Finite Group Databasebeta

← #27466219

Subgroups — #27466219

Order 13568.

788 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277408C211yesmaximal, characteristic
2#20277338C211yesmaximal, characteristic, complemented
3#20276284C211yesmaximal, characteristic, complemented
4#20276274C211yesmaximal, characteristic, complemented
5#20276247C211yesmaximal, characteristic, complemented
6#20276235C211yesmaximal, characteristic, complemented
7#20275485C211yesmaximal, characteristic, complemented
8#20275479C211yesmaximal, characteristic, complemented
9#20275035C211yesmaximal, characteristic
10#20275030C211yesmaximal, characteristic
11#20274916C211yesmaximal, characteristic, complemented
12#20274910C211yesmaximal, characteristic, complemented
13#20274726C211yesmaximal, characteristic, complemented
14#20274716C211yesmaximal, characteristic, complemented
15C53×#1798C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28#16810525C2211yescharacteristic
29#16810523—21no—
30#16810522—21no—
31#16810522—21no—
32#16810521C2211yescharacteristic
33#16810513—21no—
34#16810499—21no—
35#16810498—21no—
36#16810498—21no—
37#16810494—21no—
38#16810493C2211yescharacteristic
39#16810493C2211yescharacteristic
40#16810489C2211yescharacteristic
41#16810489C2211yescharacteristic
42#16810483C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C23×Dic53—21no—
62C22×C4×D53—21no—
63#13133191—22no—
64#13133191Q814yes—
65C4×D212—22no—
66C4×D212—22no—
67#13133176—41no—
68#13133176—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—41no—
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×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92#13133151—41no—
93#13133149—41no—
94#13133149—41no—
95#13133149—41no—
96#13133139—41no—
97#13133139—41no—
98#13133139—41no—
99#13133139—41no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#117×C53—22no—
113#117×C53—22no—
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C4⋊C8×C53—22no—
121C4⋊C8×C53—22no—
122C22⋊C8×C53—22no—
123C22⋊C8×C53—22no—
124C24×C53C2×D411yescharacteristic
125C23×D53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×Q812yes—
136C22×C4×C53#4112yes—
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—24no—
139C2×C22⋊D53—81no—
140C2×C22⋊D53—81no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—42no—
149C22×Dic53—81no—
150C22×Dic53—81no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212C2×Q812yes—
157C2×C4×D53—42no—
158C2×C4×D53—81no—
159C2×C4×D53—81no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53#4112yes—
179C42×C53#4112yes—
180C22⋊Dic53—42no—
181C22⋊Dic53—81no—
182#11719921—24no—
183#11719921—42no—
184#11719921—81no—
185#11719921—81no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
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—42no—
200C4⋊Dic53—81no—
201C4⋊Dic53—81no—
202C4⋊Dic53—81no—
203C4⋊Dic53—81no—
204C4⋊Dic53C2×Q812yes—
205#11719919—81no—
206C4×Dic53—42no—
207C4×Dic53—81no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53C22⋊Q812yes—
215C23×C53C22×D411yescharacteristic
216C22×D53—22no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53C22⋊Q812yes—
240C2×C4×C53#12312yes—
241C2×C4×C53#12312yes—
242C2×C4×C53C22×Q811yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53H(5,𝔽2)11yescharacteristic
247C2×C4×C53H(5,𝔽2)11yescharacteristic
248C22⋊D53—84no—
249C2×Dic53—22no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
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—84no—
271D212—22no—
272D212—22no—
273D212—41no—
274D212—41no—
275C4×D53—84no—
276C8×C53—41no—
277C8×C53—41no—
278C8×C53—41no—
279C8×C53—41no—
280#27688—531nomaximal, Sylow
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53C2×C22⋊Q811yescharacteristic
289C22×C53C2×#12311yescharacteristic
290C22×C53#53511yescharacteristic
291C22×C53#57611yescharacteristic
292C22×C53#57711yescharacteristic
293C22×C53#57711yescharacteristic
294C22×C53#57711yescharacteristic
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53—81no—
299C2×D53—81no—
300C2×D53—81no—
301C2×D53—81no—
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53#55311yescharacteristic
308C4×C53#55311yescharacteristic
309C4×C53#55411yescharacteristic
310C4×C53#55411yescharacteristic
311Dic53—42no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317Dic53—81no—
318#2324—531no—
319#2317—531no—
320#2101—531no—
321#2099—531no—
322#2094—531no—
323#2092—531no—
324#1949—531no—
325#1948—531no—
326#1875—531no—
327#1874—531no—
328#1854—531no—
329#1853—531no—
330#1804—531no—
331#1802—531no—
332#1798D5311yescharacteristic, complemented
333C2×C53—42no—
334C2×C53#248711yescharacteristic
335C2×C53#308111yescharacteristic
336C2×C53#308411yescharacteristic
337C2×C53#309611yescharacteristic
338C2×C53#309911yescharacteristic
339C2×C53#322111yescharacteristic
340C2×C53#322411yescharacteristic
341D53—82no—
342C2×#121—531no—
343C2×C4×D4—531no—
344C2×#117—531no—
345C2×#117C2×D5311yescharacteristic
346C22×C4⋊C4—531no—
347C2×#107—531no—
348C2×#106—531no—
349C2×#106—531no—
350C2×#106—531no—
351C2×C4⋊C8C2×D5311yescharacteristic
352C2×C4⋊C8C2×D5311yescharacteristic
353C2×#102—531no—
354C2×#102—531no—
355C2×#102—531no—
356C2×#102—531no—
357C2×C22⋊C8C2×D5311yescharacteristic
358C2×C22⋊C8C2×D5311yescharacteristic
359#390—531no—
360#390—1061no—
361#389—531no—
362#389—1061no—
363#389—1061no—
364#388—1061no—
365#386—1061no—
366#385—1061no—
367#385—1061no—
368#385—1061no—
369#384—531no—
370#384—531no—
371#382—531no—
372#382—531no—
373#380—531no—
374C4×C4⋊C4—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#336C2×D5311yescharacteristic
384#336C2×D5311yescharacteristic
385C53#2768811yescharacteristic, complemented, Sylow
386C22×D4—531no—
387C22×D4—1061no—
388C23×C4—1061no—
389C23×C4—1061no—
390C23×C4C22×D5311yescharacteristic
391C22×C8C22×D5311yescharacteristic
392C22×C8C22×D5311yescharacteristic
393C22×C8C22×D5311yescharacteristic
394C22×C8C22×D5311yescharacteristic
395#121—534no—
396#121—1062no—
397C4×D4—1062no—
398C4×D4—1062no—
399#117—22no—
400#117—22no—
401#117—2121no—
402#117—2121no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—2121no—
417C2×C4⋊C4—2121no—
418C2×C4⋊C4D21211yescharacteristic
419C2×C4⋊C4D21211yescharacteristic
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4D21211yescharacteristic
427C2×C22⋊C4D21211yescharacteristic
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42C22×D5311yescharacteristic
431C2×C42C22×D5311yescharacteristic
432#107—2121no—
433#106—2121no—
434#106—2121no—
435#106—2121no—
436C4⋊C8—22no—
437C4⋊C8—22no—
438#102—2121no—
439#102—2121no—
440#102—2121no—
441#102—2121no—
442C22⋊C8—22no—
443C22⋊C8—22no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451C24—1061no—
452C24C2×D21211yescharacteristic
453C2×D4—532no—
454C2×D4—1064no—
455C2×D4—1061no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4—1061no—
459C2×D4—1061no—
460C2×D4—4241no—
461C2×D4—4241no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—4241no—
474C22×C4—4241no—
475C22×C4—4241no—
476C22×C4—4241no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4C2×D21211yescharacteristic
480C22×C4C2×D21211yescharacteristic
481C22×C4C2×D21211yescharacteristic
482C22×C4C2×D21211yescharacteristic
483C22×C4C2×D21211yescharacteristic
484C22×C4#1171994712yes—
485C22×C4Q8×D5312yes—
486C22×C4C23×D5311yescharacteristic
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—532no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—4241no—
516C4⋊C4—4241no—
517C4⋊C4—4241no—
518C4⋊C4—4241no—
519C4⋊C4—4241no—
520C4⋊C4—2122no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—1064no—
524C22⋊C4—4241no—
525C22⋊C4—4241no—
526C22⋊C4—4241no—
527C22⋊C4—2122no—
528C22⋊C4—2122no—
529C42—21no—
530C42—21no—
531C42—4241no—
532C42—2122no—
533C42#1171994712yes—
534C42#1171994712yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—1062no—
542C23—2121no—
543C23—2121no—
544C23—2121no—
545C23—2121no—
546C23—2121no—
547C23—2121no—
548C23#1313319712yes—
549C23C22×D21211yescharacteristic
550D4—1062no—
551D4—1062no—
552D4—4244no—
553D4—2121no—
554D4—2121no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—1062no—
572C2×C4—4244no—
573C2×C4—4244no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4#1313319712yes—
595C2×C4#1313320112yes—
596C2×C4#1313320112yes—
597C2×C4C2×Q8×D5311yescharacteristic
598C2×C4C2×#1171994711yescharacteristic
599C2×C4C2×#1171994711yescharacteristic
600C2×C4C2×#1171994711yescharacteristic
601C2×C4#1313326311yescharacteristic
602C2×C4#1313326311yescharacteristic
603C8—41no—
604C8—41no—
605C8—41no—
606C8—41no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—4241no—
615C22—4241no—
616C22—4241no—
617C22—4241no—
618C22—2122no—
619C22—2122no—
620C22—2122no—
621C22C2×#1313319711yescharacteristic
622C22C2×#1313320111yescharacteristic
623C22#1681119211yescharacteristic
624C22#1681142811yescharacteristic
625C22#1681143511yescharacteristic
626C22#1681143511yescharacteristic
627C22#1681143511yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—4241no—
634C4—4241no—
635C4—4241no—
636C4—4241no—
637C4—4241no—
638C4—4241no—
639C4—2122no—
640C4#1681132511yescharacteristic
641C4#1681132511yescharacteristic
642C4#1681133111yescharacteristic
643C4#1681133111yescharacteristic
644C2—42no—
645C2—4242no—
646C2#2027903311yescharacteristic
647C2#2028454711yescharacteristic
648C2#2028457911yescharacteristic
649C2#2028467411yescharacteristic
650C2#2028470611yescharacteristic
651C2#2028577811yescharacteristic
652C2#2028581111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1798C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1798D53
hypocenterC53#27688
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. #20277338C2semidirect
3. #20276284C2semidirect
4. #20276274C2semidirect
5. #20276247C2semidirect
6. #20276235C2semidirect
7. #20275485C2semidirect
8. #20275479C2semidirect
11. #20274916C2semidirect
12. #20274910C2semidirect
13. #20274726C2semidirect
14. #20274716C2semidirect
15. C53×#1798C2semidirect
332. #1798D53semidirect
385. C53#27688semidirect

Maximal subgroups

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

subgroupindex
#202774082
#202773382
#202762842
#202762742
#202762472
#202762352
#202754852
#202754792
#202750352
#202750302
#202749162
#202749102
#202747262
#202747162
C53×#17982
#2768853