Finite Group Databasebeta

← #27402580

Subgroups — #27402580

Order 13568.

846 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810738C211yesmaximal, characteristic
2#20283596C212yesmaximal
3#20283312C211yesmaximal, characteristic, complemented
4#20275492C212yesmaximal, complemented
5#20274948C212yesmaximal, complemented
6#20274945C212yesmaximal, complemented
7#20274878C211yesmaximal, characteristic, complemented
8#20274780C211yesmaximal, characteristic, complemented
9#20274779C211yesmaximal, characteristic, complemented
10#20274596C211yesmaximal, characteristic, complemented
11C53×#2913C211yesmaximal, characteristic, complemented
12C2×#13133237C2211yescharacteristic
13C2×#13133223C2211yescharacteristic
14#16811103C2212yes—
15#16810738C418yescomplemented
16#16810733—22no—
17#16810728—22no—
18C4×#11719924—22no—
19C4×#11719923—22no—
20C2×#13133153C2211yescharacteristic
21C2×#13133152C2211yescharacteristic
22#16810666C2212yes—
23#16810662—22no—
24#16810660—21no—
25C2×#13133151C2211yescharacteristic
26#107×D53—21no—
27C2×#13133148C2212yes—
28C2×#13133144C2211yescharacteristic
29C2×#13133143C2211yescharacteristic
30#16810624C2212yes—
31#16810622C2212yes—
32#16810579C2212yes—
33C2×C8×Dic53C2211yescharacteristic
34#16810488C2212yes—
35#16810392C2212yes—
36#16810391C2212yes—
37#16810382C2212yes—
38C2×#131×C53C2211yescharacteristic
39C53×#520C2212yes—
40C53×#428—22no—
41C53×#428C2212yes—
42C53×#427—21no—
43C53×#427C2211yescharacteristic
44C2×#107×C53—21no—
45C2×#107×C53C2211yescharacteristic
46C2×#11719945C2311yescharacteristic
47#13133237C2×C414yes—
48#13133237Q814yes—
49C2×#11719925D411yescharacteristic
50C2×#11719923D411yescharacteristic
51C2×#11719924D411yescharacteristic
52C2×#11719922D411yescharacteristic
53#13133223—22no—
54#13133223C2×C414yescomplemented
55C2×#11719917C2312yes—
56C2×#11719917C2311yescharacteristic
57C2×#11719912—21no—
58C22×#88322C2311yescharacteristic
59C2×C8×D53—21no—
60C4×Dic106—22no—
61C4×D212—22no—
62#13133177—42no—
63#13133175—22no—
64C2×C4⋊Dic53C2312yes—
65C4⋊C4×D53—22no—
66C2×C4×Dic53C2311yescharacteristic
67#13133153—22no—
68#13133152C2×C414yescomplemented
69#13133151—21no—
70#13133151—21no—
71#13133150—22no—
72#13133148—22no—
73#13133148—22no—
74#13133148—22no—
75#13133144C2×C414yescomplemented
76#13133143C2×C414yescomplemented
77#13133141—22no—
78#13133140—22no—
79#13133138—22no—
80#13133137—22no—
81#13133128—22no—
82C4×#88322—22no—
83C8×Dic53C2×C414yescomplemented
84#13133124—42no—
85C2×#41×C53C2311yescharacteristic
86#131×C53—22no—
87#131×C53C2×C414yescomplemented
88C2×M16×C53—21no—
89C2×M16×C53C2311yescharacteristic
90C22×C8×C53—21no—
91C22×C8×C53C2311yescharacteristic
92C4×Q8×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—42no—
95#117×C53—22no—
96#117×C53C2312yes—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53C2312yes—
99#107×C53—21no—
100#107×C53—21no—
101#107×C53—22no—
102#107×C53D411yescharacteristic
103#107×C53D411yescharacteristic
104#106×C53—22no—
105#106×C53D412yes—
106#41×C53—21no—
107#41×C53—21no—
108#41×C53C22×C412yes—
109#41×C53C2×Q812yes—
110C2×Q8×C53C2×D411yescharacteristic
111C2×D4×C53—21no—
112C2×D4×C53C2×D411yescharacteristic
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—42no—
116C22×C4×C53C2411yescharacteristic
117C2×C22⋊D53—21no—
118C22×Dic53—21no—
119C22×Dic53—22no—
120#11719945—21no—
121#11719945—21no—
122#11719945C22×C412yes—
123#11719945C2×Q812yes—
124C2×D212C2×D411yescharacteristic
125C2×C4×D53—21no—
126C2×C4×D53—42no—
127C2×Dic106C2×D411yescharacteristic
128M16×C53—21no—
129M16×C53—24no—
130M16×C53C22×C412yes—
131C2×C8×C53—21no—
132C2×C8×C53—24no—
133C2×C8×C53C22×C412yes—
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53#4112yes—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—42no—
140C4⋊C4×C53C2×D412yes—
141C4⋊C4×C53C2×D412yes—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—42no—
144C42×C53—22no—
145C42×C53—42no—
146#11719925C4⋊C414yes—
147#11719924C4⋊C414yes—
148#11719923C4⋊C414yes—
149#11719922C4⋊C414yes—
150#11719921—42no—
151C4⋊Dic53—22no—
152C4⋊Dic53—22no—
153#11719919—42no—
154C4×Dic53—21no—
155C4×Dic53—42no—
156C4×Dic53C22×C412yes—
157#11719917—21no—
158#11719917—22no—
159#11719917—22no—
160#11719917C22×C412yes—
161C2×#88322—21no—
162C2×#88322—22no—
163C2×#88322C22×C412yes—
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166C2×#88322C2×Q812yes—
167#11719912—24no—
168C8×D53—24no—
169C23×C53—21no—
170C23×C53H(5,𝔽2)11yescharacteristic
171C22×D53—21no—
172Q8×C53—21no—
173Q8×C53C2×C4⋊C412yes—
174D4×C53—22no—
175D4×C53—22no—
176D4×C53—21no—
177D4×C53C2×C4⋊C412yes—
178C2×C4×C53—21no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—22no—
182C2×C4×C53—22no—
183C2×C4×C53—42no—
184C2×C4×C53—82no—
185C2×C4×C53—162no—
186C2×C4×C53C23×C411yescharacteristic
187C2×C4×C53C22×D411yescharacteristic
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C53C22×Q811yescharacteristic
190C2×C4×C53C2×#4111yescharacteristic
191C2×C4×C532−1+411yescharacteristic
192C22⋊D53—22no—
193C22⋊D53—22no—
194C2×Dic53—21no—
195C2×Dic53—22no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—42no—
199C2×Dic53—82no—
200D212—21no—
201D212C2×C4⋊C412yes—
202C4×D53—22no—
203C4×D53—22no—
204C4×D53—162no—
205Dic106—21no—
206Dic106C2×C4⋊C412yes—
207C8×C53—22no—
208C8×C53C4×D412yes—
209C8×C53C4×D412yes—
210#88322—22no—
211#88322—42no—
212#88322C2×C4⋊C412yes—
213#88322C2×C4⋊C412yes—
214#22691—531nomaximal, Sylow
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—22no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53#52011yescharacteristic
221C22×C53D4211yescharacteristic
222C22×C53#55411yescharacteristic
223C2×D53—22no—
224C2×D53—41no—
225C2×D53—41no—
226C4×C53—22no—
227C4×C53—42no—
228C4×C53—42no—
229C4×C53—82no—
230C4×C53C22×C4⋊C411yescharacteristic
231C4×C53C2×C4×D411yescharacteristic
232C4×C53#54711yescharacteristic
233C4×C53D4×Q811yescharacteristic
234Dic53—22no—
235Dic53—42no—
236Dic53—82no—
237C2×#443—531no—
238#2952—532no—
239#2913—531no—
240#2913D5311yescharacteristic, complemented
241#1951—532no—
242#1860—532no—
243#1860—532no—
244#1846—531no—
245#1816—531no—
246#1816—531no—
247#1772—531no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—42no—
251C2×C53D4×C4⋊C411yescharacteristic
252C2×C53#329011yescharacteristic
253C2×C53#329011yescharacteristic
254D53—42no—
255C2×#135—531no—
256C2×#131—531no—
257C2×#131C2×D5311yescharacteristic
258#520—532no—
259#520C2×D5312yes—
260#443—538no—
261#442—1062no—
262#441—1062no—
263C4×SD16—1062no—
264C4×SD16—1062no—
265C2×#108—531no—
266C2×#108—531no—
267#428—22no—
268#428—532no—
269#428—1062no—
270#428C2×D5312yes—
271#427—21no—
272#427—1061no—
273#427C2×D5311yescharacteristic
274C2×#107—21no—
275C2×#107—531no—
276C2×#107—1061no—
277C2×#107C2×D5311yescharacteristic
278C2×#106—532no—
279C2×#104—531no—
280C2×#104—531no—
281#417—532no—
282#417—532no—
283#405—532no—
284C2×C4×C8—531no—
285#382—532no—
286#343—532no—
287#343—532no—
288#339—532no—
289C53#2269111yescharacteristic, complemented, Sylow
290C2×#41—531no—
291C2×#41C22×D5311yescharacteristic
292#135—534no—
293#135—534no—
294C2×Q16—531no—
295C2×SD16—531no—
296C2×SD16—531no—
297C2×D8—531no—
298#131—22no—
299#131—534no—
300#131—1062no—
301#131C4×D5314yescomplemented
302C2×M16—21no—
303C2×M16—531no—
304C2×M16—532no—
305C2×M16—1061no—
306C2×M16C22×D5311yescharacteristic
307C22×C8—21no—
308C22×C8—531no—
309C22×C8—1061no—
310C22×C8C22×D5311yescharacteristic
311C4×Q8—22no—
312C4×Q8—1062no—
313C4×D4—22no—
314C4×D4—42no—
315C4×D4—1062no—
316C4×D4—2122no—
317#117—22no—
318#117—1062no—
319#117C22×D5312yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—532no—
322C2×C4⋊C4—1062no—
323C2×C4⋊C4C22×D5312yes—
324C2×C42—531no—
325#108—534no—
326#108—1062no—
327#107—21no—
328#107—21no—
329#107—22no—
330#107—1061no—
331#107—1061no—
332#107—1062no—
333#107C22⋊D5311yescharacteristic
334#107C22⋊D5311yescharacteristic
335#106—22no—
336#106—1062no—
337#106—1062no—
338#106—1062no—
339#106C22⋊D5312yes—
340#104—534no—
341#104—534no—
342#103—1062no—
343#103—1062no—
344#102—1062no—
345#102—1062no—
346#97—1062no—
347C4×C8—534no—
348C4×C8—1062no—
349#95—2122no—
350#41—21no—
351#41—21no—
352#41—532no—
353#41—532no—
354#41—1061no—
355#41—1061no—
356#41C2×C4×D5312yes—
357#41Q8×D5312yes—
358C2×Q8—531no—
359C2×Q8D4×D5311yescharacteristic
360C2×D4—21no—
361C2×D4—531no—
362C2×D4—1061no—
363C2×D4D4×D5311yescharacteristic
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—1061no—
368C22×C4—1061no—
369C22×C4—1062no—
370C22×C4—2122no—
371C22×C4C23×D5311yescharacteristic
372Q16—534no—
373SD16—534no—
374SD16—534no—
375D8—534no—
376M16—21no—
377M16—24no—
378M16—532no—
379M16—1061no—
380M16—1064no—
381M16—1062no—
382M16—1062no—
383M16C2×C4×D5312yes—
384C2×C8—21no—
385C2×C8—24no—
386C2×C8—532no—
387C2×C8—531no—
388C2×C8—531no—
389C2×C8—532no—
390C2×C8—1061no—
391C2×C8—1064no—
392C2×C8—1062no—
393C2×C8C2×C4×D5312yes—
394C2×C8#1171994512yes—
395C2×C8C2×C22⋊D5311yescharacteristic
396C2×C8C2×C22⋊D5311yescharacteristic
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—42no—
400C4⋊C4—1062no—
401C4⋊C4—1062no—
402C4⋊C4—2122no—
403C4⋊C4C2×C22⋊D5312yes—
404C4⋊C4C2×C22⋊D5312yes—
405C22⋊C4—22no—
406C22⋊C4—42no—
407C22⋊C4—2122no—
408C42—22no—
409C42—42no—
410C42—532no—
411C42—1061no—
412C42—2122no—
413C23—21no—
414C23—1061no—
415C23#1313326211yescharacteristic
416Q8—21no—
417Q8—532no—
418Q8—1061no—
419Q8C4⋊C4×D5312yes—
420D4—22no—
421D4—22no—
422D4—21no—
423D4—532no—
424D4—1062no—
425D4—1062no—
426D4—1061no—
427D4C4⋊C4×D5312yes—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—162no—
436C2×C4—8482no—
437C2×C4—1061no—
438C2×C4—1062no—
439C2×C4—1062no—
440C2×C4—1062no—
441C2×C4—4242no—
442C2×C4—2122no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4C22×C4×D5311yescharacteristic
446C2×C4C2×D4×D5311yescharacteristic
447C2×C4C22×C22⋊D5311yescharacteristic
448C2×C4C2×Q8×D5311yescharacteristic
449C2×C4C2×#1171994511yescharacteristic
450C2×C4#1313326511yescharacteristic
451C8—22no—
452C8—532no—
453C8—532no—
454C8—1062no—
455C8—2122no—
456C8C4×C22⋊D5312yes—
457C8C4×C22⋊D5312yes—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—1062no—
464C22—2121no—
465C22—2121no—
466C22#1681110411yescharacteristic
467C22D4×C22⋊D5311yescharacteristic
468C22#1681132811yescharacteristic
469C4—22no—
470C4—42no—
471C4—42no—
472C4—82no—
473C4—1062no—
474C4—4242no—
475C4—2122no—
476C4C2×C4⋊C4×D5311yescharacteristic
477C4C2×C4×C22⋊D5311yescharacteristic
478C4#1681127411yescharacteristic
479C4Q8×C22⋊D5311yescharacteristic
480C2—21no—
481C2—21no—
482C2—42no—
483C2—2122no—
484C2C4⋊C4×C22⋊D5311yescharacteristic
485C2#2028649711yescharacteristic
486C2#2028650111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2913C2
socleC22×C53D42
prime radicalC2×#13133237C22
cosocleC22×C4×C53C24
hypercenter#2913D53
hypocenterC53#22691
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
3. #20283312C2semidirect
4. #20275492C2semidirect
5. #20274948C2semidirect
6. #20274945C2semidirect
7. #20274878C2semidirect
8. #20274780C2semidirect
9. #20274779C2semidirect
10. #20274596C2semidirect
11. C53×#2913C2semidirect
15. #16810738C4semidirect
54. #13133223C2×C4semidirect
68. #13133152C2×C4semidirect
75. #13133144C2×C4semidirect
76. #13133143C2×C4semidirect
83. C8×Dic53C2×C4semidirect
87. #131×C53C2×C4semidirect
240. #2913D53semidirect
289. C53#22691semidirect
301. #131C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107382
#202835962
#202833122
#202754922
#202749482
#202749452
#202748782
#202747802
#202747792
#202745962
C53×#29132
#2269153