Finite Group Databasebeta

← #20279644

Subgroups — #20279644

Order 6784.

900 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133202C211yesmaximal, characteristic, complemented
2C2×#13133199C211yesmaximal, characteristic, complemented
3C2×#13133190C211yesmaximal, characteristic, complemented
4C2×#13133187C211yesmaximal, characteristic, complemented
5C2×#13133185C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
7#16810559C211yesmaximal, characteristic, complemented
8#16810558C211yesmaximal, characteristic, complemented
9#16810532C211yesmaximal, characteristic, complemented
10#16810531C211yesmaximal, characteristic, complemented
11#16810525C211yesmaximal, characteristic, complemented
12#16810523C211yesmaximal, characteristic, complemented
13#16810523C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic, complemented
15C53×#390C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C22⋊D53C2211yescharacteristic, complemented
23C22×C4×D53C2211yescharacteristic
24C22×C4×D53C2211yescharacteristic
25#13133202—24no—
26#13133199—24no—
27#13133190—24no—
28#13133187—24no—
29#13133185—24no—
30C2×C4⋊Dic53C2211yescharacteristic, complemented
31C2×#11719919C2211yescharacteristic, complemented
32C2×#11719919C2211yescharacteristic, complemented
33C2×#11719919C2211yescharacteristic, complemented
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic, complemented
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×#11719921C2211yescharacteristic, complemented
44C2×#11719921C2211yescharacteristic, complemented
45C2×#11719921C2211yescharacteristic, complemented
46C22⋊C4×D53—24no—
47C2×C4×Dic53C2211yescharacteristic, complemented
48#13133124C2211yescharacteristic, complemented
49#13133124C2211yescharacteristic, complemented
50C22×D4×C53C2211yescharacteristic
51C2×C4⋊C4×C53C2211yescharacteristic, complemented
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic, complemented
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×C53—41no—
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C2×C22⋊D53—22no—
78C2×C22⋊D53—22no—
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—41no—
86C2×C22⋊D53—41no—
87C2×C22⋊D53—41no—
88C2×C22⋊D53—41no—
89C22×Dic53C2311yescharacteristic
90C22×Dic53C2311yescharacteristic
91C22×Dic53C2311yescharacteristic
92C22×Dic53C2311yescharacteristic
93C22×Dic53C2311yescharacteristic
94C2×D212—22no—
95C2×D212—22no—
96C2×D212—22no—
97C2×D212—22no—
98C2×D212—22no—
99C2×D212—41no—
100C2×C4×D53—22no—
101C2×C4×D53—22no—
102C2×C4×D53—22no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C4×D53D414yescomplemented
106C4⋊C4×C53—22no—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C42×C53—41no—
112C22⋊Dic53—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#11719921D414yescomplemented
123#11719921D414yescomplemented
124C4⋊Dic53—22no—
125#11719919—22no—
126#11719919—22no—
127#11719919—22no—
128C4×Dic53—41no—
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#4112yes—
143C23×C53C2411yescharacteristic
144C22×D53—21no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—28no—
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—44no—
163C22×D53—44no—
164C22×D53—44no—
165C22×D53—44no—
166C22×D53—44no—
167C22×D53—44no—
168C22×D53C2×D412yes—
169C22×D53C2×D412yes—
170C22×D53C2×D412yes—
171C22×D53#4112yes—
172D4×C53—44no—
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—21no—
185C2×C4×C53C2×D412yes—
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×Dic53C2×D412yes—
205C2×Dic53C2×D412yes—
206C2×Dic53C2×D412yes—
207D212—44no—
208D212—44no—
209C4×D53—44no—
210C4×D53—44no—
211C22×C53—22no—
212C22×C53—22no—
213C22×C53—22no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53C22×D411yescharacteristic
226C22×C53C22×D411yescharacteristic
227C22×C53C22×D411yescharacteristic
228C22×C53C2×#4111yescharacteristic
229C22×C53C2×#4111yescharacteristic
230C22×C53H(5,𝔽2)11yescharacteristic
231C22×C53H(5,𝔽2)11yescharacteristic
232C2×D53—22no—
233C2×D53—22no—
234C2×D53—22no—
235C2×D53—22no—
236C2×D53—22no—
237C2×D53—22no—
238C2×D53—22no—
239C2×D53—22no—
240C2×D53—22no—
241C2×D53—416no—
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#12714yes—
254C4×C53—22no—
255C4×C53—22no—
256C4×C53—22no—
257C4×C53—41no—
258C4×C53—41no—
259Dic53—22no—
260Dic53—22no—
261Dic53—22no—
262Dic53—41no—
263Dic53—41no—
264#2539—531nomaximal, Sylow
265C2×C53—42no—
266C2×C53—81no—
267C2×C53#52511yescharacteristic
268C2×C53C2×#12711yescharacteristic
269C2×C53#53411yescharacteristic
270C2×C53#54611yescharacteristic
271C2×C53#54611yescharacteristic
272C2×C53#54611yescharacteristic
273C2×C53#54611yescharacteristic
274D53—24no—
275D53—42no—
276D53—81no—
277C2×#123—531no—
278C2×#123—531no—
279C2×#121—531no—
280C2×#120—531no—
281C2×#120—531no—
282C22×C22⋊C4—531no—
283#399—531no—
284#399—531no—
285#392—531no—
286#392—531no—
287#390—531no—
288#390—531no—
289#390—531no—
290#390D5311yescharacteristic, complemented
291C4×C22⋊C4—531no—
292C53#253911yescharacteristic, complemented, Sylow
293C25—531no—
294C22×D4—531no—
295C22×D4—531no—
296C22×D4—531no—
297C22×D4—531no—
298C22×D4—531no—
299C22×D4—531no—
300C22×D4C2×D5311yescharacteristic
301C23×C4—531no—
302C23×C4—531no—
303#123—1064no—
304#123—1064no—
305#121—1064no—
306#120—1064no—
307#120—1064no—
308C2×C4⋊C4—531no—
309C2×C4⋊C4—531no—
310C2×C4⋊C4—531no—
311C2×C4⋊C4—531no—
312C2×C4⋊C4C2×D5311yescharacteristic, complemented
313C2×C22⋊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—1064no—
326C2×C22⋊C4C2×D5311yescharacteristic
327C2×C22⋊C4C2×D5311yescharacteristic
328C2×C22⋊C4C2×D5311yescharacteristic, complemented
329C2×C22⋊C4C2×D5311yescharacteristic, complemented
330C2×C42—531no—
331C2×C42C2×D5311yescharacteristic, complemented
332#95—531no—
333#95—531no—
334C24—531no—
335C24—531no—
336C24—531no—
337C24—1062no—
338C24—1062no—
339C24—1062no—
340C24—1062no—
341C24—1062no—
342C24—2121no—
343C24—2121no—
344C24C22×D5311yescharacteristic
345C24C22×D5311yescharacteristic
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—41no—
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—
367C22×C4—531no—
368C22×C4—531no—
369C22×C4—531no—
370C22×C4—534no—
371C22×C4—531no—
372C22×C4—531no—
373C22×C4—1062no—
374C22×C4—1062no—
375C22×C4—1062no—
376C22×C4—2121no—
377C22×C4—2121no—
378C22×C4C22×D5311yescharacteristic
379C22×C4C22×D5311yescharacteristic
380C22×C4C22×D5311yescharacteristic
381C22×C4C22×D5311yescharacteristic
382C22×C4C22×D5311yescharacteristic
383C4⋊C4—22no—
384C4⋊C4—1062no—
385C4⋊C4—1062no—
386C4⋊C4—1062no—
387C4⋊C4—1062no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—534no—
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—
404C42—41no—
405C42—2121no—
406C23—21no—
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—532no—
420C23—532no—
421C23—532no—
422C23—532no—
423C23—1068no—
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—1061no—
440C23—1061no—
441C23—2124no—
442C23—2124no—
443C23—2124no—
444C23—2124no—
445C23—2124no—
446C23—2124no—
447C23#1171994512yes—
448C23C23×D5311yescharacteristic
449D4—44no—
450D4—2124no—
451D4—2124no—
452D4—2124no—
453D4—2124no—
454D4—2124no—
455D4—2124no—
456C2×C4—21no—
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—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—2124no—
484C2×C4—2124no—
485C2×C4D4×D5312yes—
486C2×C4D4×D5312yes—
487C2×C4D4×D5312yes—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—534no—
503C22—1062no—
504C22—1062no—
505C22—1062no—
506C22—1062no—
507C22—1062no—
508C22—1062no—
509C22—1062no—
510C22—1062no—
511C22—1062no—
512C22—21216no—
513C22—2121no—
514C22—2121no—
515C22—2121no—
516C22—2121no—
517C22—2121no—
518C22—2121no—
519C22—2121no—
520C22—2121no—
521C22—2121no—
522C22—2121no—
523C22—2121no—
524C22C2×D4×D5311yescharacteristic
525C22C2×D4×D5311yescharacteristic
526C22C2×D4×D5311yescharacteristic
527C22#41×D5311yescharacteristic
528C22C2×#1171994511yescharacteristic
529C22#1313326211yescharacteristic
530C22#1313326311yescharacteristic
531C4—22no—
532C4—22no—
533C4—22no—
534C4—41no—
535C4—41no—
536C4—1062no—
537C4—1062no—
538C4—1062no—
539C4—2121no—
540C4—2121no—
541C2—42no—
542C2—81no—
543C2—1064no—
544C2—4241no—
545C2—2122no—
546C2#1681113211yescharacteristic
547C2#127×D5311yescharacteristic
548C2#1681118111yescharacteristic
549C2#1681126211yescharacteristic
550C2#1681126311yescharacteristic
551C2#1681126611yescharacteristic
552C2#1681126911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2539
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13133202C2semidirect
2. C2×#13133199C2semidirect
3. C2×#13133190C2semidirect
4. C2×#13133187C2semidirect
5. C2×#13133185C2semidirect
6. C2×C22⋊C4×D53C2semidirect
7. #16810559C2semidirect
8. #16810558C2semidirect
9. #16810532C2semidirect
10. #16810531C2semidirect
11. #16810525C2semidirect
12. #16810523C2semidirect
13. #16810523C2semidirect
14. C4×#11719921C2semidirect
15. C53×#390C2semidirect
22. C22×C22⋊D53C22semidirect
30. C2×C4⋊Dic53C22semidirect
31. C2×#11719919C22semidirect
32. C2×#11719919C22semidirect
33. C2×#11719919C22semidirect
36. C2×C22⋊Dic53C22semidirect
43. C2×#11719921C22semidirect
44. C2×#11719921C22semidirect
45. C2×#11719921C22semidirect
47. C2×C4×Dic53C22semidirect
48. #13133124C22semidirect
49. #13133124C22semidirect
51. C2×C4⋊C4×C53C22semidirect
54. C2×C22⋊C4×C53C22semidirect
55. C2×C22⋊C4×C53C22semidirect
56. C2×C42×C53C22semidirect
105. C2×C4×D53D4semidirect
122. #11719921D4semidirect
123. #11719921D4semidirect
290. #390D53semidirect
292. C53#2539semidirect
312. C2×C4⋊C4C2×D53semidirect
328. C2×C22⋊C4C2×D53semidirect
329. C2×C22⋊C4C2×D53semidirect
331. C2×C42C2×D53semidirect