Finite Group Databasebeta

← #27354201

Subgroups — #27354201

Order 13568.

598 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#20284787C211yesmaximal, characteristic
2#20283823C211yesmaximal, characteristic
3#20283809C211yesmaximal, characteristic, complemented
4#20273588C211yesmaximal, characteristic, complemented
5#20273574C211yesmaximal, characteristic, complemented
6#20273567C211yesmaximal, characteristic, complemented
7#20273553C211yesmaximal, characteristic, complemented
8#20273535C211yesmaximal, characteristic, complemented
9#20273529C211yesmaximal, characteristic, complemented
10C8×Dic212C211yesmaximal, characteristic, complemented
11C8×#11719913C211yesmaximal, characteristic, complemented
12#20273500C211yesmaximal, characteristic, complemented
13#20273150C211yesmaximal, characteristic
14#20273119C211yesmaximal, characteristic, complemented
15C53×#1454C211yesmaximal, characteristic, complemented
16#16811180C2211yescharacteristic
17C4×#11719945C2211yescharacteristic
18#16810965C411yescharacteristic
19#16810965C411yescharacteristic
20#16810909C411yescharacteristic
21#16810904C411yescharacteristic
22#16810834C411yescharacteristic
23#16810829C411yescharacteristic
24#16810787C411yescharacteristic
25#16810787C411yescharacteristic
26#16810753C2211yescharacteristic
27#16810753C2211yescharacteristic
28#16810752—21no—
29#16810752—21no—
30C8×Dic106C2211yescharacteristic
31C8×Dic106C2211yescharacteristic
32C8×Dic106C2211yescharacteristic
33C8×Dic106C2211yescharacteristic
34C4×Dic212C2211yescharacteristic
35C4×Dic212C2211yescharacteristic
36C4×#11719913C2211yescharacteristic
37C4×#11719913C2211yescharacteristic
38C8×D212C2211yescharacteristic
39C8×D212C2211yescharacteristic
40#16810686—21no—
41#16810686—21no—
42#16810681—21no—
43#16810681—21no—
44#16810680C2211yescharacteristic
45#16810675—21no—
46C2×C4⋊#88322C2211yescharacteristic
47C4⋊C8×D53—21no—
48#16810376C2211yescharacteristic
49#16810376C2211yescharacteristic
50#16810374C2211yescharacteristic
51#16810374C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810355C2211yescharacteristic
54#16810355C2211yescharacteristic
55#16810355C2211yescharacteristic
56#16810355C2211yescharacteristic
57#16810355C2211yescharacteristic
58#16810352C2211yescharacteristic
59#16810352C2211yescharacteristic
60C53×#431C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C53×#431C2211yescharacteristic
63C53×#322C2211yescharacteristic
64C53×#322C2211yescharacteristic
65C53×#322C2211yescharacteristic
66C82×C53C2211yescharacteristic
67C2×#11719945C2×C411yescharacteristic
68C2×Dic212C2×C411yescharacteristic
69C2×Dic212C2×C411yescharacteristic
70C2×#11719913C2×C411yescharacteristic
71C2×#11719913C2×C411yescharacteristic
72C22×#88322—21no—
73C2×C8×D53—21no—
74C2×C8×D53—21no—
75#13133217C2×C411yescharacteristic
76#13133217C2×C411yescharacteristic
77#13133209—21no—
78#13133196—21no—
79#13133196C2×C411yescharacteristic
80#13133196C2×C411yescharacteristic
81C4×Dic106—21no—
82C4×Dic106C2×C411yescharacteristic
83C4×Dic106C2311yescharacteristic
84C4×Dic106C2311yescharacteristic
85C4×Dic106C2311yescharacteristic
86C4×D212C2×C411yescharacteristic
87C4×D212C2311yescharacteristic
88C4×C22⋊D53—21no—
89C4×C22⋊D53—21no—
90#13133176—21no—
91#13133176C2×C411yescharacteristic
92#13133174—21no—
93C42×D53—21no—
94#13133151C2×C411yescharacteristic
95#13133151C2×C411yescharacteristic
96#13133149C2×C411yescharacteristic
97#13133149C2×C411yescharacteristic
98C4⋊#88322—21no—
99C4⋊#88322C2311yescharacteristic
100C4⋊#88322C2311yescharacteristic
101C4⋊#88322C2311yescharacteristic
102C4⋊#88322C2311yescharacteristic
103#13133146—21no—
104#13133146—21no—
105#13133146—21no—
106#13133146—21no—
107#13133146—21no—
108#13133146—21no—
109#13133142C2×C411yescharacteristic
110#13133142C2×C411yescharacteristic
111#13133142C2×C411yescharacteristic
112#13133142C2×C411yescharacteristic
113#13133142C2×C411yescharacteristic
114#13133142C2×C411yescharacteristic
115#13133139C2×C411yescharacteristic
116#13133139C2×C411yescharacteristic
117C22⋊#88322—21no—
118#13133129—21no—
119#13133129—21no—
120#13133128—21no—
121#13133127—21no—
122#13133127—21no—
123C8×Dic53—21no—
124C8×Dic53—21no—
125C8×Dic53—21no—
126C8×Dic53—21no—
127C2×C42×C53C2311yescharacteristic
128C4⋊C8×C53—21no—
129C4⋊C8×C53C2×C411yescharacteristic
130C4⋊C8×C53C2×C411yescharacteristic
131C4⋊C8×C53D411yescharacteristic
132C4⋊C8×C53D411yescharacteristic
133C22⋊C8×C53—21no—
134C22⋊C8×C53C2×C411yescharacteristic
135C22⋊C8×C53C2×C411yescharacteristic
136C22⋊C8×C53D411yescharacteristic
137C22⋊C8×C53D411yescharacteristic
138C4×C8×C53C2311yescharacteristic
139C4×C8×C53C2311yescharacteristic
140C4×C8×C53C2311yescharacteristic
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C22×C4×C53C22×C411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53#4111yescharacteristic
147C2×C22⋊D53—21no—
148#11719945—22no—
149#11719945—22no—
150C2×D212C22×C411yescharacteristic
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×Dic106—21no—
155C2×Dic106C22×C411yescharacteristic
156C2×Dic106C22×C411yescharacteristic
157C2×Dic106C22×C411yescharacteristic
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53C22×C411yescharacteristic
161C2×C8×C53C22×C411yescharacteristic
162C2×C8×C53C22×C411yescharacteristic
163C2×C8×C53C22×C411yescharacteristic
164C2×C8×C53C2×D411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C2×C8×C53C2×D411yescharacteristic
168C42×C53C22×C411yescharacteristic
169C42×C53C2×D411yescharacteristic
170C42×C53#4111yescharacteristic
171C42×C53C2411yescharacteristic
172C22⋊Dic53—21no—
173C22⋊Dic53—21no—
174C22⋊Dic53—21no—
175#11719921—21no—
176#11719921—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C22×C411yescharacteristic
181C4⋊Dic53C22×C411yescharacteristic
182C4⋊Dic53C22×C411yescharacteristic
183C4⋊Dic53C22×C411yescharacteristic
184#11719919—21no—
185#11719919—21no—
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—81no—
200Dic212—22no—
201Dic212—22no—
202#11719913—22no—
203#11719913—22no—
204C8×D53—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—24no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227D212—21no—
228D212#13112yes—
229C4×D53—24no—
230C4×D53—41no—
231C4×D53—41no—
232C4×D53—81no—
233Dic106—21no—
234Dic106—21no—
235Dic106—21no—
236Dic106—21no—
237Dic106—21no—
238Dic106—81no—
239Dic106#13112yes—
240C8×C53—21no—
241C8×C53—21no—
242C8×C53—21no—
243C8×C53—21no—
244C8×C53—21no—
245C8×C53—21no—
246C8×C53—21no—
247C8×C53—21no—
248C8×C53—41no—
249C8×C53—41no—
250#88322—41no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#17921—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53C2×C4×D411yescharacteristic
259C2×D53—22no—
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—41no—
272Dic53—41no—
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276Dic53—41no—
277#3107—531no—
278#2988—531no—
279#2985—531no—
280#1587—531no—
281#1585—531no—
282#1584—531no—
283#1582—531no—
284#1579—531no—
285SD16⋊C8—531no—
286C8×Q16—531no—
287C8×SD16—531no—
288#1573—531no—
289#1488—531no—
290#1481—531no—
291#1454D5311yescharacteristic, complemented
292C2×C53—41no—
293C2×C53#292911yescharacteristic
294C2×C53#294711yescharacteristic
295C2×C53#295711yescharacteristic
296D53—42no—
297#533—531no—
298C4×#41—531no—
299#485—531no—
300#485—531no—
301#475—531no—
302#474—531no—
303#462—531no—
304#461—531no—
305#452—531no—
306#452—531no—
307#446—531no—
308#446—531no—
309#446—1061no—
310#446—1061no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—531no—
315C4×Q16—531no—
316C4×Q16—531no—
317C4×SD16—531no—
318C4×SD16—531no—
319C8×D4—531no—
320C8×D4—531no—
321C8×D4—1061no—
322C8×D4—1061no—
323#433—1061no—
324#433—1061no—
325#432—531no—
326#431—1061no—
327#431C2×D5311yescharacteristic
328#431C2×D5311yescharacteristic
329#431C2×D5311yescharacteristic
330C2×C4⋊C8—531no—
331C2×C4⋊C8—1061no—
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
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#322C2×D5311yescharacteristic
345#322C2×D5311yescharacteristic
346#322C2×D5311yescharacteristic
347C82C2×D5311yescharacteristic
348C53#1792111yescharacteristic, complemented, Sylow
349C2×#41—531no—
350C2×Q16—531no—
351C2×Q16—531no—
352C2×SD16—531no—
353C2×SD16—531no—
354C22×C8—1061no—
355C22×C8—1061no—
356C22×C8—1061no—
357C4⋊Q8—531no—
358C4⋊Q8—531no—
359#125—1061no—
360C22⋊Q8—531no—
361C22⋊Q8—531no—
362C22⋊Q8—1061no—
363C4×Q8—531no—
364C4×Q8—531no—
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—1061no—
368C4×D4—531no—
369C4×D4—531no—
370C4×D4—1061no—
371C4×D4—1061no—
372#117—531no—
373#117—1061no—
374#117—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—531no—
385C4⋊C8—531no—
386C4⋊C8—1061no—
387C4⋊C8—1061no—
388C4⋊C8—1061no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8C4×D5311yescharacteristic
394C4⋊C8C4×D5311yescharacteristic
395C4⋊C8D21211yescharacteristic
396C4⋊C8D21211yescharacteristic
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—1061no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409C22⋊C8C4×D5311yescharacteristic
410C22⋊C8C4×D5311yescharacteristic
411C22⋊C8D21211yescharacteristic
412C22⋊C8D21211yescharacteristic
413#97—1061no—
414#97—1061no—
415#97—1061no—
416C4×C8—1061no—
417C4×C8—1061no—
418C4×C8—1061no—
419C4×C8—1061no—
420C4×C8C22×D5311yescharacteristic
421C4×C8C22×D5311yescharacteristic
422C4×C8C22×D5311yescharacteristic
423C4×C8C22×D5311yescharacteristic
424C4×C8C22×D5311yescharacteristic
425C4×C8C22×D5311yescharacteristic
426#41—1062no—
427#41—1062no—
428C2×Q8—531no—
429C2×Q8—531no—
430C2×Q8—531no—
431C2×Q8—1061no—
432C2×D4—531no—
433C2×D4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4C2×C4×D5311yescharacteristic
438C22×C4C2×D21211yescharacteristic
439C22×C4#1171994511yescharacteristic
440Q16—1062no—
441Q16—1062no—
442SD16—1062no—
443SD16—1062no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—4241no—
451C2×C8—4241no—
452C2×C8—4241no—
453C2×C8C2×C4×D5311yescharacteristic
454C2×C8C2×C4×D5311yescharacteristic
455C2×C8C2×C4×D5311yescharacteristic
456C2×C8C2×C4×D5311yescharacteristic
457C2×C8C2×D21211yescharacteristic
458C2×C8C2×D21211yescharacteristic
459C2×C8C2×D21211yescharacteristic
460C2×C8C2×D21211yescharacteristic
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
485C42C2×D21211yescharacteristic
486C42#1171994511yescharacteristic
487C42C23×D5311yescharacteristic
488C23—1061no—
489C23C4×D21211yescharacteristic
490Q8—532no—
491Q8—1061no—
492Q8—1061no—
493Q8—1061no—
494Q8—1061no—
495Q8—1061no—
496Q8—4241no—
497D4—532no—
498D4—1064no—
499D4—1061no—
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×C4C4×D21211yescharacteristic
517C2×C4C4×D21211yescharacteristic
518C2×C4C4×D21211yescharacteristic
519C2×C4C22×C4×D5311yescharacteristic
520C2×C4C22×D21211yescharacteristic
521C2×C4C2×#1171994511yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—41no—
531C8—41no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—1062no—
540C22—2121no—
541C22—2121no—
542C22C2×C4×D21211yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—1062no—
548C4—2121no—
549C4—2121no—
550C4—2121no—
551C4—2121no—
552C4—2121no—
553C4—2121no—
554C4#131×D5311yescharacteristic
555C4#131×D5311yescharacteristic
556C4#1681143511yescharacteristic
557C4#1681144011yescharacteristic
558C2—41no—
559C2—2122no—
560C2#2028341111yescharacteristic
561C2#2028356011yescharacteristic
562C2#2028363311yescharacteristic
subgroupG/H
centerC2×C4C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1454C2
socleC22×C53C2×C4×D4
prime radicalC2×#11719945C2×C4
cosocleC42×C53C24
hypercenter#1454D53
hypocenterC53#17921
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
3. #20283809C2semidirect
4. #20273588C2semidirect
5. #20273574C2semidirect
6. #20273567C2semidirect
7. #20273553C2semidirect
8. #20273535C2semidirect
9. #20273529C2semidirect
10. C8×Dic212C2semidirect
11. C8×#11719913C2semidirect
12. #20273500C2semidirect
14. #20273119C2semidirect
15. C53×#1454C2semidirect
291. #1454D53semidirect
348. C53#17921semidirect

Maximal subgroups

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

subgroupindex
#202847872
#202838232
#202838092
#202735882
#202735742
#202735672
#202735532
#202735352
#202735292
C8×Dic2122
C8×#117199132
#202735002
#202731502
#202731192
C53×#14542
#1792153