Finite Group Databasebeta

← #27369482

Subgroups — #27369482

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#20285637C211yesmaximal, characteristic, complemented
2#20284530C211yesmaximal, characteristic, complemented
3#20283826C211yesmaximal, characteristic, complemented
4#20274147C212yesmaximal, complemented
5#20274140C212yesmaximal, complemented
6#20274061C212yesmaximal, complemented
7#20274054C212yesmaximal, complemented
8#20273485C211yesmaximal, characteristic, complemented
9#20273392C211yesmaximal, characteristic, complemented
10C2×#16810368C211yesmaximal, characteristic, complemented
11C53×#3035C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16810995C2212yes—
14#16810975C2212yes—
15#16810918—21no—
16#16810913—21no—
17#16810908C2212yes—
18#16810903C2212yes—
19#16810828—21no—
20#16810828—21no—
21#16810761—22no—
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#16810369C2212yes—
35#16810368C2212yes—
36#16810368C2212yes—
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×#11719923—22no—
51C2×#11719923—22no—
52C2×#11719917—21no—
53C22×#88322—22no—
54C22×#88322—21no—
55#13133217—21no—
56#13133217—21no—
57#13133217—21no—
58#13133217C2312yes—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61#13133150—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#13133140—22no—
72#13133140—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×C53SD1614yescomplemented
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—81no—
133C2×Dic106—81no—
134Q16×C53—42no—
135SD16×C53—42no—
136SD16×C53—42no—
137D8×C53—42no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—42no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53C2×D411yescharacteristic
146C4⋊C4×C53C2×D411yescharacteristic
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53—21no—
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C42×C53#4111yescharacteristic
158C42×C53C2411yescharacteristic
159#11719923—82no—
160#11719923—82no—
161C4⋊Dic53—22no—
162C4⋊Dic53—22no—
163C4⋊Dic53—41no—
164C4⋊Dic53—41no—
165#11719917—42no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—82no—
169C2×#88322—81no—
170C2×#88322C2×D412yes—
171C2×#88322C2×D412yes—
172C23×C53—21no—
173C23×C53—41no—
174C23×C53#12111yescharacteristic
175Q8×C53—21no—
176Q8×C53—41no—
177Q8×C53—41no—
178Q8×C53C2×SD1612yes—
179D4×C53—24no—
180D4×C53—21no—
181D4×C53—41no—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—81no—
186D4×C53—82no—
187D4×C53C2×SD1612yes—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—24no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—81no—
199C2×C4×C53—82no—
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53C2×SD1612yes—
204C2×C4×C53#13512yes—
205C2×C4×C53C22×D411yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C2×C4×C53C2×#4111yescharacteristic
208C2×Dic53—42no—
209C2×Dic53—82no—
210Dic106—42no—
211Dic106—42no—
212Dic106—82no—
213C8×C53—41no—
214C8×C53—41no—
215#88322—22no—
216#88322—22no—
217#88322—42no—
218#88322—42no—
219#19408—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—22no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—81no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53#46512yes—
229C22×C53C2×#12111yescharacteristic
230C4×C53—21no—
231C4×C53—22no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—41no—
235C4×C53—41no—
236C4×C53—81no—
237C4×C53#46512yes—
238C4×C53C22×SD1611yescharacteristic
239C4×C53C2×#13511yescharacteristic
240C4×C53#57611yescharacteristic
241C4×C53#57711yescharacteristic
242Dic53—82no—
243#3206—531no—
244#3079—531no—
245#3035D5311yescharacteristic
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—161no—
257C2×C53#304511yescharacteristic
258C2×C53C2×#46511yescharacteristic
259C2×C53#306911yescharacteristic
260C2×#135—21no—
261#532C2×D5311yescharacteristic
262C2×C4⋊Q8—531no—
263C4×#41C2×D5311yescharacteristic
264#492—532no—
265#487—532no—
266#477—1061no—
267#476—1061no—
268#475—532no—
269#474—532no—
270#464—21no—
271#463—21no—
272#462C2×D5311yescharacteristic
273#461—1061no—
274#461—1061no—
275#461C2×D5311yescharacteristic
276#460C2×D5311yescharacteristic
277#459C2×D5311yescharacteristic
278#448—1062no—
279C8×Q8—532no—
280C8×D4—532no—
281C8×D4—1062no—
282#431—531no—
283#431—1061no—
284#428—1061no—
285#423—531no—
286C2×C4⋊C8—531no—
287C2×C4⋊C8—1061no—
288C2×C4⋊C8C2×D5311yescharacteristic
289#417—1061no—
290#416—21no—
291#334—532no—
292#334—532no—
293#332—532no—
294#332—532no—
295#332—532no—
296#331—532no—
297C53#1940811yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C2×#41C22⋊D5311yescharacteristic
300C22×Q8—1061no—
301#135—44no—
302C2×Q16—21no—
303C2×SD16—21no—
304C2×SD16—21no—
305C2×SD16—1062no—
306C2×SD16—1062no—
307C2×D8—21no—
308C2×M16—1061no—
309C22×C8—21no—
310C22×C8—1062no—
311C22×C8—1061no—
312C4⋊Q8—532no—
313C4⋊Q8—1061no—
314C4⋊Q8—1061no—
315C4⋊Q8—1061no—
316C4⋊Q8C22×D5311yescharacteristic
317#127C22×D5311yescharacteristic
318#124—21no—
319#121—21no—
320#121—21no—
321C4×Q8C22⋊D5311yescharacteristic
322C4×Q8C22×D5311yescharacteristic
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D5311yescharacteristic
328C4×D4C22×D5311yescharacteristic
329#117—21no—
330#117C22⋊D5311yescharacteristic
331C2×C4⋊C4—1061no—
332C2×C4⋊C4—1061no—
333C2×C42—21no—
334C2×C42C22×D5311yescharacteristic
335#107—1062no—
336#106—1062no—
337C4⋊C8—21no—
338C4⋊C8—531no—
339C4⋊C8—531no—
340C4⋊C8—532no—
341C4⋊C8—532no—
342C4⋊C8—1062no—
343C4⋊C8—1061no—
344C4⋊C8—1061no—
345C4⋊C8C22×D5311yescharacteristic
346C4⋊C8C22×D5311yescharacteristic
347#103—21no—
348#103—21no—
349#103—1062no—
350#103—1062no—
351#103—1062no—
352#102—21no—
353#102—21no—
354#102—1062no—
355C22⋊C8—532no—
356C22⋊C8—1062no—
357C4×C8—532no—
358C4×C8—1062no—
359#41—22no—
360#41—42no—
361#41—42no—
362#41#1171992314yescomplemented
363C2×Q8—21no—
364C2×Q8—1062no—
365C2×Q8—4241no—
366C2×Q8—4241no—
367C2×Q8—2121no—
368C2×Q8C2×C22⋊D5311yescharacteristic
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
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—42no—
384SD16—42no—
385SD16—42no—
386SD16—4242no—
387SD16—4242no—
388D8—42no—
389M16—2122no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—42no—
393C2×C8—532no—
394C2×C8—532no—
395C2×C8—1062no—
396C2×C8—1062no—
397C2×C8—4242no—
398C2×C8—4241no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—1062no—
404C4⋊C4—1062no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C4⋊C4C2×C22⋊D5311yescharacteristic
408C4⋊C4C2×C22⋊D5311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42D4×D5311yescharacteristic
418C42#1171994711yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—21no—
422C23—41no—
423C23#1313318911yescharacteristic
424Q8—21no—
425Q8—41no—
426Q8—41no—
427Q8—4242no—
428Q8—2122no—
429Q8—2122no—
430Q8C2×#1171992312yes—
431D4—24no—
432D4—21no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—82no—
439D4C2×#1171992312yes—
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—41no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—4242no—
453C2×C4—2122no—
454C2×C4#1313318911yescharacteristic
455C2×C4#1313318911yescharacteristic
456C2×C4#1313318911yescharacteristic
457C2×C4C2×#1171992312yes—
458C2×C4#1313323712yes—
459C2×C4C2×D4×D5311yescharacteristic
460C2×C4C22×C22⋊D5311yescharacteristic
461C2×C4C2×#1171994711yescharacteristic
462C8—41no—
463C8—41no—
464C8—1062no—
465C8—1062no—
466C8—2122no—
467C8—2122no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—41no—
472C22—41no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22#1681085012yes—
477C22C2×#1313318911yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—81no—
485C4—4242no—
486C4#1681085012yes—
487C4C22×#1171992311yescharacteristic
488C4C2×#1313323711yescharacteristic
489C4#1681142511yescharacteristic
490C4#1681143011yescharacteristic
491C2—22no—
492C2—42no—
493C2—161no—
494C2#2028425211yescharacteristic
495C2C2×#1681085011yescharacteristic
496C2#2028445411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3035C2
socleC22×C53C2×#121
hypercenter#3035D53
hypocenterC53#19408
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. #20285637C2semidirect
2. #20284530C2semidirect
3. #20283826C2semidirect
4. #20274147C2semidirect
5. #20274140C2semidirect
6. #20274061C2semidirect
7. #20274054C2semidirect
8. #20273485C2semidirect
9. #20273392C2semidirect
10. C2×#16810368C2semidirect
113. #41×C53SD16semidirect
297. C53#19408semidirect
362. #41#11719923semidirect

Maximal subgroups

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

subgroupindex
#202856372
#202845302
#202838262
#202741472
#202741402
#202740612
#202740542
#202734852
#202733922
C2×#168103682
C53×#30352
#1940853