Finite Group Databasebeta

← #27447641

Subgroups — #27447641

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278417C211yesmaximal, characteristic, complemented
2#20277729C211yesmaximal, characteristic
3#20276274C211yesmaximal, characteristic, complemented
4#20276203C211yesmaximal, characteristic, complemented
5#20276102C211yesmaximal, characteristic, complemented
6#20275852C211yesmaximal, characteristic, complemented
7#20275666C211yesmaximal, characteristic, complemented
8#20275287C211yesmaximal, characteristic, complemented
9#20275143C211yesmaximal, characteristic
10#20275030C211yesmaximal, characteristic
11#20274965C211yesmaximal, characteristic, complemented
12#20274905C211yesmaximal, characteristic, complemented
13#20274716C211yesmaximal, characteristic, complemented
14C2×#16810386C211yesmaximal, characteristic, complemented
15C53×#1850C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×#13133194—21no—
18C2×#13133191C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20C2×#13133177—21no—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×C4⋊Dic53C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133151C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29#16810555—21no—
30#16810544—21no—
31#16810525C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810515—21no—
36#16810500—21no—
37#16810498—21no—
38#16810497—21no—
39#16810493C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C23×Dic53—21no—
63C23×Dic53—21no—
64C22×C4×D53—21no—
65#13133217—22no—
66#13133217—41no—
67#13133194—44no—
68#13133191—22no—
69#13133191Q814yescomplemented
70C4×D212—22no—
71C4×D212—41no—
72#13133177—44no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53—41no—
81C2×C4⋊Dic53—41no—
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×C22⋊Dic53—21no—
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×#11719921—21no—
96C2×#11719921—21no—
97C2×C4×Dic53—21no—
98#13133151—22no—
99#13133151—41no—
100#13133139—22no—
101#13133139—41no—
102#13133139—41no—
103#13133139—41no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53—22no—
111C22×C8×C53—41no—
112C22×C8×C53—41no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C4⋊C8×C53—41no—
122C4⋊C8×C53—41no—
123C22⋊C8×C53—22no—
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
126C24×C53C2×D411yescharacteristic
127C23×D53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×Q812yes—
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C2×C22⋊D53—24no—
141C2×C22⋊D53—42no—
142C2×C22⋊D53—42no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—81no—
155C22×Dic53—81no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212C2×Q812yes—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
164C2×C4×D53—81no—
165C2×Dic106—42no—
166C2×Dic106—42no—
167C2×Dic106—81no—
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×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—44no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53D814yes—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C22⋊Dic53—42no—
188#11719921—24no—
189#11719921—42no—
190#11719921—42no—
191#11719921—81no—
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—21no—
203C4⋊Dic53—42no—
204C4⋊Dic53—42no—
205C4⋊Dic53—42no—
206C4⋊Dic53—42no—
207C4⋊Dic53—81no—
208C4⋊Dic53C2×Q812yes—
209C4⋊Dic53#4112yes—
210C4⋊Dic53#4112yes—
211#11719919—42no—
212#11719919—42no—
213#11719919—42no—
214C4×Dic53—42no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53C22⋊Q812yes—
220C23×C53C2×D812yes—
221C23×C53#13712yes—
222C23×C53C22×D411yescharacteristic
223C22×D53—22no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
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×C53C2×D812yes—
248C2×C4×C53C2×D812yes—
249C2×C4×C53C22×Q811yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C22⋊D53—48no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
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—41no—
276C2×Dic53—41no—
277C2×Dic53—84no—
278C2×Dic53—84no—
279D212—22no—
280D212—22no—
281D212—41no—
282D212—41no—
283C4×D53—84no—
284Dic106—84no—
285C8×C53—41no—
286C8×C53—41no—
287C8×C53—41no—
288C8×C53—41no—
289#26408—531nomaximal, Sylow
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53#47414yes—
297C22×C53C2×C22⋊Q811yescharacteristic
298C22×C53#54611yescharacteristic
299C22×C53#54611yescharacteristic
300C22×C53C22×D811yescharacteristic
301C22×C53C2×#13711yescharacteristic
302C22×C53#57711yescharacteristic
303C22×C53#57711yescharacteristic
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—42no—
307C2×D53—81no—
308C2×D53—81no—
309C2×D53—81no—
310C2×D53—81no—
311C4×C53—22no—
312C4×C53—22no—
313C4×C53—22no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53#52911yescharacteristic
317C4×C53#53811yescharacteristic
318C4×C53#55411yescharacteristic
319C4×C53#55411yescharacteristic
320Dic53—42no—
321Dic53—42no—
322Dic53—42no—
323Dic53—81no—
324Dic53—81no—
325Dic53—81no—
326Dic53—81no—
327#2440—531no—
328#2361—531no—
329#2099—531no—
330#2086—531no—
331#2067—531no—
332#2023—531no—
333#1988—531no—
334#1918—531no—
335#1893—531no—
336#1874—531no—
337#1862—531no—
338#1852—531no—
339#1850D5311yescharacteristic, complemented
340#1802—531no—
341C2×#340—531no—
342C2×C53—24no—
343C2×C53#245011yescharacteristic
344C2×C53C2×#47411yescharacteristic
345C2×C53#308411yescharacteristic
346C2×C53#329411yescharacteristic
347C2×C53#329411yescharacteristic
348C2×C53#331611yescharacteristic
349C2×C53#332011yescharacteristic
350D53—82no—
351C23×C8C2×D5311yescharacteristic
352C2×C4⋊Q8—531no—
353C2×C22⋊Q8—1061no—
354C2×#121—531no—
355C2×C4×D4—531no—
356C2×C4×D4—1061no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4—531no—
359C2×#107—531no—
360C2×#107—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#398—1061no—
369#395—1061no—
370#390—531no—
371#389—531no—
372#389—531no—
373#389—531no—
374#388—1061no—
375#386—1061no—
376#385—1061no—
377#385—1061no—
378#384—531no—
379#380—531no—
380C4×C4⋊C4—531no—
381C4×C22⋊C4C2×D5311yescharacteristic
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#336C2×D5311yescharacteristic
393#336C2×D5311yescharacteristic
394C53#2640811yescharacteristic, complemented, Sylow
395C22×Q8—1061no—
396C22×D4—531no—
397C22×D4—1061no—
398C23×C4—1061no—
399C23×C4—1061no—
400C23×C4—1061no—
401C23×C4C22×D5311yescharacteristic
402C22×C8—22no—
403C22×C8—41no—
404C22×C8—41no—
405C22×C8C22×D5311yescharacteristic
406C22×C8C22×D5311yescharacteristic
407C22×C8C22×D5311yescharacteristic
408C22×C8C22×D5311yescharacteristic
409C4⋊Q8—1062no—
410C4⋊Q8—2121no—
411C22⋊Q8—2124no—
412#121—534no—
413#121—1062no—
414C4×D4—1062no—
415C4×D4—2121no—
416C4×D4—2124no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—1062no—
425C2×C4⋊C4—1062no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—2121no—
435C2×C4⋊C4—2121no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4D21211yescharacteristic
442C2×C22⋊C4D21211yescharacteristic
443C2×C42—1061no—
444C2×C42C22×D5311yescharacteristic
445C2×C42C22×D5311yescharacteristic
446#107—1062no—
447#107—2121no—
448C4⋊C8—41no—
449C4⋊C8—41no—
450#102—1062no—
451#102—2121no—
452#102—2121no—
453#102—2121no—
454C22⋊C8—22no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95D21211yescharacteristic
461#95D21211yescharacteristic
462C24—1061no—
463C24C2×D21211yescharacteristic
464C2×Q8—4241no—
465C2×Q8—2122no—
466C2×Q8—2122no—
467C2×D4—532no—
468C2×D4—1064no—
469C2×D4—1061no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C2×D4—1061no—
474C2×D4—2122no—
475C2×D4—2122no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—4241no—
488C22×C4—4241no—
489C22×C4—4241no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4C2×D21211yescharacteristic
497C22×C4C2×D21211yescharacteristic
498C22×C4C2×D21211yescharacteristic
499C22×C4C2×D21211yescharacteristic
500C22×C4C2×D21211yescharacteristic
501C22×C4#1171994712yes—
502C22×C4Q8×D5312yes—
503C22×C4C23×D5311yescharacteristic
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
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—44no—
517C4⋊C4—532no—
518C4⋊C4—532no—
519C4⋊C4—532no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
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—4241no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4—2122no—
536C4⋊C4—2122no—
537C4⋊C4—2122no—
538C4⋊C4—2122no—
539C22⋊C4—22no—
540C22⋊C4—1064no—
541C22⋊C4—4241no—
542C22⋊C4—2122no—
543C22⋊C4—2122no—
544C22⋊C4—2122no—
545C22⋊C4D42414yes—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—2122no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—1062no—
556C23—2121no—
557C23—2121no—
558C23—2121no—
559C23—2121no—
560C23—2121no—
561C23—2121no—
562C23#1313319712yes—
563C23C2×D42412yes—
564C23#1313325012yes—
565C23C22×D21211yescharacteristic
566Q8—4244no—
567D4—1062no—
568D4—1062no—
569D4—2128no—
570D4—2121no—
571D4—2121no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—4244no—
592C2×C4—4244no—
593C2×C4—4244no—
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—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4#1313319712yes—
613C2×C4C2×D42412yes—
614C2×C4C2×D42412yes—
615C2×C4C2×Q8×D5311yescharacteristic
616C2×C4#41×D5311yescharacteristic
617C2×C4#41×D5311yescharacteristic
618C2×C4C2×#1171994711yescharacteristic
619C2×C4#1313326211yescharacteristic
620C2×C4#1313326211yescharacteristic
621C8—41no—
622C8—41no—
623C8—41no—
624C8—41no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—4241no—
632C22—4241no—
633C22—4241no—
634C22—4241no—
635C22—2122no—
636C22—2122no—
637C22—2122no—
638C22#1681090414yes—
639C22C2×#1313319711yescharacteristic
640C22#1681126811yescharacteristic
641C22#1681126811yescharacteristic
642C22C22×D42411yescharacteristic
643C22C2×#1313325011yescharacteristic
644C22#1681143511yescharacteristic
645C22#1681143511yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—41no—
650C4—41no—
651C4—4241no—
652C4—4241no—
653C4—4241no—
654C4—4241no—
655C4—2122no—
656C4—2122no—
657C4—2122no—
658C4#1681115011yescharacteristic
659C4#1681120811yescharacteristic
660C4#1681132711yescharacteristic
661C4#1681132711yescharacteristic
662C2—24no—
663C2—4242no—
664C2#2027853711yescharacteristic
665C2C2×#1681090411yescharacteristic
666C2#2028457911yescharacteristic
667C2#2028652911yescharacteristic
668C2#2028652911yescharacteristic
669C2#2028682811yescharacteristic
670C2#2028688811yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1850C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1850D53
hypocenterC53#26408
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. #20278417C2semidirect
3. #20276274C2semidirect
4. #20276203C2semidirect
5. #20276102C2semidirect
6. #20275852C2semidirect
7. #20275666C2semidirect
8. #20275287C2semidirect
11. #20274965C2semidirect
12. #20274905C2semidirect
13. #20274716C2semidirect
14. C2×#16810386C2semidirect
15. C53×#1850C2semidirect
69. #13133191Q8semidirect
339. #1850D53semidirect
394. C53#26408semidirect

Maximal subgroups

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

subgroupindex
#202784172
#202777292
#202762742
#202762032
#202761022
#202758522
#202756662
#202752872
#202751432
#202750302
#202749652
#202749052
#202747162
C2×#168103862
C53×#18502
#2640853