Finite Group Databasebeta

← #27412248

Subgroups — #27412248

Order 13568.

1240 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719924C212yesmaximal, complemented
2C4×#13133191C211yesmaximal, characteristic
3#20276045C211yesmaximal, characteristic, complemented
4#20275958C211yesmaximal, characteristic, complemented
5#20275908C212yesmaximal, complemented
6#20275848C211yesmaximal, characteristic, complemented
7#20275798C211yesmaximal, characteristic, complemented
8#20275716C212yesmaximal
9C4×#13133141C212yesmaximal, complemented
10C4×C22⋊#88322C211yesmaximal, characteristic, complemented
11C2×C53×#533C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C2×#13133191C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic
15C2×C4×C22⋊D53—21no—
16C4×#11719924—28no—
17C2×#13133148C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133137C2212yes—
22C2×C22⋊#88322C2212yes—
23C2×C4×#88322C2212yes—
24#16810525C2211yescharacteristic
25#16810521C2211yescharacteristic
26#16810514—21no—
27#16810502—21no—
28#16810483C2211yescharacteristic
29C4×C4⋊Dic53C2211yescharacteristic
30C4×#11719921—21no—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2211yescharacteristic
34#16810378C2211yescharacteristic
35C53×#533—28no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#125×C53—21no—
39C2×C22⋊Q8×C53—21no—
40C2×C22⋊Q8×C53C2212yes—
41C2×C4×Q8×C53—21no—
42C2×C4×Q8×C53C2212yes—
43C2×#117×C53—21no—
44C22×C42×C53C2211yescharacteristic
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C22×C4×D53—21no—
48C2×#11719924—24no—
49C2×#11719924—24no—
50C2×#11719924—24no—
51C22×#88322C2312yes—
52C22×#88322C2312yes—
53#13133191—22no—
54#13133191—22no—
55C4×D212—24no—
56C4×C22⋊D53—44no—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2311yescharacteristic
60C2×#11719919—21no—
61C2×C22⋊Dic53—21no—
62C2×#11719921—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66C2×C4×Dic53—21no—
67#13133148—24no—
68#13133141—24no—
69#13133141—24no—
70#13133141—24no—
71#13133137—24no—
72C22⋊#88322—24no—
73C4×#88322—24no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77C22×Q8×C53—21no—
78C22×Q8×C53C2312yes—
79C23×C4×C53D411yescharacteristic
80C23×C4×C53D411yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C4⋊Q8×C53—24no—
83C4⋊Q8×C53—24no—
84#125×C53—28no—
85C22⋊Q8×C53—28no—
86C22⋊Q8×C53—24no—
87C22⋊Q8×C53—24no—
88C4×Q8×C53—28no—
89C4×Q8×C53—24no—
90C4×Q8×C53D418yescomplemented
91#117×C53—28no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53C2312yes—
98C2×C22⋊C4×C53—22no—
99C2×C42×C53—24no—
100C2×C42×C53—22no—
101C2×C42×C53D411yescharacteristic
102C2×C42×C53D411yescharacteristic
103C2×C42×C53C2311yescharacteristic
104C2×C42×C53C2311yescharacteristic
105C24×C53C2×D411yescharacteristic
106C23×D53—21no—
107C2×Q8×C53—24no—
108C2×Q8×C53—22no—
109C2×Q8×C53—22no—
110C2×Q8×C53—22no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—42no—
114C2×Q8×C53—42no—
115C2×Q8×C53C2×D414yes—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—41no—
127C22×C4×C53SD1614yescomplemented
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C2×C22⋊D53—24no—
135C2×C22⋊D53—42no—
136C2×C22⋊D53—42no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C2×D212—22no—
141C2×D212—22no—
142C2×D212—21no—
143C2×D212#4112yes—
144C2×C4×D53—42no—
145C2×C4×D53—42no—
146C2×C4×D53—42no—
147C4⋊C4×C53—24no—
148C4⋊C4×C53—24no—
149C4⋊C4×C53—28no—
150C4⋊C4×C53—24no—
151C4⋊C4×C53—28no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53C2×D414yes—
154C22⋊C4×C53—28no—
155C42×C53—22no—
156C42×C53—22no—
157C42×C53—28no—
158C42×C53—22no—
159C42×C53C2×D414yes—
160C22⋊Dic53—42no—
161#11719924—216no—
162#11719921—24no—
163#11719921—42no—
164#11719921—42no—
165#11719921—42no—
166C4⋊Dic53—21no—
167C4⋊Dic53—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53#4112yes—
170#11719919—42no—
171C4×Dic53—42no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322#4114yes—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53C2×SD1612yes—
185C23×C53C22×D411yescharacteristic
186C22×D53—22no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—41no—
192C22×D53—41no—
193Q8×C53—24no—
194Q8×C53—24no—
195Q8×C53—48no—
196Q8×C53—42no—
197Q8×C53—42no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—28no—
205C2×C4×C53—21no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—44no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53#12012yes—
218C2×C4×C53#12012yes—
219C2×C4×C53C2×SD1612yes—
220C2×C4×C53C2×SD1612yes—
221C2×C4×C53#13512yes—
222C2×C4×C53#13712yes—
223C2×C4×C53C22×D412yes—
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4112yes—
226C2×C4×C53C2×#4111yescharacteristic
227C22⋊D53—48no—
228C2×Dic53—22no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237D212—24no—
238D212—42no—
239C4×D53—84no—
240#88322—24no—
241#88322—42no—
242#23568—531nomaximal, Sylow
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53C2×#12011yescharacteristic
251C22×C53#52511yescharacteristic
252C22×C53#53211yescharacteristic
253C22×C53C22×SD1611yescharacteristic
254C22×C53C2×#13511yescharacteristic
255C22×C53C2×#13711yescharacteristic
256C22×C53#57511yescharacteristic
257C2×D53—42no—
258C2×D53—42no—
259C2×D53—42no—
260C2×D53—81no—
261C2×D53—81no—
262C2×D53—81no—
263C2×D53—81no—
264C4×C53—22no—
265C4×C53—22no—
266C4×C53—22no—
267C4×C53—41no—
268C4×C53—44no—
269C4×C53—44no—
270C4×C53#44814yes—
271C4×C53#52512yes—
272C4×C53#53212yes—
273Dic53—42no—
274Dic53—81no—
275Dic53—81no—
276C2×#533D5311yescharacteristic, complemented
277C2×C4×SD16—532no—
278C4×#121—531no—
279#2058—531no—
280#2043—531no—
281#2035—532no—
282#2023—531no—
283#2013—531no—
284#2000—532no—
285C4×#103—532no—
286C4×C22⋊C8—531no—
287C2×C53—42no—
288C2×C53#238811yescharacteristic
289C2×C53C2×#44811yescharacteristic
290C2×C53#300211yescharacteristic
291C2×C53#310111yescharacteristic
292C2×C53#310711yescharacteristic
293C2×C53#316711yescharacteristic
294C2×C53#317211yescharacteristic
295D53—82no—
296C22×SD16—532no—
297#533—28no—
298C2×C4⋊Q8C2×D5311yescharacteristic
299C2×C4⋊Q8C2×D5311yescharacteristic
300C2×#125—21no—
301C2×C22⋊Q8—21no—
302C2×C22⋊Q8C2×D5312yes—
303C2×#121—531no—
304C2×C4×Q8—21no—
305C2×C4×Q8C2×D5312yes—
306C2×C4×D4—531no—
307C2×C4×D4—1061no—
308C2×#117—21no—
309C22×C42C2×D5311yescharacteristic
310C4×SD16—1068no—
311C2×#106—532no—
312C2×#103—532no—
313C2×#103—532no—
314C2×#103—532no—
315C2×#102—532no—
316C2×C22⋊C8—532no—
317C2×C4×C8—532no—
318#390—531no—
319#389—531no—
320#388—1061no—
321#386—1061no—
322#380—531no—
323C4×C4⋊C4—531no—
324C4×C22⋊C4—1061no—
325#340—532no—
326#340—532no—
327#336—531no—
328#336—531no—
329C53#2356811yescharacteristic, complemented, Sylow
330C22×Q8—21no—
331C22×Q8C22×D5312yes—
332C22×D4—531no—
333C22×D4—1061no—
334C23×C4—1061no—
335C23×C4C22⋊D5311yescharacteristic
336C23×C4C22⋊D5311yescharacteristic
337C23×C4C22×D5311yescharacteristic
338C2×SD16—1064no—
339C2×SD16—1064no—
340C2×SD16—1064no—
341C22×C8—532no—
342C22×C8—532no—
343C4⋊Q8—24no—
344C4⋊Q8—24no—
345#125—28no—
346C22⋊Q8—28no—
347C22⋊Q8—24no—
348C22⋊Q8—24no—
349#121—1062no—
350#121—1062no—
351C4×Q8—28no—
352C4×Q8—24no—
353C4×Q8C22⋊D5318yescomplemented
354C4×D4—1064no—
355C4×D4—2124no—
356#117—28no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4—531no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4C22×D5312yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—1061no—
369C2×C22⋊C4—1061no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C22⋊C4—1061no—
373C2×C42—22no—
374C2×C42—24no—
375C2×C42—1061no—
376C2×C42C22⋊D5311yescharacteristic
377C2×C42C22⋊D5311yescharacteristic
378C2×C42C22×D5311yescharacteristic
379C2×C42C22×D5311yescharacteristic
380#106—1064no—
381#103—1064no—
382#103—1064no—
383#103—1064no—
384#102—1064no—
385C22⋊C8—1064no—
386C4×C8—1064no—
387#95—1061no—
388#95—1061no—
389#95—1061no—
390C24—1061no—
391C24C2×C22⋊D5311yescharacteristic
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8C2×C22⋊D5314yes—
401C2×D4—532no—
402C2×D4—1064no—
403C2×D4—1062no—
404C2×D4—1062no—
405C2×D4—1061no—
406C2×D4—2122no—
407C2×D4—2122no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4—1061no—
422C22×C4—2122no—
423C22×C4—2122no—
424C22×C4—2122no—
425C22×C4#1171992414yescomplemented
426C22×C4C2×C22⋊D5311yescharacteristic
427C22×C4C2×C22⋊D5311yescharacteristic
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C2×C22⋊D5311yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C23×D5311yescharacteristic
432SD16—10616no—
433C2×C8—534no—
434C2×C8—1062no—
435C2×C8—1062no—
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—1062no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—28no—
444C4⋊C4—22no—
445C4⋊C4—532no—
446C4⋊C4—1061no—
447C4⋊C4—1062no—
448C4⋊C4—1062no—
449C4⋊C4—2122no—
450C4⋊C4C2×C22⋊D5314yes—
451C22⋊C4—28no—
452C22⋊C4—1064no—
453C22⋊C4—2122no—
454C22⋊C4—2122no—
455C22⋊C4—2122no—
456C22⋊C4—2122no—
457C42—22no—
458C42—22no—
459C42—28no—
460C42—22no—
461C42—2122no—
462C42C2×C22⋊D5314yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—1062no—
470C23—2121no—
471C23—2121no—
472C23—2121no—
473C23—2121no—
474C23—2121no—
475C23—2121no—
476C23C2×#1171992412yes—
477C23C22×C22⋊D5311yescharacteristic
478Q8—24no—
479Q8—24no—
480Q8—48no—
481Q8—42no—
482Q8—42no—
483D4—1064no—
484D4—2128no—
485D4—2122no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—28no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—1062no—
506C2×C4—4244no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4#1313318612yes—
516C2×C4#1313318612yes—
517C2×C4C2×#1171992412yes—
518C2×C4C2×#1171992412yes—
519C2×C4#1313323712yes—
520C2×C4#1313324812yes—
521C2×C4C22×C22⋊D5312yes—
522C2×C4#41×D5311yescharacteristic
523C2×C4#41×D5312yes—
524C2×C4#41×D5311yescharacteristic
525C8—1064no—
526C8—2122no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—4241no—
535C22—4241no—
536C22—4241no—
537C22—4241no—
538C22—2122no—
539C22—2122no—
540C22—2122no—
541C22C2×#1313318611yescharacteristic
542C22#1681113011yescharacteristic
543C22#1681116911yescharacteristic
544C22C22×#1171992411yescharacteristic
545C22C2×#1313323711yescharacteristic
546C22C2×#1313324811yescharacteristic
547C22#1681142111yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—44no—
552C4—41no—
553C4—44no—
554C4—4241no—
555C4—4241no—
556C4—2122no—
557C4#1681076214yes—
558C4#1681113012yes—
559C4#1681116912yes—
560C2—42no—
561C2—4242no—
562C2#2027792811yescharacteristic
563C2C2×#1681076211yescharacteristic
564C2#2028393311yescharacteristic
565C2#2028471911yescharacteristic
566C2#2028478011yescharacteristic
567C2#2028527511yescharacteristic
568C2#2028531511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#533C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#533D53
hypocenterC53#23568
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C4×#11719924C2semidirect
3. #20276045C2semidirect
4. #20275958C2semidirect
5. #20275908C2semidirect
6. #20275848C2semidirect
7. #20275798C2semidirect
9. C4×#13133141C2semidirect
10. C4×C22⋊#88322C2semidirect
11. C2×C53×#533C2semidirect
90. C4×Q8×C53D4semidirect
127. C22×C4×C53SD16semidirect
276. C2×#533D53semidirect
329. C53#23568semidirect
353. C4×Q8C22⋊D53semidirect
425. C22×C4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199242
C4×#131331912
#202760452
#202759582
#202759082
#202758482
#202757982
#202757162
C4×#131331412
C4×C22⋊#883222
C2×C53×#5332
#2356853