Finite Group Databasebeta

← #27355480

Subgroups — #27355480

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284754C211yesmaximal, characteristic
2#20283816C211yesmaximal, characteristic
3#20283804C211yesmaximal, characteristic
4#20273621C211yesmaximal, characteristic, complemented
5#20273609C211yesmaximal, characteristic, complemented
6#20273589C211yesmaximal, characteristic, complemented
7#20273583C211yesmaximal, characteristic, complemented
8#20273568C211yesmaximal, characteristic, complemented
9#20273562C211yesmaximal, characteristic, complemented
10#20273531C211yesmaximal, characteristic, complemented
11#20273519C211yesmaximal, characteristic, complemented
12#20273493C211yesmaximal, characteristic
13#20273189C211yesmaximal, characteristic
14#20273015C211yesmaximal, characteristic
15C53×#1479C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16810958C411yescharacteristic
18#16810958C411yescharacteristic
19#16810918C411yescharacteristic
20#16810918C411yescharacteristic
21#16810833C411yescharacteristic
22#16810833C411yescharacteristic
23#16810779C411yescharacteristic
24#16810779C411yescharacteristic
25#16810754—21no—
26#16810754—21no—
27#16810754C2211yescharacteristic
28#16810754C2211yescharacteristic
29#16810753—21no—
30#16810750C2211yescharacteristic
31#16810750C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33Q8×#88322C2211yescharacteristic
34C8×Dic106—21no—
35#16810747C2211yescharacteristic
36#16810747C2211yescharacteristic
37C4×#11719925C2211yescharacteristic
38C4×#11719925C2211yescharacteristic
39C4×#11719925C2211yescharacteristic
40C4×#11719925C2211yescharacteristic
41#16810681—21no—
42#16810681—21no—
43#16810680—21no—
44#16810680C2211yescharacteristic
45#16810676—21no—
46#16810676C2211yescharacteristic
47#16810676C2211yescharacteristic
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810373C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810355C2211yescharacteristic
54#16810355C2211yescharacteristic
55#16810355C2211yescharacteristic
56#16810345C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C2×C4×Q8×C53C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C53×#431C2211yescharacteristic
63C53×Q8⋊C8C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×Q8⋊C8C2211yescharacteristic
67C2×#11719925C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719925C2×C411yescharacteristic
70C2×#11719925C2×C411yescharacteristic
71#13133217C2×C411yescharacteristic
72#13133217C2×C411yescharacteristic
73#13133209—21no—
74#13133196—21no—
75#13133196C2×C411yescharacteristic
76#13133196C2×C411yescharacteristic
77C4×Dic106—21no—
78C4×Dic106C2311yescharacteristic
79C4×Dic106C2311yescharacteristic
80#13133176—21no—
81#13133150C2×C411yescharacteristic
82#13133150C2×C411yescharacteristic
83#13133150C2×C411yescharacteristic
84#13133150C2×C411yescharacteristic
85C4⋊#88322—21no—
86C4⋊#88322—21no—
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322C2×C411yescharacteristic
90C4⋊#88322C2×C411yescharacteristic
91C4⋊#88322D411yescharacteristic
92C4⋊#88322D411yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133146—21no—
96#13133146—21no—
97#13133146—21no—
98#13133146—21no—
99#13133141C2×C411yescharacteristic
100#13133141C2×C411yescharacteristic
101#13133141C2×C411yescharacteristic
102#13133141C2×C411yescharacteristic
103#13133140C2×C411yescharacteristic
104#13133140C2×C411yescharacteristic
105#13133140C2×C411yescharacteristic
106#13133140C2×C411yescharacteristic
107C22⋊#88322—21no—
108C22⋊#88322C2×C411yescharacteristic
109C22⋊#88322C2×C411yescharacteristic
110C22⋊#88322D411yescharacteristic
111C22⋊#88322D411yescharacteristic
112#13133128—21no—
113#13133128—21no—
114#13133128—21no—
115#13133127—21no—
116#13133127—21no—
117C4×#88322—21no—
118C4×#88322—21no—
119C4×#88322C2311yescharacteristic
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C8×Dic53—21no—
124C8×Dic53—21no—
125C22×Q8×C53C2×C411yescharacteristic
126C4×Q8×C53—21no—
127C4×Q8×C53—21no—
128C4×Q8×C53C2×C411yescharacteristic
129C4×Q8×C53C2×C411yescharacteristic
130C4×Q8×C53C2311yescharacteristic
131C4×Q8×C53C2311yescharacteristic
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53C2×C411yescharacteristic
134C2×C42×C53—21no—
135C2×C42×C53C2311yescharacteristic
136C4⋊C8×C53—21no—
137C4⋊C8×C53C2311yescharacteristic
138C4⋊C8×C53C2311yescharacteristic
139C22⋊C8×C53—21no—
140C22⋊C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53—21no—
145C2×Q8×C53—22no—
146C2×Q8×C53—41no—
147C2×Q8×C53C22×C411yescharacteristic
148C2×Q8×C53C22×C411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53#4111yescharacteristic
154C2×Dic106—21no—
155C2×Dic106C22×C411yescharacteristic
156C2×Dic106C22×C411yescharacteristic
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53C22×C411yescharacteristic
167C4⋊C4×C53C22×C411yescharacteristic
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53C22×C411yescharacteristic
173C42×C53C2×D411yescharacteristic
174C42×C53#4111yescharacteristic
175C42×C53C2411yescharacteristic
176C22⋊Dic53—21no—
177C22⋊Dic53—21no—
178#11719925—22no—
179#11719925—22no—
180#11719925—41no—
181#11719925—41no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53C22×C411yescharacteristic
186C4⋊Dic53C22×C411yescharacteristic
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322C22×C411yescharacteristic
196C2×#88322C22×C411yescharacteristic
197C2×#88322C22×C411yescharacteristic
198C2×#88322C22×C411yescharacteristic
199C2×#88322C2×D411yescharacteristic
200C2×#88322C2×D411yescharacteristic
201C2×#88322C2×D411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C23×C53C4×D411yescharacteristic
204Q8×C53—21no—
205Q8×C53—21no—
206Q8×C53—21no—
207Q8×C53—42no—
208Q8×C53#13112yes—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53C4×D411yescharacteristic
221C2×C4×C53C4×D411yescharacteristic
222C2×C4×C53C4×D411yescharacteristic
223C2×C4×C53C23×C411yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230Dic106—21no—
231Dic106—21no—
232Dic106—21no—
233Dic106—21no—
234Dic106—42no—
235C8×C53—41no—
236C8×C53—41no—
237C8×C53—41no—
238C8×C53—41no—
239#88322—21no—
240#88322—21no—
241#88322—21no—
242#88322—21no—
243#88322—21no—
244#88322—21no—
245#88322—21no—
246#88322—21no—
247#88322—41no—
248#88322—41no—
249#18032—531nomaximal, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53C2×C4×D411yescharacteristic
254C4×C53—21no—
255C4×C53—22no—
256C4×C53—21no—
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—41no—
260C4×C53—41no—
261C4×C53—41no—
262C4×C53C2×#13111yescharacteristic
263C4×C53#56811yescharacteristic
264C4×C53#57811yescharacteristic
265C4×C53#57811yescharacteristic
266Dic53—41no—
267Dic53—41no—
268Dic53—41no—
269Dic53—41no—
270#3104—531no—
271#2986—531no—
272#2984—531no—
273#1593—531no—
274#1591—531no—
275#1587—531no—
276#1587—531no—
277#1584—531no—
278#1584—531no—
279#1579—531no—
280C8×Q16—531no—
281#1572—531no—
282#1497—531no—
283#1479D5311yescharacteristic
284#1454—531no—
285C2×C53—41no—
286C2×C53#293311yescharacteristic
287C2×C53#295011yescharacteristic
288C2×C53#295511yescharacteristic
289#533—531no—
290C2×C4×Q8C2×D5311yescharacteristic
291#484—531no—
292#484—531no—
293#477—531no—
294#477—531no—
295#462—531no—
296#462—531no—
297#451—531no—
298#451—531no—
299#446—531no—
300#446—531no—
301#446—531no—
302#446—531no—
303#446—1061no—
304#446—1061no—
305#446—1061no—
306C8×Q8—531no—
307C8×Q8—531no—
308C8×Q8—531no—
309C8×Q8—531no—
310C8×Q8—1061no—
311C4×Q16—531no—
312C4×Q16—531no—
313C4×Q16—531no—
314C4×Q16—531no—
315#433—1061no—
316#433—1061no—
317#432—531no—
318#432—1061no—
319#431—531no—
320#431—531no—
321#431—1061no—
322#431C2×D5311yescharacteristic
323#431C2×D5311yescharacteristic
324#335—531no—
325#335—531no—
326#334—531no—
327#334—531no—
328Q8⋊C8—531no—
329Q8⋊C8—531no—
330Q8⋊C8—531no—
331Q8⋊C8—531no—
332Q8⋊C8C2×D5311yescharacteristic
333Q8⋊C8C2×D5311yescharacteristic
334Q8⋊C8C2×D5311yescharacteristic
335Q8⋊C8C2×D5311yescharacteristic
336#322—531no—
337#322—531no—
338#322—531no—
339C82—531no—
340C53#1803211yescharacteristic, complemented, Sylow
341C22×Q8C4×D5311yescharacteristic
342C2×Q16—531no—
343C2×Q16—531no—
344C2×Q16—531no—
345C2×Q16—531no—
346C4⋊Q8—531no—
347C4⋊Q8—531no—
348#125—1061no—
349C22⋊Q8—531no—
350C22⋊Q8—531no—
351C22⋊Q8—1061no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—531no—
355C4×Q8—531no—
356C4×Q8—1061no—
357C4×Q8C4×D5311yescharacteristic
358C4×Q8C4×D5311yescharacteristic
359C4×Q8C22×D5311yescharacteristic
360C4×Q8C22×D5311yescharacteristic
361#117—1061no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C4×D5311yescharacteristic
364C2×C42—21no—
365C2×C42C22×D5311yescharacteristic
366#107—531no—
367#107—531no—
368#107—531no—
369#107—531no—
370C4⋊C8—21no—
371C4⋊C8—531no—
372C4⋊C8—531no—
373C4⋊C8—531no—
374C4⋊C8—531no—
375C4⋊C8—531no—
376C4⋊C8—531no—
377C4⋊C8—1061no—
378C4⋊C8—1061no—
379C4⋊C8—1061no—
380C4⋊C8—1061no—
381C4⋊C8—1061no—
382C4⋊C8—1061no—
383C4⋊C8—1061no—
384C4⋊C8—1061no—
385C4⋊C8C22×D5311yescharacteristic
386C4⋊C8C22×D5311yescharacteristic
387#103—531no—
388#103—531no—
389#103—531no—
390#103—531no—
391#103—531no—
392#103—531no—
393#103—531no—
394#103—531no—
395C22⋊C8—21no—
396C22⋊C8—21no—
397C22⋊C8—531no—
398C22⋊C8—531no—
399C22⋊C8—531no—
400C22⋊C8—531no—
401C22⋊C8—1061no—
402#97—1061no—
403#97—1061no—
404#97—1061no—
405#97—1061no—
406#97—1061no—
407C4×C8—21no—
408C4×C8—531no—
409C4×C8—531no—
410C4×C8—531no—
411C4×C8—531no—
412C4×C8—1061no—
413C4×C8—1061no—
414C4×C8—1061no—
415C4×C8—1061no—
416C4×C8C22×D5311yescharacteristic
417C4×C8C22×D5311yescharacteristic
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8—531no—
422C2×Q8—531no—
423C2×Q8—1061no—
424C2×Q8C2×C4×D5311yescharacteristic
425C2×Q8C2×C4×D5311yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4C2×C4×D5311yescharacteristic
429C22×C4D4×D5311yescharacteristic
430C22×C4#1171994711yescharacteristic
431Q16—1062no—
432Q16—1062no—
433Q16—2121no—
434Q16—2121no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—531no—
440C2×C8—531no—
441C2×C8—531no—
442C2×C8—531no—
443C2×C8—531no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—1061no—
448C2×C8—1061no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—531no—
455C4⋊C4—531no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4C2×C4×D5311yescharacteristic
464C4⋊C4C2×C4×D5311yescharacteristic
465C22⋊C4—1061no—
466C22⋊C4—1061no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—1061no—
472C42—1061no—
473C42C2×C4×D5311yescharacteristic
474C42D4×D5311yescharacteristic
475C42#1171994711yescharacteristic
476C42C23×D5311yescharacteristic
477C23#1313317711yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—42no—
482Q8—1061no—
483Q8—1061no—
484Q8—1061no—
485Q8—1061no—
486Q8—2122no—
487Q8#1313322312yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4#1313317711yescharacteristic
504C2×C4#1313317711yescharacteristic
505C2×C4#1313317711yescharacteristic
506C2×C4C22×C4×D5311yescharacteristic
507C2×C4C2×D4×D5311yescharacteristic
508C2×C4C2×#1171994711yescharacteristic
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—1061no—
514C8—1061no—
515C8—1061no—
516C8—1061no—
517C8—1061no—
518C8—1061no—
519C8—1061no—
520C8—1061no—
521C8—2121no—
522C8—2121no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×#1313317711yescharacteristic
527C4—21no—
528C4—22no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—2121no—
536C4—2121no—
537C4—2121no—
538C4—2121no—
539C4C2×#1313322311yescharacteristic
540C4#1681138911yescharacteristic
541C4#1681143911yescharacteristic
542C4#1681143911yescharacteristic
543C2—41no—
544C2#2028343611yescharacteristic
545C2#2028357911yescharacteristic
546C2#2028361411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1479C2
socleC22×C53C2×C4×D4
prime radicalC23×C53C4×D4
cosocleC42×C53C24
hypercenter#1479D53
hypocenterC53#18032
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
4. #20273621C2semidirect
5. #20273609C2semidirect
6. #20273589C2semidirect
7. #20273583C2semidirect
8. #20273568C2semidirect
9. #20273562C2semidirect
10. #20273531C2semidirect
11. #20273519C2semidirect
340. C53#18032semidirect

Maximal subgroups

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

subgroupindex
#202847542
#202838162
#202838042
#202736212
#202736092
#202735892
#202735832
#202735682
#202735622
#202735312
#202735192
#202734932
#202731892
#202730152
C53×#14792
#1803253