Finite Group Databasebeta

← #27370169

Subgroups — #27370169

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285640C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic
3#20283804C211yesmaximal, characteristic, complemented
4#20274295C212yesmaximal, complemented
5#20274189C212yesmaximal, complemented
6#20274171C212yesmaximal, complemented
7#20274065C212yesmaximal, complemented
8#20273498C211yesmaximal, characteristic, complemented
9#20273432C211yesmaximal, characteristic, complemented
10#20273408C211yesmaximal, characteristic
11C2×C53×#475C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16810997C2212yes—
14#16810973C2212yes—
15#16810936—21no—
16#16810918C2211yescharacteristic
17#16810918C2211yescharacteristic
18#16810825—22no—
19#16810823C2211yescharacteristic
20#16810823C2211yescharacteristic
21#16810762—21no—
22#16810762—21no—
23#16810762—21no—
24#16810754—22no—
25#16810754C2212yes—
26Q8×#88322C2212yes—
27#16810721—21no—
28#16810721—21no—
29#16810680—21no—
30#16810680C2211yescharacteristic
31#16810680C2211yescharacteristic
32#16810676—21no—
33#16810676C2211yescharacteristic
34#16810375C2212yes—
35#16810373C2212yes—
36#16810371C2212yes—
37#16810369C2212yes—
38#16810369C2212yes—
39#16810366C2212yes—
40C2×C4⋊Q8×C53C2211yescharacteristic
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#475—21no—
43C53×#475—21no—
44C53×#475C2212yes—
45C53×#475C2212yes—
46C2×#106×C53—21no—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#103×C53—21no—
49C2×#11719945—21no—
50C2×#11719924—22no—
51C2×#11719924—22no—
52C2×#11719924—21no—
53C2×#11719924—21no—
54#13133217C2311yescharacteristic
55#13133214C2311yescharacteristic
56#13133205—21no—
57#13133191—21no—
58#13133191—21no—
59C4×D212—21no—
60C4×D212—21no—
61#13133150—21no—
62#13133150—21no—
63C4⋊#88322—22no—
64C4⋊#88322—22no—
65C4⋊#88322D412yes—
66C4⋊#88322C2312yes—
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2311yescharacteristic
69#13133141—22no—
70#13133141—22no—
71#13133141—21no—
72#13133141—21no—
73#13133137—21no—
74#13133137—21no—
75C22⋊#88322—21no—
76C22⋊#88322—21no—
77C22⋊#88322D411yescharacteristic
78C22⋊#88322D411yescharacteristic
79#13133128—21no—
80#13133128—21no—
81#13133128—22no—
82C4×#88322—22no—
83C4×#88322C2312yes—
84C22×Q8×C53—21no—
85C22×Q8×C53D411yescharacteristic
86C22×C8×C53—21no—
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53C2312yes—
91C4×Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53D411yescharacteristic
94C4×Q8×C53D411yescharacteristic
95C4×Q8×C53C2311yescharacteristic
96C4×Q8×C53C2311yescharacteristic
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C42×C53—21no—
102C2×C42×C53C2311yescharacteristic
103#106×C53—22no—
104#106×C53—22no—
105C4⋊C8×C53—21no—
106C4⋊C8×C53C2312yes—
107#103×C53—22no—
108#103×C53—22no—
109C2×Q8×C53—21no—
110C2×Q8×C53—22no—
111C2×Q8×C53—22no—
112C2×Q8×C53—41no—
113C2×Q8×C53—41no—
114C2×Q8×C53—82no—
115C2×Q8×C53C2×D411yescharacteristic
116C2×Q8×C53C2×D411yescharacteristic
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—41no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C2×C22⋊D53—41no—
124#11719945—82no—
125C2×D212—21no—
126C2×D212—21no—
127C2×D212—21no—
128C2×C4×D53—41no—
129C2×Dic106—21no—
130C2×C8×C53—22no—
131C2×C8×C53—42no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
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
143C42×C53—21no—
144C42×C53—21no—
145C42×C53—21no—
146C42×C53—21no—
147C42×C53C2×D411yescharacteristic
148C42×C53C2×D411yescharacteristic
149C42×C53#4111yescharacteristic
150C42×C53C2411yescharacteristic
151#11719924—42no—
152#11719924—42no—
153#11719924—82no—
154#11719924—82no—
155#11719921—41no—
156#11719921—41no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C2×#88322—22no—
160C2×#88322—21no—
161C2×#88322—21no—
162C2×#88322C2×D412yes—
163C2×#88322C2×D412yes—
164C23×C53#12111yescharacteristic
165C22×D53—41no—
166Q8×C53—21no—
167Q8×C53—21no—
168Q8×C53—21no—
169Q8×C53—42no—
170Q8×C53—42no—
171Q8×C53—42no—
172Q8×C53—82no—
173Q8×C53#13712yes—
174C2×C4×C53—21no—
175C2×C4×C53—22no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—22no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—82no—
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C2×C4×C53C2×#4111yescharacteristic
192C22⋊D53—82no—
193C2×Dic53—41no—
194D212—41no—
195D212—41no—
196D212—42no—
197D212—81no—
198C4×D53—82no—
199Dic106—81no—
200C8×C53—42no—
201#88322—22no—
202#88322—22no—
203#88322—41no—
204#88322—41no—
205#88322—42no—
206#19469—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53#47112yes—
210C22×C53C2×#12111yescharacteristic
211C2×D53—81no—
212C2×D53—81no—
213C2×D53—81no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—22no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—41no—
220C4×C53—41no—
221C4×C53—41no—
222C4×C53—82no—
223C4×C53C2×#13711yescharacteristic
224C4×C53#57511yescharacteristic
225C4×C53#57712yes—
226Dic53—81no—
227#3206—531no—
228#3112—531no—
229C2×#475D5311yescharacteristic, complemented
230#2984—531no—
231#1704—532no—
232#1686—532no—
233#1684—532no—
234#1666—532no—
235#1573—531no—
236#1556—531no—
237#1551—531no—
238C2×C53—22no—
239C2×C53#304311yescharacteristic
240C2×C53#305611yescharacteristic
241C2×C53C2×#47111yescharacteristic
242D53—161no—
243#532—531no—
244C2×C4⋊Q8C2×D5311yescharacteristic
245C2×C4×Q8C2×D5311yescharacteristic
246#492—532no—
247#487—532no—
248#480—1061no—
249#477—531no—
250#477—531no—
251#475—21no—
252#475—21no—
253#475C2×D5312yes—
254#475C2×D5312yes—
255#461—1062no—
256#460—531no—
257#460—531no—
258#448—1061no—
259#448—1061no—
260#448—1061no—
261#446—532no—
262#446—1062no—
263C8×Q8—532no—
264#440—1061no—
265#440—1061no—
266#432—531no—
267#432—531no—
268#432—1061no—
269#431—531no—
270#431—1061no—
271C2×#106—21no—
272C2×C4⋊C8C2×D5311yescharacteristic
273C2×#103—21no—
274#335—532no—
275#334—532no—
276#333—532no—
277#332—532no—
278#332—532no—
279#331—532no—
280C53#1946911yescharacteristic, complemented, Sylow
281C2×#41—1061no—
282C22×Q8—21no—
283C22×Q8C22⋊D5311yescharacteristic
284C2×SD16—1062no—
285C2×SD16—1062no—
286C2×SD16—1061no—
287C2×SD16—1061no—
288C22×C8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—531no—
293C4⋊Q8C22×D5312yes—
294#127—531no—
295#124—1061no—
296#121—1061no—
297#121—1061no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8C22⋊D5311yescharacteristic
301C4×Q8C22⋊D5311yescharacteristic
302C4×Q8C22×D5311yescharacteristic
303C4×Q8C22×D5311yescharacteristic
304C4×D4—1061no—
305C4×D4—1061no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C22⋊D5311yescharacteristic
310C2×C42—21no—
311C2×C42C22×D5311yescharacteristic
312#107—1061no—
313#107—1061no—
314#106—22no—
315#106—22no—
316C4⋊C8—21no—
317C4⋊C8—532no—
318C4⋊C8—532no—
319C4⋊C8—531no—
320C4⋊C8—531no—
321C4⋊C8—1062no—
322C4⋊C8—1062no—
323C4⋊C8C22×D5312yes—
324#103—22no—
325#103—22no—
326#103—1062no—
327#103—1062no—
328#103—1061no—
329#103—1061no—
330#102—1061no—
331#102—1061no—
332C22⋊C8—531no—
333C22⋊C8—531no—
334C22⋊C8—1061no—
335C22⋊C8—1061no—
336#97—1062no—
337#97—1061no—
338#97—1061no—
339C4×C8—532no—
340C4×C8—1062no—
341#41—4242no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—82no—
348C2×Q8—1061no—
349C2×Q8C2×C22⋊D5311yescharacteristic
350C2×Q8C2×C22⋊D5311yescharacteristic
351C2×D4—1061no—
352C2×D4—1061no—
353C2×D4—1061no—
354C2×D4—2121no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—41no—
358C22×C4—2121no—
359C22×C4D4×D5311yescharacteristic
360C22×C4#1171994711yescharacteristic
361C22×C4C2×C22⋊D5311yescharacteristic
362SD16—4242no—
363SD16—4242no—
364SD16—2122no—
365SD16—2122no—
366C2×C8—22no—
367C2×C8—42no—
368C2×C8—532no—
369C2×C8—532no—
370C2×C8—1062no—
371C2×C8—1061no—
372C2×C8—1061no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—1061no—
383C4⋊C4—1061no—
384C4⋊C4C2×C22⋊D5311yescharacteristic
385C4⋊C4C2×C22⋊D5311yescharacteristic
386C22⋊C4—2121no—
387C22⋊C4—2121no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42D4×D5311yescharacteristic
393C42#1171994711yescharacteristic
394C42C2×C22⋊D5311yescharacteristic
395C42C23×D5311yescharacteristic
396C23—2121no—
397C23#1313318911yescharacteristic
398Q8—21no—
399Q8—21no—
400Q8—21no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404Q8—82no—
405Q8—4241no—
406Q8#1313324812yes—
407D4—4241no—
408D4—4242no—
409D4—2121no—
410D4—2121no—
411D4—2122no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—4242no—
425C2×C4—2121no—
426C2×C4#1313318911yescharacteristic
427C2×C4#1313318911yescharacteristic
428C2×C4#1313318911yescharacteristic
429C2×C4C2×D4×D5311yescharacteristic
430C2×C4C22×C22⋊D5311yescharacteristic
431C2×C4C2×#1171994711yescharacteristic
432C8—42no—
433C8—1062no—
434C8—1062no—
435C8—2121no—
436C8—2121no—
437C8—2122no—
438C22—21no—
439C22—21no—
440C22—4241no—
441C22—4241no—
442C22—4241no—
443C22#1681088812yes—
444C22C2×#1313318911yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—82no—
454C4—4241no—
455C4C2×#1313324811yescharacteristic
456C4#1681142111yescharacteristic
457C4#1681142912yes—
458C2—22no—
459C2—8481no—
460C2#2028423311yescharacteristic
461C2#2028434111yescharacteristic
462C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#475C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×#475D53
hypocenterC53#19469
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #20283804C2semidirect
4. #20274295C2semidirect
5. #20274189C2semidirect
6. #20274171C2semidirect
7. #20274065C2semidirect
8. #20273498C2semidirect
9. #20273432C2semidirect
11. C2×C53×#475C2semidirect
229. C2×#475D53semidirect
280. C53#19469semidirect

Maximal subgroups

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

subgroupindex
#202856402
#202848372
#202838042
#202742952
#202741892
#202741712
#202740652
#202734982
#202734322
#202734082
C2×C53×#4752
#1946953