Finite Group Databasebeta

← #27418404

Subgroups — #27418404

Order 13568.

1198 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×#16810910C211yesmaximal, characteristic, complemented
2C2×#16810825C211yesmaximal, characteristic, complemented
3C4×#13133191C211yesmaximal, characteristic
4#20275908C212yesmaximal, complemented
5#20275716C212yesmaximal
6#20275664C211yesmaximal, characteristic, complemented
7#20275639C211yesmaximal, characteristic, complemented
8#20275070C212yesmaximal, complemented
9#20274983C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C2×C53×#533C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C2×#13133191C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic
15C2×C4×C22⋊D53—21no—
16#16810910—24no—
17#16810825—24no—
18C2×#13133148C2212yes—
19C2×C4⋊#88322C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133141C2212yes—
22C2×#13133141C2212yes—
23C2×#13133141C2212yes—
24C2×#13133137C2212yes—
25C2×C22⋊#88322C2212yes—
26#16810525C2211yescharacteristic
27#16810521C2211yescharacteristic
28#16810514—21no—
29#16810502—21no—
30#16810483C2211yescharacteristic
31C4×C4⋊Dic53C2211yescharacteristic
32C4×#11719921—21no—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C53×#533—24no—
37C53×#533—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×C4⋊Q8×C53C2211yescharacteristic
40C2×#125×C53—21no—
41C2×C22⋊Q8×C53—21no—
42C2×C22⋊Q8×C53C2212yes—
43C2×C4×Q8×C53—21no—
44C2×C4×Q8×C53C2212yes—
45C2×#117×C53—21no—
46C22×C42×C53C2211yescharacteristic
47C22×D212C2311yescharacteristic
48C22×C22⋊D53—21no—
49C22×C4×D53—21no—
50C2×#11719924—24no—
51C2×#11719924—24no—
52C2×#11719924—42no—
53C22×#88322C2312yes—
54C22×#88322C2312yes—
55#13133191—22no—
56#13133191—22no—
57C4×D212—22no—
58C4×D212—22no—
59C4×C22⋊D53—44no—
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C4⋊Dic53C2311yescharacteristic
63C2×#11719919—21no—
64C2×C22⋊Dic53—21no—
65C2×#11719921—21no—
66C2×#11719921—21no—
67C2×#11719921—21no—
68C2×#11719921—21no—
69C2×C4×Dic53—21no—
70#13133148—24no—
71C4⋊#88322—22no—
72C4⋊#88322—22no—
73#13133141—24no—
74#13133141—24no—
75#13133141—42no—
76#13133137—24no—
77C22⋊#88322—42no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81C22×Q8×C53—21no—
82C22×Q8×C53C2312yes—
83C23×C4×C53D411yescharacteristic
84C23×C4×C53D411yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C4⋊Q8×C53—22no—
87C4⋊Q8×C53—22no—
88C4⋊Q8×C53D414yescomplemented
89C4⋊Q8×C53D414yescomplemented
90#125×C53—28no—
91C22⋊Q8×C53—28no—
92C22⋊Q8×C53—24no—
93C22⋊Q8×C53—24no—
94C4×Q8×C53—28no—
95C4×Q8×C53—24no—
96C4×Q8×C53—24no—
97#117×C53—28no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—22no—
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53C2312yes—
104C2×C22⋊C4×C53—22no—
105C2×C42×C53—22no—
106C2×C42×C53—22no—
107C2×C42×C53—22no—
108C2×C42×C53D411yescharacteristic
109C2×C42×C53D411yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C24×C53C2×D411yescharacteristic
113C23×D53—21no—
114C2×Q8×C53—24no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53—22no—
120C2×Q8×C53—42no—
121C2×Q8×C53—42no—
122C2×Q8×C53C2×D414yes—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—41no—
135C22×C4×C53—41no—
136C22×C4×C53SD1614yescomplemented
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C2×C22⋊D53—24no—
144C2×C22⋊D53—42no—
145C2×C22⋊D53—42no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212#4112yes—
155C2×C4×D53—42no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C4⋊C4×C53—28no—
159C4⋊C4×C53—28no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—24no—
162C4⋊C4×C53—24no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53C2×D414yes—
165C22⋊C4×C53—28no—
166C42×C53—28no—
167C42×C53—22no—
168C42×C53—22no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53C2×D412yes—
172C42×C53C2×D412yes—
173C22⋊Dic53—42no—
174#11719924—48no—
175#11719921—24no—
176#11719921—42no—
177#11719921—42no—
178#11719921—42no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53#4112yes—
185#11719919—42no—
186C4×Dic53—42no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53C2×SD1612yes—
200C23×C53C22×D411yescharacteristic
201C22×D53—22no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208Q8×C53—24no—
209Q8×C53—24no—
210Q8×C53—48no—
211Q8×C53—42no—
212Q8×C53—42no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—28no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—44no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53#12012yes—
236C2×C4×C53#12012yes—
237C2×C4×C53C2×SD1612yes—
238C2×C4×C53C2×SD1612yes—
239C2×C4×C53#13712yes—
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×C4×C532−1+411yescharacteristic
246C22⋊D53—48no—
247C2×Dic53—22no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256D212—22no—
257D212—22no—
258D212—41no—
259D212—41no—
260C4×D53—84no—
261#88322—42no—
262#88322—42no—
263#24151—531nomaximal, Sylow
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53C2×#12011yescharacteristic
272C22×C53#52511yescharacteristic
273C22×C53#53711yescharacteristic
274C22×C53C22×SD1611yescharacteristic
275C22×C53C2×#13711yescharacteristic
276C22×C53#57511yescharacteristic
277C22×C53#57511yescharacteristic
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—41no—
288C4×C53—44no—
289C4×C53—41no—
290C4×C53—44no—
291C4×C53#44814yes—
292C4×C53#54611yescharacteristic
293C4×C53#54711yescharacteristic
294C4×C53#54811yescharacteristic
295C4×C53#55011yescharacteristic
296Dic53—42no—
297Dic53—81no—
298Dic53—81no—
299C2×#533D5311yescharacteristic, complemented
300C2×#476—531no—
301C2×#461—531no—
302C4×#121—531no—
303#2035—532no—
304#2000—532no—
305#1987—531no—
306#1981—531no—
307#1880—532no—
308#1865—532no—
309#1797—531no—
310C2×C53—42no—
311C2×C53#240711yescharacteristic
312C2×C53C2×#44811yescharacteristic
313C2×C53#300511yescharacteristic
314C2×C53#310111yescharacteristic
315C2×C53#311611yescharacteristic
316C2×C53#324611yescharacteristic
317C2×C53#325211yescharacteristic
318D53—82no—
319C22×SD16—532no—
320#533—24no—
321#533—24no—
322C2×C4⋊Q8C2×D5311yescharacteristic
323C2×C4⋊Q8C2×D5311yescharacteristic
324C2×#125—21no—
325C2×C22⋊Q8—21no—
326C2×C22⋊Q8C2×D5312yes—
327C2×#121—531no—
328C2×C4×Q8—21no—
329C2×C4×Q8C2×D5312yes—
330C2×C4×D4—531no—
331C2×C4×D4—1061no—
332C2×#117—21no—
333C22×C42C2×D5311yescharacteristic
334#476—1064no—
335#461—1064no—
336C2×#106—532no—
337C2×C4⋊C8—531no—
338C2×C4⋊C8—531no—
339C2×#103—532no—
340C2×#103—532no—
341C2×#103—532no—
342C2×#102—532no—
343C2×C22⋊C8—532no—
344#390—531no—
345#389—531no—
346#388—1061no—
347#386—1061no—
348#380—531no—
349C4×C4⋊C4—531no—
350C4×C22⋊C4—1061no—
351#340—532no—
352#340—532no—
353#336—532no—
354C53#2415111yescharacteristic, complemented, Sylow
355C22×Q8—21no—
356C22×Q8C22×D5312yes—
357C22×D4—531no—
358C22×D4—1061no—
359C23×C4—1061no—
360C23×C4C22⋊D5311yescharacteristic
361C23×C4C22⋊D5311yescharacteristic
362C23×C4C22×D5311yescharacteristic
363C2×SD16—1064no—
364C2×SD16—1064no—
365C2×SD16—2122no—
366C22×C8—532no—
367C22×C8—532no—
368C4⋊Q8—22no—
369C4⋊Q8—22no—
370C4⋊Q8C22⋊D5314yescomplemented
371C4⋊Q8C22⋊D5314yescomplemented
372#125—28no—
373C22⋊Q8—28no—
374C22⋊Q8—24no—
375C22⋊Q8—24no—
376#121—1062no—
377#121—1062no—
378C4×Q8—28no—
379C4×Q8—24no—
380C4×Q8—24no—
381C4×D4—1062no—
382C4×D4—1062no—
383C4×D4—2124no—
384#117—28no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4C22×D5312yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—1061no—
405C2×C42C22⋊D5311yescharacteristic
406C2×C42C22⋊D5311yescharacteristic
407C2×C42C22×D5311yescharacteristic
408C2×C42C22×D5311yescharacteristic
409#106—1064no—
410C4⋊C8—1062no—
411C4⋊C8—1062no—
412#103—1064no—
413#103—1064no—
414#103—2122no—
415#102—1064no—
416C22⋊C8—2122no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420C24—1061no—
421C24C2×C22⋊D5311yescharacteristic
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8C2×C22⋊D5314yes—
431C2×D4—532no—
432C2×D4—1064no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C2×D4—1061no—
438C2×D4—2122no—
439C2×D4—2122no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4—2122no—
459C22×C4#1171992414yescomplemented
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466SD16—2128no—
467C2×C8—1062no—
468C2×C8—1062no—
469C2×C8—1062no—
470C2×C8—1062no—
471C2×C8—1062no—
472C2×C8—1062no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—532no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—2122no—
486C4⋊C4C2×C22⋊D5314yes—
487C22⋊C4—28no—
488C22⋊C4—1064no—
489C22⋊C4—2122no—
490C22⋊C4—2122no—
491C22⋊C4—2122no—
492C22⋊C4—2122no—
493C42—28no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—2122no—
499C42C2×C22⋊D5312yes—
500C42C2×C22⋊D5312yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—1062no—
508C23—2121no—
509C23—2121no—
510C23—2121no—
511C23—2121no—
512C23—2121no—
513C23—2121no—
514C23C2×#1171992412yes—
515C23C22×C22⋊D5311yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—48no—
519Q8—42no—
520Q8—42no—
521D4—1062no—
522D4—1062no—
523D4—2128no—
524D4—2121no—
525D4—2121no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—1062no—
549C2×C4—4244no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4#1313318612yes—
559C2×C4#1313318612yes—
560C2×C4C2×#1171992412yes—
561C2×C4C2×#1171992412yes—
562C2×C4#1313324812yes—
563C2×C4C22×C22⋊D5311yescharacteristic
564C2×C4C22×C22⋊D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4#1313326311yescharacteristic
568C2×C4#1313326411yescharacteristic
569C8—2122no—
570C8—2122no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—4241no—
579C22—4241no—
580C22—4241no—
581C22—4241no—
582C22—2122no—
583C22—2122no—
584C22—2122no—
585C22C2×#1313318611yescharacteristic
586C22#1681113011yescharacteristic
587C22#1681120111yescharacteristic
588C22C22×#1171992411yescharacteristic
589C22C2×#1313324811yescharacteristic
590C22#1681142111yescharacteristic
591C22#1681142111yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—44no—
596C4—41no—
597C4—44no—
598C4—4241no—
599C4—4241no—
600C4—2122no—
601C4#1681076214yes—
602C4#1681126111yescharacteristic
603C4#1681127211yescharacteristic
604C4#1681128011yescharacteristic
605C4#1681129211yescharacteristic
606C2—42no—
607C2—4242no—
608C2#2027805411yescharacteristic
609C2C2×#1681076211yescharacteristic
610C2#2028395711yescharacteristic
611C2#2028471911yescharacteristic
612C2#2028487111yescharacteristic
613C2#2028602711yescharacteristic
614C2#2028608211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#533C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#533D53
hypocenterC53#24151
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#16810910C2semidirect
2. C2×#16810825C2semidirect
4. #20275908C2semidirect
6. #20275664C2semidirect
7. #20275639C2semidirect
8. #20275070C2semidirect
9. #20274983C2semidirect
10. #20274695C2semidirect
11. C2×C53×#533C2semidirect
88. C4⋊Q8×C53D4semidirect
89. C4⋊Q8×C53D4semidirect
136. C22×C4×C53SD16semidirect
299. C2×#533D53semidirect
354. C53#24151semidirect
370. C4⋊Q8C22⋊D53semidirect
371. C4⋊Q8C22⋊D53semidirect
459. C22×C4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×#168108252
C4×#131331912
#202759082
#202757162
#202756642
#202756392
#202750702
#202749832
#202746952
C2×C53×#5332
#2415153