Finite Group Databasebeta

← #27447007

Subgroups — #27447007

Order 13568.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810923C211yesmaximal, characteristic, complemented
2C2×#16810663C211yesmaximal, characteristic, complemented
3#20278428C211yesmaximal, characteristic, complemented
4#20276248C211yesmaximal, characteristic, complemented
5#20276217C211yesmaximal, characteristic, complemented
6#20276103C211yesmaximal, characteristic, complemented
7#20276053C211yesmaximal, characteristic, complemented
8#20275478C211yesmaximal, characteristic, complemented
9#20275286C211yesmaximal, characteristic, complemented
10#20275154C211yesmaximal, characteristic
11#20275038C211yesmaximal, characteristic
12#20274976C211yesmaximal, characteristic, complemented
13#20274908C211yesmaximal, characteristic, complemented
14#20274900C211yesmaximal, characteristic, complemented
15C53×#2361C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#16810923—24no—
20#16810663—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34#16810559—21no—
35#16810546—21no—
36#16810521C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810520—21no—
39#16810506—21no—
40#16810493C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×C4×D4×C53—21no—
52C2×C4×D4×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#386—21no—
58C53×#385—21no—
59C4×C4⋊C4×C53C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C2×#11719917—22no—
62C2×#11719917—22no—
63C2×#11719917—22no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133209—22no—
69#13133209—41no—
70#13133176—22no—
71#13133176—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85#13133150—22no—
86#13133150D414yescomplemented
87#13133148—22no—
88#13133148—22no—
89#13133148—22no—
90#13133148—41no—
91C4⋊#88322—22no—
92C4⋊#88322—22no—
93#13133138—22no—
94#13133138—22no—
95#13133138—22no—
96#13133138—41no—
97C22⋊#88322—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C22×D4×C53—21no—
102C22×D4×C53C2311yescharacteristic
103C23×C4×C53—21no—
104C23×C4×C53—21no—
105C23×C4×C53C2311yescharacteristic
106#121×C53—22no—
107#121×C53Q814yes—
108C4×D4×C53—22no—
109C4×D4×C53—41no—
110C4×D4×C53—44no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—41no—
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×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53—21no—
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2311yescharacteristic
127#95×C53—21no—
128#95×C53—21no—
129#95×C53D411yescharacteristic
130#95×C53D411yescharacteristic
131C24×C53—21no—
132C24×C53C2×D411yescharacteristic
133C2×D4×C53—24no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—42no—
140C2×D4×C53—42no—
141C2×D4×C53C2×Q812yes—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—42no—
150C22×C4×C53—42no—
151C22×C4×C53—42no—
152C22×C4×C53—42no—
153C22×C4×C53—81no—
154C22×C4×C53—81no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×Q812yes—
161C22×C4×C53#4112yes—
162C22×C4×C53C2411yescharacteristic
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53C2×D412yes—
174C4⋊C4×C53C2×Q812yes—
175C22⋊C4×C53—24no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—41no—
178C22⋊C4×C53—42no—
179C22⋊C4×C53—42no—
180C22⋊C4×C53—81no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—42no—
186C22⋊Dic53—42no—
187C22⋊Dic53—81no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—42no—
196C4⋊Dic53—81no—
197C4⋊Dic53C2×D412yes—
198#11719919—42no—
199#11719919—42no—
200#11719919—42no—
201#11719919—81no—
202C4×Dic53—42no—
203#11719917—28no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322C2×D412yes—
215C2×#88322#4112yes—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—22no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53C22⋊Q812yes—
230C23×C53C22×D411yescharacteristic
231D4×C53—22no—
232D4×C53—22no—
233D4×C53—48no—
234D4×C53—41no—
235D4×C53—41no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—84no—
263C2×C4×C53—84no—
264C2×C4×C53C22⋊Q812yes—
265C2×C4×C53AGL(1,ℤ/8)12yes—
266C2×C4×C53C22×D411yescharacteristic
267C2×C4×C53C22×Q811yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53C2×#4111yescharacteristic
270C2×C4×C53H(5,𝔽2)11yescharacteristic
271C2×C4×C532−1+411yescharacteristic
272C2×Dic53—22no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284#88322—22no—
285#88322—22no—
286#88322—41no—
287#88322—41no—
288#26366—531nomaximal, Sylow
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—42no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53C2×C22⋊Q811yescharacteristic
304C22×C53D4211yescharacteristic
305C22×C53#54711yescharacteristic
306C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C53#57511yescharacteristic
308C22×C53#57711yescharacteristic
309C22×C53#57711yescharacteristic
310C4×C53—22no—
311C4×C53—22no—
312C4×C53—42no—
313C4×C53—41no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53—42no—
317C4×C53—81no—
318C4×C53#52511yescharacteristic
319C4×C53#53911yescharacteristic
320C4×C53D4×Q811yescharacteristic
321C4×C53#55411yescharacteristic
322Dic53—42no—
323Dic53—81no—
324Dic53—81no—
325Dic53—81no—
326C2×#478—531no—
327C2×#428—531no—
328#2441—531no—
329#2361D5311yescharacteristic
330#2094—531no—
331#2088—531no—
332#2067—531no—
333#2059—531no—
334#1948—531no—
335#1918—531no—
336#1895—531no—
337#1875—531no—
338#1864—531no—
339#1853—531no—
340#1851—531no—
341C2×C53—42no—
342C2×C53—82no—
343C2×C53#244911yescharacteristic
344C2×C53#307711yescharacteristic
345C2×C53#308411yescharacteristic
346C2×C53#327111yescharacteristic
347C2×C53#327211yescharacteristic
348C2×C53#334511yescharacteristic
349C2×C53#335111yescharacteristic
350C22×M16—531no—
351C2×#125—531no—
352C2×#121C2×D5311yescharacteristic
353C2×C4×D4—21no—
354C2×C4×D4C2×D5311yescharacteristic
355C2×#117—531no—
356C22×C4⋊C4C2×D5311yescharacteristic
357#478—1064no—
358#428—1064no—
359C2×#107—531no—
360C2×#107—531no—
361C2×#106—531no—
362C2×#106—531no—
363C2×#106—531no—
364C2×#106—531no—
365C2×C4⋊C8—531no—
366C2×C4⋊C8—531no—
367C2×#102—531no—
368C2×#102—531no—
369C2×#102—531no—
370C2×#102—531no—
371C2×C22⋊C8—531no—
372#399—1061no—
373#396—1061no—
374#390C2×D5311yescharacteristic
375#389—531no—
376#389—531no—
377#389—1061no—
378#389C2×D5311yescharacteristic
379#388—21no—
380#387—1061no—
381#386—21no—
382#385—21no—
383#384—531no—
384#380—531no—
385C4×C4⋊C4C2×D5311yescharacteristic
386C4×C22⋊C4C2×D5311yescharacteristic
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#336—531no—
394#336—531no—
395C53#2636611yescharacteristic, complemented, Sylow
396C22×D4—21no—
397C22×D4C22×D5311yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4C22×D5311yescharacteristic
401C2×M16—1062no—
402C2×M16—1062no—
403C2×M16—1062no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8—531no—
407C22×C8—531no—
408#125—1062no—
409#125—2121no—
410#121—22no—
411#121Dic10614yes—
412C4×D4—22no—
413C4×D4—41no—
414C4×D4—44no—
415#117—1062no—
416#117—2121no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—41no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4C22⋊D5311yescharacteristic
441C2×C22⋊C4C22⋊D5311yescharacteristic
442C2×C42—21no—
443C2×C42—1061no—
444C2×C42C22×D5311yescharacteristic
445C2×C42C22×D5311yescharacteristic
446#107—534no—
447#107—1062no—
448#106—1062no—
449#106—1062no—
450#106—1062no—
451#106—2121no—
452C4⋊C8—1062no—
453C4⋊C8—1062no—
454#102—1062no—
455#102—1062no—
456#102—1062no—
457#102—2121no—
458C22⋊C8—1062no—
459#95—21no—
460#95—21no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95C22⋊D5311yescharacteristic
465#95C22⋊D5311yescharacteristic
466C24—21no—
467C24C2×C22⋊D5311yescharacteristic
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4C2×Dic10612yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4C2×Dic10612yes—
495C22×C4#1171994512yes—
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C2×C22⋊D5311yescharacteristic
499C22×C4C2×C22⋊D5311yescharacteristic
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C23×D5311yescharacteristic
502M16—1068no—
503C2×C8—532no—
504C2×C8—532no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—532no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—4241no—
530C4⋊C4—4241no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4C2×Dic10612yes—
536C4⋊C4D4×D5312yes—
537C22⋊C4—24no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—4241no—
544C22⋊C4—2122no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C42—2122no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23#1313319612yes—
565C23C22×C22⋊D5311yescharacteristic
566D4—22no—
567D4—22no—
568D4—48no—
569D4—41no—
570D4—41no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—1062no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4#1313319612yes—
612C2×C4#1313324312yes—
613C2×C4C2×D4×D5311yescharacteristic
614C2×C4C22×Dic10611yescharacteristic
615C2×C4#41×D5311yescharacteristic
616C2×C4C2×#1171994511yescharacteristic
617C2×C4#1313326211yescharacteristic
618C2×C4#1313326411yescharacteristic
619C8—1062no—
620C8—1062no—
621C8—2121no—
622C8—2121no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C2×#1313319611yescharacteristic
638C22D4×C22⋊D5311yescharacteristic
639C22#1681127211yescharacteristic
640C22C2×#1313324311yescharacteristic
641C22#1681142111yescharacteristic
642C22#1681143111yescharacteristic
643C22#1681143411yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—42no—
651C4—81no—
652C4—4241no—
653C4—4241no—
654C4—4241no—
655C4—2122no—
656C4#1681113211yescharacteristic
657C4#1681121511yescharacteristic
658C4D4×Dic10611yescharacteristic
659C4#1681133211yescharacteristic
660C2—42no—
661C2—82no—
662C2#2027851811yescharacteristic
663C2#2028451211yescharacteristic
664C2#2028457711yescharacteristic
665C2#2028630111yescharacteristic
666C2#2028631111yescharacteristic
667C2#2028718511yescharacteristic
668C2#2028726911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2361C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2361D53
hypocenterC53#26366
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. C2×#16810923C2semidirect
2. C2×#16810663C2semidirect
3. #20278428C2semidirect
4. #20276248C2semidirect
5. #20276217C2semidirect
6. #20276103C2semidirect
7. #20276053C2semidirect
8. #20275478C2semidirect
9. #20275286C2semidirect
12. #20274976C2semidirect
13. #20274908C2semidirect
14. #20274900C2semidirect
86. #13133150D4semidirect
395. C53#26366semidirect

Maximal subgroups

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

subgroupindex
C2×#168109232
C2×#168106632
#202784282
#202762482
#202762172
#202761032
#202760532
#202754782
#202752862
#202751542
#202750382
#202749762
#202749082
#202749002
C53×#23612
#2636653