Finite Group Databasebeta

← #28248481

Subgroups — #28248481

Order 13568.

1810 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288118C211yesmaximal, characteristic
2#20283656C211yesmaximal, characteristic, complemented
3#20283654C212yesmaximal, complemented
4#20283652C211yesmaximal, characteristic, complemented
5#20283651C218yesmaximal, complemented
6#20283649C211yesmaximal, characteristic, complemented
7#20283647C212yesmaximal, complemented
8#20283644C218yesmaximal, complemented
9#20283277C211yesmaximal, characteristic
10#20283276C212yesmaximal
11C2×#16810577C211yesmaximal, characteristic
12#405×D53C212yesmaximal
13C53×#2960C211yesmaximal, characteristic, complemented
14C22×#11719912C2211yescharacteristic
15C2×M16×D53C2212yes—
16C22×C8×D53C2211yescharacteristic
17#16811253C414yes—
18#16811252C414yes—
19#16811250C414yes—
20#16811248C414yes—
21#16811180C2211yescharacteristic
22#16811177C2218yes—
23#16811176C2212yes—
24C2×#13133174C2211yescharacteristic
25#117×D53C2212yes—
26#16810754C2218yescomplemented
27#16810753C2214yescomplemented
28#16810752C2218yescomplemented
29#16810752C2218yescomplemented
30#16810751C2218yescomplemented
31#16810750C2218yescomplemented
32C8×Dic106C2214yescomplemented
33C8×Dic106C2218yescomplemented
34#16810651C2212yes—
35#16810650C2212yes—
36#16810649C2212yes—
37#16810648C2218yes—
38#16810648C2218yes—
39#16810647C2211yescharacteristic
40#16810646C2212yes—
41#16810645C2218yes—
42#16810581C2212yes—
43#16810580C2214yes—
44#16810579C2212yes—
45#16810578C2212yes—
46#16810577C2214yes—
47#16810577C2214yes—
48#16810577C2214yes—
49C2×#13133128C2211yescharacteristic
50C2×#13133127C2211yescharacteristic
51#97×D53C2214yes—
52C2×C8×Dic53C2211yescharacteristic
53C4×C8×D53C2214yes—
54C53×#533C2211yescharacteristic
55C53×#446C2214yescomplemented
56C8×Q8×C53C2214yescomplemented
57C53×#424C2212yes—
58C53×#423C2211yescharacteristic
59C53×#405C2212yes—
60C2×C4×C8×C53C2211yescharacteristic
61C22×C4×D53C2311yescharacteristic
62C2×#11719917C2312yes—
63C2×#11719912—24no—
64C2×#11719912C2314yes—
65C2×#11719912C2314yes—
66M16×D53—28no—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C2×C8×D53Q818yescomplemented
70C2×C8×D53C2314yes—
71C2×C8×D53C2312yes—
72C2×C8×D53C2312yes—
73#13133217C2×C412yes—
74#13133216C2×C414yes—
75#13133216C2×C418yes—
76#13133215C2×C418yes—
77#13133209C2×C412yes—
78#13133208C2×C414yes—
79#13133207C2×C418yes—
80#13133207C2×C418yes—
81#13133197C2×C418yes—
82#13133196C2×C414yes—
83#13133195C2×C418yes—
84#13133195C2×C418yes—
85#13133194C2×C418yes—
86#13133193C2×C418yes—
87C4×Dic106C2314yes—
88C4×Dic106C2318yes—
89#13133176C2312yes—
90#13133175C2×C414yes—
91#13133174C2×C412yes—
92#13133174C2×C412yes—
93#13133174C2314yes—
94#13133174C2314yes—
95#13133173C2×C414yes—
96C2×#11719919C2×C412yes—
97C4⋊C4×D53C2×C414yes—
98C2×#11719921C2×C412yes—
99C22⋊C4×D53C2×C414yes—
100C2×C4×Dic53C2311yescharacteristic
101C42×D53C2314yes—
102C4⋊#88322C2314yes—
103C4⋊#88322C2318yes—
104#13133146C2318yes—
105#13133146C2318yes—
106#13133146C2318yes—
107#13133128C2314yes—
108#13133128C2314yes—
109#13133127C2314yes—
110#13133127C2314yes—
111#13133127C2314yes—
112C4×#88322C2314yes—
113C8×Dic53C2314yes—
114C8×Dic53C2314yes—
115C8×Dic53C2312yes—
116C8×Dic53C2312yes—
117C2×M16×C53C2312yes—
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C4⋊Q8×C53C2×C412yes—
121#125×C53C2×C412yes—
122C22⋊Q8×C53C2×C414yes—
123C4×Q8×C53C2314yes—
124#117×C53C2312yes—
125C2×C42×C53C2311yescharacteristic
126C4⋊C8×C53C2314yes—
127C4⋊C8×C53C2318yes—
128#97×C53C2314yes—
129C4×C8×C53C2312yes—
130C4×C8×C53C2314yes—
131C4×C8×C53C2312yes—
132C23×D53C22×C411yescharacteristic
133C2×Q8×C53C22×C414yes—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2411yescharacteristic
136C22×Dic53C22×C412yes—
137C22×Dic53C22×C411yescharacteristic
138C2×C4×D53C22×C418yes—
139C2×C4×D53C22×C412yes—
140C2×C4×D53C2×Q814yes—
141C2×C4×D53#4114yes—
142C2×C4×D53C2412yes—
143C2×Dic106C22×C418yes—
144C2×Dic106C22×C414yes—
145M16×C53—24no—
146C2×C8×C53—22no—
147C2×C8×C53C2×Q814yes—
148C2×C8×C53C2412yes—
149C2×C8×C53C2414yes—
150C2×C8×C53C2411yescharacteristic
151C2×C8×C53C2411yescharacteristic
152C4⋊C4×C53C22×C414yes—
153C4⋊C4×C53C22×C418yes—
154C4⋊C4×C53C22×C414yes—
155C22⋊C4×C53C22×C414yes—
156C42×C53C22×C411yescharacteristic
157C42×C53C22×C411yescharacteristic
158C42×C53C2412yes—
159C42×C53C2414yes—
160C22⋊Dic53C22×C414yes—
161#11719921C22×C418yes—
162#11719921C22×C414yes—
163#11719921C22×C414yes—
164C4⋊Dic53C22×C414yes—
165C4⋊Dic53C22×C418yes—
166C4⋊Dic53C22×C414yes—
167#11719919C22×C418yes—
168#11719919C22×C414yes—
169#11719919C22×C414yes—
170#11719919C22×C418yes—
171#11719919C22×C418yes—
172#11719919C22×C418yes—
173C4×Dic53C22×C418yes—
174C4×Dic53C22×C412yes—
175C4×Dic53C2414yes—
176C4×Dic53C2412yes—
177#11719917—24no—
178C2×#88322—22no—
179C2×#88322C2×Q814yes—
180C2×#88322C2412yes—
181C2×#88322C2414yes—
182C2×#88322C2412yes—
183#11719912—28no—
184#11719912—28no—
185C8×D53—28no—
186C8×D53C2×Q818yes—
187C8×D53C2×Q818yes—
188C23×C53C23×C411yescharacteristic
189C22×D53—42no—
190C22×D53C4×Q814yes—
191C22×D53C23×C412yes—
192Q8×C53—44no—
193C2×C4×C53—42no—
194C2×C4×C53C23×C412yes—
195C2×C4×C53C23×C418yes—
196C2×C4×C53C23×C412yes—
197C2×C4×C53C23×C411yescharacteristic
198C2×C4×C53C22×Q812yes—
199C2×C4×C53C2×#4112yes—
200C2×C4×C53C2511yescharacteristic
201C2×Dic53—42no—
202C2×Dic53C4×Q814yes—
203C2×Dic53C23×C418yes—
204C2×Dic53C23×C414yes—
205C2×Dic53C23×C412yes—
206C4×D53—48no—
207C4×D53C4×Q818yes—
208C4×D53C22×Q814yes—
209C4×D53C2×#4114yes—
210Dic106—44no—
211Dic106—48no—
212C8×C53—24no—
213C8×C53—22no—
214C8×C53C22×Q812yes—
215C8×C53C22×Q812yes—
216#88322—24no—
217#88322—22no—
218#88322C22×Q814yes—
219#60191—531nomaximal, Sylow
220C22×C53—22no—
221C22×C53C2×C4×Q812yes—
222C22×C53C24×C411yescharacteristic
223C2×D53—24no—
224C2×D53—44no—
225C2×D53C2×C4×Q814yes—
226C4×C53—22no—
227C4×C53—28no—
228C4×C53—22no—
229C4×C53C2×C4×Q812yes—
230C4×C53C23×Q811yescharacteristic
231C4×C53C22×#4111yescharacteristic
232Dic53—28no—
233Dic53—24no—
234Dic53C2×C4×Q814yes—
235#3458—531no—
236#2961—538no—
237#2961—532no—
238#2961—531no—
239#2961—531no—
240#2960—538no—
241#2960—532no—
242#2960—531no—
243#2960D5311yescharacteristic, complemented
244#2905—532no—
245#2905—531no—
246C2×#405—532no—
247C2×#405—531no—
248C2×C53—22no—
249C2×C53C22×C4×Q811yescharacteristic
250C2×C53#357312yes—
251D53—24no—
252C22×M16—532no—
253C22×M16—531no—
254C23×C8—531no—
255#544—534no—
256#544—534no—
257#544—534no—
258#543—534no—
259#533—538no—
260#533—532no—
261#533—531no—
262#533C2×D5311yescharacteristic
263C2×#117—532no—
264C2×#117—531no—
265#446—538no—
266#446—538no—
267#446—538no—
268#446—538no—
269#446—538no—
270#446—534no—
271#446C2×D5314yescomplemented
272C8×Q8—538no—
273C8×Q8—534no—
274C8×Q8C2×D5314yescomplemented
275#424—532no—
276#424—532no—
277#424—532no—
278#424—538no—
279#424—538no—
280#424C2×D5312yes—
281#423—532no—
282#423—531no—
283#423—538no—
284#423C2×D5311yescharacteristic
285#405—532no—
286#405—532no—
287#405—532no—
288#405—534no—
289#405—534no—
290#405—534no—
291#405—534no—
292#405C2×D5312yes—
293C2×#97—531no—
294C2×#97—531no—
295C2×#97—534no—
296C2×C4×C8—531no—
297C2×C4×C8—534no—
298C2×C4×C8C2×D5311yescharacteristic
299C53#6019111yescharacteristic, complemented, Sylow
300C23×C4—531no—
301C2×M16—532no—
302C2×M16—534no—
303C2×M16—534no—
304C2×M16—1064no—
305C2×M16—1068no—
306C2×M16C22×D5312yes—
307C22×C8—531no—
308C22×C8—538no—
309C22×C8—532no—
310C22×C8—532no—
311C22×C8—531no—
312C22×C8—534no—
313C22×C8C22×D5311yescharacteristic
314C22×C8C22×D5311yescharacteristic
315C4⋊Q8—538no—
316C4⋊Q8—538no—
317C4⋊Q8—534no—
318C4⋊Q8—532no—
319C4⋊Q8C4×D5312yes—
320#125—538no—
321#125—538no—
322#125—534no—
323#125—532no—
324#125C4×D5312yes—
325C22⋊Q8—538no—
326C22⋊Q8—534no—
327C22⋊Q8—538no—
328C22⋊Q8—538no—
329C22⋊Q8—538no—
330C22⋊Q8—538no—
331C22⋊Q8C4×D5314yes—
332C4×Q8—538no—
333C4×Q8—534no—
334C4×Q8C22×D5314yes—
335#117—532no—
336#117—534no—
337#117—534no—
338#117—532no—
339#117—532no—
340#117—534no—
341#117—534no—
342#117C22×D5312yes—
343C2×C4⋊C4—532no—
344C2×C4⋊C4—534no—
345C2×C22⋊C4—532no—
346C2×C22⋊C4—534no—
347C2×C42—531no—
348C2×C42—534no—
349C2×C42C22×D5311yescharacteristic
350C4⋊C8—538no—
351C4⋊C8—538no—
352C4⋊C8—538no—
353C4⋊C8—538no—
354C4⋊C8—534no—
355C4⋊C8C22×D5318yes—
356C4⋊C8C22×D5314yes—
357#97—534no—
358#97—534no—
359#97—534no—
360#97—534no—
361#97—534no—
362#97C22×D5314yes—
363C4×C8—534no—
364C4×C8—532no—
365C4×C8—534no—
366C4×C8—532no—
367C4×C8—534no—
368C4×C8C22×D5312yes—
369C4×C8C22×D5314yes—
370C4×C8C22×D5312yes—
371C24—531no—
372C2×Q8—538no—
373C2×Q8—534no—
374C2×Q8C2×C4×D5314yes—
375C22×C4—531no—
376C22×C4—532no—
377C22×C4—534no—
378C22×C4—534no—
379C22×C4—532no—
380C22×C4—532no—
381C22×C4—538no—
382C22×C4C2×C4×D5312yes—
383C22×C4C23×D5311yescharacteristic
384M16—24no—
385M16—1068no—
386M16—1068no—
387M16—1064no—
388C2×C8—22no—
389C2×C8—534no—
390C2×C8—532no—
391C2×C8—538no—
392C2×C8—538no—
393C2×C8—532no—
394C2×C8—534no—
395C2×C8—1068no—
396C2×C8—1062no—
397C2×C8Q8×D5314yes—
398C2×C8C23×D5314yes—
399C2×C8C23×D5312yes—
400C2×C8C23×D5311yescharacteristic
401C2×C8C23×D5311yescharacteristic
402C4⋊C4—538no—
403C4⋊C4—534no—
404C4⋊C4—534no—
405C4⋊C4—534no—
406C4⋊C4—538no—
407C4⋊C4—538no—
408C4⋊C4—538no—
409C4⋊C4—538no—
410C4⋊C4—534no—
411C4⋊C4C2×C4×D5314yes—
412C4⋊C4C2×C4×D5314yes—
413C4⋊C4C2×C4×D5318yes—
414C22⋊C4—534no—
415C22⋊C4—538no—
416C22⋊C4—534no—
417C22⋊C4—534no—
418C22⋊C4C2×C4×D5314yes—
419C42—538no—
420C42—532no—
421C42—532no—
422C42—534no—
423C42C2×C4×D5311yescharacteristic
424C42C2×C4×D5311yescharacteristic
425C42C23×D5314yes—
426C42C23×D5312yes—
427C23—532no—
428C23—534no—
429C23—2122no—
430C23C22×C4×D5311yescharacteristic
431Q8—44no—
432Q8—2124no—
433Q8—2128no—
434C2×C4—42no—
435C2×C4—534no—
436C2×C4—532no—
437C2×C4—538no—
438C2×C4—534no—
439C2×C4—534no—
440C2×C4—538no—
441C2×C4—534no—
442C2×C4—2128no—
443C2×C4—2122no—
444C2×C4C22×C4×D5318yes—
445C2×C4C22×C4×D5312yes—
446C2×C4C22×C4×D5312yes—
447C2×C4C22×C4×D5311yescharacteristic
448C2×C4C2×Q8×D5312yes—
449C2×C4#41×D5312yes—
450C2×C4C24×D5311yescharacteristic
451C8—22no—
452C8—24no—
453C8—534no—
454C8—1064no—
455C8—1062no—
456C8C2×Q8×D5312yes—
457C8C2×Q8×D5312yes—
458C22—22no—
459C22—534no—
460C22—1064no—
461C22—2124no—
462C22C4×Q8×D5312yes—
463C22C23×C4×D5311yescharacteristic
464C4—22no—
465C4—22no—
466C4—28no—
467C4—534no—
468C4—1068no—
469C4—1064no—
470C4C4×Q8×D5312yes—
471C4C22×Q8×D5311yescharacteristic
472C4C2×#41×D5311yescharacteristic
473C2—22no—
474C2—1064no—
475C2C2×C4×Q8×D5311yescharacteristic
476C2#2028942612yes—
subgroupG/H
centerC2×C8C23×D53
commutator, socleC22×C53C24×C4
FrattiniC2×C4C24×D53
FittingC53×#2960C2
prime radicalC23×D53C22×C4
cosocleC2×C4×C53C25
hypercenter#2960D53
hypocenterC53#60191
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ #2960

Complements

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

NH ≅ G/Ntype
2. #20283656C2semidirect
3. #20283654C2semidirect
4. #20283652C2semidirect
5. #20283651C2semidirect
6. #20283649C2semidirect
7. #20283647C2semidirect
8. #20283644C2semidirect
13. C53×#2960C2semidirect
26. #16810754C22semidirect
27. #16810753C22semidirect
28. #16810752C22semidirect
29. #16810752C22semidirect
30. #16810751C22semidirect
31. #16810750C22semidirect
32. C8×Dic106C22semidirect
33. C8×Dic106C22semidirect
55. C53×#446C22semidirect
56. C8×Q8×C53C22semidirect
69. C2×C8×D53Q8semidirect
243. #2960D53semidirect
271. #446C2×D53semidirect
274. C8×Q8C2×D53semidirect
299. C53#60191semidirect

Maximal subgroups

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

subgroupindex
#202881182
#202836562
#202836542
#202836522
#202836512
#202836492
#202836472
#202836442
#202832772
#202832762
C2×#168105772
#405×D532
C53×#29602
#6019153