Finite Group Databasebeta

← #27306706

Subgroups — #27306706

Order 13568.

1486 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Dic53C211yesmaximal, characteristic
2#20272976C212yesmaximal, complemented
3#20272962C212yesmaximal
4#20272952C212yesmaximal
5#20272941C211yesmaximal, characteristic, complemented
6#20272941C211yesmaximal, characteristic, complemented
7#20272940C211yesmaximal, characteristic, complemented
8#20272939C212yesmaximal, complemented
9#20272938C211yesmaximal, characteristic
10#95×Dic53C211yesmaximal, characteristic, complemented
11C53×C22⋊#95C211yesmaximal, characteristic
12C22×C22⋊Dic53C2212yes—
13C22×C22⋊Dic53C2211yescharacteristic
14C22×C22⋊Dic53C2211yescharacteristic
15C22×C4×Dic53C2212yes—
16C22×C4×Dic53C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C4×C22⋊Dic53C418yescomplemented
19C2×#13133124C2211yescharacteristic
20C2×#13133124C2212yes—
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C24×C4×C53C2211yescharacteristic
32C22×C22⋊C4×C53C2212yes—
33C2×#95×C53C2211yescharacteristic
34C2×#95×C53C2211yescharacteristic
35C2×#95×C53C2212yes—
36C23×Dic53C2312yes—
37C23×Dic53C2311yescharacteristic
38C23×Dic53C2311yescharacteristic
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2311yescharacteristic
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—24no—
45C2×C22⋊Dic53—22no—
46C2×C22⋊Dic53—22no—
47C2×C22⋊Dic53—24no—
48C2×C22⋊Dic53—42no—
49C2×C22⋊Dic53C2×C414yes—
50C2×C22⋊Dic53C2×C414yes—
51C2×C4×Dic53—24no—
52C2×C4×Dic53—24no—
53C2×C4×Dic53—42no—
54C2×C4×Dic53—42no—
55C2×C4×Dic53—42no—
56C2×C4×Dic53C2×C414yes—
57C2×C4×Dic53C2×C414yes—
58#13133124—22no—
59#13133124—24no—
60#13133124—22no—
61#13133124—24no—
62#13133124—24no—
63#13133124—24no—
64#13133124—42no—
65#13133124—42no—
66#13133124—42no—
67#13133124—42no—
68#13133124C2×C414yes—
69#13133124C2×C414yes—
70C25×C53C2311yescharacteristic
71C23×C4×C53—22no—
72C23×C4×C53—22no—
73C23×C4×C53—42no—
74C23×C4×C53—42no—
75C23×C4×C53C2×C414yes—
76C23×C4×C53C2312yes—
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2312yes—
79C23×C4×C53C2311yescharacteristic
80C2×C22⋊C4×C53—24no—
81C2×C22⋊C4×C53—24no—
82C2×C22⋊C4×C53—42no—
83#95×C53—22no—
84#95×C53—22no—
85#95×C53—42no—
86C24×C53—22no—
87C24×C53—21no—
88C24×C53—21no—
89C24×C53—21no—
90C24×C53—21no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53C22×C412yes—
95C24×C53#4112yes—
96C24×C53#4112yes—
97C24×C53C2411yescharacteristic
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
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—28no—
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—44no—
119C22×C4×C53—44no—
120C22×C4×C53—48no—
121C22×C4×C53—48no—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C22×C412yes—
124C22×C4×C53#4114yes—
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
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—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
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×Dic53#4114yes—
159C22×Dic53#4112yes—
160C22×Dic53#4112yes—
161C22⋊C4×C53—48no—
162C22⋊Dic53—28no—
163C22⋊Dic53—28no—
164C22⋊Dic53—48no—
165C4×Dic53—44no—
166C4×Dic53—44no—
167C4×Dic53—48no—
168C23×C53—22no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—24no—
179C23×C53—24no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53#11714yes—
191C23×C53C23×C411yescharacteristic
192C23×C53C2×#4111yescharacteristic
193C23×C53C2×#4111yescharacteristic
194C23×C53C2×#4112yes—
195C23×C53C2×#4111yescharacteristic
196C23×C53C2×#4111yescharacteristic
197C23×C53C2×#4111yescharacteristic
198C23×C53C2×#4112yes—
199C23×C53C2×#4111yescharacteristic
200C23×C53H(5,𝔽2)12yes—
201C23×C53H(5,𝔽2)12yes—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—24no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—416no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53#11714yes—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—24no—
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×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53#11714yes—
261#13947—531nomaximal, Sylow
262C22×C53—24no—
263C22×C53—24no—
264C22×C53—24no—
265C22×C53—42no—
266C22×C53—44no—
267C22×C53—44no—
268C22×C53—44no—
269C22×C53—44no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53C2×#11711yescharacteristic
274C22×C53C2×#11711yescharacteristic
275C22×C53C2×#11711yescharacteristic
276C22×C53C4×#4111yescharacteristic
277C22×C53C4×#4111yescharacteristic
278C22×C53#51811yescharacteristic
279C22×C53#51811yescharacteristic
280C22×C53#52912yes—
281C22×C53#52911yescharacteristic
282C22×C53#52911yescharacteristic
283C22×C53#52912yes—
284C22×C53#52911yescharacteristic
285C22×C53#52911yescharacteristic
286C22×C53#53812yes—
287C22×C53#53812yes—
288C22×C53#55112yes—
289C22×C53#55112yes—
290C22×C53#55112yes—
291C22×C53#55312yes—
292C22×C53#55312yes—
293C22×C53#55312yes—
294C22×C53#55312yes—
295C22×C53#55912yes—
296C4×C53—24no—
297C4×C53—42no—
298C4×C53—44no—
299C4×C53—82no—
300Dic53—24no—
301Dic53—44no—
302Dic53—42no—
303Dic53—42no—
304Dic53—42no—
305Dic53—82no—
306C2×C4×C22⋊C4—531no—
307#1443—532no—
308#1440—532no—
309#1437—532no—
310#1434—532no—
311#1434—531no—
312#1434—531no—
313#1434—531no—
314C22⋊#95—531no—
315C22⋊#95D5311yescharacteristic
316C4×#95—531no—
317C2×C53—44no—
318C2×C53#228311yescharacteristic
319C2×C53#229811yescharacteristic
320C2×C53#231211yescharacteristic
321C2×C53#235311yescharacteristic
322C2×C53#235311yescharacteristic
323C2×C53#235311yescharacteristic
324C2×C53#235711yescharacteristic
325C2×C53#257612yes—
326C2×C53#257812yes—
327C2×C53#259512yes—
328C2×C53#259712yes—
329C24×C4C2×D5311yescharacteristic
330C22×C22⋊C4—531no—
331C22×C22⋊C4—531no—
332C22×C22⋊C4—532no—
333C22×C22⋊C4C2×D5312yes—
334C22×C42—531no—
335C22×C42—531no—
336C22×C42—532no—
337C4×C22⋊C4—538no—
338C2×#95—531no—
339C2×#95—531no—
340C2×#95—531no—
341C2×#95—532no—
342C2×#95—531no—
343C2×#95—532no—
344C2×#95—532no—
345C2×#95—532no—
346C2×#95—532no—
347C2×#95—532no—
348C2×#95—532no—
349C2×#95—532no—
350C2×#95C2×D5311yescharacteristic
351C2×#95C2×D5311yescharacteristic
352C2×#95C2×D5312yes—
353C53#1394711yescharacteristic, complemented, Sylow
354C25C22×D5311yescharacteristic
355C23×C4—22no—
356C23×C4—22no—
357C23×C4—42no—
358C23×C4—42no—
359C23×C4—531no—
360C23×C4—531no—
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—532no—
364C23×C4—532no—
365C23×C4C4×D5314yes—
366C23×C4C22×D5312yes—
367C23×C4C22×D5311yescharacteristic
368C23×C4C22×D5312yes—
369C23×C4C22×D5311yescharacteristic
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—534no—
374C2×C22⋊C4—534no—
375C2×C22⋊C4—1062no—
376C2×C22⋊C4—1062no—
377C2×C22⋊C4—1062no—
378C2×C22⋊C4—1062no—
379C2×C22⋊C4—1064no—
380C2×C22⋊C4—1064no—
381C2×C22⋊C4—2122no—
382C2×C42—534no—
383C2×C42—534no—
384C2×C42—1064no—
385C2×C42—1064no—
386C2×C42—2122no—
387C2×C42—2122no—
388C2×C42—2122no—
389#95—22no—
390#95—22no—
391#95—42no—
392#95—534no—
393#95—534no—
394#95—1064no—
395#95—1064no—
396#95—1064no—
397#95—1064no—
398#95—1062no—
399#95—1062no—
400#95—2122no—
401#95—2122no—
402#95—2122no—
403#95—2122no—
404C24—22no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24C2×C4×D5312yes—
413C24#1171994712yes—
414C24#1171994912yes—
415C24C23×D5311yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—532no—
441C22×C4—532no—
442C22×C4—532no—
443C22×C4—532no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—534no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1062no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4C2×C4×D5312yes—
475C22×C4C2×C4×D5312yes—
476C22×C4#1171994514yes—
477C22×C4#1171994712yes—
478C22×C4#1171994712yes—
479C22⋊C4—48no—
480C22⋊C4—1068no—
481C22⋊C4—1068no—
482C22⋊C4—2128no—
483C42—2124no—
484C42—2124no—
485C42—2128no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23#1313317514yes—
509C23C22×C4×D5311yescharacteristic
510C23#41×D5312yes—
511C23#41×D5311yescharacteristic
512C23#41×D5311yescharacteristic
513C23C2×#1171994711yescharacteristic
514C23C2×#1171994711yescharacteristic
515C23C2×#1171994711yescharacteristic
516C23C2×#1171994512yes—
517C23C2×#1171994911yescharacteristic
518C23#1313326212yes—
519C23#1313326212yes—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—416no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—534no—
545C2×C4—1064no—
546C2×C4—1064no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1064no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4#1313317314yes—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—44no—
590C22#117×D5311yescharacteristic
591C22C2×#1313317311yescharacteristic
592C22C2×#1313317511yescharacteristic
593C22#1681109111yescharacteristic
594C22C4×#1171994511yescharacteristic
595C22#1681109611yescharacteristic
596C22#1681109611yescharacteristic
597C22#1681115312yes—
598C22#1681115411yescharacteristic
599C22#1681115411yescharacteristic
600C22#1681115512yes—
601C22#1681115911yescharacteristic
602C22#1681115911yescharacteristic
603C22#1681120812yes—
604C22#1681120812yes—
605C22#1681129912yes—
606C22#1681130012yes—
607C22#1681130212yes—
608C22#1681131612yes—
609C22#1681131812yes—
610C22#1681132112yes—
611C22#1681132312yes—
612C22#1681135712yes—
613C4—24no—
614C4—42no—
615C4—44no—
616C4—82no—
617C4—1064no—
618C4—4242no—
619C4—2122no—
620C4—2122no—
621C4—2122no—
622C4—2124no—
623C2—44no—
624C2#2027711011yescharacteristic
625C2#2027721111yescharacteristic
626C2#2027731111yescharacteristic
627C2#2027765111yescharacteristic
628C2#2027765711yescharacteristic
629C2#2027765811yescharacteristic
630C2#2027768311yescharacteristic
631C2#2028001612yes—
632C2#2028004812yes—
633C2#2028023312yes—
634C2#2028026512yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22⋊#95D53
hypocenterC53#13947
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. #20272976C2semidirect
5. #20272941C2semidirect
6. #20272941C2semidirect
7. #20272940C2semidirect
8. #20272939C2semidirect
10. #95×Dic53C2semidirect
18. C4×C22⋊Dic53C4semidirect
353. C53#13947semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Dic532
#202729762
#202729622
#202729522
#202729412
#202729402
#202729392
#202729382
#95×Dic532
C53×C22⋊#952
#1394753