Finite Group Databasebeta

← #27369513

Subgroups — #27369513

Order 13568.

620 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20284497C211yesmaximal, characteristic
3#20283823C211yesmaximal, characteristic
4#20274170C212yesmaximal, complemented
5#20274163C212yesmaximal, complemented
6#20274082C212yesmaximal, complemented
7#20274077C212yesmaximal, complemented
8#20273490C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10C2×#16810369C211yesmaximal, characteristic, complemented
11C53×#3209C211yesmaximal, characteristic, complemented
12C4×#11719945C2211yescharacteristic
13#16810927—21no—
14#16810920—21no—
15#16810915—21no—
16#16810915C2211yescharacteristic
17#16810910—21no—
18#16810910C2211yescharacteristic
19#16810905C2211yescharacteristic
20#16810905C2212yes—
21#16810900C2211yescharacteristic
22#16810900C2212yes—
23C8×Dic106C2212yes—
24C8×D212C2212yes—
25#16810686—22no—
26#16810675—21no—
27#16810663—21no—
28#16810659—21no—
29#16810647C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31C2×C4⋊#88322C2211yescharacteristic
32C4⋊C8×D53—21no—
33#16810622—21no—
34#16810617—21no—
35#16810376C2212yes—
36#16810376C2212yes—
37#16810372C2212yes—
38#16810369C2212yes—
39#16810369C2212yes—
40#16810369C2212yes—
41C53×#533C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#494C2212yes—
44C53×#487C2212yes—
45C53×#462—21no—
46C53×#462—21no—
47C53×#451—22no—
48C53×#431C2211yescharacteristic
49C2×#11719945D411yescharacteristic
50C2×#11719917—21no—
51C22×#88322—21no—
52C22×#88322—21no—
53C2×C8×D53—22no—
54C4×Dic106D411yescharacteristic
55C4×Dic106C2311yescharacteristic
56C4×D212D411yescharacteristic
57C4×D212C2311yescharacteristic
58C4×C22⋊D53—21no—
59C4×C22⋊D53—21no—
60#13133176D411yescharacteristic
61#13133174—21no—
62C42×D53—21no—
63#13133150—21no—
64#13133150—22no—
65#13133150—21no—
66#13133148—22no—
67#13133148—21no—
68#13133148—21no—
69C4⋊#88322—21no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322C2312yes—
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2312yes—
76#13133146—22no—
77#13133140—21no—
78#13133140—21no—
79#13133140—22no—
80#13133137—21no—
81#13133137—21no—
82#13133137—22no—
83#13133129—22no—
84C8×Dic53—22no—
85C22×Q8×C53—21no—
86C2×Q16×C53—22no—
87C2×Q16×C53—22no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53C2311yescharacteristic
92C4⋊Q8×C53C2312yes—
93C4⋊Q8×C53C2311yescharacteristic
94#125×C53—21no—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97C4×Q8×C53—21no—
98C4×Q8×C53—21no—
99#117×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C42×C53C2311yescharacteristic
103C4⋊C8×C53D411yescharacteristic
104C4⋊C8×C53D411yescharacteristic
105#103×C53—22no—
106#103×C53—22no—
107C22⋊C8×C53D412yes—
108C4×C8×C53C2312yes—
109C2×Q8×C53—22no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—41no—
113C2×Q8×C53—81no—
114C2×Q8×C53—81no—
115C22×C4×C53—41no—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C2×C22⋊D53—21no—
120#11719945—22no—
121#11719945Q1614yescomplemented
122C2×D212C2×D411yescharacteristic
123C2×C4×D53—21no—
124C2×C4×D53—21no—
125C2×C4×D53—21no—
126C2×Dic106C2×D411yescharacteristic
127Q16×C53—82no—
128Q16×C53—82no—
129C2×C8×C53C2×D412yes—
130C2×C8×C53C2×D412yes—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53—41no—
141C22⋊C4×C53—41no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147C22⋊Dic53—21no—
148#11719921—21no—
149#11719921—21no—
150C4⋊Dic53C2×D411yescharacteristic
151C4⋊Dic53C2×D411yescharacteristic
152#11719919—21no—
153#11719919—21no—
154C4×Dic53—21no—
155C4×Dic53—21no—
156C4×Dic53—21no—
157#11719917—42no—
158C2×#88322—22no—
159C2×#88322—21no—
160C2×#88322—22no—
161C2×#88322—21no—
162C2×#88322—42no—
163C2×#88322—81no—
164C8×D53—82no—
165C23×C53#12111yescharacteristic
166C22×D53—21no—
167Q8×C53—42no—
168Q8×C53—41no—
169Q8×C53—42no—
170Q8×C53—41no—
171Q8×C53—82no—
172Q8×C53—81no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—41no—
176C2×C4×C53—42no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—82no—
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53C2×Q1612yes—
184C2×C4×C53#13512yes—
185C2×C4×C53C22×D411yescharacteristic
186C2×C4×C53C22×D411yescharacteristic
187C2×C4×C53C2×#4111yescharacteristic
188C22⋊D53—24no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—21no—
192D212—21no—
193D212C2×Q1612yes—
194C4×D53—24no—
195C4×D53—41no—
196C4×D53—41no—
197C4×D53—81no—
198Dic106—21no—
199Dic106C2×Q1612yes—
200C8×C53—22no—
201C8×C53—22no—
202#88322—41no—
203#88322—41no—
204#88322—42no—
205#88322—42no—
206#19411—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53#46712yes—
210C22×C53C2×#12111yescharacteristic
211C2×D53—22no—
212C2×D53—41no—
213C2×D53—41no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—82no—
218C4×C53—81no—
219C4×C53—81no—
220C4×C53#46712yes—
221C4×C53C22×Q1611yescharacteristic
222C4×C53C2×#13511yescharacteristic
223C4×C53#57611yescharacteristic
224C4×C53#57711yescharacteristic
225Dic53—22no—
226Dic53—41no—
227Dic53—41no—
228#3209D5311yescharacteristic, complemented
229#3079—531no—
230#3076—531no—
231#2988—531no—
232#1684—532no—
233#1683—532no—
234#1670—532no—
235#1669—532no—
236#1571—531no—
237#1550—531no—
238C2×#332—531no—
239C2×C53—22no—
240C2×C53#304511yescharacteristic
241C2×C53C2×#46711yescharacteristic
242C2×C53#306911yescharacteristic
243D53—42no—
244#533C2×D5311yescharacteristic
245C2×C4⋊Q8C2×D5311yescharacteristic
246C4×#41—531no—
247#494C2×D5312yes—
248#487C2×D5312yes—
249#479—1061no—
250#478—1061no—
251#477—531no—
252#477—1061no—
253#476—531no—
254#476—1061no—
255#475—531no—
256#475—532no—
257#474—531no—
258#474—532no—
259#462—21no—
260#462—21no—
261#451—22no—
262C8×Q8—532no—
263C8×D4—532no—
264C8×D4—1062no—
265#431—1061no—
266#431C2×D5311yescharacteristic
267#428—1061no—
268#427—1061no—
269#423—531no—
270C2×C4⋊C8—531no—
271C2×C4⋊C8—531no—
272C2×C4⋊C8—1061no—
273#417—1061no—
274#416—1061no—
275#335—532no—
276#335—532no—
277#333—532no—
278#332—532no—
279#332—532no—
280#332—532no—
281C53#1941111yescharacteristic, complemented, Sylow
282C2×#41—531no—
283C22×Q8—21no—
284C2×Q16—22no—
285C2×Q16—22no—
286C2×M16—1061no—
287C22×C8—1061no—
288C22×C8—1061no—
289C22×C8—1062no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8C22×D5312yes—
294C4⋊Q8C22×D5311yescharacteristic
295C4⋊Q8C22×D5311yescharacteristic
296#125—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—531no—
302C4×Q8—531no—
303C4×D4—531no—
304C4×D4—531no—
305C4×D4—1061no—
306C4×D4—1061no—
307#117—21no—
308#117—531no—
309#117—1061no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C42—1061no—
313C2×C42C22×D5311yescharacteristic
314#107—1061no—
315#107—1061no—
316#107—1062no—
317#106—1061no—
318#106—1061no—
319#106—1062no—
320C4⋊C8—532no—
321C4⋊C8—531no—
322C4⋊C8—531no—
323C4⋊C8—532no—
324C4⋊C8—1062no—
325C4⋊C8—1061no—
326C4⋊C8—1061no—
327C4⋊C8—1061no—
328C4⋊C8C22⋊D5311yescharacteristic
329C4⋊C8C22⋊D5311yescharacteristic
330#103—22no—
331#103—22no—
332#103—1061no—
333#103—1061no—
334#103—1062no—
335#102—1061no—
336#102—1061no—
337#102—1062no—
338C22⋊C8—1062no—
339C22⋊C8C22⋊D5312yes—
340C4×C8—1062no—
341C4×C8C22×D5312yes—
342#41—534no—
343#41—1062no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—41no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×Q8—531no—
351C2×D4—531no—
352C2×D4—1061no—
353C22×C4—41no—
354C22×C4—1061no—
355C22×C4—1061no—
356C22×C4—1061no—
357C22×C4D4×D5311yescharacteristic
358C22×C4#1171994711yescharacteristic
359C22×C4C2×C22⋊D5311yescharacteristic
360Q16—82no—
361Q16—82no—
362M16—2122no—
363C2×C8—1061no—
364C2×C8—1062no—
365C2×C8—1061no—
366C2×C8—1062no—
367C2×C8—4242no—
368C2×C8—4241no—
369C2×C8—2122no—
370C2×C8C2×C22⋊D5312yes—
371C2×C8C2×C22⋊D5312yes—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—531no—
383C4⋊C4—531no—
384C4⋊C4—1061no—
385C4⋊C4—1061no—
386C22⋊C4—41no—
387C22⋊C4—1061no—
388C22⋊C4—1061no—
389C22⋊C4—1061no—
390C42—41no—
391C42—1061no—
392C42—1061no—
393C42—1061no—
394C42D4×D5311yescharacteristic
395C42#1171994711yescharacteristic
396C42C2×C22⋊D5311yescharacteristic
397C42C23×D5311yescharacteristic
398C23—1061no—
399C23#1313318811yescharacteristic
400Q8—41no—
401Q8—42no—
402Q8—41no—
403Q8—42no—
404Q8—81no—
405Q8—82no—
406Q8—532no—
407Q8—1061no—
408D4—532no—
409D4—1064no—
410D4—1061no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—1061no—
419C2×C4—1064no—
420C2×C4—1061no—
421C2×C4—1061no—
422C2×C4—4241no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4#1313318811yescharacteristic
426C2×C4#1313318811yescharacteristic
427C2×C4#1313318811yescharacteristic
428C2×C4Q16×D5312yes—
429C2×C4#1313324012yes—
430C2×C4C2×D4×D5311yescharacteristic
431C2×C4C22×C22⋊D5311yescharacteristic
432C2×C4C2×#1171994711yescharacteristic
433C8—22no—
434C8—22no—
435C8—2121no—
436C8—2121no—
437C8—2122no—
438C8—2122no—
439C22—21no—
440C22—21no—
441C22—1062no—
442C22—2121no—
443C22—2121no—
444C22#1681086112yes—
445C22C2×#1313318811yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—1062no—
453C4—2121no—
454C4—2121no—
455C4#1681086112yes—
456C4C2×Q16×D5311yescharacteristic
457C4C2×#1313324011yescharacteristic
458C4#1681142611yescharacteristic
459C4#1681143411yescharacteristic
460C2—22no—
461C2—2122no—
462C2#2028425011yescharacteristic
463C2C2×#1681086111yescharacteristic
464C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3209C2
socleC22×C53C2×#121
prime radicalC2×#11719945D4
hypercenter#3209D53
hypocenterC53#19411
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
4. #20274170C2semidirect
5. #20274163C2semidirect
6. #20274082C2semidirect
7. #20274077C2semidirect
8. #20273490C2semidirect
9. #20273405C2semidirect
10. C2×#16810369C2semidirect
11. C53×#3209C2semidirect
121. #11719945Q16semidirect
228. #3209D53semidirect
281. C53#19411semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202844972
#202838232
#202741702
#202741632
#202740822
#202740772
#202734902
#202734052
C2×#168103692
C53×#32092
#1941153