Finite Group Databasebeta

← #27306818

Subgroups — #27306818

Order 13568.

1622 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic106C211yesmaximal, characteristic
2C2×#16810490C211yesmaximal, characteristic
3C2×#16810468C211yesmaximal, characteristic
4#20272985C212yesmaximal
5#20272981C212yesmaximal
6#20272949C212yesmaximal
7#20272946C212yesmaximal
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×#11719919C2211yescharacteristic
19C22×C4×Dic53C2212yes—
20C22×C4×Dic53C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22#16810490—24no—
23#16810468—24no—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C42×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2212yes—
40C22×Dic106—22no—
41C22×Dic106—22no—
42C22×Dic106—22no—
43C22×Dic106—22no—
44C22×Dic106—22no—
45C22×Dic106—22no—
46C22×Dic106C2×C414yes—
47C23×Dic53C2312yes—
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53C4×Dic106—232no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53C2×C414yes—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919C2×C414yes—
62C2×#11719919C2×C414yes—
63C2×C4×Dic53—22no—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53C2×C414yes—
69C2×C4×Dic53C2×C414yes—
70C2×C4×Dic53D414yes—
71C2×C4×Dic53D414yes—
72#13133124—22no—
73#13133124—22no—
74#13133124—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—24no—
79#13133124—24no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2312yes—
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C2×C42×C53—22no—
89C2×C42×C53—22no—
90C2×C42×C53C2×C414yes—
91#95×C53—22no—
92#95×C53—22no—
93#95×C53—24no—
94#95×C53—24no—
95C24×C53C2411yescharacteristic
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
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—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53#4114yes—
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53C22×C412yes—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53#4114yes—
163C22×Dic53#4112yes—
164C22×Dic53#4112yes—
165C2×Dic106—28no—
166C2×Dic106—28no—
167C2×Dic106—28no—
168C2×Dic106—28no—
169C2×Dic106—28no—
170C2×Dic106—28no—
171C2×Dic106—28no—
172C42×C53—28no—
173C4⋊Dic53—28no—
174#11719919—28no—
175#11719919—28no—
176C4×Dic53—216no—
177C4×Dic53—28no—
178C4×Dic53C22⋊C4116yes—
179C23×C53C23×C411yescharacteristic
180C23×C53C22×D411yescharacteristic
181C23×C53C22×D411yescharacteristic
182C23×C53C2×#4111yescharacteristic
183C23×C53C2×#4111yescharacteristic
184C23×C53C2×#4112yes—
185C23×C53C2×#4111yescharacteristic
186C23×C53C2×#4112yes—
187C23×C53C2×#4111yescharacteristic
188C23×C532−1+412yes—
189C23×C532−1+412yes—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—24no—
199C2×C4×C53—24no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53C2×C22⋊C414yes—
220C2×C4×C53#11714yes—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
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—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53C2×C22⋊C414yes—
257C2×Dic53C2×C22⋊C414yes—
258C2×Dic53#11714yes—
259Dic106—232no—
260#13961—531nomaximal, Sylow
261C22×C53C22×C22⋊C411yescharacteristic
262C22×C53C2×#11711yescharacteristic
263C22×C53C2×#11711yescharacteristic
264C22×C53C4×#4111yescharacteristic
265C22×C53C4×#4111yescharacteristic
266C22×C53#51911yescharacteristic
267C22×C53#51911yescharacteristic
268C22×C53#52512yes—
269C22×C53#52511yescharacteristic
270C22×C53#52511yescharacteristic
271C22×C53#52511yescharacteristic
272C22×C53#52511yescharacteristic
273C22×C53#53212yes—
274C22×C53#53612yes—
275C22×C53#53612yes—
276C22×C53#55212yes—
277C22×C53#55212yes—
278C22×C53#55212yes—
279C22×C53#55212yes—
280C22×C53#55612yes—
281C22×C53#55612yes—
282C22×C53#55612yes—
283C22×C53#55612yes—
284C4×C53—24no—
285C4×C53—24no—
286C4×C53—44no—
287C4×C53—42no—
288C4×C53—82no—
289Dic53—24no—
290Dic53—24no—
291Dic53—24no—
292Dic53—44no—
293Dic53—42no—
294Dic53—82no—
295C22×C4×Q8—531no—
296C2×#383—531no—
297C2×#376—531no—
298#1446—532no—
299#1445—532no—
300#1436—532no—
301#1436—532no—
302#1434—531no—
303#1434—531no—
304#1434—531no—
305#1434D5311yescharacteristic
306C2×C53#228011yescharacteristic
307C2×C53#229711yescharacteristic
308C2×C53#231011yescharacteristic
309C2×C53#235611yescharacteristic
310C2×C53#235611yescharacteristic
311C2×C53#235611yescharacteristic
312C2×C53#235611yescharacteristic
313C2×C53#259912yes—
314C2×C53#259912yes—
315C2×C53#260412yes—
316C2×C53#260412yes—
317C23×Q8—531no—
318C2×C4×Q8—538no—
319C2×C4×Q8—1064no—
320C2×C4×Q8—1064no—
321C22×C4⋊C4—531no—
322C22×C4⋊C4—531no—
323C22×C4⋊C4—531no—
324C22×C42—531no—
325C22×C42—531no—
326C22×C42—532no—
327C22×C42C2×D5311yescharacteristic
328#383—1064no—
329#376—1064no—
330C2×#95—531no—
331C2×#95—531no—
332C2×#95—532no—
333C2×#95—532no—
334C2×#95—532no—
335C2×#95—532no—
336C2×#95—532no—
337C2×#95—532no—
338C2×#95—532no—
339C2×#95—532no—
340C2×#95—532no—
341C2×#95C2×D5311yescharacteristic
342C2×#95C2×D5311yescharacteristic
343C2×#95C2×D5312yes—
344C2×#95C2×D5312yes—
345C53#1396111yescharacteristic, complemented, Sylow
346C22×Q8—534no—
347C22×Q8—1062no—
348C22×Q8—1062no—
349C22×Q8—1062no—
350C22×Q8—1062no—
351C22×Q8—1062no—
352C22×Q8—1062no—
353C23×C4—531no—
354C23×C4—531no—
355C23×C4—531no—
356C23×C4—531no—
357C23×C4—532no—
358C23×C4—532no—
359C23×C4C22×D5312yes—
360C23×C4C22×D5311yescharacteristic
361C23×C4C22×D5312yes—
362C23×C4C22×D5311yescharacteristic
363C23×C4C22×D5311yescharacteristic
364C4×Q8—10632no—
365C2×C4⋊C4—534no—
366C2×C4⋊C4—534no—
367C2×C4⋊C4—534no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1062no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1062no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—534no—
377C2×C42—534no—
378C2×C42—534no—
379C2×C42—534no—
380C2×C42—1064no—
381C2×C42—1064no—
382C2×C42—1064no—
383C2×C42—1062no—
384C2×C42—1062no—
385C2×C42C4×D5314yes—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—1064no—
391#95—1064no—
392#95—1064no—
393#95—1064no—
394#95—1064no—
395#95—1064no—
396#95—1064no—
397#95—1064no—
398#95—1062no—
399#95—1062no—
400#95—1064no—
401C24C23×D5311yescharacteristic
402C2×Q8—1068no—
403C2×Q8—1068no—
404C2×Q8—1068no—
405C2×Q8—1068no—
406C2×Q8—1068no—
407C2×Q8—1068no—
408C2×Q8—1068no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—532no—
436C22×C4—532no—
437C22×C4—532no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—532no—
441C22×C4—532no—
442C22×C4—534no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1062no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1062no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1062no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4C2×C4×D5312yes—
471C22×C4C2×C4×D5312yes—
472C22×C4C2×C4×D5312yes—
473C22×C4#1171994514yes—
474C22×C4D4×D5312yes—
475C22×C4D4×D5312yes—
476C22×C4#1171994712yes—
477C22×C4#1171994912yes—
478C4⋊C4—1068no—
479C4⋊C4—1068no—
480C4⋊C4—1068no—
481C42—28no—
482C42—5316no—
483C42—10616no—
484C42—1068no—
485C23C22×C4×D5311yescharacteristic
486C23C2×D4×D5311yescharacteristic
487C23C2×D4×D5311yescharacteristic
488C23#41×D5311yescharacteristic
489C23#41×D5311yescharacteristic
490C23#41×D5312yes—
491C23C2×#1171994711yescharacteristic
492C23C2×#1171994512yes—
493C23C2×#1171994911yescharacteristic
494C23#1313326412yes—
495C23#1313326412yes—
496Q8—10632no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—534no—
527C2×C4—534no—
528C2×C4—534no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1064no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—1064no—
545C2×C4—1064no—
546C2×C4—1064no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1064no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4C22⋊C4×D5314yes—
565C2×C4#1313317514yes—
566C22C2×C22⋊C4×D5311yescharacteristic
567C22#117×D5311yescharacteristic
568C22C2×#1313317511yescharacteristic
569C22#1681109111yescharacteristic
570C22C4×#1171994511yescharacteristic
571C22#1681110011yescharacteristic
572C22#1681110011yescharacteristic
573C22#1681113212yes—
574C22#1681113311yescharacteristic
575C22#1681113311yescharacteristic
576C22#1681113611yescharacteristic
577C22#1681113611yescharacteristic
578C22#1681117012yes—
579C22#1681119512yes—
580C22#1681119512yes—
581C22#1681130612yes—
582C22#1681130912yes—
583C22#1681131112yes—
584C22#1681131312yes—
585C22#1681134012yes—
586C22#1681134112yes—
587C22#1681134212yes—
588C22#1681134612yes—
589C4—24no—
590C4—24no—
591C4—42no—
592C4—44no—
593C4—82no—
594C4—1064no—
595C4—1064no—
596C4—1064no—
597C4—4242no—
598C4—2122no—
599C4—2124no—
600C2#2027709711yescharacteristic
601C2#2027720111yescharacteristic
602C2#2027730011yescharacteristic
603C2#2027767511yescharacteristic
604C2#2027767811yescharacteristic
605C2#2027768111yescharacteristic
606C2#2027768211yescharacteristic
607C2#2028028912yes—
608C2#2028029112yes—
609C2#2028035012yes—
610C2#2028035212yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1434D53
hypocenterC53#13961
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic106C4semidirect
345. C53#13961semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1062
C2×#168104902
C2×#168104682
#202729852
#202729812
#202729492
#202729462
#202729422
#202729392
C53×#14342
#1396153