Finite Group Databasebeta

← #27367886

Subgroups — #27367886

Order 13568.

690 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285297C211yesmaximal, characteristic, complemented
2#20284512C211yesmaximal, characteristic
3C2×D4×#88322C211yesmaximal, characteristic
4#20274168C212yesmaximal, complemented
5#20274164C212yesmaximal, complemented
6#20274112C212yesmaximal, complemented
7#20274078C212yesmaximal, complemented
8#20273473C211yesmaximal, characteristic, complemented
9#20273406C211yesmaximal, characteristic, complemented
10#20273405C211yesmaximal, characteristic, complemented
11C53×#3077C211yesmaximal, characteristic
12C23×#88322—21no—
13#16811180C2211yescharacteristic
14#16811026C2211yescharacteristic, complemented
15#16811002C2211yescharacteristic, complemented
16#16810975C2211yescharacteristic, complemented
17#16810973C2211yescharacteristic, complemented
18#16810923—22no—
19#16810913C2212yes—
20#16810903C2212yes—
21#16810862—21no—
22#16810862—21no—
23C4×#11719925—21no—
24C4×#11719925—21no—
25D4×#88322—21no—
26D4×#88322—21no—
27D4×#88322C2212yes—
28D4×#88322C2212yes—
29#16810663—21no—
30#16810647C2211yescharacteristic
31C2×C4⋊#88322C2211yescharacteristic
32#16810628—21no—
33C2×C22⋊#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yescomplemented
36#16810375C2212yescomplemented
37#16810372C2212yescomplemented
38#16810371C2212yescomplemented
39#16810369C2212yescomplemented
40#16810368C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#478—22no—
44C53×#476C2212yes—
45C53×#474C2212yes—
46C53×#428—21no—
47C53×#423C2211yescharacteristic
48C53×#418—21no—
49C2×#11719925—21no—
50C2×#11719925—21no—
51C2×#11719917—21no—
52C22×#88322—22no—
53C22×#88322—22no—
54C22×#88322—21no—
55C22×#88322—44no—
56C22×#88322D411yescharacteristic
57C22×#88322D411yescharacteristic
58#13133217C2311yescharacteristic
59#13133217C2311yescharacteristic
60#13133209—21no—
61#13133196—21no—
62#13133196—21no—
63C4×Dic106—21no—
64C4×Dic106—21no—
65#13133176—21no—
66#13133150—22no—
67#13133150—21no—
68#13133150—21no—
69#13133148—22no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322C2312yes—
73C4⋊#88322C2312yes—
74#13133141—21no—
75#13133141—21no—
76#13133140—21no—
77#13133140—21no—
78#13133138—22no—
79#13133138—22no—
80C22⋊#88322—22no—
81C22⋊#88322—22no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C22×D4×C53D411yescharacteristic
87C23×C4×C53—21no—
88C2×M16×C53—21no—
89C4⋊Q8×C53C2311yescharacteristic
90C4⋊Q8×C53C2311yescharacteristic
91#125×C53—21no—
92C22⋊Q8×C53—21no—
93C22⋊Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D412yes—
99C4×D4×C53C2312yes—
100#117×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#107×C53—22no—
105#106×C53—22no—
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#102×C53—22no—
109#102×C53—22no—
110C24×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—22no—
115C2×D4×C53Q1614yescomplemented
116C2×D4×C53C2×D412yes—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C2×Dic106—21no—
126C2×Dic106—21no—
127M16×C53—42no—
128C2×C8×C53—22no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53C2×D412yes—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—41no—
140C42×C53—41no—
141C42×C53C2×D411yescharacteristic
142C42×C53C2×D411yescharacteristic
143C42×C53#4111yescharacteristic
144C42×C53C2411yescharacteristic
145C22⋊Dic53—41no—
146#11719925—42no—
147#11719925—42no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152#11719919—41no—
153#11719919—41no—
154C4×Dic53—41no—
155#11719917—42no—
156C2×#88322—22no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—41no—
160C2×#88322—44no—
161C2×#88322—42no—
162C2×#88322—44no—
163C2×#88322C2×D411yescharacteristic
164C2×#88322C2×D411yescharacteristic
165C2×#88322C2×D411yescharacteristic
166C2×#88322C2×D411yescharacteristic
167C23×C53—22no—
168C23×C53—24no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53#12111yescharacteristic
172Q8×C53—41no—
173Q8×C53—41no—
174Q8×C53—81no—
175D4×C53—22no—
176D4×C53—24no—
177D4×C53C2×Q1614yes—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—22no—
181C2×C4×C53—41no—
182C2×C4×C53—44no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53C2×Q1612yes—
192C2×C4×C53#13512yes—
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C2×#4111yescharacteristic
196C2×Dic53—41no—
197C2×Dic53—41no—
198Dic106—41no—
199Dic106—41no—
200Dic106—81no—
201C8×C53—42no—
202#88322—21no—
203#88322—21no—
204#88322—21no—
205#88322—42no—
206#88322—42no—
207#88322AGL(1,ℤ/8)12yes—
208#19256—531nomaximal, Sylow
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—24no—
212C22×C53—48no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53#46712yes—
216C22×C53C2×#12111yescharacteristic
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—42no—
221C4×C53—81no—
222C4×C53—81no—
223C4×C53#46712yes—
224C4×C53C22×Q1611yescharacteristic
225C4×C53C2×#13511yescharacteristic
226C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C53#57511yescharacteristic
228Dic53—81no—
229Dic53—81no—
230#3170—531no—
231#3077—531no—
232#3077D5311yescharacteristic
233C2×C8×D4—531no—
234#1683—532no—
235#1683—532no—
236#1675—532no—
237#1669—532no—
238#1567—531no—
239#1550—531no—
240#1550—531no—
241C2×C53—22no—
242C2×C53—44no—
243C2×C53#303611yescharacteristic
244C2×C53C2×#46711yescharacteristic
245C2×C53#306711yescharacteristic
246C23×C8—1061no—
247#533—531no—
248#533C2×D5311yescharacteristic
249C2×C4×D4C2×D5311yescharacteristic
250#500—531no—
251#494—531no—
252#487—531no—
253#487—531no—
254#478—22no—
255#478—1062no—
256#476—532no—
257#476C2×D5312yes—
258#474—532no—
259#474C2×D5312yes—
260#467—1061no—
261#467—1061no—
262C4×Q16—1061no—
263C4×Q16—1061no—
264C8×D4—532no—
265C8×D4—532no—
266C8×D4—1061no—
267C8×D4—1061no—
268#428—21no—
269#428—1061no—
270#423—531no—
271#423C2×D5311yescharacteristic
272C2×C4⋊C8—531no—
273#418—21no—
274#418—1061no—
275C2×C22⋊C8—1061no—
276C2×C4×C8—531no—
277#335—532no—
278#335—532no—
279#333—532no—
280#333—532no—
281#332—532no—
282#332—532no—
283C53#1925611yescharacteristic, complemented, Sylow
284C22×D4C22⋊D5311yescharacteristic
285C23×C4—21no—
286C2×Q16—1061no—
287C2×Q16—1061no—
288C2×M16—21no—
289C2×M16—1061no—
290C22×C8—531no—
291C22×C8—531no—
292C22×C8—1062no—
293C22×C8—1062no—
294C22×C8—1061no—
295C22×C8—2124no—
296C4⋊Q8—531no—
297C4⋊Q8—531no—
298C4⋊Q8C22×D5311yescharacteristic
299C4⋊Q8C22×D5311yescharacteristic
300#125—21no—
301#125—1061no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304C22⋊Q8—1061no—
305C22⋊Q8—1061no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—1061no—
309C4×Q8—1061no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4C22⋊D5312yes—
313C4×D4C22×D5312yes—
314#117—21no—
315#117—1061no—
316C2×C4⋊C4C22⋊D5311yescharacteristic
317C2×C22⋊C4—21no—
318C2×C42C22×D5311yescharacteristic
319#107—22no—
320#107—1061no—
321#107—1061no—
322#107—1062no—
323#106—22no—
324#106—1062no—
325C4⋊C8—21no—
326C4⋊C8—532no—
327C4⋊C8—532no—
328C4⋊C8—1061no—
329C4⋊C8—1061no—
330C4⋊C8C22×D5312yes—
331#103—1061no—
332#103—1061no—
333#103—1061no—
334#103—1061no—
335#102—22no—
336#102—22no—
337#102—1062no—
338#102—1062no—
339C22⋊C8—1062no—
340C22⋊C8—1062no—
341C4×C8—531no—
342C4×C8—531no—
343C4×C8—531no—
344C4×C8—531no—
345C24—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—1061no—
349C2×Q8—1061no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4#1171992514yescomplemented
353C2×D4C2×C22⋊D5312yes—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4D4×D5311yescharacteristic
360C22×C4#1171994711yescharacteristic
361C22×C4C2×C22⋊D5311yescharacteristic
362Q16—2122no—
363Q16—2122no—
364M16—42no—
365M16—2122no—
366C2×C8—22no—
367C2×C8—531no—
368C2×C8—531no—
369C2×C8—531no—
370C2×C8—531no—
371C2×C8—1062no—
372C2×C8—1062no—
373C2×C8—1062no—
374C2×C8—2124no—
375C2×C8—2121no—
376C2×C8—2124no—
377C2×C8—2122no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—1061no—
386C4⋊C4—1061no—
387C4⋊C4—1061no—
388C4⋊C4—1061no—
389C4⋊C4—2121no—
390C4⋊C4—2121no—
391C4⋊C4C2×C22⋊D5312yes—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—41no—
395C22⋊C4—2121no—
396C42—41no—
397C42—2121no—
398C42D4×D5311yescharacteristic
399C42#1171994711yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C42C23×D5311yescharacteristic
402C23—22no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23#1313318911yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—81no—
410Q8—4241no—
411Q8—2121no—
412Q8—2121no—
413D4—24no—
414D4—22no—
415D4C2×#1171992514yes—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4#1313318911yescharacteristic
429C2×C4#1313318911yescharacteristic
430C2×C4#1313318911yescharacteristic
431C2×C4C2×#1171992512yes—
432C2×C4#1313323712yes—
433C2×C4C2×D4×D5311yescharacteristic
434C2×C4C22×C22⋊D5311yescharacteristic
435C2×C4C2×#1171994711yescharacteristic
436C8—42no—
437C8—532no—
438C8—1061no—
439C8—1061no—
440C8—1061no—
441C8—2122no—
442C8—2122no—
443C22—21no—
444C22—21no—
445C22—24no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22#1681086212yes—
450C22C2×#1313318911yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—4241no—
458C4—4241no—
459C4#1681086212yes—
460C4C22×#1171992511yescharacteristic
461C4C2×#1313323711yescharacteristic
462C4AGL(1,ℤ/8)×D5311yescharacteristic
463C4#1681141911yescharacteristic
464C2—22no—
465C2—44no—
466C2#2028417611yescharacteristic
467C2C2×#1681086211yescharacteristic
468C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3077C2
socleC22×C53C2×#121
prime radicalC22×D4×C53D4
hypercenter#3077D53
hypocenterC53#19256
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #20285297C2semidirect
4. #20274168C2semidirect
5. #20274164C2semidirect
6. #20274112C2semidirect
7. #20274078C2semidirect
8. #20273473C2semidirect
9. #20273406C2semidirect
10. #20273405C2semidirect
14. #16811026C22semidirect
15. #16811002C22semidirect
16. #16810975C22semidirect
17. #16810973C22semidirect
35. #16810375C22semidirect
36. #16810375C22semidirect
37. #16810372C22semidirect
38. #16810371C22semidirect
39. #16810369C22semidirect
40. #16810368C22semidirect
115. C2×D4×C53Q16semidirect
283. C53#19256semidirect
352. C2×D4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202852972
#202845122
C2×D4×#883222
#202741682
#202741642
#202741122
#202740782
#202734732
#202734062
#202734052
C53×#30772
#1925653