Finite Group Databasebeta

← #27310749

Subgroups — #27310749

Order 13568.

1414 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133207C211yesmaximal, characteristic
2C2×#16810490C211yesmaximal, characteristic
3C2×C4⋊C4×Dic53C211yesmaximal, characteristic
4#20272981C212yesmaximal
5#20272981C212yesmaximal
6#20272969C211yesmaximal, characteristic
7#20272968C211yesmaximal, characteristic
8#20272965C211yesmaximal, characteristic
9#20272949C212yesmaximal
10#20272946C212yesmaximal
11C53×#1441C211yesmaximal, characteristic
12C2×#13133207—24no—
13C2×#13133207—24no—
14C2×#13133207C418yescomplemented
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2211yescharacteristic
21C22×C4×Dic53C2212yes—
22C22×C4×Dic53C2211yescharacteristic
23#16810490—24no—
24C4⋊C4×Dic53—24no—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C23×Dic53C2312yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46#13133207—232no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—24no—
51C2×C4⋊Dic53C2×C414yes—
52C2×#11719919—22no—
53C2×#11719919—22no—
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919C2×C414yes—
61C2×#11719919C2×C414yes—
62C2×#11719919C2×C414yes—
63C2×#11719919C2×C414yes—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—42no—
66C2×C4×Dic53—42no—
67C2×C4×Dic53C2×C414yes—
68C2×C4×Dic53D414yes—
69C2×C4×Dic53D414yes—
70#13133124—22no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124—42no—
78#13133124—42no—
79#13133124—42no—
80C23×C4×C53C2312yes—
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—22no—
87C2×C4⋊C4×C53—22no—
88C2×C4⋊C4×C53—24no—
89C2×C4⋊C4×C53C2×C414yes—
90#95×C53—22no—
91#95×C53—24no—
92#95×C53—42no—
93C24×C53C2411yescharacteristic
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53#4114yes—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53C22×C412yes—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53#4114yes—
163C4⋊C4×C53—28no—
164C4⋊C4×C53—28no—
165C4⋊Dic53—28no—
166C4⋊Dic53—28no—
167#11719919—28no—
168#11719919—28no—
169#11719919—28no—
170#11719919—28no—
171C4×Dic53—48no—
172C4×Dic53C22⋊C4116yes—
173C23×C53C23×C411yescharacteristic
174C23×C53C22×D411yescharacteristic
175C23×C53C22×D411yescharacteristic
176C23×C53C2×#4112yes—
177C23×C53C2×#4112yes—
178C23×C53H(5,𝔽2)11yescharacteristic
179C23×C53H(5,𝔽2)11yescharacteristic
180C23×C532−1+412yes—
181C23×C532−1+411yescharacteristic
182C23×C532−1+412yes—
183C23×C532−1+411yescharacteristic
184C2×C4×C53—24no—
185C2×C4×C53—22no—
186C2×C4×C53—22no—
187C2×C4×C53—24no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—24no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—24no—
194C2×C4×C53—24no—
195C2×C4×C53—24no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—42no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53C2×C22⋊C414yes—
215C2×Dic53—24no—
216C2×Dic53—22no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—24no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—24no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—42no—
236C2×Dic53—41no—
237C2×Dic53—42no—
238C2×Dic53—41no—
239C2×Dic53—42no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53C2×C22⋊C414yes—
252C2×Dic53C2×C22⋊C414yes—
253#14289—531nomaximal, Sylow
254C22×C53C22×C22⋊C411yescharacteristic
255C22×C53C4×#4112yes—
256C22×C53#51912yes—
257C22×C53#52011yescharacteristic
258C22×C53#52011yescharacteristic
259C22×C53#52511yescharacteristic
260C22×C53#52511yescharacteristic
261C22×C53#53212yes—
262C22×C53#53511yescharacteristic
263C22×C53#53611yescharacteristic
264C22×C53#53612yes—
265C22×C53#53611yescharacteristic
266C22×C53#53611yescharacteristic
267C22×C53#53711yescharacteristic
268C22×C53#53711yescharacteristic
269C22×C53#53912yes—
270C22×C53#53912yes—
271C22×C53#53912yes—
272C22×C53#53912yes—
273C22×C53#54112yes—
274C22×C53#54112yes—
275C22×C53#54112yes—
276C22×C53#54112yes—
277C4×C53—24no—
278C4×C53—42no—
279C4×C53—44no—
280C4×C53—42no—
281C4×C53—82no—
282Dic53—24no—
283Dic53—24no—
284Dic53—42no—
285Dic53—44no—
286Dic53—42no—
287Dic53—82no—
288C22×#125—531no—
289C2×#383—531no—
290C2×C4×C4⋊C4—531no—
291#1445—532no—
292#1445—532no—
293#1441—531no—
294#1441—531no—
295#1441—531no—
296#1441D5311yescharacteristic
297#1436—532no—
298#1436—532no—
299C2×C53#229711yescharacteristic
300C2×C53#231011yescharacteristic
301C2×C53#231111yescharacteristic
302C2×C53#233612yes—
303C2×C53#233612yes—
304C2×C53#263311yescharacteristic
305C2×C53#263311yescharacteristic
306C2×C53#263411yescharacteristic
307C2×C53#263411yescharacteristic
308C2×C53#266412yes—
309C2×C53#266412yes—
310C2×#125—538no—
311C2×#125—1064no—
312C2×#125—1064no—
313C22×C4⋊C4—531no—
314C22×C4⋊C4—531no—
315C22×C4⋊C4—531no—
316C22×C4⋊C4—531no—
317C22×C4⋊C4—531no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4C2×D5311yescharacteristic
320C22×C4⋊C4C2×D5311yescharacteristic
321C22×C42—531no—
322C22×C42—532no—
323#383—1064no—
324C4×C4⋊C4—1064no—
325C2×#95—532no—
326C2×#95—532no—
327C2×#95—532no—
328C2×#95—532no—
329C2×#95—531no—
330C2×#95—532no—
331C2×#95—532no—
332C2×#95—532no—
333C2×#95—532no—
334C2×#95—532no—
335C2×#95C2×D5312yes—
336C2×#95C2×D5311yescharacteristic
337C2×#95C2×D5312yes—
338C53#1428911yescharacteristic, complemented, Sylow
339C23×C4—531no—
340C23×C4—531no—
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—532no—
344C23×C4—532no—
345C23×C4C22×D5312yes—
346C23×C4C22×D5311yescharacteristic
347C23×C4C22×D5311yescharacteristic
348C23×C4C22×D5312yes—
349C23×C4C22×D5311yescharacteristic
350#125—10632no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—534no—
356C2×C4⋊C4—534no—
357C2×C4⋊C4—534no—
358C2×C4⋊C4—534no—
359C2×C4⋊C4—534no—
360C2×C4⋊C4—1062no—
361C2×C4⋊C4—1062no—
362C2×C4⋊C4—1062no—
363C2×C4⋊C4—1062no—
364C2×C4⋊C4—1062no—
365C2×C4⋊C4—1062no—
366C2×C4⋊C4—1062no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1064no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1062no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4C4×D5314yes—
373C2×C42—534no—
374C2×C42—534no—
375C2×C42—534no—
376C2×C42—1064no—
377C2×C42—2122no—
378C2×C42—2122no—
379#95—22no—
380#95—24no—
381#95—42no—
382#95—1064no—
383#95—1064no—
384#95—1064no—
385#95—1064no—
386#95—1064no—
387#95—1062no—
388#95—1064no—
389#95—2122no—
390#95—2122no—
391#95—2122no—
392C24C23×D5311yescharacteristic
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—532no—
419C22×C4—532no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—532no—
423C22×C4—532no—
424C22×C4—532no—
425C22×C4—532no—
426C22×C4—534no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1062no—
430C22×C4—1061no—
431C22×C4—1062no—
432C22×C4—1061no—
433C22×C4—1062no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1061no—
442C22×C4—1062no—
443C22×C4—1061no—
444C22×C4—1062no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4C2×C4×D5312yes—
457C22×C4C2×C4×D5312yes—
458C22×C4C2×C4×D5312yes—
459C22×C4D4×D5312yes—
460C22×C4D4×D5312yes—
461C22×C4#1171994714yes—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—1068no—
465C4⋊C4—1068no—
466C4⋊C4—1068no—
467C4⋊C4—1068no—
468C4⋊C4—1068no—
469C4⋊C4—1068no—
470C42—5316no—
471C42—2128no—
472C23C22×C4×D5311yescharacteristic
473C23C2×D4×D5311yescharacteristic
474C23C2×D4×D5311yescharacteristic
475C23#41×D5312yes—
476C23C2×#1171994712yes—
477C23#1313326211yescharacteristic
478C23#1313326311yescharacteristic
479C23#1313326412yes—
480C23#1313326411yescharacteristic
481C23#1313326512yes—
482C23#1313326511yescharacteristic
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—534no—
514C2×C4—534no—
515C2×C4—1064no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1064no—
521C2×C4—1062no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—1062no—
527C2×C4—1064no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1064no—
531C2×C4—1064no—
532C2×C4—1064no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2122no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4C22⋊C4×D5314yes—
552C22C2×C22⋊C4×D5311yescharacteristic
553C22C4×#1171994712yes—
554C22#1681109912yes—
555C22#1681110311yescharacteristic
556C22#1681110411yescharacteristic
557C22#1681112911yescharacteristic
558C22#1681113111yescharacteristic
559C22#1681116812yes—
560C22#1681118911yescharacteristic
561C22#1681119311yescharacteristic
562C22#1681119611yescharacteristic
563C22#1681119612yes—
564C22#1681119811yescharacteristic
565C22#1681120211yescharacteristic
566C22#1681120311yescharacteristic
567C22#1681121712yes—
568C22#1681121812yes—
569C22#1681121912yes—
570C22#1681122112yes—
571C22#1681123012yes—
572C22#1681123212yes—
573C22#1681123312yes—
574C22#1681123412yes—
575C4—24no—
576C4—44no—
577C4—42no—
578C4—42no—
579C4—82no—
580C4—1064no—
581C4—1064no—
582C4—4242no—
583C4—2124no—
584C4—2122no—
585C4—2122no—
586C2#2027720011yescharacteristic
587C2#2027729811yescharacteristic
588C2#2027730411yescharacteristic
589C2#2027751312yes—
590C2#2027752012yes—
591C2#2028058211yescharacteristic
592C2#2028058511yescharacteristic
593C2#2028059011yescharacteristic
594C2#2028059311yescharacteristic
595C2#2028087312yes—
596C2#2028087512yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1441C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1441D53
hypocenterC53#14289
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
14. C2×#13133207C4semidirect
338. C53#14289semidirect

Maximal subgroups

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

subgroupindex
C22×#131332072
C2×#168104902
C2×C4⋊C4×Dic532
#202729812
#202729692
#202729682
#202729652
#202729492
#202729462
C53×#14412
#1428953