Finite Group Databasebeta

← #27384119

Subgroups — #27384119

Order 13568.

1358 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719949C211yesmaximal, characteristic
2C2×#16810675C211yesmaximal, characteristic, complemented
3#20277108C211yesmaximal, characteristic, complemented
4#20274808C212yesmaximal, complemented
5#20274730C211yesmaximal, characteristic, complemented
6#20274726C211yesmaximal, characteristic, complemented
7#20274722C211yesmaximal, characteristic, complemented
8#20274654C212yesmaximal
9#103×Dic53C212yesmaximal, complemented
10#20274529C212yesmaximal, complemented
11C53×#1805C211yesmaximal, characteristic, complemented
12C22×#11719949C2211yescharacteristic
13C4×#11719949—24no—
14C4×#11719949C418yescomplemented
15C2×Q8×Dic53C2211yescharacteristic
16C2×C4×D212—21no—
17C2×C4×D212C2211yescharacteristic
18C2×#13133179—21no—
19C2×#13133179C2211yescharacteristic
20C2×#13133175—21no—
21C2×#13133175C2211yescharacteristic
22C2×C42×D53—21no—
23C2×C42×D53C2211yescharacteristic
24#16810675—24no—
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133141C2212yes—
27C2×#13133139C2212yes—
28C2×#13133137C2212yes—
29C2×#13133129C2212yes—
30C2×C8×Dic53C2212yes—
31#16810490—21no—
32#16810489C2211yescharacteristic
33#16810485—21no—
34#16810480C2212yes—
35C4⋊C4×Dic53C2212yes—
36#16810470—21no—
37#16810468—21no—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#103×C53C2212yes—
44C53×#382C2211yescharacteristic
45C53×#340C2212yes—
46C2×#11719949—22no—
47C2×#11719949—22no—
48C2×#11719949—22no—
49C2×#11719949—22no—
50C2×#11719949—22no—
51C2×#11719949C2×C414yes—
52C22×D212—21no—
53C22×D212C2311yescharacteristic
54C22×C4×D53—21no—
55C22×C4×D53—21no—
56C22×C4×D53—21no—
57C22×C4×D53D411yescharacteristic
58C22×C4×D53D411yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C22×#88322C2312yes—
61Q8×Dic53—22no—
62Q8×Dic53C2×C414yes—
63C4×D212—22no—
64C4×D212—28no—
65C4×D212C2×C414yes—
66#13133179—22no—
67#13133179—28no—
68#13133179C2×C414yes—
69#13133175—28no—
70#13133175—22no—
71#13133175C2×C414yes—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2312yes—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719921—21no—
76C2×#11719921—22no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—22no—
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C42×D53—28no—
86C42×D53C2×C414yes—
87C42×D53D414yes—
88C4⋊#88322—22no—
89#13133141—24no—
90#13133139—24no—
91#13133137—24no—
92#13133129—24no—
93C8×Dic53—24no—
94#13133124—21no—
95#13133124—21no—
96#13133124—22no—
97#13133124—22no—
98C22×Q8×C53C2311yescharacteristic
99C22×C8×C53C2312yes—
100C4×Q8×C53—22no—
101C4×Q8×C53C2×C414yes—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2312yes—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—22no—
108#103×C53—24no—
109#95×C53—21no—
110#95×C53—21no—
111C23×D53—21no—
112C23×D53C2×D411yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53C22×C412yes—
117C2×Q8×C53#4112yes—
118C2×Q8×C53#4112yes—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C22×Dic53—21no—
126C22×Dic53—22no—
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130#11719949—28no—
131#11719949—28no—
132#11719949—28no—
133#11719949—28no—
134C2×D212—24no—
135C2×D212—24no—
136C2×D212—24no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212C22×C412yes—
141C2×D212#4112yes—
142C2×D212#4112yes—
143C2×C4×D53—24no—
144C2×C4×D53—24no—
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—24no—
150C2×C4×D53—24no—
151C2×C4×D53—24no—
152C2×C4×D53—24no—
153C2×C4×D53—42no—
154C2×C4×D53—44no—
155C2×C4×D53—44no—
156C2×C4×D53C22⋊C414yes—
157C2×C4×D53C22⋊C414yes—
158C2×C4×D53C22×C412yes—
159C2×C4×D53C2×D412yes—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C2×C8×C53—22no—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53C22×C412yes—
168C42×C53—24no—
169C42×C53C22×C412yes—
170C42×C53C2×D412yes—
171#11719921—24no—
172#11719921—24no—
173#11719921—24no—
174#11719921—44no—
175C4⋊Dic53—21no—
176C4⋊Dic53—24no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53C22×C412yes—
180C4×Dic53—24no—
181C4×Dic53—24no—
182C4×Dic53—44no—
183C4×Dic53C22⋊C414yes—
184C4×Dic53C22⋊C414yes—
185C4×Dic53C22×C412yes—
186C4×Dic53C2×D412yes—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—22no—
190C23×C53C22×D411yescharacteristic
191C22×D53—22no—
192C22×D53—21no—
193C22×D53—21no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53C2×C22⋊C412yes—
201Q8×C53—22no—
202Q8×C53—22no—
203Q8×C53—22no—
204Q8×C53#11714yes—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53C2×C22⋊C412yes—
216C2×C4×C53C2×C22⋊C412yes—
217C2×C4×C53C23×C411yescharacteristic
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53C2×C22⋊C412yes—
232C2×Dic53C2×C22⋊C412yes—
233C2×Dic53C2×C22⋊C412yes—
234C2×Dic53#13514yes—
235D212—216no—
236D212—22no—
237D212—22no—
238D212—22no—
239D212#11714yes—
240C4×D53—216no—
241C4×D53—22no—
242C4×D53—22no—
243C4×D53—22no—
244C4×D53—22no—
245C4×D53—28no—
246C4×D53—44no—
247C4×D53—48no—
248C4×D53—48no—
249C8×C53—42no—
250#88322—42no—
251#20736—531nomaximal, Sylow
252C22×C53C22×C22⋊C411yescharacteristic
253C22×C53#52511yescharacteristic
254C22×C53#52511yescharacteristic
255C22×C53C2×#13512yes—
256C22×C53#57512yes—
257C2×D53—22no—
258C2×D53—22no—
259C2×D53—22no—
260C2×D53—24no—
261C2×D53—42no—
262C2×D53—42no—
263C2×D53—44no—
264C2×D53—44no—
265C2×D53—42no—
266C2×D53—42no—
267C4×C53—22no—
268C4×C53—22no—
269C4×C53—24no—
270C4×C53—42no—
271C4×C53—82no—
272C4×C53C2×#11711yescharacteristic
273C4×C53C2×#11711yescharacteristic
274C4×C53#52511yescharacteristic
275C4×C53#52511yescharacteristic
276Dic53—22no—
277Dic53—22no—
278Dic53—42no—
279Dic53—82no—
280Dic53#41614yes—
281C2×C4×#41—531no—
282C2×#431—531no—
283#2283—531no—
284#1825—532no—
285#1805—531no—
286#1805D5311yescharacteristic, complemented
287#1804—531no—
288#1804—531no—
289#1788—532no—
290C4×#103—532no—
291C4×#102—532no—
292C2×C53#228011yescharacteristic
293C2×C53C2×#41611yescharacteristic
294C2×C53#289611yescharacteristic
295C2×C53#310212yes—
296C2×C53#310212yes—
297D53—42no—
298D53—44no—
299C22×#41—531no—
300C4×#41—538no—
301C4×#41—1064no—
302C2×C4×Q8—531no—
303C2×C4×Q8C2×D5311yescharacteristic
304C2×C4×D4—531no—
305C2×C4×D4—531no—
306C2×C4×D4—1061no—
307C2×C4×D4—1061no—
308C2×#117—531no—
309C2×#117—1061no—
310C22×C42—531no—
311C22×C42—1061no—
312#431—1064no—
313C2×C4⋊C8—531no—
314C2×C4⋊C8C2×D5311yescharacteristic
315C2×#103—532no—
316C2×#103C2×D5312yes—
317C2×#102—532no—
318C2×#102—532no—
319C2×C22⋊C8—532no—
320C2×C4×C8—532no—
321#383—1061no—
322#382—531no—
323#382C2×D5311yescharacteristic
324#381—1061no—
325#380—532no—
326C4×C4⋊C4—532no—
327C4×C22⋊C4—1061no—
328#376—1061no—
329#340—532no—
330#340—532no—
331#340—532no—
332#340C2×D5312yes—
333C53#2073611yescharacteristic, complemented, Sylow
334C2×#41—534no—
335C2×#41—1062no—
336C2×#41—1062no—
337C2×#41—1062no—
338C2×#41—1062no—
339C2×#41—1062no—
340C22×Q8C22×D5311yescharacteristic
341C22×D4—531no—
342C22×D4—1061no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—1061no—
347C23×C4—1061no—
348C23×C4—1061no—
349C22×C8—532no—
350C22×C8C22×D5312yes—
351C4×Q8—22no—
352C4×Q8—534no—
353C4×Q8—1062no—
354C4×Q8C4×D5314yes—
355C4×D4—534no—
356C4×D4—534no—
357C4×D4—1068no—
358C4×D4—1062no—
359C4×D4—1068no—
360C4×D4—1062no—
361#117—534no—
362#117—1068no—
363#117—1062no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—532no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4C22×D5312yes—
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C22⋊C4—1061no—
373C2×C22⋊C4—1062no—
374C2×C42—21no—
375C2×C42—531no—
376C2×C42—531no—
377C2×C42—531no—
378C2×C42—534no—
379C2×C42—534no—
380C2×C42—1068no—
381C2×C42—1061no—
382C2×C42—1061no—
383C2×C42—1062no—
384C2×C42C22×D5311yescharacteristic
385C4⋊C8—22no—
386C4⋊C8—1062no—
387#103—24no—
388#103—1064no—
389#102—1064no—
390#102—1064no—
391C22⋊C8—1064no—
392C4×C8—1064no—
393#95—21no—
394#95—21no—
395#95—1062no—
396#95—1062no—
397#95—1061no—
398#95—1061no—
399C24—531no—
400C24—1061no—
401#41—1068no—
402#41—1068no—
403#41—1068no—
404#41—1068no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×C4×D5312yes—
409C2×Q8#1171994712yes—
410C2×Q8#1171994912yes—
411C2×D4—532no—
412C2×D4—532no—
413C2×D4—532no—
414C2×D4—1064no—
415C2×D4—1064no—
416C2×D4—1064no—
417C2×D4—1061no—
418C2×D4—1061no—
419C2×D4—1061no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—534no—
427C22×C4—532no—
428C22×C4—532no—
429C22×C4—534no—
430C22×C4—1064no—
431C22×C4—1064no—
432C22×C4—1064no—
433C22×C4—1064no—
434C22×C4—1064no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1064no—
441C22×C4—1062no—
442C22×C4—2122no—
443C22×C4—2124no—
444C22×C4—2124no—
445C22×C4D4×D5311yescharacteristic
446C22×C4D4×D5311yescharacteristic
447C22×C4C23×D5311yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—1062no—
452C2×C8—1062no—
453C2×C8—1062no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—532no—
459C4⋊C4—1064no—
460C4⋊C4—1061no—
461C4⋊C4—1062no—
462C4⋊C4—1062no—
463C4⋊C4C2×C4×D5312yes—
464C22⋊C4—1064no—
465C22⋊C4—1064no—
466C22⋊C4—1064no—
467C22⋊C4—2124no—
468C42—24no—
469C42—534no—
470C42—534no—
471C42—532no—
472C42—532no—
473C42—1064no—
474C42—1064no—
475C42—2124no—
476C42C2×C4×D5312yes—
477C42D4×D5312yes—
478C23—532no—
479C23—1062no—
480C23—1062no—
481C23—1061no—
482C23—1062no—
483C23—1061no—
484C23—1062no—
485C23—1062no—
486C23—2122no—
487C23—2122no—
488C23C2×D4×D5311yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—22no—
492Q8#1313317514yes—
493D4—534no—
494D4—10616no—
495D4—1062no—
496D4—1062no—
497D4—1062no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—532no—
509C2×C4—534no—
510C2×C4—532no—
511C2×C4—532no—
512C2×C4—1068no—
513C2×C4—10616no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1062no—
521C2×C4—1062no—
522C2×C4—2124no—
523C2×C4—2128no—
524C2×C4—2128no—
525C2×C4—2122no—
526C2×C4—2122no—
527C2×C4—2122no—
528C2×C4—2122no—
529C2×C4C22⋊C4×D5312yes—
530C2×C4C22⋊C4×D5312yes—
531C2×C4C22×C4×D5311yescharacteristic
532C2×C4C2×D4×D5311yescharacteristic
533C2×C4#41×D5311yescharacteristic
534C2×C4#41×D5311yescharacteristic
535C2×C4C2×#1171994711yescharacteristic
536C2×C4C2×#1171994911yescharacteristic
537C8—42no—
538C8—2122no—
539C22—1062no—
540C22—1062no—
541C22—1064no—
542C22—1062no—
543C22—2122no—
544C22—2124no—
545C22—2122no—
546C22—2124no—
547C22—2122no—
548C22—2122no—
549C22C2×C22⋊C4×D5311yescharacteristic
550C22#1681113311yescharacteristic
551C22#1681113611yescharacteristic
552C22#135×D5312yes—
553C22#1681142212yes—
554C4—22no—
555C4—22no—
556C4—24no—
557C4—42no—
558C4—82no—
559C4—534no—
560C4—1062no—
561C4—1062no—
562C4—4242no—
563C4—2122no—
564C4#117×D5311yescharacteristic
565C4C2×#1313317511yescharacteristic
566C4#1681113311yescharacteristic
567C4#1681113611yescharacteristic
568C2—2122no—
569C2—2124no—
570C2#2027709711yescharacteristic
571C2#416×D5311yescharacteristic
572C2#2028323311yescharacteristic
573C2#2028473212yes—
574C2#2028474012yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1805C2
socleC23×C53C22×D4
prime radicalC22×#11719949C22
cosocleC22×C4×C53C24
hypercenter#1805D53
hypocenterC53#20736
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#16810675C2semidirect
3. #20277108C2semidirect
4. #20274808C2semidirect
5. #20274730C2semidirect
6. #20274726C2semidirect
7. #20274722C2semidirect
9. #103×Dic53C2semidirect
10. #20274529C2semidirect
11. C53×#1805C2semidirect
14. C4×#11719949C4semidirect
286. #1805D53semidirect
333. C53#20736semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199492
C2×#168106752
#202771082
#202748082
#202747302
#202747262
#202747222
#202746542
#103×Dic532
#202745292
C53×#18052
#2073653