Finite Group Databasebeta

← #27405897

Subgroups — #27405897

Order 13568.

898 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
1C2×#16810911C211yesmaximal, characteristic
2#20277801C211yesmaximal, characteristic, complemented
3D4×#11719919C211yesmaximal, characteristic
4#20275674C211yesmaximal, characteristic, complemented
5#20275475C211yesmaximal, characteristic, complemented
6#20275441C211yesmaximal, characteristic, complemented
7#20275440C211yesmaximal, characteristic, complemented
8#20275294C211yesmaximal, characteristic, complemented
9#20275291C211yesmaximal, characteristic, complemented
10#20274974C211yesmaximal, characteristic
11#20274576C211yesmaximal, characteristic, complemented
12#102×Dic53C211yesmaximal, characteristic
13#102×Dic53C211yesmaximal, characteristic
14C4×#13133138C211yesmaximal, characteristic
15C53×#1864C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C22×#11719919—21no—
20#16810911C418yescomplemented
21C2×#13133149C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810528—21no—
31#16810527—21no—
32#16810521C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810518—21no—
36#16810496—21no—
37#16810495—21no—
38#16810492C2211yescharacteristic
39#16810481—21no—
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×#11719919C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810474—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×Dic106—21no—
61C23×Dic53—21no—
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133216—22no—
66#13133216C2×C414yescomplemented
67D4×Dic53—22no—
68D4×Dic53—22no—
69D4×Dic53—41no—
70D4×Dic53C2×C414yes—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—44no—
82C2×#11719919D411yescharacteristic
83C2×#11719919D411yescharacteristic
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133149C2×C414yescomplemented
92#13133148—41no—
93#13133148C2×C414yescomplemented
94#13133146C2×C414yescomplemented
95#13133138C2×C414yes—
96#13133138Q814yescomplemented
97C4×#88322—41no—
98C8×Dic53—22no—
99C8×Dic53—41no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C22×D4×C53C2311yescharacteristic
109C23×C4×C53—21no—
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C4×D4×C53—22no—
113C4×D4×C53—41no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C42×C53C2311yescharacteristic
120#106×C53—41no—
121#102×C53—22no—
122#102×C53C2×C414yes—
123#95×C53—21no—
124C24×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53C22×C412yes—
129C2×D4×C53C2×Q812yes—
130C2×D4×C53#4112yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—81no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—24no—
144C22×Dic53—21no—
145C22×Dic53—42no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—81no—
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C2×Dic106—24no—
155C2×Dic106—81no—
156C2×Dic106—81no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C22×C412yes—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—81no—
169C4⋊C4×C53C22×C412yes—
170C22⋊C4×C53—42no—
171C42×C53—21no—
172C42×C53—21no—
173C22⋊Dic53—24no—
174C22⋊Dic53—42no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C22×C412yes—
181C4⋊Dic53C22×C412yes—
182C4⋊Dic53C2×Q812yes—
183#11719919—24no—
184#11719919—22no—
185#11719919—42no—
186#11719919—42no—
187#11719919—42no—
188#11719919—81no—
189#11719919—81no—
190#11719919SD1614yes—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—42no—
194C4×Dic53—81no—
195C4×Dic53—81no—
196C4×Dic53C22×C412yes—
197C4×Dic53#4112yes—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322C22×C412yes—
203C2×#88322C2×Q812yes—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53C22×D411yescharacteristic
212D4×C53—22no—
213D4×C53—22no—
214D4×C53—22no—
215D4×C53C4×Q814yes—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—84no—
231C2×C4×C53C2×SD1612yes—
232C2×C4×C53C23×C411yescharacteristic
233C2×C4×C53C22×Q811yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53H(5,𝔽2)11yescharacteristic
237C2×C4×C532−1+411yescharacteristic
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—48no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—84no—
259C2×Dic53C4×D412yes—
260C2×Dic53C4×D412yes—
261C2×Dic53C2×SD1612yes—
262C2×Dic53C2×SD1612yes—
263C2×Dic53#13512yes—
264Dic106—84no—
265C8×C53—22no—
266C8×C53—41no—
267#88322—22no—
268#88322—22no—
269#22963—531nomaximal, Sylow
270C22×C53—24no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53C2×C4×D411yescharacteristic
278C22×C53#54611yescharacteristic
279C22×C53D4×Q811yescharacteristic
280C22×C53C22×SD1611yescharacteristic
281C22×C53C2×#13511yescharacteristic
282C22×C53#57611yescharacteristic
283C22×C53#57711yescharacteristic
284C4×C53—22no—
285C4×C53—42no—
286C4×C53—41no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53C2×C4×Q811yescharacteristic
290C4×C53#52011yescharacteristic
291C4×C53#55211yescharacteristic
292C4×C53#55411yescharacteristic
293Dic53—22no—
294Dic53—22no—
295Dic53—42no—
296Dic53—42no—
297Dic53—42no—
298Dic53—41no—
299Dic53—81no—
300Dic53C4×SD1614yes—
301C2×#476—531no—
302#2369—531no—
303D4×C4⋊C4—531no—
304#1990—531no—
305#1948—531no—
306#1941—531no—
307#1941—531no—
308#1919—531no—
309#1919—531no—
310#1864—531no—
311#1864D5311yescharacteristic
312C8×C4⋊C4—531no—
313C4×#102—531no—
314C4×#102—531no—
315C4×#102—531no—
316C2×C53—44no—
317C2×C53#236011yescharacteristic
318C2×C53C2×C4×SD1611yescharacteristic
319C2×C53#294611yescharacteristic
320C2×C53#329511yescharacteristic
321C2×C53#330311yescharacteristic
322C2×C53Q8×SD1611yescharacteristic
323C2×C53#338011yescharacteristic
324C2×C4⋊Q8—531no—
325C2×C4×D4—531no—
326C2×C4×D4—531no—
327C2×C4×D4C2×D5311yescharacteristic
328C22×C4⋊C4—1061no—
329#476—538no—
330C2×#106—531no—
331C2×#106—531no—
332C2×#106—531no—
333C2×#106C2×D5311yescharacteristic
334C2×C4⋊C8—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×#102C2×D5311yescharacteristic
338C2×#102C2×D5311yescharacteristic
339C2×C4×C8—531no—
340C2×C4×C8—531no—
341C2×C4×C8—531no—
342#391—1061no—
343#391—1061no—
344#389—531no—
345#389—531no—
346#389—531no—
347#389—1061no—
348#389C2×D5311yescharacteristic
349#385—1061no—
350#385—1061no—
351#384—531no—
352#380—1061no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—1061no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340C2×D5311yescharacteristic
366#336—531no—
367#336C2×D5311yescharacteristic
368C53#2296311yescharacteristic, complemented, Sylow
369C22×Q8—1061no—
370C22×D4C22×D5311yescharacteristic
371C23×C4—21no—
372C23×C4—1061no—
373C23×C4—1061no—
374C22×C8—531no—
375C22×C8—531no—
376C22×C8C22×D5311yescharacteristic
377C22×C8C22×D5311yescharacteristic
378C4⋊Q8—534no—
379C4⋊Q8—1062no—
380C4×D4—22no—
381C4×D4—41no—
382C4×D4—534no—
383C4×D4—1062no—
384C4×D4—1062no—
385C4×D4—2121no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—2124no—
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C42—531no—
407C2×C42—531no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42C22×D5311yescharacteristic
412#106—41no—
413#106—534no—
414#106—534no—
415#106—2121no—
416C4⋊C8—534no—
417#102—22no—
418#102—534no—
419#102—534no—
420#102C4×D5314yes—
421C4×C8—1062no—
422C4×C8—2121no—
423C4×C8—2121no—
424#95—21no—
425#95—531no—
426#95—531no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433C24—21no—
434C2×Q8—1064no—
435C2×Q8—4241no—
436C2×Q8—4241no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C2×C4×D5312yes—
441C2×D4#1171994712yes—
442C2×D4Q8×D5312yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—1064no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—4241no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4D4×D5311yescharacteristic
464C22×C4D4×D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—532no—
472C2×C8—532no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8C2×C4×D5312yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—534no—
485C4⋊C4—532no—
486C4⋊C4—532no—
487C4⋊C4—532no—
488C4⋊C4—1062no—
489C4⋊C4—1064no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—4241no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4C2×C4×D5312yes—
501C22⋊C4—42no—
502C22⋊C4—1064no—
503C22⋊C4—2122no—
504C42—21no—
505C42—21no—
506C42—532no—
507C42—532no—
508C42—1061no—
509C42—1061no—
510C42—4241no—
511C42—4241no—
512C42—2122no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×D4×D5311yescharacteristic
521Q8—4244no—
522D4—22no—
523D4—22no—
524D4—22no—
525D4#1313318214yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4—532no—
542C2×C4—532no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—4244no—
557C2×C4—2128no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4SD16×D5312yes—
568C2×C4C22×C4×D5311yescharacteristic
569C2×C4C2×Q8×D5311yescharacteristic
570C2×C4#41×D5311yescharacteristic
571C2×C4C2×#1171994711yescharacteristic
572C2×C4#1313326211yescharacteristic
573C2×C4#1313326411yescharacteristic
574C8—22no—
575C8—41no—
576C8—1062no—
577C8—1062no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22C4×D4×D5311yescharacteristic
586C22#1681126011yescharacteristic
587C22#1681128611yescharacteristic
588C22C2×SD16×D5311yescharacteristic
589C22#135×D5311yescharacteristic
590C22#1681142711yescharacteristic
591C22#1681143311yescharacteristic
592C4—22no—
593C4—42no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—534no—
598C4—1062no—
599C4—1062no—
600C4—4241no—
601C4—2122no—
602C4—2122no—
603C4—2122no—
604C4—2121no—
605C4C2×#1313318211yescharacteristic
606C4#1681110211yescharacteristic
607C4#1681131011yescharacteristic
608C4#1681132711yescharacteristic
609C2—44no—
610C2#2027771511yescharacteristic
611C2C4×SD16×D5311yescharacteristic
612C2#2028354411yescharacteristic
613C2#2028653211yescharacteristic
614C2#2028663611yescharacteristic
615C2#2028757011yescharacteristic
616C2#2028757711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1864C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1864D53
hypocenterC53#22963
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #20277801C2semidirect
4. #20275674C2semidirect
5. #20275475C2semidirect
6. #20275441C2semidirect
7. #20275440C2semidirect
8. #20275294C2semidirect
9. #20275291C2semidirect
11. #20274576C2semidirect
20. #16810911C4semidirect
66. #13133216C2×C4semidirect
91. #13133149C2×C4semidirect
93. #13133148C2×C4semidirect
94. #13133146C2×C4semidirect
96. #13133138Q8semidirect
368. C53#22963semidirect

Maximal subgroups

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

subgroupindex
C2×#168109112
#202778012
D4×#117199192
#202756742
#202754752
#202754412
#202754402
#202752942
#202752912
#202749742
#202745762
#102×Dic532
C4×#131331382
C53×#18642
#2296353