Finite Group Databasebeta

← #27538748

Subgroups — #27538748

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285226C211yesmaximal, characteristic, complemented
2C4×D8×D53C211yesmaximal, characteristic
3#20276871C212yesmaximal, complemented
4#20276870C212yesmaximal, complemented
5#20276859C211yesmaximal, characteristic, complemented
6#20276853C211yesmaximal, characteristic, complemented
7#20276852C211yesmaximal, characteristic, complemented
8#20276502C212yesmaximal, complemented
9#20276469C211yesmaximal, characteristic, complemented
10#20276429C212yesmaximal
11C53×#2226C211yesmaximal, characteristic, complemented
12C2×D8×D53C2211yescharacteristic
13#16811178—21no—
14C4×D4×D53—21no—
15C2×#13133156C2212yes—
16#16811028C2211yescharacteristic, complemented
17#16811024—21no—
18#16811024C2212yescomplemented
19#16811019—21no—
20C4×D424C2211yescharacteristic
21D8×Dic53C2211yescharacteristic
22#16810705C2211yescharacteristic
23C4×#11719922—21no—
24#16810703—21no—
25#16810660—21no—
26#16810660C2212yes—
27#107×D53C2211yescharacteristic
28#102×D53—21no—
29C4×C8×D53C2211yescharacteristic
30#16810450C2212yescomplemented
31#16810449C2212yescomplemented
32#16810448C2212yescomplemented
33C4⋊#11719908C2211yescharacteristic, complemented
34#16810441C2212yescomplemented
35#16810425C2212yes—
36#16810424C2212yes—
37#16810423C2212yes—
38C53×#500C2211yescharacteristic, complemented
39C4×D8×C53C2211yescharacteristic
40C53×#367C2212yescomplemented
41C53×C4⋊C16C2211yescharacteristic, complemented
42C53×#357C2212yes—
43C2×D4×D53—21no—
44C22×C4×D53—41no—
45C2×D424C2311yescharacteristic
46C2×#11719922—21no—
47D8×D53—22no—
48D8×D53Q814yescomplemented
49C2×C8×D53D411yescharacteristic
50C2×C8×D53C2311yescharacteristic
51#13133217—21no—
52#13133208—22no—
53#13133197—41no—
54#13133193—41no—
55Q8×Dic53—41no—
56#13133182—41no—
57C4×D212—21no—
58D4×Dic53—21no—
59#13133179—21no—
60C4×C22⋊D53—41no—
61#13133177—41no—
62#13133175—22no—
63C4⋊C4×D53—21no—
64C22⋊C4×D53—41no—
65C42×D53D411yescharacteristic
66#13133156—24no—
67C2×#11719908C2312yes—
68#13133151—21no—
69#13133151C2312yes—
70#13133151C2311yescharacteristic
71#13133150—21no—
72#13133148—22no—
73#13133148—22no—
74#13133139—21no—
75#13133138—21no—
76#13133137—21no—
77C4×#88322D411yescharacteristic
78C8×Dic53D411yescharacteristic
79C8×Dic53C2311yescharacteristic
80C2×D8×C53C2311yescharacteristic
81C4⋊Q8×C53—21no—
82C4×D4×C53—21no—
83C2×C16×C53C2312yes—
84#107×C53—21no—
85#107×C53C2312yes—
86#107×C53C2311yescharacteristic
87#102×C53—21no—
88C4×C8×C53C2311yescharacteristic
89C23×D53—41no—
90C2×Q8×C53—41no—
91C2×D4×C53—21no—
92C22×C4×C53—41no—
93C2×C22⋊D53—41no—
94C22×Dic53—41no—
95D4×D53—24no—
96D4×D53—42no—
97C2×D212—21no—
98C2×C4×D53—41no—
99C2×C4×D53—41no—
100C2×C4×D53—41no—
101C2×C4×D53—84no—
102C2×C4×D53D811yescharacteristic
103C2×C4×D53D811yescharacteristic
104C2×C4×D53C2×D411yescharacteristic
105C2×Dic106—41no—
106D8×C53—21no—
107D8×C53C2×Q812yes—
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2411yescharacteristic
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C42×C53C2×D411yescharacteristic
115C22⋊Dic53—41no—
116#11719922—24no—
117#11719921—42no—
118#11719921—41no—
119#11719921—41no—
120C4⋊Dic53—21no—
121C4⋊Dic53—22no—
122C4⋊Dic53—41no—
123C4⋊Dic53—42no—
124#11719919—41no—
125#11719919—42no—
126C4×Dic53—42no—
127C4×Dic53D811yescharacteristic
128C4×Dic53D811yescharacteristic
129C4×Dic53C2×D411yescharacteristic
130C2×#88322C2×D411yescharacteristic
131C2×#88322C2×D411yescharacteristic
132D424—21no—
133D424C2×Q812yes—
134C8×D53—22no—
135C8×D53C2×Q812yes—
136C8×D53#4112yes—
137C16×C53—22no—
138#11719908—22no—
139C23×C53—41no—
140C22×D53—41no—
141C22×D53—44no—
142C22×D53—82no—
143C22×D53—82no—
144C22×D53C2×D811yescharacteristic
145Q8×C53—82no—
146D4×C53—22no—
147D4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—82no—
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C22×D411yescharacteristic
154C22⋊D53—44no—
155C2×Dic53—42no—
156C2×Dic53—41no—
157C2×Dic53—82no—
158C2×Dic53C2×D811yescharacteristic
159C2×Dic53C2×D811yescharacteristic
160C2×Dic53C2×D811yescharacteristic
161D212—22no—
162D212—41no—
163C4×D53—22no—
164C4×D53—41no—
165C4×D53—82no—
166C4×D53—82no—
167C4×D53—82no—
168C4×D53C22⋊Q812yes—
169C4×D53#13712yes—
170Dic106—82no—
171C8×C53—21no—
172C8×C53C22×Q811yescharacteristic
173C8×C53C2×#4111yescharacteristic
174#88322—21no—
175#88322C22⋊Q812yes—
176#33627—531nomaximal, Sylow
177C22×C53—42no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53C22×D811yescharacteristic
181C2×D53—21no—
182C2×D53—21no—
183C2×D53—42no—
184C2×D53—88no—
185C2×D53—81no—
186C2×D53—81no—
187C2×D53#47412yes—
188C4×C53—21no—
189C4×C53—81no—
190C4×C53—82no—
191C4×C53#50912yes—
192C4×C53C2×C22⋊Q811yescharacteristic
193C4×C53C2×#13711yescharacteristic
194Dic53—21no—
195Dic53—21no—
196Dic53—82no—
197Dic53—81no—
198Dic53#47412yes—
199#3160—531no—
200C2×C4×D8—531no—
201#2228—532no—
202#2228—532no—
203#2226—531no—
204#2226D5311yescharacteristic, complemented
205#2224—531no—
206#2224—531no—
207#2159—532no—
208#2150—531no—
209#2140—532no—
210C2×C53—82no—
211C2×C53C2×#47411yescharacteristic
212C2×C53C2×#50911yescharacteristic
213C2×C53#341411yescharacteristic
214D53—22no—
215D53—82no—
216C22×D8—531no—
217#533—1061no—
218C2×C4×D4—1061no—
219C2×M32—532no—
220#500—531no—
221#500C2×D5311yescharacteristic, complemented
222#499—532no—
223#499—1061no—
224#498—1061no—
225C4×D8—531no—
226C4×D8—531no—
227C4×D8—531no—
228C4×D8—1061no—
229C4×D8—1061no—
230C4×D8C2×D5311yescharacteristic
231#427—532no—
232#427—1061no—
233C2×#107—531no—
234C2×#102—1061no—
235C2×C4×C8—531no—
236#367—532no—
237#367C2×D5312yescomplemented
238#366—532no—
239#366—532no—
240C4⋊C16—532no—
241C4⋊C16—531no—
242C4⋊C16C2×D5311yescharacteristic, complemented
243#357—532no—
244#357—532no—
245#357—532no—
246#357C2×D5312yes—
247C53#3362711yescharacteristic, complemented, Sylow
248C22×D4—1061no—
249C23×C4—2121no—
250C2×D8—534no—
251C2×D8—531no—
252C2×D8—1062no—
253C2×D8—1061no—
254C2×D8C22×D5311yescharacteristic
255C22×C8—531no—
256C22×C8—531no—
257C4⋊Q8—21no—
258C4⋊Q8—1061no—
259#125—1062no—
260C22⋊Q8—2121no—
261C22⋊Q8—2121no—
262C4×Q8—2121no—
263C4×Q8—2121no—
264C4×D4—21no—
265C4×D4—1061no—
266C4×D4—1061no—
267C4×D4—1061no—
268C4×D4—2121no—
269C4×D4—2121no—
270#117—1062no—
271C2×C4⋊C4—1061no—
272C2×C22⋊C4—2121no—
273C2×C42—531no—
274M32—1064no—
275C2×C16—532no—
276C2×C16C22×D5312yes—
277#107—21no—
278#107—532no—
279#107—531no—
280#107—1061no—
281#107—1061no—
282#107C22×D5312yes—
283#107C22×D5311yescharacteristic
284#106—1062no—
285#106—1062no—
286#102—21no—
287#102—1061no—
288#102—1061no—
289#102—1061no—
290C4×C8—531no—
291C4×C8—531no—
292C4×C8—531no—
293C4×C8C22×D5311yescharacteristic
294C24—2121no—
295C2×Q8—41no—
296C2×Q8—2121no—
297C2×D4—21no—
298C2×D4—1064no—
299C2×D4—1061no—
300C2×D4—2122no—
301C2×D4—2121no—
302C22×C4—41no—
303C22×C4—531no—
304C22×C4—531no—
305C22×C4—531no—
306C22×C4—4244no—
307C22×C4—2121no—
308C22×C4—2121no—
309C22×C4—2121no—
310C22×C4—2121no—
311D8—21no—
312D8—532no—
313D8—1064no—
314D8—1061no—
315D8Q8×D5312yes—
316C2×C8—531no—
317C2×C8—532no—
318C2×C8—532no—
319C2×C8—531no—
320C2×C8—1062no—
321C2×C8D4×D5311yescharacteristic
322C2×C8C23×D5311yescharacteristic
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—1061no—
327C4⋊C4—1062no—
328C4⋊C4—2121no—
329C4⋊C4—2122no—
330C4⋊C4—2122no—
331C4⋊C4—2121no—
332C22⋊C4—41no—
333C22⋊C4—2121no—
334C22⋊C4—2122no—
335C22⋊C4—2121no—
336C22⋊C4—2121no—
337C42—531no—
338C42—531no—
339C42—531no—
340C42—2122no—
341C42D4×D5311yescharacteristic
342C16—22no—
343C16—1062no—
344C23—41no—
345C23—531no—
346C23—4242no—
347C23—4242no—
348C23—2124no—
349C23—2121no—
350Q8—82no—
351Q8—4242no—
352D4—22no—
353D4—41no—
354D4—1062no—
355D4—2124no—
356D4—2121no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—82no—
360C2×C4—531no—
361C2×C4—532no—
362C2×C4—532no—
363C2×C4—531no—
364C2×C4—531no—
365C2×C4—1062no—
366C2×C4—4242no—
367C2×C4—4242no—
368C2×C4—4242no—
369C2×C4—4242no—
370C2×C4—2121no—
371C2×C4—2122no—
372C2×C4—2121no—
373C2×C4D8×D5311yescharacteristic
374C2×C4D8×D5311yescharacteristic
375C2×C4C2×D4×D5311yescharacteristic
376C8—21no—
377C8—532no—
378C8—1061no—
379C8C2×Q8×D5311yescharacteristic
380C8#41×D5311yescharacteristic
381C22—42no—
382C22—81no—
383C22—81no—
384C22—532no—
385C22—1061no—
386C22—1061no—
387C22—4248no—
388C22—4241no—
389C22—4241no—
390C22—2122no—
391C22C2×D8×D5311yescharacteristic
392C4—21no—
393C4—81no—
394C4—82no—
395C4—532no—
396C4—1061no—
397C4—1061no—
398C4—4241no—
399C4—4242no—
400C4#1681106012yes—
401C4C22⋊Q8×D5311yescharacteristic
402C4#137×D5311yescharacteristic
403C2—82no—
404C2—1062no—
405C2—4242no—
406C2#474×D5311yescharacteristic
407C2C2×#1681106011yescharacteristic
408C2#2028784711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2226C2
socleC22×C53C22×D8
prime radicalC2×D8×D53C22
hypercenter#2226D53
hypocenterC53#33627
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #20285226C2semidirect
3. #20276871C2semidirect
4. #20276870C2semidirect
5. #20276859C2semidirect
6. #20276853C2semidirect
7. #20276852C2semidirect
8. #20276502C2semidirect
9. #20276469C2semidirect
11. C53×#2226C2semidirect
16. #16811028C22semidirect
18. #16811024C22semidirect
30. #16810450C22semidirect
31. #16810449C22semidirect
32. #16810448C22semidirect
33. C4⋊#11719908C22semidirect
34. #16810441C22semidirect
38. C53×#500C22semidirect
40. C53×#367C22semidirect
41. C53×C4⋊C16C22semidirect
48. D8×D53Q8semidirect
204. #2226D53semidirect
221. #500C2×D53semidirect
237. #367C2×D53semidirect
242. C4⋊C16C2×D53semidirect
247. C53#33627semidirect

Maximal subgroups

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

subgroupindex
#202852262
C4×D8×D532
#202768712
#202768702
#202768592
#202768532
#202768522
#202765022
#202764692
#202764292
C53×#22262
#3362753