Finite Group Databasebeta

← #27452179

Subgroups — #27452179

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277718C211yesmaximal, characteristic, complemented
2#20276032C211yesmaximal, characteristic, complemented
3#20275911C211yesmaximal, characteristic, complemented
4#20275853C211yesmaximal, characteristic, complemented
5#20275718C211yesmaximal, characteristic
6#20275668C211yesmaximal, characteristic, complemented
7#20275401C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic, complemented
9#20275323C211yesmaximal, characteristic, complemented
10#20275300C211yesmaximal, characteristic, complemented
11#20275073C211yesmaximal, characteristic, complemented
12#20275033C211yesmaximal, characteristic
13#20274803C211yesmaximal, characteristic, complemented
14C4×#13133140C211yesmaximal, characteristic, complemented
15C53×#2491C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810520—21no—
34#16810510—21no—
35#16810495—21no—
36C4×C4⋊Dic53C2211yescharacteristic
37C22⋊C4×Dic53—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4⋊Q8×C53C2211yescharacteristic
49C2×#124×C53—21no—
50C2×#121×C53C2211yescharacteristic
51C53×#394—21no—
52C53×#393—21no—
53C53×#390C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#388—21no—
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53—21no—
61C2×#11719923—22no—
62C2×#11719923—41no—
63C2×#11719923—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133196—22no—
69#13133196—22no—
70C4×Dic106—22no—
71C4×Dic106—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—41no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84#13133148—22no—
85C4⋊#88322—41no—
86#13133140—22no—
87#13133140—22no—
88#13133140—41no—
89#13133138—22no—
90#13133138—22no—
91#13133138—41no—
92C22⋊#88322—41no—
93C22⋊#88322—41no—
94C4×#88322—41no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99C22×Q8×C53—21no—
100C22×D4×C53—21no—
101C22×D4×C53C2311yescharacteristic
102C23×C4×C53C2311yescharacteristic
103C4⋊Q8×C53—22no—
104C4⋊Q8×C53—41no—
105#124×C53—44no—
106#121×C53—22no—
107#121×C53—41no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123#95×C53—21no—
124#95×C53—21no—
125#95×C53D411yescharacteristic
126#95×C53D411yescharacteristic
127C24×C53—21no—
128C24×C53C2×D411yescharacteristic
129C2×Q8×C53—42no—
130C2×Q8×C53—42no—
131C2×Q8×C53—81no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—42no—
139C2×D4×C53—42no—
140C2×D4×C53—81no—
141C22×C4×C53—21no—
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—21no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—81no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106#4112yes—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53#4112yes—
176C22⋊C4×C53—41no—
177C22⋊C4×C53—42no—
178C22⋊C4×C53—42no—
179C22⋊C4×C53—42no—
180C22⋊C4×C53—42no—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—81no—
183C22⋊C4×C53SD1614yescomplemented
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—42no—
189C22⋊Dic53—24no—
190#11719923—44no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—42no—
196C4⋊Dic53#4112yes—
197C4⋊Dic53#4112yes—
198#11719919—24no—
199#11719919—42no—
200#11719919—81no—
201C4×Dic53—42no—
202C4×Dic53—81no—
203C2×#88322—21no—
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×#88322—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53C2×SD1612yes—
229C23×C53C22×D411yescharacteristic
230Q8×C53—84no—
231D4×C53—41no—
232D4×C53—41no—
233D4×C53—41no—
234D4×C53—41no—
235D4×C53—84no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—22no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
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×C53C2×SD1612yes—
263C2×C4×C53C2×SD1612yes—
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53H(5,𝔽2)11yescharacteristic
269C2×C4×C53H(5,𝔽2)11yescharacteristic
270C2×Dic53—22no—
271C2×Dic53—22no—
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—84no—
283Dic106—22no—
284Dic106—22no—
285Dic106—41no—
286Dic106—41no—
287#88322—41no—
288#88322—41no—
289#88322—41no—
290#88322—41no—
291#26724—531nomaximal, Sylow
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53#52511yescharacteristic
307C22×C53#54611yescharacteristic
308C22×C53#54611yescharacteristic
309C22×C53C22×SD1611yescharacteristic
310C22×C53#57511yescharacteristic
311C22×C53#57611yescharacteristic
312C22×C53#57811yescharacteristic
313C4×C53—22no—
314C4×C53—22no—
315C4×C53—41no—
316C4×C53—42no—
317C4×C53—41no—
318C4×C53—41no—
319C4×C53—81no—
320C4×C53—81no—
321C4×C53#52911yescharacteristic
322C4×C53#53811yescharacteristic
323C4×C53#55511yescharacteristic
324C4×C53#55911yescharacteristic
325Dic53—42no—
326Dic53—42no—
327Dic53—42no—
328Dic53—81no—
329#2491D5311yescharacteristic
330#2360—531no—
331#2055—531no—
332#2035—531no—
333#2024—531no—
334#2000—531no—
335#1988—531no—
336#1934—531no—
337#1926—531no—
338#1923—531no—
339#1920—531no—
340#1880—531no—
341#1874—531no—
342#1824—531no—
343C4×#103—531no—
344C2×C53—42no—
345C2×C53—161no—
346C2×C53#245911yescharacteristic
347C2×C53#310311yescharacteristic
348C2×C53#312411yescharacteristic
349C2×C53#329511yescharacteristic
350C2×C53#329811yescharacteristic
351C2×C53#331911yescharacteristic
352C2×C53#332711yescharacteristic
353C22×SD16—531no—
354C2×C4⋊Q8C2×D5311yescharacteristic
355C2×#124—21no—
356C2×C22⋊Q8—531no—
357C2×#121C2×D5311yescharacteristic
358C2×C4×Q8—531no—
359C22×C4⋊C4—531no—
360C2×#106—531no—
361C2×C4⋊C8—531no—
362C2×#103—531no—
363C2×#103—531no—
364C2×#103—531no—
365C2×#102—531no—
366C2×#102—531no—
367C2×#102—531no—
368C2×C22⋊C8—531no—
369C2×C22⋊C8—531no—
370C2×C4×C8—531no—
371#394—21no—
372#393—21no—
373#391—531no—
374#390C2×D5311yescharacteristic
375#390C2×D5311yescharacteristic
376#389—531no—
377#389—1061no—
378#388—21no—
379#387—1061no—
380#385—1061no—
381#380C2×D5311yescharacteristic
382C4×C4⋊C4—531no—
383C4×C4⋊C4C2×D5311yescharacteristic
384C4×C22⋊C4—1061no—
385C4×C22⋊C4C2×D5311yescharacteristic
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#336—531no—
395#336—531no—
396C53#2672411yescharacteristic, complemented, Sylow
397C22×Q8—21no—
398C22×Q8—531no—
399C22×D4—21no—
400C22×D4C22×D5311yescharacteristic
401C23×C4—1061no—
402C23×C4C22×D5311yescharacteristic
403C2×SD16—1062no—
404C2×SD16—2121no—
405C2×SD16—2121no—
406C22×C8—531no—
407C22×C8—531no—
408C22×C8—531no—
409C22×C8—531no—
410C4⋊Q8—22no—
411C4⋊Q8—41no—
412#124—44no—
413C22⋊Q8—1062no—
414C22⋊Q8—1062no—
415#121—22no—
416#121—41no—
417C4×Q8—1062no—
418C4×Q8—2121no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—1062no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—2121no—
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4C22⋊D5311yescharacteristic
440C2×C22⋊C4C22⋊D5311yescharacteristic
441C2×C42—21no—
442C2×C42—1061no—
443C2×C42—1061no—
444C2×C42C22×D5311yescharacteristic
445C2×C42C22×D5311yescharacteristic
446#106—1062no—
447C4⋊C8—2121no—
448#103—1062no—
449#103—1062no—
450#103—2121no—
451#102—1062no—
452#102—1062no—
453#102—2121no—
454C22⋊C8—2121no—
455C22⋊C8—2121no—
456C4×C8—2121no—
457#95—21no—
458#95—21no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95C22⋊D5311yescharacteristic
464#95C22⋊D5311yescharacteristic
465C24—21no—
466C24C2×C22⋊D5311yescharacteristic
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—81no—
470C2×Q8—532no—
471C2×Q8—1061no—
472C2×Q8—1061no—
473C2×Q8—1061no—
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—81no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—4241no—
499C22×C4—2122no—
500C22×C4—2122no—
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C2×C22⋊D5311yescharacteristic
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C23×D5311yescharacteristic
507SD16—2124no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—532no—
527C4⋊C4—532no—
528C4⋊C4—1064no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—4241no—
534C4⋊C4—2122no—
535C4⋊C4—2122no—
536C4⋊C4#1171994712yes—
537C22⋊C4—41no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C22⋊C4—1064no—
545C22⋊C4#1171992314yescomplemented
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—42no—
551C42—4241no—
552C42—2122no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23C2×#1171992312yes—
567C23C22×C22⋊D5311yescharacteristic
568Q8—84no—
569Q8—1062no—
570Q8—1062no—
571Q8—2121no—
572Q8—2121no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—84no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—1062no—
605C2×C4—1062no—
606C2×C4—1062no—
607C2×C4—4244no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4C2×#1171992312yes—
618C2×C4C2×#1171992312yes—
619C2×C4#41×D5311yescharacteristic
620C2×C4#41×D5311yescharacteristic
621C2×C4#41×D5311yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4#1313326211yescharacteristic
624C2×C4#1313326311yescharacteristic
625C8—2121no—
626C8—2121no—
627C8—2121no—
628C8—2121no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22#1681113011yescharacteristic
644C22#1681126111yescharacteristic
645C22#1681126711yescharacteristic
646C22C22×#1171992311yescharacteristic
647C22#1681142111yescharacteristic
648C22#1681142611yescharacteristic
649C22#1681143811yescharacteristic
650C4—22no—
651C4—22no—
652C4—41no—
653C4—42no—
654C4—41no—
655C4—41no—
656C4—81no—
657C4—81no—
658C4—4241no—
659C4—2122no—
660C4—2122no—
661C4—2122no—
662C4#1681115211yescharacteristic
663C4#1681120711yescharacteristic
664C4#1681133511yescharacteristic
665C4#1681135511yescharacteristic
666C2—42no—
667C2—161no—
668C2#2027864811yescharacteristic
669C2#2028474211yescharacteristic
670C2#2028497111yescharacteristic
671C2#2028653911yescharacteristic
672C2#2028657411yescharacteristic
673C2#2028686511yescharacteristic
674C2#2028698311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2491C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2491D53
hypocenterC53#26724
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2491

Complements

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

NH ≅ G/Ntype
1. #20277718C2semidirect
2. #20276032C2semidirect
3. #20275911C2semidirect
4. #20275853C2semidirect
6. #20275668C2semidirect
7. #20275401C2semidirect
8. #20275339C2semidirect
9. #20275323C2semidirect
10. #20275300C2semidirect
11. #20275073C2semidirect
13. #20274803C2semidirect
14. C4×#13133140C2semidirect
183. C22⋊C4×C53SD16semidirect
396. C53#26724semidirect
545. C22⋊C4#11719923semidirect

Maximal subgroups

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

subgroupindex
#202777182
#202760322
#202759112
#202758532
#202757182
#202756682
#202754012
#202753392
#202753232
#202753002
#202750732
#202750332
#202748032
C4×#131331402
C53×#24912
#2672453