Finite Group Databasebeta

← #27306271

Subgroups — #27306271

Order 13568.

1602 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810483C211yesmaximal, characteristic
2C2×C4×C22⋊Dic53C211yesmaximal, characteristic
3C2×#16810468C211yesmaximal, characteristic, complemented
4#20272971C212yesmaximal, complemented
5#20272962C212yesmaximal
6#20272952C212yesmaximal
7#20272949C212yesmaximal, complemented
8#20272941C211yesmaximal, characteristic, complemented
9#20272940C211yesmaximal, characteristic, complemented
10#20272939C211yesmaximal, characteristic, complemented
11C53×C22⋊#95C211yesmaximal, characteristic
12C22×C4⋊Dic53C2212yes—
13C22×C22⋊Dic53C2212yes—
14C22×C22⋊Dic53C2211yescharacteristic
15C22×C22⋊Dic53C2211yescharacteristic
16C22×C4×Dic53C2212yes—
17C22×C4×Dic53C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810483—24no—
20C4×C22⋊Dic53C418yescomplemented
21#16810468—24no—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33C24×C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2212yes—
35C2×#95×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C23×Dic53C2312yes—
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C2×C4⋊Dic53—24no—
45C2×C4⋊Dic53—24no—
46C2×C4⋊Dic53—42no—
47C2×C22⋊Dic53—22no—
48C2×C22⋊Dic53—22no—
49C2×C22⋊Dic53—24no—
50C2×C22⋊Dic53—24no—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—42no—
53C2×C22⋊Dic53C2×C414yes—
54C2×C22⋊Dic53C2×C414yes—
55C2×C4×Dic53—22no—
56C2×C4×Dic53—24no—
57C2×C4×Dic53—22no—
58C2×C4×Dic53—24no—
59C2×C4×Dic53—24no—
60C2×C4×Dic53—42no—
61C2×C4×Dic53C2×C414yes—
62C2×C4×Dic53C2×C414yes—
63#13133124—22no—
64#13133124—24no—
65#13133124—24no—
66#13133124—24no—
67#13133124—42no—
68#13133124—42no—
69#13133124—42no—
70#13133124—42no—
71#13133124C2×C414yes—
72#13133124C2×C414yes—
73#13133124D414yes—
74C25×C53C2311yescharacteristic
75C23×C4×C53—22no—
76C23×C4×C53—24no—
77C23×C4×C53—22no—
78C23×C4×C53—42no—
79C23×C4×C53C2×C414yes—
80C23×C4×C53C2312yes—
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84C2×C22⋊C4×C53—24no—
85C2×C22⋊C4×C53—24no—
86C2×C22⋊C4×C53—42no—
87#95×C53—22no—
88#95×C53—24no—
89#95×C53—22no—
90C24×C53—21no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—21no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—21no—
97C24×C53—21no—
98C24×C53C22×C412yes—
99C24×C53#4112yes—
100C24×C53#4112yes—
101C24×C53C2411yescharacteristic
102C22×C4×C53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—21no—
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—28no—
114C22×C4×C53—28no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—44no—
123C22×C4×C53—48no—
124C22×C4×C53—48no—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C2×D412yes—
128C22×C4×C53#4114yes—
129C22×C4×C53#4114yes—
130C22×C4×C53#4112yes—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53C22×C412yes—
162C22×Dic53C22×C412yes—
163C22×Dic53C22×C412yes—
164C22×Dic53C22×C412yes—
165C22×Dic53C2×D414yes—
166C22×Dic53#4114yes—
167C22⋊C4×C53—48no—
168C22⋊Dic53—28no—
169C22⋊Dic53—28no—
170C22⋊Dic53—48no—
171C4⋊Dic53—48no—
172C4×Dic53—216no—
173C4×Dic53—28no—
174C4×Dic53—44no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—24no—
186C23×C53—22no—
187C23×C53—24no—
188C23×C53—22no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53#11714yes—
198C23×C53C23×C411yescharacteristic
199C23×C53C22×D411yescharacteristic
200C23×C53C2×#4112yes—
201C23×C53C2×#4112yes—
202C23×C53C2×#4111yescharacteristic
203C23×C53C2×#4111yescharacteristic
204C23×C53C2×#4112yes—
205C23×C53C2×#4111yescharacteristic
206C23×C53H(5,𝔽2)12yes—
207C23×C53H(5,𝔽2)11yescharacteristic
208C23×C532−1+411yescharacteristic
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—416no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53#11714yes—
233C2×C4×C53C4×D414yes—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—24no—
237C2×Dic53—24no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—42no—
257C2×Dic53—41no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—41no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53#11714yes—
270#13898—531nomaximal, Sylow
271C22×C53—24no—
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53—44no—
281C22×C53—42no—
282C22×C53C2×#11711yescharacteristic
283C22×C53C2×#11711yescharacteristic
284C22×C53C2×#11711yescharacteristic
285C22×C53C2×C4×D411yescharacteristic
286C22×C53C4×#4111yescharacteristic
287C22×C53#51811yescharacteristic
288C22×C53#52011yescharacteristic
289C22×C53#52512yes—
290C22×C53#52912yes—
291C22×C53#52912yes—
292C22×C53#52912yes—
293C22×C53#54012yes—
294C22×C53#54212yes—
295C22×C53#54611yescharacteristic
296C22×C53#54612yes—
297C22×C53#54811yescharacteristic
298C22×C53#55112yes—
299C22×C53#55112yes—
300C22×C53#55212yes—
301C22×C53#55211yescharacteristic
302C22×C53#55312yes—
303C22×C53#55311yescharacteristic
304C22×C53#55512yes—
305C4×C53—24no—
306C4×C53—24no—
307C4×C53—44no—
308C4×C53—82no—
309Dic53—24no—
310Dic53—42no—
311Dic53—42no—
312Dic53—42no—
313Dic53—44no—
314Dic53—82no—
315C2×#380—531no—
316C2×C4×C22⋊C4—531no—
317C2×#376—531no—
318#1442—532no—
319#1440—532no—
320#1437—532no—
321#1436—532no—
322#1434—531no—
323#1434—531no—
324#1434—531no—
325C22⋊#95D5311yescharacteristic
326C2×C53—44no—
327C2×C53#228311yescharacteristic
328C2×C53#229011yescharacteristic
329C2×C53#231911yescharacteristic
330C2×C53#234211yescharacteristic
331C2×C53#234611yescharacteristic
332C2×C53#235211yescharacteristic
333C2×C53#235311yescharacteristic
334C2×C53#247612yes—
335C2×C53#247712yes—
336C2×C53#253012yes—
337C2×C53#253112yes—
338C24×C4C2×D5311yescharacteristic
339C22×C4⋊C4—532no—
340C22×C22⋊C4—531no—
341C22×C22⋊C4—531no—
342C22×C22⋊C4—532no—
343C22×C22⋊C4C2×D5312yes—
344C22×C42—531no—
345C22×C42—531no—
346C22×C42—532no—
347#380—1064no—
348C4×C22⋊C4—538no—
349#376—1064no—
350C2×#95—531no—
351C2×#95—531no—
352C2×#95—531no—
353C2×#95—531no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95C2×D5311yescharacteristic
362C2×#95C2×D5311yescharacteristic
363C2×#95C2×D5312yes—
364C53#1389811yescharacteristic, complemented, Sylow
365C25C22×D5311yescharacteristic
366C23×C4—22no—
367C23×C4—24no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—531no—
371C23×C4—532no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—532no—
376C23×C4C4×D5314yes—
377C23×C4C22×D5312yes—
378C23×C4C22×D5312yes—
379C23×C4C22×D5311yescharacteristic
380C23×C4C22×D5311yescharacteristic
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—2122no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—534no—
388C2×C22⋊C4—534no—
389C2×C22⋊C4—1064no—
390C2×C22⋊C4—1062no—
391C2×C22⋊C4—1062no—
392C2×C22⋊C4—1064no—
393C2×C22⋊C4—1064no—
394C2×C22⋊C4—2122no—
395C2×C42—534no—
396C2×C42—534no—
397C2×C42—1064no—
398C2×C42—1064no—
399C2×C42—1064no—
400C2×C42—1062no—
401C2×C42—1062no—
402C2×C42—2122no—
403#95—22no—
404#95—24no—
405#95—22no—
406#95—534no—
407#95—534no—
408#95—534no—
409#95—1064no—
410#95—1064no—
411#95—1064no—
412#95—1062no—
413#95—2122no—
414#95—2122no—
415#95—2122no—
416#95—2122no—
417C24—21no—
418C24—22no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—21no—
424C24—21no—
425C24C2×C4×D5312yes—
426C24#1171994712yes—
427C24#1171994912yes—
428C24C23×D5311yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—532no—
453C22×C4—534no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—534no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1062no—
461C22×C4—1061no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1062no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4C2×C4×D5312yes—
489C22×C4C2×C4×D5312yes—
490C22×C4#1171994514yes—
491C22×C4D4×D5312yes—
492C22×C4#1171994714yes—
493C22×C4#1171994712yes—
494C4⋊C4—2128no—
495C22⋊C4—48no—
496C22⋊C4—1068no—
497C22⋊C4—1068no—
498C22⋊C4—2128no—
499C42—10616no—
500C42—1068no—
501C42—2124no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23#1313317514yes—
525C23C22×C4×D5311yescharacteristic
526C23C2×D4×D5311yescharacteristic
527C23#41×D5312yes—
528C23C2×#1171994711yescharacteristic
529C23C2×#1171994711yescharacteristic
530C23C2×#1171994712yes—
531C23C2×#1171994512yes—
532C23C2×#1171994911yescharacteristic
533C23#1313326212yes—
534C23#1313326311yescharacteristic
535C23#1313326411yescharacteristic
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—416no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—534no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1064no—
563C2×C4—1064no—
564C2×C4—1064no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2122no—
586C2×C4—2121no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2121no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4#1313317314yes—
596C2×C4#1313317714yes—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—44no—
608C22#117×D5311yescharacteristic
609C22C2×#1313317311yescharacteristic
610C22C2×#1313317511yescharacteristic
611C22C2×#1313317711yescharacteristic
612C22C4×#1171994511yescharacteristic
613C22#1681109611yescharacteristic
614C22#1681110311yescharacteristic
615C22#1681113212yes—
616C22#1681115412yes—
617C22#1681115512yes—
618C22#1681115912yes—
619C22#1681122512yes—
620C22#1681124112yes—
621C22#1681126012yes—
622C22#1681126411yescharacteristic
623C22#1681127911yescharacteristic
624C22#1681129912yes—
625C22#1681130412yes—
626C22#1681131312yes—
627C22#1681131511yescharacteristic
628C22#1681132112yes—
629C22#1681132511yescharacteristic
630C22#1681133812yes—
631C4—24no—
632C4—24no—
633C4—44no—
634C4—82no—
635C4—1064no—
636C4—4242no—
637C4—2122no—
638C4—2124no—
639C4—2122no—
640C4—2122no—
641C2—44no—
642C2#2027711011yescharacteristic
643C2#2027716111yescharacteristic
644C2#2027735511yescharacteristic
645C2#2027756611yescharacteristic
646C2#2027759811yescharacteristic
647C2#2027764611yescharacteristic
648C2#2027765111yescharacteristic
649C2#2027890512yes—
650C2#2027892012yes—
651C2#2027955112yes—
652C2#2027956612yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22⋊#95D53
hypocenterC53#13898
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#16810468C2semidirect
4. #20272971C2semidirect
7. #20272949C2semidirect
8. #20272941C2semidirect
9. #20272940C2semidirect
10. #20272939C2semidirect
20. C4×C22⋊Dic53C4semidirect
364. C53#13898semidirect

Maximal subgroups

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

subgroupindex
C2×#168104832
C2×C4×C22⋊Dic532
C2×#168104682
#202729712
#202729622
#202729522
#202729492
#202729412
#202729402
#202729392
C53×C22⋊#952
#1389853