Finite Group Databasebeta

← #27387642

Subgroups — #27387642

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#108×D53C211yesmaximal, characteristic
2#20283302C212yesmaximal
3C2×#16810660C211yesmaximal, characteristic
4#405×D53C211yesmaximal, characteristic
5#20275501C211yesmaximal, characteristic, complemented
6#20275500C211yesmaximal, characteristic, complemented
7#20275498C211yesmaximal, characteristic, complemented
8#20275494C212yesmaximal, complemented
9#20275492C212yesmaximal, complemented
10#20274452C212yesmaximal
11C53×#1952C211yesmaximal, characteristic, complemented
12C2×M16×D53C2212yes—
13C2×M16×D53C2211yescharacteristic
14C22×C8×D53C2211yescharacteristic
15C2×#13133175C2212yes—
16#117×D53C2211yescharacteristic
17C2×#13133153C2211yescharacteristic
18C2×#13133152C2212yes—
19#108×D53C418yescomplemented
20#16810666C2212yes—
21#16810665C2212yes—
22#16810664—22no—
23#16810663C2212yes—
24#16810662—22no—
25#16810660—21no—
26#16810660—21no—
27#16810660—21no—
28#16810660—21no—
29#16810660—22no—
30#16810658—22no—
31C2×#13133151C2211yescharacteristic
32C2×#13133148C2212yes—
33#16810581C2211yescharacteristic
34#16810580—21no—
35#16810579C2211yescharacteristic
36#16810578C2211yescharacteristic
37#16810577—21no—
38#97×D53—21no—
39C4×C8×D53—21no—
40#16810392C2212yes—
41#16810391C2212yes—
42#16810391C2212yes—
43C2×#108×C53C2211yescharacteristic
44C53×#428C2212yes—
45C2×#107×C53C2211yescharacteristic
46C53×#405C2211yescharacteristic
47C53×#343C2212yes—
48C22×C4×D53C2311yescharacteristic
49C2×#11719917C2312yes—
50C2×#11719917C2311yescharacteristic
51C2×#11719912—21no—
52C2×#11719912—22no—
53M16×D53—24no—
54M16×D53—22no—
55M16×D53C2×C418yes—
56M16×D53Q814yes—
57C22×#88322C2311yescharacteristic
58C2×C8×D53—21no—
59C2×C8×D53—22no—
60C2×C8×D53C2×C414yes—
61C2×C8×D53D411yescharacteristic
62C2×C8×D53D411yescharacteristic
63C2×C8×D53D411yescharacteristic
64C2×C8×D53D411yescharacteristic
65C2×C8×D53Q814yes—
66#13133176C2311yescharacteristic
67#13133175—22no—
68#13133175—22no—
69#13133175—22no—
70#13133175—22no—
71#13133175D411yescharacteristic
72#13133175D411yescharacteristic
73#13133174—21no—
74#13133173—21no—
75C2×C4⋊Dic53C2312yes—
76C4⋊C4×D53D411yescharacteristic
77C4⋊C4×D53D411yescharacteristic
78C2×#11719921—22no—
79C22⋊C4×D53—21no—
80C2×C4×Dic53—22no—
81C42×D53—21no—
82#13133153C2×C414yescomplemented
83#13133152C2×C418yescomplemented
84#13133151—21no—
85#13133151—21no—
86#13133151—22no—
87#13133150—22no—
88#13133150—22no—
89#13133149—22no—
90#13133148—22no—
91#13133148—22no—
92#13133148—22no—
93#13133148—22no—
94#13133128—21no—
95#13133127—21no—
96#13133127—21no—
97C4×#88322—21no—
98C8×Dic53—21no—
99C8×Dic53—21no—
100C2×M16×C53C2312yes—
101C2×M16×C53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103#117×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2312yes—
105#108×C53C2×C414yescomplemented
106#107×C53—21no—
107#107×C53—21no—
108#107×C53—22no—
109#106×C53—22no—
110#97×C53—21no—
111C4×C8×C53—21no—
112C23×D53#4111yescharacteristic
113C22×C4×C53—22no—
114C22×C4×C53C2411yescharacteristic
115C22×Dic53—22no—
116C22×Dic53#4111yescharacteristic
117C2×C4×D53—21no—
118C2×C4×D53—21no—
119C2×C4×D53C22×C412yes—
120C2×C4×D53C2×D411yescharacteristic
121C2×C4×D53C2×D411yescharacteristic
122C2×C4×D53C2×D411yescharacteristic
123C2×C4×D53C2×D411yescharacteristic
124C2×C4×D53C2×Q812yes—
125C2×C4×D53C2×Q812yes—
126C2×C4×D53#4112yes—
127M16×C53—22no—
128M16×C53—21no—
129M16×C53C22×C414yes—
130M16×C53C2×Q812yes—
131C2×C8×C53—21no—
132C2×C8×C53—22no—
133C2×C8×C53C22×C412yes—
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×Q812yes—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53C2×D411yescharacteristic
140C4⋊C4×C53C2×D411yescharacteristic
141C22⋊C4×C53—21no—
142C42×C53—21no—
143C22⋊Dic53—21no—
144#11719921—21no—
145#11719921—21no—
146#11719921—42no—
147#11719921—42no—
148C4⋊Dic53—22no—
149C4⋊Dic53—22no—
150C4⋊Dic53C2×D411yescharacteristic
151C4⋊Dic53C2×D411yescharacteristic
152#11719919—21no—
153#11719919—21no—
154C4×Dic53—21no—
155C4×Dic53—21no—
156C4×Dic53—21no—
157C4×Dic53—42no—
158C4×Dic53—42no—
159#11719917—22no—
160#11719917—21no—
161#11719917C22×C414yes—
162#11719917C2×Q812yes—
163C2×#88322—21no—
164C2×#88322—22no—
165C2×#88322C22×C412yes—
166C2×#88322C2×D411yescharacteristic
167C2×#88322C2×D411yescharacteristic
168C2×#88322C2×Q812yes—
169#11719912—28no—
170#11719912—24no—
171C8×D53—28no—
172C8×D53—24no—
173C8×D53C4⋊C414yes—
174C8×D53C4⋊C414yes—
175C8×D53C4⋊C414yes—
176C8×D53C4⋊C414yes—
177C23×C53C2×#4111yescharacteristic
178C22×D53—41no—
179C22×D53—41no—
180C22×D53C4×Q812yes—
181C22×D53#12111yescharacteristic
182C22×D53#12111yescharacteristic
183C22×D53#12512yes—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—42no—
187C2×C4×C53—82no—
188C2×C4×C53C23×C411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C2×C4×C53C22×Q811yescharacteristic
192C2×C4×C53C22×Q811yescharacteristic
193C2×C4×C53C2×#4111yescharacteristic
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—42no—
197C2×Dic53—82no—
198C2×Dic53C4×Q812yes—
199C2×Dic53#12111yescharacteristic
200C2×Dic53#12111yescharacteristic
201C2×Dic53#12512yes—
202C4×D53—42no—
203C4×D53—42no—
204C4×D53C2×C4⋊C412yes—
205C4×D53C2×C4⋊C412yes—
206C4×D53C4×D412yes—
207C4×D53C4×D412yes—
208C4×D53C22⋊Q812yes—
209C4×D53C22⋊Q812yes—
210C4×D53C4⋊Q812yes—
211C4×D53C4⋊Q812yes—
212C8×C53—24no—
213C8×C53—22no—
214C8×C53C2×C4⋊C412yes—
215C8×C53C2×C4⋊C412yes—
216#88322—24no—
217#88322—22no—
218#88322C2×C4⋊C412yes—
219#88322C2×C4⋊C412yes—
220#21181—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53C2×C4×Q811yescharacteristic
224C22×C53C2×#12111yescharacteristic
225C22×C53C2×#12511yescharacteristic
226C2×D53—21no—
227C2×D53—21no—
228C2×D53—21no—
229C2×D53—21no—
230C2×D53—42no—
231C2×D53—42no—
232C2×D53#38912yes—
233C2×D53#38912yes—
234C4×C53—41no—
235C4×C53—41no—
236C4×C53—82no—
237C4×C53C22×C4⋊C411yescharacteristic
238C4×C53C2×C4×D411yescharacteristic
239C4×C53C2×C22⋊Q811yescharacteristic
240C4×C53C2×C4⋊Q811yescharacteristic
241Dic53—41no—
242Dic53—41no—
243Dic53—82no—
244Dic53#38912yes—
245Dic53#38912yes—
246C22×#108—531no—
247#2911—532no—
248C2×#427—531no—
249C2×#405—531no—
250#1952—531no—
251#1952—531no—
252#1952—531no—
253#1952D5311yescharacteristic, complemented
254#1951—532no—
255#1951—532no—
256#1742—532no—
257C2×C53—21no—
258C2×C53—21no—
259C2×C53C2×#38911yescharacteristic
260C2×C53#306711yescharacteristic
261C2×C53#306711yescharacteristic
262D53—22no—
263D53—22no—
264C22×M16—531no—
265C22×M16—532no—
266C23×C8—531no—
267C2×#117—531no—
268C2×#117—532no—
269C2×#108—532no—
270C2×#108—531no—
271C2×#108—538no—
272C2×#108C2×D5311yescharacteristic
273#428—532no—
274#428—532no—
275#428—532no—
276#428—1062no—
277#428—1062no—
278#428C2×D5312yes—
279#427—1062no—
280#427—1062no—
281#427—1061no—
282#427—1061no—
283#427—1061no—
284#427—1061no—
285C2×#107—531no—
286C2×#107C2×D5311yescharacteristic
287C2×#106—532no—
288#405—531no—
289#405—531no—
290#405—531no—
291#405—1061no—
292#405—1061no—
293#405C2×D5311yescharacteristic
294C2×#97—1061no—
295C2×C4×C8—1061no—
296#343—532no—
297#343—532no—
298#343—532no—
299#343C2×D5312yes—
300C53#2118111yescharacteristic, complemented, Sylow
301C23×C4—531no—
302C2×M16—531no—
303C2×M16—532no—
304C2×M16—538no—
305C2×M16—534no—
306C2×M16—1061no—
307C2×M16—1062no—
308C2×M16—1062no—
309C2×M16—1064no—
310C2×M16C22×D5312yes—
311C2×M16C22×D5311yescharacteristic
312C22×C8—531no—
313C22×C8—534no—
314C22×C8—531no—
315C22×C8—531no—
316C22×C8—531no—
317C22×C8—531no—
318C22×C8—534no—
319C22×C8—1061no—
320C22×C8—1062no—
321C22×C8C22×D5311yescharacteristic
322#117—531no—
323#117—531no—
324#117—531no—
325#117—1061no—
326#117—1061no—
327#117—1062no—
328#117—1062no—
329#117—1062no—
330#117—1062no—
331#117C22×D5311yescharacteristic
332C2×C4⋊C4—531no—
333C2×C4⋊C4—531no—
334C2×C4⋊C4—532no—
335C2×C4⋊C4C22×D5312yes—
336C2×C22⋊C4—1061no—
337C2×C22⋊C4—1062no—
338C2×C42—1061no—
339C2×C42—1062no—
340#108—538no—
341#108—534no—
342#108C4×D5314yescomplemented
343#107—21no—
344#107—21no—
345#107—22no—
346#107—1062no—
347#107—1062no—
348#107—1061no—
349#107—1061no—
350#107—1062no—
351#106—22no—
352#106—1062no—
353#106—1062no—
354#106—1062no—
355#106—1062no—
356#106—1062no—
357#97—21no—
358#97—1061no—
359#97—1061no—
360#97—1061no—
361C4×C8—21no—
362C4×C8—1061no—
363C4×C8—1061no—
364C4×C8—1061no—
365C24—531no—
366C22×C4—22no—
367C22×C4—531no—
368C22×C4—532no—
369C22×C4—531no—
370C22×C4—531no—
371C22×C4—531no—
372C22×C4—531no—
373C22×C4—532no—
374C22×C4—532no—
375C22×C4—532no—
376C22×C4—1061no—
377C22×C4—1061no—
378C22×C4—1062no—
379C22×C4C23×D5311yescharacteristic
380M16—22no—
381M16—21no—
382M16—534no—
383M16—532no—
384M16—1068no—
385M16—1064no—
386M16—1062no—
387M16—1061no—
388M16C2×C4×D5314yes—
389M16Q8×D5312yes—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—532no—
393C2×C8—531no—
394C2×C8—531no—
395C2×C8—532no—
396C2×C8—534no—
397C2×C8—534no—
398C2×C8—534no—
399C2×C8—534no—
400C2×C8—1068no—
401C2×C8—1064no—
402C2×C8—1061no—
403C2×C8—1062no—
404C2×C8C2×C4×D5312yes—
405C2×C8D4×D5311yescharacteristic
406C2×C8D4×D5311yescharacteristic
407C2×C8Q8×D5312yes—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—531no—
411C4⋊C4—531no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1062no—
415C4⋊C4—1062no—
416C4⋊C4D4×D5311yescharacteristic
417C4⋊C4D4×D5311yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—1061no—
420C22⋊C4—1061no—
421C22⋊C4—1061no—
422C22⋊C4—2122no—
423C22⋊C4—2122no—
424C42—21no—
425C42—1061no—
426C42—1061no—
427C42—1061no—
428C42—2122no—
429C42—2122no—
430C23—531no—
431C23—531no—
432C23—532no—
433C23—532no—
434C23—2121no—
435C23—2121no—
436C23#41×D5311yescharacteristic
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—532no—
442C2×C4—531no—
443C2×C4—531no—
444C2×C4—532no—
445C2×C4—532no—
446C2×C4—532no—
447C2×C4—532no—
448C2×C4—532no—
449C2×C4—532no—
450C2×C4—532no—
451C2×C4—532no—
452C2×C4—532no—
453C2×C4—1061no—
454C2×C4—1061no—
455C2×C4—4242no—
456C2×C4—2122no—
457C2×C4—2122no—
458C2×C4—2122no—
459C2×C4C22×C4×D5311yescharacteristic
460C2×C4C2×D4×D5311yescharacteristic
461C2×C4C2×D4×D5311yescharacteristic
462C2×C4C2×Q8×D5311yescharacteristic
463C2×C4C2×Q8×D5311yescharacteristic
464C2×C4#41×D5311yescharacteristic
465C8—24no—
466C8—22no—
467C8—532no—
468C8—532no—
469C8—1064no—
470C8—1062no—
471C8C4⋊C4×D5312yes—
472C8C4⋊C4×D5312yes—
473C22—21no—
474C22—21no—
475C22—532no—
476C22—532no—
477C22—1061no—
478C22—1061no—
479C22—1061no—
480C22—1061no—
481C22—2122no—
482C22—2122no—
483C22C4×Q8×D5311yescharacteristic
484C22#121×D5311yescharacteristic
485C22#125×D5311yescharacteristic
486C4—41no—
487C4—41no—
488C4—82no—
489C4—532no—
490C4—532no—
491C4—4242no—
492C4—2121no—
493C4—2121no—
494C4C2×C4⋊C4×D5311yescharacteristic
495C4C4×D4×D5311yescharacteristic
496C4C22⋊Q8×D5311yescharacteristic
497C4C4⋊Q8×D5311yescharacteristic
498C2—21no—
499C2—21no—
500C2—1062no—
501C2—1062no—
502C2#389×D5311yescharacteristic
503C2#2028443311yescharacteristic
504C2#2028443711yescharacteristic
subgroupG/H
centerC22#121×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1952C2
socleC22×C53C2×#121
prime radicalC23×D53#41
cosocleC22×C4×C53C24
hypercenter#1952D53
hypocenterC53#21181
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
5. #20275501C2semidirect
6. #20275500C2semidirect
7. #20275498C2semidirect
8. #20275494C2semidirect
9. #20275492C2semidirect
11. C53×#1952C2semidirect
19. #108×D53C4semidirect
82. #13133153C2×C4semidirect
83. #13133152C2×C4semidirect
105. #108×C53C2×C4semidirect
253. #1952D53semidirect
300. C53#21181semidirect
342. #108C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#108×D532
#202833022
C2×#168106602
#405×D532
#202755012
#202755002
#202754982
#202754942
#202754922
#202744522
C53×#19522
#2118153