Finite Group Databasebeta

← #20278086

Subgroups — #20278086

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×#13133189C211yesmaximal, characteristic, complemented
3C2×#13133185C211yesmaximal, characteristic, complemented
4C2×#13133185C211yesmaximal, characteristic, complemented
5C22×#11719921C211yesmaximal, characteristic, complemented
6#16810540C211yesmaximal, characteristic, complemented
7#16810538C211yesmaximal, characteristic, complemented
8#16810532C211yesmaximal, characteristic, complemented
9#16810531C211yesmaximal, characteristic, complemented
10#16810503C211yesmaximal, characteristic, complemented
11#16810502C211yesmaximal, characteristic, complemented
12#16810499C211yesmaximal, characteristic, complemented
13#16810494C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic, complemented
15C2×#121×C53C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C24×D53C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C22⋊D53C2211yescharacteristic, complemented
23C22×C22⋊D53C2211yescharacteristic, complemented
24C2×D4×D53—24no—
25C2×D4×D53—24no—
26C2×D4×D53—24no—
27C23×Dic53C2211yescharacteristic, complemented
28C22×C4×D53C2211yescharacteristic
29#13133189—24no—
30#13133185—24no—
31#13133185—24no—
32C2×#11719919C2211yescharacteristic, complemented
33C2×C22⋊Dic53C2211yescharacteristic, complemented
34C2×C22⋊Dic53C2211yescharacteristic, complemented
35C2×C22⋊Dic53C2211yescharacteristic, complemented
36C2×C22⋊Dic53C2211yescharacteristic, complemented
37C2×#11719921—24no—
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic, complemented
42C2×#11719921C2211yescharacteristic, complemented
43C2×#11719921C2211yescharacteristic, complemented
44C2×#11719921C2211yescharacteristic, complemented
45C2×#11719921C2211yescharacteristic, complemented
46C2×#11719921C2211yescharacteristic, complemented
47#13133124C2211yescharacteristic, complemented
48#13133124C2211yescharacteristic, complemented
49#13133124C2211yescharacteristic, complemented
50#13133124C2211yescharacteristic, complemented
51C22×D4×C53C2211yescharacteristic
52C22×D4×C53C2211yescharacteristic, complemented
53C22×D4×C53C2211yescharacteristic, complemented
54C23×C4×C53C2211yescharacteristic, complemented
55#121×C53—24no—
56C2×C4⋊C4×C53C2211yescharacteristic, complemented
57C2×C22⋊C4×C53C2211yescharacteristic, complemented
58C2×C22⋊C4×C53C2211yescharacteristic, complemented
59C24×C53C2311yescharacteristic
60C24×C53C2311yescharacteristic
61C24×C53C2311yescharacteristic
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—22no—
71C23×D53—41no—
72C23×D53—41no—
73C23×D53—41no—
74C23×D53—41no—
75C23×D53D414yescomplemented
76C23×D53C2311yescharacteristic
77C23×D53C2311yescharacteristic
78C23×D53C2311yescharacteristic
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—41no—
87C2×D4×C53D414yescomplemented
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—41no—
91C22×C4×C53C2311yescharacteristic
92C22×C4×C53C2311yescharacteristic
93C22×C4×C53C2311yescharacteristic
94C22×C4×C53C2311yescharacteristic
95C2×C22⋊D53—22no—
96C2×C22⋊D53—22no—
97C2×C22⋊D53—22no—
98C2×C22⋊D53—22no—
99C2×C22⋊D53—22no—
100C2×C22⋊D53—22no—
101C2×C22⋊D53—22no—
102C2×C22⋊D53—22no—
103C2×C22⋊D53—22no—
104C2×C22⋊D53—22no—
105C2×C22⋊D53—22no—
106C2×C22⋊D53—41no—
107C2×C22⋊D53—41no—
108C2×C22⋊D53—41no—
109C2×C22⋊D53D414yescomplemented
110C22×Dic53—22no—
111C22×Dic53—22no—
112C22×Dic53—41no—
113C22×Dic53C2311yescharacteristic
114C22×Dic53C2311yescharacteristic
115C22×Dic53C2311yescharacteristic
116C22×Dic53C2311yescharacteristic
117C22×Dic53C2311yescharacteristic
118D4×D53—232no—
119C2×D212—22no—
120C2×D212—22no—
121C2×D212—22no—
122C2×C4×D53—22no—
123C2×C4×D53—22no—
124C2×C4×D53—22no—
125C4⋊C4×C53—22no—
126C22⋊C4×C53—22no—
127C22⋊C4×C53—22no—
128C22⋊Dic53—22no—
129C22⋊Dic53—22no—
130C22⋊Dic53—22no—
131C22⋊Dic53—41no—
132#11719921—22no—
133#11719921—22no—
134#11719921—22no—
135#11719921—22no—
136#11719921—22no—
137#11719921—22no—
138#11719921—22no—
139#11719921—41no—
140#11719921D414yescomplemented
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×C53—21no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53C2×D412yes—
160C23×C53C2×D412yes—
161C23×C53C2×D412yes—
162C23×C53C2×D412yes—
163C23×C53C2411yescharacteristic
164C22×D53—21no—
165C22×D53—21no—
166C22×D53—21no—
167C22×D53—21no—
168C22×D53—28no—
169C22×D53—28no—
170C22×D53—21no—
171C22×D53—21no—
172C22×D53—21no—
173C22×D53—21no—
174C22×D53—21no—
175C22×D53—21no—
176C22×D53—21no—
177C22×D53—21no—
178C22×D53—21no—
179C22×D53—21no—
180C22×D53—21no—
181C22×D53—21no—
182C22×D53—28no—
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—
200D4×C53—44no—
201D4×C53—44no—
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—44no—
213C2×C4×C53C2×D412yes—
214C2×C4×C53C2×D412yescomplemented
215C22⋊D53—28no—
216C22⋊D53—28no—
217C22⋊D53—44no—
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—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—44no—
234C2×Dic53C2×D412yes—
235C2×Dic53C2×D412yescomplemented
236D212—28no—
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—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53#12014yescomplemented
259C22×C53C22×D411yescharacteristic
260C22×C53C22×D411yescharacteristic
261C22×C53C22×D411yescharacteristic
262C22×C53C22×D411yescharacteristic
263C22×C53C2×#4111yescharacteristic
264C22×C53H(5,𝔽2)11yescharacteristic
265C22×C53H(5,𝔽2)11yescharacteristic
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53—22no—
272C2×D53—22no—
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—416no—
279C2×D53—416no—
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—
293Dic53—22no—
294Dic53—22no—
295Dic53—41no—
296Dic53—41no—
297Dic53—41no—
298#2410—531nomaximal, Sylow
299C2×C53—24no—
300C2×C53—42no—
301C2×C53—81no—
302C2×C53C2×#12011yescharacteristic
303C2×C53C2×#12111yescharacteristic
304C2×C53#53411yescharacteristic
305C2×C53D4211yescharacteristic
306C2×C53D4211yescharacteristic
307C2×C53#54611yescharacteristic
308C2×C53#54611yescharacteristic
309D53—24no—
310D53—42no—
311D53—42no—
312C23×D4—531no—
313C2×#121—531no—
314C2×#121D5311yescharacteristic, complemented
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#386—531no—
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—1064no—
337C22×D4—1064no—
338C22×D4—1064no—
339C22×D4C2×D5311yescharacteristic
340C22×D4C2×D5311yescharacteristic, complemented
341C22×D4C2×D5311yescharacteristic, complemented
342C23×C4—531no—
343C23×C4—531no—
344C23×C4C2×D5311yescharacteristic, complemented
345#121—24no—
346#121—1064no—
347#120—1064no—
348#120—1064no—
349C2×C4⋊C4—531no—
350C2×C4⋊C4C2×D5311yescharacteristic, complemented
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—531no—
364C2×C22⋊C4—1064no—
365C2×C22⋊C4C2×D5311yescharacteristic, complemented
366C2×C22⋊C4C2×D5311yescharacteristic, complemented
367#95—531no—
368#95—531no—
369#95—531no—
370#95—531no—
371C24—531no—
372C24—531no—
373C24—531no—
374C24—534no—
375C24—1062no—
376C24—1062no—
377C24—1062no—
378C24—1062no—
379C24—1062no—
380C24—1062no—
381C24—1062no—
382C24—1062no—
383C24—1062no—
384C24—2121no—
385C24—2121no—
386C24—2121no—
387C24—2121no—
388C24C22×D5311yescharacteristic
389C24C22×D5311yescharacteristic
390C24C22×D5311yescharacteristic
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—41no—
399C2×D4—534no—
400C2×D4—1062no—
401C2×D4—1062no—
402C2×D4—1062no—
403C2×D4—1062no—
404C2×D4—10632no—
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—1062no—
415C2×D4—2121no—
416C2×D4—2121no—
417C2×D4—2121no—
418C2×D4C22⋊D5314yescomplemented
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—1062no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—2121no—
433C22×C4C22×D5311yescharacteristic
434C22×C4C22×D5311yescharacteristic
435C22×C4C22×D5311yescharacteristic
436C22×C4C22×D5311yescharacteristic
437C4⋊C4—22no—
438C4⋊C4—1062no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—534no—
442C22⋊C4—1062no—
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—2121no—
453C22⋊C4—2121no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—532no—
472C23—532no—
473C23—532no—
474C23—532no—
475C23—532no—
476C23—1068no—
477C23—1068no—
478C23—1061no—
479C23—1061no—
480C23—1061no—
481C23—1061no—
482C23—1068no—
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—
506C23D4×D5312yes—
507C23D4×D5312yes—
508C23C2×C22⋊D5312yes—
509C23C2×C22⋊D5312yes—
510C23C23×D5311yescharacteristic
511D4—28no—
512D4—44no—
513D4—44no—
514D4—1068no—
515D4—1068no—
516D4—1068no—
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—44no—
531C2×C4—532no—
532C2×C4—532no—
533C2×C4—1068no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
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×C4D4×D5312yescomplemented
549C2×C4C2×C22⋊D5312yes—
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—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—534no—
571C22—1062no—
572C22—1062no—
573C22—1062no—
574C22—1062no—
575C22—1062no—
576C22—1062no—
577C22—1062no—
578C22—1062no—
579C22—1062no—
580C22—1062no—
581C22—1062no—
582C22—1062no—
583C22—21216no—
584C22—21216no—
585C22—2121no—
586C22—2121no—
587C22—2121no—
588C22—2121no—
589C22—2121no—
590C22—2121no—
591C22—2121no—
592C22—2121no—
593C22#1313318514yescomplemented
594C22C2×D4×D5311yescharacteristic
595C22C2×D4×D5311yescharacteristic
596C22C2×D4×D5311yescharacteristic
597C22C22×C22⋊D5311yescharacteristic
598C22#41×D5311yescharacteristic
599C22#1313326211yescharacteristic
600C22#1313326311yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—1062no—
606C4—1062no—
607C4—2121no—
608C4—2121no—
609C4—2121no—
610C2—24no—
611C2—42no—
612C2—81no—
613C2—1064no—
614C2—2122no—
615C2—2122no—
616C2C2×#1313318511yescharacteristic
617C2#121×D5311yescharacteristic
618C2#1681118111yescharacteristic
619C2D4×C22⋊D5311yescharacteristic
620C2#1681125711yescharacteristic
621C2#1681126111yescharacteristic
622C2#1681126211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2410
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×D4×D53C2semidirect
2. C2×#13133189C2semidirect
3. C2×#13133185C2semidirect
4. C2×#13133185C2semidirect
5. C22×#11719921C2semidirect
6. #16810540C2semidirect
7. #16810538C2semidirect
8. #16810532C2semidirect
9. #16810531C2semidirect
10. #16810503C2semidirect
11. #16810502C2semidirect
12. #16810499C2semidirect
13. #16810494C2semidirect
14. #16810478C2semidirect
15. C2×#121×C53C2semidirect
22. C22×C22⋊D53C22semidirect
23. C22×C22⋊D53C22semidirect
27. C23×Dic53C22semidirect
32. C2×#11719919C22semidirect
33. C2×C22⋊Dic53C22semidirect
34. C2×C22⋊Dic53C22semidirect
35. C2×C22⋊Dic53C22semidirect
36. C2×C22⋊Dic53C22semidirect
41. C2×#11719921C22semidirect
42. C2×#11719921C22semidirect
43. C2×#11719921C22semidirect
44. C2×#11719921C22semidirect
45. C2×#11719921C22semidirect
46. C2×#11719921C22semidirect
47. #13133124C22semidirect
48. #13133124C22semidirect
49. #13133124C22semidirect
50. #13133124C22semidirect
52. C22×D4×C53C22semidirect
53. C22×D4×C53C22semidirect
54. C23×C4×C53C22semidirect
56. C2×C4⋊C4×C53C22semidirect
57. C2×C22⋊C4×C53C22semidirect
58. C2×C22⋊C4×C53C22semidirect
75. C23×D53D4semidirect
87. C2×D4×C53D4semidirect
109. C2×C22⋊D53D4semidirect
140. #11719921D4semidirect
214. C2×C4×C53C2×D4semidirect
235. C2×Dic53C2×D4semidirect
258. C22×C53#120semidirect
288. C2×D53#121semidirect
314. C2×#121D53semidirect
327. C53#2410semidirect
340. C22×D4C2×D53semidirect
341. C22×D4C2×D53semidirect
344. C23×C4C2×D53semidirect
350. C2×C4⋊C4C2×D53semidirect
365. C2×C22⋊C4C2×D53semidirect
366. C2×C22⋊C4C2×D53semidirect
418. C2×D4C22⋊D53semidirect
548. C2×C4D4×D53semidirect
593. C22#13133185semidirect