Finite Group Databasebeta

← #27416721

Subgroups — #27416721

Order 13568.

1844 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810934C212yesmaximal, complemented
2C2×#16810483C211yesmaximal, characteristic, complemented
3#20275894C214yesmaximal, complemented
4#20275033C212yesmaximal, complemented
5#20274903C212yesmaximal, complemented
6#20274646C212yesmaximal, complemented
7#20274625C211yesmaximal, characteristic, complemented
8C22×#121×C53C211yesmaximal, characteristic
9C22×C4⋊Dic53C2211yescharacteristic, complemented
10C22×C4⋊Dic53C2211yescharacteristic, complemented
11C22×C22⋊Dic53—21no—
12#16810934—28no—
13C2×#13133150C2214yescomplemented
14C2×#13133138C2214yescomplemented
15C2×#13133138C2214yescomplemented
16C2×C22⋊#88322C2212yes—
17C2×C22⋊#88322C2212yescomplemented
18C2×C22⋊#88322C2212yescomplemented
19#16810483—21no—
20#16810483—21no—
21#16810483C2212yes—
22#16810483C2212yescomplemented
23C2×#13133124—21no—
24#16810385C2214yescomplemented
25C23×D4×C53—21no—
26C23×D4×C53C2211yescharacteristic
27C24×C4×C53C2211yescharacteristic
28C2×#121×C53—21no—
29C2×#121×C53—21no—
30C2×#121×C53—28no—
31C2×#121×C53C2212yes—
32C2×#121×C53C2212yes—
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53—21no—
35C23×Dic53—21no—
36C23×Dic53—21no—
37C22×#88322C2314yes—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—24no—
41C2×C4⋊Dic53—42no—
42C2×C4⋊Dic53C2312yes—
43C2×C4⋊Dic53C2312yes—
44C2×C22⋊Dic53—22no—
45C2×C22⋊Dic53—22no—
46C2×C22⋊Dic53—44no—
47#13133150—28no—
48#13133138—28no—
49#13133138—44no—
50C22⋊#88322—24no—
51C22⋊#88322—24no—
52C22⋊#88322—42no—
53#13133124—22no—
54#13133124—22no—
55C25×C53—21no—
56C25×C53D411yescharacteristic
57C22×D4×C53—28no—
58C22×D4×C53—22no—
59C22×D4×C53—22no—
60C22×D4×C53—24no—
61C22×D4×C53—21no—
62C22×D4×C53—24no—
63C22×D4×C53—42no—
64C22×D4×C53—44no—
65C22×D4×C53—44no—
66C22×D4×C53C2312yes—
67C23×C4×C53—21no—
68C23×C4×C53—24no—
69C23×C4×C53—24no—
70C23×C4×C53—42no—
71C23×C4×C53D411yescharacteristic
72C23×C4×C53D411yescharacteristic
73C23×C4×C53D411yescharacteristic
74C23×C4×C53C2311yescharacteristic
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77#121×C53—28no—
78#121×C53—28no—
79#121×C53—24no—
80#121×C53—24no—
81#121×C53—24no—
82#121×C53D418yes—
83C2×C4⋊C4×C53—24no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53C2312yes—
86C2×C22⋊C4×C53—28no—
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53—22no—
89C24×C53—22no—
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—24no—
94C24×C53—22no—
95C24×C53—41no—
96C24×C53—41no—
97C24×C53—42no—
98C24×C53—42no—
99C24×C53—42no—
100C24×C53—42no—
101C24×C53—42no—
102C24×C53—42no—
103C24×C53D814yes—
104C24×C53C2×D411yescharacteristic
105C24×C53C2×D411yescharacteristic
106C24×C53C2×D411yescharacteristic
107C2×D4×C53—24no—
108C2×D4×C53—28no—
109C2×D4×C53—28no—
110C2×D4×C53—22no—
111C2×D4×C53—24no—
112C2×D4×C53—24no—
113C2×D4×C53—416no—
114C2×D4×C53—42no—
115C2×D4×C53—432no—
116C2×D4×C53—44no—
117C2×D4×C53—44no—
118C2×D4×C53—44no—
119C2×D4×C53—44no—
120C2×D4×C53—42no—
121C2×D4×C53C2×D414yes—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—24no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—416no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D414yes—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53#4114yes—
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53C2×D414yes—
160C22⋊C4×C53—28no—
161C22⋊C4×C53—28no—
162C22⋊Dic53—44no—
163C22⋊Dic53—44no—
164C4⋊Dic53—24no—
165C4⋊Dic53—24no—
166C4⋊Dic53—42no—
167C4⋊Dic53—42no—
168C2×#88322—24no—
169C2×#88322—24no—
170C2×#88322—24no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—28no—
186C23×C53—28no—
187C23×C53—24no—
188C23×C53—44no—
189C23×C53—44no—
190C23×C53—44no—
191C23×C53—44no—
192C23×C53—44no—
193C23×C53—48no—
194C23×C53—48no—
195C23×C53—48no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—84no—
203C23×C53—84no—
204C23×C53—84no—
205C23×C53—84no—
206C23×C53#12012yes—
207C23×C53#12312yes—
208C23×C53C2×D812yes—
209C23×C53C2×D812yes—
210C23×C53C2×D812yes—
211C23×C53AGL(1,ℤ/8)12yes—
212C23×C53#13712yes—
213C23×C53C22×D411yescharacteristic
214D4×C53—28no—
215D4×C53—48no—
216D4×C53—416no—
217D4×C53—416no—
218D4×C53—44no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—28no—
224C2×C4×C53—28no—
225C2×C4×C53—21no—
226C2×C4×C53—24no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—44no—
230C2×C4×C53—44no—
231C2×C4×C53—44no—
232C2×C4×C53—48no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—44no—
236C2×C4×C53#12012yes—
237C2×C4×C53#12312yes—
238C2×C4×C53C22×D412yes—
239C2×C4×C53C2×#4112yes—
240C2×C4×C53H(5,𝔽2)12yes—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—84no—
248C2×Dic53—84no—
249#88322—44no—
250#24002—531nomaximal, Sylow
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—416no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—44no—
272C22×C53—44no—
273C22×C53—44no—
274C22×C53—816no—
275C22×C53—82no—
276C22×C53—82no—
277C22×C53—82no—
278C22×C53—82no—
279C22×C53#44714yes—
280C22×C53#48014yescomplemented
281C22×C53C2×#12011yescharacteristic
282C22×C53C2×#12311yescharacteristic
283C22×C53#53411yescharacteristic
284C22×C53C22×D811yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53C2×#13711yescharacteristic
287C22×C53#57511yescharacteristic
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—41no—
291C4×C53—44no—
292C4×C53#54614yes—
293Dic53—82no—
294Dic53—82no—
295C22×#121D5311yescharacteristic
296C2×#480—532no—
297C2×#380—531no—
298#2033—534no—
299#1874—532no—
300#1852—532no—
301#1786—532no—
302#1779—531no—
303C2×C53—24no—
304C2×C53—24no—
305C2×C53—42no—
306C2×C53—84no—
307C2×C53#240511yescharacteristic
308C2×C53C2×#44711yescharacteristic
309C2×C53#300711yescharacteristic
310C2×C53C2×#48011yescharacteristic
311C2×C53#309411yescharacteristic
312C2×C53#318611yescharacteristic
313C2×C53#319311yescharacteristic
314C23×D4—21no—
315C23×D4C2×D5311yescharacteristic
316C24×C4C2×D5311yescharacteristic
317C2×#121—21no—
318C2×#121—21no—
319C2×#121—28no—
320C2×#121C2×D5312yes—
321C2×#121C2×D5312yes—
322C22×C4⋊C4—531no—
323C22×C4⋊C4—531no—
324C22×C4⋊C4C2×D5311yescharacteristic
325C22×C22⋊C4—21no—
326C22×C22⋊C4—1061no—
327#480—1068no—
328C2×#107—534no—
329C2×#102—534no—
330C2×#102—534no—
331C2×C22⋊C8—532no—
332C2×C22⋊C8—532no—
333C2×C22⋊C8—532no—
334#380—532no—
335#380—532no—
336#380—1061no—
337#380—1061no—
338C2×#95—1061no—
339#340—534no—
340C53#2400211yescharacteristic, complemented, Sylow
341C25—21no—
342C25C22⋊D5311yescharacteristic
343C22×D4—28no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—24no—
347C22×D4—24no—
348C22×D4—21no—
349C22×D4—42no—
350C22×D4—44no—
351C22×D4—44no—
352C22×D4C22×D5312yes—
353C23×C4—21no—
354C23×C4—24no—
355C23×C4—24no—
356C23×C4—42no—
357C23×C4—1061no—
358C23×C4—1061no—
359C23×C4C22⋊D5311yescharacteristic
360C23×C4C22⋊D5311yescharacteristic
361C23×C4C22⋊D5311yescharacteristic
362C23×C4C22×D5311yescharacteristic
363C23×C4C22×D5311yescharacteristic
364C23×C4C22×D5311yescharacteristic
365C22×C8—534no—
366#121—28no—
367#121—28no—
368#121—24no—
369#121—24no—
370#121—24no—
371#121C22⋊D5318yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—532no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—2122no—
380C2×C4⋊C4C22×D5312yes—
381C2×C22⋊C4—28no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1062no—
385C2×C22⋊C4—1062no—
386C2×C22⋊C4—2124no—
387#107—1068no—
388#102—1068no—
389#102—2124no—
390C22⋊C8—1064no—
391C22⋊C8—1064no—
392C22⋊C8—2122no—
393#95—1062no—
394#95—1062no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—24no—
400C24—22no—
401C24—41no—
402C24—41no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24#1171992214yes—
410C24C2×C22⋊D5311yescharacteristic
411C24C2×C22⋊D5311yescharacteristic
412C24C2×C22⋊D5311yescharacteristic
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—416no—
420C2×D4—42no—
421C2×D4—432no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4C2×C22⋊D5314yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—416no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4#1171994714yes—
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5314yes—
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C23×D5311yescharacteristic
463C2×C8—1064no—
464C2×C8—1064no—
465C2×C8—1064no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—1064no—
469C4⋊C4—1064no—
470C4⋊C4—2122no—
471C4⋊C4—2122no—
472C4⋊C4C2×C22⋊D5314yes—
473C22⋊C4—28no—
474C22⋊C4—28no—
475C22⋊C4—2124no—
476C22⋊C4—2124no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—28no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—28no—
493C23—24no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23#1313318612yes—
513C23#1313319812yes—
514C23C2×#1171992212yes—
515C23C2×#1171992212yes—
516C23C2×#1171992212yes—
517C23#1313324312yes—
518C23#1313324712yes—
519C23C22×C22⋊D5311yescharacteristic
520D4—28no—
521D4—48no—
522D4—416no—
523D4—416no—
524D4—44no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—4244no—
543C2×C4—4244no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4#1313318612yes—
551C2×C4#1313319812yes—
552C2×C4C22×C22⋊D5312yes—
553C2×C4C2×#1171994712yes—
554C2×C4#1313326212yes—
555C8—2124no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—416no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—816no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22#1681075714yes—
585C22#1681093414yescomplemented
586C22C2×#1313318611yescharacteristic
587C22C2×#1313319811yescharacteristic
588C22#1681118211yescharacteristic
589C22C22×#1171992211yescharacteristic
590C22C2×#1313324311yescharacteristic
591C22C2×#1313324711yescharacteristic
592C22#1681142111yescharacteristic
593C4—22no—
594C4—22no—
595C4—41no—
596C4—44no—
597C4—4242no—
598C4—4242no—
599C4#1681126714yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2—84no—
604C2#2027803811yescharacteristic
605C2C2×#1681075711yescharacteristic
606C2#2028397511yescharacteristic
607C2C2×#1681093411yescharacteristic
608C2#2028465111yescharacteristic
609C2#2028546311yescharacteristic
610C2#2028552111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC22×#121×C53C2
socleC23×C53C22×D4
hypercenterC22×#121D53
hypocenterC53#24002
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16810934C2semidirect
2. C2×#16810483C2semidirect
3. #20275894C2semidirect
4. #20275033C2semidirect
5. #20274903C2semidirect
6. #20274646C2semidirect
7. #20274625C2semidirect
9. C22×C4⋊Dic53C22semidirect
10. C22×C4⋊Dic53C22semidirect
13. C2×#13133150C22semidirect
14. C2×#13133138C22semidirect
15. C2×#13133138C22semidirect
17. C2×C22⋊#88322C22semidirect
18. C2×C22⋊#88322C22semidirect
22. #16810483C22semidirect
24. #16810385C22semidirect
280. C22×C53#480semidirect
340. C53#24002semidirect
585. C22#16810934semidirect

Maximal subgroups

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

subgroupindex
C2×#168109342
C2×#168104832
#202758942
#202750332
#202749032
#202746462
#202746252
C22×#121×C532
#2400253