Finite Group Databasebeta

← #20282711

Subgroups — #20282711

Order 6784.

712 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133187C211yesmaximal, characteristic, complemented
2C2×#13133185C211yesmaximal, characteristic, complemented
3#16810551C211yesmaximal, characteristic, complemented
4#16810550C211yesmaximal, characteristic, complemented
5#16810549C211yesmaximal, characteristic, complemented
6#16810541C211yesmaximal, characteristic, complemented
7#16810540C211yesmaximal, characteristic, complemented
8#16810539C211yesmaximal, characteristic, complemented
9#16810539C211yesmaximal, characteristic, complemented
10#16810538C211yesmaximal, characteristic, complemented
11#16810532C211yesmaximal, characteristic, complemented
12#16810532C211yesmaximal, characteristic, complemented
13#16810531C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic, complemented
15C53×#394C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C22⋊D53C2211yescharacteristic, complemented
23#13133187—24no—
24#13133185—24no—
25C2×C4⋊Dic53C2211yescharacteristic, complemented
26C2×#11719919C2211yescharacteristic, complemented
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic, complemented
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic, complemented
39C2×#11719921C2211yescharacteristic, complemented
40C2×#11719921C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46C22×D4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic, complemented
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic, complemented
51C2×C22⋊C4×C53C2211yescharacteristic, complemented
52#95×C53C2211yescharacteristic, complemented
53C24×C53C2311yescharacteristic
54C24×C53C2311yescharacteristic
55C23×D53—41no—
56C23×D53—41no—
57C23×D53—41no—
58C23×D53—41no—
59C23×D53—41no—
60C23×D53—41no—
61C23×D53D414yescomplemented
62C23×D53C2311yescharacteristic
63C23×D53C2311yescharacteristic
64C23×D53C2311yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C2×C22⋊D53—22no—
74C2×C22⋊D53—22no—
75C2×C22⋊D53—22no—
76C2×C22⋊D53—41no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53—41no—
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C2×C22⋊D53—41no—
82C2×C22⋊D53—41no—
83C2×C22⋊D53—41no—
84C2×C22⋊D53—41no—
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C4⋊C4×C53—41no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊Dic53—22no—
102C22⋊Dic53—41no—
103C22⋊Dic53—41no—
104#11719921—22no—
105#11719921—22no—
106#11719921—22no—
107#11719921—22no—
108#11719921—41no—
109#11719921—41no—
110#11719921—41no—
111#11719921—41no—
112#11719921—41no—
113#11719921—41no—
114#11719921—41no—
115C4⋊Dic53—41no—
116#11719919—41no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53C2×D412yes—
131C23×C53C2411yescharacteristic
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—21no—
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—21no—
149C22×D53—21no—
150C22×D53—21no—
151C22×D53—44no—
152C22×D53—44no—
153C22×D53—44no—
154C22×D53—44no—
155C22×D53—44no—
156C22×D53—44no—
157C22×D53—44no—
158C22×D53C2×D412yes—
159C22×D53C2×D412yes—
160D4×C53—44no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C22⋊D53—44no—
177C22⋊D53—44no—
178C22⋊D53—44no—
179C22⋊D53—44no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—21no—
192C2×Dic53—21no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195D212—44no—
196D212—44no—
197C22×C53—22no—
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53C22×D411yescharacteristic
212C22×C53H(5,𝔽2)11yescharacteristic
213C22×C53H(5,𝔽2)11yescharacteristic
214C22×C53H(5,𝔽2)11yescharacteristic
215C22×C53H(5,𝔽2)11yescharacteristic
216C22×C53H(5,𝔽2)11yescharacteristic
217C22×C53H(5,𝔽2)11yescharacteristic
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53—22no—
223C2×D53—22no—
224C2×D53—416no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C2×D53—41no—
229C2×D53—41no—
230C2×D53—41no—
231C2×D53—41no—
232C2×D53—41no—
233C2×D53—41no—
234C2×D53—41no—
235C2×D53—41no—
236C2×D53—41no—
237C2×D53—41no—
238C2×D53—41no—
239C2×D53—41no—
240C4×C53—41no—
241C4×C53—41no—
242C4×C53—41no—
243C4×C53—41no—
244C4×C53—41no—
245Dic53—41no—
246Dic53—41no—
247Dic53—41no—
248Dic53—41no—
249Dic53—41no—
250#2812—531nomaximal, Sylow
251C2×C53—42no—
252C2×C53—81no—
253C2×C53#53411yescharacteristic
254C2×C53#53411yescharacteristic
255C2×C53#53411yescharacteristic
256C2×C53#56011yescharacteristic
257C2×C53#56011yescharacteristic
258C2×C53#56011yescharacteristic
259C2×C53H(3,𝔽4)11yescharacteristic
260D53—42no—
261D53—42no—
262D53—81no—
263C2×#120—531no—
264C2×#120—531no—
265#397—531no—
266#397—531no—
267#397—531no—
268#394—531no—
269#394—531no—
270#394—531no—
271#394—531no—
272#394—531no—
273#394D5311yescharacteristic, complemented
274#392—531no—
275#392—531no—
276#392—531no—
277#379—531no—
278C53#281211yescharacteristic, complemented, Sylow
279C25—531no—
280C22×D4—531no—
281C22×D4—531no—
282C22×D4—531no—
283C22×D4—531no—
284C22×D4—531no—
285C22×D4—531no—
286C22×D4C2×D5311yescharacteristic
287#120—1064no—
288#120—1064no—
289C2×C4⋊C4—531no—
290C2×C4⋊C4—531no—
291C2×C4⋊C4C2×D5311yescharacteristic, complemented
292C2×C22⋊C4—531no—
293C2×C22⋊C4—531no—
294C2×C22⋊C4—531no—
295C2×C22⋊C4—531no—
296C2×C22⋊C4—531no—
297C2×C22⋊C4—531no—
298C2×C22⋊C4—531no—
299C2×C22⋊C4—531no—
300C2×C22⋊C4—531no—
301C2×C22⋊C4—531no—
302C2×C22⋊C4—531no—
303C2×C22⋊C4—531no—
304C2×C22⋊C4—531no—
305C2×C22⋊C4—531no—
306C2×C22⋊C4C2×D5311yescharacteristic
307C2×C22⋊C4C2×D5311yescharacteristic
308C2×C22⋊C4C2×D5311yescharacteristic, complemented
309C2×C22⋊C4C2×D5311yescharacteristic, complemented
310#95—531no—
311#95—531no—
312#95—531no—
313#95—531no—
314#95—531no—
315#95C2×D5311yescharacteristic, complemented
316C24—531no—
317C24—531no—
318C24—534no—
319C24—531no—
320C24—2121no—
321C24—2121no—
322C24—2121no—
323C24—2121no—
324C24—2121no—
325C24—2121no—
326C24C22×D5311yescharacteristic
327C24C22×D5311yescharacteristic
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—1062no—
332C2×D4—1062no—
333C2×D4—1062no—
334C2×D4—1062no—
335C2×D4—1062no—
336C2×D4—2121no—
337C2×D4—2121no—
338C2×D4—2121no—
339C2×D4—2121no—
340C2×D4—2121no—
341C2×D4—2121no—
342C2×D4—2121no—
343C2×D4—2121no—
344C2×D4—2121no—
345C2×D4—2121no—
346C2×D4—2121no—
347C2×D4—2121no—
348C2×D4—2121no—
349C22×C4—531no—
350C22×C4—531no—
351C22×C4—531no—
352C22×C4—531no—
353C22×C4—531no—
354C22×C4C22×D5311yescharacteristic
355C22×C4C22×D5311yescharacteristic
356C22×C4C22×D5311yescharacteristic
357C22×C4C22×D5311yescharacteristic
358C22×C4C22×D5311yescharacteristic
359C4⋊C4—41no—
360C4⋊C4—2121no—
361C4⋊C4—2121no—
362C22⋊C4—22no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—1062no—
367C22⋊C4—1062no—
368C22⋊C4—1062no—
369C22⋊C4—1062no—
370C22⋊C4—1062no—
371C22⋊C4—2121no—
372C22⋊C4—2121no—
373C22⋊C4—2121no—
374C22⋊C4—2121no—
375C22⋊C4—2121no—
376C22⋊C4—2121no—
377C22⋊C4—2121no—
378C22⋊C4—2121no—
379C22⋊C4—2121no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—532no—
394C23—532no—
395C23—1061no—
396C23—1061no—
397C23—1061no—
398C23—1061no—
399C23—1061no—
400C23—1061no—
401C23—1061no—
402C23—1061no—
403C23—1061no—
404C23—1061no—
405C23—1061no—
406C23—1061no—
407C23—1061no—
408C23—1061no—
409C23—1061no—
410C23—1061no—
411C23—1061no—
412C23—1061no—
413C23—1061no—
414C23—2124no—
415C23—2124no—
416C23—2124no—
417C23—2124no—
418C23—2124no—
419C23—2124no—
420C23—2124no—
421C23D4×D5312yes—
422C23C23×D5311yescharacteristic
423D4—44no—
424D4—2124no—
425D4—2124no—
426D4—2124no—
427D4—2124no—
428D4—2124no—
429D4—2124no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—1061no—
446C2×C4—1061no—
447C2×C4—1061no—
448C2×C4—1061no—
449C2×C4—1061no—
450C2×C4—1061no—
451C2×C4—1061no—
452C2×C4—1061no—
453C2×C4—1061no—
454C2×C4—1061no—
455C2×C4—1061no—
456C2×C4—1061no—
457C2×C4—1061no—
458C2×C4—1061no—
459C2×C4—1061no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—1062no—
475C22—1062no—
476C22—1062no—
477C22—1062no—
478C22—1062no—
479C22—1062no—
480C22—21216no—
481C22—2121no—
482C22—2121no—
483C22—2121no—
484C22—2121no—
485C22—2121no—
486C22—2121no—
487C22—2121no—
488C22—2121no—
489C22—2121no—
490C22—2121no—
491C22—2121no—
492C22—2121no—
493C22—2121no—
494C22—2121no—
495C22—2121no—
496C22C2×D4×D5311yescharacteristic
497C22#1313326211yescharacteristic
498C22#1313326211yescharacteristic
499C22#1313326211yescharacteristic
500C22#1313326311yescharacteristic
501C22#1313326311yescharacteristic
502C22#1313326311yescharacteristic
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—2121no—
509C4—2121no—
510C4—2121no—
511C4—2121no—
512C4—2121no—
513C2—42no—
514C2—81no—
515C2—4241no—
516C2—2122no—
517C2—2122no—
518C2#1681118111yescharacteristic
519C2#1681118311yescharacteristic
520C2#1681118411yescharacteristic
521C2#1681136111yescharacteristic
522C2#1681136111yescharacteristic
523C2#1681136311yescharacteristic
524C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2812
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133187C2semidirect
2. C2×#13133185C2semidirect
3. #16810551C2semidirect
4. #16810550C2semidirect
5. #16810549C2semidirect
6. #16810541C2semidirect
7. #16810540C2semidirect
8. #16810539C2semidirect
9. #16810539C2semidirect
10. #16810538C2semidirect
11. #16810532C2semidirect
12. #16810532C2semidirect
13. #16810531C2semidirect
14. #16810478C2semidirect
15. C53×#394C2semidirect
22. C22×C22⋊D53C22semidirect
25. C2×C4⋊Dic53C22semidirect
26. C2×#11719919C22semidirect
29. C2×C22⋊Dic53C22semidirect
38. C2×#11719921C22semidirect
39. C2×#11719921C22semidirect
40. C2×#11719921C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
47. C2×C4⋊C4×C53C22semidirect
50. C2×C22⋊C4×C53C22semidirect
51. C2×C22⋊C4×C53C22semidirect
52. #95×C53C22semidirect
61. C23×D53D4semidirect
273. #394D53semidirect
278. C53#2812semidirect
291. C2×C4⋊C4C2×D53semidirect
308. C2×C22⋊C4C2×D53semidirect
309. C2×C22⋊C4C2×D53semidirect
315. #95C2×D53semidirect