Finite Group Databasebeta

← #27370123

Subgroups — #27370123

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285833C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3#20283750C211yesmaximal, characteristic, complemented
4#20274246C212yesmaximal, complemented
5#20274236C212yesmaximal, complemented
6#20274122C212yesmaximal, complemented
7#20274112C212yesmaximal, complemented
8#20273501C211yesmaximal, characteristic, complemented
9#20273440C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3109C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16811026C2212yescomplemented
14#16810969C2212yescomplemented
15#16810934—21no—
16#16810934—21no—
17#16810913—21no—
18#16810913—21no—
19#16810913C2212yes—
20#16810913C2212yes—
21#16810903—22no—
22#16810701C2212yes—
23#16810695—22no—
24D4×#88322C2212yes—
25#16810680C2211yescharacteristic
26#16810676C2211yescharacteristic
27C2×#13133148—21no—
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133138—21no—
30#16810587—21no—
31C2×C22⋊#88322—21no—
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810372C2212yescomplemented
35#16810371C2212yescomplemented
36#16810371C2212yescomplemented
37#16810367C2212yescomplemented
38C53×#532C2211yescharacteristic
39C2×C4×D4×C53C2211yescharacteristic
40C53×#481—21no—
41C53×#476C2211yescharacteristic
42C53×#476C2211yescharacteristic
43C53×#459C2211yescharacteristic
44C53×#459C2211yescharacteristic
45C53×#447—21no—
46C53×#442—21no—
47C53×#442—21no—
48C53×#432C2211yescharacteristic
49C22×Dic106—21no—
50C2×#11719917—22no—
51C22×#88322—22no—
52C22×#88322—21no—
53#13133217—21no—
54#13133217—21no—
55#13133217—21no—
56#13133217C2312yes—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—21no—
59#13133150—22no—
60#13133150—22no—
61#13133148—22no—
62#13133148—22no—
63C4⋊#88322—21no—
64C4⋊#88322D412yes—
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133138—22no—
68#13133138—22no—
69#13133138—22no—
70#13133138—22no—
71C22⋊#88322—22no—
72C22⋊#88322—22no—
73C22⋊#88322—21no—
74C22⋊#88322D411yescharacteristic
75C22⋊#88322D411yescharacteristic
76#13133128—21no—
77C4×#88322C2312yes—
78C2×#41×C53—21no—
79C22×D4×C53D411yescharacteristic
80C23×C4×C53—21no—
81C2×D8×C53—21no—
82C2×D8×C53—21no—
83C4⋊Q8×C53C2311yescharacteristic
84#127×C53C2311yescharacteristic
85#124×C53—21no—
86#121×C53—21no—
87#121×C53—21no—
88C4×D4×C53—21no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53D411yescharacteristic
93C4×D4×C53D411yescharacteristic
94C4×D4×C53C2311yescharacteristic
95C4×D4×C53C2311yescharacteristic
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C22⋊C4×C53—21no—
98C2×C42×C53C2311yescharacteristic
99#106×C53—21no—
100#106×C53—21no—
101C4⋊C8×C53C2311yescharacteristic
102C4⋊C8×C53C2311yescharacteristic
103#102×C53—21no—
104#102×C53—21no—
105#102×C53—21no—
106#102×C53—21no—
107C22⋊C8×C53—21no—
108#97×C53—21no—
109C24×C53—21no—
110#41×C53—82no—
111C2×Q8×C53—21no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53C2×D411yescharacteristic
120C2×D4×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—41no—
127C22×C4×C53—82no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53#4111yescharacteristic
131C22×Dic53—41no—
132C2×Dic106—22no—
133C2×Dic106—41no—
134C2×Dic106—82no—
135D8×C53—42no—
136D8×C53—42no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53C2×D411yescharacteristic
143C4⋊C4×C53C2×D411yescharacteristic
144C22⋊C4×C53—21no—
145C22⋊C4×C53—21no—
146C22⋊C4×C53—21no—
147C22⋊C4×C53—21no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154C4⋊Dic53—22no—
155C4⋊Dic53—22no—
156C4⋊Dic53—41no—
157C4⋊Dic53—41no—
158#11719917—82no—
159C2×#88322—22no—
160C2×#88322—22no—
161C2×#88322—42no—
162C2×#88322—82no—
163C2×#88322C2×D412yes—
164C2×#88322C2×D412yes—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—42no—
168C23×C53—42no—
169C23×C53—41no—
170C23×C53—81no—
171C23×C53#12111yescharacteristic
172Q8×C53—81no—
173D4×C53—21no—
174D4×C53—21no—
175D4×C53—21no—
176D4×C53—42no—
177D4×C53—41no—
178D4×C53—41no—
179D4×C53—42no—
180D4×C53—81no—
181D4×C53—82no—
182D4×C53#13712yes—
183C2×C4×C53—22no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—22no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—41no—
192C2×C4×C53—81no—
193C2×C4×C53—81no—
194C2×C4×C53—82no—
195C2×C4×C53—81no—
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C2×#4111yescharacteristic
202C2×Dic53—42no—
203C2×Dic53—82no—
204Dic106—82no—
205Dic106—82no—
206C8×C53—41no—
207C8×C53—41no—
208#88322—22no—
209#88322—22no—
210#88322—42no—
211#88322—42no—
212#19465—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—22no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—84no—
222C22×C53—81no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53#47212yes—
226C22×C53C2×#12111yescharacteristic
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—81no—
234C4×C53C2×#13711yescharacteristic
235C4×C53#57511yescharacteristic
236C4×C53#57712yes—
237Dic53—82no—
238#3226—531no—
239#3109D5311yescharacteristic
240C2×#476—531no—
241#2974—531no—
242#1697—532no—
243#1695—532no—
244#1677—532no—
245#1675—532no—
246#1573—531no—
247#1558—531no—
248#1551—531no—
249C2×C53—22no—
250C2×C53—42no—
251C2×C53—81no—
252C2×C53—161no—
253C2×C53#304211yescharacteristic
254C2×C53#305611yescharacteristic
255C2×C53C2×#47211yescharacteristic
256#532C2×D5311yescharacteristic
257C2×C4⋊Q8—531no—
258C2×C4×D4C2×D5311yescharacteristic
259#500—532no—
260#486—532no—
261#481—21no—
262#480—1061no—
263#480—1061no—
264#476—532no—
265#476—532no—
266#476—1061no—
267#476—1061no—
268#476C2×D5311yescharacteristic
269#476C2×D5311yescharacteristic
270#474—1062no—
271#459C2×D5311yescharacteristic
272#459C2×D5311yescharacteristic
273#447—21no—
274#442—21no—
275#442—21no—
276#436—532no—
277#435—1062no—
278C8×D4—532no—
279#432—531no—
280#432C2×D5311yescharacteristic
281#431—531no—
282C2×#106—1061no—
283C2×C4⋊C8—531no—
284C2×#102—1061no—
285#407—1061no—
286C2×C22⋊C8—1061no—
287#335—532no—
288#334—532no—
289#333—532no—
290#333—532no—
291#333—532no—
292#331—532no—
293C53#1946511yescharacteristic, complemented, Sylow
294C2×#41—21no—
295C22×Q8—1061no—
296C22×D4C22⋊D5311yescharacteristic
297C23×C4—21no—
298C2×D8—21no—
299C2×D8—21no—
300C2×M16—1062no—
301C22×C8—1062no—
302C22×C8—1061no—
303C4⋊Q8—532no—
304C4⋊Q8—1061no—
305C4⋊Q8—1061no—
306C4⋊Q8—1061no—
307C4⋊Q8C22×D5311yescharacteristic
308#127C22×D5311yescharacteristic
309#124—21no—
310#121—21no—
311#121—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4C22⋊D5311yescharacteristic
317C4×D4C22⋊D5311yescharacteristic
318C4×D4C22×D5311yescharacteristic
319C4×D4C22×D5311yescharacteristic
320C2×C4⋊C4—1061no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4C22⋊D5311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C42C22×D5311yescharacteristic
325#107—1062no—
326#107—1062no—
327#106—21no—
328#106—21no—
329#106—1062no—
330#106—1062no—
331C4⋊C8—532no—
332C4⋊C8—532no—
333C4⋊C8—532no—
334C4⋊C8—1061no—
335C4⋊C8C22×D5311yescharacteristic
336C4⋊C8C22×D5311yescharacteristic
337#102—21no—
338#102—21no—
339#102—21no—
340#102—21no—
341#102—1062no—
342#102—1062no—
343#102—1062no—
344#102—1062no—
345C22⋊C8—21no—
346C22⋊C8—531no—
347C22⋊C8—531no—
348C22⋊C8—1062no—
349C22⋊C8—1062no—
350C22⋊C8—1061no—
351#97—21no—
352#97—1061no—
353C4×C8—532no—
354C24—21no—
355#41—82no—
356C2×Q8—21no—
357C2×Q8—1062no—
358C2×Q8—4242no—
359C2×Q8—2121no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4C2×C22⋊D5311yescharacteristic
368C2×D4C2×C22⋊D5311yescharacteristic
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—41no—
375C22×C4—82no—
376C22×C4—2121no—
377C22×C4D4×D5311yescharacteristic
378C22×C4#1171994711yescharacteristic
379C22×C4C2×C22⋊D5311yescharacteristic
380D8—42no—
381D8—42no—
382M16—4242no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—532no—
386C2×C8—532no—
387C2×C8—1062no—
388C2×C8—1062no—
389C2×C8—4242no—
390C2×C8—2122no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—1062no—
395C4⋊C4—1062no—
396C4⋊C4—2121no—
397C4⋊C4—2121no—
398C4⋊C4C2×C22⋊D5311yescharacteristic
399C4⋊C4C2×C22⋊D5311yescharacteristic
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C42D4×D5311yescharacteristic
407C42#1171994711yescharacteristic
408C42C2×C22⋊D5311yescharacteristic
409C42C23×D5311yescharacteristic
410C23—21no—
411C23—21no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—81no—
416C23#1313318911yescharacteristic
417Q8—81no—
418Q8—4242no—
419Q8—4242no—
420D4—21no—
421D4—21no—
422D4—21no—
423D4—42no—
424D4—41no—
425D4—41no—
426D4—42no—
427D4—81no—
428D4—82no—
429D4#1313324712yes—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—4242no—
444C2×C4—2122no—
445C2×C4#1313318911yescharacteristic
446C2×C4#1313318911yescharacteristic
447C2×C4#1313318911yescharacteristic
448C2×C4C2×D4×D5311yescharacteristic
449C2×C4C22×C22⋊D5311yescharacteristic
450C2×C4C2×#1171994711yescharacteristic
451C8—41no—
452C8—41no—
453C8—1062no—
454C8—1062no—
455C8—2122no—
456C8—2122no—
457C22—21no—
458C22—21no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—84no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22#1681089112yes—
470C22C2×#1313318911yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—41no—
476C4—41no—
477C4—81no—
478C4—4242no—
479C4C2×#1313324711yescharacteristic
480C4#1681142111yescharacteristic
481C4#1681143312yes—
482C2—22no—
483C2—42no—
484C2—81no—
485C2—161no—
486C2#2028422911yescharacteristic
487C2#2028434711yescharacteristic
488C2C2×#1681089111yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3109C2
socleC22×C53C2×#121
hypercenter#3109D53
hypocenterC53#19465
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #20285833C2semidirect
2. C2×#16810913C2semidirect
3. #20283750C2semidirect
4. #20274246C2semidirect
5. #20274236C2semidirect
6. #20274122C2semidirect
7. #20274112C2semidirect
8. #20273501C2semidirect
9. #20273440C2semidirect
10. #20273410C2semidirect
13. #16811026C22semidirect
14. #16810969C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810372C22semidirect
35. #16810371C22semidirect
36. #16810371C22semidirect
37. #16810367C22semidirect
293. C53#19465semidirect

Maximal subgroups

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

subgroupindex
#202858332
C2×#168109132
#202837502
#202742462
#202742362
#202741222
#202741122
#202735012
#202734402
#202734102
C53×#31092
#1946553