Finite Group Databasebeta

← #27301667

Subgroups — #27301667

Order 13568.

640 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#20283198C211yesmaximal, characteristic, complemented
2#20276407C211yesmaximal, characteristic, complemented
3#20276365C211yesmaximal, characteristic, complemented
4#20274690C211yesmaximal, characteristic, complemented
5#20272844C211yesmaximal, characteristic, complemented
6#20272838C211yesmaximal, characteristic, complemented
7C53×#3010C211yesmaximal, characteristic
8#16811390—21no—
9#16810640—21no—
10#16810632—21no—
11#16810607—22no—
12#16810607C2211yescharacteristic
13#16810607C2211yescharacteristic
14#16810607C2211yescharacteristic
15C2×#13133136—22no—
16C2×#13133136—21no—
17#16810597C2211yescharacteristic
18#16810420—22no—
19#16810410—22no—
20C2×2−1+4×C53C2211yescharacteristic
21C2×AGL(1,ℤ/8)×C53C411yescharacteristic, complemented
22C22×SD16×C53C411yescharacteristic, complemented
23C53×#535C2211yescharacteristic
24C53×#449—21no—
25C53×#449—21no—
26C53×#448—21no—
27C53×#448—21no—
28C53×#419C411yescharacteristic, complemented
29C53×#416C411yescharacteristic, complemented
30C53×#408C2211yescharacteristic
31#13133224—42no—
32#13133224—42no—
33C2×#11719917—24no—
34C2×#11719917—21no—
35C2×#11719917—21no—
36C2×#11719917—21no—
37#13133176—21no—
38#13133144—42no—
39#13133141—42no—
40#13133136—24no—
41#13133136—21no—
42#13133136—21no—
43#13133136D412yes—
44#13133134—21no—
45#13133134—21no—
46#13133134D412yes—
47#13133132—21no—
48#13133132—21no—
492−1+4×C53—42no—
502−1+4×C53—42no—
51C2×#41×C53—21no—
52C2×#41×C53—21no—
53C2×#41×C53—21no—
54C2×#41×C53C2×C411yescharacteristic
55C2×#41×C53D411yescharacteristic
56C2×#41×C53D411yescharacteristic
57C2×#41×C53C2311yescharacteristic
58C22×Q8×C53—21no—
59C22×Q8×C53C2×C411yescharacteristic
60C22×Q8×C53D411yescharacteristic
61C22×Q8×C53D411yescharacteristic
62C22×D4×C53C2×C411yescharacteristic
63AGL(1,ℤ/8)×C53—42no—
64C2×SD16×C53—21no—
65C2×SD16×C53—21no—
66C2×SD16×C53—21no—
67C2×SD16×C53—42no—
68C2×D8×C53—21no—
69C2×M16×C53C2×C411yescharacteristic
70C22×C8×C53C2×C411yescharacteristic
71#127×C53—21no—
72#124×C53—21no—
73#121×C53—21no—
74#121×C53—21no—
75#120×C53—21no—
76#120×C53—21no—
77#117×C53C2×C411yescharacteristic
78#103×C53—21no—
79#103×C53—21no—
80#103×C53—21no—
81#102×C53—21no—
82C22⋊C8×C53—21no—
83C22⋊C8×C53—21no—
84C24×C53—21no—
85#41×C53—22no—
86#41×C53—22no—
87#41×C53—22no—
88#41×C53—22no—
89#41×C53—41no—
90#41×C53—41no—
91#41×C53—41no—
92#41×C53—41no—
93#41×C53—44no—
94#41×C53—44no—
95#41×C53—44no—
96C2×Q8×C53—21no—
97C2×Q8×C53—21no—
98C2×Q8×C53—21no—
99C2×Q8×C53—22no—
100C2×Q8×C53—22no—
101C2×Q8×C53—41no—
102C2×Q8×C53—41no—
103C2×Q8×C53—41no—
104C2×Q8×C53—41no—
105C2×Q8×C53—44no—
106C2×Q8×C53C22⋊C411yescharacteristic
107C2×Q8×C53C22⋊C411yescharacteristic
108C2×Q8×C53C22⋊C411yescharacteristic
109C2×Q8×C53C2×D411yescharacteristic
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53C22⋊C411yescharacteristic
120C2×D4×C53C2×D411yescharacteristic
121C2×D4×C53#4111yescharacteristic
122C2×D4×C53#4111yescharacteristic
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132SD16×C53—42no—
133SD16×C53—42no—
134SD16×C53—42no—
135D8×C53—42no—
136M16×C53—41no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—41no—
140C4⋊C4×C53—21no—
141C22⋊C4×C53—21no—
142C22⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C42×C53—21no—
145C22⋊Dic53—41no—
146C4⋊Dic53—41no—
147C4×Dic53—41no—
148#11719917—22no—
149#11719917—22no—
150#11719917—44no—
151#11719917—41no—
152#11719917—41no—
153#11719917—41no—
154#11719917—41no—
155C2×#88322—44no—
156C2×#88322—41no—
157C2×#88322—41no—
158C2×#88322—41no—
159C23×C53—21no—
160C23×C53—41no—
161C23×C53—81no—
162C23×C53—161no—
163C23×C53C4×D411yescharacteristic
164C23×C53#12111yescharacteristic
165C23×C53#12311yescharacteristic
166Q8×C53—21no—
167Q8×C53—21no—
168Q8×C53—21no—
169Q8×C53—21no—
170Q8×C53—21no—
171Q8×C53—41no—
172Q8×C53—42no—
173Q8×C53—42no—
174Q8×C53—42no—
175Q8×C53#9912yes—
176D4×C53—24no—
177D4×C53—21no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—81no—
186D4×C53—161no—
187D4×C53—161no—
188D4×C53#9912yes—
189C2×C4×C53—24no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—41no—
205C2×C4×C53C2×C22⋊C411yescharacteristic
206C2×C4×C53C4×D411yescharacteristic
207C2×C4×C53#12011yescharacteristic
208C2×C4×C53#12111yescharacteristic
209C2×Dic53—81no—
210C8×C53—41no—
211C8×C53—41no—
212#88322—42no—
213#88322—42no—
214#88322—81no—
215#13098—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—22no—
219C22×C53—21no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53—81no—
227C22×C53—162no—
228C22×C53#38611yescharacteristic
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—22no—
232C4×C53—21no—
233C4×C53—41no—
234C4×C53—41no—
235C4×C53—81no—
236C4×C53C2×#9911yescharacteristic
237C4×C53C2×#9911yescharacteristic
238Dic53—161no—
239#3010D5311yescharacteristic
240#2889—531no—
241#2134—531no—
242#2122—531no—
243#1796—531no—
244#1395—531no—
245#1392—531no—
246C2×C53—21no—
247C2×C53—41no—
248C2×C53—41no—
249C2×C53—41no—
250C2×C53—42no—
251C2×C53—161no—
252C2×C53#189811yescharacteristic
253C2×2−1+4C2×D5311yescharacteristic
254C2×AGL(1,ℤ/8)Dic5311yescharacteristic, complemented
255C22×SD16Dic5311yescharacteristic, complemented
256#568—1061no—
257#535C2×D5311yescharacteristic
258#449—21no—
259#449—21no—
260#448—21no—
261#448—21no—
262#421—1061no—
263#419—1061no—
264#419Dic5311yescharacteristic, complemented
265#416Dic5311yescharacteristic, complemented
266#413—531no—
267#413—531no—
268#413—531no—
269#413—1062no—
270C2×#101—1062no—
271C2×#101—1061no—
272#410—531no—
273#408C2×D5311yescharacteristic
274#355—1062no—
275AΣL(1,𝔽16)—1062no—
276C53#1309811yescharacteristic, complemented, Sylow
2772−1+4—42no—
2782−1+4—42no—
279C2×#41—21no—
280C2×#41—21no—
281C2×#41—21no—
282C2×#41C2×Dic5311yescharacteristic
283C2×#41C22⋊D5311yescharacteristic
284C2×#41C22⋊D5311yescharacteristic
285C2×#41C22×D5311yescharacteristic
286C22×Q8—21no—
287C22×Q8C2×Dic5311yescharacteristic
288C22×Q8C22⋊D5311yescharacteristic
289C22×Q8C22⋊D5311yescharacteristic
290C22×D4C2×Dic5311yescharacteristic
291AGL(1,ℤ/8)—42no—
292C2×SD16—21no—
293C2×SD16—21no—
294C2×SD16—21no—
295C2×SD16—42no—
296C2×D8—21no—
297#131—2122no—
298#131—2122no—
299C2×M16—1064no—
300C2×M16—1061no—
301C2×M16—1061no—
302C2×M16—1061no—
303C2×M16C2×Dic5311yescharacteristic
304C22×C8C2×Dic5311yescharacteristic
305#127—21no—
306#124—21no—
307#121—21no—
308#121—21no—
309#120—21no—
310#120—21no—
311#117—1061no—
312#117C2×Dic5311yescharacteristic
313#104—2122no—
314#103—21no—
315#103—21no—
316#103—21no—
317#103—2122no—
318#102—21no—
319#101—532no—
320#101—1064no—
321#101—1061no—
322#101—1061no—
323#100—532no—
324#100—1061no—
325#100—1061no—
326#99—1061no—
327#99—1061no—
328C22⋊C8—21no—
329C22⋊C8—21no—
330C24—21no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—44no—
340#41—44no—
341#41—44no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—44no—
352C2×Q8C22⋊Dic5311yescharacteristic
353C2×Q8C22⋊Dic5311yescharacteristic
354C2×Q8C22⋊Dic5311yescharacteristic
355C2×Q8D4×D5311yescharacteristic
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4C22⋊Dic5311yescharacteristic
366C2×D4D4×D5311yescharacteristic
367C2×D4#1171994711yescharacteristic
368C2×D4#1171994711yescharacteristic
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4C22×Dic5311yescharacteristic
376C22×C4C2×C22⋊D5311yescharacteristic
377C22×C4C2×C22⋊D5311yescharacteristic
378SD16—42no—
379SD16—42no—
380SD16—42no—
381D8—42no—
382M16—41no—
383M16—1062no—
384M16—1062no—
385M16—2121no—
386M16—2121no—
387M16—2124no—
388M16—2121no—
389M16—2121no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—41no—
393C2×C8—2121no—
394C2×C8—2124no—
395C2×C8—2121no—
396C2×C8—2121no—
397C4⋊C4—21no—
398C4⋊C4—2121no—
399C22⋊C4—21no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—2121no—
403C42—21no—
404C42—2121no—
405C23—21no—
406C23—41no—
407C23—81no—
408C23—161no—
409C23D4×Dic5311yescharacteristic
410C23#1313318911yescharacteristic
411C23#1313319811yescharacteristic
412Q8—21no—
413Q8—21no—
414Q8—21no—
415Q8—21no—
416Q8—21no—
417Q8—41no—
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8#1313313212yes—
422D4—24no—
423D4—21no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—42no—
428D4—42no—
429D4—41no—
430D4—41no—
431D4—81no—
432D4—161no—
433D4—161no—
434D4#1313313212yes—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—4241no—
452C2×C4C2×C22⋊Dic5311yescharacteristic
453C2×C4D4×Dic5311yescharacteristic
454C2×C4#1313318511yescharacteristic
455C2×C4#1313318911yescharacteristic
456C8—41no—
457C8—41no—
458C8—4241no—
459C8—2122no—
460C8—2122no—
461C22—21no—
462C22—21no—
463C22—22no—
464C22—21no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—162no—
473C22#1681050311yescharacteristic
474C4—21no—
475C4—21no—
476C4—22no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—81no—
481C4—8481no—
482C4C2×#1313313211yescharacteristic
483C4C2×#1313313211yescharacteristic
484C2—21no—
485C2—41no—
486C2—41no—
487C2—41no—
488C2—42no—
489C2—161no—
490C2#2027516911yescharacteristic
subgroupG/H
centerC2#20275169
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
Fitting, prime radicalC53×#3010C2
socleC2×C53#1898
cosocleC2×#41×C53C23
hypercenter#3010D53
hypocenterC53#13098
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3010

Complements

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

NH ≅ G/Ntype
1. #20283198C2semidirect
2. #20276407C2semidirect
3. #20276365C2semidirect
4. #20274690C2semidirect
5. #20272844C2semidirect
6. #20272838C2semidirect
21. C2×AGL(1,ℤ/8)×C53C4semidirect
22. C22×SD16×C53C4semidirect
28. C53×#419C4semidirect
29. C53×#416C4semidirect
254. C2×AGL(1,ℤ/8)Dic53semidirect
255. C22×SD16Dic53semidirect
264. #419Dic53semidirect
265. #416Dic53semidirect
276. C53#13098semidirect

Maximal subgroups

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

subgroupindex
#202831982
#202764072
#202763652
#202746902
#202728442
#202728382
C53×#30102
#1309853