Finite Group Databasebeta

← #27368031

Subgroups — #27368031

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284871C212yesmaximal
2C8×#11719945C211yesmaximal, characteristic
3#20274116C211yesmaximal, characteristic, complemented
4#20274111C211yesmaximal, characteristic, complemented
5#20274106C211yesmaximal, characteristic, complemented
6#20274101C211yesmaximal, characteristic, complemented
7#20274094C212yesmaximal, complemented
8#20274087C212yesmaximal, complemented
9#20273478C211yesmaximal, characteristic, complemented
10#20273425C212yesmaximal, complemented
11C53×#3160C211yesmaximal, characteristic, complemented
12C4×#11719945C2211yescharacteristic
13#16810961—22no—
14#16810915C2212yes—
15#16810910C2212yes—
16#16810830C2212yes—
17#16810825C2212yes—
18#16810783—22no—
19C8×Dic106C2211yescharacteristic
20C8×Dic106C2211yescharacteristic
21#16810728—22no—
22#16810721—22no—
23C8×D212C2211yescharacteristic
24C8×D212C2211yescharacteristic
25C8×C22⋊D53—21no—
26C8×C22⋊D53—21no—
27#16810680C2212yes—
28#16810676C2211yescharacteristic
29#16810674—21no—
30C4×C8×D53—21no—
31#16810376C2211yescharacteristic
32#16810376C2211yescharacteristic
33#16810374C2212yes—
34#16810372C2212yes—
35#16810372C2212yes—
36#16810369C2212yes—
37#16810369C2212yes—
38C53×#533C2212yes—
39C53×#500C2211yescharacteristic
40C53×#500C2211yescharacteristic
41C53×#499—21no—
42C53×#499—21no—
43C53×#498C2212yes—
44C53×#427—21no—
45C53×#427—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×#11719945D411yescharacteristic
48C2×#11719925—22no—
49C2×#11719924—22no—
50C2×C8×D53—21no—
51C2×C8×D53—21no—
52C4×Dic106D411yescharacteristic
53C4×Dic106C2311yescharacteristic
54C4×D212D411yescharacteristic
55C4×D212C2311yescharacteristic
56C4×C22⋊D53—21no—
57C4×C22⋊D53—21no—
58#13133176D411yescharacteristic
59#13133174—21no—
60C42×D53—21no—
61#13133150—22no—
62#13133148—22no—
63C4⋊#88322C2312yes—
64C4⋊#88322C2312yes—
65C4⋊#88322C2311yescharacteristic
66C4⋊#88322C2311yescharacteristic
67#13133146—21no—
68#13133146—21no—
69#13133141—22no—
70#13133141—22no—
71#13133140—22no—
72#13133137—22no—
73C22⋊#88322—22no—
74C22⋊#88322—21no—
75#13133129—21no—
76#13133129—21no—
77#13133128—22no—
78C4×#88322—21no—
79C8×Dic53—21no—
80C8×Dic53—21no—
81C22×C8×C53D411yescharacteristic
82C22×C8×C53D411yescharacteristic
83C4⋊Q8×C53C2312yes—
84C4⋊Q8×C53C2312yes—
85#125×C53—22no—
86C22⋊Q8×C53—22no—
87C22⋊Q8×C53—22no—
88C4×Q8×C53—22no—
89C4×Q8×C53—22no—
90#117×C53—22no—
91C2×C42×C53C2311yescharacteristic
92#107×C53—21no—
93#107×C53—21no—
94#107×C53—21no—
95#107×C53—21no—
96#106×C53—22no—
97#106×C53—22no—
98C4×C8×C53D411yescharacteristic
99C4×C8×C53D411yescharacteristic
100C4×C8×C53C2311yescharacteristic
101C4×C8×C53C2311yescharacteristic
102C2×Q8×C53—22no—
103C2×Q8×C53—22no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C2×C22⋊D53—21no—
108#11719945—22no—
109#11719945—22no—
110C2×D212C2×D411yescharacteristic
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×C4×D53—21no—
114C2×Dic106C2×D411yescharacteristic
115C2×C8×C53—22no—
116C2×C8×C53—22no—
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53C2×D411yescharacteristic
120C2×C8×C53C2×D411yescharacteristic
121C4⋊C4×C53—22no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—42no—
126C4⋊C4×C53—42no—
127C22⋊C4×C53—42no—
128C42×C53—42no—
129C42×C53C2×D411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C42×C53#4111yescharacteristic
132C42×C53C2411yescharacteristic
133C22⋊Dic53—21no—
134#11719925—44no—
135#11719924—44no—
136#11719921—21no—
137#11719921—21no—
138C4⋊Dic53C2×D411yescharacteristic
139C4⋊Dic53C2×D411yescharacteristic
140#11719919—21no—
141#11719919—21no—
142C4×Dic53—21no—
143C4×Dic53—21no—
144C4×Dic53—21no—
145C2×#88322—21no—
146C2×#88322—21no—
147C2×#88322—22no—
148C2×#88322—22no—
149C8×D53—42no—
150C8×D53—42no—
151C23×C53#12111yescharacteristic
152C22×D53—21no—
153Q8×C53—42no—
154Q8×C53—42no—
155Q8×C53—82no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53C22×D411yescharacteristic
167C2×C4×C53C22×D411yescharacteristic
168C2×C4×C53C2×#4111yescharacteristic
169C22⋊D53—24no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173D212—21no—
174D212#13512yes—
175C4×D53—24no—
176C4×D53—42no—
177C4×D53—41no—
178C4×D53—41no—
179Dic106—21no—
180Dic106#13512yes—
181C8×C53—21no—
182C8×C53—21no—
183C8×C53#13712yes—
184C8×C53#13712yes—
185#88322—42no—
186#88322—42no—
187#88322—41no—
188#88322—41no—
189#19273—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53C2×#12111yescharacteristic
194C2×D53—22no—
195C2×D53—41no—
196C2×D53—41no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—82no—
202C4×C53—82no—
203C4×C53C2×#13511yescharacteristic
204C4×C53C2×#13511yescharacteristic
205C4×C53C2×#13711yescharacteristic
206C4×C53C2×#13711yescharacteristic
207Dic53—22no—
208Dic53—41no—
209Dic53—41no—
210#3160D5311yescharacteristic, complemented
211#3116—532no—
212C8×#41—531no—
213#1676—531no—
214#1675—531no—
215#1674—531no—
216#1673—531no—
217#1672—532no—
218#1671—532no—
219#1568—531no—
220#1554—532no—
221C2×C53—41no—
222C2×C53#303911yescharacteristic
223C2×C53#305011yescharacteristic
224C2×C53#306711yescharacteristic
225D53—42no—
226#533C2×D5312yes—
227C4×#41—531no—
228#500C2×D5311yescharacteristic
229#500C2×D5311yescharacteristic
230#499—21no—
231#499—21no—
232#498C2×D5312yes—
233#485—1062no—
234#477—532no—
235#476—532no—
236#462—532no—
237#461—532no—
238#452—1062no—
239C8×Q8—531no—
240C8×Q8—531no—
241#441—1062no—
242#440—1062no—
243C8×D4—531no—
244C8×D4—531no—
245C8×D4—1061no—
246C8×D4—1061no—
247#432—532no—
248#431—531no—
249#431—1061no—
250#427—21no—
251#427—21no—
252C2×C4×C8—1061no—
253C2×C4×C8C2×D5311yescharacteristic
254#335—531no—
255#335—531no—
256#334—532no—
257#333—532no—
258#333—532no—
259#332—532no—
260#332—532no—
261C53#1927311yescharacteristic, complemented, Sylow
262C2×#41—531no—
263C2×Q16—1062no—
264C2×SD16—1062no—
265C22×C8—1061no—
266C22×C8—1061no—
267C22×C8C22⋊D5311yescharacteristic
268C22×C8C22⋊D5311yescharacteristic
269C4⋊Q8C22×D5312yes—
270C4⋊Q8C22×D5312yes—
271#125—22no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—531no—
277C4×Q8—531no—
278C4×D4—531no—
279C4×D4—531no—
280C4×D4—1061no—
281C4×D4—1061no—
282#117—22no—
283#117—531no—
284#117—1061no—
285C2×C42—1061no—
286C2×C42C22×D5311yescharacteristic
287#107—21no—
288#107—21no—
289#107—21no—
290#107—21no—
291#107—1062no—
292#106—22no—
293#106—22no—
294#106—1062no—
295C4⋊C8—531no—
296C4⋊C8—531no—
297C4⋊C8—532no—
298C4⋊C8—532no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301#103—1062no—
302#103—1062no—
303#103—1062no—
304#102—1062no—
305C22⋊C8—1062no—
306C22⋊C8—1061no—
307C22⋊C8—1061no—
308C22⋊C8—1061no—
309#97—1062no—
310C4×C8—1061no—
311C4×C8—1061no—
312C4×C8—1061no—
313C4×C8C22⋊D5311yescharacteristic
314C4×C8C22⋊D5311yescharacteristic
315C4×C8C22×D5311yescharacteristic
316C4×C8C22×D5311yescharacteristic
317#41—1062no—
318#41—1062no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—531no—
322C2×D4—531no—
323C2×D4—1061no—
324C22×C4—1061no—
325C22×C4—1061no—
326C22×C4—1061no—
327C22×C4D4×D5311yescharacteristic
328C22×C4#1171994711yescharacteristic
329C22×C4C2×C22⋊D5311yescharacteristic
330Q16—2124no—
331SD16—2124no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—1062no—
335C2×C8—1062no—
336C2×C8—1061no—
337C2×C8—1061no—
338C2×C8—2122no—
339C2×C8—2122no—
340C2×C8C2×C22⋊D5311yescharacteristic
341C2×C8C2×C22⋊D5311yescharacteristic
342C2×C8C2×C22⋊D5311yescharacteristic
343C2×C8C2×C22⋊D5311yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—531no—
351C4⋊C4—531no—
352C4⋊C4—1061no—
353C4⋊C4—1061no—
354C22⋊C4—42no—
355C22⋊C4—1061no—
356C22⋊C4—1061no—
357C22⋊C4—1061no—
358C42—42no—
359C42—1061no—
360C42—1061no—
361C42—1061no—
362C42D4×D5311yescharacteristic
363C42#1171994711yescharacteristic
364C42C2×C22⋊D5311yescharacteristic
365C42C23×D5311yescharacteristic
366C23—1061no—
367C23#1313318811yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—82no—
371Q8—532no—
372Q8—1061no—
373D4—532no—
374D4—1064no—
375D4—1061no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—1064no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—2122no—
388C2×C4—2121no—
389C2×C4—2121no—
390C2×C4#1313318811yescharacteristic
391C2×C4#1313318811yescharacteristic
392C2×C4#1313318811yescharacteristic
393C2×C4C2×D4×D5311yescharacteristic
394C2×C4C22×C22⋊D5311yescharacteristic
395C2×C4C2×#1171994711yescharacteristic
396C8—21no—
397C8—21no—
398C8—2122no—
399C8—2122no—
400C8—2121no—
401C8—2121no—
402C8#1313324812yes—
403C8#1313324812yes—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—1062no—
408C22—2121no—
409C22—2121no—
410C22C2×#1313318811yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—82no—
416C4—82no—
417C4—1062no—
418C4—2121no—
419C4—2121no—
420C4#135×D5311yescharacteristic
421C4#135×D5311yescharacteristic
422C4C2×#1313324811yescharacteristic
423C4C2×#1313324811yescharacteristic
424C2—41no—
425C2—2122no—
426C2#2028419811yescharacteristic
427C2#2028429211yescharacteristic
428C2#2028443111yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3160C2
socleC22×C53C2×#121
prime radicalC2×#11719945D4
hypercenter#3160D53
hypocenterC53#19273
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
3. #20274116C2semidirect
4. #20274111C2semidirect
5. #20274106C2semidirect
6. #20274101C2semidirect
7. #20274094C2semidirect
8. #20274087C2semidirect
9. #20273478C2semidirect
10. #20273425C2semidirect
11. C53×#3160C2semidirect
210. #3160D53semidirect
261. C53#19273semidirect

Maximal subgroups

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

subgroupindex
#202848712
C8×#117199452
#202741162
#202741112
#202741062
#202741012
#202740942
#202740872
#202734782
#202734252
C53×#31602
#1927353