Finite Group Databasebeta

← #27369556

Subgroups — #27369556

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285840C211yesmaximal, characteristic, complemented
2#20284530C211yesmaximal, characteristic, complemented
3#20283826C211yesmaximal, characteristic, complemented
4#20274129C212yesmaximal, complemented
5#20274122C212yesmaximal, complemented
6#20274095C212yesmaximal, complemented
7#20274088C212yesmaximal, complemented
8#20273485C211yesmaximal, characteristic, complemented
9#20273392C211yesmaximal, characteristic, complemented
10C2×#16810371C211yesmaximal, characteristic, complemented
11C53×#3035C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16811019C2212yes—
14#16810969C2212yes—
15#16810939—22no—
16#16810918C2212yes—
17#16810913—21no—
18#16810913—21no—
19#16810913C2212yes—
20#16810908—21no—
21#16810903—21no—
22Q8×#88322C2212yes—
23D4×#88322—22no—
24D4×#88322C2212yes—
25#16810676—21no—
26#16810676C2211yescharacteristic
27#16810666—21no—
28#16810647C2211yescharacteristic
29C2×C4⋊#88322—21no—
30C2×C4⋊#88322C2211yescharacteristic
31#16810624—21no—
32#16810373C2212yes—
33#16810373C2212yes—
34#16810371C2212yes—
35#16810371C2212yes—
36#16810369C2212yes—
37#16810367C2212yes—
38C2×#135×C53—21no—
39C53×#532C2211yescharacteristic
40C4×#41×C53C2211yescharacteristic
41C53×#464—21no—
42C53×#463—21no—
43C53×#462C2211yescharacteristic
44C53×#461C2211yescharacteristic
45C53×#460C2211yescharacteristic
46C53×#459C2211yescharacteristic
47C2×C4⋊C8×C53C2211yescharacteristic
48C53×#416—21no—
49C22×Dic106—21no—
50C2×#11719917—21no—
51C22×#88322—22no—
52C22×#88322—21no—
53#13133217—21no—
54#13133217—21no—
55#13133217—21no—
56#13133217C2312yes—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—21no—
59#13133150—22no—
60#13133148—22no—
61#13133148—22no—
62#13133148—22no—
63C4⋊#88322—21no—
64C4⋊#88322—21no—
65C4⋊#88322—22no—
66C4⋊#88322D411yescharacteristic
67C4⋊#88322D411yescharacteristic
68C4⋊#88322C2312yes—
69C4⋊#88322C2312yes—
70#13133141—22no—
71#13133138—22no—
72#13133138—22no—
73#13133138—22no—
74C22⋊#88322—22no—
75C22⋊#88322D412yes—
76C4×#88322—22no—
77C4×#88322C2312yes—
78C2×#41×C53—21no—
79C2×#41×C53D411yescharacteristic
80#135×C53—44no—
81C2×Q16×C53—21no—
82C2×SD16×C53—21no—
83C2×SD16×C53—21no—
84C2×D8×C53—21no—
85C22×C8×C53—21no—
86C4⋊Q8×C53C2311yescharacteristic
87#127×C53C2311yescharacteristic
88#124×C53—21no—
89#121×C53—21no—
90#121×C53—21no—
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53C2311yescharacteristic
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53D411yescharacteristic
98C4×D4×C53C2311yescharacteristic
99#117×C53—21no—
100#117×C53D411yescharacteristic
101C2×C42×C53—21no—
102C2×C42×C53C2311yescharacteristic
103C4⋊C8×C53—21no—
104C4⋊C8×C53C2311yescharacteristic
105C4⋊C8×C53C2311yescharacteristic
106#103×C53—21no—
107#103×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110#41×C53—22no—
111#41×C53—42no—
112#41×C53—42no—
113#41×C53SD1614yes—
114C2×Q8×C53—21no—
115C2×Q8×C53C2×D411yescharacteristic
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C2×D4×C53C2×D411yescharacteristic
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—41no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53#4111yescharacteristic
129C22×Dic53—41no—
130C2×Dic106—22no—
131C2×Dic106—41no—
132C2×Dic106—82no—
133Q16×C53—42no—
134SD16×C53—42no—
135SD16×C53—42no—
136D8×C53—42no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—42no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53C2×D411yescharacteristic
145C4⋊C4×C53C2×D411yescharacteristic
146C22⋊C4×C53—21no—
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C42×C53—21no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53C2×D411yescharacteristic
155C42×C53C2×D411yescharacteristic
156C42×C53#4111yescharacteristic
157C42×C53C2411yescharacteristic
158C4⋊Dic53—22no—
159C4⋊Dic53—22no—
160C4⋊Dic53—41no—
161C4⋊Dic53—41no—
162#11719917—42no—
163C2×#88322—22no—
164C2×#88322—22no—
165C2×#88322—82no—
166C2×#88322—81no—
167C2×#88322C2×D412yes—
168C2×#88322C2×D412yes—
169C23×C53—21no—
170C23×C53—41no—
171C23×C53#12111yescharacteristic
172Q8×C53—21no—
173Q8×C53—41no—
174Q8×C53—41no—
175Q8×C53C2×SD1612yes—
176D4×C53—24no—
177D4×C53—21no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—41no—
182D4×C53—81no—
183D4×C53—82no—
184D4×C53C2×SD1612yes—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—24no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—81no—
196C2×C4×C53—82no—
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53C2×SD1612yes—
201C2×C4×C53#13512yes—
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C2×#4111yescharacteristic
205C2×Dic53—42no—
206C2×Dic53—82no—
207Dic106—82no—
208Dic106—82no—
209C8×C53—41no—
210C8×C53—41no—
211#88322—22no—
212#88322—22no—
213#88322—42no—
214#88322—42no—
215#19414—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—22no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—81no—
222C22×C53—81no—
223C22×C53—81no—
224C22×C53#46512yes—
225C22×C53C2×#12111yescharacteristic
226C4×C53—21no—
227C4×C53—22no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—81no—
233C4×C53#46512yes—
234C4×C53C22×SD1611yescharacteristic
235C4×C53C2×#13511yescharacteristic
236C4×C53#57711yescharacteristic
237C4×C53#57811yescharacteristic
238Dic53—82no—
239#3227—531no—
240#3079—531no—
241#3035D5311yescharacteristic
242#2988—531no—
243#1678—532no—
244#1677—532no—
245#1672—532no—
246#1671—532no—
247#1570—531no—
248#1547—531no—
249C2×#333—531no—
250C2×C53—22no—
251C2×C53—42no—
252C2×C53—161no—
253C2×C53#304611yescharacteristic
254C2×C53C2×#46511yescharacteristic
255C2×C53#306811yescharacteristic
256C2×#135—21no—
257#532C2×D5311yescharacteristic
258C2×C4⋊Q8—531no—
259C4×#41C2×D5311yescharacteristic
260#498—532no—
261#486—532no—
262#481—1062no—
263#477—532no—
264#476—532no—
265#476—1061no—
266#476—1061no—
267#475—1061no—
268#474—1061no—
269#464—21no—
270#463—21no—
271#462C2×D5311yescharacteristic
272#461C2×D5311yescharacteristic
273#460C2×D5311yescharacteristic
274#459C2×D5311yescharacteristic
275C8×Q8—532no—
276C8×D4—532no—
277C8×D4—1062no—
278#431—531no—
279#431—1061no—
280#428—1061no—
281#423—531no—
282C2×C4⋊C8—531no—
283C2×C4⋊C8—1061no—
284C2×C4⋊C8C2×D5311yescharacteristic
285#417—1061no—
286#416—21no—
287#334—532no—
288#334—532no—
289#333—532no—
290#333—532no—
291#332—532no—
292#331—532no—
293C53#1941411yescharacteristic, complemented, Sylow
294C2×#41—21no—
295C2×#41C22⋊D5311yescharacteristic
296C22×Q8—1061no—
297#135—44no—
298C2×Q16—21no—
299C2×SD16—21no—
300C2×SD16—21no—
301C2×D8—21no—
302C2×M16—1061no—
303C22×C8—21no—
304C22×C8—1062no—
305C22×C8—1061no—
306C4⋊Q8—532no—
307C4⋊Q8—1061no—
308C4⋊Q8—1061no—
309C4⋊Q8—1061no—
310C4⋊Q8C22×D5311yescharacteristic
311#127C22×D5311yescharacteristic
312#124—21no—
313#121—21no—
314#121—21no—
315C4×Q8C22⋊D5311yescharacteristic
316C4×Q8C22×D5311yescharacteristic
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4C22⋊D5311yescharacteristic
322C4×D4C22×D5311yescharacteristic
323#117—21no—
324#117C22⋊D5311yescharacteristic
325C2×C4⋊C4—1061no—
326C2×C4⋊C4—1061no—
327C2×C42—21no—
328C2×C42C22×D5311yescharacteristic
329#107—1062no—
330#106—1062no—
331#106—1062no—
332#106—1062no—
333C4⋊C8—21no—
334C4⋊C8—531no—
335C4⋊C8—531no—
336C4⋊C8—532no—
337C4⋊C8—532no—
338C4⋊C8—1062no—
339C4⋊C8—1061no—
340C4⋊C8—1061no—
341C4⋊C8C22×D5311yescharacteristic
342C4⋊C8C22×D5311yescharacteristic
343#103—21no—
344#103—21no—
345#103—1062no—
346#102—21no—
347#102—21no—
348#102—1062no—
349#102—1062no—
350#102—1062no—
351C22⋊C8—532no—
352C22⋊C8—1062no—
353C4×C8—532no—
354C4×C8—1062no—
355#41—22no—
356#41—42no—
357#41—42no—
358#41#1171992314yes—
359C2×Q8—21no—
360C2×Q8—1062no—
361C2×Q8—4242no—
362C2×Q8—2121no—
363C2×Q8C2×C22⋊D5311yescharacteristic
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4C2×C22⋊D5311yescharacteristic
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—2121no—
375C22×C4D4×D5311yescharacteristic
376C22×C4#1171994711yescharacteristic
377C22×C4C2×C22⋊D5311yescharacteristic
378Q16—42no—
379SD16—42no—
380SD16—42no—
381D8—42no—
382M16—2122no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—42no—
386C2×C8—532no—
387C2×C8—532no—
388C2×C8—1062no—
389C2×C8—1062no—
390C2×C8—4242no—
391C2×C8—4241no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—1062no—
397C4⋊C4—1062no—
398C4⋊C4—2121no—
399C4⋊C4—2121no—
400C4⋊C4C2×C22⋊D5311yescharacteristic
401C4⋊C4C2×C22⋊D5311yescharacteristic
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42D4×D5311yescharacteristic
411C42#1171994711yescharacteristic
412C42C2×C22⋊D5311yescharacteristic
413C42C23×D5311yescharacteristic
414C23—21no—
415C23—41no—
416C23#1313318911yescharacteristic
417Q8—21no—
418Q8—41no—
419Q8—41no—
420Q8—4242no—
421Q8—4242no—
422Q8C2×#1171992312yes—
423D4—24no—
424D4—21no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—81no—
430D4—82no—
431D4C2×#1171992312yes—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—4242no—
445C2×C4—2122no—
446C2×C4#1313318911yescharacteristic
447C2×C4#1313318911yescharacteristic
448C2×C4#1313318911yescharacteristic
449C2×C4C2×#1171992312yes—
450C2×C4#1313323712yes—
451C2×C4C2×D4×D5311yescharacteristic
452C2×C4C22×C22⋊D5311yescharacteristic
453C2×C4C2×#1171994711yescharacteristic
454C8—41no—
455C8—41no—
456C8—1062no—
457C8—1062no—
458C8—2122no—
459C8—2122no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22#1681085012yes—
469C22C2×#1313318911yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—4242no—
478C4#1681085012yes—
479C4C22×#1171992311yescharacteristic
480C4C2×#1313323711yescharacteristic
481C4#1681143011yescharacteristic
482C4#1681143711yescharacteristic
483C2—22no—
484C2—42no—
485C2—161no—
486C2#2028426311yescharacteristic
487C2C2×#1681085011yescharacteristic
488C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3035C2
socleC22×C53C2×#121
hypercenter#3035D53
hypocenterC53#19414
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #20285840C2semidirect
2. #20284530C2semidirect
3. #20283826C2semidirect
4. #20274129C2semidirect
5. #20274122C2semidirect
6. #20274095C2semidirect
7. #20274088C2semidirect
8. #20273485C2semidirect
9. #20273392C2semidirect
10. C2×#16810371C2semidirect
293. C53#19414semidirect

Maximal subgroups

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

subgroupindex
#202858402
#202845302
#202838262
#202741292
#202741222
#202740952
#202740882
#202734852
#202733922
C2×#168103712
C53×#30352
#1941453