Finite Group Databasebeta

← #27306354

Subgroups — #27306354

Order 13568.

2006 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810483C211yesmaximal, characteristic
2C2×#16810481C211yesmaximal, characteristic
3C2×#16810481C211yesmaximal, characteristic
4C2×C22⋊C4×Dic53C211yesmaximal, characteristic
5C22×#13133124C211yesmaximal, characteristic
6#20272979C212yesmaximal, complemented
7#20272973C212yesmaximal, complemented
8#20272949C212yesmaximal, complemented
9#20272946C212yesmaximal, complemented
10#20272937C211yesmaximal, characteristic
11C53×C22⋊#95C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C22×C4⋊Dic53C2212yes—
15C22×#11719919C2212yes—
16C22×#11719919C2212yes—
17C22×C22⋊Dic53C2211yescharacteristic
18C22×C22⋊Dic53C2211yescharacteristic
19C22×C4×Dic53C2212yes—
20#16810483—24no—
21#16810481—24no—
22#16810481—24no—
23C22⋊C4×Dic53—24no—
24C2×#13133124C418yescomplemented
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2211yescharacteristic
35C24×C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2211yescharacteristic
37C22×C22⋊C4×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2212yes—
40C23×Dic53—22no—
41C23×Dic53—22no—
42C23×Dic53—24no—
43C23×Dic53—42no—
44C23×Dic53—42no—
45C23×Dic53—42no—
46C23×Dic53C2×C414yes—
47C23×Dic53C2×C414yes—
48C23×Dic53D414yes—
49C23×Dic53Q814yes—
50C23×Dic53C2312yes—
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2312yes—
53C23×Dic53C2311yescharacteristic
54C23×Dic53C2311yescharacteristic
55C23×Dic53C2311yescharacteristic
56C2×C4⋊Dic53—24no—
57C2×C4⋊Dic53—24no—
58C2×C4⋊Dic53—42no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919—42no—
63C2×#11719919—42no—
64C2×#11719919—42no—
65C2×C22⋊Dic53—22no—
66C2×C22⋊Dic53—22no—
67C2×C22⋊Dic53—22no—
68C2×C22⋊Dic53—22no—
69C2×C22⋊Dic53—22no—
70C2×C22⋊Dic53D414yes—
71C2×C4×Dic53—24no—
72C2×C4×Dic53—42no—
73C2×C4×Dic53—42no—
74#13133124—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—24no—
79#13133124—42no—
80#13133124C2×C414yes—
81#13133124C2×C414yes—
82#13133124C2×C414yes—
83#13133124C2×C414yes—
84C25×C53C2311yescharacteristic
85C23×C4×C53—22no—
86C23×C4×C53—22no—
87C23×C4×C53—24no—
88C23×C4×C53—42no—
89C23×C4×C53C2×C414yes—
90C23×C4×C53C2312yes—
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2312yes—
93C23×C4×C53C2311yescharacteristic
94C2×C22⋊C4×C53—22no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—22no—
97C2×C22⋊C4×C53—22no—
98C2×C22⋊C4×C53—22no—
99C2×C22⋊C4×C53Q814yes—
100#95×C53—24no—
101#95×C53—42no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—22no—
106C24×C53C22×C412yes—
107C24×C53C2×D412yes—
108C24×C53C2×D412yes—
109C24×C53C2×Q812yes—
110C24×C53C2×Q812yes—
111C24×C53#4112yes—
112C24×C53#4112yes—
113C24×C53C2411yescharacteristic
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—28no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—28no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—44no—
137C22×C4×C53—48no—
138C22×C4×C53—48no—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C22×C412yes—
141C22×C4×C53C2×Q814yes—
142C22×C4×C53#4114yes—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—28no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—21no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—28no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—44no—
170C22×Dic53—44no—
171C22×Dic53—44no—
172C22×Dic53—48no—
173C22×Dic53—48no—
174C22×Dic53—48no—
175C22×Dic53—48no—
176C22×Dic53C4⋊C4116yescomplemented
177C22×Dic53C22×C412yes—
178C22×Dic53C22×C412yes—
179C22×Dic53C22×C412yes—
180C22×Dic53C22×C412yes—
181C22×Dic53C2×D414yes—
182C22×Dic53C2×D412yes—
183C22×Dic53C2×D412yes—
184C22×Dic53C2×Q812yes—
185C22×Dic53C2×Q812yes—
186C22×Dic53#4114yes—
187C22⋊C4×C53—28no—
188C22⋊C4×C53—44no—
189C22⋊Dic53—28no—
190C22⋊Dic53—44no—
191C4⋊Dic53—48no—
192#11719919—48no—
193#11719919—48no—
194C4×Dic53—48no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—24no—
200C23×C53—24no—
201C23×C53—24no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—24no—
205C23×C53—24no—
206C23×C53—24no—
207C23×C53—24no—
208C23×C53—24no—
209C23×C53—24no—
210C23×C53—24no—
211C23×C53C2×C4⋊C414yes—
212C23×C53C4×D414yes—
213C23×C53C4×Q814yes—
214C23×C53#12114yes—
215C23×C53C22⋊Q814yes—
216C23×C53#12514yes—
217C23×C53C4⋊Q814yes—
218C23×C53C23×C411yescharacteristic
219C23×C53C22×D411yescharacteristic
220C23×C53C22×D411yescharacteristic
221C23×C53C22×Q811yescharacteristic
222C23×C53C22×Q811yescharacteristic
223C23×C53C2×#4112yes—
224C23×C53C2×#4112yes—
225C23×C53C2×#4111yescharacteristic
226C23×C53C2×#4111yescharacteristic
227C23×C53H(5,𝔽2)12yes—
228C23×C53H(5,𝔽2)12yes—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—416no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53#11714yes—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—24no—
272C2×Dic53—416no—
273C2×Dic53—416no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53C2×C4⋊C414yes—
288C2×Dic53C2×C4⋊C414yes—
289C2×Dic53#11714yes—
290#13908—531nomaximal, Sylow
291C22×C53—28no—
292C22×C53—28no—
293C22×C53—24no—
294C22×C53—24no—
295C22×C53—24no—
296C22×C53—24no—
297C22×C53—24no—
298C22×C53—24no—
299C22×C53—28no—
300C22×C53—28no—
301C22×C53#38918yes—
302C22×C53C22×C4⋊C411yescharacteristic
303C22×C53C2×#11711yescharacteristic
304C22×C53C2×#11711yescharacteristic
305C22×C53C2×C4×D411yescharacteristic
306C22×C53C2×C4×Q811yescharacteristic
307C22×C53#51811yescharacteristic
308C22×C53#51811yescharacteristic
309C22×C53C2×#12111yescharacteristic
310C22×C53C2×C22⋊Q811yescharacteristic
311C22×C53C2×#12511yescharacteristic
312C22×C53C2×C4⋊Q811yescharacteristic
313C22×C53#53212yes—
314C22×C53#53312yes—
315C22×C53#53412yes—
316C22×C53#54312yes—
317C22×C53#54612yes—
318C22×C53#54612yes—
319C22×C53#55112yes—
320C22×C53#55112yes—
321C22×C53#55312yes—
322C22×C53#55312yes—
323C22×C53#55412yes—
324C22×C53#55412yes—
325C4×C53—24no—
326C4×C53—44no—
327C4×C53—42no—
328C4×C53—82no—
329Dic53—24no—
330Dic53—24no—
331Dic53—24no—
332Dic53—42no—
333Dic53—44no—
334Dic53—82no—
335C2×#380—531no—
336C2×#380—531no—
337C2×#380—531no—
338C2×C4×C22⋊C4—531no—
339C22×#95—531no—
340#1444—532no—
341#1442—532no—
342#1436—532no—
343#1436—532no—
344C22⋊#95—531no—
345C22⋊#95D5311yescharacteristic
346C2×C53—28no—
347C2×C53#228511yescharacteristic
348C2×C53C2×#38911yescharacteristic
349C2×C53#231511yescharacteristic
350C2×C53#234211yescharacteristic
351C2×C53#234211yescharacteristic
352C2×C53#234511yescharacteristic
353C2×C53#234511yescharacteristic
354C2×C53#254312yes—
355C2×C53#254912yes—
356C2×C53#256412yes—
357C2×C53#256912yes—
358C24×C4—531no—
359C24×C4—531no—
360C24×C4C2×D5311yescharacteristic
361C22×C4⋊C4—532no—
362C22×C4⋊C4—532no—
363C22×C4⋊C4—532no—
364C22×C22⋊C4—531no—
365C22×C22⋊C4—531no—
366C22×C22⋊C4C2×D5311yescharacteristic
367C22×C22⋊C4C2×D5311yescharacteristic
368C22×C42—532no—
369#380—1064no—
370#380—1064no—
371#380—1064no—
372C4×C22⋊C4—1064no—
373C2×#95—538no—
374C2×#95—531no—
375C2×#95—531no—
376C2×#95—531no—
377C2×#95—531no—
378C2×#95—532no—
379C2×#95—532no—
380C2×#95—532no—
381C2×#95—532no—
382C2×#95—532no—
383C2×#95—532no—
384C2×#95C2×D5312yes—
385C2×#95C2×D5312yes—
386C53#1390811yescharacteristic, complemented, Sylow
387C25C22×D5311yescharacteristic
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—24no—
391C23×C4—42no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—531no—
395C23×C4—534no—
396C23×C4—534no—
397C23×C4—534no—
398C23×C4—532no—
399C23×C4—531no—
400C23×C4—534no—
401C23×C4—532no—
402C23×C4—1064no—
403C23×C4—1062no—
404C23×C4—1062no—
405C23×C4—2122no—
406C23×C4—2122no—
407C23×C4—2122no—
408C23×C4C4×D5314yes—
409C23×C4C22×D5312yes—
410C23×C4C22×D5312yes—
411C23×C4C22×D5311yescharacteristic
412C23×C4C22×D5311yescharacteristic
413C2×C4⋊C4—1064no—
414C2×C4⋊C4—1064no—
415C2×C4⋊C4—1064no—
416C2×C4⋊C4—1064no—
417C2×C4⋊C4—1064no—
418C2×C4⋊C4—2122no—
419C2×C4⋊C4—2122no—
420C2×C4⋊C4—2122no—
421C2×C4⋊C4—2122no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—534no—
428C2×C22⋊C4—1062no—
429C2×C22⋊C4—1062no—
430C2×C22⋊C4—1062no—
431C2×C22⋊C4—1062no—
432C2×C22⋊C4—1062no—
433C2×C22⋊C4Dic10614yes—
434C2×C42—1064no—
435C2×C42—2122no—
436C2×C42—2122no—
437#95—24no—
438#95—42no—
439#95—534no—
440#95—534no—
441#95—534no—
442#95—534no—
443#95—1064no—
444#95—1064no—
445#95—1064no—
446#95—1064no—
447#95—1064no—
448#95—2122no—
449C24—22no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24C2×Dic10612yes—
454C24C2×C4×D5312yes—
455C24#1171994512yes—
456C24D4×D5312yes—
457C24D4×D5312yes—
458C24#1171994712yes—
459C24Q8×D5312yes—
460C24C23×D5311yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—28no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—28no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—532no—
487C22×C4—532no—
488C22×C4—532no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—532no—
493C22×C4—534no—
494C22×C4—5316no—
495C22×C4—532no—
496C22×C4—534no—
497C22×C4—1068no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1068no—
506C22×C4—1061no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1061no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—2124no—
524C22×C4—2124no—
525C22×C4—2124no—
526C22×C4—2128no—
527C22×C4—2128no—
528C22×C4—2128no—
529C22×C4—2128no—
530C22×C4C2×Dic10614yes—
531C22×C4C2×C4×D5312yes—
532C22×C4C2×C4×D5312yes—
533C22×C4#1171994714yes—
534C4⋊C4—2128no—
535C4⋊C4—2128no—
536C4⋊C4—2128no—
537C22⋊C4—28no—
538C22⋊C4—44no—
539C22⋊C4—1068no—
540C22⋊C4—2124no—
541C42—2128no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23C4⋊C4×D5314yes—
559C23#1313317714yes—
560C23C4×Dic10614yes—
561C23#1313319014yes—
562C23#1313319414yes—
563C23#1313320714yes—
564C23#1313321614yes—
565C23C22×C4×D5311yescharacteristic
566C23C2×D4×D5311yescharacteristic
567C23C2×D4×D5311yescharacteristic
568C23C2×Q8×D5311yescharacteristic
569C23C22×Dic10611yescharacteristic
570C23#41×D5312yes—
571C23C2×#1171994711yescharacteristic
572C23C2×#1171994712yes—
573C23C2×#1171994511yescharacteristic
574C23#1313326212yes—
575C23#1313326312yes—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—416no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—534no—
601C2×C4—534no—
602C2×C4—534no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1062no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1064no—
612C2×C4—1064no—
613C2×C4—1064no—
614C2×C4—1064no—
615C2×C4—1064no—
616C2×C4—1062no—
617C2×C4—1062no—
618C2×C4—1062no—
619C2×C4—1064no—
620C2×C4—1064no—
621C2×C4—21216no—
622C2×C4—21216no—
623C2×C4—2122no—
624C2×C4—2122no—
625C2×C4—2122no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4#1313317314yes—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—24no—
644C22—28no—
645C22—28no—
646C22—28no—
647C22#1681052018yes—
648C22C2×C4⋊C4×D5311yescharacteristic
649C22#117×D5311yescharacteristic
650C22C2×#1313317311yescharacteristic
651C22C2×#1313317711yescharacteristic
652C22C2×C4×Dic10611yescharacteristic
653C22#1681109611yescharacteristic
654C22#1681109711yescharacteristic
655C22C2×#1313319011yescharacteristic
656C22C2×#1313319411yescharacteristic
657C22C2×#1313320711yescharacteristic
658C22C2×#1313321611yescharacteristic
659C22#1681116812yes—
660C22#1681117512yes—
661C22#1681118112yes—
662C22#1681124812yes—
663C22#1681126012yes—
664C22#1681126512yes—
665C22#1681129612yes—
666C22#1681130412yes—
667C22#1681131912yes—
668C22#1681132212yes—
669C22#1681133012yes—
670C22#1681133212yes—
671C4—24no—
672C4—42no—
673C4—44no—
674C4—82no—
675C4—1064no—
676C4—1064no—
677C4—1064no—
678C4—4242no—
679C4—2122no—
680C4—2124no—
681C2—28no—
682C2#2027711711yescharacteristic
683C2C2×#1681052011yescharacteristic
684C2#2027732811yescharacteristic
685C2#2027755711yescharacteristic
686C2#2027756611yescharacteristic
687C2#2027759011yescharacteristic
688C2#2027759311yescharacteristic
689C2#2027967912yes—
690C2#2027972812yes—
691C2#2027989912yes—
692C2#2027994112yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C22⋊#95C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC22⋊#95D53
hypocenterC53#13908
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
6. #20272979C2semidirect
7. #20272973C2semidirect
8. #20272949C2semidirect
9. #20272946C2semidirect
24. C2×#13133124C4semidirect
176. C22×Dic53C4⋊C4semidirect
386. C53#13908semidirect

Maximal subgroups

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

subgroupindex
C2×#168104832
C2×#168104812
C2×C22⋊C4×Dic532
C22×#131331242
#202729792
#202729732
#202729492
#202729462
#202729372
C53×C22⋊#952
#1390853