Finite Group Databasebeta

← #27369562

Subgroups — #27369562

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#20285843C211yesmaximal, characteristic, complemented
2#20284170C211yesmaximal, characteristic
3#20283826C211yesmaximal, characteristic, complemented
4#20274133C212yesmaximal, complemented
5#20274126C212yesmaximal, complemented
6#20274099C212yesmaximal, complemented
7#20274092C212yesmaximal, complemented
8#20273487C211yesmaximal, characteristic, complemented
9#20273393C211yesmaximal, characteristic, complemented
10C2×#16810372C211yesmaximal, characteristic, complemented
11C53×#3079C211yesmaximal, characteristic, complemented
12C2×#13133237—21no—
13#16811173C2211yescharacteristic
14#16811019C2212yes—
15#16810971C2212yes—
16#16810941—22no—
17#16810910—21no—
18#16810910—21no—
19#16810845—21no—
20#16810838—21no—
21#16810833C2211yescharacteristic
22#16810828C2211yescharacteristic
23#16810823C2211yescharacteristic
24#16810818C2211yescharacteristic
25Q8×#88322C2212yes—
26D4×#88322—22no—
27D4×#88322C2212yes—
28#16810676—21no—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322—21no—
31C2×C4⋊#88322C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810617—21no—
34#16810373C2212yes—
35#16810373C2212yes—
36#16810372C2212yes—
37#16810372C2212yes—
38#16810370C2212yes—
39#16810366C2212yes—
40C2×C4⋊Q8×C53C2211yescharacteristic
41C4×#41×C53C2211yescharacteristic
42C53×#477C2212yes—
43C53×#476C2212yes—
44C53×#475—21no—
45C53×#474—21no—
46C53×#428—21no—
47C53×#423C2211yescharacteristic
48C53×#417—21no—
49C2×#11719945—21no—
50#13133237—44no—
51C2×#11719925—21no—
52C2×#11719923—21no—
53C2×#11719924—21no—
54C2×#11719922—21no—
55C22×#88322—22no—
56C22×#88322—21no—
57C22×#88322—21no—
58#13133217C2311yescharacteristic
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65#13133148—22no—
66#13133148—22no—
67C4⋊#88322—21no—
68C4⋊#88322—21no—
69C4⋊#88322—22no—
70C4⋊#88322D411yescharacteristic
71C4⋊#88322D411yescharacteristic
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2312yes—
75#13133140—21no—
76#13133140—21no—
77#13133137—21no—
78#13133137—21no—
79#13133137—22no—
80#13133137—22no—
81C22⋊#88322—22no—
82C22⋊#88322D412yes—
83C4×#88322—22no—
84C4×#88322C2312yes—
85C2×#41×C53D411yescharacteristic
86C22×Q8×C53—21no—
87C2×M16×C53—21no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53C2312yes—
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53D411yescharacteristic
97C4×D4×C53C2311yescharacteristic
98#117×C53—21no—
99#117×C53D411yescharacteristic
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C42×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#107×C53—22no—
105#106×C53—22no—
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#103×C53—22no—
109#102×C53—22no—
110#41×C53—22no—
111#41×C53SD1614yes—
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×Q8×C53—82no—
115C2×Q8×C53C2×D411yescharacteristic
116C2×D4×C53—21no—
117C2×D4×C53C2×D411yescharacteristic
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—41no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C2×C22⋊D53—41no—
126#11719945—42no—
127#11719945—42no—
128C2×D212—21no—
129C2×D212—21no—
130C2×D212—21no—
131C2×C4×D53—41no—
132C2×Dic106—21no—
133M16×C53—42no—
134C2×C8×C53—22no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53C2×D411yescharacteristic
142C4⋊C4×C53C2×D411yescharacteristic
143C22⋊C4×C53—21no—
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53—21no—
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C42×C53#4111yescharacteristic
152C42×C53C2411yescharacteristic
153#11719925—42no—
154#11719924—42no—
155#11719923—42no—
156#11719922—42no—
157#11719921—41no—
158#11719921—41no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C2×#88322—22no—
162C2×#88322—21no—
163C2×#88322—21no—
164C2×#88322—42no—
165C2×#88322—82no—
166C2×#88322—81no—
167C2×#88322C2×D412yes—
168C2×#88322C2×D412yes—
169C23×C53—21no—
170C23×C53#12111yescharacteristic
171C22×D53—41no—
172Q8×C53—21no—
173Q8×C53—82no—
174Q8×C53—82no—
175Q8×C53C2×SD1612yes—
176D4×C53—24no—
177D4×C53—21no—
178D4×C53C2×SD1612yes—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—24no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—42no—
189C2×C4×C53—81no—
190C2×C4×C53—82no—
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53C2×SD1612yes—
195C2×C4×C53#13512yes—
196C2×C4×C53C22×D411yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C2×#4111yescharacteristic
199C22⋊D53—82no—
200C2×Dic53—41no—
201D212—41no—
202D212—41no—
203D212—41no—
204D212—41no—
205D212—81no—
206C4×D53—82no—
207Dic106—41no—
208Dic106—41no—
209C8×C53—42no—
210#88322—22no—
211#88322—22no—
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—82no—
233C4×C53#46512yes—
234C4×C53C22×SD1611yescharacteristic
235C4×C53C2×#13511yescharacteristic
236C4×C53#57711yescharacteristic
237C4×C53#57811yescharacteristic
238Dic53—81no—
239#3227—531no—
240#3079D5311yescharacteristic, complemented
241#3035—531no—
242#2988—531no—
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⋊Q8C2×D5311yescharacteristic
259C4×#41C2×D5311yescharacteristic
260#498—532no—
261#486—532no—
262#481—1062no—
263#477C2×D5312yes—
264#476—1061no—
265#476—1061no—
266#476C2×D5312yes—
267#475—21no—
268#474—21no—
269#464—1061no—
270#463—1061no—
271#462—531no—
272#461—531no—
273#460—531no—
274#459—531no—
275C8×Q8—532no—
276C8×D4—532no—
277C8×D4—1062no—
278#431—531no—
279#431—1061no—
280#428—21no—
281#423C2×D5311yescharacteristic
282C2×C4⋊C8—531no—
283C2×C4⋊C8—531no—
284C2×C4⋊C8—1061no—
285#417—21no—
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×#41C22⋊D5311yescharacteristic
296C22×Q8—21no—
297#135—2124no—
298C2×Q16—1061no—
299C2×SD16—1061no—
300C2×SD16—1061no—
301C2×D8—1061no—
302C2×M16—21no—
303C22×C8—1062no—
304C22×C8—1061no—
305C22×C8—1061no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—531no—
310C4⋊Q8C22×D5312yes—
311#127—531no—
312#124—1061no—
313#121—1061no—
314#121—1061no—
315C4×Q8C22⋊D5311yescharacteristic
316C4×Q8C22×D5311yescharacteristic
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—1061no—
320C4×D4—1061no—
321C4×D4C22⋊D5311yescharacteristic
322C4×D4C22×D5311yescharacteristic
323#117—21no—
324#117C22⋊D5311yescharacteristic
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C42—21no—
328C2×C42C22×D5311yescharacteristic
329#107—22no—
330#106—22no—
331#106—1062no—
332#106—1062no—
333C4⋊C8—21no—
334C4⋊C8—531no—
335C4⋊C8—531no—
336C4⋊C8—532no—
337C4⋊C8—531no—
338C4⋊C8—531no—
339C4⋊C8—1062no—
340C4⋊C8—1061no—
341C4⋊C8—1061no—
342C4⋊C8C22×D5312yes—
343#103—22no—
344#103—1061no—
345#103—1061no—
346#102—22no—
347#102—1062no—
348#102—1062no—
349#102—1061no—
350#102—1061no—
351C22⋊C8—532no—
352C22⋊C8—1062no—
353C4×C8—532no—
354C4×C8—1062no—
355#41—22no—
356#41—2122no—
357#41—2122no—
358#41#1171992414yes—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—82no—
362C2×Q8—1061no—
363C2×Q8C2×C22⋊D5311yescharacteristic
364C2×D4—21no—
365C2×D4—1061no—
366C2×D4—1061no—
367C2×D4—1061no—
368C2×D4—2121no—
369C2×D4C2×C22⋊D5311yescharacteristic
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—2121no—
375C22×C4D4×D5311yescharacteristic
376C22×C4#1171994711yescharacteristic
377C22×C4C2×C22⋊D5311yescharacteristic
378Q16—2122no—
379SD16—2122no—
380SD16—2122no—
381D8—2122no—
382M16—42no—
383C2×C8—22no—
384C2×C8—532no—
385C2×C8—532no—
386C2×C8—1062no—
387C2×C8—1061no—
388C2×C8—1061no—
389C2×C8—4242no—
390C2×C8—4241no—
391C2×C8—2122no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—1061no—
399C4⋊C4—1061no—
400C4⋊C4C2×C22⋊D5311yescharacteristic
401C4⋊C4C2×C22⋊D5311yescharacteristic
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—2121no—
406C22⋊C4—2121no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42D4×D5311yescharacteristic
411C42#1171994711yescharacteristic
412C42C2×C22⋊D5311yescharacteristic
413C42C23×D5311yescharacteristic
414C23—21no—
415C23—2121no—
416C23#1313318911yescharacteristic
417Q8—21no—
418Q8—82no—
419Q8—82no—
420Q8—2121no—
421Q8—2121no—
422Q8C2×#1171992412yes—
423D4—24no—
424D4—21no—
425D4—4241no—
426D4—4242no—
427D4—2121no—
428D4—2121no—
429D4—2121no—
430D4—2121no—
431D4C2×#1171992412yes—
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—42no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—4242no—
445C2×C4—2121no—
446C2×C4#1313318911yescharacteristic
447C2×C4#1313318911yescharacteristic
448C2×C4#1313318911yescharacteristic
449C2×C4C2×#1171992412yes—
450C2×C4#1313323712yes—
451C2×C4C2×D4×D5311yescharacteristic
452C2×C4C22×C22⋊D5311yescharacteristic
453C2×C4C2×#1171994711yescharacteristic
454C8—42no—
455C8—1062no—
456C8—1062no—
457C8—2121no—
458C8—2121no—
459C8—2122no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—4241no—
466C22—4241no—
467C22—4241no—
468C22#1681085412yes—
469C22C2×#1313318911yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—82no—
477C4—4241no—
478C4#1681085412yes—
479C4C22×#1171992411yescharacteristic
480C4C2×#1313323711yescharacteristic
481C4#1681143211yescharacteristic
482C4#1681143911yescharacteristic
483C2—22no—
484C2—42no—
485C2—8481no—
486C2#2028426911yescharacteristic
487C2C2×#1681085411yescharacteristic
488C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3079C2
socleC22×C53C2×#121
prime radical#20284170C2
hypercenter#3079D53
hypocenterC53#19414
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3079

Complements

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

NH ≅ G/Ntype
1. #20285843C2semidirect
3. #20283826C2semidirect
4. #20274133C2semidirect
5. #20274126C2semidirect
6. #20274099C2semidirect
7. #20274092C2semidirect
8. #20273487C2semidirect
9. #20273393C2semidirect
10. C2×#16810372C2semidirect
11. C53×#3079C2semidirect
240. #3079D53semidirect
293. C53#19414semidirect

Maximal subgroups

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

subgroupindex
#202858432
#202841702
#202838262
#202741332
#202741262
#202740992
#202740922
#202734872
#202733932
C2×#168103722
C53×#30792
#1941453