Finite Group Databasebeta

← #27433154

Subgroups — #27433154

Order 13568.

844 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810965C211yesmaximal, characteristic, complemented
2#20279012C211yesmaximal, characteristic
3#20278166C211yesmaximal, characteristic
4#20276213C211yesmaximal, characteristic, complemented
5#20276138C211yesmaximal, characteristic, complemented
6#20276109C211yesmaximal, characteristic, complemented
7#20275919C211yesmaximal, characteristic, complemented
8#20275448C211yesmaximal, characteristic, complemented
9#20275391C211yesmaximal, characteristic, complemented
10#20275148C211yesmaximal, characteristic, complemented
11#20275148C211yesmaximal, characteristic, complemented
12#20274905C211yesmaximal, characteristic, complemented
13#20274748C211yesmaximal, characteristic, complemented
14#20274435C211yesmaximal, characteristic, complemented
15C2×C53×#436C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810965—24no—
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29#16810556—21no—
30#16810551—21no—
31#16810546—21no—
32#16810537—21no—
33#16810529C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810511—21no—
38#16810504—21no—
39#16810503—21no—
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810481—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C22×M16×C53C2211yescharacteristic
55C2×C4×D4×C53C2211yescharacteristic
56C53×#436—24no—
57C2×C4⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63#13133196—22no—
64#13133196—22no—
65#13133196—22no—
66#13133196—41no—
67#13133176—22no—
68#13133176—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—22no—
71C2×C4⋊Dic53—41no—
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×C4×Dic53—21no—
92#13133151—22no—
93#13133151—22no—
94#13133151—41no—
95#13133149—22no—
96#13133142—22no—
97#13133142—22no—
98#13133142—22no—
99#13133142—41no—
100#13133124—21no—
101#13133124—21no—
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—22no—
115C2×M16×C53—22no—
116C2×M16×C53—41no—
117C2×M16×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×C53—22no—
129C22⋊C8×C53—22no—
130C22⋊C8×C53D414yes—
131C4×C8×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×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53#4112yes—
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—
162C22×Dic53—42no—
163C22×Dic53—42no—
164C22×Dic53—81no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171M16×C53—44no—
172M16×C53—44no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53C2×D412yes—
183C2×C8×C53C2×D412yes—
184C2×C8×C53#4112yes—
185C4⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—22no—
188C42×C53—21no—
189C42×C53—21no—
190C22⋊Dic53—42no—
191C22⋊Dic53—42no—
192C22⋊Dic53—42no—
193C22⋊Dic53—42no—
194C22⋊Dic53—81no—
195C22⋊Dic53—81no—
196C22⋊Dic53—81no—
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—21no—
210C4⋊Dic53—42no—
211C4⋊Dic53#4112yes—
212#11719919—42no—
213#11719919—42no—
214#11719919—42no—
215#11719919—81no—
216#11719919—81no—
217C4×Dic53—42no—
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#12112yes—
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—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
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—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53#12112yes—
253C2×C4×C53#13712yes—
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—
284C2×Dic53—84no—
285Dic106—41no—
286Dic106—41no—
287Dic106—41no—
288Dic106—41no—
289C8×C53—22no—
290C8×C53—22no—
291C8×C53—22no—
292C8×C53—41no—
293#25261—531nomaximal, Sylow
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—22no—
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×C53—41no—
308C22×C53C2×#12111yescharacteristic
309C22×C53#54611yescharacteristic
310C22×C53#54711yescharacteristic
311C22×C53C2×#13711yescharacteristic
312C22×C53#57511yescharacteristic
313C22×C53#57611yescharacteristic
314C22×C53#57811yescharacteristic
315C4×C53—22no—
316C4×C53—22no—
317C4×C53—41no—
318C4×C53—41no—
319C4×C53#52911yescharacteristic
320C4×C53#53911yescharacteristic
321C4×C53#54611yescharacteristic
322C4×C53#54711yescharacteristic
323Dic53—42no—
324Dic53—81no—
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328Dic53—81no—
329Dic53—81no—
330Dic53—81no—
331C2×#485—531no—
332C2×#436D5311yescharacteristic
333#2485—531no—
334#2417—531no—
335#2088—531no—
336#2075—531no—
337#2068—531no—
338#2036—531no—
339#1942—531no—
340#1933—531no—
341#1894—531no—
342#1894—531no—
343#1852—531no—
344#1809—531no—
345#1736—531no—
346C2×C53—42no—
347C2×C53—81no—
348C2×C53#242511yescharacteristic
349C2×C53#305611yescharacteristic
350C2×C53#306111yescharacteristic
351C2×C53#332211yescharacteristic
352C2×C53#332811yescharacteristic
353C2×C53#335211yescharacteristic
354C2×C53#335711yescharacteristic
355C22×M16C2×D5311yescharacteristic
356C22×M16C2×D5311yescharacteristic
357C2×C22⋊Q8—531no—
358C2×C22⋊Q8—531no—
359C2×C4×D4C2×D5311yescharacteristic
360C2×#117—531no—
361C22×C4⋊C4—531no—
362#485—1064no—
363#436—24no—
364C2×#107—531no—
365C2×#107—531no—
366C2×#107—531no—
367C2×#106—531no—
368C2×C4⋊C8C2×D5311yescharacteristic
369C2×#103—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×C22⋊C8C2×D5311yescharacteristic
374C2×C22⋊C8C2×D5311yescharacteristic
375C2×C4×C8C2×D5311yescharacteristic
376#398—1061no—
377#397—1061no—
378#396—1061no—
379#394—1061no—
380#391—531no—
381#389—531no—
382#389—531no—
383#389—531no—
384#388—1061no—
385#387—1061no—
386#386—1061no—
387#380—531no—
388#380—531no—
389#380—531no—
390#380—531no—
391#380—1061no—
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399#340—531no—
400C53#2526111yescharacteristic, complemented, Sylow
401C22×Q8—531no—
402C22×D4D21211yescharacteristic
403C23×C4—1061no—
404C23×C4C22×D5311yescharacteristic
405C23×C4C22×D5311yescharacteristic
406C2×M16—22no—
407C2×M16—22no—
408C2×M16—22no—
409C2×M16—22no—
410C2×M16—41no—
411C2×M16—41no—
412C22×C8C22×D5311yescharacteristic
413C22×C8C22×D5311yescharacteristic
414C22×C8C22×D5311yescharacteristic
415C22×C8C22×D5311yescharacteristic
416C22⋊Q8—1062no—
417C22⋊Q8—1062no—
418C22⋊Q8—1062no—
419C22⋊Q8—2121no—
420C4×D4—22no—
421C4×D4—22no—
422#117—1062no—
423#117—1062no—
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—1062no—
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⋊C4—2121no—
439C2×C4⋊C4D21211yescharacteristic
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×C42C22×D5311yescharacteristic
451#107—1062no—
452#107—1062no—
453#107—2121no—
454#106—1062no—
455C4⋊C8—22no—
456#103—1062no—
457#103—1062no—
458#103—1062no—
459#103—2121no—
460C22⋊C8—22no—
461C22⋊C8C22⋊D5314yes—
462C4×C8—22no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472C24C2×D21211yescharacteristic
473C24C2×D21211yescharacteristic
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×Q8—1061no—
478C2×Q8—1061no—
479C2×Q8—1061no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—41no—
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—1061no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—4241no—
502C22×C4—2122no—
503C22×C4—2122no—
504C22×C4C2×D21211yescharacteristic
505C22×C4C2×D21211yescharacteristic
506C22×C4C2×D21211yescharacteristic
507C22×C4C2×D21211yescharacteristic
508C22×C4#1171994512yes—
509C22×C4C2×C22⋊D5312yes—
510C22×C4C23×D5311yescharacteristic
511M16—44no—
512M16—44no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8#1171994712yes—
523C2×C8C2×C22⋊D5312yes—
524C2×C8C2×C22⋊D5312yes—
525C4⋊C4—22no—
526C4⋊C4—532no—
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—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—4241no—
541C4⋊C4—4241no—
542C4⋊C4—2122no—
543C4⋊C4—2122no—
544C4⋊C4—2122no—
545C4⋊C4—2122no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—4241no—
549C22⋊C4—4241no—
550C22⋊C4—4241no—
551C22⋊C4—2122no—
552C22⋊C4—2122no—
553C22⋊C4—2122no—
554C22⋊C4—2122no—
555C42—21no—
556C42—21no—
557C42—2122no—
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—21no—
571C23#1313319112yes—
572C23C22×D21211yescharacteristic
573Q8—2121no—
574Q8—2121no—
575Q8—2121no—
576Q8—2121no—
577D4—44no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
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—1062no—
597C2×C4—4244no—
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—2121no—
620C2×C4—2121no—
621C2×C4#1313319112yes—
622C2×C4#1313325012yes—
623C2×C4C22×C22⋊D5311yescharacteristic
624C2×C4#41×D5311yescharacteristic
625C2×C4C2×#1171994711yescharacteristic
626C2×C4C2×#1171994511yescharacteristic
627C2×C4#1313326211yescharacteristic
628C2×C4#1313326411yescharacteristic
629C8—22no—
630C8—22no—
631C8—22no—
632C8—41no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22C2×#1313319111yescharacteristic
648C22#1681126811yescharacteristic
649C22#1681127711yescharacteristic
650C22C2×#1313325011yescharacteristic
651C22#1681142311yescharacteristic
652C22#1681142811yescharacteristic
653C22#1681144011yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—4241no—
659C4—4241no—
660C4—4241no—
661C4—4241no—
662C4—4241no—
663C4—4241no—
664C4—4241no—
665C4—2122no—
666C4#1681115311yescharacteristic
667C4#1681121511yescharacteristic
668C4#1681126711yescharacteristic
669C4#1681127211yescharacteristic
670C2—42no—
671C2—81no—
672C2#2027826011yescharacteristic
673C2#2028434511yescharacteristic
674C2#2028438511yescharacteristic
675C2#2028691811yescharacteristic
676C2#2028699911yescharacteristic
677C2#2028728911yescharacteristic
678C2#2028734211yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#436C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×#436D53
hypocenterC53#25261
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. C2×#16810965C2semidirect
4. #20276213C2semidirect
5. #20276138C2semidirect
6. #20276109C2semidirect
7. #20275919C2semidirect
8. #20275448C2semidirect
9. #20275391C2semidirect
10. #20275148C2semidirect
11. #20275148C2semidirect
12. #20274905C2semidirect
13. #20274748C2semidirect
14. #20274435C2semidirect
400. C53#25261semidirect

Maximal subgroups

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

subgroupindex
C2×#168109652
#202790122
#202781662
#202762132
#202761382
#202761092
#202759192
#202754482
#202753912
#202751482
#202749052
#202747482
#202744352
C2×C53×#4362
#2526153