Finite Group Databasebeta

← #27434390

Subgroups — #27434390

Order 13568.

832 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810931C211yesmaximal, characteristic, complemented
2#20279881C211yesmaximal, characteristic
3#20279637C211yesmaximal, characteristic
4#20276148C211yesmaximal, characteristic, complemented
5#20276138C211yesmaximal, characteristic, complemented
6#20275919C211yesmaximal, characteristic, complemented
7#20275919C211yesmaximal, characteristic, complemented
8#20275684C211yesmaximal, characteristic, complemented
9#20275474C211yesmaximal, characteristic, complemented
10#20275148C211yesmaximal, characteristic, complemented
11#20275138C211yesmaximal, characteristic, complemented
12#20274748C211yesmaximal, characteristic, complemented
13#20274744C211yesmaximal, characteristic, complemented
14#20274546C211yesmaximal, characteristic, complemented
15C2×C53×#435C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810931—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29#16810561—21no—
30#16810546—21no—
31#16810546—21no—
32#16810537—21no—
33#16810526—21no—
34#16810519—21no—
35#16810517—21no—
36#16810517—21no—
37#16810493C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810483C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×C4×D4×C53C2211yescharacteristic
56C53×#435—24no—
57C2×C4⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C2×#97×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62#13133209—22no—
63#13133209—22no—
64#13133196—22no—
65#13133196—22no—
66#13133196—41no—
67#13133196—41no—
68C4×Dic106—22no—
69C4×Dic106—22no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94#13133151—22no—
95#13133151—41no—
96#13133151—41no—
97#13133149—22no—
98#13133142—22no—
99#13133142—22no—
100#13133142—22no—
101#13133142—22no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×D4×C53D411yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C23×C4×C53C2311yescharacteristic
112C2×M16×C53—22no—
113C2×M16×C53—22no—
114C2×M16×C53—41no—
115C22×C8×C53—22no—
116C22×C8×C53—22no—
117C22×C8×C53—41no—
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C4×D4×C53—22no—
123C4×D4×C53—22no—
124C2×C4⋊C4×C53D411yescharacteristic
125C2×C22⋊C4×C53D411yescharacteristic
126C2×C22⋊C4×C53D411yescharacteristic
127C2×C42×C53C2311yescharacteristic
128C4⋊C8×C53D414yes—
129C22⋊C8×C53—22no—
130C22⋊C8×C53—22no—
131#97×C53—22no—
132C24×C53C2×D411yescharacteristic
133C24×C53C2×D411yescharacteristic
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—41no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
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×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106#4112yes—
168M16×C53—44no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—44no—
179C2×C8×C53C2×D412yes—
180C2×C8×C53C2×D412yes—
181C2×C8×C53#4112yes—
182C4⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C42×C53C2×D412yes—
186C42×C53#4112yes—
187C22⋊Dic53—42no—
188C22⋊Dic53—42no—
189C22⋊Dic53—42no—
190C22⋊Dic53—42no—
191C22⋊Dic53—81no—
192C22⋊Dic53—81no—
193C22⋊Dic53—81no—
194C22⋊Dic53—81no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53—42no—
210C4⋊Dic53—81no—
211#11719919—42no—
212#11719919—42no—
213#11719919—42no—
214#11719919—42no—
215#11719919—42no—
216C4×Dic53—42no—
217C4×Dic53—81no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53#13712yes—
232C23×C53C22×D411yescharacteristic
233D4×C53—44no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
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—21no—
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×C53—41no—
252C2×C4×C53#12112yes—
253C2×C4×C53#12112yes—
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53H(5,𝔽2)11yescharacteristic
259C2×C4×C532−1+411yescharacteristic
260C2×Dic53—22no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
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—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284Dic106—22no—
285Dic106—22no—
286Dic106—41no—
287Dic106—41no—
288C8×C53—22no—
289C8×C53—22no—
290C8×C53—22no—
291C8×C53—41no—
292#25353—531nomaximal, Sylow
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—41no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53C2×#12111yescharacteristic
308C22×C53#54611yescharacteristic
309C22×C53#54711yescharacteristic
310C22×C53C2×#13711yescharacteristic
311C22×C53#57511yescharacteristic
312C22×C53#57611yescharacteristic
313C22×C53#57811yescharacteristic
314C4×C53—22no—
315C4×C53—22no—
316C4×C53—41no—
317C4×C53—41no—
318C4×C53#52911yescharacteristic
319C4×C53#54211yescharacteristic
320C4×C53#54611yescharacteristic
321C4×C53#54811yescharacteristic
322Dic53—42no—
323Dic53—81no—
324Dic53—81no—
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328Dic53—81no—
329Dic53—81no—
330C2×#479—531no—
331C2×#435D5311yescharacteristic
332#2562—531no—
333#2538—531no—
334#2077—531no—
335#2075—531no—
336#2036—531no—
337#2036—531no—
338#1992—531no—
339#1947—531no—
340#1894—531no—
341#1892—531no—
342#1809—531no—
343#1808—531no—
344#1763—531no—
345C2×C53—42no—
346C2×C53—81no—
347C2×C53#242711yescharacteristic
348C2×C53#305611yescharacteristic
349C2×C53#306111yescharacteristic
350C2×C53#330911yescharacteristic
351C2×C53#331211yescharacteristic
352C2×C53#334611yescharacteristic
353C2×C53#334911yescharacteristic
354C22×M16C2×D5311yescharacteristic
355C23×C8C2×D5311yescharacteristic
356C2×#125—531no—
357C2×C22⋊Q8—531no—
358C2×C22⋊Q8—531no—
359C2×C4×Q8—531no—
360C2×C4×D4C2×D5311yescharacteristic
361#479—1064no—
362#435—24no—
363C2×#107—531no—
364C2×#107—531no—
365C2×#107—531no—
366C2×#106—531no—
367C2×C4⋊C8C2×D5311yescharacteristic
368C2×#103—531no—
369C2×#103—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×C22⋊C8C2×D5311yescharacteristic
373C2×C22⋊C8C2×D5311yescharacteristic
374C2×#97C2×D5311yescharacteristic
375#400—1061no—
376#396—1061no—
377#396—1061no—
378#394—1061no—
379#390—1061no—
380#389—1061no—
381#388—1061no—
382#388—1061no—
383#384—531no—
384#380—531no—
385#380—531no—
386#380—531no—
387#380—531no—
388#380—531no—
389#380—531no—
390C4×C4⋊C4—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399C53#2535311yescharacteristic, complemented, Sylow
400C22×Q8—531no—
401C22×D4D21211yescharacteristic
402C23×C4C22×D5311yescharacteristic
403C23×C4C22×D5311yescharacteristic
404C2×M16—22no—
405C2×M16—22no—
406C2×M16—41no—
407C22×C8—22no—
408C22×C8—22no—
409C22×C8—41no—
410C22×C8C22×D5311yescharacteristic
411C22×C8C22×D5311yescharacteristic
412C22×C8C22×D5311yescharacteristic
413C22×C8C22×D5311yescharacteristic
414#125—1062no—
415#125—1062no—
416C22⋊Q8—1062no—
417C22⋊Q8—1062no—
418C22⋊Q8—2121no—
419C22⋊Q8—2121no—
420C4×Q8—1062no—
421C4×Q8—1062no—
422C4×D4—22no—
423C4×D4—22no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4D21211yescharacteristic
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C22⋊C4—1061no—
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4D21211yescharacteristic
448C2×C22⋊C4D21211yescharacteristic
449C2×C42—1061no—
450C2×C42—1061no—
451C2×C42C22×D5311yescharacteristic
452#107—1062no—
453#107—2121no—
454#107—2121no—
455#106—1062no—
456C4⋊C8C22⋊D5314yes—
457#103—1062no—
458#103—1062no—
459#103—1062no—
460#103—1062no—
461C22⋊C8—22no—
462C22⋊C8—22no—
463#97—22no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471C24C2×D21211yescharacteristic
472C24C2×D21211yescharacteristic
473C2×Q8—532no—
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×Q8—1061no—
478C2×Q8—1061no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—41no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
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—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4C2×D21211yescharacteristic
503C22×C4C2×D21211yescharacteristic
504C22×C4C2×D21211yescharacteristic
505C22×C4C2×D21211yescharacteristic
506C22×C4C23×D5311yescharacteristic
507M16—44no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—44no—
518C2×C8#1171994712yes—
519C2×C8C2×C22⋊D5312yes—
520C2×C8C2×C22⋊D5312yes—
521C4⋊C4—22no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—4241no—
537C4⋊C4—2122no—
538C4⋊C4—2122no—
539C4⋊C4—2122no—
540C4⋊C4—2122no—
541C4⋊C4—2122no—
542C4⋊C4—2122no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—4241no—
546C22⋊C4—4241no—
547C22⋊C4—4241no—
548C22⋊C4—4241no—
549C22⋊C4—2122no—
550C22⋊C4—2122no—
551C22⋊C4—2122no—
552C22⋊C4—2122no—
553C42—4241no—
554C42—2122no—
555C42#1171994512yes—
556C42C2×C22⋊D5312yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#1313325012yes—
571C23C22×D21211yescharacteristic
572Q8—1062no—
573Q8—1062no—
574Q8—2121no—
575Q8—2121no—
576D4—44no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—1062no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
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—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4#1313319112yes—
620C2×C4#1313319112yes—
621C2×C4C22×C22⋊D5311yescharacteristic
622C2×C4#41×D5311yescharacteristic
623C2×C4C2×#1171994711yescharacteristic
624C2×C4C2×#1171994511yescharacteristic
625C2×C4#1313326211yescharacteristic
626C2×C4#1313326411yescharacteristic
627C8—22no—
628C8—22no—
629C8—22no—
630C8—41no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22C2×#1313319111yescharacteristic
646C22#1681126811yescharacteristic
647C22#1681127711yescharacteristic
648C22C2×#1313325011yescharacteristic
649C22#1681142311yescharacteristic
650C22#1681142811yescharacteristic
651C22#1681144011yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—4241no—
657C4—4241no—
658C4—4241no—
659C4—4241no—
660C4—4241no—
661C4—4241no—
662C4—4241no—
663C4—2122no—
664C4#1681115511yescharacteristic
665C4#1681123811yescharacteristic
666C4#1681126711yescharacteristic
667C4#1681128011yescharacteristic
668C2—42no—
669C2—81no—
670C2#2027828611yescharacteristic
671C2#2028434511yescharacteristic
672C2#2028438511yescharacteristic
673C2#2028672311yescharacteristic
674C2#2028676811yescharacteristic
675C2#2028720211yescharacteristic
676C2#2028724711yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#435C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#435D53
hypocenterC53#25353
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
1. C2×#16810931C2semidirect
4. #20276148C2semidirect
5. #20276138C2semidirect
6. #20275919C2semidirect
7. #20275919C2semidirect
8. #20275684C2semidirect
9. #20275474C2semidirect
10. #20275148C2semidirect
11. #20275138C2semidirect
12. #20274748C2semidirect
13. #20274744C2semidirect
14. #20274546C2semidirect
399. C53#25353semidirect

Maximal subgroups

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

subgroupindex
C2×#168109312
#202798812
#202796372
#202761482
#202761382
#202759192
#202756842
#202754742
#202751482
#202751382
#202747482
#202747442
#202745462
C2×C53×#4352
#2535353