Finite Group Databasebeta

← #27315421

Subgroups — #27315421

Order 13568.

2842 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133201C211yesmaximal, characteristic
2C23×C4×Dic53C211yesmaximal, characteristic, complemented
3C2×#16810502C211yesmaximal, characteristic
4C2×#16810500C211yesmaximal, characteristic
5C2×#16810484C211yesmaximal, characteristic, complemented
6#20272951C214yesmaximal, complemented
7#20272949C214yesmaximal, complemented
8#20272937C211yesmaximal, characteristic, complemented
9C53×C22⋊#95C211yesmaximal, characteristic, complemented
10C23×C22⋊D53C2211yescharacteristic
11C24×Dic53C2211yescharacteristic
12C24×Dic53C2211yescharacteristic
13C2×#13133201—24no—
14C2×#13133201—24no—
15C2×#13133201C418yescomplemented
16C22×C4⋊Dic53C2212yes—
17C22×C22⋊Dic53C2211yescharacteristic
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20C22×#11719921C2212yes—
21C22×C4×Dic53—24no—
22C22×C4×Dic53—24no—
23C22×C4×Dic53—24no—
24C22×C4×Dic53C2214yes—
25#16810502—24no—
26#16810500—24no—
27#16810484—24no—
28C2×#13133124C2214yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2214yes—
32C24×C4×C53C2211yescharacteristic
33C22×C22⋊C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2211yescharacteristic
35C2×#95×C53C2214yes—
36C24×D53C2311yescharacteristic
37C22×C22⋊D53—22no—
38C22×C22⋊D53—22no—
39C22×C22⋊D53—22no—
40C22×C22⋊D53—22no—
41C22×C22⋊D53—22no—
42C22×C22⋊D53—22no—
43C22×C22⋊D53C2×C414yes—
44C23×Dic53—22no—
45C23×Dic53—22no—
46C23×Dic53—22no—
47C23×Dic53—22no—
48C23×Dic53—22no—
49C23×Dic53—22no—
50C23×Dic53—28no—
51C23×Dic53C2×C414yes—
52C23×Dic53D414yes—
53C23×Dic53D414yes—
54C23×Dic53D414yes—
55C23×Dic53C2312yes—
56C23×Dic53C2311yescharacteristic
57C23×Dic53C2312yes—
58C23×Dic53C2312yes—
59#13133201—232no—
60C2×C4⋊Dic53—24no—
61C2×C4⋊Dic53—24no—
62C2×C4⋊Dic53C2×C418yescomplemented
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—22no—
66C2×#11719921—24no—
67C2×#11719921—28no—
68C2×#11719921—24no—
69C2×#11719921—24no—
70C2×#11719921—24no—
71C2×#11719921—28no—
72C2×#11719921C2×C418yescomplemented
73C2×C4×Dic53—232no—
74C2×C4×Dic53—28no—
75C2×C4×Dic53—28no—
76C2×C4×Dic53—28no—
77#13133124—24no—
78#13133124—24no—
79#13133124—28no—
80#13133124—28no—
81C25×C53C2311yescharacteristic
82C23×C4×C53—22no—
83C23×C4×C53—22no—
84C23×C4×C53—28no—
85C23×C4×C53D414yes—
86C23×C4×C53C2312yes—
87C23×C4×C53C2312yes—
88C23×C4×C53C2312yes—
89C2×C22⋊C4×C53—22no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53—22no—
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53C2×C414yes—
95#95×C53—28no—
96C24×C53—24no—
97C24×C53—24no—
98C24×C53C22×C412yes—
99C24×C53C2×D412yes—
100C24×C53C2×D412yes—
101C24×C53C2×D412yes—
102C24×C53C2×D412yes—
103C24×C53#4112yes—
104C24×C53#4112yes—
105C24×C53C2411yescharacteristic
106C23×D53—21no—
107C23×D53—21no—
108C23×D53—21no—
109C23×D53—21no—
110C23×D53—24no—
111C23×D53—24no—
112C23×D53—21no—
113C23×D53—21no—
114C23×D53C22×C412yes—
115C22×C4×C53—216no—
116C22×C4×C53—216no—
117C22×C4×C53—28no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—24no—
121C22×C4×C53—216no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—24no—
125C22×C4×C53—24no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—24no—
129C22×C4×C53—22no—
130C22×C4×C53—24no—
131C22×C4×C53C22×C414yes—
132C22×C4×C53C2×D414yes—
133C22×C4×C53#4118yes—
134C2×C22⋊D53—28no—
135C2×C22⋊D53—28no—
136C2×C22⋊D53—28no—
137C2×C22⋊D53—44no—
138C2×C22⋊D53—44no—
139C2×C22⋊D53—44no—
140C2×C22⋊D53—44no—
141C22×Dic53—216no—
142C22×Dic53—216no—
143C22×Dic53—28no—
144C22×Dic53—28no—
145C22×Dic53—28no—
146C22×Dic53—28no—
147C22×Dic53—28no—
148C22×Dic53—28no—
149C22×Dic53—28no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—24no—
157C22×Dic53—216no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—24no—
167C22×Dic53—24no—
168C22×Dic53—22no—
169C22×Dic53C22⋊C4116yescomplemented
170C22×Dic53C22×C412yes—
171C22×Dic53C22×C414yes—
172C22×Dic53C2×D414yes—
173C22×Dic53C2×D414yes—
174C22×Dic53C2×D414yes—
175C22×Dic53#4118yes—
176C22⋊C4×C53—28no—
177C22⋊C4×C53—44no—
178C22⋊Dic53—44no—
179#11719921—216no—
180#11719921—48no—
181#11719921—48no—
182C4⋊Dic53—216no—
183C4×Dic53—232no—
184C23×C53—28no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—24no—
191C23×C53—28no—
192C23×C53—28no—
193C23×C53—28no—
194C23×C53C2×C22⋊C414yes—
195C23×C53C4×D414yes—
196C23×C53C4×D414yes—
197C23×C53#12014yes—
198C23×C53#12114yes—
199C23×C53#12114yes—
200C23×C53#12314yes—
201C23×C53C23×C411yescharacteristic
202C23×C53C22×D411yescharacteristic
203C23×C53C22×D411yescharacteristic
204C23×C53C22×D411yescharacteristic
205C23×C53C22×D411yescharacteristic
206C23×C53C2×#4114yes—
207C23×C53C2×#4114yes—
208C23×C53C2×#4111yescharacteristic
209C23×C53C2×#4111yescharacteristic
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—28no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—42no—
223C22×D53—42no—
224C22×D53—44no—
225C22×D53—44no—
226C22×D53—44no—
227C2×C4×C53—232no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—28no—
231C2×C4×C53—28no—
232C2×C4×C53—24no—
233C2×C4×C53—28no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—44no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—44no—
241C2×C4×C53—44no—
242C2×C4×C53#12318yescomplemented
243C22⋊D53—416no—
244C2×Dic53—232no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—28no—
254C2×Dic53—28no—
255C2×Dic53—232no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—44no—
267C2×Dic53—44no—
268C2×Dic53C2×C22⋊C418yes—
269C2×Dic53#12318yescomplemented
270#14706—531nomaximal, Sylow
271C22×C53—24no—
272C22×C53—24no—
273C22×C53—28no—
274C22×C53—28no—
275C22×C53—28no—
276C22×C53—28no—
277C22×C53—28no—
278C22×C53—28no—
279C22×C53#38618yes—
280C22×C53C22×C22⋊C411yescharacteristic
281C22×C53C2×C4×D411yescharacteristic
282C22×C53C2×C4×D411yescharacteristic
283C22×C53C4×#4114yes—
284C22×C53C2×#12011yescharacteristic
285C22×C53C2×#12111yescharacteristic
286C22×C53C2×#12111yescharacteristic
287C22×C53C2×#12312yes—
288C22×C53C2×#12312yes—
289C22×C53C2×#12311yescharacteristic
290C22×C53#52512yes—
291C22×C53#52512yes—
292C22×C53#52512yes—
293C22×C53#52512yes—
294C22×C53#52914yes—
295C22×C53#52914yes—
296C22×C53#53214yes—
297C2×D53—42no—
298C2×D53—44no—
299C2×D53—44no—
300C2×D53—44no—
301C2×D53—81no—
302C2×D53—81no—
303C2×D53—82no—
304C2×D53—82no—
305C2×D53—82no—
306C4×C53—28no—
307C4×C53—44no—
308C4×C53—82no—
309Dic53—28no—
310Dic53—28no—
311Dic53—42no—
312Dic53—82no—
313C22×#123—531no—
314C23×C42—531no—
315C2×#386—531no—
316C2×#386—531no—
317C2×#381—531no—
318#1437—534no—
319#1436—534no—
320C22⋊#95—531no—
321C22⋊#95D5311yescharacteristic, complemented
322C2×C53—28no—
323C2×C53C2×#38611yescharacteristic
324C2×C53#229712yes—
325C2×C53C4×#12312yes—
326C2×C53C4×#12312yes—
327C2×C53#284911yescharacteristic
328C2×C53#284911yescharacteristic
329C2×C53#285312yes—
330C2×C53#285412yes—
331C2×C53#285412yes—
332D53—82no—
333C23×D4—531no—
334C24×C4—531no—
335C24×C4—531no—
336C24×C4C2×D5311yescharacteristic
337C2×#123—538no—
338C2×#123—1064no—
339C2×#123—1064no—
340C22×C4⋊C4—532no—
341C22×C22⋊C4—532no—
342C22×C22⋊C4—531no—
343C22×C22⋊C4—532no—
344C22×C22⋊C4—532no—
345C22×C22⋊C4C2×D5311yescharacteristic
346C22×C22⋊C4C2×D5311yescharacteristic
347C22×C42—534no—
348C22×C42—1064no—
349C22×C42—1064no—
350C22×C42—1064no—
351#386—1064no—
352#386—1064no—
353#381—1064no—
354C2×#95—534no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—534no—
358C2×#95C2×D5314yes—
359C53#1470611yescharacteristic, complemented, Sylow
360C25—531no—
361C25C22×D5311yescharacteristic
362C22×D4—534no—
363C22×D4—1062no—
364C22×D4—1062no—
365C22×D4—1062no—
366C22×D4—1062no—
367C22×D4—1062no—
368C22×D4—1062no—
369C23×C4—22no—
370C23×C4—22no—
371C23×C4—28no—
372C23×C4—534no—
373C23×C4—534no—
374C23×C4—534no—
375C23×C4—532no—
376C23×C4—532no—
377C23×C4—534no—
378C23×C4—531no—
379C23×C4—532no—
380C23×C4—1068no—
381C23×C4—1062no—
382C23×C4—1062no—
383C23×C4—1062no—
384C23×C4—1062no—
385C23×C4—1062no—
386C23×C4—1062no—
387C23×C4D21214yes—
388C23×C4C22×D5312yes—
389C23×C4C22×D5312yes—
390C23×C4C22×D5312yes—
391#123—10632no—
392C2×C4⋊C4—538no—
393C2×C4⋊C4—1064no—
394C2×C4⋊C4—1064no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—538no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1068no—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—1062no—
406C2×C22⋊C4—1062no—
407C2×C22⋊C4—1062no—
408C2×C22⋊C4—1064no—
409C2×C22⋊C4—1068no—
410C2×C22⋊C4C4×D5314yes—
411C2×C42—1068no—
412C2×C42—1068no—
413C2×C42—10632no—
414C2×C42—1068no—
415#95—28no—
416#95—1068no—
417#95—1068no—
418#95—1064no—
419#95—1064no—
420C24—24no—
421C24—24no—
422C24—532no—
423C24—1061no—
424C24—1061no—
425C24—1061no—
426C24—1061no—
427C24—1064no—
428C24—1064no—
429C24—1061no—
430C24—1061no—
431C24C2×C4×D5312yes—
432C24C2×D21212yes—
433C24#1171994512yes—
434C24D4×D5312yes—
435C24D4×D5312yes—
436C24D4×D5312yes—
437C24#1171994912yes—
438C24C23×D5311yescharacteristic
439C2×D4—1068no—
440C2×D4—1068no—
441C2×D4—1068no—
442C2×D4—2124no—
443C2×D4—2124no—
444C2×D4—2124no—
445C2×D4—2124no—
446C22×C4—216no—
447C22×C4—216no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—216no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—5316no—
463C22×C4—534no—
464C22×C4—534no—
465C22×C4—534no—
466C22×C4—534no—
467C22×C4—538no—
468C22×C4—532no—
469C22×C4—10616no—
470C22×C4—10616no—
471C22×C4—10616no—
472C22×C4—1068no—
473C22×C4—1068no—
474C22×C4—1068no—
475C22×C4—1068no—
476C22×C4—1068no—
477C22×C4—1068no—
478C22×C4—1068no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1064no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1064no—
495C22×C4—1064no—
496C22×C4—1062no—
497C22×C4C2×C4×D5314yes—
498C22×C4C2×D21214yes—
499C22×C4#1171994718yes—
500C4⋊C4—10616no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—10616no—
504C22⋊C4—2128no—
505C22⋊C4—2124no—
506C22⋊C4—2128no—
507C42—10632no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—1062no—
519C23—1062no—
520C23—1062no—
521C23—1068no—
522C23—2121no—
523C23—2121no—
524C23—2121no—
525C23—2122no—
526C23—2122no—
527C23—2121no—
528C23—2122no—
529C23—2122no—
530C23—2122no—
531C23—2122no—
532C23—2124no—
533C23—2124no—
534C23—2124no—
535C23C22⋊C4×D5314yes—
536C23#1313317914yes—
537C23C4×D21214yes—
538C23#1313318714yes—
539C23#1313319014yes—
540C23#1313319214yes—
541C23#1313320214yes—
542C23C22×C4×D5311yescharacteristic
543C23C2×D4×D5311yescharacteristic
544C23C2×D4×D5311yescharacteristic
545C23C2×D4×D5311yescharacteristic
546C23C22×D21211yescharacteristic
547C23#41×D5314yes—
548C23C2×#1171994714yes—
549C23C2×#1171994511yescharacteristic
550C23C2×#1171994911yescharacteristic
551D4—21216no—
552C2×C4—232no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—538no—
568C2×C4—538no—
569C2×C4—10632no—
570C2×C4—10632no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1068no—
580C2×C4—1068no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2124no—
588C2×C4—2124no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4#1313320118yescomplemented
594C22—24no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—4241no—
603C22—4241no—
604C22—4242no—
605C22—4242no—
606C22—4242no—
607C22—2122no—
608C22—2124no—
609C22—2124no—
610C22—2124no—
611C22#1681050118yes—
612C22C2×C22⋊C4×D5311yescharacteristic
613C22C2×#1313317911yescharacteristic
614C22C2×C4×D21211yescharacteristic
615C22C4×#1171994714yes—
616C22C2×#1313318711yescharacteristic
617C22C2×#1313319011yescharacteristic
618C22C2×#1313319211yescharacteristic
619C22#123×D5312yes—
620C22C2×#1313320112yes—
621C22C2×#1313320211yescharacteristic
622C22#1681112912yes—
623C22#1681112912yes—
624C22#1681113112yes—
625C22#1681113512yes—
626C22#1681115314yes—
627C22#1681115914yes—
628C22#1681116814yes—
629C4—28no—
630C4—44no—
631C4—82no—
632C4—1068no—
633C4—1068no—
634C4—4242no—
635C4—2122no—
636C2—28no—
637C2—4242no—
638C2C2×#1681050111yescharacteristic
639C2#2027720012yes—
640C2#2027723412yes—
641C2C4×#1313320112yes—
642C2#2028300811yescharacteristic
643C2#2028300911yescharacteristic
644C2#2028304012yes—
645C2#2028304312yes—
646C2#2028304712yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC23×C22⋊D53C22
hypercenterC22⋊#95D53
hypocenterC53#14706
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C23×C4×Dic53C2semidirect
5. C2×#16810484C2semidirect
6. #20272951C2semidirect
7. #20272949C2semidirect
8. #20272937C2semidirect
9. C53×C22⋊#95C2semidirect
15. C2×#13133201C4semidirect
62. C2×C4⋊Dic53C2×C4semidirect
72. C2×#11719921C2×C4semidirect
169. C22×Dic53C22⋊C4semidirect
242. C2×C4×C53#123semidirect
269. C2×Dic53#123semidirect
321. C22⋊#95D53semidirect
359. C53#14706semidirect
593. C2×C4#13133201semidirect

Maximal subgroups

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

subgroupindex
C22×#131332012
C23×C4×Dic532
C2×#168105022
C2×#168105002
C2×#168104842
#202729512
#202729492
#202729372
C53×C22⋊#952
#1470653