Finite Group Databasebeta

← #27449594

Subgroups — #27449594

Order 13568.

872 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810934C211yesmaximal, characteristic
2C2×C4×#11719923C211yesmaximal, characteristic, complemented
3#20277718C211yesmaximal, characteristic, complemented
4#20276266C211yesmaximal, characteristic, complemented
5#20276032C211yesmaximal, characteristic, complemented
6#20275918C211yesmaximal, characteristic, complemented
7#20275718C211yesmaximal, characteristic
8#20275481C211yesmaximal, characteristic, complemented
9#20275445C211yesmaximal, characteristic, complemented
10#20275332C211yesmaximal, characteristic, complemented
11#20275323C211yesmaximal, characteristic, complemented
12#20274976C211yesmaximal, characteristic, complemented
13#20274968C211yesmaximal, characteristic, complemented
14#20274907C211yesmaximal, characteristic, complemented
15C53×#2420C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810934—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×C22⋊#88322C2211yescharacteristic
35C2×C22⋊#88322C2211yescharacteristic
36C2×C4×#88322C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810520—21no—
40#16810510—21no—
41#16810495—21no—
42C4×C4⋊Dic53C2211yescharacteristic
43C22⋊C4×Dic53—21no—
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#397—21no—
53C53×#394—21no—
54C53×#390C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#386—21no—
57C53×#382C2211yescharacteristic
58C53×#381—21no—
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C2×#11719923—22no—
64C2×#11719923—22no—
65C2×#11719923—22no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133196—22no—
71#13133196—22no—
72C4×Dic106—22no—
73C4×Dic106—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—41no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86#13133150—22no—
87#13133150—22no—
88#13133150—22no—
89#13133148—22no—
90#13133148—41no—
91C4⋊#88322—22no—
92#13133140—22no—
93#13133140—22no—
94#13133140—41no—
95#13133138—22no—
96#13133138—22no—
97#13133138D414yes—
98C22⋊#88322—22no—
99C22⋊#88322—22no—
100C4×#88322—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×D4×C53—21no—
106C22×D4×C53C2311yescharacteristic
107C23×C4×C53—21no—
108C23×C4×C53C2311yescharacteristic
109#121×C53—22no—
110#121×C53—41no—
111C4×D4×C53—22no—
112C4×D4×C53—41no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
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×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53—21no—
133C24×C53C2×D411yescharacteristic
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×C53—81no—
142C2×D4×C53C2×D412yes—
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×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—42no—
153C22×C4×C53—42no—
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×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—42no—
166C22×Dic53—42no—
167C22×Dic53—81no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C2×Dic106—21no—
172C2×Dic106—21no—
173C2×Dic106#4112yes—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—81no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—42no—
184C22⋊C4×C53—42no—
185C22⋊C4×C53—81no—
186C22⋊C4×C53—81no—
187C22⋊C4×C53—81no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—21no—
192C22⋊Dic53—24no—
193#11719923—28no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—42no—
199C4⋊Dic53C2×D412yes—
200C4⋊Dic53#4112yes—
201#11719919—24no—
202#11719919—42no—
203#11719919—81no—
204C4×Dic53—42no—
205C4×Dic53—81no—
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—
215C2×#88322—21no—
216C2×#88322C2×D412yes—
217C2×#88322#4112yes—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—22no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53#13512yes—
232C23×C53C22×D411yescharacteristic
233D4×C53—22no—
234D4×C53—22no—
235D4×C53—41no—
236D4×C53—41no—
237D4×C53—84no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
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—21no—
251C2×C4×C53—21no—
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×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—84no—
266C2×C4×C53#13712yes—
267C2×C4×C53C22×D411yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53C2×#4111yescharacteristic
270C2×C4×C53C2×#4111yescharacteristic
271C2×C4×C53H(5,𝔽2)11yescharacteristic
272C2×C4×C532−1+411yescharacteristic
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—84no—
286Dic106—22no—
287Dic106—22no—
288Dic106—41no—
289Dic106—41no—
290#88322—22no—
291#88322—22no—
292#88322—41no—
293#88322—41no—
294#26550—531nomaximal, Sylow
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53#52511yescharacteristic
310C22×C53D4211yescharacteristic
311C22×C53#54711yescharacteristic
312C22×C53C2×#13511yescharacteristic
313C22×C53C2×#13711yescharacteristic
314C22×C53#57611yescharacteristic
315C22×C53#57711yescharacteristic
316C4×C53—22no—
317C4×C53—41no—
318C4×C53—41no—
319C4×C53—41no—
320C4×C53—41no—
321C4×C53—81no—
322C4×C53—81no—
323C4×C53—81no—
324C4×C53#52511yescharacteristic
325C4×C53#53911yescharacteristic
326C4×C53#54711yescharacteristic
327C4×C53#55911yescharacteristic
328Dic53—42no—
329Dic53—42no—
330Dic53—42no—
331Dic53—81no—
332C2×#480—531no—
333C2×C4×SD16—531no—
334#2420D5311yescharacteristic
335#2360—531no—
336#2097—531no—
337#2055—531no—
338#2036—531no—
339#2000—531no—
340#1949—531no—
341#1942—531no—
342#1925—531no—
343#1923—531no—
344#1864—531no—
345#1863—531no—
346#1853—531no—
347C2×C53—42no—
348C2×C53—82no—
349C2×C53#245511yescharacteristic
350C2×C53#310811yescharacteristic
351C2×C53#311911yescharacteristic
352C2×C53#326611yescharacteristic
353C2×C53#328011yescharacteristic
354C2×C53#333711yescharacteristic
355C2×C53#335711yescharacteristic
356C22×SD16—531no—
357C2×C22⋊Q8—531no—
358C2×#121C2×D5311yescharacteristic
359C2×C4×Q8—531no—
360C2×C4×D4C2×D5311yescharacteristic
361C22×C4⋊C4—531no—
362#480—1064no—
363C4×SD16—1064no—
364C2×#107—531no—
365C2×#107—531no—
366C2×#107—531no—
367C2×#106—531no—
368C2×#106—531no—
369C2×C4⋊C8—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#102—531no—
374C2×#102—531no—
375C2×#102—531no—
376C2×C22⋊C8—531no—
377C2×C22⋊C8—531no—
378C2×C4×C8—531no—
379#397—21no—
380#394—21no—
381#391—531no—
382#390C2×D5311yescharacteristic
383#389—531no—
384#389—1061no—
385#389C2×D5311yescharacteristic
386#387—1061no—
387#386—21no—
388#385—1061no—
389#382C2×D5311yescharacteristic
390#381—21no—
391#380C2×D5311yescharacteristic
392C4×C4⋊C4—531no—
393C4×C22⋊C4—1061no—
394C4×C22⋊C4C2×D5311yescharacteristic
395#340—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399#336—531no—
400#336—531no—
401C53#2655011yescharacteristic, complemented, Sylow
402C22×Q8—531no—
403C22×D4—21no—
404C22×D4C22×D5311yescharacteristic
405C23×C4—21no—
406C23×C4—1061no—
407C23×C4C22×D5311yescharacteristic
408C2×SD16—1062no—
409C2×SD16—1062no—
410C2×SD16—1062no—
411C22×C8—531no—
412C22×C8—531no—
413C22×C8—531no—
414C22×C8—531no—
415C22⋊Q8—1062no—
416C22⋊Q8—1062no—
417#121—22no—
418#121—41no—
419C4×Q8—1062no—
420C4×Q8—2121no—
421C4×D4—22no—
422C4×D4—41no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—2121no—
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4C22⋊D5311yescharacteristic
443C2×C22⋊C4C22⋊D5311yescharacteristic
444C2×C42—1061no—
445C2×C42—1061no—
446C2×C42C22×D5311yescharacteristic
447C2×C42C22×D5311yescharacteristic
448#107—1062no—
449#107—1062no—
450#107—1062no—
451#106—1062no—
452#106—2121no—
453C4⋊C8—1062no—
454#103—1062no—
455#103—1062no—
456#103—2121no—
457#102—534no—
458#102—1062no—
459#102—1062no—
460C22⋊C8—1062no—
461C22⋊C8—1062no—
462C4×C8—1062no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95C22⋊D5311yescharacteristic
472#95C22⋊D5311yescharacteristic
473C24—21no—
474C24C2×C22⋊D5311yescharacteristic
475C2×Q8—532no—
476C2×Q8—1061no—
477C2×Q8—1061no—
478C2×Q8—1061no—
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—81no—
489C2×D4D4×D5312yes—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—81no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—4241no—
507C22×C4—2122no—
508C22×C4—2122no—
509C22×C4C2×C22⋊D5311yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511C22×C4C2×C22⋊D5311yescharacteristic
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C2×C22⋊D5311yescharacteristic
514C22×C4C23×D5311yescharacteristic
515SD16—1068no—
516C2×C8—532no—
517C2×C8—532no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—81no—
535C4⋊C4—532no—
536C4⋊C4—532no—
537C4⋊C4—1064no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—4241no—
543C4⋊C4—2122no—
544C4⋊C4—2122no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—81no—
551C22⋊C4—81no—
552C22⋊C4—1064no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—4241no—
558C42—2122no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23#1313323712yes—
573C23C22×C22⋊D5311yescharacteristic
574Q8—1062no—
575Q8—1062no—
576Q8—2121no—
577Q8—2121no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582D4—84no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—84no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—4244no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4#1313324712yes—
625C2×C4C2×D4×D5311yescharacteristic
626C2×C4#41×D5311yescharacteristic
627C2×C4#41×D5311yescharacteristic
628C2×C4#41×D5311yescharacteristic
629C2×C4#1313326211yescharacteristic
630C2×C4#1313326411yescharacteristic
631C8—1062no—
632C8—1062no—
633C8—2121no—
634C8—2121no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22#1681113011yescharacteristic
650C22D4×C22⋊D5311yescharacteristic
651C22#1681127211yescharacteristic
652C22C2×#1313323711yescharacteristic
653C22C2×#1313324711yescharacteristic
654C22#1681142611yescharacteristic
655C22#1681143411yescharacteristic
656C4—22no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—4241no—
665C4—2122no—
666C4—2122no—
667C4—2122no—
668C4#1681113111yescharacteristic
669C4#1681121611yescharacteristic
670C4#1681127111yescharacteristic
671C4#1681135711yescharacteristic
672C2—42no—
673C2—82no—
674C2#2027859411yescharacteristic
675C2#2028479111yescharacteristic
676C2#2028490811yescharacteristic
677C2#2028622111yescharacteristic
678C2#2028639611yescharacteristic
679C2#2028708811yescharacteristic
680C2#2028733511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2420C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2420D53
hypocenterC53#26550
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2420

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719923C2semidirect
3. #20277718C2semidirect
4. #20276266C2semidirect
5. #20276032C2semidirect
6. #20275918C2semidirect
8. #20275481C2semidirect
9. #20275445C2semidirect
10. #20275332C2semidirect
11. #20275323C2semidirect
12. #20274976C2semidirect
13. #20274968C2semidirect
14. #20274907C2semidirect
401. C53#26550semidirect

Maximal subgroups

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

subgroupindex
C2×#168109342
C2×C4×#117199232
#202777182
#202762662
#202760322
#202759182
#202757182
#202754812
#202754452
#202753322
#202753232
#202749762
#202749682
#202749072
C53×#24202
#2655053