Finite Group Databasebeta

← #27447017

Subgroups — #27447017

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×#16810925C211yesmaximal, characteristic, complemented
2C2×#16810666C211yesmaximal, characteristic, complemented
3#20278422C211yesmaximal, characteristic, complemented
4#20277735C211yesmaximal, characteristic
5#20276246C211yesmaximal, characteristic, complemented
6#20276218C211yesmaximal, characteristic, complemented
7#20276099C211yesmaximal, characteristic, complemented
8#20276052C211yesmaximal, characteristic, complemented
9#20275477C211yesmaximal, characteristic, complemented
10#20275289C211yesmaximal, characteristic, complemented
11#20275154C211yesmaximal, characteristic
12#20274974C211yesmaximal, characteristic, complemented
13#20274910C211yesmaximal, characteristic, complemented
14#20274897C211yesmaximal, characteristic, complemented
15C53×#1875C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×D4×Dic53—21no—
19C2×D4×Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810925—24no—
22#16810666—24no—
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133146C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33#16810559—21no—
34#16810546—21no—
35#16810526C2211yescharacteristic
36#16810522C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810521—21no—
39#16810519C2211yescharacteristic
40#16810516—21no—
41#16810507—21no—
42#16810503—21no—
43#16810498—21no—
44#16810492C2211yescharacteristic
45#16810483C2211yescharacteristic
46C4⋊C4×Dic53C2211yescharacteristic
47C4×C22⋊Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#121×C53C2211yescharacteristic
55C2×#117×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63C2×#11719917—22no—
64C2×#11719917—22no—
65C2×#11719917—22no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133208—22no—
69#13133208—41no—
70D4×Dic53—22no—
71D4×Dic53—41no—
72D4×Dic53—44no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—41no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53D411yescharacteristic
92C2×C22⋊Dic53D411yescharacteristic
93C2×C4×Dic53—21no—
94C2×C4×Dic53C2311yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96#13133150—22no—
97#13133150D414yescomplemented
98#13133149—22no—
99#13133149—41no—
100#13133148—22no—
101#13133148—22no—
102#13133146—22no—
103#13133146—22no—
104#13133138—22no—
105#13133138—22no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124D411yescharacteristic
112#13133124D411yescharacteristic
113C22×D4×C53—21no—
114C22×D4×C53C2311yescharacteristic
115C23×C4×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118#121×C53—22no—
119#121×C53Q814yes—
120#117×C53—22no—
121#117×C53—41no—
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C22⋊C4×C53—21no—
126C2×C22⋊C4×C53—21no—
127C2×C42×C53—21no—
128#102×C53—22no—
129#102×C53—41no—
130C22⋊C8×C53—22no—
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×C53C2×D411yescharacteristic
150C22×C4×C53C2×Q812yes—
151C22×C4×C53#4112yes—
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53—42no—
160C22×Dic53—42no—
161C22×Dic53—81no—
162C22×Dic53—81no—
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C22×Dic53C2×D411yescharacteristic
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53C2×D412yes—
178C4⋊C4×C53C2×Q812yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—42no—
181C42×C53—42no—
182C22⋊Dic53—22no—
183C22⋊Dic53—41no—
184C22⋊Dic53—42no—
185C22⋊Dic53—42no—
186C22⋊Dic53—81no—
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—42no—
197C4⋊Dic53—42no—
198C4⋊Dic53—42no—
199C4⋊Dic53C2×D412yes—
200#11719919—42no—
201#11719919—42no—
202#11719919—81no—
203#11719919—81no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—21no—
208C4×Dic53—42no—
209#11719917—28no—
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—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53C22⋊Q812yes—
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—21no—
259C2×Dic53—21no—
260C2×Dic53—22no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
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—84no—
282C2×Dic53—84no—
283C2×Dic53AGL(1,ℤ/8)12yes—
284C8×C53—41no—
285C8×C53—41no—
286#88322—22no—
287#88322—22no—
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—42no—
312C4×C53—42no—
313C4×C53—81no—
314C4×C53#52511yescharacteristic
315C4×C53#53911yescharacteristic
316C4×C53D4×Q811yescharacteristic
317C4×C53#55411yescharacteristic
318Dic53—22no—
319Dic53—41no—
320Dic53—41no—
321Dic53—41no—
322Dic53—42no—
323Dic53—81no—
324Dic53—81no—
325Dic53—81no—
326C2×#478—531no—
327C2×#428—531no—
328#2441—531no—
329#2361—531no—
330#2094—531no—
331#2088—531no—
332#2067—531no—
333#2059—531no—
334#1948—531no—
335#1918—531no—
336#1895—531no—
337#1875D5311yescharacteristic
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—531no—
354C2×C4×D4—1061no—
355C2×#117C2×D5311yescharacteristic
356C22×C4⋊C4—531no—
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×#102C2×D5311yescharacteristic
370C2×#102C2×D5311yescharacteristic
371C2×C22⋊C8C2×D5311yescharacteristic
372#399—1061no—
373#396—1061no—
374#390—531no—
375#389—531no—
376#389—531no—
377#389—531no—
378#389—1061no—
379#388—1061no—
380#387—1061no—
381#386—1061no—
382#385—1061no—
383#384—531no—
384#380—531no—
385C4×C4⋊C4—531no—
386C4×C22⋊C4—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340C2×D5311yescharacteristic
392#340C2×D5311yescharacteristic
393#336—531no—
394#336—531no—
395C53#2636611yescharacteristic, complemented, Sylow
396C22×D4—21no—
397C22×D4C22×D5311yescharacteristic
398C23×C4—1061no—
399C23×C4—1061no—
400C23×C4C22×D5311yescharacteristic
401C2×M16—1062no—
402C2×M16—1062no—
403C2×M16—1062no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8C22×D5311yescharacteristic
407C22×C8C22×D5311yescharacteristic
408#125—1062no—
409#125—2121no—
410#121—22no—
411#121Dic10614yes—
412C4×D4—1062no—
413C4×D4—2121no—
414C4×D4—2124no—
415#117—22no—
416#117—41no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—1062no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—2121no—
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—531no—
437C2×C22⋊C4—531no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C42—21no—
443C2×C42—531no—
444C2×C42—531no—
445C2×C42—1061no—
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—22no—
455#102—41no—
456#102—1062no—
457#102—1062no—
458C22⋊C8—22no—
459#95—531no—
460#95—531no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466C24—21no—
467C24D4×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—531no—
485C22×C4—531no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—4241no—
493C22×C4—4241no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4C2×Dic10612yes—
499C22×C4D4×D5311yescharacteristic
500C22×C4#1171994712yes—
501C22×C4C23×D5311yescharacteristic
502M16—1068no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—532no—
510C2×C8—532no—
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—532no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—4241no—
528C4⋊C4—4241no—
529C4⋊C4—2122no—
530C4⋊C4—2122no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4C2×Dic10612yes—
536C4⋊C4D4×D5312yes—
537C22⋊C4—24no—
538C22⋊C4—42no—
539C22⋊C4—1062no—
540C22⋊C4—4241no—
541C22⋊C4—4241no—
542C22⋊C4—2122no—
543C22⋊C4—2121no—
544C22⋊C4—2122no—
545C42—42no—
546C42—1061no—
547C42—1061no—
548C42—1061no—
549C42—1061no—
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#1313319412yes—
565C23C2×D4×D5311yescharacteristic
566D4—22no—
567D4—22no—
568D4—48no—
569D4—41no—
570D4—41no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—532no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1062no—
599C2×C4—4244no—
600C2×C4—4244no—
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—2121no—
612C2×C4#1313319412yes—
613C2×C4C2×D4×D5311yescharacteristic
614C2×C4C22×Dic10611yescharacteristic
615C2×C4#41×D5311yescharacteristic
616C2×C4C2×#1171994711yescharacteristic
617C2×C4#1313326211yescharacteristic
618C2×C4#1313326411yescharacteristic
619C8—41no—
620C8—41no—
621C8—1062no—
622C8—1062no—
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×#1313319411yescharacteristic
638C22#1681125511yescharacteristic
639C22#1681127111yescharacteristic
640C22AGL(1,ℤ/8)×D5311yescharacteristic
641C22#1681141911yescharacteristic
642C22#1681143011yescharacteristic
643C22#1681143011yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—1062no—
649C4—4241no—
650C4—4241no—
651C4—4241no—
652C4—2121no—
653C4—2121no—
654C4—2121no—
655C4—2122no—
656C4#1681113311yescharacteristic
657C4#1681121811yescharacteristic
658C4D4×Dic10611yescharacteristic
659C4#1681133211yescharacteristic
660C2—42no—
661C2—82no—
662C2#2027852811yescharacteristic
663C2#2028450911yescharacteristic
664C2#2028457511yescharacteristic
665C2#2028629811yescharacteristic
666C2#2028630811yescharacteristic
667C2#2028718811yescharacteristic
668C2#2028726811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1875C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#1875D53
hypocenterC53#26366
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
1. C2×#16810925C2semidirect
2. C2×#16810666C2semidirect
3. #20278422C2semidirect
5. #20276246C2semidirect
6. #20276218C2semidirect
7. #20276099C2semidirect
8. #20276052C2semidirect
9. #20275477C2semidirect
10. #20275289C2semidirect
12. #20274974C2semidirect
13. #20274910C2semidirect
14. #20274897C2semidirect
97. #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×#168109252
C2×#168106662
#202784222
#202777352
#202762462
#202762182
#202760992
#202760522
#202754772
#202752892
#202751542
#202749742
#202749102
#202748972
C53×#18752
#2636653