Finite Group Databasebeta

← #20278251

Subgroups — #20278251

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×#13133191C211yesmaximal, characteristic, complemented
2C2×#13133190C211yesmaximal, characteristic, complemented
3C2×#13133186C211yesmaximal, characteristic, complemented
4C2×#13133186C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊D53C211yesmaximal, characteristic, complemented
6#16810546C211yesmaximal, characteristic, complemented
7#16810538C211yesmaximal, characteristic, complemented
8#16810538C211yesmaximal, characteristic, complemented
9#16810532C211yesmaximal, characteristic, complemented
10#16810523C211yesmaximal, characteristic, complemented
11#16810516C211yesmaximal, characteristic, complemented
12#16810502C211yesmaximal, characteristic, complemented
13#16810499C211yesmaximal, characteristic, complemented
14#16810496C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C53C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic, complemented
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C22⋊D53C2211yescharacteristic
23C22×C4×D53C2211yescharacteristic, complemented
24#13133191—24no—
25#13133190—24no—
26#13133186—24no—
27#13133186—24no—
28C4×C22⋊D53—24no—
29C2×C4⋊Dic53C2211yescharacteristic, complemented
30C2×#11719919C2211yescharacteristic, complemented
31C2×#11719919C2211yescharacteristic, complemented
32C2×#11719919C2211yescharacteristic, complemented
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×C22⋊Dic53C2211yescharacteristic
39C2×#11719921C2211yescharacteristic, complemented
40C2×#11719921C2211yescharacteristic, complemented
41C2×#11719921C2211yescharacteristic, 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
50C23×C4×C53C2211yescharacteristic
51C23×C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53—24no—
53C2×C22⋊C4×C53C2211yescharacteristic
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
67C23×D53C2311yescharacteristic
68C23×D53C2311yescharacteristic
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C22×C4×C53C2311yescharacteristic
78C22×C4×C53C2311yescharacteristic
79C2×C22⋊D53—22no—
80C2×C22⋊D53—22no—
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—22no—
90C2×C22⋊D53—22no—
91C2×C22⋊D53—22no—
92C2×C22⋊D53—41no—
93C2×C22⋊D53—41no—
94C2×C22⋊D53—41no—
95C2×C22⋊D53—41no—
96C2×C22⋊D53D414yescomplemented
97C22×Dic53C2311yescharacteristic
98C22×Dic53C2311yescharacteristic
99C22×Dic53C2311yescharacteristic
100C22×Dic53C2311yescharacteristic
101C22×Dic53C2311yescharacteristic
102C22×Dic53C2311yescharacteristic
103C2×D212—22no—
104C2×D212—22no—
105C2×D212—41no—
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—41no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53D414yescomplemented
113C22⋊Dic53—22no—
114C22⋊Dic53—22no—
115C22⋊Dic53—22no—
116C22⋊Dic53—22no—
117C22⋊Dic53—22no—
118C22⋊Dic53D414yescomplemented
119#11719921—22no—
120#11719921—22no—
121#11719921—22no—
122#11719921—22no—
123#11719921—41no—
124C4⋊Dic53—22no—
125#11719919—22no—
126#11719919—22no—
127#11719919—41no—
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—44no—
147C23×C53—44no—
148C23×C53—44no—
149C23×C53—44no—
150C23×C53—44no—
151C23×C53—44no—
152C23×C53C2×D412yes—
153C23×C53C2×D412yes—
154C23×C53C2×D412yes—
155C23×C53#4112yes—
156C23×C53#4112yes—
157C23×C53C2411yescharacteristic
158C22×D53—21no—
159C22×D53—21no—
160C22×D53—21no—
161C22×D53—21no—
162C22×D53—21no—
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×D53C2×D412yes—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
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—44no—
183C2×C4×C53—44no—
184C2×C4×C53C2×D412yes—
185C2×C4×C53C2×D412yes—
186C22⋊D53—28no—
187C22⋊D53—44no—
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×Dic53C2×D412yes—
208C2×Dic53C2×D412yes—
209C2×Dic53C2×D412yes—
210D212—44no—
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#12114yescomplemented
234C22×C53C22×D411yescharacteristic
235C22×C53C22×D411yescharacteristic
236C22×C53C22×D411yescharacteristic
237C22×C53C2×#4111yescharacteristic
238C22×C53C2×#4111yescharacteristic
239C22×C53H(5,𝔽2)11yescharacteristic
240C22×C53H(5,𝔽2)11yescharacteristic
241C2×D53—22no—
242C2×D53—22no—
243C2×D53—22no—
244C2×D53—41no—
245C2×D53—41no—
246C2×D53—41no—
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—42no—
269C2×C53C2×#12111yescharacteristic
270C2×C53#52511yescharacteristic
271C2×C53#53811yescharacteristic
272C2×C53D4211yescharacteristic
273C2×C53D4211yescharacteristic
274C2×C53#54611yescharacteristic
275C2×C53#54611yescharacteristic
276D53—42no—
277D53—81no—
278C2×#121—531no—
279C2×#121—531no—
280C2×#120—531no—
281C2×#120—531no—
282C2×C4×D4—531no—
283C22×C22⋊C4D5311yescharacteristic, complemented
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×D4—531no—
300C22×D4—531no—
301C22×D4—531no—
302C23×C4—531no—
303C23×C4C2×D5311yescharacteristic
304C23×C4C2×D5311yescharacteristic
305#121—1064no—
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⋊C4—531no—
314C2×C22⋊C4—24no—
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⋊C4C2×D5311yescharacteristic
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—
343C24—531no—
344C24C22×D5311yescharacteristic
345C24C22×D5311yescharacteristic
346C24C22×D5311yescharacteristic
347C2×D4—534no—
348C2×D4—1062no—
349C2×D4—1062no—
350C2×D4—1062no—
351C2×D4—1062no—
352C2×D4—1062no—
353C2×D4—1062no—
354C2×D4—1062no—
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×D4—2121no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—531no—
375C22×C4—531no—
376C22×C4—531no—
377C22×C4—531no—
378C22×C4—531no—
379C22×C4—531no—
380C22×C4—1062no—
381C22×C4—1062no—
382C22×C4—2121no—
383C22×C4C22×D5311yescharacteristic
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—22no—
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—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—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—532no—
431C23—1061no—
432C23—1061no—
433C23—1061no—
434C23—1061no—
435C23—1061no—
436C23—1061no—
437C23—1061no—
438C23—1061no—
439C23—1061no—
440C23—1061no—
441C23—1061no—
442C23—1061no—
443C23—1061no—
444C23#1171994512yes—
445C23#1171994512yes—
446C23D4×D5312yes—
447C23D4×D5312yes—
448C23C2×C22⋊D5312yes—
449C23C23×D5311yescharacteristic
450D4—1068no—
451D4—2124no—
452D4—2124no—
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—44no—
469C2×C4—532no—
470C2×C4—532no—
471C2×C4—532no—
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—1061no—
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—1062no—
512C22—1062no—
513C22—1062no—
514C22—2121no—
515C22—2121no—
516C22—2121no—
517C22—2121no—
518C22—2121no—
519C22—2121no—
520C22—2121no—
521C22—2121no—
522C22—2121no—
523C22—2121no—
524C22—2121no—
525C22#1313319014yescomplemented
526C22C2×D4×D5311yescharacteristic
527C22C2×D4×D5311yescharacteristic
528C22C22×C22⋊D5311yescharacteristic
529C22C2×#1171994511yescharacteristic
530C22C2×#1171994511yescharacteristic
531C22#1313326211yescharacteristic
532C22#1313326211yescharacteristic
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—42no—
546C2—4241no—
547C2—2122no—
548C2C2×#1313319011yescharacteristic
549C2#1681113411yescharacteristic
550C2#1681121111yescharacteristic
551C2D4×C22⋊D5311yescharacteristic
552C2D4×C22⋊D5311yescharacteristic
553C2#1681126311yescharacteristic
554C2#1681126311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C22⋊C4×C53C2
hypercenterC22×C22⋊C4D53
hypocenterC53#2424
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13133191C2semidirect
2. C2×#13133190C2semidirect
3. C2×#13133186C2semidirect
4. C2×#13133186C2semidirect
5. C2×C4×C22⋊D53C2semidirect
6. #16810546C2semidirect
7. #16810538C2semidirect
8. #16810538C2semidirect
9. #16810532C2semidirect
10. #16810523C2semidirect
11. #16810516C2semidirect
12. #16810502C2semidirect
13. #16810499C2semidirect
14. #16810496C2semidirect
15. C22×C22⋊C4×C53C2semidirect
16. C22×D212C22semidirect
23. C22×C4×D53C22semidirect
29. C2×C4⋊Dic53C22semidirect
30. C2×#11719919C22semidirect
31. C2×#11719919C22semidirect
32. C2×#11719919C22semidirect
39. C2×#11719921C22semidirect
40. C2×#11719921C22semidirect
41. C2×#11719921C22semidirect
42. C2×#11719921C22semidirect
43. C2×#11719921C22semidirect
44. C2×C4×Dic53C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
48. #13133124C22semidirect
96. C2×C22⋊D53D4semidirect
112. C22⋊C4×C53D4semidirect
118. C22⋊Dic53D4semidirect
233. C22×C53#121semidirect
283. C22×C22⋊C4D53semidirect
293. C53#2424semidirect
405. C22⋊C4C22⋊D53semidirect
525. C22#13133190semidirect