Finite Group Databasebeta

← #27315310

Subgroups — #27315310

Order 13568.

2430 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Dic53C211yesmaximal, characteristic
2C2×#16810504C212yesmaximal, complemented
3C2×#16810495C211yesmaximal, characteristic
4#20272947C212yesmaximal, complemented
5#20272946C212yesmaximal, complemented
6#20272939C212yesmaximal, complemented
7#20272938C211yesmaximal, characteristic
8#20272937C211yesmaximal, characteristic
9#20272937C211yesmaximal, characteristic
10#20272937C211yesmaximal, characteristic
11C2×C53×#385C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×C22⋊Dic53C2211yescharacteristic
16C22×C22⋊Dic53C2211yescharacteristic
17C22×C4×Dic53—28no—
18C22×C4×Dic53—24no—
19C22×C4×Dic53C2212yes—
20C22×C4×Dic53C2212yes—
21#16810504—28no—
22#16810495C418yescomplemented
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C24×C4×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2211yescharacteristic
37C53×#385—24no—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2211yescharacteristic
40C23×Dic53—22no—
41C23×Dic53—22no—
42C23×Dic53—22no—
43C23×Dic53—22no—
44C23×Dic53—24no—
45C23×Dic53—24no—
46C23×Dic53—24no—
47C23×Dic53—24no—
48C23×Dic53C2×C414yes—
49C23×Dic53Q814yes—
50C23×Dic53C2312yes—
51C23×Dic53C2312yes—
52C23×Dic53C2312yes—
53C23×Dic53C2312yes—
54C2×#11719919—24no—
55C2×#11719919—24no—
56C2×#11719919C2×C418yescomplemented
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—22no—
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53C2×C414yes—
63C2×C4×Dic53—232no—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53—24no—
69C2×C4×Dic53D418yes—
70#13133124—24no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124C2×C418yescomplemented
79C25×C53C2311yescharacteristic
80C23×C4×C53—22no—
81C23×C4×C53—22no—
82C23×C4×C53—24no—
83C23×C4×C53—24no—
84C23×C4×C53D414yes—
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53—24no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—24no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—22no—
97C2×C22⋊C4×C53—22no—
98C2×C22⋊C4×C53—22no—
99C2×C22⋊C4×C53—22no—
100C2×C22⋊C4×C53C2×C414yes—
101#95×C53—22no—
102#95×C53—22no—
103C24×C53—22no—
104C24×C53—22no—
105C24×C53—22no—
106C24×C53—22no—
107C24×C53C22×C412yes—
108C24×C53C2×D412yes—
109C24×C53C2×Q812yes—
110C24×C53#4112yes—
111C24×C53#4112yes—
112C24×C53#4112yes—
113C24×C53#4112yes—
114C24×C53C2411yescharacteristic
115C22×C4×C53—216no—
116C22×C4×C53—216no—
117C22×C4×C53—28no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—28no—
125C22×C4×C53—28no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—22no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53C22×C412yes—
145C22×C4×C53C22×C412yes—
146C22×C4×C53C2×D414yes—
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2×D412yes—
149C22×C4×C53#4114yes—
150C22×Dic53—28no—
151C22×Dic53—28no—
152C22×Dic53—216no—
153C22×Dic53—216no—
154C22×Dic53—28no—
155C22×Dic53—28no—
156C22×Dic53—28no—
157C22×Dic53—28no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—216no—
167C22×Dic53—216no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53—22no—
173C22×Dic53—22no—
174C22×Dic53—22no—
175C22×Dic53—22no—
176C22×Dic53—22no—
177C22×Dic53—22no—
178C22×Dic53—22no—
179C22×Dic53—22no—
180C22×Dic53—22no—
181C22×Dic53C22×C414yes—
182C22×Dic53C22×C414yes—
183C22×Dic53C2×D414yes—
184C22×Dic53C2×D414yes—
185C22×Dic53C2×Q814yes—
186C22×Dic53#4114yes—
187C22×Dic53#4114yes—
188C4⋊C4×C53—44no—
189C4⋊C4×C53—44no—
190C22⋊C4×C53—28no—
191C22⋊C4×C53—44no—
192C22⋊Dic53—28no—
193C22⋊Dic53—44no—
194#11719919—216no—
195C4×Dic53—216no—
196C4×Dic53—216no—
197C23×C53—24no—
198C23×C53—24no—
199C23×C53—22no—
200C23×C53—24no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—22no—
204C23×C53—24no—
205C23×C53—24no—
206C23×C53—24no—
207C23×C53—24no—
208C23×C53—24no—
209C23×C53—24no—
210C23×C53—24no—
211C23×C53—24no—
212C23×C53—24no—
213C23×C53#11714yes—
214C23×C53C4×D414yes—
215C23×C53C4×Q814yes—
216C23×C53C22⋊Q814yes—
217C23×C53#12314yes—
218C23×C53#12514yes—
219C23×C53#12614yes—
220C23×C53C23×C411yescharacteristic
221C23×C53C22×D412yes—
222C23×C53C22×D411yescharacteristic
223C23×C53C22×Q811yescharacteristic
224C23×C53C2×#4112yes—
225C23×C53C2×#4112yes—
226C23×C53C2×#4111yescharacteristic
227C23×C53C2×#4111yescharacteristic
228C23×C53C2×#4112yes—
229C23×C53C2×#4111yescharacteristic
230C23×C53C2×#4111yescharacteristic
231C2×C4×C53—232no—
232C2×C4×C53—22no—
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—24no—
240C2×C4×C53—24no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—24no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53#12014yes—
261C2×C4×C53#12314yes—
262C2×Dic53—232no—
263C2×Dic53—232no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—24no—
272C2×Dic53—24no—
273C2×Dic53—24no—
274C2×Dic53—24no—
275C2×Dic53—24no—
276C2×Dic53—24no—
277C2×Dic53—24no—
278C2×Dic53—24no—
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×Dic53—42no—
288C2×Dic53—42no—
289C2×Dic53—42no—
290C2×Dic53C4×D418yes—
291C2×Dic53C22⋊Q818yes—
292#14695—531nomaximal, Sylow
293C22×C53—24no—
294C22×C53—24no—
295C22×C53—28no—
296C22×C53—28no—
297C22×C53—24no—
298C22×C53—24no—
299C22×C53—28no—
300C22×C53—24no—
301C22×C53—24no—
302C22×C53—28no—
303C22×C53#38718yescomplemented
304C22×C53C2×#11711yescharacteristic
305C22×C53C2×C4×D412yes—
306C22×C53C2×C4×D411yescharacteristic
307C22×C53C2×C4×Q811yescharacteristic
308C22×C53C4×#4112yes—
309C22×C53C2×#12011yescharacteristic
310C22×C53C2×C22⋊Q812yes—
311C22×C53C2×C22⋊Q811yescharacteristic
312C22×C53C2×#12311yescharacteristic
313C22×C53C2×#12311yescharacteristic
314C22×C53#52512yes—
315C22×C53#52512yes—
316C22×C53#52511yescharacteristic
317C22×C53#52511yescharacteristic
318C22×C53#52512yes—
319C22×C53#52512yes—
320C22×C53C2×#12511yescharacteristic
321C22×C53C2×#12611yescharacteristic
322C22×C53#52912yes—
323C22×C53#52912yes—
324C22×C53#52912yes—
325C22×C53#52912yes—
326C22×C53#53312yes—
327C4×C53—24no—
328C4×C53—24no—
329C4×C53—42no—
330C4×C53—42no—
331C4×C53—81no—
332C4×C53—81no—
333Dic53—28no—
334Dic53—28no—
335Dic53—44no—
336Dic53—82no—
337C23×C42—531no—
338C2×#387—532no—
339C2×#385—531no—
340C2×#385D5311yescharacteristic
341#1436—532no—
342#1436—532no—
343#1434—532no—
344C22⋊#95—531no—
345C22⋊#95—531no—
346C22⋊#95—531no—
347C22⋊#95—531no—
348C2×C53—28no—
349C2×C53C2×#38711yescharacteristic
350C2×C53#229012yes—
351C2×C53C4×#12011yescharacteristic
352C2×C53C4×#12311yescharacteristic
353C2×C53C4×C22⋊Q812yes—
354C2×C53#284811yescharacteristic
355C2×C53#284911yescharacteristic
356C2×C53#285112yes—
357C2×C53#285212yes—
358C2×C53#285311yescharacteristic
359C2×C53#285511yescharacteristic
360C24×C4—531no—
361C24×C4—531no—
362C24×C4C2×D5311yescharacteristic
363C22×C4⋊C4—532no—
364C22×C4⋊C4C2×D5311yescharacteristic
365C22×C4⋊C4C2×D5311yescharacteristic
366C22×C22⋊C4—531no—
367C22×C22⋊C4—531no—
368C22×C22⋊C4C2×D5311yescharacteristic
369C22×C22⋊C4C2×D5311yescharacteristic
370C22×C42—532no—
371C22×C42—532no—
372C22×C42—1068no—
373C22×C42—1064no—
374#387—1068no—
375#385—24no—
376#385—538no—
377C2×#95—532no—
378C2×#95—532no—
379C2×#95—532no—
380C2×#95—532no—
381C2×#95—532no—
382C2×#95—532no—
383C2×#95—532no—
384C2×#95—532no—
385C2×#95—532no—
386C2×#95C2×D5311yescharacteristic
387C2×#95C2×D5311yescharacteristic
388C53#1469511yescharacteristic, complemented, Sylow
389C25C22×D5311yescharacteristic
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—532no—
395C23×C4—532no—
396C23×C4—534no—
397C23×C4—534no—
398C23×C4—532no—
399C23×C4—532no—
400C23×C4—1064no—
401C23×C4—1064no—
402C23×C4—1064no—
403C23×C4—1064no—
404C23×C4—1062no—
405C23×C4—1062no—
406C23×C4—1062no—
407C23×C4—1062no—
408C23×C4C22⋊D5314yes—
409C23×C4C22×D5311yescharacteristic
410C23×C4C22×D5311yescharacteristic
411C23×C4C22×D5311yescharacteristic
412C23×C4C22×D5311yescharacteristic
413C23×C4C22×D5311yescharacteristic
414C23×C4C22×D5311yescharacteristic
415C2×C4⋊C4—24no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—22no—
419C2×C4⋊C4—538no—
420C2×C4⋊C4—1064no—
421C2×C4⋊C4—1064no—
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⋊C4C4×D5314yes—
434C2×C42—538no—
435C2×C42—1064no—
436C2×C42—1064no—
437C2×C42—10632no—
438C2×C42—1064no—
439C2×C42—1064no—
440C2×C42—1064no—
441#95—22no—
442#95—22no—
443#95—538no—
444#95—1064no—
445#95—1064no—
446#95—1064no—
447#95—1064no—
448#95—1064no—
449#95—1064no—
450#95—1064no—
451#95—1064no—
452C24—22no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24C2×C4×D5312yes—
457C24#1171994512yes—
458C24#1171994712yes—
459C24#1171994712yes—
460C24Q8×D5312yes—
461C24#1171994912yes—
462C24C2×C22⋊D5312yes—
463C24C23×D5311yescharacteristic
464C22×C4—216no—
465C22×C4—216no—
466C22×C4—28no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—534no—
494C22×C4—534no—
495C22×C4—534no—
496C22×C4—534no—
497C22×C4—534no—
498C22×C4—534no—
499C22×C4—534no—
500C22×C4—10616no—
501C22×C4—10616no—
502C22×C4—1068no—
503C22×C4—1068no—
504C22×C4—1068no—
505C22×C4—1068no—
506C22×C4—10616no—
507C22×C4—10616no—
508C22×C4—1068no—
509C22×C4—1068no—
510C22×C4—1062no—
511C22×C4—1062no—
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—1062no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—1062no—
531C22×C4C2×C4×D5312yes—
532C22×C4C2×C4×D5312yes—
533C22×C4D4×D5314yes—
534C22×C4#1171994714yes—
535C22×C4C2×C22⋊D5312yes—
536C22×C4C2×C22⋊D5312yes—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—10616no—
540C22⋊C4—28no—
541C22⋊C4—44no—
542C22⋊C4—1068no—
543C22⋊C4—2124no—
544C42—10616no—
545C42—10616no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—24no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23#1313317514yes—
563C23C4×C22⋊D5314yes—
564C23#1313318214yes—
565C23#1313319314yes—
566C23#1313319814yes—
567C23#1313320714yes—
568C23#1313321214yes—
569C23C22×C4×D5311yescharacteristic
570C23C2×D4×D5312yes—
571C23C22×C22⋊D5311yescharacteristic
572C23C2×Q8×D5311yescharacteristic
573C23#41×D5312yes—
574C23#41×D5312yes—
575C23C2×#1171994711yescharacteristic
576C23C2×#1171994711yescharacteristic
577C23C2×#1171994712yes—
578C23C2×#1171994511yescharacteristic
579C23C2×#1171994911yescharacteristic
580C2×C4—232no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—538no—
610C2×C4—538no—
611C2×C4—10632no—
612C2×C4—10632no—
613C2×C4—1064no—
614C2×C4—1064no—
615C2×C4—1064no—
616C2×C4—1064no—
617C2×C4—1064no—
618C2×C4—1064no—
619C2×C4—1064no—
620C2×C4—1064no—
621C2×C4—1064no—
622C2×C4—1064no—
623C2×C4—1064no—
624C2×C4—1064no—
625C2×C4—1064no—
626C2×C4—1064no—
627C2×C4—1064no—
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—2122no—
637C2×C4—2122no—
638C2×C4—2122no—
639C2×C4#1313318514yes—
640C2×C4#1313319814yes—
641C22—24no—
642C22—24no—
643C22—28no—
644C22—28no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—28no—
651C22#1681050418yescomplemented
652C22C2×#1313317511yescharacteristic
653C22C4×D4×D5312yes—
654C22C2×C4×C22⋊D5311yescharacteristic
655C22C2×#1313318211yescharacteristic
656C22C4×#1171994712yes—
657C22C2×#1313318511yescharacteristic
658C22C22⋊Q8×D5312yes—
659C22C2×#1313319311yescharacteristic
660C22C2×#1313319811yescharacteristic
661C22C2×#1313319811yescharacteristic
662C22#1681112912yes—
663C22#1681112912yes—
664C22#1681113011yescharacteristic
665C22#1681113011yescharacteristic
666C22#1681113212yes—
667C22#1681113612yes—
668C22C2×#1313320711yescharacteristic
669C22C2×#1313321211yescharacteristic
670C22#1681115012yes—
671C22#1681115012yes—
672C22#1681115312yes—
673C22#1681115912yes—
674C22#1681117512yes—
675C4—24no—
676C4—24no—
677C4—42no—
678C4—42no—
679C4—81no—
680C4—81no—
681C4—1068no—
682C4—1068no—
683C4—4242no—
684C4—2124no—
685C2—28no—
686C2C2×#1681050411yescharacteristic
687C2#2027715112yes—
688C2C4×#1313318511yescharacteristic
689C2C4×#1313319811yescharacteristic
690C2#2027724712yes—
691C2#2028299911yescharacteristic
692C2#2028300511yescharacteristic
693C2#2028302112yes—
694C2#2028303212yes—
695C2#2028303411yescharacteristic
696C2#2028304911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#385C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×#385D53
hypocenterC53#14695
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
2. C2×#16810504C2semidirect
4. #20272947C2semidirect
5. #20272946C2semidirect
6. #20272939C2semidirect
22. #16810495C4semidirect
56. C2×#11719919C2×C4semidirect
78. #13133124C2×C4semidirect
303. C22×C53#387semidirect
388. C53#14695semidirect
651. C22#16810504semidirect

Maximal subgroups

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

subgroupindex
C23×C4×Dic532
C2×#168105042
C2×#168104952
#202729472
#202729462
#202729392
#202729382
#202729372
C2×C53×#3852
#1469553