Finite Group Databasebeta

← #20278245

Subgroups — #20278245

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×#13133189C211yesmaximal, characteristic, complemented
3C2×#13133187C211yesmaximal, characteristic, complemented
4C2×#13133185C211yesmaximal, characteristic, complemented
5C22×#11719921C211yesmaximal, characteristic, complemented
6#16810545C211yesmaximal, characteristic, complemented
7#16810540C211yesmaximal, characteristic, complemented
8#16810539C211yesmaximal, characteristic, complemented
9#16810532C211yesmaximal, characteristic, complemented
10#16810525C211yesmaximal, characteristic, complemented
11#16810514C211yesmaximal, characteristic, complemented
12#16810502C211yesmaximal, characteristic, complemented
13#16810502C211yesmaximal, characteristic, complemented
14#16810498C211yesmaximal, characteristic, complemented
15C2×C4×D4×C53C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C22⋊D53C2211yescharacteristic, complemented
23C23×Dic53C2211yescharacteristic
24#13133191—24no—
25#13133189—24no—
26#13133187—24no—
27#13133185—24no—
28C2×C4⋊Dic53C2211yescharacteristic, complemented
29C2×C4⋊Dic53C2211yescharacteristic, complemented
30C2×#11719919C2211yescharacteristic, complemented
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic, complemented
33C2×#11719921—24no—
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic, complemented
43C2×#11719921C2211yescharacteristic, complemented
44C2×#11719921C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47#13133124C2211yescharacteristic, complemented
48#13133124C2211yescharacteristic, complemented
49C22×D4×C53C2211yescharacteristic
50C23×C4×C53C2211yescharacteristic
51C23×C4×C53C2211yescharacteristic, complemented
52C4×D4×C53—24no—
53C2×C4⋊C4×C53C2211yescharacteristic, complemented
54C2×C22⋊C4×C53C2211yescharacteristic
55C2×C22⋊C4×C53C2211yescharacteristic, complemented
56C2×C42×C53C2211yescharacteristic, complemented
57C24×C53C2311yescharacteristic
58C24×C53C2311yescharacteristic
59C23×D53—22no—
60C23×D53—22no—
61C23×D53—22no—
62C23×D53—22no—
63C23×D53—22no—
64C23×D53—41no—
65C23×D53—41no—
66C23×D53C2311yescharacteristic
67C23×D53C2311yescharacteristic
68C23×D53C2311yescharacteristic
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53D414yescomplemented
72C22×C4×C53—22no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53C2311yescharacteristic
79C22×C4×C53C2311yescharacteristic
80C22×C4×C53C2311yescharacteristic
81C22×C4×C53C2311yescharacteristic
82C22×C4×C53C2311yescharacteristic
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—
95C22×Dic53—22no—
96C22×Dic53—22no—
97C22×Dic53—41no—
98C22×Dic53C2311yescharacteristic
99C22×Dic53C2311yescharacteristic
100C22×Dic53C2311yescharacteristic
101C22×Dic53C2311yescharacteristic
102C22×Dic53C2311yescharacteristic
103C2×D212—22no—
104C2×D212—22no—
105C2×D212—22no—
106C2×D212—22no—
107C2×D212—41no—
108C2×D212—41no—
109C4⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53D414yescomplemented
112C42×C53—22no—
113C22⋊Dic53—22no—
114C22⋊Dic53—22no—
115#11719921—22no—
116#11719921—22no—
117#11719921—22no—
118#11719921—22no—
119#11719921—22no—
120#11719921—22no—
121#11719921—22no—
122#11719921—22no—
123#11719921—22no—
124#11719921—41no—
125#11719921D414yescomplemented
126C4⋊Dic53—22no—
127C4⋊Dic53—41no—
128#11719919—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×C53C2×D412yes—
140C23×C53C2×D412yes—
141C23×C53C2×D412yes—
142C23×C53#4112yes—
143C23×C53C2411yescharacteristic
144C22×D53—28no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—21no—
149C22×D53—21no—
150C22×D53—21no—
151C22×D53—21no—
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—
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—21no—
183C2×C4×C53—21no—
184C2×C4×C53—44no—
185C2×C4×C53—44no—
186C2×C4×C53C2×D412yes—
187C2×C4×C53C2×D412yes—
188C2×C4×C53C2×D412yes—
189C2×C4×C53C2×D412yes—
190C22⋊D53—44no—
191C22⋊D53—44no—
192C22⋊D53—44no—
193C22⋊D53—44no—
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×Dic53—44no—
209C2×Dic53C2×D412yes—
210D212—44no—
211D212—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—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53#12114yescomplemented
226C22×C53C22×D411yescharacteristic
227C22×C53C22×D411yescharacteristic
228C22×C53C22×D411yescharacteristic
229C22×C53C2×#4111yescharacteristic
230C22×C53C2×#4111yescharacteristic
231C22×C53H(5,𝔽2)11yescharacteristic
232C22×C53H(5,𝔽2)11yescharacteristic
233C2×D53—22no—
234C2×D53—22no—
235C2×D53—22no—
236C2×D53—22no—
237C2×D53—22no—
238C2×D53—22no—
239C2×D53—416no—
240C2×D53—41no—
241C2×D53—41no—
242C2×D53—41no—
243C2×D53—41no—
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—22no—
258C4×C53—22no—
259C4×C53—41no—
260Dic53—22no—
261Dic53—41no—
262Dic53—41no—
263Dic53—41no—
264Dic53—41no—
265#2424—531nomaximal, Sylow
266C2×C53—24no—
267C2×C53—42no—
268C2×C53C2×#12111yescharacteristic
269C2×C53#52511yescharacteristic
270C2×C53#53811yescharacteristic
271C2×C53D4211yescharacteristic
272C2×C53D4211yescharacteristic
273C2×C53#54611yescharacteristic
274C2×C53#54611yescharacteristic
275D53—42no—
276D53—42no—
277D53—81no—
278C2×#121—531no—
279C2×#121—531no—
280C2×#120—531no—
281C2×#120—531no—
282C2×C4×D4D5311yescharacteristic, complemented
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×D4—531no—
300C22×D4—531no—
301C22×D4C2×D5311yescharacteristic
302C23×C4—531no—
303C23×C4C2×D5311yescharacteristic
304C23×C4C2×D5311yescharacteristic, complemented
305#121—1064no—
306#121—1064no—
307#120—1064no—
308#120—1064no—
309C4×D4—24no—
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×C42C2×D5311yescharacteristic, complemented
331#95—531no—
332#95—531no—
333#95—531no—
334#95—531no—
335C24—531no—
336C24—531no—
337C24—531no—
338C24—1062no—
339C24—1062no—
340C24—1062no—
341C24—1062no—
342C24—1062no—
343C24—2121no—
344C24—2121no—
345C24C22×D5311yescharacteristic
346C24C22×D5311yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
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—2121no—
363C2×D4—2121no—
364C2×D4—2121no—
365C2×D4—2121no—
366C2×D4—2121no—
367C2×D4D21214yescomplemented
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—1062no—
380C22×C4—1062no—
381C22×C4—2121no—
382C22×C4C22×D5311yescharacteristic
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—22no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—532no—
418C23—532no—
419C23—1068no—
420C23—1061no—
421C23—1061no—
422C23—1061no—
423C23—1061no—
424C23—1061no—
425C23—1061no—
426C23—1061no—
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—
445C23C2×D21212yes—
446C23C2×D21212yes—
447C23#1171994512yes—
448C23C2×C22⋊D5312yes—
449C23C23×D5311yescharacteristic
450D4—28no—
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—21no—
468C2×C4—44no—
469C2×C4—44no—
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—2124no—
486C2×C4C2×D21212yes—
487C2×C4D4×D5312yes—
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—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—1062no—
504C22—1062no—
505C22—1062no—
506C22—1062no—
507C22—1062no—
508C22—1062no—
509C22—21216no—
510C22—2121no—
511C22—2121no—
512C22—2121no—
513C22—2121no—
514C22—2121no—
515C22—2121no—
516C22—2121no—
517C22—2121no—
518C22—2121no—
519C22—2121no—
520C22—2121no—
521C22—2121no—
522C22—2121no—
523C22—2121no—
524C22—2121no—
525C22#1313319114yescomplemented
526C22C2×D4×D5311yescharacteristic
527C22C22×C22⋊D5311yescharacteristic
528C22C22×D21211yescharacteristic
529C22#41×D5311yescharacteristic
530C22C2×#1171994511yescharacteristic
531C22#1313326211yescharacteristic
532C22#1313326311yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—1062no—
539C4—2121no—
540C4—2121no—
541C4—2121no—
542C4—2121no—
543C2—24no—
544C2—42no—
545C2—4241no—
546C2—2122no—
547C2—2122no—
548C2C2×#1313319111yescharacteristic
549C2#1681113211yescharacteristic
550C2#1681120611yescharacteristic
551C2D4×C22⋊D5311yescharacteristic
552C2D4×D21211yescharacteristic
553C2#1681126111yescharacteristic
554C2#1681126811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C4×D4×C53C2
hypercenterC2×C4×D4D53
hypocenterC53#2424
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133191C2semidirect
2. C2×#13133189C2semidirect
3. C2×#13133187C2semidirect
4. C2×#13133185C2semidirect
5. C22×#11719921C2semidirect
6. #16810545C2semidirect
7. #16810540C2semidirect
8. #16810539C2semidirect
9. #16810532C2semidirect
10. #16810525C2semidirect
11. #16810514C2semidirect
12. #16810502C2semidirect
13. #16810502C2semidirect
14. #16810498C2semidirect
15. C2×C4×D4×C53C2semidirect
22. C22×C22⋊D53C22semidirect
28. C2×C4⋊Dic53C22semidirect
29. C2×C4⋊Dic53C22semidirect
30. C2×#11719919C22semidirect
32. C2×C22⋊Dic53C22semidirect
42. C2×#11719921C22semidirect
43. C2×#11719921C22semidirect
44. C2×#11719921C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
48. #13133124C22semidirect
51. C23×C4×C53C22semidirect
53. C2×C4⋊C4×C53C22semidirect
55. C2×C22⋊C4×C53C22semidirect
56. C2×C42×C53C22semidirect
71. C2×D4×C53D4semidirect
111. C22⋊C4×C53D4semidirect
125. #11719921D4semidirect
225. C22×C53#121semidirect
282. C2×C4×D4D53semidirect
293. C53#2424semidirect
304. C23×C4C2×D53semidirect
313. C2×C4⋊C4C2×D53semidirect
329. C2×C22⋊C4C2×D53semidirect
330. C2×C42C2×D53semidirect
367. C2×D4D212semidirect
405. C22⋊C4C22⋊D53semidirect
525. C22#13133191semidirect