Finite Group Databasebeta

← #27405892

Subgroups — #27405892

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810901C211yesmaximal, characteristic
2#20277774C211yesmaximal, characteristic, complemented
3D4×#11719919C211yesmaximal, characteristic
4#20275674C211yesmaximal, characteristic, complemented
5#20275472C211yesmaximal, characteristic, complemented
6#20275448C211yesmaximal, characteristic, complemented
7#20275447C211yesmaximal, characteristic, complemented
8#20275286C211yesmaximal, characteristic, complemented
9#20275285C211yesmaximal, characteristic, complemented
10#20274964C211yesmaximal, characteristic
11#20274962C211yesmaximal, characteristic
12C8×#11719919C211yesmaximal, characteristic, complemented
13#102×Dic53C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C4×#102×C53C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×C4×Dic106—21no—
18C2×D4×Dic53C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20C22×#11719919—21no—
21#16810901C418yescomplemented
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810527—21no—
32#16810522C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810510—21no—
37#16810507—21no—
38#16810496—21no—
39#16810495—21no—
40#16810492C2211yescharacteristic
41#16810481—21no—
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×#11719919C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106—21no—
61C23×Dic53—21no—
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133216—22no—
66#13133216C2×C414yescomplemented
67C4×Dic106—44no—
68D4×Dic53—22no—
69D4×Dic53—22no—
70D4×Dic53—41no—
71D4×Dic53C2×C414yes—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—44no—
84C2×#11719919D411yescharacteristic
85C2×#11719919D411yescharacteristic
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133151—41no—
93#13133151C2×C414yescomplemented
94#13133150—41no—
95#13133150C2×C414yescomplemented
96#13133146C2×C414yescomplemented
97#13133138—22no—
98#13133138C2×C414yes—
99C8×Dic53—22no—
100C8×Dic53—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×D4×C53C2311yescharacteristic
110C23×C4×C53—21no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C4×D4×C53—22no—
114C4×D4×C53—41no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#102×C53C2×C414yes—
122#102×C53Q814yescomplemented
123C4×C8×C53—41no—
124#95×C53—21no—
125C24×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53C22×C412yes—
130C2×D4×C53C2×Q812yes—
131C2×D4×C53#4112yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—81no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—24no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—81no—
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C2×Dic106—24no—
156C2×Dic106—42no—
157C2×Dic106—42no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C22×C412yes—
163C2×C8×C53C2×Q812yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53C22×C412yes—
169C4⋊C4×C53C2×Q812yes—
170C22⋊C4×C53—42no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—42no—
174C22⋊Dic53—24no—
175C22⋊Dic53—42no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53C22×C412yes—
184C4⋊Dic53C22×C412yes—
185#11719919—24no—
186#11719919—22no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—81no—
192#11719919D814yes—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53C22×C412yes—
198C4×Dic53#4112yes—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322C22×C412yes—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53C22×D411yescharacteristic
213D4×C53—22no—
214D4×C53—22no—
215D4×C53—22no—
216D4×C53C4×Q814yescomplemented
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
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—41no—
231C2×C4×C53—84no—
232C2×C4×C53C2×D812yes—
233C2×C4×C53C23×C411yescharacteristic
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53H(5,𝔽2)11yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—48no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—84no—
260C2×Dic53C4×D412yes—
261C2×Dic53C4×D412yes—
262C2×Dic53C2×D812yes—
263C2×Dic53C2×D812yes—
264C2×Dic53#13512yes—
265Dic106—48no—
266C8×C53—22no—
267C8×C53—22no—
268#88322—22no—
269#88322—41no—
270#22962—531nomaximal, Sylow
271C22×C53—24no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53C2×C4×D411yescharacteristic
279C22×C53#54611yescharacteristic
280C22×C53D4×Q811yescharacteristic
281C22×C53C22×D811yescharacteristic
282C22×C53C2×#13511yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57811yescharacteristic
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—81no—
290C4×C53C2×C4×Q811yescharacteristic
291C4×C53#52011yescharacteristic
292C4×C53#55211yescharacteristic
293C4×C53#55411yescharacteristic
294Dic53—22no—
295Dic53—22no—
296Dic53—42no—
297Dic53—42no—
298Dic53—41no—
299Dic53—81no—
300Dic53—81no—
301Dic53C4×D814yes—
302C2×#474—531no—
303#2365—531no—
304D4×C4⋊C4—531no—
305#1990—531no—
306#1947—531no—
307#1942—531no—
308#1942—531no—
309#1918—531no—
310#1918—531no—
311#1862—531no—
312#1862—531no—
313C8×C4⋊C4—531no—
314C4×#102—531no—
315C4×#102—531no—
316C4×#102D5311yescharacteristic
317C2×C53—44no—
318C2×C53#236011yescharacteristic
319C2×C53C2×C4×D811yescharacteristic
320C2×C53#294711yescharacteristic
321C2×C53#329411yescharacteristic
322C2×C53#330211yescharacteristic
323C2×C53Q8×D811yescharacteristic
324C2×C53#338111yescharacteristic
325C2×C4⋊Q8—531no—
326C2×C4×Q8—1061no—
327C2×C4×D4—531no—
328C2×C4×D4—531no—
329C2×C4×D4C2×D5311yescharacteristic
330C22×C4⋊C4—1061no—
331#474—538no—
332C2×#107—531no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#107—531no—
336C2×C4⋊C8—531no—
337C2×#102—531no—
338C2×#102—531no—
339C2×#102C2×D5311yescharacteristic
340C2×#102C2×D5311yescharacteristic
341C2×C4×C8—531no—
342C2×C4×C8—531no—
343C2×C4×C8C2×D5311yescharacteristic
344#391—1061no—
345#389—531no—
346#389—531no—
347#389—531no—
348#389—531no—
349#387—1061no—
350#387—1061no—
351#385—1061no—
352#385—1061no—
353#384—531no—
354#380—1061no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4C2×D5311yescharacteristic
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340C2×D5311yescharacteristic
366#340C2×D5311yescharacteristic
367#336—531no—
368#336—531no—
369C53#2296211yescharacteristic, complemented, Sylow
370C22×Q8—1061no—
371C22×D4C22×D5311yescharacteristic
372C23×C4—21no—
373C23×C4—1061no—
374C23×C4—1061no—
375C22×C8—531no—
376C22×C8—531no—
377C22×C8C22×D5311yescharacteristic
378C22×C8C22×D5311yescharacteristic
379C4⋊Q8—534no—
380C4⋊Q8—1062no—
381C4×Q8—2124no—
382C4×D4—22no—
383C4×D4—41no—
384C4×D4—534no—
385C4×D4—1062no—
386C4×D4—1062no—
387C4×D4—2121no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—2124no—
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—21no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42C22×D5311yescharacteristic
414#107—534no—
415#107—534no—
416#107—2121no—
417#107—2121no—
418C4⋊C8—534no—
419#102—534no—
420#102—1062no—
421#102Dic10614yescomplemented
422#102C4×D5314yes—
423C4×C8—41no—
424C4×C8—1062no—
425C4×C8—2121no—
426#95—21no—
427#95—531no—
428#95—531no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435C24—21no—
436C2×Q8—1064no—
437C2×Q8—2122no—
438C2×Q8—2122no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×Dic10612yes—
443C2×D4C2×C4×D5312yes—
444C2×D4#1171994512yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—1064no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—4241no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4D4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C22×C4C23×D5311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—532no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8C2×Dic10612yes—
479C2×C8C2×C4×D5312yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—534no—
485C4⋊C4—532no—
486C4⋊C4—532no—
487C4⋊C4—1062no—
488C4⋊C4—1064no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4C2×Dic10612yes—
502C4⋊C4C2×C4×D5312yes—
503C22⋊C4—42no—
504C22⋊C4—1064no—
505C22⋊C4—2122no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42—532no—
510C42—532no—
511C42—1061no—
512C42—1061no—
513C42—2122no—
514C42—2122no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×D4×D5311yescharacteristic
523Q8—2128no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4C4×Dic10614yescomplemented
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—4244no—
558C2×C4—2128no—
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×C4—2121no—
568C2×C4—2121no—
569C2×C4D8×D5312yes—
570C2×C4C22×C4×D5311yescharacteristic
571C2×C4C22×Dic10611yescharacteristic
572C2×C4#41×D5311yescharacteristic
573C2×C4C2×#1171994511yescharacteristic
574C2×C4#1313326211yescharacteristic
575C2×C4#1313326411yescharacteristic
576C8—22no—
577C8—22no—
578C8—1062no—
579C8—2121no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22C4×D4×D5311yescharacteristic
588C22#1681126311yescharacteristic
589C22D4×Dic10611yescharacteristic
590C22C2×D8×D5311yescharacteristic
591C22#135×D5311yescharacteristic
592C22#1681142911yescharacteristic
593C22#1681143711yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—534no—
600C4—1062no—
601C4—1062no—
602C4—4241no—
603C4—4241no—
604C4—2122no—
605C4—2122no—
606C4—2121no—
607C4C2×C4×Dic10611yescharacteristic
608C4#1681110211yescharacteristic
609C4#1681130611yescharacteristic
610C4#1681133211yescharacteristic
611C2—44no—
612C2#2027772111yescharacteristic
613C2C4×D8×D5311yescharacteristic
614C2#2028355111yescharacteristic
615C2#2028652411yescharacteristic
616C2#2028662811yescharacteristic
617C2D8×Dic10611yescharacteristic
618C2#2028759111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#102×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterC4×#102D53
hypocenterC53#22962
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #20277774C2semidirect
4. #20275674C2semidirect
5. #20275472C2semidirect
6. #20275448C2semidirect
7. #20275447C2semidirect
8. #20275286C2semidirect
9. #20275285C2semidirect
12. C8×#11719919C2semidirect
21. #16810901C4semidirect
66. #13133216C2×C4semidirect
93. #13133151C2×C4semidirect
95. #13133150C2×C4semidirect
96. #13133146C2×C4semidirect
122. #102×C53Q8semidirect
216. D4×C53C4×Q8semidirect
369. C53#22962semidirect
421. #102Dic106semidirect
527. D4C4×Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168109012
#202777742
D4×#117199192
#202756742
#202754722
#202754482
#202754472
#202752862
#202752852
#202749642
#202749622
C8×#117199192
#102×Dic532
C4×#102×C532
#2296253