Finite Group Databasebeta

← #27384088

Subgroups — #27384088

Order 13568.

1802 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D53C211yesmaximal, characteristic
2C2×#16810674C211yesmaximal, characteristic, complemented
3#20277119C211yesmaximal, characteristic, complemented
4#20274800C212yesmaximal, complemented
5#20274714C212yesmaximal, complemented
6#20274713C212yesmaximal, complemented
7#20274644C212yesmaximal
8C4×#13133137C211yesmaximal, characteristic, complemented
9C4×#13133139C211yesmaximal, characteristic, complemented
10C4×#13133138C211yesmaximal, characteristic, complemented
11C4×#102×C53C211yesmaximal, characteristic, complemented
12C22×D4×D53C2211yescharacteristic
13C23×C4×D53—21no—
14C2×C4×D212C2211yescharacteristic
15C2×D4×Dic53C2211yescharacteristic
16C2×#13133179C2211yescharacteristic
17C2×C4×C22⋊D53—21no—
18C2×#13133177—21no—
19C4×D4×D53—24no—
20C4×D4×D53C418yescomplemented
21C2×#13133174—21no—
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C22⋊C4×D53—21no—
24C2×C42×D53C2211yescharacteristic
25#16810674—24no—
26C2×#13133146C2212yescomplemented
27C2×#13133139C2212yes—
28C2×#13133138C2212yes—
29C2×#13133137C2212yes—
30C2×#13133129C2212yes—
31C2×C4×#88322C2211yescharacteristic, complemented
32#16810492C2212yescomplemented
33#16810487—21no—
34#16810482—21no—
35#16810480C2212yes—
36C4×#11719919—21no—
37C4×C4⋊Dic53C2211yescharacteristic, complemented
38#16810386C2212yescomplemented
39#16810385C2212yescomplemented
40#16810385C2212yescomplemented
41C2×C4×D4×C53C2211yescharacteristic
42C2×#102×C53C2212yes—
43C2×C4×C8×C53C2211yescharacteristic, complemented
44C4×C4⋊C4×C53C2211yescharacteristic, complemented
45C53×#340C2212yescomplemented
46C24×D53—21no—
47C22×D212C2311yescharacteristic
48C22×C22⋊D53—21no—
49C2×D4×D53—24no—
50C2×D4×D53—24no—
51C2×D4×D53—22no—
52C2×D4×D53C2×C414yes—
53C23×Dic53—21no—
54C22×C4×D53—21no—
55C22×C4×D53—21no—
56C22×C4×D53—21no—
57C22×C4×D53—28no—
58C22×C4×D53—44no—
59C22×C4×D53—44no—
60C22×C4×D53D411yescharacteristic
61C22×C4×D53D411yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2312yes—
64C4×D212—22no—
65C4×D212C2×C414yes—
66D4×Dic53—22no—
67D4×Dic53C2×C414yes—
68#13133179—22no—
69#13133179C2×C414yes—
70C4×C22⋊D53—28no—
71#13133177—28no—
72#13133174—44no—
73C2×C4⋊Dic53C2312yes—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C4⋊C4×D53—22no—
79C4⋊C4×D53C2×C414yes—
80C2×C22⋊Dic53—21no—
81C2×#11719921—22no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C22⋊C4×D53—28no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89C42×D53C2×C414yes—
90C42×D53D414yes—
91#13133146—24no—
92#13133139—24no—
93#13133138—24no—
94#13133137—24no—
95#13133129—24no—
96C4×#88322—22no—
97#13133124—21no—
98#13133124—22no—
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53—21no—
101C22×C8×C53C2312yes—
102C4×D4×C53—22no—
103C4×D4×C53C2×C414yes—
104C2×C4⋊C4×C53C2312yes—
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109#102×C53—24no—
110C4×C8×C53—22no—
111#95×C53—21no—
112C24×C53—21no—
113C23×D53—28no—
114C23×D53—21no—
115C23×D53—24no—
116C23×D53—44no—
117C23×D53—42no—
118C23×D53—42no—
119C23×D53C2×D411yescharacteristic
120C2×D4×C53—21no—
121C2×D4×C53—22no—
122C2×D4×C53C22×C412yes—
123C2×D4×C53#4114yes—
124C22×C4×C53—21no—
125C22×C4×C53—24no—
126C22×C4×C53—22no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C2×C22⋊D53—28no—
133C2×C22⋊D53—24no—
134C22×Dic53—21no—
135C22×Dic53—24no—
136C22×Dic53—22no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142D4×D53—216no—
143D4×D53—28no—
144D4×D53—28no—
145C2×D212—21no—
146C2×D212—22no—
147C2×D212C22×C412yes—
148C2×D212#4114yes—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—24no—
152C2×C4×D53—24no—
153C2×C4×D53—24no—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53—432no—
162C2×C4×D53C22⋊C414yes—
163C2×C4×D53C22⋊C414yes—
164C2×C4×D53D814yes—
165C2×C4×D53SD1614yes—
166C2×C4×D53C22×C412yes—
167C2×C4×D53C2×D412yes—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53C22×C412yes—
175C22⋊C4×C53—24no—
176C42×C53—42no—
177C42×C53C22×C412yes—
178C42×C53C2×D412yes—
179C22⋊Dic53—24no—
180#11719921—24no—
181#11719921—24no—
182#11719921—44no—
183C4⋊Dic53—21no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53C22×C412yes—
187#11719919—24no—
188#11719919—44no—
189#11719919—44no—
190C4×Dic53—42no—
191C4×Dic53C22⋊C414yes—
192C4×Dic53C22⋊C414yes—
193C4×Dic53C22×C412yes—
194C4×Dic53C2×D412yes—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322—22no—
198C23×C53—22no—
199C23×C53—24no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—42no—
203C23×C53C22×D411yescharacteristic
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—216no—
207C22×D53—22no—
208C22×D53—24no—
209C22×D53—416no—
210C22×D53—416no—
211C22×D53—48no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—42no—
215C22×D53—48no—
216C22×D53C2×C22⋊C412yes—
217C22×D53C2×D812yes—
218C22×D53C2×SD1612yes—
219D4×C53—22no—
220D4×C53—24no—
221D4×C53#11714yescomplemented
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—48no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53C2×C22⋊C412yes—
231C2×C4×C53C2×C22⋊C412yes—
232C2×C4×C53C2×D812yes—
233C2×C4×C53C2×SD1612yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C2×#4112yes—
237C2×C4×C53C2×#4112yes—
238C22⋊D53—216no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—48no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53C2×C22⋊C412yes—
249C2×Dic53C2×C22⋊C412yes—
250C2×Dic53C2×C22⋊C412yes—
251C2×Dic53C2×D812yes—
252C2×Dic53C2×SD1612yes—
253D212—22no—
254D212—24no—
255D212#11714yescomplemented
256C4×D53—22no—
257C4×D53—22no—
258C4×D53—24no—
259C4×D53—44no—
260C4×D53—48no—
261C4×D53—48no—
262C4×D53—48no—
263C4×D53#102116yescomplemented
264C8×C53—42no—
265#88322—42no—
266#20732—531nomaximal, Sylow
267C22×C53—24no—
268C22×C53—44no—
269C22×C53—44no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53C22×C22⋊C411yescharacteristic
273C22×C53#52512yes—
274C22×C53C22×D811yescharacteristic
275C22×C53C22×SD1611yescharacteristic
276C22×C53#57512yes—
277C2×D53—24no—
278C2×D53—24no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—24no—
282C2×D53—432no—
283C2×D53—44no—
284C2×D53—44no—
285C2×D53—42no—
286C2×D53—42no—
287C2×D53C2×#10214yes—
288C4×C53—22no—
289C4×C53—42no—
290C4×C53—82no—
291C4×C53C2×#10214yes—
292C4×C53C2×#11711yescharacteristic
293C4×C53C2×#11711yescharacteristic
294C4×C53#52512yes—
295Dic53—22no—
296Dic53—22no—
297Dic53—42no—
298Dic53—82no—
299Dic53C2×#10214yes—
300C22×C4×D4—531no—
301C2×#431—531no—
302#2285—531no—
303#1823—532no—
304#1802—532no—
305#1802—532no—
306#1786—532no—
307C4×#102—531no—
308C4×#102—531no—
309C4×#102—531no—
310C4×#102D5311yescharacteristic, complemented
311C2×C53—44no—
312C2×C53#228011yescharacteristic
313C2×C53C22×#10211yescharacteristic
314C2×C53#289611yescharacteristic
315C2×C53#310012yes—
316C2×C53#310112yes—
317D53—24no—
318D53—44no—
319C23×D4—531no—
320C24×C4—1061no—
321C2×C4×D4—531no—
322C2×C4×D4—538no—
323C2×C4×D4—531no—
324C2×C4×D4—531no—
325C2×C4×D4—1061no—
326C2×C4×D4—1064no—
327C2×C4×D4—1061no—
328C2×C4×D4C2×D5311yescharacteristic
329C2×#117—1061no—
330C22×C4⋊C4—531no—
331C22×C22⋊C4—1061no—
332C22×C42—531no—
333#431—1064no—
334C2×C4⋊C8—532no—
335C2×#102—532no—
336C2×#102—532no—
337C2×#102—532no—
338C2×#102C2×D5312yes—
339C2×C22⋊C8—532no—
340C2×C4×C8—531no—
341C2×C4×C8C2×D5311yescharacteristic, complemented
342#384—532no—
343#382—1061no—
344#380—532no—
345#380—1061no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—1061no—
348C4×C4⋊C4C2×D5311yescharacteristic, complemented
349#340—532no—
350#340—532no—
351#340—532no—
352#340C2×D5312yescomplemented
353C53#2073211yescharacteristic, complemented, Sylow
354C25—1061no—
355C22×D4—534no—
356C22×D4—531no—
357C22×D4—1062no—
358C22×D4—1061no—
359C22×D4—1064no—
360C22×D4—1064no—
361C22×D4C22×D5311yescharacteristic
362C23×C4—21no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—1061no—
367C23×C4—1068no—
368C23×C4—1061no—
369C23×C4—1061no—
370C23×C4—1061no—
371C23×C4—2124no—
372C23×C4—2124no—
373C22×C8—532no—
374C22×C8C22×D5312yes—
375C4×D4—22no—
376C4×D4—534no—
377C4×D4—534no—
378C4×D4—534no—
379C4×D4—1068no—
380C4×D4—1062no—
381C4×D4—1068no—
382C4×D4—1062no—
383C4×D4—1062no—
384C4×D4C4×D5314yes—
385#117—2124no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—534no—
388C2×C4⋊C4—532no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4C22×D5312yes—
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1068no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1062no—
401C2×C42—21no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—534no—
406C2×C42—534no—
407C2×C42—1061no—
408C2×C42C22×D5311yescharacteristic
409C4⋊C8—1064no—
410#102—24no—
411#102—1064no—
412#102—1064no—
413#102—1064no—
414C22⋊C8—1064no—
415C4×C8—22no—
416C4×C8—1062no—
417#95—21no—
418#95—1062no—
419#95—1061no—
420C24—21no—
421C24—531no—
422C24—1064no—
423C24—1068no—
424C24—1061no—
425C24—2122no—
426C24—2122no—
427C24—2124no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—532no—
431C2×D4—534no—
432C2×D4—1064no—
433C2×D4—1068no—
434C2×D4—10616no—
435C2×D4—1068no—
436C2×D4—1068no—
437C2×D4—1061no—
438C2×D4—1062no—
439C2×D4C2×C4×D5312yes—
440C2×D4#1171994514yes—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—531no—
447C22×C4—534no—
448C22×C4—534no—
449C22×C4—534no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—534no—
455C22×C4—1064no—
456C22×C4—1064no—
457C22×C4—1064no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1061no—
461C22×C4—1064no—
462C22×C4—1062no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4—21232no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4D4×D5311yescharacteristic
474C22×C4D4×D5311yescharacteristic
475C22×C4C23×D5311yescharacteristic
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—1062no—
480C2×C8—1062no—
481C2×C8—1062no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—532no—
486C4⋊C4—1064no—
487C4⋊C4—1061no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—2124no—
491C4⋊C4—2124no—
492C4⋊C4C2×C4×D5312yes—
493C22⋊C4—24no—
494C22⋊C4—1064no—
495C22⋊C4—1064no—
496C22⋊C4—1064no—
497C22⋊C4—2124no—
498C42—42no—
499C42—534no—
500C42—534no—
501C42—532no—
502C42—532no—
503C42—2122no—
504C42C2×C4×D5312yes—
505C42D4×D5312yes—
506C23—22no—
507C23—24no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—532no—
512C23—532no—
513C23—532no—
514C23—10616no—
515C23—1062no—
516C23—1062no—
517C23—1062no—
518C23—1064no—
519C23—2128no—
520C23—21216no—
521C23—21216no—
522C23—2128no—
523C23—2121no—
524C23—2121no—
525C23—2122no—
526C23C2×D4×D5311yescharacteristic
527D4—22no—
528D4—24no—
529D4—534no—
530D4—10616no—
531D4—1062no—
532D4—1064no—
533D4#1313317414yescomplemented
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—532no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—5316no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1064no—
553C2×C4—1062no—
554C2×C4—2124no—
555C2×C4—2128no—
556C2×C4—2128no—
557C2×C4—2128no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2128no—
564C2×C4C22⋊C4×D5312yes—
565C2×C4C22⋊C4×D5312yes—
566C2×C4D8×D5312yes—
567C2×C4SD16×D5312yes—
568C2×C4C22×C4×D5311yescharacteristic
569C2×C4C2×D4×D5311yescharacteristic
570C2×C4#41×D5312yes—
571C2×C4C2×#1171994512yes—
572C8—42no—
573C8—2122no—
574C22—24no—
575C22—44no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—534no—
580C22—1064no—
581C22—1064no—
582C22—1062no—
583C22—1062no—
584C22—1064no—
585C22—21232no—
586C22—2124no—
587C22—2124no—
588C22—2122no—
589C22—2122no—
590C22C2×C22⋊C4×D5311yescharacteristic
591C22#1681113212yes—
592C22C2×D8×D5311yescharacteristic
593C22C2×SD16×D5311yescharacteristic
594C22#1681142012yes—
595C4—22no—
596C4—42no—
597C4—82no—
598C4—534no—
599C4—1062no—
600C4—1062no—
601C4—4242no—
602C4—2122no—
603C4#102×D5314yes—
604C4#117×D5311yescharacteristic
605C4C2×#1313317411yescharacteristic
606C4#1681113212yes—
607C2—44no—
608C2—1064no—
609C2—2124no—
610C2#2027709611yescharacteristic
611C2C2×#102×D5311yescharacteristic
612C2#2028323011yescharacteristic
613C2#2028471312yes—
614C2#2028472112yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#102×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×D53C22
cosocleC22×C4×C53C24
hypercenterC4×#102D53
hypocenterC53#20732
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16810674C2semidirect
3. #20277119C2semidirect
4. #20274800C2semidirect
5. #20274714C2semidirect
6. #20274713C2semidirect
8. C4×#13133137C2semidirect
9. C4×#13133139C2semidirect
10. C4×#13133138C2semidirect
11. C4×#102×C53C2semidirect
20. C4×D4×D53C4semidirect
26. C2×#13133146C22semidirect
31. C2×C4×#88322C22semidirect
32. #16810492C22semidirect
37. C4×C4⋊Dic53C22semidirect
38. #16810386C22semidirect
39. #16810385C22semidirect
40. #16810385C22semidirect
43. C2×C4×C8×C53C22semidirect
44. C4×C4⋊C4×C53C22semidirect
45. C53×#340C22semidirect
221. D4×C53#117semidirect
255. D212#117semidirect
263. C4×D53#102semidirect
310. C4×#102D53semidirect
341. C2×C4×C8C2×D53semidirect
348. C4×C4⋊C4C2×D53semidirect
352. #340C2×D53semidirect
353. C53#20732semidirect
533. D4#13133174semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×D532
C2×#168106742
#202771192
#202748002
#202747142
#202747132
#202746442
C4×#131331372
C4×#131331392
C4×#131331382
C4×#102×C532
#2073253