Finite Group Databasebeta

← #20278088

Subgroups — #20278088

Order 6784.

1246 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D53C211yesmaximal, characteristic, complemented
2C2×#13133191C211yesmaximal, characteristic, complemented
3C2×#13133189C211yesmaximal, characteristic, complemented
4C2×#13133187C211yesmaximal, characteristic, complemented
5C2×#13133187C211yesmaximal, characteristic, complemented
6C22×#11719921C211yesmaximal, characteristic, complemented
7#16810541C211yesmaximal, characteristic, complemented
8#16810539C211yesmaximal, characteristic, complemented
9#16810533C211yesmaximal, characteristic, complemented
10#16810532C211yesmaximal, characteristic, complemented
11#16810501C211yesmaximal, characteristic, complemented
12#16810500C211yesmaximal, characteristic, complemented
13#16810497C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic, complemented
15C53×#386C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C24×D53C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C22×D212C2211yescharacteristic, complemented
21C22×D212C2211yescharacteristic, complemented
22C22×C22⋊D53C2211yescharacteristic
23C22×C22⋊D53C2211yescharacteristic
24C22×C22⋊D53C2211yescharacteristic, complemented
25C22×C22⋊D53C2211yescharacteristic, complemented
26C2×D4×D53—24no—
27C2×D4×D53—24no—
28C2×D4×D53—24no—
29C23×Dic53C2211yescharacteristic, complemented
30C22×C4×D53C2211yescharacteristic
31#13133191—24no—
32#13133189—24no—
33#13133187—24no—
34#13133187—24no—
35C2×C4⋊Dic53C2211yescharacteristic, complemented
36C2×#11719919C2211yescharacteristic, complemented
37C2×C22⋊Dic53C2211yescharacteristic, complemented
38C2×#11719921—24no—
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic, complemented
43C2×#11719921C2211yescharacteristic, complemented
44C2×#11719921C2211yescharacteristic, complemented
45C2×#11719921C2211yescharacteristic, complemented
46C2×#11719921C2211yescharacteristic, complemented
47C2×#11719921C2211yescharacteristic, complemented
48C2×#11719921C2211yescharacteristic, complemented
49C2×#11719921C2211yescharacteristic, complemented
50#13133124C2211yescharacteristic, complemented
51#13133124C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C23×C4×C53C2211yescharacteristic, complemented
54C2×C22⋊C4×C53C2211yescharacteristic, complemented
55C2×C22⋊C4×C53C2211yescharacteristic, complemented
56C2×C22⋊C4×C53C2211yescharacteristic, complemented
57#95×C53C2211yescharacteristic, complemented
58#95×C53C2211yescharacteristic, complemented
59C24×C53C2311yescharacteristic
60C24×C53C2311yescharacteristic
61C23×D53—22no—
62C23×D53—22no—
63C23×D53—22no—
64C23×D53—22no—
65C23×D53—22no—
66C23×D53—22no—
67C23×D53—22no—
68C23×D53—22no—
69C23×D53—22no—
70C23×D53—41no—
71C23×D53—41no—
72C23×D53—41no—
73C23×D53—41no—
74C23×D53D414yescomplemented
75C23×D53C2311yescharacteristic
76C23×D53C2311yescharacteristic
77C23×D53C2311yescharacteristic
78C23×D53C2311yescharacteristic
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53D414yescomplemented
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—41no—
85C22×C4×C53C2311yescharacteristic
86C22×C4×C53C2311yescharacteristic
87C22×C4×C53C2311yescharacteristic
88C22×C4×C53C2311yescharacteristic
89C22×C4×C53C2311yescharacteristic
90C2×C22⋊D53—22no—
91C2×C22⋊D53—22no—
92C2×C22⋊D53—22no—
93C2×C22⋊D53—22no—
94C2×C22⋊D53—22no—
95C2×C22⋊D53—22no—
96C2×C22⋊D53—22no—
97C2×C22⋊D53—22no—
98C2×C22⋊D53—22no—
99C2×C22⋊D53—22no—
100C2×C22⋊D53—41no—
101C2×C22⋊D53D414yescomplemented
102C22×Dic53—22no—
103C22×Dic53—22no—
104C22×Dic53—41no—
105C22×Dic53C2311yescharacteristic
106C22×Dic53C2311yescharacteristic
107C22×Dic53C2311yescharacteristic
108C22×Dic53C2311yescharacteristic
109D4×D53—232no—
110C2×D212—22no—
111C2×D212—22no—
112C2×D212—22no—
113C2×D212—22no—
114C2×D212—22no—
115C2×D212—22no—
116C2×D212—22no—
117C2×D212—22no—
118C2×D212—22no—
119C2×D212—41no—
120C2×D212—41no—
121C2×D212—41no—
122C2×C4×D53—22no—
123C2×C4×D53—22no—
124C2×C4×D53—22no—
125C22⋊C4×C53—22no—
126C22⋊C4×C53—22no—
127C22⋊C4×C53—41no—
128C22⋊Dic53—22no—
129#11719921—22no—
130#11719921—22no—
131#11719921—22no—
132#11719921—22no—
133#11719921—22no—
134#11719921—22no—
135#11719921—22no—
136#11719921—22no—
137#11719921—22no—
138#11719921—41no—
139#11719921D414yescomplemented
140C4⋊Dic53—22no—
141#11719919—22no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53C2×D412yes—
153C23×C53C2×D412yes—
154C23×C53C2×D412yes—
155C23×C53C2×D412yes—
156C23×C53C2411yescharacteristic
157C22×D53—21no—
158C22×D53—21no—
159C22×D53—21no—
160C22×D53—21no—
161C22×D53—28no—
162C22×D53—28no—
163C22×D53—21no—
164C22×D53—21no—
165C22×D53—21no—
166C22×D53—21no—
167C22×D53—21no—
168C22×D53—21no—
169C22×D53—21no—
170C22×D53—21no—
171C22×D53—21no—
172C22×D53—21no—
173C22×D53—21no—
174C22×D53—21no—
175C22×D53—28no—
176C22×D53—21no—
177C22×D53—21no—
178C22×D53—21no—
179C22×D53—21no—
180C22×D53—21no—
181C22×D53—21no—
182C22×D53—21no—
183C22×D53—44no—
184C22×D53—44no—
185C22×D53—44no—
186C22×D53—44no—
187C22×D53—44no—
188C22×D53—44no—
189C22×D53—44no—
190C22×D53—44no—
191C22×D53—44no—
192C22×D53—44no—
193C22×D53—44no—
194C22×D53C2×D412yes—
195C22×D53C2×D412yes—
196C22×D53C2×D412yes—
197C22×D53C2×D412yes—
198C22×D53#4112yes—
199D4×C53—28no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—44no—
214C2×C4×C53C2×D412yes—
215C2×C4×C53C2×D412yescomplemented
216C22⋊D53—28no—
217C22⋊D53—28no—
218C22⋊D53—44no—
219C22⋊D53—44no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—44no—
231C2×Dic53C2×D412yes—
232C2×Dic53C2×D412yescomplemented
233D212—28no—
234D212—44no—
235D212—44no—
236D212—44no—
237C4×D53—28no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53#12014yescomplemented
252C22×C53C22×D411yescharacteristic
253C22×C53C22×D411yescharacteristic
254C22×C53C22×D411yescharacteristic
255C22×C53C22×D411yescharacteristic
256C22×C53C2×#4111yescharacteristic
257C22×C53H(5,𝔽2)11yescharacteristic
258C22×C53H(5,𝔽2)11yescharacteristic
259C2×D53—22no—
260C2×D53—22no—
261C2×D53—22no—
262C2×D53—22no—
263C2×D53—22no—
264C2×D53—22no—
265C2×D53—22no—
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53—416no—
272C2×D53—416no—
273C2×D53—41no—
274C2×D53—41no—
275C2×D53—41no—
276C2×D53—41no—
277C2×D53—41no—
278C2×D53—41no—
279C2×D53—41no—
280C2×D53—41no—
281C2×D53—41no—
282C2×D53—41no—
283C2×D53—41no—
284C2×D53—41no—
285C2×D53—41no—
286C2×D53—41no—
287C2×D53—41no—
288C2×D53#12114yescomplemented
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—41no—
292C4×C53—41no—
293C4×C53—41no—
294Dic53—22no—
295Dic53—22no—
296Dic53—41no—
297Dic53—41no—
298#2410—531nomaximal, Sylow
299C2×C53—24no—
300C2×C53—42no—
301C2×C53C2×#12011yescharacteristic
302C2×C53C2×#12111yescharacteristic
303C2×C53#53411yescharacteristic
304C2×C53D4211yescharacteristic
305C2×C53D4211yescharacteristic
306C2×C53#54611yescharacteristic
307C2×C53#54611yescharacteristic
308D53—24no—
309D53—42no—
310D53—42no—
311D53—81no—
312C23×D4—531no—
313C2×#121—531no—
314C2×#121—531no—
315C2×#120—531no—
316C2×#120—531no—
317C22×C22⋊C4—531no—
318#394—531no—
319#394—531no—
320#392—531no—
321#392—531no—
322#386—531no—
323#386—531no—
324#386D5311yescharacteristic, complemented
325#385—531no—
326#379—531no—
327C53#241011yescharacteristic, complemented, Sylow
328C25—531no—
329C25—531no—
330C22×D4—531no—
331C22×D4—531no—
332C22×D4—531no—
333C22×D4—531no—
334C22×D4—531no—
335C22×D4—531no—
336C22×D4—531no—
337C22×D4—531no—
338C22×D4—1064no—
339C22×D4—1064no—
340C22×D4—1064no—
341C22×D4C2×D5311yescharacteristic
342C23×C4—531no—
343C23×C4—531no—
344C23×C4C2×D5311yescharacteristic, complemented
345#121—1064no—
346#121—1064no—
347#120—1064no—
348#120—1064no—
349C2×C4⋊C4—531no—
350C2×C4⋊C4—531no—
351C2×C22⋊C4—531no—
352C2×C22⋊C4—531no—
353C2×C22⋊C4—531no—
354C2×C22⋊C4—531no—
355C2×C22⋊C4—531no—
356C2×C22⋊C4—531no—
357C2×C22⋊C4—531no—
358C2×C22⋊C4—531no—
359C2×C22⋊C4—531no—
360C2×C22⋊C4—531no—
361C2×C22⋊C4—531no—
362C2×C22⋊C4—531no—
363C2×C22⋊C4—1064no—
364C2×C22⋊C4C2×D5311yescharacteristic, complemented
365C2×C22⋊C4C2×D5311yescharacteristic, complemented
366C2×C22⋊C4C2×D5311yescharacteristic, complemented
367#95—531no—
368#95—531no—
369#95C2×D5311yescharacteristic, complemented
370#95C2×D5311yescharacteristic, complemented
371C24—531no—
372C24—531no—
373C24—531no—
374C24—534no—
375C24—531no—
376C24—1062no—
377C24—1062no—
378C24—1062no—
379C24—1062no—
380C24—1062no—
381C24—1062no—
382C24—1062no—
383C24—1062no—
384C24—1062no—
385C24—2121no—
386C24—2121no—
387C24—2121no—
388C24—2121no—
389C24C22×D5311yescharacteristic
390C24C22×D5311yescharacteristic
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—534no—
394C2×D4—1062no—
395C2×D4—1062no—
396C2×D4—1062no—
397C2×D4—1062no—
398C2×D4—1062no—
399C2×D4—1062no—
400C2×D4—1062no—
401C2×D4—1062no—
402C2×D4—1062no—
403C2×D4—10632no—
404C2×D4—1062no—
405C2×D4—1062no—
406C2×D4—1062no—
407C2×D4—1062no—
408C2×D4—1062no—
409C2×D4—1062no—
410C2×D4—1062no—
411C2×D4—1062no—
412C2×D4—1062no—
413C2×D4—1062no—
414C2×D4—2121no—
415C2×D4—2121no—
416C2×D4—2121no—
417C2×D4—2121no—
418C2×D4D21214yescomplemented
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—2121no—
432C22×C4C22×D5311yescharacteristic
433C22×C4C22×D5311yescharacteristic
434C22×C4C22×D5311yescharacteristic
435C22×C4C22×D5311yescharacteristic
436C22×C4C22×D5311yescharacteristic
437C4⋊C4—1062no—
438C4⋊C4—1062no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—41no—
442C22⋊C4—534no—
443C22⋊C4—1062no—
444C22⋊C4—1062no—
445C22⋊C4—1062no—
446C22⋊C4—1062no—
447C22⋊C4—1062no—
448C22⋊C4—1062no—
449C22⋊C4—1062no—
450C22⋊C4—1062no—
451C22⋊C4—1062no—
452C22⋊C4—1062no—
453C22⋊C4—2121no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—532no—
465C23—532no—
466C23—532no—
467C23—532no—
468C23—532no—
469C23—1068no—
470C23—1068no—
471C23—1061no—
472C23—1061no—
473C23—1061no—
474C23—1061no—
475C23—1068no—
476C23—1061no—
477C23—1061no—
478C23—1061no—
479C23—1061no—
480C23—1061no—
481C23—1061no—
482C23—1061no—
483C23—1061no—
484C23—1061no—
485C23—1061no—
486C23—1061no—
487C23—1061no—
488C23—1061no—
489C23—1061no—
490C23—1061no—
491C23—1061no—
492C23—1061no—
493C23—1061no—
494C23—1061no—
495C23—2124no—
496C23—2124no—
497C23—2124no—
498C23—2124no—
499C23—2124no—
500C23—2124no—
501C23—2124no—
502C23—2124no—
503C23—2124no—
504C23—2124no—
505C23—2124no—
506C23C2×D21212yes—
507C23C2×D21212yes—
508C23D4×D5312yes—
509C23D4×D5312yes—
510C23C23×D5311yescharacteristic
511D4—28no—
512D4—1068no—
513D4—1068no—
514D4—1068no—
515D4—2124no—
516D4—2124no—
517D4—2124no—
518D4—2124no—
519D4—2124no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—44no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—1068no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—2124no—
548C2×C4C2×D21212yes—
549C2×C4D4×D5312yescomplemented
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—534no—
564C22—1062no—
565C22—1062no—
566C22—1062no—
567C22—1062no—
568C22—1062no—
569C22—1062no—
570C22—1062no—
571C22—1062no—
572C22—1062no—
573C22—1062no—
574C22—1062no—
575C22—1062no—
576C22—21216no—
577C22—21216no—
578C22—2121no—
579C22—2121no—
580C22—2121no—
581C22—2121no—
582C22—2121no—
583C22—2121no—
584C22—2121no—
585C22—2121no—
586C22—2121no—
587C22—2121no—
588C22—2121no—
589C22—2121no—
590C22—2121no—
591C22—2121no—
592C22—2121no—
593C22#1313318714yescomplemented
594C22C2×D4×D5311yescharacteristic
595C22C2×D4×D5311yescharacteristic
596C22C2×D4×D5311yescharacteristic
597C22C22×D21211yescharacteristic
598C22#41×D5311yescharacteristic
599C22#1313326211yescharacteristic
600C22#1313326311yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—1062no—
607C4—1062no—
608C4—2121no—
609C4—2121no—
610C2—24no—
611C2—42no—
612C2—1064no—
613C2—4241no—
614C2—2122no—
615C2—2122no—
616C2C2×#1313318711yescharacteristic
617C2#121×D5311yescharacteristic
618C2#1681118111yescharacteristic
619C2#1681125511yescharacteristic
620C2D4×D21211yescharacteristic
621C2#1681126511yescharacteristic
622C2#1681126811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2410
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×D4×D53C2semidirect
2. C2×#13133191C2semidirect
3. C2×#13133189C2semidirect
4. C2×#13133187C2semidirect
5. C2×#13133187C2semidirect
6. C22×#11719921C2semidirect
7. #16810541C2semidirect
8. #16810539C2semidirect
9. #16810533C2semidirect
10. #16810532C2semidirect
11. #16810501C2semidirect
12. #16810500C2semidirect
13. #16810497C2semidirect
14. #16810478C2semidirect
15. C53×#386C2semidirect
20. C22×D212C22semidirect
21. C22×D212C22semidirect
24. C22×C22⋊D53C22semidirect
25. C22×C22⋊D53C22semidirect
29. C23×Dic53C22semidirect
35. C2×C4⋊Dic53C22semidirect
36. C2×#11719919C22semidirect
37. C2×C22⋊Dic53C22semidirect
42. C2×#11719921C22semidirect
43. C2×#11719921C22semidirect
44. C2×#11719921C22semidirect
45. C2×#11719921C22semidirect
46. C2×#11719921C22semidirect
47. C2×#11719921C22semidirect
48. C2×#11719921C22semidirect
49. C2×#11719921C22semidirect
50. #13133124C22semidirect
51. #13133124C22semidirect
53. C23×C4×C53C22semidirect
54. C2×C22⋊C4×C53C22semidirect
55. C2×C22⋊C4×C53C22semidirect
56. C2×C22⋊C4×C53C22semidirect
57. #95×C53C22semidirect
58. #95×C53C22semidirect
74. C23×D53D4semidirect
81. C2×D4×C53D4semidirect
101. C2×C22⋊D53D4semidirect
139. #11719921D4semidirect
215. C2×C4×C53C2×D4semidirect
232. C2×Dic53C2×D4semidirect
251. C22×C53#120semidirect
288. C2×D53#121semidirect
324. #386D53semidirect
327. C53#2410semidirect
344. C23×C4C2×D53semidirect
364. C2×C22⋊C4C2×D53semidirect
365. C2×C22⋊C4C2×D53semidirect
366. C2×C22⋊C4C2×D53semidirect
369. #95C2×D53semidirect
370. #95C2×D53semidirect
418. C2×D4D212semidirect
549. C2×C4D4×D53semidirect
593. C22#13133187semidirect