Finite Group Databasebeta

← #27366986

Subgroups — #27366986

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284750C211yesmaximal, characteristic, complemented
2C2×#16810820C211yesmaximal, characteristic
3M16×Dic106C211yesmaximal, characteristic, complemented
4#20274280C212yesmaximal, complemented
5#20274252C212yesmaximal, complemented
6#20274156C212yesmaximal, complemented
7#20274128C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10#20273383C211yesmaximal, characteristic
11C53×#3138C211yesmaximal, characteristic
12C22×#11719923—21no—
13C2×C4×Dic106C2211yescharacteristic
14#16810956—21no—
15#16810915C2211yescharacteristic, complemented
16#16810915C2211yescharacteristic, complemented
17#16810830C2211yescharacteristic, complemented
18#16810830C2211yescharacteristic, complemented
19#16810820—21no—
20#16810820—21no—
21#16810820C2212yes—
22#16810820C2212yes—
23#16810778—21no—
24#16810753C2212yescomplemented
25#16810751—22no—
26C8×Dic106C2212yescomplemented
27C4×#11719925—21no—
28C4×#11719925—21no—
29#16810676C2211yescharacteristic
30#16810647C2211yescharacteristic
31#16810646—21no—
32C2×C4⋊#88322C2211yescharacteristic
33C2×#13133138—21no—
34M16×Dic53—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810372C2212yescomplemented
38#16810369C2212yescomplemented
39#16810367C2212yes—
40#16810367C2212yes—
41C53×#533C2211yescharacteristic
42C2×#127×C53C2211yescharacteristic
43C53×#490—22no—
44C53×#486C2212yes—
45C53×#486C2212yes—
46C53×#418—21no—
47C53×#418—21no—
48C4×M16×C53C2211yescharacteristic
49C22×Dic106D411yescharacteristic
50C2×#11719925—21no—
51C2×#11719925—21no—
52C2×#11719923—22no—
53C2×#11719923—22no—
54C2×#11719923—44no—
55C2×#11719917—21no—
56C22×#88322—21no—
57C4×Dic106—21no—
58C4×Dic106—21no—
59C4×Dic106D411yescharacteristic
60C4×Dic106D411yescharacteristic
61C4×Dic106C2311yescharacteristic
62C4×Dic106C2311yescharacteristic
63C2×C4⋊Dic53D411yescharacteristic
64C2×#11719919—21no—
65C2×C4×Dic53—21no—
66#13133150—21no—
67#13133150—21no—
68C4⋊#88322—21no—
69C4⋊#88322—21no—
70C4⋊#88322C2312yes—
71C4⋊#88322C2312yes—
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74#13133146—22no—
75#13133146—22no—
76#13133141—21no—
77#13133141—21no—
78#13133140—21no—
79#13133140—21no—
80#13133138—22no—
81#13133138—22no—
82C22⋊#88322—21no—
83#13133127—22no—
84C4×#88322—21no—
85C8×Dic53—22no—
86C22×D4×C53—21no—
87C22×D4×C53—21no—
88C22×D4×C53—21no—
89C2×M16×C53D411yescharacteristic
90C2×M16×C53D411yescharacteristic
91C4⋊Q8×C53C2311yescharacteristic
92C4⋊Q8×C53C2311yescharacteristic
93#127×C53—21no—
94#127×C53—21no—
95#127×C53—21no—
96#127×C53C2312yes—
97#125×C53—21no—
98C22⋊Q8×C53—21no—
99C22⋊Q8×C53—21no—
100C4×Q8×C53—21no—
101C4×Q8×C53—21no—
102#117×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#102×C53—22no—
105#102×C53—22no—
106#102×C53—22no—
107#102×C53—22no—
108#97×C53D412yes—
109C4×C8×C53C2312yes—
110C24×C53—41no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—42no—
117C2×D4×C53—42no—
118C2×D4×C53—41no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—42no—
122C2×D4×C53—42no—
123C2×D4×C53—82no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53#4111yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C2×Dic106—22no—
130C2×Dic106—21no—
131C2×Dic106—41no—
132C2×Dic106C2×D411yescharacteristic
133C2×Dic106C2×D411yescharacteristic
134M16×C53—22no—
135M16×C53—22no—
136C2×C8×C53C2×D412yes—
137C2×C8×C53C2×D412yes—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C42×C53—41no—
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C42×C53#4111yescharacteristic
149C42×C53C2411yescharacteristic
150#11719925—42no—
151#11719925—42no—
152#11719923—44no—
153#11719923—44no—
154C4⋊Dic53—21no—
155C4⋊Dic53C2×D411yescharacteristic
156C4⋊Dic53C2×D411yescharacteristic
157#11719919—21no—
158#11719919—21no—
159#11719919—21no—
160#11719919—21no—
161C4×Dic53—21no—
162C4×Dic53—21no—
163C4×Dic53—21no—
164C4×Dic53—21no—
165#11719917—42no—
166C2×#88322—22no—
167C2×#88322—21no—
168C2×#88322—21no—
169C2×#88322—22no—
170C2×#88322—42no—
171C23×C53—42no—
172C23×C53—82no—
173C23×C53—82no—
174C23×C53—82no—
175C23×C53#12111yescharacteristic
176Q8×C53—41no—
177Q8×C53—41no—
178Q8×C53—81no—
179D4×C53—42no—
180D4×C53—42no—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—82no—
184D4×C53—82no—
185D4×C53—82no—
186D4×C53—82no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12111yescharacteristic
195C2×C4×C53#13512yes—
196C2×C4×C53#13712yes—
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C2×#4111yescharacteristic
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—42no—
205C2×Dic53—81no—
206Dic106—21no—
207Dic106—21no—
208Dic106—21no—
209Dic106—42no—
210Dic106AGL(1,ℤ/8)12yes—
211C8×C53—22no—
212C8×C53—22no—
213#88322—42no—
214#88322—42no—
215#88322—41no—
216#88322—41no—
217#19174—531nomaximal, Sylow
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—82no—
221C22×C53—82no—
222C22×C53—82no—
223C22×C53—164no—
224C22×C53#46412yes—
225C22×C53C2×#12111yescharacteristic
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—81no—
230C4×C53—81no—
231C4×C53#46412yes—
232C4×C53C2×#13511yescharacteristic
233C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
234C4×C53C2×#13711yescharacteristic
235C4×C53#57511yescharacteristic
236Dic53—22no—
237Dic53—41no—
238Dic53—41no—
239Dic53—41no—
240#3138D5311yescharacteristic
241#3104—531no—
242C2×#460—531no—
243Q8×M16—531no—
244#1702—532no—
245#1698—532no—
246#1682—532no—
247#1678—532no—
248#1565—531no—
249#1550—531no—
250#1545—531no—
251C2×C53—22no—
252C2×C53—162no—
253C2×C53C2×#46411yescharacteristic
254C2×C53#305511yescharacteristic
255C2×C53#306811yescharacteristic
256C22×SD16—1061no—
257#533C2×D5311yescharacteristic
258C2×#127C2×D5311yescharacteristic
259C2×C4×Q8—531no—
260#490—22no—
261#486C2×D5312yes—
262#486C2×D5312yes—
263#484—1061no—
264#477—531no—
265#477—531no—
266#462—531no—
267#462—531no—
268#460—532no—
269#460—532no—
270#460—1061no—
271#460—1061no—
272#451—1061no—
273#446—532no—
274#446—1062no—
275C8×Q8—532no—
276C4×Q16—1061no—
277C4×Q16—1061no—
278#431—531no—
279#423—531no—
280#423—1061no—
281C2×C4⋊C8—531no—
282#418—21no—
283#418—21no—
284C2×#102—1061no—
285C4×M16—1061no—
286C4×M16C2×D5311yescharacteristic
287#335—532no—
288#334—532no—
289#333—532no—
290#332—532no—
291#331—532no—
292#331—532no—
293C53#1917411yescharacteristic, complemented, Sylow
294C22×Q8—531no—
295C22×D4—21no—
296C22×D4—21no—
297C22×D4—21no—
298C2×Q16—1061no—
299C2×Q16—1061no—
300C2×SD16—1062no—
301C2×SD16—1062no—
302C2×SD16—2124no—
303C2×M16—1061no—
304C2×M16C22⋊D5311yescharacteristic
305C2×M16C22⋊D5311yescharacteristic
306C22×C8—1061no—
307C4⋊Q8C22×D5311yescharacteristic
308C4⋊Q8C22×D5311yescharacteristic
309#127—21no—
310#127—21no—
311#127—21no—
312#127C22×D5312yes—
313#125—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—531no—
319C4×Q8—531no—
320C4×Q8—531no—
321C4×Q8—531no—
322C4×Q8—1061no—
323C4×Q8—1061no—
324#117—21no—
325C2×C4⋊C4—531no—
326C2×C4⋊C4—1061no—
327C2×C42—1061no—
328C2×C42C22×D5311yescharacteristic
329#107—1061no—
330#107—1061no—
331C4⋊C8—532no—
332C4⋊C8—531no—
333C4⋊C8—531no—
334C4⋊C8—532no—
335C4⋊C8—1062no—
336C4⋊C8—1062no—
337C4⋊C8—1061no—
338C4⋊C8—1061no—
339#103—1061no—
340#103—1061no—
341#103—1061no—
342#103—1061no—
343#102—22no—
344#102—22no—
345#102—22no—
346#102—22no—
347#102—1062no—
348#102—1062no—
349C22⋊C8—1061no—
350#97—1062no—
351#97C22⋊D5312yes—
352C4×C8—1062no—
353C4×C8—1061no—
354C4×C8C22×D5312yes—
355C24—41no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—531no—
359C2×Q8—531no—
360C2×Q8—1062no—
361C2×Q8—1061no—
362C2×Q8—2121no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—82no—
374C22×C4—1061no—
375C22×C4—1061no—
376C22×C4D4×D5311yescharacteristic
377C22×C4#1171994711yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379Q16—2122no—
380Q16—2122no—
381SD16—2124no—
382SD16—2124no—
383M16—22no—
384M16—22no—
385M16—2122no—
386C2×C8—1061no—
387C2×C8—1061no—
388C2×C8—1062no—
389C2×C8—1062no—
390C2×C8—2122no—
391C2×C8C2×C22⋊D5312yes—
392C2×C8C2×C22⋊D5312yes—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—531no—
400C4⋊C4—531no—
401C4⋊C4—1061no—
402C4⋊C4—1061no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4—1061no—
406C22⋊C4—41no—
407C42—41no—
408C42—1061no—
409C42—1061no—
410C42—1061no—
411C42—1061no—
412C42D4×D5311yescharacteristic
413C42#1171994711yescharacteristic
414C42C2×C22⋊D5311yescharacteristic
415C42C23×D5311yescharacteristic
416C23—42no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23#1313318811yescharacteristic
421Q8—41no—
422Q8—41no—
423Q8—81no—
424Q8—532no—
425Q8—1061no—
426Q8—1061no—
427Q8—1061no—
428Q8—2122no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—1061no—
443C2×C4—1061no—
444C2×C4—1061no—
445C2×C4—1061no—
446C2×C4—4241no—
447C2×C4—2122no—
448C2×C4#1313318811yescharacteristic
449C2×C4#1313318811yescharacteristic
450C2×C4#1313318811yescharacteristic
451C2×C4#1313323712yes—
452C2×C4#1313324712yes—
453C2×C4C2×D4×D5311yescharacteristic
454C2×C4C22×C22⋊D5311yescharacteristic
455C2×C4C2×#1171994711yescharacteristic
456C8—22no—
457C8—22no—
458C8—2122no—
459C8—2122no—
460C8—2121no—
461C8—2121no—
462C22—21no—
463C22—21no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22#1681084212yes—
469C22C2×#1313318811yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—81no—
474C4—81no—
475C4—1062no—
476C4—2121no—
477C4—2121no—
478C4—2121no—
479C4#1681084212yes—
480C4C2×#1313323711yescharacteristic
481C4AGL(1,ℤ/8)×D5311yescharacteristic
482C4C2×#1313324711yescharacteristic
483C4#1681141911yescharacteristic
484C2—22no—
485C2—162no—
486C2C2×#1681084211yescharacteristic
487C2#2028432811yescharacteristic
488C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3138C2
socleC22×C53C2×#121
prime radicalC2×#127×C53C22
hypercenter#3138D53
hypocenterC53#19174
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #20284750C2semidirect
3. M16×Dic106C2semidirect
4. #20274280C2semidirect
5. #20274252C2semidirect
6. #20274156C2semidirect
7. #20274128C2semidirect
8. #20273467C2semidirect
9. #20273405C2semidirect
15. #16810915C22semidirect
16. #16810915C22semidirect
17. #16810830C22semidirect
18. #16810830C22semidirect
24. #16810753C22semidirect
26. C8×Dic106C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810372C22semidirect
38. #16810369C22semidirect
293. C53#19174semidirect

Maximal subgroups

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

subgroupindex
#202847502
C2×#168108202
M16×Dic1062
#202742802
#202742522
#202741562
#202741282
#202734672
#202734052
#202733832
C53×#31382
#1917453