Finite Group Databasebeta

← #27420174

Subgroups — #27420174

Order 13568.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811180C211yesmaximal, characteristic, complemented
2C2×#16810914C211yesmaximal, characteristic, complemented
3C2×#16810904C211yesmaximal, characteristic, complemented
4C4×#13133191C211yesmaximal, characteristic
5#20276054C212yesmaximal, complemented
6#20275862C212yesmaximal, complemented
7#20275666C211yesmaximal, characteristic, complemented
8#20275662C211yesmaximal, characteristic, complemented
9#20275035C212yesmaximal
10#20274916C212yesmaximal, complemented
11C53×#1797C211yesmaximal, characteristic, complemented
12#16811180—24no—
13#16811180—24no—
14C2×#13133217C2211yescharacteristic
15C2×#13133217C2211yescharacteristic
16C2×#13133209C2212yes—
17C2×#13133196—21no—
18C2×#13133196—21no—
19C2×#13133191C2211yescharacteristic
20C2×C4×Dic106—21no—
21C2×C4×Dic106—21no—
22C2×C4×D212C2211yescharacteristic
23C2×C4×C22⋊D53—21no—
24C2×#13133176C2212yes—
25#16810914—24no—
26#16810904—24no—
27C2×#13133151C2212yes—
28C2×#13133149C2212yes—
29C2×#13133139C2212yes—
30C2×#13133139C2212yes—
31#16810525C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810514—21no—
34#16810502—21no—
35#16810483C2211yescharacteristic
36C4×C4⋊Dic53C2211yescharacteristic
37C4×#11719921—21no—
38#16810386C2212yes—
39#16810386C2212yes—
40#16810386C2212yes—
41#16810386C2212yes—
42C22×C42×C53C2211yescharacteristic
43C2×C4⋊C8×C53C2211yescharacteristic
44C2×C4⋊C8×C53C2211yescharacteristic
45C2×C22⋊C8×C53C2212yes—
46C53×#336C2212yes—
47C22×Dic106—21no—
48C22×Dic106—21no—
49C22×D212C2311yescharacteristic
50C22×C22⋊D53—21no—
51C22×C4×D53—21no—
52#13133217—22no—
53#13133217—22no—
54#13133217D414yescomplemented
55#13133217D414yescomplemented
56#13133209—24no—
57#13133209—24no—
58#13133196—28no—
59#13133196—28no—
60#13133191—22no—
61#13133191Q814yescomplemented
62C4×Dic106—28no—
63C4×Dic106—28no—
64C4×D212—22no—
65C4×D212—22no—
66C4×C22⋊D53—44no—
67#13133176—24no—
68#13133176—24no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2312yes—
74C2×C4⋊Dic53C2312yes—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—22no—
77C2×#11719919—22no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—22no—
80C2×C22⋊Dic53—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×C4×Dic53—22no—
86C2×C4×Dic53—21no—
87#13133151—24no—
88#13133149—24no—
89#13133139—24no—
90#13133139—24no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94C23×C4×C53D411yescharacteristic
95C23×C4×C53D411yescharacteristic
96C23×C4×C53C2311yescharacteristic
97C22×C8×C53C2312yes—
98C22×C8×C53C2312yes—
99C2×C42×C53—22no—
100C2×C42×C53—22no—
101C2×C42×C53D411yescharacteristic
102C2×C42×C53D411yescharacteristic
103C2×C42×C53C2311yescharacteristic
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—22no—
106C4⋊C8×C53—22no—
107C22⋊C8×C53—24no—
108C24×C53C2×D411yescharacteristic
109C23×D53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×Q812yes—
126C22×C4×C53#4112yes—
127C22×C4×C53C2411yescharacteristic
128C2×C22⋊D53—24no—
129C2×C22⋊D53—42no—
130C2×C22⋊D53—42no—
131C22×Dic53—21no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212C2×Q812yes—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C2×C4×D53—42no—
145C2×Dic106—24no—
146C2×Dic106—24no—
147C2×Dic106—42no—
148C2×Dic106—42no—
149C2×Dic106—42no—
150C2×Dic106—42no—
151C2×C8×C53—22no—
152C2×C8×C53—22no—
153C2×C8×C53—22no—
154C2×C8×C53—22no—
155C2×C8×C53—22no—
156C2×C8×C53—22no—
157C42×C53—22no—
158C42×C53—22no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53C2×D412yes—
162C42×C53C2×D412yes—
163C22⋊Dic53—28no—
164C22⋊Dic53—42no—
165#11719921—24no—
166#11719921—42no—
167#11719921—42no—
168#11719921—42no—
169C4⋊Dic53—24no—
170C4⋊Dic53—24no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53C2×D414yes—
179C4⋊Dic53C2×D414yes—
180C4⋊Dic53C2×Q812yes—
181#11719919—28no—
182#11719919—28no—
183#11719919—42no—
184C4×Dic53—28no—
185C4×Dic53—42no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53C22⋊Q812yes—
193C23×C53C22×D411yescharacteristic
194C22×D53—22no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—44no—
216C2×C4×C53—44no—
217C2×C4×C53#12012yes—
218C2×C4×C53#12012yes—
219C2×C4×C53C22⋊Q812yes—
220C2×C4×C53#13512yes—
221C2×C4×C53AGL(1,ℤ/8)12yes—
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C22×Q811yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×C4×C532−1+411yescharacteristic
228C22⋊D53—48no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—22no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—41no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244D212—22no—
245D212—22no—
246D212—41no—
247D212—41no—
248C4×D53—84no—
249Dic106—48no—
250Dic106—48no—
251C8×C53—42no—
252C8×C53—42no—
253#24300—531nomaximal, Sylow
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×#12011yescharacteristic
262C22×C53C2×C22⋊Q811yescharacteristic
263C22×C53#53611yescharacteristic
264C22×C53C2×#13511yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57511yescharacteristic
268C2×D53—42no—
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—81no—
272C2×D53—81no—
273C2×D53—81no—
274C2×D53—81no—
275C4×C53—22no—
276C4×C53—22no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53#44914yes—
280C4×C53#54811yescharacteristic
281C4×C53#54811yescharacteristic
282C4×C53D4×Q811yescharacteristic
283C4×C53D4×Q811yescharacteristic
284Dic53—42no—
285Dic53—44no—
286Dic53—44no—
287Dic53—81no—
288Dic53—81no—
289C2×#533—531no—
290C2×#476—531no—
291C2×#474—531no—
292C4×#121—531no—
293#2059—532no—
294#2025—532no—
295#1988—531no—
296#1987—531no—
297#1875—532no—
298#1854—532no—
299#1797D5311yescharacteristic, complemented
300C2×C53—42no—
301C2×C53#240911yescharacteristic
302C2×C53C2×#44911yescharacteristic
303C2×C53#300511yescharacteristic
304C2×C53#307611yescharacteristic
305C2×C53#307711yescharacteristic
306C2×C53#323311yescharacteristic
307C2×C53#323511yescharacteristic
308D53—82no—
309#533—1064no—
310#533—1064no—
311C2×C4⋊Q8—531no—
312C2×C4⋊Q8—531no—
313C2×#125—532no—
314C2×C22⋊Q8—1061no—
315C2×C22⋊Q8—1061no—
316C2×#121—531no—
317C2×C4×Q8—1061no—
318C2×C4×Q8—1061no—
319C2×C4×D4—531no—
320C2×C4×D4—1061no—
321C2×#117—532no—
322C22×C42C2×D5311yescharacteristic
323#476—1064no—
324#474—1064no—
325C2×#107—532no—
326C2×#106—532no—
327C2×C4⋊C8C2×D5311yescharacteristic
328C2×C4⋊C8C2×D5311yescharacteristic
329C2×#102—532no—
330C2×#102—532no—
331C2×C22⋊C8C2×D5312yes—
332#390—531no—
333#389—531no—
334#388—1061no—
335#386—1061no—
336#380—531no—
337C4×C4⋊C4—531no—
338C4×C22⋊C4—1061no—
339#340—532no—
340#340—532no—
341#340—532no—
342#340—532no—
343#336C2×D5312yes—
344C53#2430011yescharacteristic, complemented, Sylow
345C22×Q8—1061no—
346C22×Q8—1061no—
347C22×D4—531no—
348C22×D4—1061no—
349C23×C4—1061no—
350C23×C4D21211yescharacteristic
351C23×C4D21211yescharacteristic
352C23×C4C22×D5311yescharacteristic
353C22×C8C22×D5312yes—
354C22×C8C22×D5312yes—
355C4⋊Q8—534no—
356C4⋊Q8—534no—
357C4⋊Q8—1062no—
358C4⋊Q8—1062no—
359#125—1064no—
360#125—1064no—
361C22⋊Q8—1068no—
362C22⋊Q8—1068no—
363#121—534no—
364#121—1062no—
365C4×Q8—1068no—
366C4×Q8—1068no—
367C4×D4—1062no—
368C4×D4—1062no—
369C4×D4—2124no—
370#117—1064no—
371#117—1064no—
372C2×C4⋊C4—532no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—1062no—
391C2×C42—1061no—
392C2×C42D21211yescharacteristic
393C2×C42D21211yescharacteristic
394C2×C42C22×D5311yescharacteristic
395C2×C42C22×D5311yescharacteristic
396#107—1064no—
397#106—1064no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400#102—1064no—
401#102—1064no—
402C22⋊C8—24no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406C24—1061no—
407C24C2×D21211yescharacteristic
408C2×Q8—1064no—
409C2×Q8—1064no—
410C2×Q8—2122no—
411C2×Q8—2122no—
412C2×Q8—2122no—
413C2×Q8—2122no—
414C2×D4—532no—
415C2×D4—1064no—
416C2×D4—1061no—
417C2×D4—1061no—
418C2×D4—1061no—
419C2×D4—1061no—
420C2×D4—1061no—
421C2×D4—2122no—
422C2×D4—2122no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—1061no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—2122no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4C2×D21211yescharacteristic
442C22×C4C2×D21211yescharacteristic
443C22×C4C2×D21211yescharacteristic
444C22×C4C2×D21211yescharacteristic
445C22×C4C2×D21211yescharacteristic
446C22×C4#1171994712yes—
447C22×C4Q8×D5312yes—
448C22×C4C23×D5311yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C4⋊C4—534no—
456C4⋊C4—534no—
457C4⋊C4—532no—
458C4⋊C4—1068no—
459C4⋊C4—1068no—
460C4⋊C4—1064no—
461C4⋊C4—1064no—
462C4⋊C4—1062no—
463C4⋊C4—1062no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—2122no—
470C22⋊C4—1068no—
471C22⋊C4—1064no—
472C22⋊C4—2122no—
473C22⋊C4—2122no—
474C22⋊C4—2122no—
475C22⋊C4—2122no—
476C42—22no—
477C42—22no—
478C42—21no—
479C42—21no—
480C42—1068no—
481C42—2122no—
482C42D4×D5312yes—
483C42D4×D5312yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—1062no—
491C23—2121no—
492C23—2121no—
493C23—2121no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23#1313319712yes—
498C23C22×D21211yescharacteristic
499Q8—2128no—
500Q8—2128no—
501D4—1062no—
502D4—1062no—
503D4—2128no—
504D4—2121no—
505D4—2121no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1062no—
525C2×C4—4244no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2122no—
533C2×C4—2121no—
534C2×C4—2122no—
535C2×C4—2121no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4#1313318712yes—
539C2×C4#1313318712yes—
540C2×C4#1313319712yes—
541C2×C4#1313323912yes—
542C2×C4#1313324512yes—
543C2×C4C2×D4×D5311yescharacteristic
544C2×C4C2×D4×D5311yescharacteristic
545C2×C4C2×Q8×D5311yescharacteristic
546C2×C4C2×#1171994711yescharacteristic
547C2×C4#1313326411yescharacteristic
548C2×C4#1313326411yescharacteristic
549C8—42no—
550C8—42no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—4241no—
559C22—4241no—
560C22—4241no—
561C22—4241no—
562C22—2122no—
563C22—2122no—
564C22—2122no—
565C22C2×#1313318711yescharacteristic
566C22C2×#1313319711yescharacteristic
567C22#1681119911yescharacteristic
568C22C2×#1313323911yescharacteristic
569C22C2×#1313324511yescharacteristic
570C22#1681142311yescharacteristic
571C22#1681142311yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—4241no—
577C4—4241no—
578C4—2124no—
579C4—2124no—
580C4—2122no—
581C4#1681077014yes—
582C4#1681127911yescharacteristic
583C4#1681127911yescharacteristic
584C4#1681128611yescharacteristic
585C4#1681128611yescharacteristic
586C2—42no—
587C2—4242no—
588C2#2027808111yescharacteristic
589C2C2×#1681077011yescharacteristic
590C2#2028396011yescharacteristic
591C2#2028450511yescharacteristic
592C2#2028451411yescharacteristic
593C2#2028589611yescharacteristic
594C2#2028591511yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1797C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1797D53
hypocenterC53#24300
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
1. C2×#16811180C2semidirect
2. C2×#16810914C2semidirect
3. C2×#16810904C2semidirect
5. #20276054C2semidirect
6. #20275862C2semidirect
7. #20275666C2semidirect
8. #20275662C2semidirect
10. #20274916C2semidirect
11. C53×#1797C2semidirect
54. #13133217D4semidirect
55. #13133217D4semidirect
61. #13133191Q8semidirect
299. #1797D53semidirect
344. C53#24300semidirect

Maximal subgroups

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

subgroupindex
C2×#168111802
C2×#168109142
C2×#168109042
C4×#131331912
#202760542
#202758622
#202756662
#202756622
#202750352
#202749162
C53×#17972
#2430053