125×C53—21no—
37C53×#520—21no—
38C4×#41×C53C2211yescharacteristic
39C2×#11719917—21no—
40C2×C4×Dic53—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43#13133144—24no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46#13133128—21no—
47#13133128—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50#13133124—21no—
51#13133124—21no—
52C2×#41×C53—21no—
53C2×#41×C53D411yescharacteristic
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53C2×C411yescharacteristic
56C4⋊Q8×C53C2×C411yescharacteristic
57#126×C53—22no—
58#126×C53—22no—
59#126×C53—22no—
60#126×C53—21no—
61#125×C53—22no—
62#125×C53—22no—
63#125×C53—22no—
64#125×C53—21no—
65#125×C53—21no—
66#125×C53—22no—
67#125×C53—21no—
68#125×C53—21no—
69#125×C53D411yescharacteristic
70#124×C53D411yescharacteristic
71#123×C53—22no—
72#123×C53—22no—
73#123×C53—22no—
74#123×C53—22no—
75#123×C53—21no—
76C22⋊Q8×C53—21no—
77C22⋊Q8×C53—21no—
78C22⋊Q8×C53—22no—
79C22⋊Q8×C53—22no—
80C22⋊Q8×C53—21no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53D411yescharacteristic
83C22⋊Q8×C53D412yes—
84#121×C53—22no—
85#121×C53—21no—
86#121×C53D411yescharacteristic
87C4×Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53—21no—
90C4×Q8×C53C2×C411yescharacteristic
91C4×Q8×C53C2×C411yescharacteristic
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53D412yes—
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53C2×C411yescharacteristic
102C4×D4×C53C2×C411yescharacteristic
103C4×D4×C53D411yescharacteristic
104#117×C53—22no—
105#117×C53—22no—
106#117×C53—21no—
107#117×C53—21no—
108#117×C53—21no—
109#117×C53D411yescharacteristic
110C2×C4⋊C4×C53—22no—
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117#41×C53—22no—
118#41×C53—41no—
119#41×C53—42no—
120#41×C53—42no—
121C2×Q8×C53—21no—
122C2×Q8×C53C22⋊C411yescharacteristic
123C2×Q8×C53C22⋊C411yescharacteristic
124C2×Q8×C53C22⋊C412yes—
125C2×D4×C53—22no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53C22⋊C411yescharacteristic
129C2×D4×C53C22⋊C411yescharacteristic
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×Dic53—41no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53C22⋊C411yescharacteristic
161C4⋊C4×C53C22⋊C411yescharacteristic
162C4⋊C4×C53C22⋊C411yescharacteristic
163C4⋊C4×C53C22⋊C411yescharacteristic
164C4⋊C4×C53C22⋊C412yes—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—21no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C42×C53—22no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—22no—
183C42×C53C22×C411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C42×C53C2×D411yescharacteristic
186C42×C53C2×D411yescharacteristic
187C4×Dic53—22no—
188C4×Dic53—41no—
189#11719917—22no—
190#11719917—41no—
191C2×#88322—41no—
192C2×#88322—41no—
193C2×#88322—41no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53#12011yescharacteristic
197Q8×C53—21no—
198Q8×C53—21no—
199Q8×C53—42no—
200Q8×C53—41no—
201Q8×C53—41no—
202Q8×C53—41no—
203D4×C53—21no—
204D4×C53—41no—
205D4×C53—42no—
206D4×C53—44no—
207D4×C53—81no—
208D4×C53#10412yes—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—44no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—81no—
227C2×C4×C53—82no—
228C2×C4×C53—82no—
229C2×C4×C53C2×C22⋊C411yescharacteristic
230C2×C4×C53C2×C22⋊C411yescharacteristic
231C2×C4×C53C2×C22⋊C411yescharacteristic
232C2×C4×C53#12011yescharacteristic
233C2×C4×C53#12011yescharacteristic
234C2×C4×C53#12011yescharacteristic
235C2×Dic53—42no—
236C2×Dic53—81no—
237C2×Dic53—81no—
238#88322—42no—
239#88322—81no—
240#88322—81no—
241#8415—531nomaximal, Sylow
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—22no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53#37911yescharacteristic
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—21no—
257C4×C53—41no—
258C4×C53—41no—
259C4×C53—42no—
260C4×C53—41no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—42no—
265C4×C53C2×#10411yescharacteristic
266C4×C53#42111yescharacteristic
267Dic53—82no—
268#3515D5311yescharacteristic
269#1759—531no—
270#1529—531no—
271#1506—531no—
272#1500—531no—
273#1488—531no—
274#1485—531no—
275C2×C53—41no—
276C2×C53—42no—
277C2×C53—81no—
278C2×C53#178911yescharacteristic
279C2×C53#179411yescharacteristic
280C2×C53#179611yescharacteristic
281#557Dic5311yescharacteristic, complemented
282#554Dic5311yescharacteristic, complemented
283#554Dic5311yescharacteristic, complemented
284#553—21no—
285#552—21no—
286#552—22no—
287#552—21no—
288#547—21no—
289#547Dic5311yescharacteristic, complemented
290#541—21no—
291#539—22no—
292#533C2×D5311yescharacteristic
293#529—21no—
294#529C2×D5311yescharacteristic
295C2×#125—21no—
296#520—21no—
297C4×#41C2×D5311yescharacteristic
298#432—531no—
299#431—531no—
300C2×#104—1061no—
301C4×M16—531no—
302#376—531no—
303#339—1061no—
304#330—1061no—
305#329—1061no—
306#328—1061no—
307#328—1062no—
308#327—1061no—
309Q8⋊C8—1062no—
310Q8⋊C8—1061no—
311D4⋊C8—1061no—
312C53#841511yescharacteristic, complemented, Sylow
313C2×#41—21no—
314C2×#41C22⋊D5311yescharacteristic
315C2×M16—1061no—
316C4⋊Q8—21no—
317C4⋊Q8C2×Dic5311yescharacteristic
318C4⋊Q8C2×Dic5311yescharacteristic
319#126—22no—
320#126—22no—
321#126—22no—
322#126—21no—
323#125—22no—
324#125—22no—
325#125—22no—
326#125—21no—
327#125—21no—
328#125—22no—
329#125—21no—
330#125—21no—
331#125C22⋊D5311yescharacteristic
332#124C22⋊D5311yescharacteristic
333#123—22no—
334#123—22no—
335#123—22no—
336#123—21no—
337#123—22no—
338C22⋊Q8—21no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8C22⋊D5311yescharacteristic
345C22⋊Q8C22⋊D5312yes—
346#121—22no—
347#121—21no—
348#121C22⋊D5311yescharacteristic
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8C2×Dic5311yescharacteristic
353C4×Q8C2×Dic5311yescharacteristic
354C4×Q8C22⋊D5311yescharacteristic
355C4×Q8C22⋊D5312yes—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4C2×Dic5311yescharacteristic
364C4×D4C2×Dic5311yescharacteristic
365C4×D4C22⋊D5311yescharacteristic
366#117—22no—
367#117—22no—
368#117—21no—
369#117—21no—
370#117—21no—
371#117C22⋊D5311yescharacteristic
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C42—21no—
378C2×C42—1061no—
379C2×C42C22×D5311yescharacteristic
380C4⋊C8—1061no—
381C4⋊C8—1061no—
382#104—1064no—
383C22⋊C8—1061no—
384C22⋊C8—1061no—
385#97—1061no—
386#97—1061no—
387C4×C8—1061no—
388C4×C8—1061no—
389#95—1061no—
390#95—1061no—
391#41—22no—
392#41—41no—
393#41—42no—
394#41—42no—
395C2×Q8—21no—
396C2×Q8C22⋊Dic5311yescharacteristic
397C2×Q8C22⋊Dic5311yescharacteristic
398C2×Q8C22⋊Dic5312yes—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4C22⋊Dic5311yescharacteristic
403C2×D4C22⋊Dic5311yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—2121no—
412C22×C4C2×C22⋊D5311yescharacteristic
413C22×C4C2×C22⋊D5311yescharacteristic
414C22×C4C2×C22⋊D5311yescharacteristic
415M16—1062no—
416M16—2121no—
417C2×C8—2121no—
418C2×C8—2121no—
419C2×C8—2121no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4C22⋊Dic5311yescharacteristic
440C4⋊C4C22⋊Dic5311yescharacteristic
441C4⋊C4C22⋊Dic5311yescharacteristic
442C4⋊C4C22⋊Dic5311yescharacteristic
443C4⋊C4C22⋊Dic5312yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C42—22no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—22no—
462C42—1062no—
463C42—2121no—
464C42C22×Dic5311yescharacteristic
465C42C2×C22⋊D5311yescharacteristic
466C42C2×C22⋊D5311yescharacteristic
467C42C2×C22⋊D5311yescharacteristic
468C23—21no—
469C23—21no—
470C23#1313318611yescharacteristic
471Q8—21no—
472Q8—21no—
473Q8—41no—
474Q8—42no—
475Q8—41no—
476Q8—41no—
477D4—21no—
478D4—41no—
479D4—42no—
480D4—44no—
481D4—81no—
482D4#1313314412yes—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—4241no—
504C2×C4—4241no—
505C2×C4—2122no—
506C2×C4C2×C22⋊Dic5311yescharacteristic
507C2×C4C2×C22⋊Dic5311yescharacteristic
508C2×C4C2×C22⋊Dic5311yescharacteristic
509C2×C4#1313318611yescharacteristic
510C2×C4#1313318611yescharacteristic
511C2×C4#1313318611yescharacteristic
512C8—4241no—
513C8—4241no—
514C8—2122no—
515C22—21no—
516C22—21no—
517C22—22no—
518C22—21no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22#1681047911yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—42no—
538C4—4242no—
539C4C2×#1313314411yescharacteristic
540C4#1681064011yescharacteristic
541C2—41no—
542C2—42no—
543C2—81no—
544C2#2027465911yescharacteristic
545C2#2027467911yescharacteristic
546C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3515C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3515D53
hypocenterC53#8415
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3515

Complements

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

NH ≅ G/Ntype
1. #20274525C2semidirect
2. #20273322C2semidirect
3. #20273227C2semidirect
4. #20273201C2semidirect
5. #20273149C2semidirect
6. #20273136C2semidirect
22. C53×#557C4semidirect
23. C53×#554C4semidirect
24. C53×#554C4semidirect
30. C53×#547C4semidirect
281. #557Dic53semidirect
282. #554Dic53semidirect
283. #554Dic53semidirect
289. #547Dic53semidirect
312. C53#8415semidirect

Maximal subgroups

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

subgroupindex
#202745252
#202733222
#202732272
#202732012
#202731492
#202731362
C53×#35152
#841553