Finite Group Databasebeta

← #27541849

Subgroups — #27541849

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287583C211yesmaximal, characteristic, complemented
2#20276956C211yesmaximal, characteristic, complemented
3#20276951C211yesmaximal, characteristic, complemented
4#20276946C211yesmaximal, characteristic, complemented
5#20276879C211yesmaximal, characteristic, complemented
6#20276872C211yesmaximal, characteristic, complemented
7#20276852C211yesmaximal, characteristic, complemented
8#20276767C211yesmaximal, characteristic, complemented
9#20276767C211yesmaximal, characteristic, complemented
10#20276747C211yesmaximal, characteristic
11#20276624C211yesmaximal, characteristic, complemented
12#20276589C211yesmaximal, characteristic, complemented
13#20276577C211yesmaximal, characteristic, complemented
14C4×#13133158C211yesmaximal, characteristic, complemented
15C53×#3396C211yesmaximal, characteristic, complemented
16#16811333—21no—
17C2×#13133160C2211yescharacteristic
18C2×#13133158C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20#16811028C2211yescharacteristic
21#16811025C2211yescharacteristic
22#16811025C2211yescharacteristic
23#16810924—21no—
24#16810924—21no—
25#16810914—21no—
26C4×D424C2211yescharacteristic
27C4×D424C2211yescharacteristic
28C4×D424C2211yescharacteristic
29#16810450C2211yescharacteristic
30#16810448C2211yescharacteristic
31#16810448C2211yescharacteristic
32C4⋊#11719908C2211yescharacteristic
33C4⋊#11719908C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810424C2211yescharacteristic
39#16810424C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810423C2211yescharacteristic
45#16810399C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#559—21no—
49C53×#495C2211yescharacteristic
50C53×#495C2211yescharacteristic
51C53×#493C2211yescharacteristic
52C53×#463—21no—
53C53×#463—21no—
54C53×#460—21no—
55C8×Q8×C53C2211yescharacteristic
56C4×Q16×C53C2211yescharacteristic
57C4×D8×C53C2211yescharacteristic
58C4×D8×C53C2211yescharacteristic
59C2×D424C2311yescharacteristic
60#13133217—21no—
61#13133209—21no—
62#13133209—21no—
63#13133195—41no—
64#13133195—41no—
65#13133193—41no—
66C4×D212—21no—
67C4×D212—21no—
68C4×D212—21no—
69C4⋊C4×D53—41no—
70#13133160—22no—
71#13133158—22no—
72#13133158—22no—
73C2×#11719908C2311yescharacteristic
74C2×#11719908C2311yescharacteristic
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77#13133151—21no—
78#13133151C2311yescharacteristic
79#13133151C2311yescharacteristic
80#13133151C2311yescharacteristic
81#13133149—21no—
82#13133149—21no—
83#13133139—21no—
84#13133139—21no—
85#13133139—21no—
86C2×Q16×C53C2311yescharacteristic
87C2×SD16×C53—21no—
88C2×SD16×C53—21no—
89C2×D8×C53—21no—
90C2×D8×C53C2311yescharacteristic
91C2×D8×C53C2311yescharacteristic
92#127×C53—21no—
93#124×C53—21no—
94#124×C53—21no—
95#124×C53—41no—
96#123×C53—41no—
97#123×C53—41no—
98C4×Q8×C53—21no—
99C4×Q8×C53D411yescharacteristic
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—41no—
103#107×C53C2311yescharacteristic
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106C4⋊C8×C53D411yescharacteristic
107#103×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110C4×C8×C53C2311yescharacteristic
111C4×C8×C53C2311yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53D811yescharacteristic
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—41no—
121C2×D212—21no—
122C2×C4×D53—41no—
123C2×C4×D53—41no—
124C2×C4×D53—41no—
125C2×Dic106—41no—
126Q16×C53—21no—
127Q16×C53—21no—
128SD16×C53—81no—
129SD16×C53—81no—
130D8×C53—21no—
131D8×C53—21no—
132D8×C53—21no—
133D8×C53—21no—
134D8×C53—81no—
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2411yescharacteristic
139C4⋊C4×C53—21no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53D811yescharacteristic
142C4⋊C4×C53D811yescharacteristic
143C4⋊C4×C53D811yescharacteristic
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53—41no—
151C42×C53C2×D411yescharacteristic
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D411yescharacteristic
154#11719921—41no—
155#11719921—41no—
156#11719921—41no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—41no—
161#11719919—41no—
162#11719919—41no—
163#11719919—41no—
164#11719919—41no—
165D424—21no—
166D424#4112yes—
167#11719908—21no—
168#11719908—21no—
169#11719908—21no—
170#11719908—21no—
171C23×C53—41no—
172C23×C53—41no—
173C22×D53—41no—
174Q8×C53—41no—
175Q8×C53—41no—
176Q8×C53—41no—
177D4×C53—41no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—82no—
182D4×C53—161no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—161no—
186C2×C4×C53—161no—
187C2×C4×C53C2×D811yescharacteristic
188C2×C4×C53C2×D811yescharacteristic
189C2×C4×C53C2×D811yescharacteristic
190C2×C4×C53C2×D811yescharacteristic
191C2×C4×C53C2×D811yescharacteristic
192C2×C4×C53C2×D811yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197D212—22no—
198D212—41no—
199C4×D53—82no—
200C4×D53—161no—
201C4×D53—161no—
202Dic106—161no—
203C8×C53—21no—
204C8×C53—21no—
205C8×C53—21no—
206C8×C53C2×#4111yescharacteristic
207C8×C532−1+411yescharacteristic
208#33937—531nomaximal, Sylow
209C22×C53—81no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53—81no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53C22×D811yescharacteristic
216C2×D53—42no—
217C2×D53—81no—
218C2×D53—81no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—81no—
226C4×C53—81no—
227C4×C53#54811yescharacteristic
228C4×C53#57711yescharacteristic
229Dic53—81no—
230Dic53—81no—
231Dic53—81no—
232#3396D5311yescharacteristic, complemented
233#3380—531no—
234#2244—531no—
235#2243—531no—
236#2242—531no—
237#2229—531no—
238#2228—531no—
239#2224—531no—
240#2210—531no—
241#2210—531no—
242#2206—531no—
243#2183—531no—
244#2177—531no—
245#2175—531no—
246C4×D16—531no—
247C2×C53—161no—
248C2×C53—161no—
249C2×C53#335811yescharacteristic
250C2×C53#341411yescharacteristic
251C2×C53#341511yescharacteristic
252D53—82no—
253#559—21no—
254#554—1061no—
255C2×SD32—531no—
256C2×D16—531no—
257C2×D16—531no—
258#500—531no—
259#499—531no—
260#499—531no—
261#495C2×D5311yescharacteristic
262#495C2×D5311yescharacteristic
263#493C2×D5311yescharacteristic
264#478—1061no—
265#478—1061no—
266#476—1061no—
267#463—21no—
268#463—21no—
269#460—21no—
270C8×Q8C2×D5311yescharacteristic
271C4×Q16C2×D5311yescharacteristic
272C4×D8—531no—
273C4×D8—531no—
274C4×D8—531no—
275C4×D8C2×D5311yescharacteristic
276C4×D8C2×D5311yescharacteristic
277#367—531no—
278#366—531no—
279#366—531no—
280C4⋊C16—531no—
281C4⋊C16—531no—
282C4⋊C16—531no—
283#358—531no—
284#358—531no—
285#358—531no—
286#357—531no—
287#357—531no—
288#357—531no—
289#357—531no—
290#357—531no—
291#357—531no—
292#357—531no—
293#346—531no—
294#346—531no—
295C4×C16—531no—
296C53#3393711yescharacteristic, complemented, Sylow
297C2×Q16C22×D5311yescharacteristic
298C2×SD16—21no—
299C2×SD16—21no—
300C2×D8—21no—
301C2×D8—531no—
302C2×D8C22×D5311yescharacteristic
303C2×D8C22×D5311yescharacteristic
304C4⋊Q8—1061no—
305#127—21no—
306#125—1061no—
307#125—1061no—
308#124—21no—
309#124—21no—
310#124—41no—
311#123—41no—
312#123—41no—
313C22⋊Q8—2121no—
314C22⋊Q8—2121no—
315C22⋊Q8—2121no—
316C4×Q8—21no—
317C4×Q8C22⋊D5311yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—41no—
321C4×D4—1061no—
322C4×D4—1061no—
323C4×D4—1061no—
324C2×C4⋊C4—2121no—
325SD32—1062no—
326D16—1062no—
327D16—1062no—
328C2×C16—531no—
329C2×C16—531no—
330C2×C16—531no—
331C2×C16—531no—
332#107—531no—
333#107—531no—
334#107—531no—
335#107—1061no—
336#107C22×D5311yescharacteristic
337#106—1061no—
338#106—1061no—
339C4⋊C8C22⋊D5311yescharacteristic
340C4⋊C8C22⋊D5311yescharacteristic
341C4⋊C8C22⋊D5311yescharacteristic
342#103—21no—
343#102—21no—
344#102—21no—
345#102—1061no—
346#102—1061no—
347#102—1061no—
348C4×C8C22×D5311yescharacteristic
349C4×C8C22×D5311yescharacteristic
350C4×C8C22×D5311yescharacteristic
351C2×Q8—21no—
352C2×Q8—2121no—
353C2×Q8#1171992211yescharacteristic
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—1061no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—2121no—
362C22×C4—2121no—
363C22×C4—2121no—
364Q16—21no—
365Q16—21no—
366SD16—81no—
367SD16—81no—
368D8—21no—
369D8—21no—
370D8—21no—
371D8—21no—
372D8—81no—
373D8—532no—
374D8—1061no—
375C2×C8C2×C22⋊D5311yescharacteristic
376C2×C8C2×C22⋊D5311yescharacteristic
377C2×C8C2×C22⋊D5311yescharacteristic
378C2×C8C23×D5311yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—41no—
381C4⋊C4—1061no—
382C4⋊C4—1061no—
383C4⋊C4—1061no—
384C4⋊C4—2121no—
385C4⋊C4—2121no—
386C4⋊C4—2121no—
387C4⋊C4—2121no—
388C4⋊C4—2121no—
389C4⋊C4#1171992211yescharacteristic
390C4⋊C4#1171992211yescharacteristic
391C4⋊C4#1171992211yescharacteristic
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—2121no—
399C22⋊C4—2121no—
400C22⋊C4—2121no—
401C42—41no—
402C42C2×C22⋊D5311yescharacteristic
403C42C2×C22⋊D5311yescharacteristic
404C42C2×C22⋊D5311yescharacteristic
405C16—1061no—
406C16—1061no—
407C16—1061no—
408C16—1061no—
409C23—41no—
410C23—41no—
411C23—2121no—
412Q8—41no—
413Q8—41no—
414Q8—41no—
415Q8—8481no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—82no—
421D4—161no—
422D4—1062no—
423D4—2121no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—161no—
428C2×C4—8481no—
429C2×C4—8481no—
430C2×C4—4242no—
431C2×C4—2121no—
432C2×C4—2121no—
433C2×C4—2121no—
434C2×C4C2×#1171992211yescharacteristic
435C2×C4C2×#1171992211yescharacteristic
436C2×C4C2×#1171992211yescharacteristic
437C2×C4C2×#1171992211yescharacteristic
438C2×C4C2×#1171992211yescharacteristic
439C2×C4C2×#1171992211yescharacteristic
440C2×C4C22×C22⋊D5311yescharacteristic
441C8—21no—
442C8—21no—
443C8—21no—
444C8#41×D5311yescharacteristic
445C8#1313326411yescharacteristic
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—4241no—
453C22—4241no—
454C22—2122no—
455C22C22×#1171992211yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—81no—
463C4—81no—
464C4—4241no—
465C4—4241no—
466C4—4241no—
467C4#1681128011yescharacteristic
468C4#1681143411yescharacteristic
469C2—161no—
470C2—161no—
471C2—4242no—
472C2#2028734611yescharacteristic
473C2#2028784911yescharacteristic
474C2#2028785411yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3396C2
socleC22×C53C22×D8
prime radical#20276747C2
hypercenter#3396D53
hypocenterC53#33937
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
1. #20287583C2semidirect
2. #20276956C2semidirect
3. #20276951C2semidirect
4. #20276946C2semidirect
5. #20276879C2semidirect
6. #20276872C2semidirect
7. #20276852C2semidirect
8. #20276767C2semidirect
9. #20276767C2semidirect
11. #20276624C2semidirect
12. #20276589C2semidirect
13. #20276577C2semidirect
14. C4×#13133158C2semidirect
15. C53×#3396C2semidirect
232. #3396D53semidirect
296. C53#33937semidirect

Maximal subgroups

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

subgroupindex
#202875832
#202769562
#202769512
#202769462
#202768792
#202768722
#202768522
#202767672
#202767472
#202766242
#202765892
#202765772
C4×#131331582
C53×#33962
#3393753