Finite Group Databasebeta

← #27305402

Subgroups — #27305402

Order 13568.

1862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic106C211yesmaximal, characteristic
2C2×#16810527C212yesmaximal
3C2×#16810474C211yesmaximal, characteristic
4C2×#16810468C211yesmaximal, characteristic
5#20272949C212yesmaximal
6#20272946C212yesmaximal
7#20272943C212yesmaximal
8#20272942C211yesmaximal, characteristic
9#20272939C211yesmaximal, characteristic
10#20272939C211yesmaximal, characteristic
11C53×#1434C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C2×C4×Dic106—24no—
14C2×C4×Dic106—24no—
15C2×C4×Dic106C418yescomplemented
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2212yes—
19C22×#11719919C2211yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21C22×C4×Dic53C2212yes—
22C22×C4×Dic53C2212yes—
23C22×C4×Dic53C2211yescharacteristic
24#16810527C4116yescomplemented
25#16810474—24no—
26#16810468—24no—
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C22×C42×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C2×#95×C53C2212yes—
41C22×Dic106—22no—
42C22×Dic106—22no—
43C22×Dic106—24no—
44C22×Dic106C2×C418yes—
45C22×Dic106C2×C414yes—
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2312yes—
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C4×Dic106—232no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53C2×C414yes—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919C2×C418yes—
63C2×#11719919C2×C414yes—
64C2×#11719919C2×C414yes—
65C2×C4×Dic53—22no—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53—24no—
69C2×C4×Dic53—24no—
70C2×C4×Dic53—42no—
71C2×C4×Dic53—42no—
72C2×C4×Dic53C2×C418yes—
73C2×C4×Dic53C2×C414yes—
74C2×C4×Dic53C2×C414yes—
75C2×C4×Dic53D414yes—
76C2×C4×Dic53D414yes—
77#13133124—22no—
78#13133124—22no—
79#13133124—24no—
80#13133124—42no—
81#13133124—42no—
82#13133124—42no—
83#13133124C2×C418yes—
84#13133124C2×C418yes—
85#13133124C2×C418yes—
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2312yes—
88C23×C4×C53C2312yes—
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C2×C42×C53—22no—
92C2×C42×C53—22no—
93C2×C42×C53C2×C414yes—
94#95×C53—22no—
95#95×C53—22no—
96#95×C53—42no—
97#95×C53C2×C418yes—
98C24×C53C2411yescharacteristic
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53C22×C414yes—
123C22×C4×C53C22×C414yes—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C22×C414yes—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—22no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53C22×C414yes—
156C22×Dic53C22×C414yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C22×C414yes—
159C22×Dic53C22×C412yes—
160C22×Dic53C22×C412yes—
161C22×Dic53C22×C412yes—
162C22×Dic53C22×C414yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×D412yes—
165C22×Dic53C2×D412yes—
166C22×Dic53C2×D412yes—
167C22×Dic53#4114yes—
168C2×Dic106—28no—
169C2×Dic106—28no—
170C2×Dic106—216no—
171C2×Dic106—216no—
172C2×Dic106C42116yescomplemented
173C42×C53—28no—
174C4⋊Dic53—28no—
175#11719919—28no—
176#11719919—28no—
177#11719919—216no—
178C4×Dic53—28no—
179C4×Dic53—216no—
180C4×Dic53—48no—
181C4×Dic53C22⋊C4116yes—
182C23×C53C23×C411yescharacteristic
183C23×C53C23×C412yes—
184C23×C53C22×D411yescharacteristic
185C23×C53C22×D411yescharacteristic
186C23×C53C2×#4112yes—
187C23×C53H(5,𝔽2)11yescharacteristic
188C23×C53H(5,𝔽2)11yescharacteristic
189C23×C532−1+412yes—
190C23×C532−1+412yes—
191C23×C532−1+411yescharacteristic
192C23×C532−1+411yescharacteristic
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—24no—
196C2×C4×C53—24no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—22no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—22no—
210C2×C4×C53—41no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—42no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53C2×C4214yes—
223C2×C4×C53C2×C22⋊C414yes—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—24no—
238C2×Dic53—24no—
239C2×Dic53—22no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—22no—
243C2×Dic53—41no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53C2×C4214yes—
256C2×Dic53C2×C4214yes—
257C2×Dic53C2×C22⋊C414yes—
258C2×Dic53C2×C22⋊C414yes—
259C2×Dic53#11714yes—
260C2×Dic53C4×D418yes—
261C2×Dic53C4×D418yes—
262Dic106—232no—
263#13786—531nomaximal, Sylow
264C22×C53C22×C4211yescharacteristic
265C22×C53C22×C22⋊C411yescharacteristic
266C22×C53C2×#11711yescharacteristic
267C22×C53C2×C4×D412yes—
268C22×C53C2×C4×D412yes—
269C22×C53#51912yes—
270C22×C53#51911yescharacteristic
271C22×C53#51911yescharacteristic
272C22×C53#51912yes—
273C22×C53#52012yes—
274C22×C53#52011yescharacteristic
275C22×C53#52012yes—
276C22×C53#52011yescharacteristic
277C22×C53#53511yescharacteristic
278C22×C53#53612yes—
279C22×C53#53612yes—
280C22×C53#53611yescharacteristic
281C22×C53#53711yescharacteristic
282C22×C53#53711yescharacteristic
283C22×C53#54112yes—
284C22×C53#54112yes—
285C22×C53#54212yes—
286C22×C53#54212yes—
287C4×C53—24no—
288C4×C53—24no—
289C4×C53—42no—
290C4×C53—44no—
291C4×C53—44no—
292Dic53—24no—
293Dic53—24no—
294Dic53—42no—
295Dic53—44no—
296Dic53—44no—
297Dic53C4×C22⋊C418yes—
298C22×C4×Q8—531no—
299C2×#391—532no—
300C2×C4×C4⋊C4—531no—
301C2×#376—531no—
302#1436—532no—
303#1436—532no—
304C4⋊#95—532no—
305#1434—531no—
306#1434—531no—
307#1434—531no—
308#1434D5311yescharacteristic
309C2×C53C2×C4×C22⋊C411yescharacteristic
310C2×C53#227611yescharacteristic
311C2×C53#227611yescharacteristic
312C2×C53#231011yescharacteristic
313C2×C53#231111yescharacteristic
314C2×C53#232011yescharacteristic
315C2×C53#232011yescharacteristic
316C2×C53#232312yes—
317C2×C53#232312yes—
318C2×C53#232612yes—
319C2×C53#232712yes—
320C23×Q8—531no—
321C2×C4×Q8—538no—
322C2×C4×Q8—1064no—
323C2×C4×Q8—1064no—
324C22×C4⋊C4—531no—
325C22×C4⋊C4—531no—
326C22×C4⋊C4—532no—
327C22×C4⋊C4—531no—
328C22×C42—531no—
329C22×C42—531no—
330C22×C42—532no—
331C22×C42—532no—
332C22×C42C2×D5311yescharacteristic
333#391—5316no—
334C4×C4⋊C4—1064no—
335#376—1064no—
336C2×#95—532no—
337C2×#95—532no—
338C2×#95—532no—
339C2×#95—531no—
340C2×#95—531no—
341C2×#95—532no—
342C2×#95—532no—
343C2×#95—532no—
344C2×#95—532no—
345C2×#95C2×D5311yescharacteristic
346C2×#95C2×D5311yescharacteristic
347C2×#95C2×D5312yes—
348C2×#95C2×D5312yes—
349C53#1378611yescharacteristic, complemented, Sylow
350C22×Q8—534no—
351C22×Q8—538no—
352C22×Q8—1062no—
353C22×Q8—1062no—
354C22×Q8—1064no—
355C23×C4—531no—
356C23×C4—531no—
357C23×C4—531no—
358C23×C4—531no—
359C23×C4—532no—
360C23×C4—532no—
361C23×C4C22×D5311yescharacteristic
362C23×C4C22×D5312yes—
363C23×C4C22×D5312yes—
364C23×C4C22×D5311yescharacteristic
365C23×C4C22×D5311yescharacteristic
366C4×Q8—10632no—
367C2×C4⋊C4—534no—
368C2×C4⋊C4—534no—
369C2×C4⋊C4—534no—
370C2×C4⋊C4—538no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1062no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1064no—
378C2×C4⋊C4—1064no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—534no—
382C2×C42—534no—
383C2×C42—534no—
384C2×C42—534no—
385C2×C42—538no—
386C2×C42—1064no—
387C2×C42—1064no—
388C2×C42—1062no—
389C2×C42—1062no—
390C2×C42—1064no—
391C2×C42—2122no—
392C2×C42—2122no—
393C2×C42C4×D5314yes—
394#95—22no—
395#95—22no—
396#95—42no—
397#95—538no—
398#95—538no—
399#95—538no—
400#95—1064no—
401#95—1062no—
402#95—1062no—
403#95—2122no—
404#95—2122no—
405#95—2122no—
406#95C4×D5318yes—
407C24C23×D5311yescharacteristic
408C2×Q8—5316no—
409C2×Q8—1068no—
410C2×Q8—1068no—
411C2×Q8—10616no—
412C2×Q8—10616no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—532no—
437C22×C4—534no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—534no—
441C22×C4—534no—
442C22×C4—532no—
443C22×C4—532no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—534no—
447C22×C4—534no—
448C22×C4—532no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1062no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1061no—
463C22×C4—1062no—
464C22×C4—1061no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1061no—
472C22×C4—1062no—
473C22×C4—1061no—
474C22×C4C2×C4×D5314yes—
475C22×C4C2×C4×D5314yes—
476C22×C4C2×C4×D5312yes—
477C22×C4C2×C4×D5314yes—
478C22×C4C2×C4×D5312yes—
479C22×C4C2×C4×D5312yes—
480C22×C4D4×D5312yes—
481C22×C4D4×D5312yes—
482C4⋊C4—1068no—
483C4⋊C4—1068no—
484C4⋊C4—1068no—
485C4⋊C4—10616no—
486C42—28no—
487C42—5316no—
488C42—10616no—
489C42—1068no—
490C42—2128no—
491C23C22×C4×D5311yescharacteristic
492C23C22×C4×D5312yes—
493C23C2×D4×D5311yescharacteristic
494C23C2×D4×D5311yescharacteristic
495C23#41×D5312yes—
496C23#1313326211yescharacteristic
497C23#1313326311yescharacteristic
498C23#1313326412yes—
499C23#1313326412yes—
500C23#1313326411yescharacteristic
501C23#1313326511yescharacteristic
502Q8—10632no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—534no—
533C2×C4—534no—
534C2×C4—534no—
535C2×C4—538no—
536C2×C4—538no—
537C2×C4—534no—
538C2×C4—534no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—1064no—
542C2×C4—1064no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1064no—
551C2×C4—1064no—
552C2×C4—1064no—
553C2×C4—1064no—
554C2×C4—1062no—
555C2×C4—1064no—
556C2×C4—1064no—
557C2×C4—1062no—
558C2×C4—2122no—
559C2×C4—2121no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4C42×D5314yes—
571C2×C4C22⋊C4×D5314yes—
572C22C2×C42×D5311yescharacteristic
573C22C2×C22⋊C4×D5311yescharacteristic
574C22#117×D5311yescharacteristic
575C22C4×D4×D5312yes—
576C22C4×D4×D5312yes—
577C22#1681109912yes—
578C22#1681110011yescharacteristic
579C22#1681110011yescharacteristic
580C22#1681110012yes—
581C22#1681110212yes—
582C22#1681110311yescharacteristic
583C22#1681110312yes—
584C22#1681110411yescharacteristic
585C22#1681118911yescharacteristic
586C22#1681119512yes—
587C22#1681119512yes—
588C22#1681119611yescharacteristic
589C22#1681120211yescharacteristic
590C22#1681120311yescharacteristic
591C22#1681123012yes—
592C22#1681123212yes—
593C22#1681123712yes—
594C22#1681123912yes—
595C4—24no—
596C4—24no—
597C4—42no—
598C4—44no—
599C4—44no—
600C4—538no—
601C4—1064no—
602C4—1064no—
603C4—2122no—
604C4—2124no—
605C4—2124no—
606C2C4×C22⋊C4×D5311yescharacteristic
607C2#2027707811yescharacteristic
608C2#2027707911yescharacteristic
609C2#2027730011yescharacteristic
610C2#2027730411yescharacteristic
611C2#2027736311yescharacteristic
612C2#2027736511yescharacteristic
613C2#2027739412yes—
614C2#2027739612yes—
615C2#2027742412yes—
616C2#2027743012yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC53×#1434C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1434D53
hypocenterC53#13786
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic106C4semidirect
24. #16810527C4semidirect
172. C2×Dic106C42semidirect
349. C53#13786semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1062
C2×#168105272
C2×#168104742
C2×#168104682
#202729492
#202729462
#202729432
#202729422
#202729392
C53×#14342
#1378653