Finite Group Databasebeta

← #27539230

Subgroups — #27539230

Order 13568.

534 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285374C211yesmaximal, characteristic, complemented
2#20276917C211yesmaximal, characteristic, complemented
3#20276912C211yesmaximal, characteristic, complemented
4#20276879C211yesmaximal, characteristic, complemented
5#20276872C211yesmaximal, characteristic, complemented
6#20276765C211yesmaximal, characteristic, complemented
7#20276760C211yesmaximal, characteristic, complemented
8#20276655C211yesmaximal, characteristic
9#20276655C211yesmaximal, characteristic
10#20276584C211yesmaximal, characteristic, complemented
11#20276579C211yesmaximal, characteristic, complemented
12C4×#13133163C211yesmaximal, characteristic, complemented
13C4×#13133161C211yesmaximal, characteristic, complemented
14#20276522C211yesmaximal, characteristic, complemented
15C53×#2951C211yesmaximal, characteristic
16#16811235—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133163C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20C2×#13133161C2211yescharacteristic
21#16811025C2211yescharacteristic
22#16811003C2211yescharacteristic
23#16810992—21no—
24#16810881—21no—
25#16810864C2211yescharacteristic
26#16810864C2211yescharacteristic
27#16810731—21no—
28C4×Dic212C2211yescharacteristic
29C4×Dic212C2211yescharacteristic
30#16810450C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35C4⋊#11719908C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44C22⋊#11719908C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C2×#135×C53C2211yescharacteristic
49C53×#520—21no—
50C53×#440—21no—
51C53×#440—21no—
52C4×Q16×C53C2211yescharacteristic
53C4×Q16×C53C2211yescharacteristic
54C4×D8×C53C2211yescharacteristic
55C4×D8×C53C2211yescharacteristic
56C2×#107×C53C2211yescharacteristic
57C53×#417—21no—
58C53×#405C2211yescharacteristic
59C2×Dic212—21no—
60C2×Dic212C2311yescharacteristic
61C2×Dic212C2311yescharacteristic
62#13133217—21no—
63#13133210—41no—
64#13133210—41no—
65#13133209—21no—
66#13133207—41no—
67#13133207—41no—
68#13133196—21no—
69#13133196—21no—
70C4×Dic106—21no—
71C4×Dic106—21no—
72#13133163—22no—
73#13133163—22no—
74#13133161—22no—
75#13133161—22no—
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80#13133151C2311yescharacteristic
81#13133151C2311yescharacteristic
82#13133149—21no—
83#13133142—21no—
84#13133142—21no—
85#13133142—21no—
86#13133142—21no—
87C2×#41×C53—21no—
88#135×C53—22no—
89#135×C53—22no—
90C2×Q16×C53C2311yescharacteristic
91C2×SD16×C53—21no—
92C2×D8×C53C2311yescharacteristic
93C2×M16×C53D411yescharacteristic
94C22×C8×C53C2311yescharacteristic
95C4×Q8×C53—21no—
96C4×Q8×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—41no—
100C4×D4×C53—41no—
101#117×C53—41no—
102#117×C53D411yescharacteristic
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—41no—
105#107×C53—21no—
106#107×C53C2311yescharacteristic
107#107×C53C2311yescharacteristic
108#103×C53—21no—
109#103×C53—21no—
110#102×C53—21no—
111#102×C53—21no—
112#97×C53D411yescharacteristic
113#97×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116#41×C53—24no—
117#41×C53—42no—
118C2×Q8×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C22×C4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53C2×D411yescharacteristic
126C2×Dic106—21no—
127C2×Dic106—21no—
128Q16×C53—21no—
129Q16×C53#4112yes—
130SD16×C53—24no—
131D8×C53—21no—
132D8×C53#4112yes—
133M16×C53—41no—
134C2×C8×C53—21no—
135C2×C8×C53—21no—
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2411yescharacteristic
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53D811yescharacteristic
146C4⋊C4×C53D811yescharacteristic
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53D811yescharacteristic
150C22⋊C4×C53D811yescharacteristic
151C42×C53—41no—
152C42×C53—41no—
153C42×C53C2×D411yescharacteristic
154C42×C53C2×D411yescharacteristic
155C22⋊Dic53—41no—
156C22⋊Dic53—41no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—41no—
160C4⋊Dic53—41no—
161#11719919—41no—
162#11719919—41no—
163#11719919—41no—
164#11719919—41no—
165#11719919—41no—
166#11719919—41no—
167C4×Dic53—41no—
168C4×Dic53—41no—
169Dic212—21no—
170Dic212—21no—
171Dic212—21no—
172Dic212—21no—
173Dic212—42no—
174#11719908—21no—
175#11719908—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53—41no—
179C23×C53C2×D811yescharacteristic
180Q8×C53—22no—
181Q8×C53—41no—
182D4×C53—22no—
183D4×C53—44no—
184D4×C53—41no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—44no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—161no—
192C2×C4×C53—161no—
193C2×C4×C53—161no—
194C2×C4×C53C2×D811yescharacteristic
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204Dic106—41no—
205Dic106—41no—
206Dic106—41no—
207Dic106—41no—
208C8×C53—21no—
209C8×C53—21no—
210C8×C53—21no—
211C8×C53C2×#4111yescharacteristic
212C8×C53C2×#4111yescharacteristic
213#33680—531nomaximal, Sylow
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—42no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53C22×D811yescharacteristic
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—42no—
227C4×C53—81no—
228C4×C53—81no—
229C4×C53#52511yescharacteristic
230C4×C53#57511yescharacteristic
231Dic53—81no—
232Dic53—81no—
233Dic53—81no—
234Dic53—81no—
235#3177—531no—
236#2951D5311yescharacteristic
237#2236—531no—
238#2235—531no—
239#2229—531no—
240#2228—531no—
241#2209—531no—
242#2208—531no—
243#2189—531no—
244#2189—531no—
245#2176—531no—
246#2175—531no—
247C4×Q32—531no—
248C4×SD32—531no—
249#2164—531no—
250C2×C53—41no—
251C2×C53—82no—
252C2×C53#310011yescharacteristic
253C2×C53#341611yescharacteristic
254C2×C53#341711yescharacteristic
255C2×#135C2×D5311yescharacteristic
256#541—1061no—
257#520—21no—
258C2×Q32—531no—
259C2×Q32—531no—
260C2×SD32—531no—
261C2×SD32—531no—
262#499—531no—
263#494—531no—
264#491—1061no—
265#470—1061no—
266#467—531no—
267#467—531no—
268#441—1061no—
269#440—21no—
270#440—21no—
271C4×Q16—531no—
272C4×Q16—531no—
273C4×Q16C2×D5311yescharacteristic
274C4×Q16C2×D5311yescharacteristic
275C4×D8C2×D5311yescharacteristic
276C4×D8C2×D5311yescharacteristic
277C2×#107C2×D5311yescharacteristic
278#417—21no—
279#405C2×D5311yescharacteristic
280#367—531no—
281#367—531no—
282#366—531no—
283#366—531no—
284C4⋊C16—531no—
285C4⋊C16—531no—
286#358—531no—
287#358—531no—
288#358—531no—
289#358—531no—
290#358—531no—
291#358—531no—
292#357—531no—
293#357—531no—
294C22⋊C16—531no—
295C22⋊C16—531no—
296#346—531no—
297C4×C16—531no—
298C53#3368011yescharacteristic, complemented, Sylow
299C2×#41—21no—
300#135—22no—
301#135—22no—
302C2×Q16—531no—
303C2×Q16—531no—
304C2×Q16—1061no—
305C2×Q16C22×D5311yescharacteristic
306C2×SD16—21no—
307C2×D8C22×D5311yescharacteristic
308C2×M16C22⋊D5311yescharacteristic
309C22×C8C22×D5311yescharacteristic
310C4⋊Q8—1061no—
311#126—2121no—
312#126—2121no—
313#125—1061no—
314#125—2121no—
315#125—2121no—
316C22⋊Q8—1061no—
317C22⋊Q8—1061no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—1061no—
321C4×Q8—1061no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—41no—
326#117—41no—
327#117C22⋊D5311yescharacteristic
328C2×C4⋊C4—21no—
329C2×C4⋊C4—41no—
330Q32—1062no—
331Q32—1062no—
332SD32—1062no—
333SD32—1062no—
334C2×C16—531no—
335C2×C16—531no—
336C2×C16—531no—
337C2×C16—531no—
338#107—21no—
339#107—531no—
340#107—531no—
341#107C22×D5311yescharacteristic
342#107C22×D5311yescharacteristic
343#106—1061no—
344#103—21no—
345#103—21no—
346#103—1061no—
347#103—1061no—
348#103—1061no—
349#103—1061no—
350#102—21no—
351#102—21no—
352#97C22⋊D5311yescharacteristic
353#97C22⋊D5311yescharacteristic
354C4×C8C22×D5311yescharacteristic
355C4×C8C22×D5311yescharacteristic
356#41—24no—
357#41—42no—
358C2×Q8—21no—
359C2×Q8—1061no—
360C2×Q8—1061no—
361C2×D4—21no—
362C2×D4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4C2×C22⋊D5311yescharacteristic
368Q16—21no—
369Q16—1061no—
370Q16—1061no—
371Q16—1061no—
372Q16—1061no—
373Q16—2122no—
374Q16#1171994512yes—
375SD16—24no—
376D8—21no—
377D8#1171994512yes—
378M16—41no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8C2×C22⋊D5311yescharacteristic
382C2×C8C2×C22⋊D5311yescharacteristic
383C2×C8C2×C22⋊D5311yescharacteristic
384C2×C8C23×D5311yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—1061no—
391C4⋊C4—1061no—
392C4⋊C4—2121no—
393C4⋊C4—2121no—
394C4⋊C4—2121no—
395C4⋊C4—2121no—
396C4⋊C4—2121no—
397C4⋊C4—2121no—
398C4⋊C4—2121no—
399C4⋊C4—2121no—
400C4⋊C4#1171992211yescharacteristic
401C4⋊C4#1171992211yescharacteristic
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—2121no—
405C22⋊C4—2121no—
406C22⋊C4#1171992211yescharacteristic
407C22⋊C4#1171992211yescharacteristic
408C42—41no—
409C42—41no—
410C42—2121no—
411C42—2121no—
412C42C2×C22⋊D5311yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C16—1061no—
415C16—1061no—
416C16—1061no—
417C16—1061no—
418C23—41no—
419C23C2×#1171992211yescharacteristic
420Q8—22no—
421Q8—41no—
422Q8—2121no—
423Q8—2121no—
424Q8—2121no—
425Q8—2121no—
426D4—22no—
427D4—44no—
428D4—41no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—161no—
436C2×C4—161no—
437C2×C4—161no—
438C2×C4—2121no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4C2×#1171992211yescharacteristic
443C2×C4C2×#1171992211yescharacteristic
444C2×C4C2×#1171992211yescharacteristic
445C2×C4C2×#1171992211yescharacteristic
446C2×C4C2×#1171992211yescharacteristic
447C2×C4C22×C22⋊D5311yescharacteristic
448C8—21no—
449C8—21no—
450C8—21no—
451C8C2×#1171994511yescharacteristic
452C8C2×#1171994511yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22C22×#1171992211yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—81no—
467C4—81no—
468C4—4241no—
469C4—4241no—
470C4—4241no—
471C4—4241no—
472C4#1681113411yescharacteristic
473C4#1681142111yescharacteristic
474C2—41no—
475C2—82no—
476C2#2028471511yescharacteristic
477C2#2028786211yescharacteristic
478C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2951C2
socleC22×C53C22×D8
prime radicalC2×#135×C53C22
hypercenter#2951D53
hypocenterC53#33680
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #20285374C2semidirect
2. #20276917C2semidirect
3. #20276912C2semidirect
4. #20276879C2semidirect
5. #20276872C2semidirect
6. #20276765C2semidirect
7. #20276760C2semidirect
10. #20276584C2semidirect
11. #20276579C2semidirect
12. C4×#13133163C2semidirect
13. C4×#13133161C2semidirect
14. #20276522C2semidirect
298. C53#33680semidirect

Maximal subgroups

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

subgroupindex
#202853742
#202769172
#202769122
#202768792
#202768722
#202767652
#202767602
#202766552
#202765842
#202765792
C4×#131331632
C4×#131331612
#202765222
C53×#29512
#3368053