Finite Group Databasebeta

← #27542177

Subgroups — #27542177

Order 13568.

504 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287704C211yesmaximal, characteristic, complemented
2#20276987C211yesmaximal, characteristic, complemented
3#20276984C211yesmaximal, characteristic, complemented
4#20276974C211yesmaximal, characteristic
5#20276777C211yesmaximal, characteristic, complemented
6#20276774C211yesmaximal, characteristic, complemented
7#20276770C211yesmaximal, characteristic
8#20276767C211yesmaximal, characteristic
9#20276765C211yesmaximal, characteristic, complemented
10#20276762C211yesmaximal, characteristic, complemented
11#20276624C211yesmaximal, characteristic, complemented
12#20276579C211yesmaximal, characteristic, complemented
13#20276577C211yesmaximal, characteristic, complemented
14C4×#13133163C211yesmaximal, characteristic, complemented
15C53×#3393C211yesmaximal, characteristic, complemented
16#16811346—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133163C2211yescharacteristic
19C2×#13133163C2211yescharacteristic
20C2×#13133163C2211yescharacteristic
21C2×#13133161C2211yescharacteristic
22C2×#13133161C2211yescharacteristic
23C2×#13133160C2211yescharacteristic
24C2×#13133160C2211yescharacteristic
25C2×#13133158C2211yescharacteristic
26#16811007C2211yescharacteristic
27#16811007C2211yescharacteristic
28#16811003C2211yescharacteristic
29#16810846—21no—
30#16810846—21no—
31#16810829—21no—
32C4×Dic212C2211yescharacteristic
33C4×Dic212C2211yescharacteristic
34C4×D424C2211yescharacteristic
35#16810448C2211yescharacteristic
36C4⋊#11719908C2211yescharacteristic
37C4⋊#11719908C2211yescharacteristic
38C4⋊#11719908C2211yescharacteristic
39#16810427C2211yescharacteristic
40#16810427C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810426C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810423C2211yescharacteristic
45#16810399C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#556—21no—
49C53×#495C2211yescharacteristic
50C53×#495C2211yescharacteristic
51C53×#494C2211yescharacteristic
52C53×#464—21no—
53C53×#464—21no—
54C53×#461—21no—
55C8×Q8×C53C2211yescharacteristic
56C4×Q16×C53C2211yescharacteristic
57C4×Q16×C53C2211yescharacteristic
58C4×D8×C53C2211yescharacteristic
59C2×Dic212—21no—
60C2×Dic212C2311yescharacteristic
61C2×Dic212C2311yescharacteristic
62C2×#11719913—21no—
63C2×#11719913—21no—
64C2×D424C2311yescharacteristic
65#13133217—21no—
66#13133212—41no—
67#13133212—41no—
68#13133205—21no—
69#13133205—21no—
70#13133193—41no—
71C4×Dic106—21no—
72C4×Dic106—21no—
73C4×D212—21no—
74#13133175—41no—
75#13133163—22no—
76#13133163—41no—
77#13133163—41no—
78#13133163—41no—
79#13133161—22no—
80#13133161—41no—
81#13133160—22no—
82#13133160—41no—
83#13133158—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133142—21no—
90#13133142—21no—
91#13133139—21no—
92C2×Q16×C53—21no—
93C2×Q16×C53C2311yescharacteristic
94C2×Q16×C53C2311yescharacteristic
95C2×SD16×C53—21no—
96C2×SD16×C53—21no—
97C2×D8×C53C2311yescharacteristic
98C4⋊Q8×C53—21no—
99#126×C53—41no—
100#126×C53—41no—
101#124×C53—21no—
102#124×C53—21no—
103C22⋊Q8×C53—41no—
104C4×Q8×C53—21no—
105C4×Q8×C53—21no—
106C4×Q8×C53D411yescharacteristic
107C4×D4×C53—21no—
108#117×C53—41no—
109#107×C53C2311yescharacteristic
110C4⋊C8×C53D411yescharacteristic
111C4⋊C8×C53D411yescharacteristic
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#103×C53—21no—
115#102×C53—21no—
116C4×C8×C53C2311yescharacteristic
117C4×C8×C53C2311yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53D811yescharacteristic
122C2×D4×C53—21no—
123C22×C4×C53—41no—
124C2×D212—21no—
125C2×C4×D53—41no—
126C2×Dic106—21no—
127C2×Dic106—21no—
128Q16×C53—21no—
129Q16×C53—21no—
130Q16×C53—21no—
131Q16×C53—21no—
132Q16×C53—81no—
133SD16×C53—81no—
134SD16×C53—81no—
135D8×C53—21no—
136D8×C53—21no—
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2×D411yescharacteristic
140C2×C8×C53C2411yescharacteristic
141C4⋊C4×C53—21no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53D811yescharacteristic
147C4⋊C4×C53D811yescharacteristic
148C4⋊C4×C53D811yescharacteristic
149C22⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C42×C53—41no—
155C42×C53—41no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159#11719921—41no—
160#11719921—41no—
161#11719921—41no—
162#11719921—41no—
163#11719921—41no—
164C4⋊Dic53—21no—
165C4⋊Dic53—41no—
166C4⋊Dic53—41no—
167#11719919—41no—
168#11719919—41no—
169C4×Dic53—41no—
170C4×Dic53—41no—
171Dic212—21no—
172Dic212—21no—
173Dic212—21no—
174Dic212—21no—
175Dic212—81no—
176D424—21no—
177D424—21no—
178#11719913—81no—
179#11719913—81no—
180#11719908—21no—
181#11719908—21no—
182#11719908—21no—
183#11719908#4112yes—
184C23×C53—41no—
185C22×D53—41no—
186Q8×C53—41no—
187Q8×C53—41no—
188Q8×C53—41no—
189Q8×C53—41no—
190Q8×C53—41no—
191D4×C53—41no—
192D4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—82no—
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C2×D811yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207D212—41no—
208D212—41no—
209C4×D53—82no—
210Dic106—41no—
211Dic106—41no—
212Dic106—41no—
213Dic106—41no—
214C8×C53—21no—
215C8×C53—21no—
216C8×C53—21no—
217C8×C53C2×#4111yescharacteristic
218C8×C53H(5,𝔽2)11yescharacteristic
219#33969—531nomaximal, Sylow
220C22×C53—81no—
221C22×C53—81no—
222C22×C53—81no—
223C22×C53C22×D811yescharacteristic
224C2×D53—81no—
225C2×D53—81no—
226C2×D53—81no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—81no—
234C4×C53—81no—
235C4×C53—81no—
236C4×C53#55011yescharacteristic
237C4×C53#57611yescharacteristic
238Dic53—81no—
239Dic53—81no—
240Dic53—81no—
241#3393—531no—
242#3393D5311yescharacteristic, complemented
243#2251—531no—
244#2251—531no—
245#2249—531no—
246#2211—531no—
247#2211—531no—
248#2210—531no—
249#2210—531no—
250#2209—531no—
251#2209—531no—
252#2183—531no—
253#2175—531no—
254#2175—531no—
255C4×Q32—531no—
256C2×C53—161no—
257C2×C53#339111yescharacteristic
258C2×C53#341411yescharacteristic
259C2×C53#341711yescharacteristic
260D53—161no—
261#556—21no—
262#556—1061no—
263C2×Q32—531no—
264C2×Q32—531no—
265C2×Q32—531no—
266C2×Q32—531no—
267C2×SD32—531no—
268C2×SD32—531no—
269C2×SD32—531no—
270C2×SD32—531no—
271C2×D16—531no—
272#495—531no—
273#495—531no—
274#495C2×D5311yescharacteristic
275#495C2×D5311yescharacteristic
276#494—531no—
277#494C2×D5311yescharacteristic
278#464—21no—
279#464—21no—
280#464—1061no—
281#464—1061no—
282#461—21no—
283#461—1061no—
284C8×Q8C2×D5311yescharacteristic
285C4×Q16—531no—
286C4×Q16—531no—
287C4×Q16C2×D5311yescharacteristic
288C4×Q16C2×D5311yescharacteristic
289C4×D8—531no—
290C4×D8C2×D5311yescharacteristic
291#366—531no—
292C4⋊C16—531no—
293C4⋊C16—531no—
294C4⋊C16—531no—
295#358—531no—
296#358—531no—
297#358—531no—
298#358—531no—
299#357—531no—
300#357—531no—
301#346—531no—
302#346—531no—
303C4×C16—531no—
304C53#3396911yescharacteristic, complemented, Sylow
305C2×Q16—21no—
306C2×Q16—531no—
307C2×Q16—531no—
308C2×Q16—1061no—
309C2×Q16C22×D5311yescharacteristic
310C2×Q16C22×D5311yescharacteristic
311C2×SD16—21no—
312C2×SD16—21no—
313C2×SD16—1061no—
314C2×SD16—1061no—
315C2×D8—531no—
316C2×D8C22×D5311yescharacteristic
317C4⋊Q8—21no—
318C4⋊Q8—1061no—
319#126—41no—
320#126—41no—
321#126—2121no—
322#126—2121no—
323#124—21no—
324#124—21no—
325#124—1061no—
326#124—1061no—
327C22⋊Q8—41no—
328C22⋊Q8—2121no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—1061no—
332C4×Q8—1061no—
333C4×Q8C22⋊D5311yescharacteristic
334C4×D4—21no—
335C4×D4—1061no—
336#117—41no—
337#117—2121no—
338Q32—1062no—
339Q32—2121no—
340Q32—2121no—
341Q32—2121no—
342SD32—1062no—
343SD32—1062no—
344SD32—2121no—
345SD32—2121no—
346D16—2121no—
347C2×C16—531no—
348C2×C16—531no—
349C2×C16—531no—
350C2×C16—531no—
351#107—531no—
352#107C22×D5311yescharacteristic
353C4⋊C8C22⋊D5311yescharacteristic
354C4⋊C8C22⋊D5311yescharacteristic
355C4⋊C8C22⋊D5311yescharacteristic
356#103—21no—
357#103—21no—
358#103—1061no—
359#103—1061no—
360#102—21no—
361#102—1061no—
362C4×C8C22×D5311yescharacteristic
363C4×C8C22×D5311yescharacteristic
364C4×C8C22×D5311yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—1061no—
368C2×Q8—1061no—
369C2×Q8#1171992211yescharacteristic
370C2×D4—21no—
371C2×D4—1061no—
372C22×C4—41no—
373C22×C4—2121no—
374Q16—21no—
375Q16—21no—
376Q16—21no—
377Q16—21no—
378Q16—81no—
379Q16—1061no—
380Q16—1061no—
381Q16—1061no—
382Q16—1061no—
383Q16—4241no—
384SD16—81no—
385SD16—81no—
386SD16—4241no—
387SD16—4241no—
388D8—21no—
389D8—21no—
390D8—1061no—
391D8—1061no—
392C2×C8C2×C22⋊D5311yescharacteristic
393C2×C8C2×C22⋊D5311yescharacteristic
394C2×C8C2×C22⋊D5311yescharacteristic
395C2×C8C23×D5311yescharacteristic
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—1061no—
402C4⋊C4—2121no—
403C4⋊C4—2121no—
404C4⋊C4—2121no—
405C4⋊C4—2121no—
406C4⋊C4#1171992211yescharacteristic
407C4⋊C4#1171992211yescharacteristic
408C4⋊C4#1171992211yescharacteristic
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—2121no—
415C22⋊C4—2121no—
416C22⋊C4—2121no—
417C22⋊C4—2121no—
418C22⋊C4—2121no—
419C42—41no—
420C42—41no—
421C42—2121no—
422C42—2121no—
423C42C2×C22⋊D5311yescharacteristic
424C42C2×C22⋊D5311yescharacteristic
425C42C2×C22⋊D5311yescharacteristic
426C16—532no—
427C16—1061no—
428C16—1061no—
429C16—1061no—
430C23—41no—
431C23—2121no—
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436Q8—41no—
437Q8—2121no—
438Q8—2121no—
439Q8—2121no—
440Q8—2121no—
441D4—41no—
442D4—41no—
443D4—2121no—
444D4—2121no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—82no—
449C2×C4—4242no—
450C2×C4—2121no—
451C2×C4—2121no—
452C2×C4—2121no—
453C2×C4C2×#1171992211yescharacteristic
454C2×C4C2×#1171992211yescharacteristic
455C2×C4C2×#1171992211yescharacteristic
456C2×C4C2×#1171992211yescharacteristic
457C2×C4C2×#1171992211yescharacteristic
458C2×C4C2×#1171992211yescharacteristic
459C2×C4C22×C22⋊D5311yescharacteristic
460C8—21no—
461C8—21no—
462C8—21no—
463C8#41×D5311yescharacteristic
464C8#1313326311yescharacteristic
465C22—81no—
466C22—81no—
467C22—81no—
468C22—4241no—
469C22—4241no—
470C22—4241no—
471C22C22×#1171992211yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—4241no—
482C4—4241no—
483C4—4241no—
484C4#1681129211yescharacteristic
485C4#1681142611yescharacteristic
486C2—161no—
487C2—8481no—
488C2#2028768711yescharacteristic
489C2#2028784911yescharacteristic
490C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3393C2
socleC22×C53C22×D8
prime radicalC2×#13133158C22
hypercenter#3393D53
hypocenterC53#33969
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3393

Complements

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

NH ≅ G/Ntype
1. #20287704C2semidirect
2. #20276987C2semidirect
3. #20276984C2semidirect
5. #20276777C2semidirect
6. #20276774C2semidirect
9. #20276765C2semidirect
10. #20276762C2semidirect
11. #20276624C2semidirect
12. #20276579C2semidirect
13. #20276577C2semidirect
14. C4×#13133163C2semidirect
15. C53×#3393C2semidirect
242. #3393D53semidirect
304. C53#33969semidirect

Maximal subgroups

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

subgroupindex
#202877042
#202769872
#202769842
#202769742
#202767772
#202767742
#202767702
#202767672
#202767652
#202767622
#202766242
#202765792
#202765772
C4×#131331632
C53×#33932
#3396953