Finite Group Databasebeta

← #27366909

Subgroups — #27366909

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284719C211yesmaximal, characteristic
2C2×#16810900C211yesmaximal, characteristic
3M16×D212C211yesmaximal, characteristic
4#20274287C212yesmaximal, complemented
5#20274259C212yesmaximal, complemented
6#20274163C212yesmaximal, complemented
7#20274135C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10#20273387C211yesmaximal, characteristic, complemented
11C53×#3139C211yesmaximal, characteristic, complemented
12C2×M16×D53—21no—
13C2×C4×D212C2211yescharacteristic
14#16810951—21no—
15#16810910C2211yescharacteristic
16#16810910C2211yescharacteristic
17#16810900—21no—
18#16810900—21no—
19#16810900C2212yes—
20#16810900C2212yes—
21#16810825C2211yescharacteristic
22#16810825C2211yescharacteristic
23#16810773—21no—
24C4×#11719924—21no—
25C4×#11719924—21no—
26#16810702C2212yes—
27#16810696—22no—
28C8×D212C2212yes—
29#16810676C2211yescharacteristic
30C2×#13133150—21no—
31#16810647C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33C2×#13133137—21no—
34#16810586—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810372C2212yescomplemented
38#16810369C2212yescomplemented
39#16810369C2212yescomplemented
40#16810369C2212yescomplemented
41C53×#533C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#491—22no—
44C53×#487C2212yescomplemented
45C53×#487C2212yescomplemented
46C53×#419—21no—
47C53×#419—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#13133150—22no—
67#13133150—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×C53C2312yes—
93C4⋊Q8×C53C2311yescharacteristic
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#103×C53—22no—
105#103×C53—22no—
106#103×C53—22no—
107#103×C53—22no—
108#97×C53D412yes—
109C4×C8×C53C2312yes—
110C23×D53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—41no—
115C2×Q8×C53—42no—
116C2×Q8×C53—42no—
117C22×C4×C53—41no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C22×Dic53—21no—
122C2×D212—22no—
123C2×D212—21no—
124C2×D212—41no—
125C2×D212C2×D411yescharacteristic
126C2×D212C2×D411yescharacteristic
127C2×C4×D53—21no—
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—42no—
132C2×C4×D53—42no—
133M16×C53—22no—
134M16×C53—22no—
135C2×C8×C53C2×D412yes—
136C2×C8×C53C2×D412yes—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C42×C53—41no—
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C42×C53#4111yescharacteristic
152C42×C53C2411yescharacteristic
153#11719924—42no—
154#11719924—42no—
155#11719921—21no—
156#11719921—21no—
157#11719921—21no—
158#11719921—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53C2×D411yescharacteristic
161C4⋊Dic53C2×D411yescharacteristic
162#11719917—42no—
163C2×#88322—21no—
164C2×#88322—21no—
165C2×#88322—22no—
166C2×#88322—22no—
167C2×#88322—42no—
168#11719912—44no—
169C8×D53—44no—
170C23×C53#12111yescharacteristic
171C22×D53—21no—
172C22×D53—21no—
173C22×D53—42no—
174C22×D53—42no—
175C22×D53—81no—
176Q8×C53—42no—
177Q8×C53—42no—
178Q8×C53—41no—
179Q8×C53—41no—
180Q8×C53—81no—
181Q8×C53—82no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—41no—
185C2×C4×C53—42no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—82no—
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#13512yes—
193C2×C4×C53AGL(1,ℤ/8)12yes—
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C2×#4111yescharacteristic
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—42no—
200D212—21no—
201D212—21no—
202D212—21no—
203D212—42no—
204D212#13712yes—
205C4×D53—41no—
206C4×D53—41no—
207C4×D53—41no—
208C4×D53—41no—
209C4×D53—42no—
210C4×D53—42no—
211C8×C53—22no—
212C8×C53—22no—
213#88322—42no—
214#88322—41no—
215#88322—42no—
216#88322—41no—
217#19167—531nomaximal, Sylow
218C22×C53—21no—
219C22×C53—21no—
220C22×C53#46312yes—
221C22×C53C2×#12111yescharacteristic
222C2×D53—22no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C2×D53—84no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—82no—
233C4×C53—81no—
234C4×C53—81no—
235C4×C53#46312yes—
236C4×C53C2×#13511yescharacteristic
237C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C53C2×#13711yescharacteristic
239C4×C53#57511yescharacteristic
240Dic53—41no—
241Dic53—41no—
242#3139D5311yescharacteristic, complemented
243#3101—531no—
244C2×#474—531no—
245D4×M16—531no—
246#1703—532no—
247#1699—532no—
248#1683—532no—
249#1679—532no—
250#1565—531no—
251#1550—531no—
252#1546—531no—
253C2×C53—22no—
254C2×C53C2×#46311yescharacteristic
255C2×C53#305611yescharacteristic
256C2×C53#306911yescharacteristic
257D53—42no—
258D53—81no—
259C22×M16—1061no—
260#533C2×D5311yescharacteristic
261C2×C4⋊Q8C2×D5311yescharacteristic
262C2×C4×D4—531no—
263#491—22no—
264#487C2×D5312yescomplemented
265#487C2×D5312yescomplemented
266#483—1061no—
267#476—531no—
268#476—531no—
269#474—532no—
270#474—532no—
271#474—1061no—
272#474—1061no—
273#461—531no—
274#461—531no—
275#450—1061no—
276C4×SD16—1061no—
277C4×SD16—1061no—
278#436—532no—
279#435—1062no—
280C8×D4—532no—
281#431—531no—
282C2×#107—1061no—
283#423—531no—
284C2×C4⋊C8—531no—
285#419—21no—
286#419—21no—
287C2×#102—1061no—
288#407—1061no—
289C4×M16C2×D5311yescharacteristic
290#335—532no—
291#334—532no—
292#333—532no—
293#332—532no—
294#332—532no—
295#332—532no—
296C53#1916711yescharacteristic, complemented, Sylow
297C22×Q8—21no—
298C22×D4—531no—
299C23×C4—1061no—
300C2×SD16—1061no—
301C2×SD16—1061no—
302C2×M16—1061no—
303C2×M16—1062no—
304C2×M16—2124no—
305C2×M16C22⋊D5311yescharacteristic
306C2×M16C22⋊D5311yescharacteristic
307C22×C8—1061no—
308C22×C8—1062no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8C22×D5312yes—
313C4⋊Q8C22×D5311yescharacteristic
314C4⋊Q8C22×D5311yescharacteristic
315#125—21no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×D4—531no—
321C4×D4—531no—
322C4×D4—531no—
323C4×D4—531no—
324C4×D4—1061no—
325C4×D4—1061no—
326#117—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—531no—
330C2×C22⋊C4—1061no—
331C2×C42C22×D5311yescharacteristic
332#107—1062no—
333#107—1062no—
334#106—1061no—
335#106—1061no—
336C4⋊C8—532no—
337C4⋊C8—531no—
338C4⋊C8—531no—
339C4⋊C8—532no—
340C4⋊C8—1061no—
341C4⋊C8—1061no—
342#103—22no—
343#103—22no—
344#103—22no—
345#103—22no—
346#103—1061no—
347#103—1061no—
348#102—1062no—
349#102—1062no—
350#102—1061no—
351#102—1061no—
352C22⋊C8—1061no—
353C22⋊C8—1062no—
354C22⋊C8—1062no—
355#97C22⋊D5312yes—
356C4×C8—1061no—
357C4×C8C22×D5312yes—
358C24—1061no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×D4—531no—
366C2×D4—531no—
367C2×D4—1062no—
368C2×D4—1061no—
369C2×D4—2121no—
370C22×C4—41no—
371C22×C4—1061no—
372C22×C4—1061no—
373C22×C4—1061no—
374C22×C4—1061no—
375C22×C4—1061no—
376C22×C4—2122no—
377C22×C4—2122no—
378C22×C4D4×D5311yescharacteristic
379C22×C4#1171994711yescharacteristic
380C22×C4C2×C22⋊D5311yescharacteristic
381SD16—2122no—
382SD16—2122no—
383M16—22no—
384M16—22no—
385M16—2124no—
386M16—2122no—
387C2×C8—1061no—
388C2×C8—1061no—
389C2×C8—1062no—
390C2×C8—1062no—
391C2×C8—2124no—
392C2×C8—2122no—
393C2×C8C2×C22⋊D5312yes—
394C2×C8C2×C22⋊D5312yes—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—531no—
406C4⋊C4—531no—
407C4⋊C4—1061no—
408C22⋊C4—41no—
409C22⋊C4—1061no—
410C22⋊C4—1061no—
411C22⋊C4—1061no—
412C22⋊C4—1061no—
413C42—41no—
414C42D4×D5311yescharacteristic
415C42#1171994711yescharacteristic
416C42C2×C22⋊D5311yescharacteristic
417C42C23×D5311yescharacteristic
418C23—1061no—
419C23—1061no—
420C23—4241no—
421C23—2122no—
422C23—2122no—
423C23#1313318811yescharacteristic
424Q8—42no—
425Q8—42no—
426Q8—41no—
427Q8—41no—
428Q8—81no—
429Q8—82no—
430D4—532no—
431D4—1061no—
432D4—1061no—
433D4—1061no—
434D4—2122no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—1061no—
443C2×C4—1061no—
444C2×C4—2122no—
445C2×C4—2122no—
446C2×C4—2122no—
447C2×C4—2121no—
448C2×C4—2121no—
449C2×C4—2121no—
450C2×C4—2121no—
451C2×C4#1313318811yescharacteristic
452C2×C4#1313318811yescharacteristic
453C2×C4#1313318811yescharacteristic
454C2×C4#1313323712yes—
455C2×C4#1313324212yes—
456C2×C4C2×D4×D5311yescharacteristic
457C2×C4C22×C22⋊D5311yescharacteristic
458C2×C4C2×#1171994711yescharacteristic
459C8—22no—
460C8—22no—
461C8—2122no—
462C8—2121no—
463C8—2122no—
464C8—2121no—
465C22—21no—
466C22—21no—
467C22—1062no—
468C22—4244no—
469C22—2121no—
470C22—2121no—
471C22—2121no—
472C22—2121no—
473C22—2121no—
474C22#1681083512yes—
475C22C2×#1313318811yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—81no—
480C4—81no—
481C4—82no—
482C4—2121no—
483C4—2121no—
484C4#1681083512yes—
485C4C2×#1313323711yescharacteristic
486C4C2×#1313324211yescharacteristic
487C4#137×D5311yescharacteristic
488C4#1681142211yescharacteristic
489C2—22no—
490C2—4241no—
491C2—2122no—
492C2C2×#1681083511yescharacteristic
493C2#2028433911yescharacteristic
494C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3139C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#3139D53
hypocenterC53#19167
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
4. #20274287C2semidirect
5. #20274259C2semidirect
6. #20274163C2semidirect
7. #20274135C2semidirect
8. #20273467C2semidirect
9. #20273405C2semidirect
10. #20273387C2semidirect
11. C53×#3139C2semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810372C22semidirect
38. #16810369C22semidirect
39. #16810369C22semidirect
40. #16810369C22semidirect
44. C53×#487C22semidirect
45. C53×#487C22semidirect
242. #3139D53semidirect
264. #487C2×D53semidirect
265. #487C2×D53semidirect
296. C53#19167semidirect

Maximal subgroups

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

subgroupindex
#202847192
C2×#168109002
M16×D2122
#202742872
#202742592
#202741632
#202741352
#202734672
#202734052
#202733872
C53×#31392
#1916753