Finite Group Databasebeta

← #27305917

Subgroups — #27305917

Order 13568.

1790 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810548C211yesmaximal, characteristic
2C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×Dic53C212yesmaximal
4C2×C22⋊C4×Dic53C211yesmaximal, characteristic
5#20272984C211yesmaximal, characteristic, complemented
6#20272976C212yesmaximal, complemented
7#20272974C211yesmaximal, characteristic, complemented
8#20272962C212yesmaximal
9#95×Dic53C212yesmaximal, complemented
10#95×Dic53C211yesmaximal, characteristic, complemented
11C53×#1439C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×C22⋊Dic53C2211yescharacteristic
15C22×C22⋊Dic53C2212yes—
16C22×C4×Dic53C2211yescharacteristic
17C22×C4×Dic53C2212yes—
18C22×C4×Dic53C2212yes—
19C22×C4×Dic53C2211yescharacteristic
20#16810548C418yescomplemented
21C4⋊C4×Dic53—24no—
22C22⋊C4×Dic53—28no—
23C22⋊C4×Dic53—24no—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2212yes—
34C22×C4⋊C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2212yes—
36C22×C22⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C23×Dic53—22no—
40C23×Dic53—22no—
41C23×Dic53—24no—
42C23×Dic53—24no—
43C23×Dic53D414yes—
44C23×Dic53C2312yes—
45C23×Dic53C2312yes—
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2311yescharacteristic
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53C2×C414yescomplemented
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—22no—
55C2×C22⋊Dic53—24no—
56C2×C22⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53C2×C418yes—
59C2×C4×Dic53—24no—
60C2×C4×Dic53—24no—
61C2×C4×Dic53—22no—
62C2×C4×Dic53—24no—
63C2×C4×Dic53—24no—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53D414yes—
69#13133124—22no—
70#13133124—24no—
71#13133124—24no—
72#13133124—22no—
73#13133124—24no—
74#13133124—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124C2×C414yescomplemented
78#13133124C2×C418yescomplemented
79C25×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2312yes—
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—24no—
86C2×C22⋊C4×C53—24no—
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53—24no—
89C2×C22⋊C4×C53—24no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53C2×C414yes—
92#95×C53—24no—
93#95×C53—22no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—21no—
99C24×C53—21no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53C22×C412yes—
103C24×C53C2×D412yes—
104C24×C53#4112yes—
105C24×C53C2411yescharacteristic
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
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—22no—
123C22×C4×C53—22no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53#4114yes—
128C22×C4×C53#4114yes—
129C22×C4×C53#4112yes—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—21no—
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—22no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—28no—
155C22×Dic53—216no—
156C22×Dic53—44no—
157C22×Dic53—44no—
158C22×Dic53—48no—
159C22×Dic53C22×C414yes—
160C22×Dic53C22×C414yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D412yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×D412yes—
165C22×Dic53#4112yes—
166C22×Dic53#4112yes—
167C22×Dic53#4114yes—
168C22×Dic53#4114yes—
169C4⋊C4×C53—28no—
170C22⋊C4×C53—216no—
171C22⋊C4×C53—28no—
172C22⋊Dic53—28no—
173C22⋊Dic53—216no—
174C4⋊Dic53—28no—
175C4×Dic53—28no—
176C4×Dic53—48no—
177C4×Dic53—48no—
178C4×Dic53—44no—
179C23×C53—22no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—24no—
185C23×C53—24no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—22no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—41no—
201C23×C53C4×D414yes—
202C23×C53C23×C411yescharacteristic
203C23×C53C22×D411yescharacteristic
204C23×C53C22×D411yescharacteristic
205C23×C53C2×#4111yescharacteristic
206C23×C53C2×#4112yes—
207C23×C53C2×#4112yes—
208C23×C53C2×#4111yescharacteristic
209C23×C53C2×#4111yescharacteristic
210C23×C53C2×#4112yes—
211C23×C53C2×#4112yes—
212C23×C53C2×#4111yescharacteristic
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—22no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—41no—
236C2×C4×C53C4×D414yes—
237C2×Dic53—24no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—22no—
244C2×Dic53—24no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—24no—
249C2×Dic53—22no—
250C2×Dic53—42no—
251C2×Dic53—41no—
252C2×Dic53—42no—
253C2×Dic53—41no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—416no—
267C2×Dic53#12114yes—
268C2×Dic53#12318yes—
269C2×Dic53#12314yes—
270C2×Dic53#12414yes—
271C2×Dic53#12618yes—
272C4×#397—531nomaximal, Sylow
273C22×C53—24no—
274C22×C53—28no—
275C22×C53—44no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—44no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—44no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53C2×C4×D411yescharacteristic
285C22×C53C2×C4×D411yescharacteristic
286C22×C53C4×#4112yes—
287C22×C53C4×#4112yes—
288C22×C53C4×#4111yescharacteristic
289C22×C53C2×#12111yescharacteristic
290C22×C53C2×#12311yescharacteristic
291C22×C53C2×#12312yes—
292C22×C53#52512yes—
293C22×C53#52511yescharacteristic
294C22×C53#52511yescharacteristic
295C22×C53C2×#12411yescharacteristic
296C22×C53C2×#12612yes—
297C22×C53#52912yes—
298C22×C53#52912yes—
299C22×C53#52912yes—
300C22×C53#52912yes—
301C22×C53#52912yes—
302C22×C53#52912yes—
303C22×C53#52912yes—
304C22×C53#52911yescharacteristic
305C22×C53#53211yescharacteristic
306C22×C53#53211yescharacteristic
307C4×C53—24no—
308C4×C53—44no—
309C4×C53—44no—
310C4×C53—42no—
311Dic53—42no—
312Dic53—42no—
313Dic53—42no—
314Dic53—44no—
315Dic53—44no—
316Dic53#39718yescomplemented
317C2×#397—531no—
318C2×C4×C4⋊C4—531no—
319C2×C4×C22⋊C4—532no—
320C2×C4×C22⋊C4—531no—
321#1445—531no—
322#1443—532no—
323#1442—531no—
324#1440—532no—
325#1439D5311yescharacteristic
326C4×#95—531no—
327C4×#95—532no—
328C2×C53—44no—
329C2×C53C4×#12111yescharacteristic
330C2×C53C4×#12312yes—
331C2×C53C4×#12311yescharacteristic
332C2×C53C4×#12411yescharacteristic
333C2×C53C4×#12612yes—
334C2×C53C2×#39711yescharacteristic
335C2×C53#239211yescharacteristic
336C2×C53#239512yes—
337C2×C53#239511yescharacteristic
338C2×C53#239811yescharacteristic
339C2×C53#240112yes—
340C24×C4—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4C2×D5311yescharacteristic
343C22×C22⋊C4—532no—
344C22×C22⋊C4—531no—
345C22×C22⋊C4C2×D5312yes—
346C22×C22⋊C4C2×D5311yescharacteristic
347C22×C42—531no—
348C22×C42—531no—
349C22×C42—532no—
350C22×C42—532no—
351#397—538no—
352C4×C4⋊C4—1064no—
353C4×C22⋊C4—1064no—
354C4×C22⋊C4—1068no—
355C2×#95—531no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—531no—
363C2×#95—531no—
364C2×#95—532no—
365C2×#95C2×D5312yes—
366C2×#95C2×D5311yescharacteristic
367C53C4×#39711yescharacteristic, complemented, Sylow
368C25C22×D5311yescharacteristic
369C23×C4—531no—
370C23×C4—534no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—1062no—
377C23×C4—1062no—
378C23×C4—1064no—
379C23×C4—1064no—
380C23×C4C22×D5311yescharacteristic
381C23×C4C22×D5312yes—
382C23×C4C22×D5312yes—
383C23×C4C22×D5311yescharacteristic
384C2×C4⋊C4—22no—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—534no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—538no—
395C2×C22⋊C4—1062no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4—1064no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4C4×D5314yes—
401C2×C42—534no—
402C2×C42—1064no—
403C2×C42—1062no—
404C2×C42—1064no—
405C2×C42—1064no—
406C2×C42—1064no—
407C2×C42—1064no—
408C2×C42—1064no—
409C2×C42—1064no—
410C2×C42—1064no—
411#95—24no—
412#95—22no—
413#95—534no—
414#95—538no—
415#95—1064no—
416#95—1064no—
417#95—1064no—
418#95—1062no—
419#95—1064no—
420#95—1062no—
421#95—1064no—
422#95—1064no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—22no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24C2×C4×D5312yes—
432C24D4×D5312yes—
433C24#1171994912yes—
434C24C23×D5311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—534no—
457C22×C4—534no—
458C22×C4—534no—
459C22×C4—534no—
460C22×C4—532no—
461C22×C4—532no—
462C22×C4—532no—
463C22×C4—1068no—
464C22×C4—10616no—
465C22×C4—1061no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—2124no—
490C22×C4—2124no—
491C22×C4—2128no—
492C22×C4C2×C4×D5312yes—
493C22×C4C2×C4×D5312yes—
494C22×C4D4×D5312yes—
495C22×C4#1171994714yes—
496C22×C4#1171994714yes—
497C22×C4#1171994712yes—
498C4⋊C4—28no—
499C4⋊C4—1068no—
500C22⋊C4—216no—
501C22⋊C4—28no—
502C22⋊C4—1068no—
503C22⋊C4—10616no—
504C42—1068no—
505C42—2128no—
506C42—2128no—
507C42—2124no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23#1313317914yes—
531C23C22×C4×D5311yescharacteristic
532C23C2×D4×D5311yescharacteristic
533C23C2×D4×D5311yescharacteristic
534C23#41×D5312yes—
535C23#41×D5312yes—
536C23#41×D5311yescharacteristic
537C23#41×D5311yescharacteristic
538C23C2×#1171994711yescharacteristic
539C23C2×#1171994712yes—
540C23C2×#1171994712yes—
541C23C2×#1171994911yescharacteristic
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—534no—
566C2×C4—534no—
567C2×C4—538no—
568C2×C4—534no—
569C2×C4—538no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1064no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1064no—
582C2×C4—1062no—
583C2×C4—21216no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4#1313317714yes—
601C22—24no—
602C22—28no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22C2×#1313317711yescharacteristic
613C22C2×#1313317911yescharacteristic
614C22C4×#1171994712yes—
615C22C4×#1171994712yes—
616C22#1681109111yescharacteristic
617C22#121×D5311yescharacteristic
618C22#123×D5312yes—
619C22#123×D5311yescharacteristic
620C22#1681112912yes—
621C22#1681112911yescharacteristic
622C22#1681113611yescharacteristic
623C22#124×D5311yescharacteristic
624C22#126×D5312yes—
625C22#1681115012yes—
626C22#1681115212yes—
627C22#1681115412yes—
628C22#1681115412yes—
629C22#1681115412yes—
630C22#1681115911yescharacteristic
631C22#1681115912yes—
632C22#1681115912yes—
633C22#1681116811yescharacteristic
634C22#1681117111yescharacteristic
635C4—24no—
636C4—42no—
637C4—44no—
638C4—44no—
639C4—538no—
640C4—2122no—
641C4—2122no—
642C4—2124no—
643C4—2122no—
644C4—2124no—
645C2—44no—
646C2#2027722111yescharacteristic
647C2#2027723412yes—
648C2#2027724011yescharacteristic
649C2#2027726211yescharacteristic
650C2#2027726912yes—
651C2#397×D5311yescharacteristic
652C2#2027795011yescharacteristic
653C2#2027797112yes—
654C2#2027798011yescharacteristic
655C2#2027799711yescharacteristic
656C2#2027801712yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1439C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53C4×#397
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic53C2semidirect
5. #20272984C2semidirect
6. #20272976C2semidirect
7. #20272974C2semidirect
9. #95×Dic53C2semidirect
10. #95×Dic53C2semidirect
20. #16810548C4semidirect
52. C2×C4⋊Dic53C2×C4semidirect
77. #13133124C2×C4semidirect
78. #13133124C2×C4semidirect
316. Dic53#397semidirect
367. C53C4×#397semidirect

Maximal subgroups

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

subgroupindex
C2×#168105482
C2×C4⋊C4×Dic532
C2×C22⋊C4×Dic532
#202729842
#202729762
#202729742
#202729622
#95×Dic532
C53×#14392
C4×#39753