Finite Group Databasebeta

← #27369346

Subgroups — #27369346

Order 13568.

650 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#20285853C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic
3#20284723C211yesmaximal, characteristic, complemented
4#20274152C212yesmaximal, complemented
5#20274124C212yesmaximal, complemented
6#20274114C212yesmaximal, complemented
7#20274090C212yesmaximal, complemented
8#20273495C211yesmaximal, characteristic, complemented
9#20273423C211yesmaximal, characteristic, complemented
10#20273375C211yesmaximal, characteristic, complemented
11C53×#2972C211yesmaximal, characteristic, complemented
12#16811180C2211yescharacteristic
13#16811173C2211yescharacteristic
14#16811028C2211yescharacteristic, complemented
15#16811021C2211yescharacteristic, complemented
16#16810970C2211yescharacteristic, complemented
17#16810970C2211yescharacteristic, complemented
18#16810953—21no—
19#16810948—22no—
20#16810941—21no—
21#16810924—22no—
22#16810913C2212yes—
23#16810913C2212yes—
24#16810828C2212yes—
25#16810818C2212yes—
26#16810774—21no—
27#16810757—21no—
28#16810733—22no—
29C4×#11719923—22no—
30#16810680C2211yescharacteristic
31#16810676C2211yescharacteristic
32#16810376C2212yescomplemented
33#16810374C2212yescomplemented
34#16810371C2212yescomplemented
35#16810371C2212yescomplemented
36#16810368C2212yescomplemented
37#16810366C2212yescomplemented
38C2×C4×D4×C53C2211yescharacteristic
39C8×D4×C53—22no—
40C8×D4×C53C2212yes—
41C8×D4×C53C2212yes—
42C53×#431C2211yescharacteristic
43C53×#431C2211yescharacteristic
44C2×C22⋊C8×C53—22no—
45C2×#11719945—21no—
46C2×#11719923—22no—
47C2×#11719922—22no—
48#13133217C2311yescharacteristic
49#13133217C2311yescharacteristic
50#13133217C2311yescharacteristic
51#13133214C2311yescharacteristic
52#13133209—21no—
53#13133205—21no—
54#13133196—21no—
55#13133196—21no—
56#13133191—21no—
57#13133191—21no—
58C4×Dic106—21no—
59C4×Dic106—21no—
60C4×D212—21no—
61C4×D212—21no—
62#13133176—21no—
63#13133151—22no—
64#13133149—22no—
65#13133148—22no—
66#13133148—22no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133140—22no—
70#13133139—22no—
71#13133139—22no—
72#13133138—22no—
73#13133138—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76C22⋊#88322—21no—
77#13133128—21no—
78C4×#88322—21no—
79C22×D4×C53D411yescharacteristic
80C23×C4×C53—21no—
81C22×C8×C53—22no—
82C22×C8×C53—22no—
83C4×D4×C53—21no—
84C4×D4×C53—21no—
85C4×D4×C53D412yes—
86C4×D4×C53C2312yes—
87C2×C4⋊C4×C53D411yescharacteristic
88C2×C22⋊C4×C53—21no—
89C2×C42×C53C2311yescharacteristic
90C4⋊C8×C53D412yes—
91C4⋊C8×C53C2312yes—
92C22⋊C8×C53—22no—
93C22⋊C8×C53—22no—
94C22⋊C8×C53—21no—
95C22⋊C8×C53D412yes—
96C4×C8×C53—21no—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99C24×C53—21no—
100C2×D4×C53—24no—
101C2×D4×C53—21no—
102C2×D4×C53C2×D412yes—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—82no—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C2×C22⋊D53—41no—
111#11719945—82no—
112C2×D212—21no—
113C2×D212—21no—
114C2×D212—21no—
115C2×C4×D53—41no—
116C2×Dic106—21no—
117C2×Dic106—21no—
118C2×Dic106—21no—
119C2×C8×C53—22no—
120C2×C8×C53—82no—
121C2×C8×C53—82no—
122C2×C8×C53C2×D412yes—
123C2×C8×C53C2×D412yes—
124C4⋊C4×C53—21no—
125C4⋊C4×C53C2×D412yes—
126C22⋊C4×C53—22no—
127C22⋊C4×C53—22no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53C2411yescharacteristic
132C22⋊Dic53—41no—
133#11719923—44no—
134#11719922—44no—
135#11719921—41no—
136#11719921—41no—
137C4⋊Dic53—21no—
138C4⋊Dic53—21no—
139C4⋊Dic53—21no—
140C4⋊Dic53—21no—
141C4⋊Dic53—21no—
142C4⋊Dic53—21no—
143#11719919—41no—
144#11719919—41no—
145C4×Dic53—41no—
146C2×#88322—22no—
147C2×#88322—22no—
148C23×C53—24no—
149C23×C53—22no—
150C23×C53—81no—
151C23×C53—81no—
152C23×C53#12111yescharacteristic
153C22×D53—41no—
154D4×C53—24no—
155D4×C53—22no—
156D4×C53#13514yes—
157C2×C4×C53—22no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—82no—
164C2×C4×C53—81no—
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53C22×D411yescharacteristic
169C2×C4×C53C22×D411yescharacteristic
170C2×C4×C53C2×#4111yescharacteristic
171C22⋊D53—82no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175D212—42no—
176D212—42no—
177D212—81no—
178C4×D53—82no—
179Dic106—42no—
180Dic106—81no—
181Dic106—81no—
182C8×C53—22no—
183C8×C53—22no—
184C8×C53—42no—
185#88322—42no—
186#88322—42no—
187#19394—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—24no—
191C22×C53—21no—
192C22×C53—42no—
193C22×C53—42no—
194C22×C53—84no—
195C22×C53C2×#12111yescharacteristic
196C2×D53—81no—
197C2×D53—81no—
198C2×D53—81no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—42no—
204C4×C53C2×#13512yes—
205C4×C53#57712yes—
206Dic53—81no—
207Dic53—81no—
208Dic53—81no—
209#3228—531no—
210#3109—531no—
211#3101—531no—
212#2972D5311yescharacteristic, complemented
213#1681—532no—
214#1677—532no—
215#1675—532no—
216#1671—532no—
217#1572—531no—
218#1554—531no—
219#1543—531no—
220C2×C53—41no—
221C2×C53—44no—
222C2×C53#304211yescharacteristic
223C2×C53#305011yescharacteristic
224C2×C53#306811yescharacteristic
225D53—161no—
226#533—531no—
227#532—531no—
228C2×C4×D4C2×D5311yescharacteristic
229#500—531no—
230#498—531no—
231#486—531no—
232#486—531no—
233#483—1061no—
234#482—1062no—
235#481—1061no—
236#478—1062no—
237#476—532no—
238#476—532no—
239#461—532no—
240#459—532no—
241#450—1061no—
242#447—1061no—
243#442—1062no—
244C4×SD16—1062no—
245C8×D4—22no—
246C8×D4C2×D5312yes—
247C8×D4C2×D5312yes—
248#432—531no—
249#431—531no—
250#431C2×D5311yescharacteristic
251#431C2×D5311yescharacteristic
252C2×C22⋊C8—22no—
253#335—532no—
254#334—532no—
255#333—532no—
256#333—532no—
257#332—532no—
258#331—532no—
259C53#1939411yescharacteristic, complemented, Sylow
260C2×#41—1061no—
261C22×D4D21211yescharacteristic
262C23×C4—21no—
263C2×SD16—1062no—
264C2×D8—1062no—
265C22×C8—22no—
266C22×C8—22no—
267C4⋊Q8—531no—
268C4⋊Q8—531no—
269C4⋊Q8—531no—
270#127—531no—
271#125—1061no—
272#124—1061no—
273C22⋊Q8—1061no—
274C22⋊Q8—1061no—
275#121—1061no—
276#121—1061no—
277C4×Q8—1061no—
278C4×Q8—1061no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—1061no—
282C4×D4—1061no—
283C4×D4D21212yes—
284C4×D4C22×D5312yes—
285#117—1061no—
286C2×C4⋊C4D21211yescharacteristic
287C2×C22⋊C4—21no—
288C2×C42C22×D5311yescharacteristic
289#107—1062no—
290#106—1062no—
291#106—1062no—
292#106—1062no—
293C4⋊C8—532no—
294C4⋊C8—532no—
295C4⋊C8C22⋊D5312yes—
296C4⋊C8C22×D5312yes—
297#103—1062no—
298#102—1062no—
299#102—1062no—
300#102—1062no—
301#102—1062no—
302#102—1062no—
303C22⋊C8—21no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—1061no—
307C22⋊C8—1061no—
308C22⋊C8C22⋊D5312yes—
309#97—1061no—
310C4×C8—21no—
311C4×C8—1061no—
312C4×C8C22×D5311yescharacteristic
313C4×C8C22×D5311yescharacteristic
314C24—21no—
315#41—4242no—
316C2×Q8—1061no—
317C2×Q8—1061no—
318C2×Q8—1061no—
319C2×D4—24no—
320C2×D4—21no—
321C2×D4—1061no—
322C2×D4—1061no—
323C2×D4—1061no—
324C2×D4—2121no—
325C2×D4C2×D21212yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—82no—
330C22×C4—2121no—
331C22×C4C2×D21211yescharacteristic
332C22×C4#1171994511yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334SD16—2124no—
335D8—2124no—
336C2×C8—22no—
337C2×C8—82no—
338C2×C8—82no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8C2×C22⋊D5312yes—
342C2×C8C2×C22⋊D5312yes—
343C4⋊C4—21no—
344C4⋊C4—1061no—
345C4⋊C4—1061no—
346C4⋊C4—1061no—
347C4⋊C4—1061no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C4⋊C4—2121no—
351C4⋊C4—2121no—
352C4⋊C4C2×D21212yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—2121no—
356C22⋊C4—2121no—
357C22⋊C4—2121no—
358C42—2121no—
359C42C2×D21211yescharacteristic
360C42#1171994511yescharacteristic
361C42C2×C22⋊D5311yescharacteristic
362C42C23×D5311yescharacteristic
363C23—24no—
364C23—22no—
365C23—81no—
366C23—81no—
367C23—2121no—
368C23#1313319111yescharacteristic
369Q8—4241no—
370Q8—4241no—
371Q8—2122no—
372D4—24no—
373D4—22no—
374D4—4241no—
375D4—4242no—
376D4—2122no—
377D4—2122no—
378D4#1313323914yes—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—4242no—
388C2×C4—2121no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4#1313319111yescharacteristic
392C2×C4#1313319111yescharacteristic
393C2×C4#1313319111yescharacteristic
394C2×C4C22×C22⋊D5311yescharacteristic
395C2×C4C22×D21211yescharacteristic
396C2×C4C2×#1171994511yescharacteristic
397C8—22no—
398C8—22no—
399C8—42no—
400C8—2122no—
401C8—2122no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—24no—
406C22—42no—
407C22—42no—
408C22—84no—
409C22—4241no—
410C22—4241no—
411C22—4241no—
412C22C2×#1313319111yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—4241no—
419C4—4241no—
420C4—4241no—
421C4C2×#1313323912yes—
422C4#1681143412yes—
423C2—41no—
424C2—44no—
425C2—8481no—
426C2#2028422711yescharacteristic
427C2#2028429611yescharacteristic
428C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2972C2
socleC22×C53C2×#121
prime radical#20284806C2
hypercenter#2972D53
hypocenterC53#19394
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #20285853C2semidirect
3. #20284723C2semidirect
4. #20274152C2semidirect
5. #20274124C2semidirect
6. #20274114C2semidirect
7. #20274090C2semidirect
8. #20273495C2semidirect
9. #20273423C2semidirect
10. #20273375C2semidirect
11. C53×#2972C2semidirect
14. #16811028C22semidirect
15. #16811021C22semidirect
16. #16810970C22semidirect
17. #16810970C22semidirect
32. #16810376C22semidirect
33. #16810374C22semidirect
34. #16810371C22semidirect
35. #16810371C22semidirect
36. #16810368C22semidirect
37. #16810366C22semidirect
212. #2972D53semidirect
259. C53#19394semidirect

Maximal subgroups

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

subgroupindex
#202858532
#202848062
#202847232
#202741522
#202741242
#202741142
#202740902
#202734952
#202734232
#202733752
C53×#29722
#1939453