Finite Group Databasebeta

← #27446966

Subgroups — #27446966

Order 13568.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810924C211yesmaximal, characteristic, complemented
2C2×#16810665C211yesmaximal, characteristic, complemented
3#20278439C211yesmaximal, characteristic, complemented
4#20277751C211yesmaximal, characteristic
5#20276235C211yesmaximal, characteristic, complemented
6#20276213C211yesmaximal, characteristic, complemented
7#20276088C211yesmaximal, characteristic, complemented
8#20276054C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275287C211yesmaximal, characteristic, complemented
11#20275153C211yesmaximal, characteristic
12#20275030C211yesmaximal, characteristic
13#20274965C211yesmaximal, characteristic, complemented
14#20274916C211yesmaximal, characteristic, complemented
15C53×#1851C211yesmaximal, characteristic, complemented
16C2×#13133209C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810924—24no—
22#16810665—24no—
23C2×#13133151C2211yescharacteristic
24C2×#13133151C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133139C2211yescharacteristic
30C2×#13133139C2211yescharacteristic
31C2×#13133139C2211yescharacteristic
32C2×#13133139C2211yescharacteristic
33#16810565—21no—
34#16810557—21no—
35#16810525C2211yescharacteristic
36#16810523—21no—
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810512—21no—
41#16810499—21no—
42#16810495—21no—
43#16810489C2211yescharacteristic
44#16810483C2211yescharacteristic
45#16810481—21no—
46C4×C4⋊Dic53C2211yescharacteristic
47#16810470—21no—
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C22×M16×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C22⋊D53—21no—
63C23×Dic53—21no—
64C22×C4×D53—21no—
65#13133209—22no—
66#13133209—41no—
67#13133191—22no—
68#13133191Q814yes—
69C4×D212—22no—
70C4×D212—41no—
71#13133176—22no—
72#13133176—41no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—41no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
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×#11719919—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96#13133151—22no—
97#13133151—41no—
98#13133151D414yescomplemented
99#13133149—22no—
100#13133149—22no—
101#13133149—22no—
102#13133139—22no—
103#13133139—22no—
104#13133139—22no—
105#13133139—41no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113C23×C4×C53C2311yescharacteristic
114C2×M16×C53—22no—
115C2×M16×C53—22no—
116C2×M16×C53—22no—
117C22×C8×C53C2311yescharacteristic
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C42×C53C2311yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C4⋊C8×C53—22no—
126C4⋊C8×C53—22no—
127C22⋊C8×C53—22no—
128#95×C53D411yescharacteristic
129#95×C53D411yescharacteristic
130C24×C53C2×D411yescharacteristic
131C23×D53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×Q812yes—
142C22×C4×C53#4112yes—
143C22×C4×C53C2411yescharacteristic
144C2×C22⋊D53—24no—
145C2×C22⋊D53—81no—
146C2×C22⋊D53—81no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—81no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212C2×Q812yes—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C4×D53—81no—
166M16×C53—28no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
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×C53C2×D412yes—
178C2×C8×C53#4112yes—
179C22⋊C4×C53—41no—
180C22⋊C4×C53—41no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C22⋊Dic53—42no—
186C22⋊Dic53—42no—
187#11719921—24no—
188#11719921—42no—
189#11719921—42no—
190#11719921—81no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—42no—
203C4⋊Dic53C2×D412yes—
204C4⋊Dic53C2×D412yes—
205C4⋊Dic53C2×Q812yes—
206#11719919—42no—
207#11719919—42no—
208#11719919—81no—
209#11719919—81no—
210#11719919—81no—
211C4×Dic53—42no—
212C4×Dic53—81no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53C22⋊Q812yes—
220C23×C53C22×D411yescharacteristic
221C22×D53—22no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
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—21no—
245C2×C4×C53—21no—
246C2×C4×C53C22⋊Q812yes—
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×Q811yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C22⋊D53—84no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
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—84no—
276D212—22no—
277D212—22no—
278D212—41no—
279D212—41no—
280C4×D53—84no—
281C8×C53—22no—
282C8×C53—22no—
283C8×C53—41no—
284C8×C53—41no—
285#26363—531nomaximal, Sylow
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53C2×C22⋊Q811yescharacteristic
294C22×C53D4211yescharacteristic
295C22×C53#54711yescharacteristic
296C22×C53#57511yescharacteristic
297C22×C53#57511yescharacteristic
298C22×C53#57611yescharacteristic
299C22×C53#57711yescharacteristic
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C4×C53—22no—
308C4×C53—41no—
309C4×C53—41no—
310C4×C53—41no—
311C4×C53—41no—
312C4×C53#52511yescharacteristic
313C4×C53#53911yescharacteristic
314C4×C53D4×Q811yescharacteristic
315C4×C53#55411yescharacteristic
316Dic53—42no—
317Dic53—42no—
318Dic53—42no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322Dic53—81no—
323C2×#478—531no—
324C2×#428—531no—
325#2442—531no—
326#2363—531no—
327#2092—531no—
328#2088—531no—
329#2065—531no—
330#2059—531no—
331#1949—531no—
332#1918—531no—
333#1895—531no—
334#1874—531no—
335#1862—531no—
336#1854—531no—
337#1851D5311yescharacteristic, complemented
338C2×C53—42no—
339C2×C53#244911yescharacteristic
340C2×C53#308011yescharacteristic
341C2×C53#308111yescharacteristic
342C2×C53#327111yescharacteristic
343C2×C53#327311yescharacteristic
344C2×C53#334811yescharacteristic
345C2×C53#335511yescharacteristic
346D53—82no—
347C22×M16C2×D5311yescharacteristic
348C2×#125—531no—
349C2×#121—531no—
350C2×C4×D4—531no—
351C2×#117—531no—
352C22×C4⋊C4—531no—
353#478—1064no—
354#428—1064no—
355C2×#107—531no—
356C2×#107—531no—
357C2×#107—531no—
358C2×#106—531no—
359C2×#106—531no—
360C2×#106—531no—
361C2×C4⋊C8C2×D5311yescharacteristic
362C2×C4⋊C8C2×D5311yescharacteristic
363C2×#102—531no—
364C2×#102—531no—
365C2×#102—531no—
366C2×#102—531no—
367C2×C22⋊C8C2×D5311yescharacteristic
368#400—1061no—
369#398—1061no—
370#390—531no—
371#390—1061no—
372#389—531no—
373#389—531no—
374#389—531no—
375#388—1061no—
376#386—1061no—
377#385—1061no—
378#382—531no—
379#380—531no—
380#380—1061no—
381C4×C4⋊C4—531no—
382C4×C22⋊C4—1061no—
383C4×C22⋊C4C2×D5311yescharacteristic
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#336C2×D5311yescharacteristic
391#336C2×D5311yescharacteristic
392C53#2636311yescharacteristic, complemented, Sylow
393C22×D4—531no—
394C22×D4—1061no—
395C23×C4—1061no—
396C23×C4—1061no—
397C23×C4C22×D5311yescharacteristic
398C2×M16—22no—
399C2×M16—22no—
400C2×M16—22no—
401C22×C8C22×D5311yescharacteristic
402C22×C8C22×D5311yescharacteristic
403C22×C8C22×D5311yescharacteristic
404C22×C8C22×D5311yescharacteristic
405#125—1062no—
406#125—2121no—
407#121—534no—
408#121—1062no—
409C4×D4—1062no—
410C4×D4—2121no—
411#117—1062no—
412#117—2121no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—1062no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—2121no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4D21211yescharacteristic
435C2×C22⋊C4D21211yescharacteristic
436C2×C42—1061no—
437C2×C42—1061no—
438C2×C42C22×D5311yescharacteristic
439C2×C42C22×D5311yescharacteristic
440#107—534no—
441#107—1062no—
442#107—2121no—
443#106—1062no—
444#106—1062no—
445#106—1062no—
446C4⋊C8—22no—
447C4⋊C8—22no—
448#102—1062no—
449#102—1062no—
450#102—1062no—
451#102—2121no—
452C22⋊C8—22no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95D21211yescharacteristic
461#95D21211yescharacteristic
462C24—1061no—
463C24C2×D21211yescharacteristic
464C2×D4—532no—
465C2×D4—1064no—
466C2×D4—1061no—
467C2×D4—1061no—
468C2×D4—1061no—
469C2×D4—1061no—
470C2×D4—1061no—
471C2×D4—4241no—
472C2×D4—4241no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—4241no—
485C22×C4—4241no—
486C22×C4—2122no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4C2×D21211yescharacteristic
491C22×C4C2×D21211yescharacteristic
492C22×C4C2×D21211yescharacteristic
493C22×C4C2×D21211yescharacteristic
494C22×C4C2×D21211yescharacteristic
495C22×C4#1171994712yes—
496C22×C4Q8×D5312yes—
497C22×C4C23×D5311yescharacteristic
498M16—28no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8D4×D5312yes—
510C2×C8#1171994712yes—
511C4⋊C4—532no—
512C4⋊C4—532no—
513C4⋊C4—532no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—4241no—
526C4⋊C4—4241no—
527C4⋊C4—4241no—
528C4⋊C4—2122no—
529C4⋊C4—2122no—
530C4⋊C4—2122no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—1064no—
534C22⋊C4—4241no—
535C22⋊C4—2122no—
536C22⋊C4—2122no—
537C22⋊C4—2122no—
538C22⋊C4—2122no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—4241no—
544C42—2122no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—1062no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23#1313319712yes—
559C23C22×D21211yescharacteristic
560D4—1062no—
561D4—1062no—
562D4—4244no—
563D4—2121no—
564D4—2121no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—4244no—
587C2×C4—4244no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
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#1313319712yes—
607C2×C4C2×D4×D5311yescharacteristic
608C2×C4C2×Q8×D5311yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4C2×#1171994711yescharacteristic
611C2×C4#1313326311yescharacteristic
612C2×C4#1313326411yescharacteristic
613C8—22no—
614C8—22no—
615C8—41no—
616C8—41no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—4241no—
625C22—4241no—
626C22—4241no—
627C22—4241no—
628C22—2122no—
629C22—2122no—
630C22—2122no—
631C22C2×#1313319711yescharacteristic
632C22D4×D21211yescharacteristic
633C22#1681127711yescharacteristic
634C22#1681142311yescharacteristic
635C22#1681142311yescharacteristic
636C22#1681142811yescharacteristic
637C22#1681143511yescharacteristic
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—4241no—
644C4—4241no—
645C4—4241no—
646C4—4241no—
647C4—2122no—
648C4—2122no—
649C4—2122no—
650C4#1681112911yescharacteristic
651C4#1681121711yescharacteristic
652C4#1681128611yescharacteristic
653C4#1681133111yescharacteristic
654C2—42no—
655C2—4242no—
656C2#2027852211yescharacteristic
657C2#2028453811yescharacteristic
658C2#2028454711yescharacteristic
659C2#2028630511yescharacteristic
660C2#2028632711yescharacteristic
661C2#2028723211yescharacteristic
662C2#2028732011yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1851C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1851D53
hypocenterC53#26363
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. C2×#16810924C2semidirect
2. C2×#16810665C2semidirect
3. #20278439C2semidirect
5. #20276235C2semidirect
6. #20276213C2semidirect
7. #20276088C2semidirect
8. #20276054C2semidirect
9. #20275485C2semidirect
10. #20275287C2semidirect
13. #20274965C2semidirect
14. #20274916C2semidirect
15. C53×#1851C2semidirect
98. #13133151D4semidirect
337. #1851D53semidirect
392. C53#26363semidirect

Maximal subgroups

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

subgroupindex
C2×#168109242
C2×#168106652
#202784392
#202777512
#202762352
#202762132
#202760882
#202760542
#202754852
#202752872
#202751532
#202750302
#202749652
#202749162
C53×#18512
#2636353