Finite Group Databasebeta

← #20278257

Subgroups — #20278257

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×#13133189C211yesmaximal, characteristic, complemented
2C2×#13133185C211yesmaximal, characteristic, complemented
3C2×#13133185C211yesmaximal, characteristic, complemented
4C2×D4×Dic53C211yesmaximal, characteristic, complemented
5C22×#11719921C211yesmaximal, characteristic, complemented
6#16810545C211yesmaximal, characteristic, complemented
7#16810539C211yesmaximal, characteristic, complemented
8#16810538C211yesmaximal, characteristic, complemented
9#16810531C211yesmaximal, characteristic, complemented
10#16810523C211yesmaximal, characteristic, complemented
11#16810513C211yesmaximal, characteristic, complemented
12#16810503C211yesmaximal, characteristic, complemented
13#16810500C211yesmaximal, characteristic, complemented
14#16810496C211yesmaximal, characteristic, complemented
15C2×#121×C53C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C23×Dic53C2211yescharacteristic
22C23×Dic53C2211yescharacteristic, complemented
23#13133189—24no—
24#13133185—24no—
25#13133185—24no—
26D4×Dic53—24no—
27C2×C4⋊Dic53C2211yescharacteristic, complemented
28C2×#11719919C2211yescharacteristic, complemented
29C2×#11719919C2211yescharacteristic, complemented
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic, complemented
33C2×C22⋊Dic53C2211yescharacteristic, complemented
34C2×C22⋊Dic53C2211yescharacteristic, complemented
35C2×#11719921—24no—
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×#11719921C2211yescharacteristic
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
53#121×C53—24no—
54C2×C4⋊C4×C53C2211yescharacteristic, complemented
55C2×C22⋊C4×C53C2211yescharacteristic, complemented
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—22no—
76C2×D4×C53—41no—
77C2×D4×C53D414yescomplemented
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—41no—
94C2×C22⋊D53—41no—
95C2×C22⋊D53—41no—
96C2×C22⋊D53—41no—
97C22×Dic53—22no—
98C22×Dic53—22no—
99C22×Dic53—22no—
100C22×Dic53—22no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53C2311yescharacteristic
104C22×Dic53C2311yescharacteristic
105C22×Dic53C2311yescharacteristic
106C22×Dic53C2311yescharacteristic
107C22×Dic53C2311yescharacteristic
108C22×Dic53C2311yescharacteristic
109C4⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊Dic53—22no—
113C22⋊Dic53—22no—
114C22⋊Dic53—22no—
115C22⋊Dic53—41no—
116C22⋊Dic53D414yescomplemented
117#11719921—22no—
118#11719921—22no—
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×C53C2×D412yes—
147C23×C53C2×D412yes—
148C23×C53C2×D412yes—
149C23×C53#4112yes—
150C23×C53C2411yescharacteristic
151C22×D53—28no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—21no—
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53—21no—
160C22×D53—21no—
161C22×D53—21no—
162C22×D53—21no—
163C22×D53—21no—
164C22×D53—44no—
165C22×D53—44no—
166C22×D53—44no—
167C22×D53—44no—
168C22×D53—44no—
169C22×D53—44no—
170C22×D53C2×D412yes—
171C22×D53#4112yes—
172D4×C53—28no—
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—21no—
185C2×C4×C53—44no—
186C2×C4×C53C2×D412yes—
187C2×C4×C53C2×D412yes—
188C22⋊D53—44no—
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—44no—
208C2×Dic53—44no—
209C2×Dic53C2×D412yes—
210C2×Dic53C2×D412yes—
211C2×Dic53C2×D412yes—
212C22×C53—22no—
213C22×C53—22no—
214C22×C53—22no—
215C22×C53—22no—
216C22×C53—22no—
217C22×C53—22no—
218C22×C53—22no—
219C22×C53—22no—
220C22×C53—22no—
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#12114yescomplemented
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—416no—
247C2×D53—41no—
248C2×D53—41no—
249C2×D53—41no—
250C2×D53—41no—
251C2×D53—41no—
252C2×D53—41no—
253C2×D53—41no—
254C2×D53—41no—
255C4×C53—22no—
256C4×C53—22no—
257C4×C53—41no—
258C4×C53—41no—
259Dic53—22no—
260Dic53—22no—
261Dic53—22no—
262Dic53—41no—
263Dic53—41no—
264Dic53—41no—
265#2424—531nomaximal, Sylow
266C2×C53—24no—
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—42no—
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
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, complemented
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—22no—
354C2×D4—41no—
355C2×D4—1062no—
356C2×D4—1062no—
357C2×D4—1062no—
358C2×D4—1062no—
359C2×D4—1062no—
360C2×D4—1062no—
361C2×D4—1062no—
362C2×D4—1062no—
363C2×D4—2121no—
364C2×D4—2121no—
365C2×D4—2121no—
366C2×D4—2121no—
367C2×D4C22⋊D5314yescomplemented
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—22no—
393C22⋊C4—534no—
394C22⋊C4—534no—
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—1062no—
405C22⋊C4—2121no—
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—532no—
425C23—532no—
426C23—1068no—
427C23—1061no—
428C23—1061no—
429C23—1061no—
430C23—1061no—
431C23—1061no—
432C23—1061no—
433C23—1061no—
434C23—1061no—
435C23—1061no—
436C23—1061no—
437C23—1061no—
438C23—1061no—
439C23—2124no—
440C23—2124no—
441C23—2124no—
442C23—2124no—
443C23—2124no—
444C23—2124no—
445C23D4×D5312yes—
446C23#1171994712yes—
447C23C2×C22⋊D5312yes—
448C23C2×C22⋊D5312yes—
449C23C23×D5311yescharacteristic
450D4—28no—
451D4—44no—
452D4—44no—
453D4—2124no—
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—21no—
467C2×C4—44no—
468C2×C4—532no—
469C2×C4—532no—
470C2×C4—532no—
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—1061no—
486C2×C4—2124no—
487C2×C4—2124no—
488C2×C4D4×D5312yes—
489C2×C4C2×C22⋊D5312yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—1062no—
511C22—1062no—
512C22—1062no—
513C22—1062no—
514C22—1062no—
515C22—1062no—
516C22—21216no—
517C22—2121no—
518C22—2121no—
519C22—2121no—
520C22—2121no—
521C22—2121no—
522C22—2121no—
523C22—2121no—
524C22—2121no—
525C22#1313318914yescomplemented
526C22C2×D4×D5311yescharacteristic
527C22C2×D4×D5311yescharacteristic
528C22C22×C22⋊D5311yescharacteristic
529C22#41×D5311yescharacteristic
530C22C2×#1171994711yescharacteristic
531C22#1313326211yescharacteristic
532C22#1313326311yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—1062no—
538C4—1062no—
539C4—1062no—
540C4—2121no—
541C4—2121no—
542C4—2121no—
543C2—24no—
544C2—42no—
545C2—81no—
546C2—2122no—
547C2—2122no—
548C2C2×#1313318911yescharacteristic
549C2#1681113311yescharacteristic
550C2#1681121011yescharacteristic
551C2D4×C22⋊D5311yescharacteristic
552C2#1681125711yescharacteristic
553C2#1681126111yescharacteristic
554C2#1681126211yescharacteristic
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×#13133189C2semidirect
2. C2×#13133185C2semidirect
3. C2×#13133185C2semidirect
4. C2×D4×Dic53C2semidirect
5. C22×#11719921C2semidirect
6. #16810545C2semidirect
7. #16810539C2semidirect
8. #16810538C2semidirect
9. #16810531C2semidirect
10. #16810523C2semidirect
11. #16810513C2semidirect
12. #16810503C2semidirect
13. #16810500C2semidirect
14. #16810496C2semidirect
15. C2×#121×C53C2semidirect
22. C23×Dic53C22semidirect
27. C2×C4⋊Dic53C22semidirect
28. C2×#11719919C22semidirect
29. C2×#11719919C22semidirect
32. C2×C22⋊Dic53C22semidirect
33. C2×C22⋊Dic53C22semidirect
34. C2×C22⋊Dic53C22semidirect
44. C2×C4×Dic53C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
48. #13133124C22semidirect
51. C22×D4×C53C22semidirect
54. C2×C4⋊C4×C53C22semidirect
55. C2×C22⋊C4×C53C22semidirect
56. C2×C22⋊C4×C53C22semidirect
77. C2×D4×C53D4semidirect
116. C22⋊Dic53D4semidirect
124. #11719921D4semidirect
232. C22×C53#121semidirect
279. C2×#121D53semidirect
293. C53#2424semidirect
301. C22×D4C2×D53semidirect
313. C2×C4⋊C4C2×D53semidirect
328. C2×C22⋊C4C2×D53semidirect
329. C2×C22⋊C4C2×D53semidirect
367. C2×D4C22⋊D53semidirect
525. C22#13133189semidirect