Finite Group Databasebeta

← #20278248

Subgroups — #20278248

Order 6784.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133188C211yesmaximal, characteristic, complemented
2C2×#13133185C211yesmaximal, characteristic, complemented
3C2×#13133185C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊D53C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
6#16810545C211yesmaximal, characteristic, complemented
7#16810537C211yesmaximal, characteristic, complemented
8#16810537C211yesmaximal, characteristic, complemented
9#16810531C211yesmaximal, characteristic, complemented
10#16810523C211yesmaximal, characteristic, complemented
11#16810516C211yesmaximal, characteristic, complemented
12#16810502C211yesmaximal, characteristic, complemented
13#16810499C211yesmaximal, characteristic, complemented
14#16810494C211yesmaximal, characteristic, complemented
15C2×#121×C53C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic
23#13133188—24no—
24#13133185—24no—
25#13133185—24no—
26C4×C22⋊D53—24no—
27C2×C4⋊Dic53C2211yescharacteristic, complemented
28C2×#11719919C2211yescharacteristic, complemented
29C2×#11719919C2211yescharacteristic, complemented
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic, complemented
34C2×C22⋊Dic53C2211yescharacteristic, complemented
35C2×C22⋊Dic53C2211yescharacteristic, complemented
36C2×C22⋊Dic53C2211yescharacteristic, complemented
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C22⋊C4×D53—24no—
44C2×C4×Dic53C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47#13133124C2211yescharacteristic, complemented
48#13133124C2211yescharacteristic, complemented
49C22×D4×C53C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51C22×D4×C53C2211yescharacteristic, complemented
52C23×C4×C53C2211yescharacteristic, complemented
53#121×C53—24no—
54C2×C4⋊C4×C53C2211yescharacteristic, complemented
55C2×C22⋊C4×C53C2211yescharacteristic
56C2×C22⋊C4×C53C2211yescharacteristic, complemented
57C24×C53C2311yescharacteristic
58C24×C53C2311yescharacteristic
59C24×C53C2311yescharacteristic
60C23×D53—22no—
61C23×D53—22no—
62C23×D53—22no—
63C23×D53—22no—
64C23×D53—22no—
65C23×D53—41no—
66C23×D53—41no—
67C23×D53C2311yescharacteristic
68C23×D53C2311yescharacteristic
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—22no—
73C2×D4×C53—22no—
74C2×D4×C53—22no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53—41no—
81C22×C4×C53C2311yescharacteristic
82C22×C4×C53C2311yescharacteristic
83C22×C4×C53C2311yescharacteristic
84C22×C4×C53C2311yescharacteristic
85C2×C22⋊D53—22no—
86C2×C22⋊D53—22no—
87C2×C22⋊D53—22no—
88C2×C22⋊D53—22no—
89C2×C22⋊D53—22no—
90C2×C22⋊D53—22no—
91C2×C22⋊D53—22no—
92C2×C22⋊D53—22no—
93C2×C22⋊D53—22no—
94C2×C22⋊D53—41no—
95C2×C22⋊D53—41no—
96C2×C22⋊D53D414yescomplemented
97C22×Dic53C2311yescharacteristic
98C22×Dic53C2311yescharacteristic
99C22×Dic53C2311yescharacteristic
100C22×Dic53C2311yescharacteristic
101C22×Dic53C2311yescharacteristic
102C22×Dic53C2311yescharacteristic
103C2×C4×D53—22no—
104C2×C4×D53—22no—
105C2×C4×D53—22no—
106C2×C4×D53—22no—
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C4⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53D414yescomplemented
112C22⋊Dic53—22no—
113C22⋊Dic53—22no—
114C22⋊Dic53—22no—
115C22⋊Dic53—22no—
116C22⋊Dic53—22no—
117C22⋊Dic53—22no—
118C22⋊Dic53—41no—
119#11719921—22no—
120#11719921—22no—
121#11719921—22no—
122#11719921—22no—
123#11719921—22no—
124#11719921D414yescomplemented
125C4⋊Dic53—22no—
126#11719919—22no—
127#11719919—41no—
128C4×Dic53—22no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53C2×D412yes—
149C23×C53C2×D412yes—
150C23×C53C2411yescharacteristic
151C22×D53—21no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—28no—
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53—21no—
160C22×D53—21no—
161C22×D53—21no—
162C22×D53—44no—
163C22×D53—44no—
164C22×D53—44no—
165C22×D53—44no—
166C22×D53—44no—
167C22×D53—44no—
168C22×D53C2×D412yes—
169C22×D53C2×D412yes—
170C22×D53#4112yes—
171C22×D53#4112yes—
172D4×C53—44no—
173D4×C53—44no—
174D4×C53—44no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—44no—
185C2×C4×C53C2×D412yes—
186C2×C4×C53C2×D412yes—
187C2×C4×C53C2×D412yes—
188C22⋊D53—28no—
189C22⋊D53—44no—
190C22⋊D53—44no—
191C22⋊D53—44no—
192C2×Dic53—21no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—21no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53C2×D412yes—
209C2×Dic53C2×D412yes—
210C4×D53—44no—
211C4×D53—44no—
212C22×C53—22no—
213C22×C53—22no—
214C22×C53—22no—
215C22×C53—22no—
216C22×C53—22no—
217C22×C53—22no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—41no—
226C22×C53—41no—
227C22×C53—41no—
228C22×C53—41no—
229C22×C53—41no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—41no—
233C22×C53C22×D411yescharacteristic
234C22×C53C22×D411yescharacteristic
235C22×C53C22×D411yescharacteristic
236C22×C53C2×#4111yescharacteristic
237C22×C53C2×#4111yescharacteristic
238C22×C53H(5,𝔽2)11yescharacteristic
239C22×C53H(5,𝔽2)11yescharacteristic
240C2×D53—22no—
241C2×D53—22no—
242C2×D53—22no—
243C2×D53—22no—
244C2×D53—22no—
245C2×D53—22no—
246C2×D53—22no—
247C2×D53—22no—
248C2×D53—22no—
249C2×D53—416no—
250C2×D53—41no—
251C2×D53—41no—
252C2×D53—41no—
253C2×D53—41no—
254C2×D53#12114yescomplemented
255C4×C53—22no—
256C4×C53—22no—
257C4×C53—22no—
258C4×C53—41no—
259Dic53—22no—
260Dic53—22no—
261Dic53—41no—
262Dic53—41no—
263Dic53—41no—
264Dic53—41no—
265#2424—531nomaximal, Sylow
266C2×C53—42no—
267C2×C53—42no—
268C2×C53—81no—
269C2×C53C2×#12111yescharacteristic
270C2×C53#52511yescharacteristic
271C2×C53#53811yescharacteristic
272C2×C53D4211yescharacteristic
273C2×C53D4211yescharacteristic
274C2×C53#54611yescharacteristic
275C2×C53#54611yescharacteristic
276D53—24no—
277D53—42no—
278C2×#121—531no—
279C2×#121D5311yescharacteristic, complemented
280C2×#120—531no—
281C2×#120—531no—
282C2×C4×D4—531no—
283C22×C22⋊C4—531no—
284#396—531no—
285#394—531no—
286#394—531no—
287#392—531no—
288#390—531no—
289#388—531no—
290#386—531no—
291#386—531no—
292#385—531no—
293C53#242411yescharacteristic, complemented, Sylow
294C25—531no—
295C22×D4—531no—
296C22×D4—531no—
297C22×D4—531no—
298C22×D4—531no—
299C22×D4C2×D5311yescharacteristic
300C22×D4C2×D5311yescharacteristic
301C22×D4C2×D5311yescharacteristic, complemented
302C23×C4—531no—
303C23×C4—531no—
304C23×C4C2×D5311yescharacteristic, complemented
305#121—24no—
306#121—1064no—
307#120—1064no—
308#120—1064no—
309C4×D4—1064no—
310C2×C4⋊C4—531no—
311C2×C4⋊C4—531no—
312C2×C4⋊C4—531no—
313C2×C4⋊C4C2×D5311yescharacteristic, complemented
314C2×C22⋊C4—531no—
315C2×C22⋊C4—531no—
316C2×C22⋊C4—531no—
317C2×C22⋊C4—531no—
318C2×C22⋊C4—531no—
319C2×C22⋊C4—531no—
320C2×C22⋊C4—531no—
321C2×C22⋊C4—531no—
322C2×C22⋊C4—531no—
323C2×C22⋊C4—531no—
324C2×C22⋊C4—531no—
325C2×C22⋊C4—531no—
326C2×C22⋊C4—531no—
327C2×C22⋊C4—1064no—
328C2×C22⋊C4C2×D5311yescharacteristic
329C2×C22⋊C4C2×D5311yescharacteristic, complemented
330C2×C42—531no—
331#95—531no—
332#95—531no—
333#95—531no—
334#95—531no—
335C24—531no—
336C24—531no—
337C24—1062no—
338C24—1062no—
339C24—1062no—
340C24—1062no—
341C24—1062no—
342C24—2121no—
343C24—2121no—
344C24C22×D5311yescharacteristic
345C24C22×D5311yescharacteristic
346C24C22×D5311yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—534no—
357C2×D4—1062no—
358C2×D4—1062no—
359C2×D4—1062no—
360C2×D4—1062no—
361C2×D4—1062no—
362C2×D4—1062no—
363C2×D4—1062no—
364C2×D4—1062no—
365C2×D4—1062no—
366C2×D4—2121no—
367C2×D4—2121no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—531no—
372C22×C4—531no—
373C22×C4—531no—
374C22×C4—531no—
375C22×C4—531no—
376C22×C4—531no—
377C22×C4—1062no—
378C22×C4—1062no—
379C22×C4—1062no—
380C22×C4—1062no—
381C22×C4—2121no—
382C22×C4—2121no—
383C22×C4C22×D5311yescharacteristic
384C22×C4C22×D5311yescharacteristic
385C22×C4C22×D5311yescharacteristic
386C22×C4C22×D5311yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—1062no—
389C4⋊C4—1062no—
390C4⋊C4—2121no—
391C22⋊C4—22no—
392C22⋊C4—534no—
393C22⋊C4—1062no—
394C22⋊C4—1062no—
395C22⋊C4—1062no—
396C22⋊C4—1062no—
397C22⋊C4—1062no—
398C22⋊C4—1062no—
399C22⋊C4—1062no—
400C22⋊C4—1062no—
401C22⋊C4—1062no—
402C22⋊C4—1062no—
403C22⋊C4—1062no—
404C22⋊C4—2121no—
405C22⋊C4C22⋊D5314yescomplemented
406C42—1062no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—532no—
427C23—532no—
428C23—532no—
429C23—532no—
430C23—1068no—
431C23—1061no—
432C23—1061no—
433C23—1061no—
434C23—1061no—
435C23—1061no—
436C23—1061no—
437C23—1061no—
438C23—1061no—
439C23—1061no—
440C23—1061no—
441C23—2124no—
442C23—2124no—
443C23—2124no—
444C23—2124no—
445C23—2124no—
446C23—2124no—
447C23D4×D5312yes—
448C23C2×C22⋊D5312yes—
449C23C23×D5311yescharacteristic
450D4—44no—
451D4—44no—
452D4—44no—
453D4—1068no—
454D4—2124no—
455D4—2124no—
456D4—2124no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—44no—
467C2×C4—532no—
468C2×C4—532no—
469C2×C4—1061no—
470C2×C4—1061no—
471C2×C4—1061no—
472C2×C4—1061no—
473C2×C4—1061no—
474C2×C4—1061no—
475C2×C4—1061no—
476C2×C4—1061no—
477C2×C4—1061no—
478C2×C4—1061no—
479C2×C4—1061no—
480C2×C4—1061no—
481C2×C4—1061no—
482C2×C4—1061no—
483C2×C4—1061no—
484C2×C4—1061no—
485C2×C4—2124no—
486C2×C4—2124no—
487C2×C4D4×D5312yes—
488C2×C4C2×C22⋊D5312yes—
489C2×C4C2×C22⋊D5312yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—534no—
512C22—1062no—
513C22—1062no—
514C22—1062no—
515C22—1062no—
516C22—1062no—
517C22—1062no—
518C22—1062no—
519C22—1062no—
520C22—1062no—
521C22—21216no—
522C22—2121no—
523C22—2121no—
524C22—2121no—
525C22—2121no—
526C22C2×D4×D5311yescharacteristic
527C22C2×D4×D5311yescharacteristic
528C22C22×C22⋊D5311yescharacteristic
529C22#41×D5311yescharacteristic
530C22#41×D5311yescharacteristic
531C22#1313326211yescharacteristic
532C22#1313326211yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—41no—
537C4—1062no—
538C4—1062no—
539C4—2121no—
540C4—2121no—
541C4—2121no—
542C4—2121no—
543C2—42no—
544C2—42no—
545C2—81no—
546C2—1064no—
547C2—2122no—
548C2#121×D5311yescharacteristic
549C2#1681113011yescharacteristic
550C2#1681120511yescharacteristic
551C2D4×C22⋊D5311yescharacteristic
552C2D4×C22⋊D5311yescharacteristic
553C2#1681126211yescharacteristic
554C2#1681127011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2424
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133188C2semidirect
2. C2×#13133185C2semidirect
3. C2×#13133185C2semidirect
4. C2×C4×C22⋊D53C2semidirect
5. C2×C22⋊C4×D53C2semidirect
6. #16810545C2semidirect
7. #16810537C2semidirect
8. #16810537C2semidirect
9. #16810531C2semidirect
10. #16810523C2semidirect
11. #16810516C2semidirect
12. #16810502C2semidirect
13. #16810499C2semidirect
14. #16810494C2semidirect
15. C2×#121×C53C2semidirect
27. C2×C4⋊Dic53C22semidirect
28. C2×#11719919C22semidirect
29. C2×#11719919C22semidirect
33. C2×C22⋊Dic53C22semidirect
34. C2×C22⋊Dic53C22semidirect
35. C2×C22⋊Dic53C22semidirect
36. C2×C22⋊Dic53C22semidirect
44. C2×C4×Dic53C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
48. #13133124C22semidirect
51. C22×D4×C53C22semidirect
52. C23×C4×C53C22semidirect
54. C2×C4⋊C4×C53C22semidirect
56. C2×C22⋊C4×C53C22semidirect
96. C2×C22⋊D53D4semidirect
111. C22⋊C4×C53D4semidirect
124. #11719921D4semidirect
254. C2×D53#121semidirect
279. C2×#121D53semidirect
293. C53#2424semidirect
301. C22×D4C2×D53semidirect
304. C23×C4C2×D53semidirect
313. C2×C4⋊C4C2×D53semidirect
329. C2×C22⋊C4C2×D53semidirect
405. C22⋊C4C22⋊D53semidirect