Finite Group Databasebeta

← #27369488

Subgroups — #27369488

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285640C211yesmaximal, characteristic, complemented
2#20284170C211yesmaximal, characteristic
3#20283826C211yesmaximal, characteristic, complemented
4#20274143C212yesmaximal, complemented
5#20274136C212yesmaximal, complemented
6#20274065C212yesmaximal, complemented
7#20274058C212yesmaximal, complemented
8#20273487C211yesmaximal, characteristic, complemented
9#20273393C211yesmaximal, characteristic, complemented
10C2×#16810369C211yesmaximal, characteristic, complemented
11C53×#3079C211yesmaximal, characteristic, complemented
12C2×#13133237—21no—
13#16811173C2211yescharacteristic
14#16810997C2212yes—
15#16810973C2212yes—
16#16810845—21no—
17#16810838—21no—
18#16810833C2211yescharacteristic
19#16810828C2211yescharacteristic
20#16810825—21no—
21#16810825—21no—
22#16810823C2211yescharacteristic
23#16810818C2211yescharacteristic
24#16810762—22no—
25Q8×#88322C2212yes—
26D4×#88322—22no—
27D4×#88322C2212yes—
28#16810676—21no—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322—21no—
31C2×C4⋊#88322C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810617—21no—
34#16810373C2212yes—
35#16810373C2212yes—
36#16810370C2212yes—
37#16810369C2212yes—
38#16810369C2212yes—
39#16810366C2212yes—
40C2×C4⋊Q8×C53C2211yescharacteristic
41C4×#41×C53C2211yescharacteristic
42C53×#477—21no—
43C53×#476—21no—
44C53×#475C2212yes—
45C53×#474C2212yes—
46C53×#428—21no—
47C53×#423C2211yescharacteristic
48C53×#417—21no—
49C2×#11719945—21no—
50#13133237—44no—
51C2×#11719925—21no—
52C2×#11719923—21no—
53C2×#11719924—22no—
54C2×#11719924—22no—
55C2×#11719924—21no—
56C2×#11719922—21no—
57C22×#88322—22no—
58C22×#88322—21no—
59C22×#88322—21no—
60#13133217C2311yescharacteristic
61#13133214C2311yescharacteristic
62#13133205—21no—
63#13133191—21no—
64#13133191—21no—
65C4×D212—21no—
66C4×D212—21no—
67C4⋊#88322—21no—
68C4⋊#88322—21no—
69C4⋊#88322—22no—
70C4⋊#88322D411yescharacteristic
71C4⋊#88322D411yescharacteristic
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2312yes—
75#13133141—22no—
76#13133141—22no—
77#13133140—21no—
78#13133140—21no—
79#13133137—21no—
80#13133137—21no—
81C22⋊#88322—22no—
82C22⋊#88322D412yes—
83C4×#88322—22no—
84C4×#88322C2312yes—
85C2×#41×C53D411yescharacteristic
86C22×Q8×C53—21no—
87C2×M16×C53—21no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53C2312yes—
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53D411yescharacteristic
97C4×D4×C53C2311yescharacteristic
98#117×C53—21no—
99#117×C53D411yescharacteristic
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C42×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#107×C53—22no—
105#106×C53—22no—
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#103×C53—22no—
109#102×C53—22no—
110#41×C53—22no—
111#41×C53SD1614yescomplemented
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×Q8×C53—81no—
115C2×Q8×C53—81no—
116C2×Q8×C53C2×D411yescharacteristic
117C2×D4×C53—21no—
118C2×D4×C53C2×D411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—41no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53#4111yescharacteristic
126C2×C22⋊D53—41no—
127#11719945—42no—
128#11719945—42no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—21no—
132C2×C4×D53—41no—
133C2×Dic106—21no—
134M16×C53—42no—
135C2×C8×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53—41no—
142C4⋊C4×C53C2×D411yescharacteristic
143C4⋊C4×C53C2×D411yescharacteristic
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154#11719925—42no—
155#11719924—42no—
156#11719924—82no—
157#11719924—82no—
158#11719923—42no—
159#11719922—42no—
160#11719921—41no—
161#11719921—41no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C2×#88322—22no—
165C2×#88322—21no—
166C2×#88322—21no—
167C2×#88322—42no—
168C2×#88322—82no—
169C2×#88322—81no—
170C2×#88322C2×D412yes—
171C2×#88322C2×D412yes—
172C23×C53—21no—
173C23×C53#12111yescharacteristic
174C22×D53—41no—
175Q8×C53—21no—
176Q8×C53—42no—
177Q8×C53—42no—
178Q8×C53—82no—
179Q8×C53C2×SD1612yes—
180D4×C53—24no—
181D4×C53—21no—
182D4×C53C2×SD1612yes—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—24no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—42no—
193C2×C4×C53—81no—
194C2×C4×C53—82no—
195C2×C4×C53#12111yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53C2×SD1612yes—
199C2×C4×C53#13512yes—
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C2×#4111yescharacteristic
203C22⋊D53—82no—
204C2×Dic53—41no—
205D212—41no—
206D212—41no—
207D212—41no—
208D212—41no—
209D212—81no—
210C4×D53—82no—
211Dic106—41no—
212Dic106—41no—
213C8×C53—42no—
214#88322—22no—
215#88322—22no—
216#88322—41no—
217#88322—41no—
218#88322—42no—
219#19408—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—22no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53#46512yes—
226C22×C53C2×#12111yescharacteristic
227C2×D53—81no—
228C2×D53—81no—
229C2×D53—81no—
230C4×C53—21no—
231C4×C53—22no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—41no—
235C4×C53—41no—
236C4×C53—82no—
237C4×C53#46512yes—
238C4×C53C22×SD1611yescharacteristic
239C4×C53C2×#13511yescharacteristic
240C4×C53#57611yescharacteristic
241C4×C53#57711yescharacteristic
242Dic53—81no—
243#3206—531no—
244#3079D5311yescharacteristic, complemented
245#3035—531no—
246#2988—531no—
247#1680—532no—
248#1679—532no—
249#1666—532no—
250#1665—532no—
251#1570—531no—
252#1547—531no—
253C2×#332—531no—
254C2×C53—22no—
255C2×C53—42no—
256C2×C53#304511yescharacteristic
257C2×C53C2×#46511yescharacteristic
258C2×C53#306911yescharacteristic
259D53—161no—
260C2×#135—1061no—
261#532—531no—
262C2×C4⋊Q8C2×D5311yescharacteristic
263C4×#41C2×D5311yescharacteristic
264#492—532no—
265#487—532no—
266#477—21no—
267#476—21no—
268#475C2×D5312yes—
269#474C2×D5312yes—
270#464—1061no—
271#463—1061no—
272#462—531no—
273#461—531no—
274#461—1061no—
275#461—1061no—
276#460—531no—
277#459—531no—
278#448—1062no—
279C8×Q8—532no—
280C8×D4—532no—
281C8×D4—1062no—
282#431—531no—
283#431—1061no—
284#428—21no—
285#423C2×D5311yescharacteristic
286C2×C4⋊C8—531no—
287C2×C4⋊C8—531no—
288C2×C4⋊C8—1061no—
289#417—21no—
290#416—1061no—
291#334—532no—
292#334—532no—
293#332—532no—
294#332—532no—
295#332—532no—
296#331—532no—
297C53#1940811yescharacteristic, complemented, Sylow
298C2×#41—1061no—
299C2×#41C22⋊D5311yescharacteristic
300C22×Q8—21no—
301#135—2124no—
302C2×Q16—1061no—
303C2×SD16—1062no—
304C2×SD16—1062no—
305C2×SD16—1061no—
306C2×SD16—1061no—
307C2×D8—1061no—
308C2×M16—21no—
309C22×C8—1062no—
310C22×C8—1061no—
311C22×C8—1061no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—531no—
316C4⋊Q8C22×D5312yes—
317#127—531no—
318#124—1061no—
319#121—1061no—
320#121—1061no—
321C4×Q8C22⋊D5311yescharacteristic
322C4×Q8C22×D5311yescharacteristic
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—1061no—
326C4×D4—1061no—
327C4×D4C22⋊D5311yescharacteristic
328C4×D4C22×D5311yescharacteristic
329#117—21no—
330#117C22⋊D5311yescharacteristic
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C42—21no—
334C2×C42C22×D5311yescharacteristic
335#107—22no—
336#106—22no—
337C4⋊C8—21no—
338C4⋊C8—531no—
339C4⋊C8—531no—
340C4⋊C8—532no—
341C4⋊C8—531no—
342C4⋊C8—531no—
343C4⋊C8—1062no—
344C4⋊C8—1061no—
345C4⋊C8—1061no—
346C4⋊C8C22×D5312yes—
347#103—22no—
348#103—1062no—
349#103—1062no—
350#103—1061no—
351#103—1061no—
352#102—22no—
353#102—1061no—
354#102—1061no—
355C22⋊C8—532no—
356C22⋊C8—1062no—
357C4×C8—532no—
358C4×C8—1062no—
359#41—22no—
360#41—2122no—
361#41—2122no—
362#41#1171992414yescomplemented
363C2×Q8—22no—
364C2×Q8—41no—
365C2×Q8—81no—
366C2×Q8—81no—
367C2×Q8—1061no—
368C2×Q8C2×C22⋊D5311yescharacteristic
369C2×D4—21no—
370C2×D4—1061no—
371C2×D4—1061no—
372C2×D4—1061no—
373C2×D4—2121no—
374C2×D4C2×C22⋊D5311yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—2121no—
380C22×C4D4×D5311yescharacteristic
381C22×C4#1171994711yescharacteristic
382C22×C4C2×C22⋊D5311yescharacteristic
383Q16—2122no—
384SD16—4242no—
385SD16—4242no—
386SD16—2122no—
387SD16—2122no—
388D8—2122no—
389M16—42no—
390C2×C8—22no—
391C2×C8—532no—
392C2×C8—532no—
393C2×C8—1062no—
394C2×C8—1061no—
395C2×C8—1061no—
396C2×C8—4242no—
397C2×C8—4241no—
398C2×C8—2122no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4C2×C22⋊D5311yescharacteristic
408C4⋊C4C2×C22⋊D5311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—2121no—
413C22⋊C4—2121no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42D4×D5311yescharacteristic
418C42#1171994711yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—21no—
422C23—2121no—
423C23#1313318911yescharacteristic
424Q8—21no—
425Q8—42no—
426Q8—42no—
427Q8—82no—
428Q8—2121no—
429Q8—2121no—
430Q8C2×#1171992412yes—
431D4—24no—
432D4—21no—
433D4—4241no—
434D4—4242no—
435D4—2121no—
436D4—2121no—
437D4—2121no—
438D4—2121no—
439D4C2×#1171992412yes—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—4242no—
453C2×C4—2121no—
454C2×C4#1313318911yescharacteristic
455C2×C4#1313318911yescharacteristic
456C2×C4#1313318911yescharacteristic
457C2×C4C2×#1171992412yes—
458C2×C4#1313323712yes—
459C2×C4C2×D4×D5311yescharacteristic
460C2×C4C22×C22⋊D5311yescharacteristic
461C2×C4C2×#1171994711yescharacteristic
462C8—42no—
463C8—1062no—
464C8—1062no—
465C8—2121no—
466C8—2121no—
467C8—2122no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—41no—
472C22—41no—
473C22—4241no—
474C22—4241no—
475C22—4241no—
476C22#1681085412yes—
477C22C2×#1313318911yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—82no—
485C4—4241no—
486C4#1681085412yes—
487C4C22×#1171992411yescharacteristic
488C4C2×#1313323711yescharacteristic
489C4#1681142711yescharacteristic
490C4#1681143311yescharacteristic
491C2—22no—
492C2—42no—
493C2—8481no—
494C2#2028425811yescharacteristic
495C2C2×#1681085411yescharacteristic
496C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3079C2
socleC22×C53C2×#121
prime radical#20284170C2
hypercenter#3079D53
hypocenterC53#19408
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3079

Complements

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

NH ≅ G/Ntype
1. #20285640C2semidirect
3. #20283826C2semidirect
4. #20274143C2semidirect
5. #20274136C2semidirect
6. #20274065C2semidirect
7. #20274058C2semidirect
8. #20273487C2semidirect
9. #20273393C2semidirect
10. C2×#16810369C2semidirect
11. C53×#3079C2semidirect
111. #41×C53SD16semidirect
244. #3079D53semidirect
297. C53#19408semidirect
362. #41#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856402
#202841702
#202838262
#202741432
#202741362
#202740652
#202740582
#202734872
#202733932
C2×#168103692
C53×#30792
#1940853