Finite Group Databasebeta

← #27276843

Subgroups — #27276843

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic53C211yesmaximal, characteristic
2#20274693C211yesmaximal, characteristic, complemented
3C2×#16810357C211yesmaximal, characteristic
4#20272943C211yesmaximal, characteristic, complemented
5#20272595C211yesmaximal, characteristic, complemented
6#20272594C211yesmaximal, characteristic, complemented
7C2×C53×#327C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9C2×D4×Dic53C411yescharacteristic, complemented
10C22×#11719919—21no—
11C22×C4×Dic53C2211yescharacteristic
12C2×#13133146—21no—
13C2×C22⋊#88322C2211yescharacteristic
14#16810526C411yescharacteristic, complemented
15#16810522C411yescharacteristic, complemented
16#16810516—21no—
17#16810503—21no—
18#16810483C2211yescharacteristic
19C4⋊C4×Dic53C411yescharacteristic, complemented
20C22⋊C4×Dic53—21no—
21C2×#13133124—21no—
22C2×#13133124C2211yescharacteristic
23#16810377—21no—
24#16810357—22no—
25C2×#121×C53C2211yescharacteristic
26C2×C22⋊C8×C53C2211yescharacteristic
27C2×#95×C53C2211yescharacteristic
28C53×#327—22no—
29C23×Dic53—21no—
30C23×Dic53—21no—
31C23×Dic53D411yescharacteristic
32C23×Dic53D411yescharacteristic
33C22×#88322—21no—
34D4×Dic53—42no—
35D4×Dic53—42no—
36D4×Dic53—42no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—24no—
41C2×#11719919—42no—
42C2×#11719919—42no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53—21no—
46C2×C22⋊Dic53—21no—
47C2×C4×Dic53—21no—
48C2×C4×Dic53—42no—
49C2×C4×Dic53C2×C411yescharacteristic
50C2×C4×Dic53C2×C411yescharacteristic
51C2×C4×Dic53D411yescharacteristic
52C2×C4×Dic53D411yescharacteristic
53#13133146—42no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59#13133124—21no—
60#13133124—21no—
61#13133124—42no—
62C22×D4×C53—21no—
63C22×D4×C53C2×C411yescharacteristic
64C23×C4×C53—21no—
65C23×C4×C53C2311yescharacteristic
66C22×C8×C53—21no—
67#121×C53—21no—
68#121×C53—21no—
69#121×C53D412yes—
70#121×C53Q812yes—
71C2×C4⋊C4×C53C2×C411yescharacteristic
72C2×C22⋊C4×C53—21no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77C24×C53—21no—
78C24×C53C2×D411yescharacteristic
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53C4⋊C412yescomplemented
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53C22×C411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×Q811yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×Dic53—21no—
106C22×Dic53—21no—
107C22×Dic53—22no—
108C22×Dic53—22no—
109C22×Dic53—22no—
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—41no—
113C22×Dic53—41no—
114C22×Dic53—42no—
115C22×Dic53—42no—
116C22×Dic53—41no—
117C22×Dic53—41no—
118C22×Dic53—41no—
119C22×Dic53—41no—
120C22×Dic53—41no—
121C22×Dic53—41no—
122C22×Dic53—42no—
123C22×Dic53C22⋊C411yescharacteristic
124C22×Dic53C22⋊C411yescharacteristic
125C22×Dic53C22⋊C411yescharacteristic
126C22×Dic53C22⋊C411yescharacteristic
127C22×Dic53D812yes—
128C22×Dic53SD1612yes—
129C2×C8×C53—41no—
130C2×C8×C53—41no—
131C2×C8×C53—41no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53C4⋊C412yescomplemented
134C22⋊C4×C53—22no—
135C22⋊C4×C53—22no—
136C22⋊Dic53—41no—
137C22⋊Dic53—41no—
138C22⋊Dic53—42no—
139C22⋊Dic53—42no—
140C22⋊Dic53—42no—
141C4⋊Dic53—42no—
142C4⋊Dic53—41no—
143C4⋊Dic53—41no—
144#11719919—48no—
145C4×Dic53—22no—
146C4×Dic53—22no—
147C4×Dic53—41no—
148C4×Dic53—41no—
149C4×Dic53—42no—
150C2×#88322—41no—
151C2×#88322—41no—
152C2×#88322—41no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—22no—
158C23×C53—41no—
159C23×C53—41no—
160C23×C53—41no—
161C23×C53—41no—
162C23×C53—41no—
163C23×C53—41no—
164C23×C53C2×C22⋊C411yescharacteristic
165C23×C53#12111yescharacteristic
166C23×C53C22⋊Q811yescharacteristic
167C23×C53C2×D811yescharacteristic
168C23×C53C2×SD1611yescharacteristic
169C23×C53AGL(1,ℤ/8)11yescharacteristic
170C23×C53#13711yescharacteristic
171D4×C53—42no—
172D4×C53—42no—
173D4×C53—42no—
174D4×C53—42no—
175D4×C53—42no—
176D4×C53—42no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—22no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—81no—
192C2×C4×C53—81no—
193C2×C4×C53—81no—
194C2×C4×C53—81no—
195C2×C4×C53C2×C4⋊C411yescharacteristic
196C2×C4×C53#11711yescharacteristic
197C2×C4×C53#12111yescharacteristic
198C2×C4×C53C22⋊Q811yescharacteristic
199C2×Dic53—24no—
200C2×Dic53—24no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—42no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—41no—
220C2×Dic53—84no—
221C2×Dic53—81no—
222C2×Dic53—81no—
223C2×Dic53—81no—
224C2×Dic53—81no—
225C2×Dic53#9912yes—
226C2×Dic53#10212yes—
227C2×Dic53#10212yes—
228C2×Dic53#10412yes—
229C8×C53—81no—
230#88322—81no—
231#8577—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—81no—
248C22×C53—81no—
249C22×C53—81no—
250C22×C53—81no—
251C22×C53#38011yescharacteristic
252C22×C53C2×#9911yescharacteristic
253C22×C53#41011yescharacteristic
254C22×C53C2×#10211yescharacteristic
255C22×C53#41711yescharacteristic
256C22×C53C2×#10411yescharacteristic
257C22×C53#42111yescharacteristic
258C22×C53#45911yescharacteristic
259C22×C53#46111yescharacteristic
260C22×C53#47411yescharacteristic
261C22×C53#47611yescharacteristic
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—42no—
267C4×C53—82no—
268Dic53—22no—
269Dic53—22no—
270Dic53—22no—
271Dic53—22no—
272Dic53—82no—
273Dic53—81no—
274Dic53—81no—
275C4×#121—531no—
276#1797—531no—
277C2×#327—531no—
278C2×#327D5311yescharacteristic
279C4⋊#95—531no—
280#1294—531no—
281#1294—531no—
282C2×C53—21no—
283C2×C53—21no—
284C2×C53—21no—
285C2×C53—21no—
286C2×C53—82no—
287C2×C53#150211yescharacteristic
288C2×C53#150411yescharacteristic
289C2×C53#151211yescharacteristic
290C2×C53#151911yescharacteristic
291C2×C53#179911yescharacteristic
292C2×C53#180211yescharacteristic
293C2×C53#180611yescharacteristic
294C2×#121C2×D5311yescharacteristic
295C2×C4×D4—531no—
296C2×C4×D4—1061no—
297C22×C4⋊C4—1061no—
298C22×C42—531no—
299C2×C4⋊C8—1061no—
300C2×C22⋊C8—531no—
301C2×C22⋊C8C2×D5311yescharacteristic
302#390—531no—
303#389—531no—
304#388—1061no—
305#386—1061no—
306#380—531no—
307C4×C4⋊C4—531no—
308C4×C22⋊C4—1061no—
309C2×#95—531no—
310C2×#95—1061no—
311C2×#95C2×D5311yescharacteristic
312#336—1061no—
313#327—22no—
314#327—1062no—
315C53#857711yescharacteristic, complemented, Sylow
316C22×D4—21no—
317C22×D4C4×D5311yescharacteristic
318C23×C4—21no—
319C23×C4—531no—
320C23×C4—531no—
321C23×C4—1061no—
322C23×C4—1061no—
323C23×C4C22×D5311yescharacteristic
324C22×C8—21no—
325C22×C8—1061no—
326#121—21no—
327#121—21no—
328#121Dic10612yes—
329#121C22⋊D5312yes—
330C4×D4—2122no—
331C4×D4—2122no—
332C4×D4—2122no—
333C2×C4⋊C4—531no—
334C2×C4⋊C4—531no—
335C2×C4⋊C4—1061no—
336C2×C4⋊C4—1064no—
337C2×C4⋊C4—2122no—
338C2×C4⋊C4—2122no—
339C2×C4⋊C4C4×D5311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1061no—
342C2×C22⋊C4—1061no—
343C2×C22⋊C4—1061no—
344C2×C22⋊C4—1061no—
345C2×C42—531no—
346C2×C42—531no—
347C2×C42—531no—
348C2×C42—531no—
349C2×C42—1061no—
350C2×C42—2122no—
351C4⋊C8—2122no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—1061no—
355C22⋊C8—1061no—
356#95—21no—
357#95—21no—
358#95—1061no—
359#95—1061no—
360#95—1061no—
361#95—1061no—
362#95—1061no—
363#95—2122no—
364C24—21no—
365C24D4×D5311yescharacteristic
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4#1171991912yescomplemented
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—531no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—531no—
390C22×C4—532no—
391C22×C4—532no—
392C22×C4—1062no—
393C22×C4—1062no—
394C22×C4—1061no—
395C22×C4—1061no—
396C22×C4—1062no—
397C22×C4—1061no—
398C22×C4—1061no—
399C22×C4—2122no—
400C22×C4—2121no—
401C22×C4—2121no—
402C22×C4—2122no—
403C22×C4—2122no—
404C22×C4—2121no—
405C22×C4—2121no—
406C22×C4—2121no—
407C22×C4—2121no—
408C22×C4—2121no—
409C22×C4—2121no—
410C22×C4C2×Dic10611yescharacteristic
411C22×C4C2×C4×D5311yescharacteristic
412C22×C4D4×D5311yescharacteristic
413C22×C4#1171994711yescharacteristic
414C22×C4#1171994711yescharacteristic
415C22×C4C2×C22⋊D5311yescharacteristic
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—2121no—
420C2×C8—2121no—
421C2×C8—2121no—
422C4⋊C4—21no—
423C4⋊C4—2121no—
424C4⋊C4—2121no—
425C4⋊C4—2122no—
426C4⋊C4—2128no—
427C4⋊C4#1171991912yescomplemented
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—2122no—
431C22⋊C4—2122no—
432C22⋊C4—2122no—
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C42—1062no—
436C42—1062no—
437C42—2121no—
438C42—2121no—
439C42—2122no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C22⋊C4×D5311yescharacteristic
452C23#1313318911yescharacteristic
453C23#1313319411yescharacteristic
454C23D8×D5311yescharacteristic
455C23SD16×D5311yescharacteristic
456C23#1313324111yescharacteristic
457C23#1313324611yescharacteristic
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—532no—
483C2×C4—532no—
484C2×C4—532no—
485C2×C4—532no—
486C2×C4—1064no—
487C2×C4—1064no—
488C2×C4—1061no—
489C2×C4—1061no—
490C2×C4—1061no—
491C2×C4—1061no—
492C2×C4—1061no—
493C2×C4—1061no—
494C2×C4—1061no—
495C2×C4—1061no—
496C2×C4—4244no—
497C2×C4—4241no—
498C2×C4—4241no—
499C2×C4—4241no—
500C2×C4—4241no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4C2×#1171991911yescharacteristic
513C2×C4#1313317311yescharacteristic
514C2×C4#1313318911yescharacteristic
515C2×C4#1313319411yescharacteristic
516C8—81no—
517C8—4241no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22#1681048111yescharacteristic
538C22#99×D5311yescharacteristic
539C22#1681059511yescharacteristic
540C22#102×D5311yescharacteristic
541C22#1681062111yescharacteristic
542C22#104×D5311yescharacteristic
543C22#1681063811yescharacteristic
544C22#1681081611yescharacteristic
545C22#1681082611yescharacteristic
546C22#1681090111yescharacteristic
547C22#1681091111yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—82no—
554C4—1062no—
555C4—1062no—
556C4—1062no—
557C4—1062no—
558C4—4242no—
559C4—4241no—
560C4—4241no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—82no—
566C2#2027320811yescharacteristic
567C2#2027321611yescharacteristic
568C2#2027325011yescharacteristic
569C2#2027328111yescharacteristic
570C2#2027470111yescharacteristic
571C2#2027471311yescharacteristic
572C2#2027473311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#327C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenterC2×#327D53
hypocenterC53#8577
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. #20274693C2semidirect
4. #20272943C2semidirect
5. #20272595C2semidirect
6. #20272594C2semidirect
9. C2×D4×Dic53C4semidirect
14. #16810526C4semidirect
15. #16810522C4semidirect
19. C4⋊C4×Dic53C4semidirect
88. C2×D4×C53C4⋊C4semidirect
133. C4⋊C4×C53C4⋊C4semidirect
315. C53#8577semidirect
375. C2×D4#11719919semidirect
427. C4⋊C4#11719919semidirect

Maximal subgroups

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

subgroupindex
#121×Dic532
#202746932
C2×#168103572
#202729432
#202725952
#202725942
C2×C53×#3272
#857753