Finite Group Databasebeta

← #27420173

Subgroups — #27420173

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×#16810913C211yesmaximal, characteristic, complemented
3C2×#16810903C211yesmaximal, characteristic, complemented
4#20276053C212yesmaximal, complemented
5#20275861C212yesmaximal, complemented
6#20275667C211yesmaximal, characteristic, complemented
7#20275663C211yesmaximal, characteristic, complemented
8#20275038C212yesmaximal
9#20274912C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C4×#121×C53C211yesmaximal, characteristic
12#16811180—24no—
13#16811180—24no—
14C2×#13133217C2211yescharacteristic
15C2×#13133217C2211yescharacteristic
16C2×#13133209C2212yes—
17C2×#13133196—21no—
18C2×#13133196—21no—
19C2×C4×Dic106—21no—
20C2×C4×Dic106—21no—
21C2×#13133176C2212yes—
22#16810913—24no—
23#16810903—24no—
24C2×#13133150C2212yes—
25C2×#13133148C2212yes—
26C2×C4⋊#88322C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133138C2212yes—
29C2×#13133138C2212yes—
30C2×C22⋊#88322C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C2×#121×C53C2211yescharacteristic
37C2×C4×D4×C53—21no—
38C2×C4×D4×C53C2211yescharacteristic
39C22×C42×C53C2211yescharacteristic
40C53×#390C2211yescharacteristic
41C53×#389C2211yescharacteristic
42C53×#388—21no—
43C53×#386—21no—
44C53×#380C2211yescharacteristic
45C4×C4⋊C4×C53C2211yescharacteristic
46C4×C22⋊C4×C53—21no—
47C22×Dic106—21no—
48C22×Dic106—21no—
49C22×#88322C2312yes—
50C22×#88322C2312yes—
51#13133217—22no—
52#13133217—22no—
53#13133217D414yescomplemented
54#13133217D414yescomplemented
55#13133209—24no—
56#13133209—24no—
57#13133196—28no—
58#13133196—28no—
59C4×Dic106—28no—
60C4×Dic106—28no—
61#13133176—24no—
62#13133176—24no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2312yes—
66C2×C4⋊Dic53C2312yes—
67C2×#11719919—22no—
68C2×#11719919—22no—
69C2×C22⋊Dic53—22no—
70C2×C4×Dic53—22no—
71#13133150—24no—
72#13133148—24no—
73C4⋊#88322—22no—
74C4⋊#88322—22no—
75#13133138—24no—
76#13133138—24no—
77C22⋊#88322—24no—
78C22×D4×C53—21no—
79C22×D4×C53C2311yescharacteristic
80C23×C4×C53—21no—
81C23×C4×C53D411yescharacteristic
82C23×C4×C53D411yescharacteristic
83C23×C4×C53C2311yescharacteristic
84#121×C53—22no—
85#121×C53Q814yescomplemented
86C4×D4×C53—22no—
87C4×D4×C53—22no—
88C4×D4×C53—44no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53C2311yescharacteristic
92C2×C4⋊C4×C53C2311yescharacteristic
93C2×C22⋊C4×C53—21no—
94C2×C22⋊C4×C53—21no—
95C2×C22⋊C4×C53—21no—
96C2×C22⋊C4×C53—21no—
97C2×C22⋊C4×C53—21no—
98C2×C42×C53—22no—
99C2×C42×C53—22no—
100C2×C42×C53—21no—
101C2×C42×C53D411yescharacteristic
102C2×C42×C53D411yescharacteristic
103C2×C42×C53C2311yescharacteristic
104C2×C42×C53C2311yescharacteristic
105#95×C53—21no—
106#95×C53—21no—
107#95×C53—21no—
108C24×C53—21no—
109C24×C53C2×D411yescharacteristic
110C2×D4×C53—24no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—42no—
117C2×D4×C53—42no—
118C2×D4×C53C2×Q812yes—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×Q812yes—
141C22×C4×C53#4112yes—
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—22no—
144C22×Dic53—22no—
145C2×Dic106—24no—
146C2×Dic106—24no—
147C2×Dic106—42no—
148C2×Dic106—42no—
149C2×Dic106—42no—
150C2×Dic106—42no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53C2×Q812yes—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—42no—
162C22⋊C4×C53—42no—
163C42×C53—22no—
164C42×C53—22no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—42no—
168C42×C53C2×D412yes—
169C42×C53C2×D412yes—
170C22⋊Dic53—28no—
171C4⋊Dic53—24no—
172C4⋊Dic53—24no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53C2×D414yes—
176C4⋊Dic53C2×D414yes—
177#11719919—28no—
178#11719919—28no—
179C4×Dic53—28no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—22no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53C22⋊Q812yes—
200C23×C53C22×D411yescharacteristic
201D4×C53—22no—
202D4×C53—22no—
203D4×C53—48no—
204D4×C53—41no—
205D4×C53—41no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—44no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—44no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—84no—
232C2×C4×C53#12012yes—
233C2×C4×C53#12012yes—
234C2×C4×C53C22⋊Q812yes—
235C2×C4×C53#13512yes—
236C2×C4×C53AGL(1,ℤ/8)12yes—
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C22×Q811yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249Dic106—48no—
250Dic106—48no—
251#88322—42no—
252#88322—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×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53C2×#12011yescharacteristic
269C22×C53C2×C22⋊Q811yescharacteristic
270C22×C53#53611yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C53#57511yescharacteristic
274C22×C53#57511yescharacteristic
275C4×C53—22no—
276C4×C53—22no—
277C4×C53—42no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53#44914yes—
283C4×C53#54811yescharacteristic
284C4×C53#54811yescharacteristic
285C4×C53D4×Q811yescharacteristic
286C4×C53D4×Q811yescharacteristic
287Dic53—44no—
288Dic53—44no—
289C2×#533—531no—
290C2×#476—531no—
291C2×#474—531no—
292C4×#121D5311yescharacteristic
293#2059—532no—
294#2025—532no—
295#1988—531no—
296#1987—531no—
297#1875—532no—
298#1854—532no—
299#1797—531no—
300C2×C53—42no—
301C2×C53—82no—
302C2×C53#240911yescharacteristic
303C2×C53C2×#44911yescharacteristic
304C2×C53#300511yescharacteristic
305C2×C53#307611yescharacteristic
306C2×C53#307711yescharacteristic
307C2×C53#323311yescharacteristic
308C2×C53#323511yescharacteristic
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×#121C2×D5311yescharacteristic
317C2×C4×Q8—1061no—
318C2×C4×Q8—1061no—
319C2×C4×D4—21no—
320C2×C4×D4C2×D5311yescharacteristic
321C2×#117—532no—
322C22×C42C2×D5311yescharacteristic
323#476—1064no—
324#474—1064no—
325C2×#107—532no—
326C2×#106—532no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×#102—532no—
330C2×#102—532no—
331C2×C22⋊C8—532no—
332#390C2×D5311yescharacteristic
333#389C2×D5311yescharacteristic
334#388—21no—
335#386—21no—
336#380C2×D5311yescharacteristic
337C4×C4⋊C4C2×D5311yescharacteristic
338C4×C22⋊C4—21no—
339#340—532no—
340#340—532no—
341#340—532no—
342#340—532no—
343#336—532no—
344C53#2430011yescharacteristic, complemented, Sylow
345C22×Q8—1061no—
346C22×Q8—1061no—
347C22×D4—21no—
348C22×D4C22×D5311yescharacteristic
349C23×C4—21no—
350C23×C4C22⋊D5311yescharacteristic
351C23×C4C22⋊D5311yescharacteristic
352C23×C4C22×D5311yescharacteristic
353C22×C8—532no—
354C22×C8—532no—
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—22no—
364#121Dic10614yescomplemented
365C4×Q8—1068no—
366C4×Q8—1068no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—44no—
370#117—1064no—
371#117—1064no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1062no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—21no—
391C2×C42—1062no—
392C2×C42C22⋊D5311yescharacteristic
393C2×C42C22⋊D5311yescharacteristic
394C2×C42C22×D5311yescharacteristic
395C2×C42C22×D5311yescharacteristic
396#107—1064no—
397#106—1064no—
398C4⋊C8—1062no—
399C4⋊C8—1062no—
400#102—1064no—
401#102—1064no—
402C22⋊C8—1064no—
403#95—21no—
404#95—21no—
405#95—21no—
406C24—21no—
407C24C2×C22⋊D5311yescharacteristic
408C2×Q8—1064no—
409C2×Q8—1064no—
410C2×Q8—2122no—
411C2×Q8—2122no—
412C2×Q8—2122no—
413C2×Q8—2122no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4C2×Dic10612yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4C2×Dic10612yes—
442C22×C4#1171994512yes—
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C23×D5311yescharacteristic
449C2×C8—1062no—
450C2×C8—1062no—
451C2×C8—1062no—
452C2×C8—1062no—
453C2×C8—1062no—
454C2×C8—1062no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—534no—
462C4⋊C4—534no—
463C4⋊C4—1068no—
464C4⋊C4—1068no—
465C4⋊C4—1064no—
466C4⋊C4—1064no—
467C4⋊C4—1062no—
468C4⋊C4—1062no—
469C4⋊C4C2×Dic10612yes—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—1068no—
476C42—22no—
477C42—22no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—1068no—
482C42D4×D5312yes—
483C42D4×D5312yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#1313319612yes—
498C23C22×C22⋊D5311yescharacteristic
499Q8—2128no—
500Q8—2128no—
501D4—22no—
502D4—22no—
503D4—48no—
504D4—41no—
505D4—41no—
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—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—1064no—
533C2×C4—1064no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4#1313318512yes—
539C2×C4#1313318512yes—
540C2×C4#1313319612yes—
541C2×C4#1313323712yes—
542C2×C4#1313324312yes—
543C2×C4C2×D4×D5311yescharacteristic
544C2×C4C2×D4×D5311yescharacteristic
545C2×C4C22×Dic10611yescharacteristic
546C2×C4C2×#1171994511yescharacteristic
547C2×C4#1313326411yescharacteristic
548C2×C4#1313326411yescharacteristic
549C8—2122no—
550C8—2122no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C2×#1313318511yescharacteristic
566C22C2×#1313319611yescharacteristic
567C22#1681119411yescharacteristic
568C22C2×#1313323711yescharacteristic
569C22C2×#1313324311yescharacteristic
570C22#1681142111yescharacteristic
571C22#1681142111yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—2124no—
580C4—2124no—
581C4#1681076614yes—
582C4#1681128211yescharacteristic
583C4#1681128211yescharacteristic
584C4D4×Dic10611yescharacteristic
585C4D4×Dic10611yescharacteristic
586C2—42no—
587C2—82no—
588C2#2027808011yescharacteristic
589C2C2×#1681076611yescharacteristic
590C2#2028395511yescharacteristic
591C2#2028450311yescharacteristic
592C2#2028451211yescharacteristic
593C2#2028589011yescharacteristic
594C2#2028590911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×#121×C53C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC4×#121D53
hypocenterC53#24300
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16811180C2semidirect
2. C2×#16810913C2semidirect
3. C2×#16810903C2semidirect
4. #20276053C2semidirect
5. #20275861C2semidirect
6. #20275667C2semidirect
7. #20275663C2semidirect
9. #20274912C2semidirect
10. #20274695C2semidirect
53. #13133217D4semidirect
54. #13133217D4semidirect
85. #121×C53Q8semidirect
344. C53#24300semidirect
364. #121Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168111802
C2×#168109132
C2×#168109032
#202760532
#202758612
#202756672
#202756632
#202750382
#202749122
#202746952
C4×#121×C532
#2430053