Finite Group Databasebeta

← #27447205

Subgroups — #27447205

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810927C211yesmaximal, characteristic, complemented
2C2×C4×#11719923C211yesmaximal, characteristic
3C4⋊C4×Dic106C211yesmaximal, characteristic, complemented
4#20276243C211yesmaximal, characteristic
5#20276191C211yesmaximal, characteristic, complemented
6#20276103C211yesmaximal, characteristic
7#20276050C211yesmaximal, characteristic, complemented
8#20275445C211yesmaximal, characteristic, complemented
9#20275274C211yesmaximal, characteristic, complemented
10#20275073C211yesmaximal, characteristic
11#20275073C211yesmaximal, characteristic
12#20274976C211yesmaximal, characteristic
13#20274968C211yesmaximal, characteristic, complemented
14#20274968C211yesmaximal, characteristic, complemented
15C53×#2454C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810927—24no—
21C4×#11719923—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810520—21no—
38#16810520—21no—
39#16810493C2211yescharacteristic
40#16810492—21no—
41C4⋊C4×Dic53—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#399—21no—
53C53×#396—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#385—21no—
59C53×#380—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719923Q814yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C4×Dic106—22no—
70C4×Dic106—22no—
71C4×Dic106—22no—
72C4×Dic106—22no—
73C4×Dic106—41no—
74C4×Dic106—41no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88#13133150—22no—
89#13133150—22no—
90#13133150—41no—
91#13133148—22no—
92#13133148—22no—
93C4⋊#88322—22no—
94#13133140—22no—
95#13133140—22no—
96#13133140D414yes—
97#13133138—22no—
98#13133138—22no—
99#13133138—41no—
100C4×#88322—22no—
101#13133124—21no—
102#13133124—21no—
103C22×D4×C53C2311yescharacteristic
104C23×C4×C53—21no—
105#125×C53—22no—
106#125×C53—41no—
107C4×D4×C53—22no—
108C4×D4×C53—41no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C42×C53C2311yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53D411yescharacteristic
129#95×C53D411yescharacteristic
130C24×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53C2×Q812yes—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—42no—
141C22×C4×C53—42no—
142C22×C4×C53—81no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106C2×D412yes—
158C2×Dic106C2×Q812yes—
159C2×Dic106#4112yes—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—81no—
170C4⋊C4×C53—81no—
171C4⋊C4×C53C2×D412yes—
172C22⋊C4×C53—42no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—81no—
175C22⋊C4×C53—81no—
176C22⋊C4×C53—81no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183#11719923—28no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—42no—
190C4⋊Dic53—81no—
191C4⋊Dic53#4112yes—
192#11719919—24no—
193#11719919—42no—
194#11719919—42no—
195#11719919—81no—
196#11719919—81no—
197C4×Dic53—24no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322C2×D412yes—
211C2×#88322C2×Q812yes—
212C23×C53—22no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53C22×D411yescharacteristic
220D4×C53—22no—
221D4×C53—22no—
222D4×C53—41no—
223D4×C53—41no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—84no—
250C2×C4×C53#13512yes—
251C2×C4×C53#13712yes—
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C22×Q811yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C532−1+411yescharacteristic
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270Dic106—22no—
271Dic106—22no—
272Dic106—22no—
273Dic106C22⋊Q814yes—
274#88322—22no—
275#88322—22no—
276#88322—41no—
277#88322—41no—
278#26379—531nomaximal, Sylow
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53#52511yescharacteristic
287C22×C53D4211yescharacteristic
288C22×C53D4×Q811yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×#13711yescharacteristic
291C22×C53#57611yescharacteristic
292C22×C53#57711yescharacteristic
293C4×C53—22no—
294C4×C53—22no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—41no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53C2×C22⋊Q811yescharacteristic
303C4×C53#53911yescharacteristic
304C4×C53#54711yescharacteristic
305C4×C53#55411yescharacteristic
306Dic53—24no—
307Dic53—42no—
308Dic53—81no—
309Dic53—81no—
310C2×#479—531no—
311C2×C4×SD16—531no—
312#2454D5311yescharacteristic
313Q8×C4⋊C4—531no—
314#2093—531no—
315#2084—531no—
316#2067—531no—
317#2059—531no—
318#1942—531no—
319#1916—531no—
320#1880—531no—
321#1880—531no—
322#1864—531no—
323#1863—531no—
324#1863—531no—
325C2×C53—82no—
326C2×C53#244911yescharacteristic
327C2×C53#310811yescharacteristic
328C2×C53#311911yescharacteristic
329C2×C53#326611yescharacteristic
330C2×C53#328711yescharacteristic
331C2×C53#338111yescharacteristic
332C2×C53#338511yescharacteristic
333C22×SD16—531no—
334C2×#125C2×D5311yescharacteristic
335C2×C4×Q8—531no—
336C2×C4×Q8—531no—
337C2×C4×Q8—531no—
338C2×C4×D4C2×D5311yescharacteristic
339#479—1064no—
340C4×SD16—1064no—
341C2×#107—531no—
342C2×#107—531no—
343C2×#107—531no—
344C2×#106—531no—
345C2×#106—531no—
346C2×C4⋊C8—531no—
347C2×#103—531no—
348C2×#103—531no—
349C2×#103—531no—
350C2×#102—531no—
351C2×#102—531no—
352C2×#102—531no—
353C2×C4×C8—531no—
354#399—21no—
355#396—21no—
356#390C2×D5311yescharacteristic
357#390C2×D5311yescharacteristic
358#389—531no—
359#389—531no—
360#389—1061no—
361#389—1061no—
362#389C2×D5311yescharacteristic
363#389C2×D5311yescharacteristic
364#385—21no—
365#384—531no—
366#384—1061no—
367#380—21no—
368C4×C4⋊C4—1061no—
369C4×C4⋊C4C2×D5311yescharacteristic
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#336—531no—
375#336—531no—
376#336—531no—
377#336—531no—
378C53#2637911yescharacteristic, complemented, Sylow
379C22×Q8—531no—
380C22×D4C22×D5311yescharacteristic
381C23×C4—21no—
382C2×SD16—534no—
383C2×SD16—1062no—
384C2×SD16—1062no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8—531no—
388C22×C8—531no—
389#125—22no—
390#125—41no—
391C4×Q8—1062no—
392C4×Q8—1062no—
393C4×Q8—1062no—
394C4×Q8—1062no—
395C4×Q8—2121no—
396C4×Q8—2121no—
397C4×D4—22no—
398C4×D4—41no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4C22⋊D5311yescharacteristic
414C2×C4⋊C4C22⋊D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C42—1061no—
424C2×C42—1061no—
425C2×C42—1061no—
426C2×C42C22×D5311yescharacteristic
427C2×C42C22×D5311yescharacteristic
428C2×C42C22×D5311yescharacteristic
429#107—1062no—
430#107—1062no—
431#107—2121no—
432#106—1062no—
433#106—1062no—
434C4⋊C8—1062no—
435#103—534no—
436#103—1062no—
437#103—1062no—
438#102—1062no—
439#102—1062no—
440#102—2121no—
441C4×C8—1062no—
442#95—21no—
443#95—21no—
444#95—1061no—
445#95—1061no—
446#95C22⋊D5311yescharacteristic
447#95C22⋊D5311yescharacteristic
448C24—21no—
449C2×Q8—532no—
450C2×Q8—532no—
451C2×Q8—532no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4Q8×D5312yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—81no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—1068no—
479C2×C8—532no—
480C2×C8—532no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—532no—
503C4⋊C4—1064no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—4241no—
510C4⋊C4—4241no—
511C4⋊C4—4241no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4D4×D5312yes—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—1064no—
528C42—2122no—
529C42—2122no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C22×C22⋊D5311yescharacteristic
538Q8—534no—
539Q8—1062no—
540Q8—1062no—
541Q8—1062no—
542D4—22no—
543D4—22no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—84no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4#1313323712yes—
585C2×C4#1313324712yes—
586C2×C4C2×D4×D5311yescharacteristic
587C2×C4C2×Q8×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4#1313326211yescharacteristic
591C2×C4#1313326511yescharacteristic
592C8—1062no—
593C8—1062no—
594C8—2121no—
595C8—2121no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1681113011yescharacteristic
604C22D4×C22⋊D5311yescharacteristic
605C22Q8×C22⋊D5311yescharacteristic
606C22C2×#1313323711yescharacteristic
607C22C2×#1313324711yescharacteristic
608C22#1681142611yescharacteristic
609C22#1681143411yescharacteristic
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—1064no—
620C4—4241no—
621C4—4241no—
622C4—2122no—
623C4C22⋊Q8×D5311yescharacteristic
624C4#1681121911yescharacteristic
625C4#1681127511yescharacteristic
626C4#1681132711yescharacteristic
627C2—82no—
628C2#2027852111yescharacteristic
629C2#2028479111yescharacteristic
630C2#2028490811yescharacteristic
631C2#2028622111yescharacteristic
632C2#2028647111yescharacteristic
633C2#2028759011yescharacteristic
634C2#2028762511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2454C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2454D53
hypocenterC53#26379
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2454

Complements

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

NH ≅ G/Ntype
1. C2×#16810927C2semidirect
3. C4⋊C4×Dic106C2semidirect
5. #20276191C2semidirect
7. #20276050C2semidirect
8. #20275445C2semidirect
9. #20275274C2semidirect
13. #20274968C2semidirect
14. #20274968C2semidirect
378. C53#26379semidirect

Maximal subgroups

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

subgroupindex
C2×#168109272
C2×C4×#117199232
C4⋊C4×Dic1062
#202762432
#202761912
#202761032
#202760502
#202754452
#202752742
#202750732
#202749762
#202749682
C53×#24542
#2637953