Finite Group Databasebeta

← #27391799

Subgroups — #27391799

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810921C211yesmaximal, characteristic
2#20277455C211yesmaximal, characteristic, complemented
3#20277076C211yesmaximal, characteristic
4#20275669C211yesmaximal, characteristic, complemented
5#20275475C211yesmaximal, characteristic, complemented
6#20275290C211yesmaximal, characteristic, complemented
7#20275287C211yesmaximal, characteristic, complemented
8#20274976C211yesmaximal, characteristic
9#20274713C211yesmaximal, characteristic
10#20274615C211yesmaximal, characteristic, complemented
11#20274603C211yesmaximal, characteristic, complemented
12#20274571C211yesmaximal, characteristic, complemented
13#20274542C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C53×#1864C211yesmaximal, characteristic
16C2×#13133208C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19#16810921C418yescomplemented
20C2×#13133151C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810519—21no—
31#16810518—21no—
32#16810508—21no—
33#16810505—21no—
34#16810492C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53—21no—
43#16810469C2211yescharacteristic
44C2×#13133124—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133208—22no—
65#13133208C2×C414yescomplemented
66D4×Dic53—22no—
67D4×Dic53—22no—
68D4×Dic53C2×C414yes—
69D4×Dic53Q814yes—
70C2×C4⋊Dic53—21no—
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×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133151C2×C414yescomplemented
89#13133148—22no—
90#13133148C2×C414yescomplemented
91#13133146—22no—
92#13133146C2×C414yescomplemented
93#13133138—22no—
94#13133138C2×C414yes—
95#13133127—22no—
96C8×Dic53—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107C22×D4×C53C2311yescharacteristic
108C23×C4×C53—21no—
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C4×D4×C53—22no—
112C4×D4×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C42×C53C2311yescharacteristic
119#106×C53—22no—
120#102×C53—22no—
121#102×C53C2×C414yes—
122#95×C53—21no—
123C24×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53C22×C412yes—
128C2×D4×C53C2×Q812yes—
129C2×D4×C53#4112yes—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—24no—
133C22×C4×C53—21no—
134C22×C4×C53—81no—
135C22×C4×C53—81no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—24no—
143C22×Dic53—24no—
144C22×Dic53—21no—
145C22×Dic53—81no—
146C22×Dic53—81no—
147C22×Dic53—81no—
148C22×Dic53—81no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53C22×C412yes—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—81no—
164C4⋊C4×C53C22×C412yes—
165C4⋊C4×C53#4112yes—
166C22⋊C4×C53—24no—
167C42×C53—21no—
168C42×C53—21no—
169C22⋊Dic53—24no—
170C22⋊Dic53—24no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—24no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53C22×C412yes—
178C4⋊Dic53C22×C412yes—
179C4⋊Dic53C2×Q812yes—
180#11719919—24no—
181#11719919—81no—
182#11719919—81no—
183#11719919—81no—
184C4×Dic53—81no—
185C4×Dic53—81no—
186C4×Dic53—81no—
187C4×Dic53—81no—
188C4×Dic53—81no—
189C4×Dic53C22×C412yes—
190C4×Dic53C2×Q812yes—
191C4×Dic53#4112yes—
192C4×Dic53#4112yes—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322C22×C412yes—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53C22×D411yescharacteristic
207D4×C53—22no—
208D4×C53—22no—
209D4×C53—22no—
210D4×C53C4×Q814yes—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—84no—
227C2×C4×C53C23×C411yescharacteristic
228C2×C4×C53C22×Q811yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—84no—
254C2×Dic53—84no—
255C2×Dic53C4×D412yes—
256C2×Dic53C4×D412yes—
257C2×Dic53C22⋊Q812yes—
258C2×Dic53C22⋊Q812yes—
259C8×C53—22no—
260C8×C53—41no—
261#88322—22no—
262#88322—41no—
263#21619—531nomaximal, Sylow
264C22×C53—24no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53C2×C4×D411yescharacteristic
272C22×C53C2×C22⋊Q811yescharacteristic
273C22×C53#52511yescharacteristic
274C22×C53#57611yescharacteristic
275C22×C53#57711yescharacteristic
276C22×C53#57711yescharacteristic
277C22×C53#57711yescharacteristic
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—42no—
281C4×C53—42no—
282C4×C53—81no—
283C4×C53C2×C4×Q811yescharacteristic
284C4×C53C4×#4111yescharacteristic
285C4×C53#52911yescharacteristic
286C4×C53#53311yescharacteristic
287Dic53—22no—
288Dic53—22no—
289Dic53—22no—
290Dic53—22no—
291Dic53—42no—
292Dic53—42no—
293Dic53—42no—
294Dic53—81no—
295C2×#478—531no—
296#2330—531no—
297#2275—531no—
298#1989—531no—
299#1948—531no—
300#1919—531no—
301#1918—531no—
302#1864—531no—
303#1864D5311yescharacteristic
304#1802—531no—
305#1776—531no—
306C4×#106—531no—
307#1768—531no—
308#1762—531no—
309C4×#102—531no—
310C2×C53—44no—
311C2×C53C4×C22⋊Q811yescharacteristic
312C2×C53#294611yescharacteristic
313C2×C53#295211yescharacteristic
314C2×C53#308111yescharacteristic
315C2×C53#308411yescharacteristic
316C2×C53#311911yescharacteristic
317C2×C53#312311yescharacteristic
318C2×#125—531no—
319C2×C4×D4—531no—
320C2×C4×D4—531no—
321C2×C4×D4C2×D5311yescharacteristic
322#478—538no—
323C2×#107—531no—
324C2×#106—531no—
325C2×#106—531no—
326C2×#106C2×D5311yescharacteristic
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×#102—531no—
330C2×#102—531no—
331C2×#102C2×D5311yescharacteristic
332C2×#102C2×D5311yescharacteristic
333C2×#97—531no—
334C2×C4×C8—531no—
335#389—531no—
336#389—1061no—
337#389—1061no—
338#389C2×D5311yescharacteristic
339#387—1061no—
340#387—1061no—
341#384—531no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C22⋊C4—1061no—
348C4×C22⋊C4—1061no—
349C4×C22⋊C4—1061no—
350#376—531no—
351C2×#95—1061no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340C2×D5311yescharacteristic
360#336—531no—
361#336C2×D5311yescharacteristic
362C53#2161911yescharacteristic, complemented, Sylow
363C22×D4C22×D5311yescharacteristic
364C23×C4—21no—
365C23×C4—1061no—
366C23×C4—1061no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8C22×D5311yescharacteristic
370C22×C8C22×D5311yescharacteristic
371#125—534no—
372#125—1062no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4—534no—
376C4×D4—534no—
377C4×D4—1062no—
378C4×D4—1062no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C42—531no—
396C2×C42—531no—
397C2×C42—1061no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42—1061no—
401C2×C42—1061no—
402C2×C42C22×D5311yescharacteristic
403#107—534no—
404#106—22no—
405#106—534no—
406#106—1062no—
407C4⋊C8—534no—
408C4⋊C8—1062no—
409#102—22no—
410#102—534no—
411#102—1062no—
412#102C4×D5314yes—
413#97—1062no—
414C4×C8—1062no—
415#95—21no—
416#95—531no—
417#95—531no—
418#95—531no—
419#95—531no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C4×D5312yes—
431C2×D4#1171994712yes—
432C2×D4Q8×D5312yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—1064no—
444C22×C4—1064no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—4241no—
450C22×C4—4241no—
451C22×C4—4241no—
452C22×C4—4241no—
453C22×C4D4×D5311yescharacteristic
454C22×C4D4×D5311yescharacteristic
455C22×C4C23×D5311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—532no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8C2×C4×D5312yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—81no—
473C4⋊C4—532no—
474C4⋊C4—532no—
475C4⋊C4—532no—
476C4⋊C4—1064no—
477C4⋊C4—1064no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—4241no—
484C4⋊C4—4241no—
485C4⋊C4—4241no—
486C4⋊C4C2×C4×D5312yes—
487C4⋊C4#1171994712yes—
488C22⋊C4—24no—
489C22⋊C4—1064no—
490C22⋊C4—1064no—
491C42—21no—
492C42—21no—
493C42—532no—
494C42—532no—
495C42—532no—
496C42—532no—
497C42—4241no—
498C42—4241no—
499C42—4241no—
500C42—4241no—
501C42—4241no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×D4×D5311yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4#1313318214yes—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—532no—
531C2×C4—532no—
532C2×C4—532no—
533C2×C4—532no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—4244no—
546C2×C4—4244no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4C22×C4×D5311yescharacteristic
557C2×C4C2×Q8×D5311yescharacteristic
558C2×C4#41×D5311yescharacteristic
559C2×C4#41×D5311yescharacteristic
560C2×C4C2×#1171994711yescharacteristic
561C2×C4C2×#1171994711yescharacteristic
562C8—22no—
563C8—41no—
564C8—1062no—
565C8—2121no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C4×D4×D5311yescharacteristic
574C22C22⋊Q8×D5311yescharacteristic
575C22#1681112911yescharacteristic
576C22#1681142511yescharacteristic
577C22#1681142911yescharacteristic
578C22#1681143311yescharacteristic
579C22#1681143311yescharacteristic
580C4—41no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—81no—
585C4—1062no—
586C4—1062no—
587C4—1062no—
588C4—1062no—
589C4—4241no—
590C4—2122no—
591C4—2122no—
592C4—2122no—
593C4C2×#1313318211yescharacteristic
594C4C4×#1171994711yescharacteristic
595C4#1681115011yescharacteristic
596C4#1681117511yescharacteristic
597C2—44no—
598C2#2027724711yescharacteristic
599C2#2028354211yescharacteristic
600C2#2028359711yescharacteristic
601C2#2028454311yescharacteristic
602C2#2028457211yescharacteristic
603C2#2028490711yescharacteristic
604C2#2028496111yescharacteristic
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#21619
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. #20277455C2semidirect
4. #20275669C2semidirect
5. #20275475C2semidirect
6. #20275290C2semidirect
7. #20275287C2semidirect
10. #20274615C2semidirect
11. #20274603C2semidirect
12. #20274571C2semidirect
19. #16810921C4semidirect
65. #13133208C2×C4semidirect
88. #13133151C2×C4semidirect
90. #13133148C2×C4semidirect
92. #13133146C2×C4semidirect
362. C53#21619semidirect

Maximal subgroups

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

subgroupindex
C2×#168109212
#202774552
#202770762
#202756692
#202754752
#202752902
#202752872
#202749762
#202747132
#202746152
#202746032
#202745712
#202745422
#102×Dic532
C53×#18642
#2161953