Finite Group Databasebeta

← #27371982

Subgroups — #27371982

Order 13568.

836 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285635C211yesmaximal, characteristic
2#20285275C211yesmaximal, characteristic, complemented
3C2×#16810997C211yesmaximal, characteristic, complemented
4#20285111C211yesmaximal, characteristic, complemented
5#20285091C211yesmaximal, characteristic, complemented
6#20274371C212yesmaximal, complemented
7#20274357C212yesmaximal, complemented
8#20274318C212yesmaximal, complemented
9#20274308C212yesmaximal, complemented
10#20272996C211yesmaximal, characteristic, complemented
11C53×#3230C211yesmaximal, characteristic, complemented
12C22×#11719924—22no—
13#16811173C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16811019C2211yescharacteristic, complemented
16#16811000C2212yes—
17#16810997—22no—
18#16810997C2212yes—
19#16810997C2212yes—
20#16810997C2211yescharacteristic, complemented
21#16810989—22no—
22#16810985—22no—
23#16810975C2212yescomplemented
24#16810973C2211yescharacteristic, complemented
25#16810971C2212yes—
26#16810971C2212yes—
27#16810971C2211yescharacteristic, complemented
28#16810971C2212yescomplemented
29#16810970C2212yes—
30#16810854—22no—
31#16810819—22no—
32#16810759—22no—
33C4×#11719924—22no—
34#16810627—22no—
35#16810622—22no—
36C4×#11719917C2211yescharacteristic
37C2×C4×#88322C2211yescharacteristic
38#16810345C2212yescomplemented
39C8×#88322C2212yescomplemented
40C53×#533C2211yescharacteristic
41C2×C4⋊Q8×C53C2211yescharacteristic
42C53×#500C2212yescomplemented
43C53×#487C2212yescomplemented
44C53×#484—22no—
45C53×#477—22no—
46C53×#431C2211yescharacteristic
47C2×#11719945—21no—
48C22×D212—21no—
49C22×D212—21no—
50C22×D212—21no—
51C2×#11719924—22no—
52C2×#11719924—24no—
53C2×#11719924—24no—
54C2×#11719924—48no—
55C2×D424—24no—
56C2×#11719917D412yes—
57C22×#88322D412yes—
58#13133217C2311yescharacteristic
59#13133214—21no—
60#13133214—21no—
61#13133214—21no—
62#13133214C2312yes—
63#13133214C2311yescharacteristic
64#13133205—21no—
65#13133191—21no—
66#13133191—21no—
67C4×D212—21no—
68C4×D212—21no—
69#13133148—22no—
70#13133141—22no—
71#13133140—24no—
72#13133139—24no—
73#13133137—22no—
74#13133137—24no—
75#13133137—24no—
76#13133137—24no—
77#13133128D412yes—
78C4×#88322D412yes—
79C4×#88322C2311yescharacteristic
80C4×#88322C2311yescharacteristic
81C4×#88322C2312yes—
82C22×Q8×C53—21no—
83C4⋊Q8×C53—21no—
84C4⋊Q8×C53—21no—
85C4⋊Q8×C53—21no—
86C4⋊Q8×C53C2311yescharacteristic
87C4⋊Q8×C53C2312yes—
88C4⋊Q8×C53C2311yescharacteristic
89#125×C53—21no—
90C22⋊Q8×C53—21no—
91C22⋊Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94#117×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C42×C53C2311yescharacteristic
98#107×C53—24no—
99C4⋊C8×C53D412yes—
100#103×C53—24no—
101C22⋊C8×C53D412yes—
102C4×C8×C53C2312yes—
103C23×D53—41no—
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×Q8×C53—22no—
107C2×Q8×C53—44no—
108C2×Q8×C53—41no—
109C22×C4×C53—41no—
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C2×C22⋊D53—41no—
114#11719945—44no—
115C2×D212—21no—
116C2×D212—22no—
117C2×D212—21no—
118C2×D212—22no—
119C2×D212—21no—
120C2×D212—22no—
121C2×D212—44no—
122C2×D212—44no—
123C2×D212—41no—
124C2×D212—41no—
125C2×D212—41no—
126C2×D212—82no—
127C2×C4×D53—41no—
128C2×Dic106—21no—
129C2×C8×C53C2×D414yes—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53—41no—
140C22⋊C4×C53—41no—
141C42×C53—41no—
142C42×C53C2×D411yescharacteristic
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53C2411yescharacteristic
146#11719924—48no—
147#11719924—44no—
148#11719924—48no—
149#11719921—41no—
150#11719921—41no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153#11719917—24no—
154C2×#88322—24no—
155C2×#88322C2×D412yes—
156C2×#88322C2×D414yes—
157C2×#88322C2×D412yes—
158D424—84no—
159C23×C53#12711yescharacteristic
160C22×D53—41no—
161C22×D53—42no—
162C22×D53—82no—
163C22×D53—82no—
164C22×D53—82no—
165Q8×C53—44no—
166Q8×C53—42no—
167Q8×C53—42no—
168Q8×C53—82no—
169C2×C4×C53—22no—
170C2×C4×C53—22no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—82no—
175C2×C4×C53#12711yescharacteristic
176C2×C4×C53#12711yescharacteristic
177C2×C4×C53#12711yescharacteristic
178C2×C4×C53#13514yes—
179C2×C4×C53C22×D411yescharacteristic
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C22⋊D53—82no—
183C2×Dic53—41no—
184D212—44no—
185D212—42no—
186D212—44no—
187D212—42no—
188D212—44no—
189D212—42no—
190D212—82no—
191D212—82no—
192D212—82no—
193C4×D53—82no—
194Dic106—42no—
195C8×C53—24no—
196#88322—24no—
197#88322—22no—
198#88322AGL(1,ℤ/8)14yes—
199#19625—531nomaximal, Sylow
200C22×C53—21no—
201C22×C53—21no—
202C22×C53#49512yes—
203C22×C53C2×#12711yescharacteristic
204C2×D53—81no—
205C2×D53—82no—
206C2×D53—82no—
207C2×D53—81no—
208C2×D53—82no—
209C2×D53—81no—
210C2×D53—164no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—82no—
214C4×C53—81no—
215C4×C53—81no—
216C4×C53#49512yes—
217C4×C53C2×#13512yes—
218C4×C53C2×AGL(1,ℤ/8)12yes—
219C4×C53#57712yes—
220Dic53—81no—
221#3230D5311yescharacteristic, complemented
222#3205—531no—
223#3167—531no—
224C2×#492—531no—
225#3141—531no—
226#3138—531no—
227#1719—532no—
228#1717—532no—
229#1708—532no—
230#1706—532no—
231C8×M16—531no—
232C2×C53—22no—
233C2×C53C2×#49511yescharacteristic
234C2×C53#315111yescharacteristic
235C2×C53#315411yescharacteristic
236D53—161no—
237D53—162no—
238C22×SD16—1062no—
239#533C2×D5311yescharacteristic
240#532—531no—
241C2×C4⋊Q8C2×D5311yescharacteristic
242C2×#127—531no—
243#500C2×D5312yescomplemented
244#498—531no—
245#493—532no—
246#492—532no—
247#492—532no—
248#492—531no—
249#492—1062no—
250#490—1062no—
251#489—1062no—
252#487—532no—
253#487—531no—
254#487C2×D5312yescomplemented
255#486—532no—
256#486—532no—
257#486—532no—
258#486—531no—
259#486—532no—
260#484—22no—
261#477—22no—
262#465—1062no—
263#459—1062no—
264#447—1062no—
265C4×SD16—1062no—
266#431C2×D5311yescharacteristic
267#418—1062no—
268#417—1062no—
269C4×M16—531no—
270C2×C4×C8—531no—
271#322—532no—
272C82—532no—
273C53#1962511yescharacteristic, complemented, Sylow
274C2×#41—1061no—
275C22×Q8—21no—
276C22×D4—1061no—
277C22×D4—1061no—
278C22×D4—1061no—
279C2×SD16—1064no—
280C2×SD16—1064no—
281C2×SD16—1062no—
282C2×SD16—2128no—
283C2×D8—1064no—
284C2×M16—532no—
285C22×C8—532no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—531no—
290C4⋊Q8C22×D5312yes—
291C4⋊Q8C22×D5311yescharacteristic
292C4⋊Q8C22×D5311yescharacteristic
293#127—532no—
294#127—531no—
295#127—1061no—
296#127—1061no—
297#127—1061no—
298#125—21no—
299#124—1061no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302#121—1061no—
303#121—1061no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×D4—1061no—
307C4×D4—1061no—
308#117—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C42C22×D5311yescharacteristic
312#107—24no—
313#106—1062no—
314C4⋊C8C22⋊D5312yes—
315#103—24no—
316#103—1064no—
317#103—1062no—
318#102—1064no—
319#102—1064no—
320#102—1064no—
321#102—1062no—
322#102—1064no—
323C22⋊C8C22⋊D5312yes—
324#97—532no—
325C4×C8—532no—
326C4×C8—531no—
327C4×C8—532no—
328C4×C8—531no—
329C4×C8C22×D5312yes—
330C24—2121no—
331#41—2124no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—21no—
335C2×Q8—44no—
336C2×Q8—41no—
337C2×Q8—1061no—
338C2×D4—1061no—
339C2×D4—1062no—
340C2×D4—1061no—
341C2×D4—1062no—
342C2×D4—1062no—
343C2×D4—1061no—
344C2×D4—4242no—
345C2×D4—2124no—
346C2×D4—2121no—
347C2×D4—2121no—
348C2×D4—2121no—
349C2×D4—2121no—
350C2×D4—2124no—
351C22×C4—41no—
352C22×C4—2121no—
353C22×C4D4×D5311yescharacteristic
354C22×C4D4×D5311yescharacteristic
355C22×C4C2×C22⋊D5311yescharacteristic
356SD16—2128no—
357SD16—2124no—
358SD16—2128no—
359D8—4244no—
360M16—1064no—
361C2×C8—532no—
362C2×C8—532no—
363C2×C8—534no—
364C2×C8—1064no—
365C2×C8C2×C22⋊D5314yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—1061no—
377C4⋊C4—1061no—
378C22⋊C4—41no—
379C22⋊C4—2121no—
380C22⋊C4—2121no—
381C42—41no—
382C42D4×D5311yescharacteristic
383C42D4×D5311yescharacteristic
384C42C2×C22⋊D5311yescharacteristic
385C42C23×D5311yescharacteristic
386C23—4242no—
387C23—4242no—
388C23—4242no—
389C23—2121no—
390C23—2122no—
391C23#1313321311yescharacteristic
392Q8—44no—
393Q8—42no—
394Q8—42no—
395Q8—82no—
396Q8—2122no—
397D4—4242no—
398D4—4242no—
399D4—4242no—
400D4—4242no—
401D4—2124no—
402D4—2122no—
403D4—2124no—
404D4—2122no—
405D4—2122no—
406D4—2124no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—4242no—
414C2×C4—2121no—
415C2×C4#1313321311yescharacteristic
416C2×C4#1313321311yescharacteristic
417C2×C4#1313321311yescharacteristic
418C2×C4#1313324014yes—
419C2×C4C2×D4×D5311yescharacteristic
420C2×C4C2×D4×D5311yescharacteristic
421C2×C4C22×C22⋊D5311yescharacteristic
422C8—24no—
423C8—534no—
424C8—1064no—
425C8—1062no—
426C22—21no—
427C22—21no—
428C22—8484no—
429C22—4242no—
430C22—4241no—
431C22—4241no—
432C22—4242no—
433C22—4242no—
434C22—4241no—
435C22#1681100812yes—
436C22C2×#1313321311yescharacteristic
437C4—21no—
438C4—21no—
439C4—81no—
440C4—82no—
441C4—81no—
442C4—4241no—
443C4#1681100812yes—
444C4C2×#1313324012yes—
445C4AGL(1,ℤ/8)×D5312yes—
446C4#1681143112yes—
447C2—22no—
448C2—8482no—
449C2—8481no—
450C2C2×#1681100811yescharacteristic
451C2#2028516311yescharacteristic
452C2#2028519311yescharacteristic
subgroupG/H
centerC22C2×#13133213
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3230C2
socleC22×C53C2×#127
prime radical#20285635C2
hypercenter#3230D53
hypocenterC53#19625
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
2. #20285275C2semidirect
3. C2×#16810997C2semidirect
4. #20285111C2semidirect
5. #20285091C2semidirect
6. #20274371C2semidirect
7. #20274357C2semidirect
8. #20274318C2semidirect
9. #20274308C2semidirect
10. #20272996C2semidirect
11. C53×#3230C2semidirect
15. #16811019C22semidirect
20. #16810997C22semidirect
23. #16810975C22semidirect
24. #16810973C22semidirect
27. #16810971C22semidirect
28. #16810971C22semidirect
38. #16810345C22semidirect
39. C8×#88322C22semidirect
42. C53×#500C22semidirect
43. C53×#487C22semidirect
221. #3230D53semidirect
243. #500C2×D53semidirect
254. #487C2×D53semidirect
273. C53#19625semidirect

Maximal subgroups

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

subgroupindex
#202856352
#202852752
C2×#168109972
#202851112
#202850912
#202743712
#202743572
#202743182
#202743082
#202729962
C53×#32302
#1962553