Finite Group Databasebeta

← #27273762

Subgroups — #27273762

Order 13568.

1058 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810585C211yesmaximal, characteristic
2C2×#16810585C211yesmaximal, characteristic
3C4×C22⋊C4×D53C211yesmaximal, characteristic
4#20272563C211yesmaximal, characteristic, complemented
5#20272562C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C53×#1282C211yesmaximal, characteristic, complemented
8C23×C4×D53C2211yescharacteristic
9C22×#11719912—21no—
10C22×#11719912C412yescomplemented
11C2×C22⋊C4×D53C412yes—
12C2×C42×D53—21no—
13#16810585—21no—
14#16810585—21no—
15#16810585—21no—
16#16810585—21no—
17#16810585—21no—
18#16810585—21no—
19#16810585C412yescomplemented
20#16810585C412yescomplemented
21C2×C22⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23C2×#13133129—21no—
24C2×#13133129C412yescomplemented
25C22⋊C4×Dic53C412yes—
26C4×#11719921—21no—
27C4×C22⋊Dic53C2211yescharacteristic
28#16810470—21no—
29#95×D53—21no—
30C2×C22⋊C8×C53C2211yescharacteristic
31C2×C22⋊C8×C53C2211yescharacteristic
32C4×C22⋊C4×C53C2211yescharacteristic
33C24×D53C2×C411yescharacteristic
34C23×Dic53C2×C411yescharacteristic
35C22×C4×D53—22no—
36C22×C4×D53—44no—
37C22×C4×D53C2×C411yescharacteristic
38C22×C4×D53C2×C411yescharacteristic
39C22×C4×D53C2×C411yescharacteristic
40C22×C4×D53C2×C411yescharacteristic
41C22×C4×D53D411yescharacteristic
42C22×C4×D53D411yescharacteristic
43C22×C4×D53D411yescharacteristic
44C22×C4×D53D411yescharacteristic
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53Q811yescharacteristic
48C22×C4×D53Q811yescharacteristic
49C2×#11719912—22no—
50C2×#11719912—22no—
51C2×#11719912—22no—
52C2×#11719912—22no—
53C2×#11719912—44no—
54C2×#11719912—44no—
55C22×#88322—21no—
56C22×#88322C2×C412yescomplemented
57C2×C22⋊Dic53C2×C412yes—
58C2×#11719921—22no—
59C22⋊C4×D53—24no—
60C22⋊C4×D53—24no—
61C2×C4×Dic53—21no—
62C2×C4×Dic53—22no—
63C42×D53—44no—
64C22⋊#88322—21no—
65C22⋊#88322—21no—
66C22⋊#88322—21no—
67C22⋊#88322C2×C412yescomplemented
68#13133129—22no—
69#13133129—22no—
70#13133129—22no—
71#13133129—22no—
72#13133124—21no—
73#13133124—22no—
74C23×C4×C53C2311yescharacteristic
75C22×C8×C53—21no—
76C22×C8×C53C2×C412yescomplemented
77C2×C22⋊C4×C53C2×C412yes—
78C2×C42×C53—21no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53—21no—
81C22⋊C8×C53—21no—
82C22⋊C8×C53C2×C412yescomplemented
83#95×C53—21no—
84C24×C53C22×C411yescharacteristic
85C23×D53—22no—
86C23×D53—22no—
87C23×D53—42no—
88C23×D53—42no—
89C23×D53C4211yescharacteristic
90C23×D53C4211yescharacteristic
91C23×D53C22⋊C411yescharacteristic
92C23×D53C22⋊C411yescharacteristic
93C23×D53C4⋊C411yescharacteristic
94C23×D53C4⋊C411yescharacteristic
95C22×C4×C53—22no—
96C22×C4×C53—42no—
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×Q811yescharacteristic
103C22×Dic53—22no—
104C22×Dic53—42no—
105C22×Dic53C4211yescharacteristic
106C22×Dic53C4211yescharacteristic
107C22×Dic53C22⋊C411yescharacteristic
108C22×Dic53C22⋊C411yescharacteristic
109C22×Dic53C4⋊C411yescharacteristic
110C22×Dic53C4⋊C411yescharacteristic
111C2×C4×D53—24no—
112C2×C4×D53—24no—
113C2×C4×D53—24no—
114C2×C4×D53—28no—
115C2×C4×D53—42no—
116C2×C4×D53—42no—
117C2×C4×D53—42no—
118C2×C4×D53—42no—
119C2×C4×D53—42no—
120C2×C4×D53—42no—
121C2×C4×D53—42no—
122C2×C4×D53—42no—
123C2×C4×D53—42no—
124C2×C4×D53—44no—
125C2×C4×D53—44no—
126C2×C4×D53C22⋊C411yescharacteristic
127C2×C4×D53C22⋊C411yescharacteristic
128C2×C4×D53C22⋊C411yescharacteristic
129C2×C4×D53C22⋊C411yescharacteristic
130C2×C4×D53C22⋊C411yescharacteristic
131C2×C4×D53C22⋊C411yescharacteristic
132C2×C4×D53C22⋊C411yescharacteristic
133C2×C4×D53C22⋊C411yescharacteristic
134C2×C4×D53C4⋊C411yescharacteristic
135C2×C4×D53C4⋊C411yescharacteristic
136C2×C4×D53C4⋊C411yescharacteristic
137C2×C4×D53C4⋊C411yescharacteristic
138C2×C4×D53C4⋊C411yescharacteristic
139C2×C4×D53C4⋊C411yescharacteristic
140C2×C4×D53C4⋊C411yescharacteristic
141C2×C4×D53C4⋊C411yescharacteristic
142C2×C8×C53—22no—
143C2×C8×C53—22no—
144C2×C8×C53—42no—
145C2×C8×C53—42no—
146C22⋊C4×C53—22no—
147C22⋊C4×C53M1614yescomplemented
148C42×C53—42no—
149C22⋊Dic53—22no—
150C22⋊Dic53M1614yescomplemented
151#11719921—28no—
152C4×Dic53—44no—
153C4×Dic53—42no—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322—42no—
157C2×#88322—42no—
158#11719912—44no—
159#11719912—44no—
160#11719912—44no—
161#11719912—44no—
162C23×C53—41no—
163C23×C53—41no—
164C23×C53C2×C4211yescharacteristic
165C23×C53C2×C22⋊C411yescharacteristic
166C23×C53C2×C4⋊C411yescharacteristic
167C23×C53C2×M1612yes—
168C23×C53C2×M1612yes—
169C22×D53—22no—
170C22×D53—22no—
171C22×D53—22no—
172C22×D53—21no—
173C22×D53—21no—
174C22×D53—21no—
175C22×D53—21no—
176C22×D53—22no—
177C22×D53—22no—
178C22×D53—22no—
179C22×D53—22no—
180C22×D53—22no—
181C22×D53—22no—
182C22×D53—22no—
183C22×D53—41no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—48no—
190C22×D53—48no—
191C22×D53#9511yescharacteristic
192C22×D53#9511yescharacteristic
193C22×D53#9511yescharacteristic
194C22×D53#9511yescharacteristic
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—24no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53C2×C22⋊C411yescharacteristic
205C2×C4×C53C2×C22⋊C411yescharacteristic
206C2×C4×C53C2×C4⋊C411yescharacteristic
207C2×C4×C53C2×C4⋊C411yescharacteristic
208C2×Dic53—22no—
209C2×Dic53—22no—
210C2×Dic53—22no—
211C2×Dic53—24no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53#9511yescharacteristic
218C2×Dic53#9511yescharacteristic
219C2×Dic53#9511yescharacteristic
220C2×Dic53#9511yescharacteristic
221C4×D53—22no—
222C4×D53—22no—
223C4×D53—22no—
224C4×D53—22no—
225C4×D53—22no—
226C4×D53—22no—
227C4×D53—22no—
228C4×D53—22no—
229C4×D53—22no—
230C4×D53—41no—
231C4×D53—41no—
232C4×D53—41no—
233C4×D53—41no—
234C4×D53—41no—
235C4×D53—41no—
236C4×D53—416no—
237C4×D53#9914yes—
238C8×C53—42no—
239C8×C53—42no—
240#88322—42no—
241#88322—42no—
242#7929—531nomaximal, Sylow
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—42no—
250C22×C53—42no—
251C22×C53C2×#9511yescharacteristic
252C22×C53C4×M1611yescharacteristic
253C22×C53C4×M1611yescharacteristic
254C22×C53#40712yes—
255C22×C53#42312yes—
256C2×D53—28no—
257C2×D53—21no—
258C2×D53—21no—
259C2×D53—21no—
260C2×D53—21no—
261C2×D53—21no—
262C2×D53—21no—
263C2×D53—21no—
264C2×D53—21no—
265C2×D53—21no—
266C2×D53—21no—
267C2×D53—42no—
268C2×D53—42no—
269C2×D53—44no—
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—44no—
273C2×D53—44no—
274C2×D53#34212yes—
275C2×D53#34412yes—
276C4×C53—21no—
277C4×C53—21no—
278C4×C53—21no—
279C4×C53—44no—
280C4×C53C2×#9912yes—
281Dic53—21no—
282Dic53—21no—
283Dic53—21no—
284Dic53—44no—
285Dic53#34212yes—
286C2×#407—531no—
287C2×#407—531no—
288C2×C4×C22⋊C4—531no—
289#1282—531no—
290#1282—531no—
291#1282—531no—
292#1282D5311yescharacteristic, complemented
293C2×C53—22no—
294C2×C53—21no—
295C2×C53—21no—
296C2×C53—22no—
297C2×C53#172811yescharacteristic
298C2×C53C2×#34211yescharacteristic
299C2×C53C2×#34411yescharacteristic
300D53—22no—
301D53—22no—
302D53—41no—
303D53—41no—
304C24×C4—531no—
305C22×M16—532no—
306C22×M16—1061no—
307C22×C22⋊C4—532no—
308C22×C42—1061no—
309#407—532no—
310#407—532no—
311#407—1061no—
312#407—1061no—
313#407—1061no—
314#407—1061no—
315#407—1061no—
316#407—1061no—
317C2×C22⋊C8—531no—
318C2×C22⋊C8—531no—
319C2×C22⋊C8—532no—
320C2×C22⋊C8—1061no—
321C2×C22⋊C8C2×D5311yescharacteristic
322C2×C22⋊C8C2×D5311yescharacteristic
323C4×C22⋊C4—531no—
324C4×C22⋊C4—532no—
325C4×C22⋊C4—1061no—
326C4×C22⋊C4—1061no—
327C4×C22⋊C4C2×D5311yescharacteristic
328C2×#95—1061no—
329C53#792911yescharacteristic, complemented, Sylow
330C25—531no—
331C23×C4—531no—
332C23×C4—531no—
333C23×C4—531no—
334C23×C4—531no—
335C23×C4—531no—
336C23×C4—531no—
337C23×C4—531no—
338C23×C4—531no—
339C23×C4—531no—
340C23×C4—531no—
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—531no—
344C23×C4—1062no—
345C23×C4—2124no—
346C23×C4C22×D5311yescharacteristic
347C2×M16—1062no—
348C2×M16—1062no—
349C2×M16—1062no—
350C2×M16—1062no—
351C2×M16—2124no—
352C2×M16—2124no—
353C22×C8—21no—
354C22×C8—532no—
355C22×C8—1061no—
356C22×C8C4×D5312yescomplemented
357C2×C22⋊C4—532no—
358C2×C22⋊C4—1064no—
359C2×C22⋊C4—1062no—
360C2×C22⋊C4—1064no—
361C2×C22⋊C4C4×D5312yes—
362C2×C42—21no—
363C2×C42—1062no—
364C2×C42—1061no—
365C2×C42—2124no—
366C22⋊C8—21no—
367C22⋊C8—21no—
368C22⋊C8—21no—
369C22⋊C8—532no—
370C22⋊C8—1062no—
371C22⋊C8—1062no—
372C22⋊C8—1062no—
373C22⋊C8—1062no—
374C22⋊C8—1061no—
375C22⋊C8—1061no—
376C22⋊C8—1061no—
377C22⋊C8C4×D5312yescomplemented
378#95—21no—
379#95—1062no—
380#95—1061no—
381C24—531no—
382C24—531no—
383C24—531no—
384C24—531no—
385C24—531no—
386C24—531no—
387C24—1062no—
388C24—1062no—
389C24—2122no—
390C24—2122no—
391C24C2×C4×D5311yescharacteristic
392C22×C4—22no—
393C22×C4—42no—
394C22×C4—531no—
395C22×C4—531no—
396C22×C4—531no—
397C22×C4—531no—
398C22×C4—531no—
399C22×C4—531no—
400C22×C4—531no—
401C22×C4—531no—
402C22×C4—531no—
403C22×C4—531no—
404C22×C4—531no—
405C22×C4—531no—
406C22×C4—531no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—531no—
410C22×C4—531no—
411C22×C4—531no—
412C22×C4—531no—
413C22×C4—531no—
414C22×C4—531no—
415C22×C4—531no—
416C22×C4—1064no—
417C22×C4—1064no—
418C22×C4—1064no—
419C22×C4—1068no—
420C22×C4—1062no—
421C22×C4—2122no—
422C22×C4—2122no—
423C22×C4—2122no—
424C22×C4—2122no—
425C22×C4—2122no—
426C22×C4—2122no—
427C22×C4—2122no—
428C22×C4—2122no—
429C22×C4—2122no—
430C22×C4—2124no—
431C22×C4—2124no—
432C22×C4—2122no—
433C22×C4C2×C4×D5311yescharacteristic
434C22×C4C2×C4×D5311yescharacteristic
435C22×C4D4×D5311yescharacteristic
436C22×C4D4×D5311yescharacteristic
437C22×C4D4×D5311yescharacteristic
438C22×C4Q8×D5311yescharacteristic
439M16—2124no—
440M16—2124no—
441M16—2124no—
442M16—2124no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—2122no—
450C2×C8—2122no—
451C22⋊C4—22no—
452C22⋊C4—534no—
453C22⋊C4—1068no—
454C22⋊C4—1062no—
455C22⋊C4#1171991214yescomplemented
456C42—42no—
457C42—2122no—
458C42—2124no—
459C23—41no—
460C23—41no—
461C23—531no—
462C23—531no—
463C23—531no—
464C23—531no—
465C23—1062no—
466C23—1062no—
467C23—1062no—
468C23—1062no—
469C23—1062no—
470C23—1062no—
471C23—1062no—
472C23—1061no—
473C23—1061no—
474C23—1061no—
475C23—1061no—
476C23—1062no—
477C23—1062no—
478C23—1062no—
479C23—2121no—
480C23—2121no—
481C23—2121no—
482C23—2121no—
483C23—2121no—
484C23—2121no—
485C23—2128no—
486C23—2128no—
487C23C42×D5311yescharacteristic
488C23C22⋊C4×D5311yescharacteristic
489C23C4⋊C4×D5311yescharacteristic
490C23M16×D5312yes—
491C23C2×#1171991212yes—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—531no—
502C2×C4—534no—
503C2×C4—531no—
504C2×C4—531no—
505C2×C4—531no—
506C2×C4—1062no—
507C2×C4—1062no—
508C2×C4—1062no—
509C2×C4—1062no—
510C2×C4—1062no—
511C2×C4—1062no—
512C2×C4—1062no—
513C2×C4—1062no—
514C2×C4—1062no—
515C2×C4—1062no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—1064no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—21216no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4C22⋊C4×D5311yescharacteristic
532C2×C4C22⋊C4×D5311yescharacteristic
533C2×C4C4⋊C4×D5311yescharacteristic
534C2×C4C4⋊C4×D5311yescharacteristic
535C8—42no—
536C8—42no—
537C8—2122no—
538C8—2122no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—42no—
546C22—42no—
547C22—532no—
548C22—532no—
549C22—1068no—
550C22—1061no—
551C22—1061no—
552C22—1061no—
553C22—1061no—
554C22—1061no—
555C22—1061no—
556C22—1061no—
557C22—1061no—
558C22—1061no—
559C22—1061no—
560C22—2122no—
561C22—2122no—
562C22—2122no—
563C22—2122no—
564C22—2124no—
565C22—2124no—
566C22—2124no—
567C22#95×D5311yescharacteristic
568C22C4×#1171991211yescharacteristic
569C22#1681057511yescharacteristic
570C22#1681058512yes—
571C22#1681064512yes—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—44no—
576C4—532no—
577C4—1061no—
578C4—1061no—
579C4—1061no—
580C4—2124no—
581C4#99×D5312yes—
582C2—22no—
583C2—21no—
584C2—21no—
585C2—22no—
586C2—1062no—
587C2—1062no—
588C2—2121no—
589C2—2121no—
590C2#2027440711yescharacteristic
591C2#342×D5311yescharacteristic
592C2#344×D5311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1282C2
socleC22×C53C2×#95
prime radicalC24×D53C2×C4
cosocleC23×C4×C53C23
hypercenter#1282D53
hypocenterC53#7929
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
4. #20272563C2semidirect
5. #20272562C2semidirect
6. #20272561C2semidirect
7. C53×#1282C2semidirect
10. C22×#11719912C4semidirect
19. #16810585C4semidirect
20. #16810585C4semidirect
24. C2×#13133129C4semidirect
56. C22×#88322C2×C4semidirect
67. C22⋊#88322C2×C4semidirect
76. C22×C8×C53C2×C4semidirect
82. C22⋊C8×C53C2×C4semidirect
147. C22⋊C4×C53M16semidirect
150. C22⋊Dic53M16semidirect
292. #1282D53semidirect
329. C53#7929semidirect
356. C22×C8C4×D53semidirect
377. C22⋊C8C4×D53semidirect
455. C22⋊C4#11719912semidirect

Maximal subgroups

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

subgroupindex
C2×#168105852
C4×C22⋊C4×D532
#202725632
#202725622
#202725612
C53×#12822
#792953