Finite Group Databasebeta

← #27369558

Subgroups — #27369558

Order 13568.

662 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#20285845C211yesmaximal, characteristic, complemented
2#20284530C211yesmaximal, characteristic, complemented
3#20284170C211yesmaximal, characteristic
4#20274131C212yesmaximal, complemented
5#20274124C212yesmaximal, complemented
6#20274097C212yesmaximal, complemented
7#20274090C212yesmaximal, complemented
8#20273486C211yesmaximal, characteristic, complemented
9#20273390C211yesmaximal, characteristic, complemented
10C2×#16810371C211yesmaximal, characteristic, complemented
11C53×#2988C211yesmaximal, characteristic, complemented
12C2×#13133237—21no—
13#16811173C2211yescharacteristic
14C2×#13133217C2211yescharacteristic
15#16811021C2212yes—
16#16810970C2212yes—
17#16810942—22no—
18#16810918C2212yes—
19#16810914—21no—
20#16810914—21no—
21#16810913C2212yes—
22#16810908—21no—
23#16810903—21no—
24#16810845—21no—
25#16810838—21no—
26#16810833C2211yescharacteristic
27#16810828C2211yescharacteristic
28#16810823C2211yescharacteristic
29#16810818C2211yescharacteristic
30#16810666—21no—
31#16810647C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810624—21no—
34#16810617—21no—
35#16810374C2212yes—
36#16810374C2212yes—
37#16810371C2212yes—
38#16810371C2212yes—
39#16810368C2212yes—
40#16810366C2212yes—
41C4×#41×C53C2211yescharacteristic
42C8×Q8×C53C2212yes—
43C8×D4×C53—22no—
44C8×D4×C53C2212yes—
45C53×#431—21no—
46C53×#431C2211yescharacteristic
47C2×C4⋊C8×C53—21no—
48C2×C4⋊C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50C22×Dic106—21no—
51#13133237—44no—
52C2×#11719925—21no—
53C2×#11719923—21no—
54C2×#11719924—21no—
55C2×#11719922—21no—
56C2×#11719917—21no—
57C22×#88322—21no—
58#13133217—21no—
59#13133217—21no—
60#13133217—21no—
61#13133217C2311yescharacteristic
62#13133217C2312yes—
63#13133214C2311yescharacteristic
64#13133205—21no—
65#13133191—21no—
66#13133191—21no—
67C4×D212—21no—
68C4×D212—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71#13133150—22no—
72#13133149—22no—
73#13133149—22no—
74#13133148—22no—
75C4⋊#88322—21no—
76C4⋊#88322—21no—
77C4⋊#88322C2311yescharacteristic
78C4⋊#88322C2311yescharacteristic
79C4⋊#88322C2312yes—
80#13133141—22no—
81#13133140—21no—
82#13133140—21no—
83#13133139—22no—
84#13133139—22no—
85#13133138—22no—
86#13133137—21no—
87#13133137—21no—
88C2×#41×C53D411yescharacteristic
89C22×C8×C53—22no—
90C22×C8×C53—21no—
91C4×Q8×C53D411yescharacteristic
92C4×Q8×C53C2311yescharacteristic
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53D411yescharacteristic
96C4×D4×C53C2311yescharacteristic
97#117×C53—21no—
98#117×C53D411yescharacteristic
99C2×C42×C53—21no—
100C2×C42×C53C2311yescharacteristic
101C4⋊C8×C53—22no—
102C4⋊C8×C53—21no—
103C4⋊C8×C53D411yescharacteristic
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53C2312yes—
106C22⋊C8×C53—22no—
107C22⋊C8×C53D412yes—
108C4×C8×C53—22no—
109C4×C8×C53C2312yes—
110#41×C53—22no—
111#41×C53SD1614yes—
112C2×Q8×C53C2×D411yescharacteristic
113C2×D4×C53—21no—
114C2×D4×C53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C2×C22⋊D53—41no—
122C22×Dic53—41no—
123#11719945—42no—
124#11719945—42no—
125C2×D212—21no—
126C2×D212—21no—
127C2×D212—21no—
128C2×C4×D53—41no—
129C2×Dic106—21no—
130C2×Dic106—22no—
131C2×Dic106—41no—
132C2×Dic106—82no—
133C2×C8×C53—22no—
134C2×C8×C53—82no—
135C2×C8×C53—81no—
136C2×C8×C53C2×D412yes—
137C2×C8×C53C2×D412yes—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53C2×D411yescharacteristic
141C4⋊C4×C53C2×D411yescharacteristic
142C22⋊C4×C53—21no—
143C22⋊C4×C53—21no—
144C22⋊C4×C53—21no—
145C42×C53—21no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53#4111yescharacteristic
151C42×C53C2411yescharacteristic
152#11719925—42no—
153#11719924—42no—
154#11719923—42no—
155#11719922—42no—
156#11719921—41no—
157#11719921—41no—
158C4⋊Dic53—22no—
159C4⋊Dic53—22no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—41no—
163C4⋊Dic53—41no—
164#11719917—42no—
165C2×#88322—21no—
166C2×#88322—21no—
167C2×#88322—22no—
168C2×#88322—42no—
169C23×C53—21no—
170C23×C53#12111yescharacteristic
171C22×D53—41no—
172Q8×C53—21no—
173Q8×C53C2×SD1612yes—
174D4×C53—24no—
175D4×C53—21no—
176D4×C53C2×SD1612yes—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—24no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—81no—
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53C2×SD1612yes—
191C2×C4×C53#13512yes—
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C2×#4111yescharacteristic
195C22⋊D53—82no—
196C2×Dic53—42no—
197C2×Dic53—41no—
198C2×Dic53—82no—
199D212—41no—
200D212—41no—
201D212—41no—
202D212—41no—
203D212—81no—
204C4×D53—82no—
205Dic106—41no—
206Dic106—41no—
207Dic106—82no—
208Dic106—82no—
209C8×C53—22no—
210C8×C53—22no—
211C8×C53—42no—
212#88322—41no—
213#88322—41no—
214#88322—42no—
215#19414—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—22no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53#46512yes—
222C22×C53C2×#12111yescharacteristic
223C2×D53—81no—
224C2×D53—81no—
225C2×D53—81no—
226C4×C53—21no—
227C4×C53—22no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53#46512yes—
233C4×C53C22×SD1611yescharacteristic
234C4×C53C2×#13511yescharacteristic
235C4×C53#57711yescharacteristic
236C4×C53#57811yescharacteristic
237Dic53—81no—
238Dic53—82no—
239#3227—531no—
240#3079—531no—
241#3035—531no—
242#2988D5311yescharacteristic, complemented
243#1678—532no—
244#1677—532no—
245#1672—532no—
246#1671—532no—
247#1570—531no—
248#1547—531no—
249C2×#333—531no—
250C2×C53—22no—
251C2×C53—42no—
252C2×C53#304611yescharacteristic
253C2×C53C2×#46511yescharacteristic
254C2×C53#306811yescharacteristic
255D53—161no—
256C2×#135—1061no—
257#532—531no—
258C2×C4⋊Q8—531no—
259C4×#41C2×D5311yescharacteristic
260#498—532no—
261#486—532no—
262#481—1062no—
263#477—532no—
264#476—532no—
265#476—1061no—
266#476—1061no—
267#475—1061no—
268#474—1061no—
269#464—1061no—
270#463—1061no—
271#462—531no—
272#461—531no—
273#460—531no—
274#459—531no—
275C8×Q8C2×D5312yes—
276C8×D4—22no—
277C8×D4C2×D5312yes—
278#431—21no—
279#431C2×D5311yescharacteristic
280#428—1061no—
281#423—531no—
282C2×C4⋊C8—21no—
283C2×C4⋊C8—531no—
284C2×C4⋊C8C2×D5311yescharacteristic
285#417—1061no—
286#416—1061no—
287#334—532no—
288#334—532no—
289#333—532no—
290#333—532no—
291#332—532no—
292#331—532no—
293C53#1941411yescharacteristic, complemented, Sylow
294C2×#41—1061no—
295C2×#41D21211yescharacteristic
296C22×Q8—1061no—
297#135—2124no—
298C2×Q16—1061no—
299C2×SD16—1061no—
300C2×SD16—1061no—
301C2×D8—1061no—
302C2×M16—1061no—
303C22×C8—21no—
304C22×C8—22no—
305C22×C8—1061no—
306C4⋊Q8—532no—
307C4⋊Q8—531no—
308C4⋊Q8—1061no—
309C4⋊Q8—1061no—
310C4⋊Q8—1061no—
311#127—531no—
312#124—1061no—
313#121—1061no—
314#121—1061no—
315C4×Q8D21211yescharacteristic
316C4×Q8C22×D5311yescharacteristic
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—1061no—
320C4×D4—1061no—
321C4×D4D21211yescharacteristic
322C4×D4C22×D5311yescharacteristic
323#117—21no—
324#117D21211yescharacteristic
325C2×C4⋊C4—1061no—
326C2×C4⋊C4—1061no—
327C2×C42—21no—
328C2×C42C22×D5311yescharacteristic
329#107—1062no—
330#106—1062no—
331#106—1062no—
332#106—1062no—
333C4⋊C8—22no—
334C4⋊C8—21no—
335C4⋊C8—532no—
336C4⋊C8—531no—
337C4⋊C8—531no—
338C4⋊C8—1061no—
339C4⋊C8—1061no—
340C4⋊C8C22⋊D5311yescharacteristic
341C4⋊C8C22⋊D5311yescharacteristic
342C4⋊C8C22×D5312yes—
343#103—1062no—
344#103—1061no—
345#103—1061no—
346#102—1062no—
347#102—1062no—
348#102—1062no—
349#102—1061no—
350#102—1061no—
351C22⋊C8—22no—
352C22⋊C8C22⋊D5312yes—
353C4×C8—22no—
354C4×C8C22×D5312yes—
355#41—22no—
356#41—2122no—
357#41—2122no—
358#41#1171991314yes—
359C2×Q8—1061no—
360C2×Q8—1062no—
361C2×Q8—4242no—
362C2×Q8—2121no—
363C2×Q8C2×D21211yescharacteristic
364C2×D4—21no—
365C2×D4—1061no—
366C2×D4—1061no—
367C2×D4—1061no—
368C2×D4—2121no—
369C2×D4C2×D21211yescharacteristic
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—2121no—
374C22×C4—2121no—
375C22×C4C2×D21211yescharacteristic
376C22×C4#1171994511yescharacteristic
377C22×C4C2×C22⋊D5311yescharacteristic
378Q16—2122no—
379SD16—2122no—
380SD16—2122no—
381D8—2122no—
382M16—2122no—
383C2×C8—22no—
384C2×C8—82no—
385C2×C8—81no—
386C2×C8—1062no—
387C2×C8—1061no—
388C2×C8—1061no—
389C2×C8—2122no—
390C2×C8C2×C22⋊D5312yes—
391C2×C8C2×C22⋊D5312yes—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—1062no—
395C4⋊C4—1061no—
396C4⋊C4—1062no—
397C4⋊C4—1061no—
398C4⋊C4—2121no—
399C4⋊C4—2121no—
400C4⋊C4C2×D21211yescharacteristic
401C4⋊C4C2×D21211yescharacteristic
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—2121no—
406C22⋊C4—2121no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42C2×D21211yescharacteristic
411C42#1171994511yescharacteristic
412C42C2×C22⋊D5311yescharacteristic
413C42C23×D5311yescharacteristic
414C23—21no—
415C23—2121no—
416C23#1313319111yescharacteristic
417Q8—21no—
418Q8—4242no—
419Q8—4242no—
420Q8—2121no—
421Q8—2121no—
422Q8C2×#1171991312yes—
423D4—24no—
424D4—21no—
425D4—4241no—
426D4—4242no—
427D4—2121no—
428D4—2121no—
429D4—2121no—
430D4—2121no—
431D4C2×#1171991312yes—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4—4242no—
443C2×C4—4242no—
444C2×C4—2121no—
445C2×C4—2122no—
446C2×C4#1313319111yescharacteristic
447C2×C4#1313319111yescharacteristic
448C2×C4#1313319111yescharacteristic
449C2×C4C2×#1171991312yes—
450C2×C4#1313323912yes—
451C2×C4C22×C22⋊D5311yescharacteristic
452C2×C4C22×D21211yescharacteristic
453C2×C4C2×#1171994511yescharacteristic
454C8—22no—
455C8—22no—
456C8—42no—
457C8—2121no—
458C8—2121no—
459C8—2122no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—4241no—
466C22—4241no—
467C22—4241no—
468C22#1681085212yes—
469C22C2×#1313319111yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—4242no—
477C4—4241no—
478C4#1681085212yes—
479C4C22×#1171991311yescharacteristic
480C4C2×#1313323911yescharacteristic
481C4#1681143411yescharacteristic
482C4#1681143811yescharacteristic
483C2—22no—
484C2—42no—
485C2—8481no—
486C2#2028426511yescharacteristic
487C2C2×#1681085211yescharacteristic
488C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2988C2
socleC22×C53C2×#121
prime radical#20284170C2
hypercenter#2988D53
hypocenterC53#19414
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #20285845C2semidirect
2. #20284530C2semidirect
4. #20274131C2semidirect
5. #20274124C2semidirect
6. #20274097C2semidirect
7. #20274090C2semidirect
8. #20273486C2semidirect
9. #20273390C2semidirect
10. C2×#16810371C2semidirect
11. C53×#2988C2semidirect
242. #2988D53semidirect
293. C53#19414semidirect

Maximal subgroups

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

subgroupindex
#202858452
#202845302
#202841702
#202741312
#202741242
#202740972
#202740902
#202734862
#202733902
C2×#168103712
C53×#29882
#1941453