Finite Group Databasebeta

← #27366887

Subgroups — #27366887

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284719C211yesmaximal, characteristic
2C2×#16810910C211yesmaximal, characteristic
3M16×D212C211yesmaximal, characteristic
4#20274235C212yesmaximal, complemented
5#20274211C212yesmaximal, complemented
6#20274111C212yesmaximal, complemented
7#20274087C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10#20273396C211yesmaximal, characteristic, complemented
11C53×#3165C211yesmaximal, characteristic, complemented
12C2×M16×D53—21no—
13C2×C4×D212C2211yescharacteristic
14#16810951—21no—
15#16810910—21no—
16#16810910—21no—
17#16810910C2212yes—
18#16810910C2212yes—
19#16810910C2211yescharacteristic
20#16810910C2211yescharacteristic
21#16810825C2211yescharacteristic
22#16810825C2211yescharacteristic
23#16810773—21no—
24C4×#11719924—21no—
25C4×#11719924—21no—
26#16810702C2212yes—
27#16810696—22no—
28C8×D212C2212yes—
29#16810676C2211yescharacteristic
30C2×#13133148—21no—
31#16810647C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33C2×#13133137—21no—
34#16810586—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810372C2212yescomplemented
38#16810372C2212yescomplemented
39#16810372C2212yescomplemented
40#16810369C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#501—22no—
44C53×#500C2212yescomplemented
45C53×#498C2212yescomplemented
46C53×#428—21no—
47C53×#428—21no—
48C4×M16×C53C2211yescharacteristic
49C22×D212D411yescharacteristic
50C22×C4×D53—21no—
51C2×#11719924—21no—
52C2×#11719924—21no—
53C2×#11719917—21no—
54C2×#11719912—22no—
55M16×D53—44no—
56C22×#88322—21no—
57C2×C8×D53—22no—
58C4×D212—21no—
59C4×D212—21no—
60C4×D212D411yescharacteristic
61C4×D212D411yescharacteristic
62C4×D212C2311yescharacteristic
63C4×D212C2311yescharacteristic
64C2×C4⋊Dic53D411yescharacteristic
65C2×#11719921—21no—
66#13133148—22no—
67#13133148—22no—
68#13133148—21no—
69#13133148—21no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2312yes—
75C4⋊#88322C2312yes—
76#13133141—21no—
77#13133141—21no—
78#13133137—21no—
79#13133137—21no—
80#13133137—22no—
81#13133137—22no—
82C22⋊#88322—21no—
83#13133129—22no—
84#13133129—22no—
85C4×#88322—21no—
86C22×Q8×C53—21no—
87C2×M16×C53D411yescharacteristic
88C2×M16×C53D411yescharacteristic
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53—21no—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53C2311yescharacteristic
93C4⋊Q8×C53C2312yes—
94C4⋊Q8×C53C2311yescharacteristic
95#125×C53—21no—
96C22⋊Q8×C53—21no—
97C22⋊Q8×C53—21no—
98C4×Q8×C53—21no—
99C4×Q8×C53—21no—
100#117×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#107×C53—22no—
105#107×C53—22no—
106#106×C53—22no—
107#106×C53—22no—
108#97×C53D412yes—
109C4×C8×C53C2312yes—
110C23×D53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53—41no—
115C2×Q8×C53—82no—
116C22×C4×C53—41no—
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C22×Dic53—21no—
121C2×D212—22no—
122C2×D212—21no—
123C2×D212—41no—
124C2×D212C2×D411yescharacteristic
125C2×D212C2×D411yescharacteristic
126C2×C4×D53—21no—
127C2×C4×D53—21no—
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—42no—
131C2×C4×D53—42no—
132M16×C53—22no—
133M16×C53SD1614yescomplemented
134C2×C8×C53C2×D412yes—
135C2×C8×C53C2×D412yes—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C42×C53—41no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53#4111yescharacteristic
151C42×C53C2411yescharacteristic
152#11719924—42no—
153#11719924—42no—
154#11719921—21no—
155#11719921—21no—
156#11719921—21no—
157#11719921—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53C2×D411yescharacteristic
160C4⋊Dic53C2×D411yescharacteristic
161#11719917—42no—
162C2×#88322—21no—
163C2×#88322—21no—
164C2×#88322—22no—
165C2×#88322—22no—
166C2×#88322—42no—
167#11719912—44no—
168C8×D53—44no—
169C23×C53#12111yescharacteristic
170C22×D53—21no—
171C22×D53—21no—
172C22×D53—42no—
173C22×D53—42no—
174C22×D53—81no—
175Q8×C53—41no—
176Q8×C53—41no—
177Q8×C53—82no—
178Q8×C53—81no—
179Q8×C53—82no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—41no—
183C2×C4×C53—42no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—82no—
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53C2×SD1612yes—
191C2×C4×C53#13712yes—
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C2×#4111yescharacteristic
195C2×Dic53—21no—
196C2×Dic53—21no—
197C2×Dic53—42no—
198D212—21no—
199D212—21no—
200D212—21no—
201D212—42no—
202D212#13712yes—
203C4×D53—41no—
204C4×D53—41no—
205C4×D53—41no—
206C4×D53—41no—
207C4×D53—42no—
208C4×D53—42no—
209C8×C53—22no—
210C8×C53C2×SD1614yes—
211#88322—42no—
212#88322—41no—
213#88322—42no—
214#88322—41no—
215#19165—531nomaximal, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53#46112yes—
219C22×C53C2×#12111yescharacteristic
220C2×D53—22no—
221C2×D53—41no—
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—84no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—81no—
231C4×C53—82no—
232C4×C53—81no—
233C4×C53#46112yes—
234C4×C53C22×SD1611yescharacteristic
235C4×C53C2×#13711yescharacteristic
236C4×C53C2×#13711yescharacteristic
237C4×C53#57511yescharacteristic
238Dic53—41no—
239Dic53—41no—
240#3165D5311yescharacteristic, complemented
241#3101—531no—
242C2×#476—531no—
243D4×M16—531no—
244#1695—532no—
245#1691—532no—
246#1675—532no—
247#1671—532no—
248#1565—531no—
249#1550—531no—
250#1548—531no—
251C2×C53—22no—
252C2×C53C2×#46111yescharacteristic
253C2×C53#305611yescharacteristic
254C2×C53#306811yescharacteristic
255D53—42no—
256D53—81no—
257C22×M16—1061no—
258#533C2×D5311yescharacteristic
259C2×C4⋊Q8C2×D5311yescharacteristic
260C2×C4×D4—531no—
261#501—22no—
262#500C2×D5312yescomplemented
263#498C2×D5312yescomplemented
264#483—1061no—
265#476—531no—
266#476—531no—
267#476—532no—
268#476—532no—
269#476—1061no—
270#476—1061no—
271#461—531no—
272#461—531no—
273#450—1061no—
274C4×SD16—1061no—
275C4×SD16—1061no—
276#436—532no—
277#435—1062no—
278C8×D4—532no—
279#431—531no—
280#428—21no—
281#428—21no—
282C2×#106—1061no—
283#423—531no—
284C2×C4⋊C8—531no—
285C2×#102—1061no—
286#407—1061no—
287C4×M16C2×D5311yescharacteristic
288#335—532no—
289#334—532no—
290#333—532no—
291#333—532no—
292#333—532no—
293#332—532no—
294C53#1916511yescharacteristic, complemented, Sylow
295C22×Q8—21no—
296C22×D4—531no—
297C23×C4—1061no—
298C2×SD16—1061no—
299C2×SD16—1061no—
300C2×M16—1061no—
301C2×M16—1062no—
302C2×M16—2124no—
303C2×M16C22⋊D5311yescharacteristic
304C2×M16C22⋊D5311yescharacteristic
305C22×C8—1061no—
306C22×C8—1062no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8C22×D5312yes—
311C4⋊Q8C22×D5311yescharacteristic
312C4⋊Q8C22×D5311yescharacteristic
313#125—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×D4—531no—
319C4×D4—531no—
320C4×D4—531no—
321C4×D4—531no—
322C4×D4—1061no—
323C4×D4—1061no—
324#117—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—531no—
328C2×C22⋊C4—1061no—
329C2×C42C22×D5311yescharacteristic
330#107—22no—
331#107—22no—
332#106—22no—
333#106—22no—
334#106—1062no—
335#106—1062no—
336#106—1061no—
337#106—1061no—
338C4⋊C8—532no—
339C4⋊C8—532no—
340C4⋊C8—531no—
341C4⋊C8—531no—
342C4⋊C8—1061no—
343C4⋊C8—1061no—
344#103—1061no—
345#103—1061no—
346#102—1062no—
347#102—1062no—
348#102—1061no—
349#102—1061no—
350C22⋊C8—1061no—
351C22⋊C8—1062no—
352C22⋊C8—1062no—
353#97C22⋊D5312yes—
354C4×C8—1061no—
355C4×C8C22×D5312yes—
356C24—1061no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—82no—
362C2×D4—531no—
363C2×D4—531no—
364C2×D4—1062no—
365C2×D4—1061no—
366C2×D4—2121no—
367C22×C4—41no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—1061no—
371C22×C4—1061no—
372C22×C4—1061no—
373C22×C4—2122no—
374C22×C4—2122no—
375C22×C4D4×D5311yescharacteristic
376C22×C4#1171994711yescharacteristic
377C22×C4C2×C22⋊D5311yescharacteristic
378SD16—2122no—
379SD16—2122no—
380M16—22no—
381M16—2122no—
382M16—2124no—
383M16#1171992414yescomplemented
384C2×C8—1061no—
385C2×C8—1061no—
386C2×C8—1062no—
387C2×C8—1062no—
388C2×C8—2122no—
389C2×C8—2124no—
390C2×C8C2×C22⋊D5312yes—
391C2×C8C2×C22⋊D5312yes—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—531no—
403C4⋊C4—531no—
404C4⋊C4—1061no—
405C22⋊C4—41no—
406C22⋊C4—1061no—
407C22⋊C4—1061no—
408C22⋊C4—1061no—
409C22⋊C4—1061no—
410C42—41no—
411C42D4×D5311yescharacteristic
412C42#1171994711yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C42C23×D5311yescharacteristic
415C23—1061no—
416C23—1061no—
417C23—4241no—
418C23—2122no—
419C23—2122no—
420C23#1313318811yescharacteristic
421Q8—41no—
422Q8—41no—
423Q8—81no—
424Q8—82no—
425Q8—82no—
426D4—532no—
427D4—1061no—
428D4—1061no—
429D4—1061no—
430D4—2122no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—82no—
438C2×C4—1061no—
439C2×C4—1061no—
440C2×C4—2122no—
441C2×C4—2122no—
442C2×C4—2122no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4#1313318811yescharacteristic
448C2×C4#1313318811yescharacteristic
449C2×C4#1313318811yescharacteristic
450C2×C4C2×#1171992412yes—
451C2×C4#1313324812yes—
452C2×C4C2×D4×D5311yescharacteristic
453C2×C4C22×C22⋊D5311yescharacteristic
454C2×C4C2×#1171994711yescharacteristic
455C8—22no—
456C8—2122no—
457C8—2121no—
458C8—2122no—
459C8—2121no—
460C8C2×#1171992414yes—
461C22—21no—
462C22—21no—
463C22—1062no—
464C22—4244no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22—2121no—
469C22—2121no—
470C22#1681082512yes—
471C22C2×#1313318811yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—82no—
476C4—81no—
477C4—81no—
478C4—2121no—
479C4—2121no—
480C4#1681082512yes—
481C4C22×#1171992411yescharacteristic
482C4#137×D5311yescharacteristic
483C4C2×#1313324811yescharacteristic
484C4#1681142211yescharacteristic
485C2—22no—
486C2—4241no—
487C2—2122no—
488C2C2×#1681082511yescharacteristic
489C2#2028433911yescharacteristic
490C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3165C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#3165D53
hypocenterC53#19165
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #20274235C2semidirect
5. #20274211C2semidirect
6. #20274111C2semidirect
7. #20274087C2semidirect
8. #20273467C2semidirect
9. #20273405C2semidirect
10. #20273396C2semidirect
11. C53×#3165C2semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810372C22semidirect
38. #16810372C22semidirect
39. #16810372C22semidirect
40. #16810369C22semidirect
44. C53×#500C22semidirect
45. C53×#498C22semidirect
133. M16×C53SD16semidirect
240. #3165D53semidirect
262. #500C2×D53semidirect
263. #498C2×D53semidirect
294. C53#19165semidirect
383. M16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202847192
C2×#168109102
M16×D2122
#202742352
#202742112
#202741112
#202740872
#202734672
#202734052
#202733962
C53×#31652
#1916553