Finite Group Databasebeta

← #27314799

Subgroups — #27314799

Order 13568.

1434 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810498C212yesmaximal
2#20272980C212yesmaximal, complemented
3#20272977C212yesmaximal, complemented
4#20272972C212yesmaximal, complemented
5#20272968C212yesmaximal, complemented
6#20272964C212yesmaximal
7#20272960C211yesmaximal, characteristic
8#20272937C211yesmaximal, characteristic
9C2×C53×#397C211yesmaximal, characteristic
10C24×Dic53C2211yescharacteristic
11C22×C4⋊Dic53C2212yes—
12C22×C4⋊Dic53C2212yes—
13C22×C4⋊Dic53C2211yescharacteristic
14C22×C22⋊Dic53C2211yescharacteristic
15C22×C22⋊Dic53C2212yes—
16#16810498—28no—
17C2×#13133124C2212yes—
18C2×#13133124C2212yes—
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C22×C4⋊C4×C53C2211yescharacteristic
28C22×C22⋊C4×C53C2212yes—
29C22×C22⋊C4×C53C2211yescharacteristic
30C53×#397C418yescomplemented
31C2×#95×C53C2211yescharacteristic
32C2×#95×C53C2212yes—
33C23×Dic53—42no—
34C23×Dic53—42no—
35C23×Dic53—41no—
36C23×Dic53—41no—
37C23×Dic53D414yes—
38C23×Dic53C2312yes—
39C23×Dic53C2312yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C2×C4⋊Dic53—24no—
43C2×C4⋊Dic53—24no—
44C2×C4⋊Dic53—42no—
45C2×C4⋊Dic53—42no—
46C2×C4⋊Dic53—42no—
47C2×C4⋊Dic53—42no—
48C2×C4⋊Dic53—42no—
49C2×C4⋊Dic53—41no—
50C2×C22⋊Dic53—24no—
51C2×C22⋊Dic53—22no—
52C2×C22⋊Dic53—41no—
53C2×C22⋊Dic53—41no—
54C2×C22⋊Dic53—42no—
55C2×C22⋊Dic53—42no—
56#13133124—24no—
57#13133124—24no—
58#13133124—42no—
59#13133124—42no—
60#13133124—41no—
61#13133124—42no—
62#13133124—42no—
63#13133124—42no—
64#13133124—42no—
65#13133124—42no—
66C25×C53C2311yescharacteristic
67C23×C4×C53C2312yes—
68C23×C4×C53C2311yescharacteristic
69C23×C4×C53C2312yes—
70C23×C4×C53C2311yescharacteristic
71C2×C4⋊C4×C53—41no—
72C2×C4⋊C4×C53—41no—
73C2×C4⋊C4×C53C2×C414yescomplemented
74C2×C22⋊C4×C53—24no—
75C2×C22⋊C4×C53—24no—
76C2×C22⋊C4×C53—41no—
77C2×C22⋊C4×C53—41no—
78C2×C22⋊C4×C53C2×C418yes—
79C2×C22⋊C4×C53C2×C414yes—
80#95×C53C2×C414yescomplemented
81#95×C53C2×C418yescomplemented
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—21no—
85C24×C53—22no—
86C24×C53—22no—
87C24×C53—22no—
88C24×C53—22no—
89C24×C53—21no—
90C24×C53C22×C412yes—
91C24×C53C2×D412yes—
92C24×C53#4112yes—
93C24×C53C2411yescharacteristic
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53C22×C414yes—
115C22×C4×C53C22×C412yes—
116C22×C4×C53C22×C414yes—
117C22×C4×C53C22×C412yes—
118C22×Dic53—22no—
119C22×Dic53—22no—
120C22×Dic53—22no—
121C22×Dic53—22no—
122C22×Dic53—22no—
123C22×Dic53—22no—
124C22×Dic53—22no—
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—48no—
146C22×Dic53—44no—
147C22×Dic53—44no—
148C22×Dic53—48no—
149C22×Dic53—44no—
150C22×Dic53C2×D414yes—
151C4⋊C4×C53—44no—
152C22⋊C4×C53—216no—
153C22⋊C4×C53—44no—
154C22⋊Dic53—48no—
155C22⋊Dic53—44no—
156C4⋊Dic53—48no—
157C4⋊Dic53—48no—
158C4⋊Dic53—44no—
159C23×C53—22no—
160C23×C53—24no—
161C23×C53—24no—
162C23×C53—22no—
163C23×C53—22no—
164C23×C53—24no—
165C23×C53—24no—
166C23×C53—22no—
167C23×C53—22no—
168C23×C53—24no—
169C23×C53—24no—
170C23×C53—22no—
171C23×C53—41no—
172C23×C53—42no—
173C23×C53—41no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—41no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—41no—
181C23×C53C4×D414yes—
182C23×C53C23×C411yescharacteristic
183C23×C53C22×D411yescharacteristic
184C23×C53C2×#4111yescharacteristic
185C23×C53H(5,𝔽2)11yescharacteristic
186C23×C53H(5,𝔽2)11yescharacteristic
187C23×C53H(5,𝔽2)11yescharacteristic
188C23×C53H(5,𝔽2)12yes—
189C23×C53H(5,𝔽2)12yes—
190C23×C532−1+412yes—
191C23×C532−1+412yes—
192C23×C532−1+411yescharacteristic
193C2×C4×C53—24no—
194C2×C4×C53—24no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—24no—
198C2×C4×C53—24no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
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—41no—
212C2×C4×C53—41no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×Dic53—24no—
218C2×Dic53—24no—
219C2×Dic53—24no—
220C2×Dic53—416no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53—42no—
230C2×Dic53—42no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246#14627—531nomaximal, Sylow
247C22×C53—28no—
248C22×C53—24no—
249C22×C53—42no—
250C22×C53—44no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—44no—
256C22×C53—42no—
257C22×C53—44no—
258C22×C53C2×C4×D411yescharacteristic
259C22×C53#51811yescharacteristic
260C22×C53#52012yes—
261C22×C53#52012yes—
262C22×C53#52011yescharacteristic
263C22×C53#53411yescharacteristic
264C22×C53#53412yes—
265C22×C53#53612yes—
266C22×C53#53711yescharacteristic
267C22×C53#53811yescharacteristic
268C22×C53#53912yes—
269C22×C53#53912yes—
270C22×C53#53911yescharacteristic
271C22×C53#56011yescharacteristic
272C22×C53#56012yes—
273C22×C53H(3,𝔽4)11yescharacteristic
274C22×C53#56212yes—
275C22×C53#56212yes—
276C22×C53#56311yescharacteristic
277C22×C53#56312yes—
278C22×C53#56312yes—
279C22×C53#56312yes—
280C22×C53#56311yescharacteristic
281C4×C53—44no—
282C4×C53—44no—
283C4×C53—42no—
284C4×C53—42no—
285Dic53—44no—
286Dic53—82no—
287Dic53—82no—
288Dic53—82no—
289C2×#397D5311yescharacteristic
290C2×#385—532no—
291#1444—532no—
292#1443—532no—
293#1442—532no—
294#1441—532no—
295#1440—532no—
296#1439—531no—
297C22⋊#95—531no—
298C2×C53—44no—
299C2×C53#232212yes—
300C2×C53#232511yescharacteristic
301C2×C53#237611yescharacteristic
302C2×C53#237911yescharacteristic
303C2×C53#238312yes—
304C2×C53#281211yescharacteristic
305C2×C53#281412yes—
306C2×C53#281711yescharacteristic
307C2×C53#282712yes—
308C2×C53#283411yescharacteristic
309C2×C53#283511yescharacteristic
310C24×C4—531no—
311C22×C4⋊C4—532no—
312C22×C4⋊C4—532no—
313C22×C4⋊C4—531no—
314C22×C4⋊C4C2×D5311yescharacteristic
315C22×C22⋊C4—531no—
316C22×C22⋊C4—532no—
317C22×C22⋊C4C2×D5312yes—
318C22×C22⋊C4C2×D5311yescharacteristic
319#397Dic5318yescomplemented
320#385—1068no—
321C2×#95—532no—
322C2×#95—532no—
323C2×#95—532no—
324C2×#95—532no—
325C2×#95—532no—
326C2×#95—531no—
327C2×#95—532no—
328C2×#95—532no—
329C2×#95—532no—
330C2×#95—532no—
331C2×#95C2×D5311yescharacteristic
332C2×#95C2×D5312yes—
333C53#1462711yescharacteristic, complemented, Sylow
334C25C22×D5311yescharacteristic
335C23×C4—532no—
336C23×C4—534no—
337C23×C4—532no—
338C23×C4—532no—
339C23×C4—532no—
340C23×C4—2121no—
341C23×C4—2122no—
342C23×C4—2122no—
343C23×C4—2121no—
344C23×C4C22×D5312yes—
345C23×C4C22×D5312yes—
346C23×C4C22×D5311yescharacteristic
347C23×C4C22×D5311yescharacteristic
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—1064no—
351C2×C4⋊C4—1064no—
352C2×C4⋊C4—2122no—
353C2×C4⋊C4—2122no—
354C2×C4⋊C4—2122no—
355C2×C4⋊C4—2122no—
356C2×C4⋊C4—2121no—
357C2×C4⋊C4—2122no—
358C2×C4⋊C4C2×Dic5314yescomplemented
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—41no—
362C2×C22⋊C4—41no—
363C2×C22⋊C4—1064no—
364C2×C22⋊C4—1062no—
365C2×C22⋊C4—2121no—
366C2×C22⋊C4—2121no—
367C2×C22⋊C4—2122no—
368C2×C22⋊C4—2122no—
369C2×C22⋊C4C2×Dic5318yes—
370C2×C22⋊C4C2×Dic5314yes—
371#95—1064no—
372#95—1064no—
373#95—2121no—
374#95—2122no—
375#95—2122no—
376#95—2122no—
377#95—2122no—
378#95—2122no—
379#95—2122no—
380#95—2122no—
381#95C2×Dic5314yescomplemented
382#95C2×Dic5318yescomplemented
383C24—21no—
384C24—21no—
385C24—21no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—21no—
391C24D4×D5312yes—
392C24#1171994712yes—
393C24C22×Dic5312yes—
394C24C23×D5311yescharacteristic
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—534no—
416C22×C4—1062no—
417C22×C4—1062no—
418C22×C4—1062no—
419C22×C4—1062no—
420C22×C4—1062no—
421C22×C4—1062no—
422C22×C4—1062no—
423C22×C4—1062no—
424C22×C4—1062no—
425C22×C4—1062no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—2124no—
444C22×C4—2128no—
445C22×C4—2124no—
446C22×C4—2128no—
447C22×C4—2124no—
448C22×C4C22×Dic5314yes—
449C22×C4C22×Dic5312yes—
450C22×C4C22×Dic5314yes—
451C22×C4C22×Dic5312yes—
452C4⋊C4—44no—
453C4⋊C4—2128no—
454C4⋊C4—2128no—
455C4⋊C4—2124no—
456C22⋊C4—216no—
457C22⋊C4—44no—
458C22⋊C4—2128no—
459C22⋊C4—2124no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23D4×Dic5314yes—
483C23C23×Dic5311yescharacteristic
484C23C2×D4×D5311yescharacteristic
485C23C2×#1171994711yescharacteristic
486C23#1313326211yescharacteristic
487C23#1313326211yescharacteristic
488C23#1313326312yes—
489C23#1313326311yescharacteristic
490C23#1313326312yes—
491C23#1313326412yes—
492C23#1313326412yes—
493C23#1313326411yescharacteristic
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—1064no—
519C2×C4—1064no—
520C2×C4—1064no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4—2122no—
526C2×C4—2122no—
527C2×C4—2122no—
528C2×C4—2122no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—21216no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C22—28no—
548C22—24no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—44no—
558C22C2×D4×Dic5311yescharacteristic
559C22#1681109811yescharacteristic
560C22#1681110612yes—
561C22#1681110612yes—
562C22#1681110611yescharacteristic
563C22#1681118111yescharacteristic
564C22#1681118412yes—
565C22#1681119512yes—
566C22#1681120011yescharacteristic
567C22#1681121011yescharacteristic
568C22#1681121712yes—
569C22#1681121712yes—
570C22#1681121811yescharacteristic
571C22#1681136311yescharacteristic
572C22#1681136412yes—
573C22#1681136511yescharacteristic
574C22#1681136712yes—
575C22#1681137012yes—
576C22#1681137311yescharacteristic
577C22#1681137512yes—
578C22#1681137512yes—
579C22#1681137512yes—
580C22#1681137611yescharacteristic
581C4—42no—
582C4—42no—
583C4—44no—
584C4—44no—
585C4—4242no—
586C4—4242no—
587C4—4242no—
588C4—2124no—
589C2—44no—
590C2#2027738712yes—
591C2#2027742011yescharacteristic
592C2#2027786311yescharacteristic
593C2#2027788911yescharacteristic
594C2#2027790712yes—
595C2#2028271311yescharacteristic
596C2#2028273412yes—
597C2#2028275311yescharacteristic
598C2#2028282512yes—
599C2#2028290011yescharacteristic
600C2#2028290511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#397C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×#397D53
hypocenterC53#14627
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. #20272980C2semidirect
3. #20272977C2semidirect
4. #20272972C2semidirect
5. #20272968C2semidirect
30. C53×#397C4semidirect
73. C2×C4⋊C4×C53C2×C4semidirect
80. #95×C53C2×C4semidirect
81. #95×C53C2×C4semidirect
319. #397Dic53semidirect
333. C53#14627semidirect
358. C2×C4⋊C4C2×Dic53semidirect
381. #95C2×Dic53semidirect
382. #95C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104982
#202729802
#202729772
#202729722
#202729682
#202729642
#202729602
#202729372
C2×C53×#3972
#1462753