Finite Group Databasebeta

← #27354165

Subgroups — #27354165

Order 13568.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284797C211yesmaximal, characteristic
2#20283860C211yesmaximal, characteristic
3#20283795C211yesmaximal, characteristic, complemented
4#20273592C211yesmaximal, characteristic, complemented
5#20273585C211yesmaximal, characteristic, complemented
6#20273564C211yesmaximal, characteristic, complemented
7#20273557C211yesmaximal, characteristic, complemented
8#20273533C211yesmaximal, characteristic, complemented
9#20273526C211yesmaximal, characteristic, complemented
10#20273521C211yesmaximal, characteristic, complemented
11#20273514C211yesmaximal, characteristic, complemented
12#20273143C211yesmaximal, characteristic, complemented
13#20273121C211yesmaximal, characteristic
14#20273015C211yesmaximal, characteristic, complemented
15C53×#1574C211yesmaximal, characteristic, complemented
16#16811157C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18#16810954C411yescharacteristic
19#16810937C411yescharacteristic
20#16810931C411yescharacteristic
21#16810931C411yescharacteristic
22#16810846C411yescharacteristic
23#16810846C411yescharacteristic
24#16810781C411yescharacteristic
25#16810764C411yescharacteristic
26#16810750C2211yescharacteristic
27C8×Dic106—21no—
28C8×Dic106—21no—
29#16810747C2211yescharacteristic
30#16810747C2211yescharacteristic
31#16810747C2211yescharacteristic
32#16810747C2211yescharacteristic
33#16810747C2211yescharacteristic
34C4×Dic212C2211yescharacteristic
35C4×Dic212C2211yescharacteristic
36C4×#11719913C2211yescharacteristic
37C4×#11719913C2211yescharacteristic
38#16810700C2211yescharacteristic
39#16810699—21no—
40#16810692—21no—
41#16810688C2211yescharacteristic
42#16810683—21no—
43#16810676C2211yescharacteristic
44#16810676C2211yescharacteristic
45#16810673—21no—
46#16810673—21no—
47#16810648—21no—
48#16810353C2211yescharacteristic
49#16810353C2211yescharacteristic
50#16810353C2211yescharacteristic
51#16810353C2211yescharacteristic
52#16810353C2211yescharacteristic
53#16810353C2211yescharacteristic
54#16810350C2211yescharacteristic
55#16810350C2211yescharacteristic
56#16810345C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#431C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C53×#423C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×#335C2211yescharacteristic
65C53×#334C2211yescharacteristic
66C53×#334C2211yescharacteristic
67C22×Dic106C2×C411yescharacteristic
68C2×Dic212C2×C411yescharacteristic
69C2×Dic212C2×C411yescharacteristic
70C2×#11719913C2×C411yescharacteristic
71C2×#11719913C2×C411yescharacteristic
72C2×#11719912—21no—
73C2×C8×D53—21no—
74#13133211—21no—
75#13133209C2×C411yescharacteristic
76#13133205C2×C411yescharacteristic
77#13133200—21no—
78#13133196C2×C411yescharacteristic
79#13133191C2×C411yescharacteristic
80C4×Dic106—21no—
81C4×Dic106—21no—
82C4×Dic106C2×C411yescharacteristic
83C4×Dic106C2×C411yescharacteristic
84C4×Dic106C2311yescharacteristic
85C4×Dic106C2311yescharacteristic
86C4×Dic106C2311yescharacteristic
87C4×D212C2311yescharacteristic
88C4×C22⋊D53—21no—
89#13133174—21no—
90C2×C4⋊Dic53C2×C411yescharacteristic
91C2×#11719919—21no—
92C2×C4×Dic53—21no—
93#13133151C2×C411yescharacteristic
94#13133151C2×C411yescharacteristic
95#13133149C2×C411yescharacteristic
96#13133149C2×C411yescharacteristic
97C4⋊#88322—21no—
98C4⋊#88322D411yescharacteristic
99C4⋊#88322D411yescharacteristic
100#13133146—21no—
101#13133146—21no—
102#13133146—21no—
103#13133146—21no—
104#13133146—21no—
105#13133146—21no—
106#13133142C2×C411yescharacteristic
107#13133142C2×C411yescharacteristic
108#13133142C2×C411yescharacteristic
109#13133142C2×C411yescharacteristic
110#13133142C2×C411yescharacteristic
111#13133142C2×C411yescharacteristic
112#13133139C2×C411yescharacteristic
113#13133139C2×C411yescharacteristic
114C22⋊#88322—21no—
115C22⋊#88322D411yescharacteristic
116C22⋊#88322D411yescharacteristic
117#13133129—21no—
118#13133129—21no—
119#13133127—21no—
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C4×#88322C2311yescharacteristic
124C8×Dic53—21no—
125C8×Dic53—21no—
126C8×Dic53—21no—
127C8×Dic53—21no—
128C8×Dic53—21no—
129C2×M16×C53—21no—
130C2×C42×C53C2311yescharacteristic
131C4⋊C8×C53—21no—
132C4⋊C8×C53C2×C411yescharacteristic
133C4⋊C8×C53C2×C411yescharacteristic
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138C22⋊C8×C53—21no—
139C22⋊C8×C53C2×C411yescharacteristic
140C22⋊C8×C53C2×C411yescharacteristic
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C22×C4×C53C22×C411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53#4111yescharacteristic
147C2×C22⋊D53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C2×D212C22×C411yescharacteristic
151C2×C4×D53—21no—
152C2×Dic106—21no—
153C2×Dic106—22no—
154C2×Dic106—22no—
155C2×Dic106C22×C411yescharacteristic
156C2×Dic106C22×C411yescharacteristic
157C2×Dic106C22×C411yescharacteristic
158M16×C53—81no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C22×C411yescharacteristic
164C2×C8×C53C22×C411yescharacteristic
165C2×C8×C53C22×C411yescharacteristic
166C2×C8×C53C22×C411yescharacteristic
167C42×C53C22×C411yescharacteristic
168C42×C53C2×D411yescharacteristic
169C42×C53#4111yescharacteristic
170C42×C53C2411yescharacteristic
171C22⋊Dic53—21no—
172#11719921—21no—
173#11719921—21no—
174#11719921—21no—
175#11719921—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53C22×C411yescharacteristic
178C4⋊Dic53C22×C411yescharacteristic
179C4⋊Dic53C22×C411yescharacteristic
180C4⋊Dic53C22×C411yescharacteristic
181#11719919—21no—
182#11719919—21no—
183#11719919—21no—
184#11719919—21no—
185#11719919—21no—
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322C2×D411yescharacteristic
197C2×#88322C2×D411yescharacteristic
198C2×#88322C2×D411yescharacteristic
199C2×#88322C2×D411yescharacteristic
200Dic212—22no—
201Dic212—22no—
202#11719913—22no—
203#11719913—22no—
204#11719912—81no—
205C8×D53—81no—
206C23×C53C4×D411yescharacteristic
207C22×D53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—41no—
213C2×C4×C53C4×D411yescharacteristic
214C2×C4×C53C4×D411yescharacteristic
215C2×C4×C53C4×D411yescharacteristic
216C2×C4×C53C23×C411yescharacteristic
217C2×C4×C53C22×D411yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C22⋊D53—81no—
220C2×Dic53—24no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—81no—
229D212—21no—
230D212—21no—
231C4×D53—41no—
232C4×D53—41no—
233Dic106—24no—
234Dic106—21no—
235Dic106—21no—
236Dic106—21no—
237Dic106—21no—
238Dic106#13112yes—
239Dic106#13112yes—
240C8×C53—21no—
241C8×C53—21no—
242C8×C53—21no—
243C8×C53—21no—
244C8×C53—41no—
245C8×C53—41no—
246C8×C53—41no—
247C8×C53—41no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—21no—
252#88322—41no—
253#88322—41no—
254#17919—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53C2×C4×D411yescharacteristic
259C2×D53—41no—
260C2×D53—41no—
261C2×D53—41no—
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53C2×#13111yescharacteristic
267C4×C53C2×#13111yescharacteristic
268C4×C53#57711yescharacteristic
269C4×C53#57811yescharacteristic
270Dic53—22no—
271Dic53—22no—
272Dic53—41no—
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276Dic53—41no—
277#3108—531no—
278#2993—531no—
279#2982—531no—
280#1588—531no—
281#1587—531no—
282#1584—531no—
283#1583—531no—
284#1579—531no—
285SD16⋊C8—531no—
286C8×Q16—531no—
287C8×SD16—531no—
288#1574D5311yescharacteristic, complemented
289#1487—531no—
290#1482—531no—
291#1454—531no—
292C2×C53—41no—
293C2×C53#292911yescharacteristic
294C2×C53#294711yescharacteristic
295C2×C53#295711yescharacteristic
296D53—81no—
297#529—531no—
298C2×C4×Q8—531no—
299#483—531no—
300#480—531no—
301#479—531no—
302#479—531no—
303#464—531no—
304#464—531no—
305#451—531no—
306#448—531no—
307#446—531no—
308C8×Q8—531no—
309C8×Q8—531no—
310C8×Q8—531no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—1061no—
314C8×Q8—1061no—
315C4×Q16—531no—
316C4×Q16—531no—
317C4×SD16—531no—
318C4×SD16—531no—
319#436—531no—
320#436—1061no—
321#435—1061no—
322C8×D4—531no—
323#433—1061no—
324#431—531no—
325#431—531no—
326#431—1061no—
327#431—1061no—
328#431C2×D5311yescharacteristic
329#431C2×D5311yescharacteristic
330#424—1061no—
331#423C2×D5311yescharacteristic
332#335C2×D5311yescharacteristic
333#335C2×D5311yescharacteristic
334#334C2×D5311yescharacteristic
335#334C2×D5311yescharacteristic
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342D4⋊C8—531no—
343D4⋊C8—531no—
344#322—531no—
345#322—531no—
346#322—531no—
347C82—531no—
348C53#1791911yescharacteristic, complemented, Sylow
349C22×Q8—531no—
350C2×Q16—531no—
351C2×Q16—531no—
352C2×SD16—531no—
353C2×SD16—531no—
354C2×M16—21no—
355C2×M16—1061no—
356C22×C8—1061no—
357#126—1061no—
358#125—531no—
359#124—531no—
360#123—1061no—
361C22⋊Q8—531no—
362#121—531no—
363C4×Q8—531no—
364C4×Q8—531no—
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—531no—
368C4×Q8—1061no—
369C4×Q8—1061no—
370C4×D4—531no—
371C4×D4—1061no—
372#117—1061no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—1061no—
375C2×C42—1061no—
376C2×C42C22×D5311yescharacteristic
377#107—531no—
378#107—531no—
379#106—531no—
380#106—531no—
381C4⋊C8—21no—
382C4⋊C8—531no—
383C4⋊C8—531no—
384C4⋊C8—1061no—
385C4⋊C8—1061no—
386C4⋊C8—1061no—
387C4⋊C8—1061no—
388C4⋊C8—1061no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8C4×D5311yescharacteristic
392C4⋊C8C4×D5311yescharacteristic
393C4⋊C8C22×D5311yescharacteristic
394C4⋊C8C22×D5311yescharacteristic
395C4⋊C8C22×D5311yescharacteristic
396C4⋊C8C22×D5311yescharacteristic
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#102—531no—
404#102—531no—
405C22⋊C8—21no—
406C22⋊C8—531no—
407C22⋊C8—531no—
408C22⋊C8—1061no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8C4×D5311yescharacteristic
412C22⋊C8C4×D5311yescharacteristic
413#97—1061no—
414C4×C8—21no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—1061no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8C22×D5311yescharacteristic
425C4×C8C22×D5311yescharacteristic
426C2×Q8—531no—
427C2×Q8—531no—
428C2×Q8—531no—
429C2×Q8—1062no—
430C2×Q8—1062no—
431C2×Q8—1061no—
432C2×D4—531no—
433C2×D4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4C2×C4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4#1171994911yescharacteristic
440Q16—1062no—
441Q16—1062no—
442SD16—1062no—
443SD16—1062no—
444M16—81no—
445M16—4241no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—4241no—
457C2×C8C2×C4×D5311yescharacteristic
458C2×C8C2×C4×D5311yescharacteristic
459C2×C8C2×C4×D5311yescharacteristic
460C2×C8C2×C4×D5311yescharacteristic
461C4⋊C4—531no—
462C4⋊C4—531no—
463C4⋊C4—531no—
464C4⋊C4—531no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C22⋊C4—1061no—
475C22⋊C4—1061no—
476C22⋊C4—1061no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C42—1061no—
480C42—1061no—
481C42—1061no—
482C42—1061no—
483C42—1061no—
484C42C2×C4×D5311yescharacteristic
485C42D4×D5311yescharacteristic
486C42#1171994911yescharacteristic
487C42C23×D5311yescharacteristic
488C23—1061no—
489C23#1313317911yescharacteristic
490Q8—532no—
491Q8—532no—
492Q8—1064no—
493Q8—1061no—
494Q8—1061no—
495Q8—1061no—
496Q8—1061no—
497D4—1061no—
498D4—1061no—
499D4—4241no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—1064no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—4241no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4#1313317911yescharacteristic
517C2×C4#1313317911yescharacteristic
518C2×C4#1313317911yescharacteristic
519C2×C4C22×C4×D5311yescharacteristic
520C2×C4C2×D4×D5311yescharacteristic
521C2×C4C2×#1171994911yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—1061no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—2121no—
540C22—2121no—
541C22—2121no—
542C22C2×#1313317911yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—1062no—
548C4—1062no—
549C4—2121no—
550C4—2121no—
551C4—2121no—
552C4—2121no—
553C4—2121no—
554C4#131×D5311yescharacteristic
555C4#131×D5311yescharacteristic
556C4#1681143311yescharacteristic
557C4#1681143911yescharacteristic
558C2—41no—
559C2—4241no—
560C2#2028340811yescharacteristic
561C2#2028355611yescharacteristic
562C2#2028363111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1574C2
socleC22×C53C2×C4×D4
prime radical#13133191C2×C4
cosocleC42×C53C24
hypercenter#1574D53
hypocenterC53#17919
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
3. #20283795C2semidirect
4. #20273592C2semidirect
5. #20273585C2semidirect
6. #20273564C2semidirect
7. #20273557C2semidirect
8. #20273533C2semidirect
9. #20273526C2semidirect
10. #20273521C2semidirect
11. #20273514C2semidirect
12. #20273143C2semidirect
14. #20273015C2semidirect
15. C53×#1574C2semidirect
288. #1574D53semidirect
348. C53#17919semidirect

Maximal subgroups

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

subgroupindex
#202847972
#202838602
#202837952
#202735922
#202735852
#202735642
#202735572
#202735332
#202735262
#202735212
#202735142
#202731432
#202731212
#202730152
C53×#15742
#1791953