Finite Group Databasebeta

← #20278247

Subgroups — #20278247

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×#13133188C211yesmaximal, characteristic, complemented
3C2×#13133186C211yesmaximal, characteristic, complemented
4C2×#13133177C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic53C211yesmaximal, characteristic, complemented
6#16810546C211yesmaximal, characteristic, complemented
7#16810539C211yesmaximal, characteristic, complemented
8#16810537C211yesmaximal, characteristic, complemented
9#16810531C211yesmaximal, characteristic, complemented
10#16810526C211yesmaximal, characteristic, complemented
11#16810511C211yesmaximal, characteristic, complemented
12#16810503C211yesmaximal, characteristic, complemented
13#16810499C211yesmaximal, characteristic, complemented
14#16810498C211yesmaximal, characteristic, complemented
15C2×#120×C53C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic, complemented
20C23×Dic53C2211yescharacteristic
21C23×Dic53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic, complemented
23#13133189—24no—
24#13133188—24no—
25#13133186—24no—
26#13133177—24no—
27C2×C4⋊Dic53C2211yescharacteristic, complemented
28C2×C4⋊Dic53C2211yescharacteristic, complemented
29C2×#11719919C2211yescharacteristic, complemented
30C2×#11719919C2211yescharacteristic, complemented
31C2×C22⋊Dic53—24no—
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×C22⋊Dic53C2211yescharacteristic
39C2×C22⋊Dic53C2211yescharacteristic, complemented
40C2×C22⋊Dic53C2211yescharacteristic, complemented
41C2×C22⋊Dic53C2211yescharacteristic, complemented
42C2×#11719921C2211yescharacteristic, complemented
43C2×#11719921C2211yescharacteristic, complemented
44C2×C4×Dic53C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47#13133124C2211yescharacteristic, complemented
48#13133124C2211yescharacteristic, complemented
49C25×C53C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51C22×D4×C53C2211yescharacteristic
52C22×D4×C53C2211yescharacteristic
53#120×C53—24no—
54C2×C22⋊C4×C53C2211yescharacteristic
55C2×C22⋊C4×C53C2211yescharacteristic
56C2×C22⋊C4×C53C2211yescharacteristic
57C24×C53—22no—
58C24×C53—22no—
59C24×C53—22no—
60C24×C53—22no—
61C24×C53—22no—
62C24×C53—41no—
63C24×C53—41no—
64C24×C53C2311yescharacteristic
65C24×C53C2311yescharacteristic
66C24×C53C2311yescharacteristic
67C24×C53C2311yescharacteristic
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×C53—41no—
78C22×C4×C53C2311yescharacteristic
79C22×C4×C53C2311yescharacteristic
80C22×C4×C53C2311yescharacteristic
81C2×C22⋊D53—22no—
82C2×C22⋊D53—22no—
83C2×C22⋊D53—22no—
84C2×C22⋊D53—22no—
85C2×C22⋊D53—22no—
86C2×C22⋊D53—22no—
87C2×C22⋊D53—22no—
88C2×C22⋊D53—22no—
89C2×C22⋊D53—41no—
90C2×C22⋊D53—41no—
91C2×C22⋊D53—41no—
92C2×C22⋊D53D414yescomplemented
93C22×Dic53—22no—
94C22×Dic53—22no—
95C22×Dic53—22no—
96C22×Dic53—22no—
97C22×Dic53—41no—
98C22×Dic53—41no—
99C22×Dic53C2311yescharacteristic
100C22×Dic53C2311yescharacteristic
101C22×Dic53C2311yescharacteristic
102C22×Dic53C2311yescharacteristic
103C22×Dic53C2311yescharacteristic
104C22×Dic53C2311yescharacteristic
105C22×Dic53C2311yescharacteristic
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—41no—
109C22⋊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—22no—
119C22⋊Dic53—22no—
120C22⋊Dic53—41no—
121C22⋊Dic53D414yescomplemented
122#11719921—22no—
123#11719921—22no—
124C4⋊Dic53—22no—
125C4⋊Dic53—41no—
126#11719919—22no—
127#11719919—22no—
128C4×Dic53—22no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—28no—
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×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—21no—
153C23×C53—44no—
154C23×C53—44no—
155C23×C53—44no—
156C23×C53—44no—
157C23×C53—44no—
158C23×C53—44no—
159C23×C53C2×D412yes—
160C23×C53C2×D412yes—
161C23×C53C2×D412yes—
162C23×C53#4112yes—
163C23×C53#4112yes—
164C23×C53C2411yescharacteristic
165C22×D53—21no—
166C22×D53—21no—
167C22×D53—21no—
168C22×D53—21no—
169C22×D53—21no—
170C22×D53—21no—
171C22×D53C2×D412yes—
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×C53C2×D412yes—
183C2×C4×C53C2×D412yes—
184C22⋊D53—28no—
185C22⋊D53—44no—
186C22⋊D53—44no—
187C22⋊D53—44no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—21no—
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—44no—
207C2×Dic53—44no—
208C2×Dic53C2×D412yes—
209C2×Dic53C2×D412yes—
210C2×Dic53C2×D412yes—
211C4×D53—44no—
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—22no—
222C22×C53—22no—
223C22×C53—22no—
224C22×C53—416no—
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×C53—41no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—41no—
239C22×C53—41no—
240C22×C53#12114yescomplemented
241C22×C53C22×D411yescharacteristic
242C22×C53C22×D411yescharacteristic
243C22×C53C22×D411yescharacteristic
244C22×C53C2×#4111yescharacteristic
245C22×C53C2×#4111yescharacteristic
246C22×C53H(5,𝔽2)11yescharacteristic
247C22×C53H(5,𝔽2)11yescharacteristic
248C2×D53—22no—
249C2×D53—22no—
250C2×D53—22no—
251C2×D53—41no—
252C2×D53—41no—
253C2×D53—41no—
254C2×D53—41no—
255C4×C53—22no—
256C4×C53—22no—
257C4×C53—41no—
258Dic53—22no—
259Dic53—22no—
260Dic53—22no—
261Dic53—41no—
262Dic53—41no—
263Dic53—41no—
264Dic53—41no—
265#2424—531nomaximal, Sylow
266C2×C53—24no—
267C2×C53—42no—
268C2×C53—42no—
269C2×C53—81no—
270C2×C53C2×#12111yescharacteristic
271C2×C53#52511yescharacteristic
272C2×C53#53811yescharacteristic
273C2×C53D4211yescharacteristic
274C2×C53D4211yescharacteristic
275C2×C53#54611yescharacteristic
276C2×C53#54611yescharacteristic
277D53—42no—
278C2×#121—531no—
279C2×#121—531no—
280C2×#120—531no—
281C2×#120D5311yescharacteristic, complemented
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
294C25C2×D5311yescharacteristic
295C22×D4—531no—
296C22×D4—531no—
297C22×D4—531no—
298C22×D4—531no—
299C22×D4C2×D5311yescharacteristic
300C22×D4C2×D5311yescharacteristic
301C22×D4C2×D5311yescharacteristic
302C23×C4—531no—
303C23×C4—531no—
304C23×C4—531no—
305#121—1064no—
306#121—1064no—
307#120—24no—
308#120—1064no—
309C4×D4—1064no—
310C2×C4⋊C4—531no—
311C2×C4⋊C4—531no—
312C2×C4⋊C4—531no—
313C2×C4⋊C4—531no—
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—1064no—
327C2×C22⋊C4C2×D5311yescharacteristic
328C2×C22⋊C4C2×D5311yescharacteristic
329C2×C22⋊C4C2×D5311yescharacteristic
330C2×C42—531no—
331#95—531no—
332#95—531no—
333#95—531no—
334#95—531no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—531no—
343C24C22×D5311yescharacteristic
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—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—2121no—
366C2×D4—2121no—
367C2×D4—2121no—
368C22×C4—531no—
369C22×C4—531no—
370C22×C4—531no—
371C22×C4—531no—
372C22×C4—531no—
373C22×C4—531no—
374C22×C4—531no—
375C22×C4—1062no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—1062no—
379C22×C4—1062no—
380C22×C4—1062no—
381C22×C4—2121no—
382C22×C4—2121no—
383C22×C4—2121no—
384C22×C4C22×D5311yescharacteristic
385C22×C4C22×D5311yescharacteristic
386C22×C4C22×D5311yescharacteristic
387C4⋊C4—1062no—
388C4⋊C4—1062no—
389C4⋊C4—1062no—
390C4⋊C4—2121no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—534no—
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—28no—
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—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—532no—
438C23—1061no—
439C23—1061no—
440C23—1061no—
441C23—1061no—
442C23—1061no—
443C23—1061no—
444C23D4×D5312yes—
445C23D4×D5312yes—
446C23#1171994712yes—
447C23#1171994712yes—
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—532no—
465C2×C4—532no—
466C2×C4—532no—
467C2×C4—1061no—
468C2×C4—1061no—
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×C4—2124no—
488C2×C4C2×C22⋊D5312yes—
489C2×C4C2×C22⋊D5312yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—416no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—1062no—
519C22—1062no—
520C22—1062no—
521C22—2121no—
522C22—2121no—
523C22—2121no—
524C22—2121no—
525C22#1313318814yescomplemented
526C22C2×D4×D5311yescharacteristic
527C22C2×D4×D5311yescharacteristic
528C22C22×C22⋊D5311yescharacteristic
529C22C2×#1171994711yescharacteristic
530C22C2×#1171994711yescharacteristic
531C22#1313326211yescharacteristic
532C22#1313326211yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—1062no—
537C4—1062no—
538C4—1062no—
539C4—2121no—
540C4—2121no—
541C4—2121no—
542C4—2121no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—81no—
547C2—2122no—
548C2C2×#1313318811yescharacteristic
549C2#1681112911yescharacteristic
550C2#1681120811yescharacteristic
551C2#1681125511yescharacteristic
552C2D4×C22⋊D5311yescharacteristic
553C2#1681126011yescharacteristic
554C2#1681126711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#120×C53C2
hypercenterC2×#120D53
hypocenterC53#2424
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. C2×#13133188C2semidirect
3. C2×#13133186C2semidirect
4. C2×#13133177C2semidirect
5. C22×C22⋊Dic53C2semidirect
6. #16810546C2semidirect
7. #16810539C2semidirect
8. #16810537C2semidirect
9. #16810531C2semidirect
10. #16810526C2semidirect
11. #16810511C2semidirect
12. #16810503C2semidirect
13. #16810499C2semidirect
14. #16810498C2semidirect
15. C2×#120×C53C2semidirect
19. C22×C22⋊D53C22semidirect
22. C22×C4×D53C22semidirect
27. C2×C4⋊Dic53C22semidirect
28. C2×C4⋊Dic53C22semidirect
29. C2×#11719919C22semidirect
30. C2×#11719919C22semidirect
39. C2×C22⋊Dic53C22semidirect
40. C2×C22⋊Dic53C22semidirect
41. C2×C22⋊Dic53C22semidirect
42. C2×#11719921C22semidirect
43. C2×#11719921C22semidirect
44. C2×C4×Dic53C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
48. #13133124C22semidirect
92. C2×C22⋊D53D4semidirect
111. C22⋊C4×C53D4semidirect
121. C22⋊Dic53D4semidirect
240. C22×C53#121semidirect
281. C2×#120D53semidirect
293. C53#2424semidirect
405. C22⋊C4C22⋊D53semidirect
525. C22#13133188semidirect