Finite Group Databasebeta

← #27484939

Subgroups — #27484939

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133191C211yesmaximal, characteristic
2#20277198C211yesmaximal, characteristic, complemented
3#20276213C211yesmaximal, characteristic, complemented
4#20276208C211yesmaximal, characteristic, complemented
5#20276102C211yesmaximal, characteristic, complemented
6#20276088C211yesmaximal, characteristic, complemented
7#20275301C211yesmaximal, characteristic, complemented
8#20275287C211yesmaximal, characteristic, complemented
9#20275035C211yesmaximal, characteristic
10#20275035C211yesmaximal, characteristic
11#20274979C211yesmaximal, characteristic, complemented
12#20274965C211yesmaximal, characteristic, complemented
13#20274916C211yesmaximal, characteristic, complemented
14#20274916C211yesmaximal, characteristic, complemented
15C53×#1797C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×C22⋊D53—21no—
19C2×#13133176C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29#16810525C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810514—21no—
36#16810502—21no—
37#16810496—21no—
38#16810496—21no—
39#16810483C2211yescharacteristic
40C4×#11719919—21no—
41C4×#11719919—21no—
42C4×C4⋊Dic53C2211yescharacteristic
43C4×#11719921—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C22×C42×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—
61C22×C4×D53—21no—
62#13133191—22no—
63#13133191Q814yes—
64C4×D212—41no—
65C4×D212—41no—
66C4×C22⋊D53—44no—
67#13133176—22no—
68#13133176—22no—
69#13133176—41no—
70#13133176—41no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—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—
92C2×C4×Dic53—21no—
93#13133151—22no—
94#13133151—41no—
95#13133149—22no—
96#13133149—41no—
97#13133139—22no—
98#13133139—22no—
99#13133139—41no—
100#13133139—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C23×C4×C53D411yescharacteristic
109C23×C4×C53D411yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C42×C53—41no—
116C2×C42×C53—41no—
117C2×C42×C53D411yescharacteristic
118C2×C42×C53D411yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C4⋊C8×C53—41no—
122C4⋊C8×C53—41no—
123C22⋊C8×C53—22no—
124C22⋊C8×C53—22no—
125C24×C53C2×D411yescharacteristic
126C23×D53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—41no—
134C22×C4×C53—41no—
135C22×C4×C53—41no—
136C22×C4×C53—41no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×Q812yes—
143C22×C4×C53#4112yes—
144C22×C4×C53C2411yescharacteristic
145C2×C22⋊D53—24no—
146C2×C22⋊D53—42no—
147C2×C22⋊D53—42no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212C2×Q812yes—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
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—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—41no—
181C42×C53—41no—
182C22⋊Dic53—24no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185#11719921—24no—
186#11719921—42no—
187#11719921—42no—
188#11719921—42no—
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—21no—
200C4⋊Dic53C2×Q812yes—
201C4⋊Dic53#4112yes—
202C4⋊Dic53#4112yes—
203#11719919—42no—
204#11719919—42no—
205#11719919—42no—
206#11719919—42no—
207#11719919—42no—
208C4×Dic53—24no—
209C4×Dic53—42no—
210C4×Dic53—42no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53C22⋊Q812yes—
218C23×C53C22×D411yescharacteristic
219C22×D53—22no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
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—44no—
243C2×C4×C53—44no—
244C2×C4×C53C22⋊Q812yes—
245C2×C4×C53#13512yes—
246C2×C4×C53AGL(1,ℤ/8)12yes—
247C2×C4×C53C22×Q811yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C22⋊D53—48no—
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—
275D212—22no—
276D212—22no—
277D212—41no—
278D212—41no—
279C4×D53—84no—
280C8×C53—41no—
281C8×C53—41no—
282C8×C53—41no—
283C8×C53—41no—
284#29236—531nomaximal, Sylow
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53C2×C22⋊Q811yescharacteristic
293C22×C53#52511yescharacteristic
294C22×C53#53411yescharacteristic
295C22×C53C2×#13511yescharacteristic
296C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C53#57511yescharacteristic
298C22×C53#57511yescharacteristic
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—81no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—41no—
310C4×C53—41no—
311C4×C53#55311yescharacteristic
312C4×C53#55311yescharacteristic
313C4×C53#55411yescharacteristic
314C4×C53#55411yescharacteristic
315Dic53—42no—
316Dic53—42no—
317Dic53—42no—
318Dic53—81no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322C4×#121—531no—
323#2296—531no—
324#2088—531no—
325#2087—531no—
326#2067—531no—
327#2065—531no—
328#1920—531no—
329#1918—531no—
330#1875—531no—
331#1875—531no—
332#1864—531no—
333#1862—531no—
334#1854—531no—
335#1854—531no—
336#1797D5311yescharacteristic, complemented
337C2×C53—42no—
338C2×C53#257911yescharacteristic
339C2×C53#307611yescharacteristic
340C2×C53#307711yescharacteristic
341C2×C53#310611yescharacteristic
342C2×C53#311111yescharacteristic
343C2×C53#319111yescharacteristic
344C2×C53#319411yescharacteristic
345D53—82no—
346C2×#121—531no—
347C2×C4×D4—531no—
348C2×C4×D4—1061no—
349C2×#117—531no—
350C2×#117—531no—
351C22×C42C2×D5311yescharacteristic
352C2×#107—531no—
353C2×#107—531no—
354C2×#106—531no—
355C2×#106—531no—
356C2×C4⋊C8C2×D5311yescharacteristic
357C2×C4⋊C8C2×D5311yescharacteristic
358C2×#102—531no—
359C2×#102—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×C22⋊C8C2×D5311yescharacteristic
363C2×C22⋊C8C2×D5311yescharacteristic
364#390—531no—
365#389—531no—
366#389—531no—
367#389—531no—
368#389—531no—
369#389—531no—
370#388—1061no—
371#386—1061no—
372#385—1061no—
373#385—1061no—
374#380—531no—
375C4×C4⋊C4—531no—
376C4×C4⋊C4—1061no—
377C4×C4⋊C4—1061no—
378C4×C22⋊C4—1061no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#336C2×D5311yescharacteristic
388#336C2×D5311yescharacteristic
389C53#2923611yescharacteristic, complemented, Sylow
390C22×D4—531no—
391C22×D4—1061no—
392C23×C4—1061no—
393C23×C4D21211yescharacteristic
394C23×C4D21211yescharacteristic
395C23×C4C22×D5311yescharacteristic
396C22×C8C22×D5311yescharacteristic
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400#121—534no—
401#121—1062no—
402C4×D4—2121no—
403C4×D4—2121no—
404C4×D4—2124no—
405#117—1062no—
406#117—1062no—
407#117—2121no—
408#117—2121no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C42—41no—
429C2×C42—41no—
430C2×C42—1061no—
431C2×C42—1061no—
432C2×C42—1061no—
433C2×C42D21211yescharacteristic
434C2×C42D21211yescharacteristic
435C2×C42C22×D5311yescharacteristic
436C2×C42C22×D5311yescharacteristic
437#107—1062no—
438#107—2121no—
439#106—1062no—
440#106—2121no—
441C4⋊C8—41no—
442C4⋊C8—41no—
443#102—1062no—
444#102—1062no—
445#102—2121no—
446#102—2121no—
447C22⋊C8—22no—
448C22⋊C8—22no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456C24—1061no—
457C24C2×D21211yescharacteristic
458C2×D4—532no—
459C2×D4—1064no—
460C2×D4—1061no—
461C2×D4—1061no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—2122no—
466C2×D4—2122no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4C2×D21211yescharacteristic
488C22×C4C2×D21211yescharacteristic
489C22×C4C2×D21211yescharacteristic
490C22×C4C2×D21211yescharacteristic
491C22×C4C2×D21211yescharacteristic
492C22×C4#1171994712yes—
493C22×C4Q8×D5312yes—
494C22×C4C23×D5311yescharacteristic
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C4⋊C4—532no—
508C4⋊C4—532no—
509C4⋊C4—532no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C22⋊C4—1064no—
527C22⋊C4—1064no—
528C22⋊C4—2122no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—41no—
538C42—41no—
539C42—1064no—
540C42—2122no—
541C42—2122no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—1062no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23#1313319712yes—
556C23C22×D21211yescharacteristic
557D4—1062no—
558D4—1062no—
559D4—2128no—
560D4—2121no—
561D4—2121no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—4244no—
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—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4#1313319712yes—
603C2×C4#1313323912yes—
604C2×C4#1313324512yes—
605C2×C4C2×Q8×D5311yescharacteristic
606C2×C4#41×D5311yescharacteristic
607C2×C4#41×D5311yescharacteristic
608C2×C4C2×#1171994711yescharacteristic
609C2×C4#1313326211yescharacteristic
610C2×C4#1313326211yescharacteristic
611C8—41no—
612C8—41no—
613C8—41no—
614C8—41no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—4241no—
623C22—4241no—
624C22—4241no—
625C22—4241no—
626C22—2122no—
627C22—2122no—
628C22—2122no—
629C22C2×#1313319711yescharacteristic
630C22#1681113511yescharacteristic
631C22#1681118511yescharacteristic
632C22C2×#1313323911yescharacteristic
633C22C2×#1313324511yescharacteristic
634C22#1681142311yescharacteristic
635C22#1681142311yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—41no—
641C4—4241no—
642C4—4241no—
643C4—4241no—
644C4—4241no—
645C4—2122no—
646C4—2122no—
647C4—2122no—
648C4#1681131611yescharacteristic
649C4#1681131611yescharacteristic
650C4#1681132711yescharacteristic
651C4#1681132711yescharacteristic
652C2—42no—
653C2—4242no—
654C2#2028006411yescharacteristic
655C2#2028450511yescharacteristic
656C2#2028451411yescharacteristic
657C2#2028477611yescharacteristic
658C2#2028482611yescharacteristic
659C2#2028550811yescharacteristic
660C2#2028553111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1797C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1797D53
hypocenterC53#29236
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #20277198C2semidirect
3. #20276213C2semidirect
4. #20276208C2semidirect
5. #20276102C2semidirect
6. #20276088C2semidirect
7. #20275301C2semidirect
8. #20275287C2semidirect
11. #20274979C2semidirect
12. #20274965C2semidirect
13. #20274916C2semidirect
14. #20274916C2semidirect
15. C53×#1797C2semidirect
336. #1797D53semidirect
389. C53#29236semidirect

Maximal subgroups

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

subgroupindex
C4×#131331912
#202771982
#202762132
#202762082
#202761022
#202760882
#202753012
#202752872
#202750352
#202749792
#202749652
#202749162
C53×#17972
#2923653