Finite Group Databasebeta

← #20280919

Subgroups — #20280919

Order 6784.

816 conjugacy classes of subgroups.

The subgroup lattice has 542 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
4#16810559C211yesmaximal, characteristic, complemented
5#16810551C211yesmaximal, characteristic, complemented
6#16810548C211yesmaximal, characteristic, complemented
7#16810538C211yesmaximal, characteristic, complemented
8#16810537C211yesmaximal, characteristic, complemented
9#16810531C211yesmaximal, characteristic, complemented
10#16810503C211yesmaximal, characteristic, complemented
11#16810503C211yesmaximal, characteristic, complemented
12#16810499C211yesmaximal, characteristic, complemented
13#16810499C211yesmaximal, characteristic, complemented
14#16810479C211yesmaximal, characteristic, complemented
15C2×#120×C53C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic, complemented
20C23×Dic53C2211yescharacteristic, complemented
21C22×C4×D53C2211yescharacteristic, complemented
22#13133189—24no—
23#13133188—24no—
24#13133186—24no—
25C2×C4⋊Dic53C2211yescharacteristic, complemented
26C2×#11719919C2211yescharacteristic, complemented
27C2×#11719919C2211yescharacteristic, complemented
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic, complemented
38C2×C22⋊Dic53C2211yescharacteristic, complemented
39C2×#11719921C2211yescharacteristic, complemented
40C2×#11719921C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47C25×C53C2211yescharacteristic
48C22×D4×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51#120×C53—24no—
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53—22no—
56C24×C53—22no—
57C24×C53—22no—
58C24×C53—41no—
59C24×C53—41no—
60C24×C53—41no—
61C24×C53—41no—
62C24×C53C2311yescharacteristic
63C24×C53C2311yescharacteristic
64C24×C53C2311yescharacteristic
65C24×C53C2311yescharacteristic
66C23×D53C2311yescharacteristic
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53D414yescomplemented
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—41no—
86C2×C22⋊D53—41no—
87C2×C22⋊D53—41no—
88C2×C22⋊D53—41no—
89C2×C22⋊D53—41no—
90C2×C22⋊D53D414yescomplemented
91C22×Dic53—22no—
92C22×Dic53—41no—
93C22×Dic53—41no—
94C22×Dic53C2311yescharacteristic
95C22×Dic53C2311yescharacteristic
96C22×Dic53C2311yescharacteristic
97C22×Dic53C2311yescharacteristic
98C22×Dic53C2311yescharacteristic
99C22×Dic53C2311yescharacteristic
100C22×Dic53C2311yescharacteristic
101C2×C4×D53—22no—
102C2×C4×D53—41no—
103C2×C4×D53—41no—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊Dic53—22no—
108C22⋊Dic53—22no—
109C22⋊Dic53—22no—
110C22⋊Dic53—22no—
111C22⋊Dic53—22no—
112C22⋊Dic53—22no—
113C22⋊Dic53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53—41no—
116C22⋊Dic53—41no—
117C22⋊Dic53—41no—
118#11719921—22no—
119#11719921—41no—
120C4⋊Dic53—22no—
121#11719919—22no—
122#11719919—41no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
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—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—44no—
148C23×C53—44no—
149C23×C53—44no—
150C23×C53—44no—
151C23×C53—44no—
152C23×C53—44no—
153C23×C53—44no—
154C23×C53C2×D412yes—
155C23×C53C2×D412yes—
156C23×C53C2×D412yes—
157C23×C53#4112yes—
158C23×C53C2411yescharacteristic
159C22×D53—21no—
160C22×D53—21no—
161C22×D53—21no—
162C22×D53—21no—
163C22×D53—21no—
164C22×D53—21no—
165C22×D53C2×D412yes—
166D4×C53—44no—
167D4×C53—44no—
168D4×C53—44no—
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—
176C2×C4×C53—21no—
177C2×C4×C53C2×D412yes—
178C22⋊D53—44no—
179C22⋊D53—44no—
180C22⋊D53—44no—
181C22⋊D53—44no—
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—
195C2×Dic53—21no—
196C2×Dic53—21no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—44no—
203C2×Dic53C2×D412yes—
204C4×D53—44no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53—22no—
209C22×C53—22no—
210C22×C53—22no—
211C22×C53—22no—
212C22×C53—22no—
213C22×C53—22no—
214C22×C53—22no—
215C22×C53—22no—
216C22×C53—22no—
217C22×C53—416no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
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×C53C22×D411yescharacteristic
235C22×C53C22×D411yescharacteristic
236C22×C53C2×#4111yescharacteristic
237C22×C53H(5,𝔽2)11yescharacteristic
238C22×C53H(5,𝔽2)11yescharacteristic
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—
248C4×C53—22no—
249C4×C53—41no—
250C4×C53—41no—
251Dic53—22no—
252Dic53—41no—
253Dic53—41no—
254Dic53—41no—
255Dic53—41no—
256Dic53—41no—
257Dic53—41no—
258#2668—531nomaximal, Sylow
259C2×C53—42no—
260C2×C53—42no—
261C2×C53—42no—
262C2×C53—42no—
263C2×C53#53411yescharacteristic
264C2×C53#53411yescharacteristic
265C2×C53#53811yescharacteristic
266C2×C53D4211yescharacteristic
267C2×C53#54611yescharacteristic
268C2×C53#54611yescharacteristic
269C2×C53#56011yescharacteristic
270D53—42no—
271C2×#121—531no—
272C2×#121—531no—
273C2×#120—531no—
274C2×#120D5311yescharacteristic, complemented
275#399—531no—
276#397—531no—
277#397—531no—
278#394—531no—
279#394—531no—
280#392—531no—
281#386—531no—
282#386—531no—
283#386—531no—
284#386—531no—
285#379—531no—
286C53#266811yescharacteristic, complemented, Sylow
287C25C2×D5311yescharacteristic
288C22×D4—531no—
289C22×D4—531no—
290C22×D4—531no—
291C22×D4—531no—
292C22×D4C2×D5311yescharacteristic
293C22×D4C2×D5311yescharacteristic
294C22×D4C2×D5311yescharacteristic
295C23×C4—531no—
296C23×C4—531no—
297#121—1064no—
298#121—1064no—
299#120—24no—
300#120—1064no—
301C2×C4⋊C4—531no—
302C2×C4⋊C4—531no—
303C2×C4⋊C4—531no—
304C2×C22⋊C4—531no—
305C2×C22⋊C4—531no—
306C2×C22⋊C4—531no—
307C2×C22⋊C4—531no—
308C2×C22⋊C4—531no—
309C2×C22⋊C4—531no—
310C2×C22⋊C4—531no—
311C2×C22⋊C4—531no—
312C2×C22⋊C4—531no—
313C2×C22⋊C4—531no—
314C2×C22⋊C4—531no—
315C2×C22⋊C4—531no—
316C2×C22⋊C4—531no—
317C2×C22⋊C4C2×D5311yescharacteristic
318C2×C22⋊C4C2×D5311yescharacteristic
319C2×C22⋊C4C2×D5311yescharacteristic
320#95—531no—
321#95—531no—
322#95—531no—
323#95—531no—
324#95—531no—
325#95—531no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—41no—
330C24—41no—
331C24—41no—
332C24—41no—
333C24—531no—
334C24C22×D5311yescharacteristic
335C24C22×D5311yescharacteristic
336C24C22×D5311yescharacteristic
337C24C22×D5311yescharacteristic
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—534no—
347C2×D4—1062no—
348C2×D4—1062no—
349C2×D4—1062no—
350C2×D4—1062no—
351C2×D4—1062no—
352C2×D4—1062no—
353C2×D4—2121no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4—2121no—
358C2×D4C22⋊D5314yescomplemented
359C22×C4—531no—
360C22×C4—531no—
361C22×C4—531no—
362C22×C4—531no—
363C22×C4—531no—
364C22×C4—531no—
365C22×C4—531no—
366C22×C4—1062no—
367C22×C4—1062no—
368C22×C4—2121no—
369C22×C4—2121no—
370C22×C4—2121no—
371C22×C4—2121no—
372C22×C4C22×D5311yescharacteristic
373C22×C4C22×D5311yescharacteristic
374C22×C4C22×D5311yescharacteristic
375C4⋊C4—1062no—
376C4⋊C4—1062no—
377C4⋊C4—2121no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—1062no—
382C22⋊C4—1062no—
383C22⋊C4—1062no—
384C22⋊C4—1062no—
385C22⋊C4—1062no—
386C22⋊C4—1062no—
387C22⋊C4—1062no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C22⋊C4—2121no—
392C22⋊C4—2121no—
393C22⋊C4—2121no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
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—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—532no—
426C23—1061no—
427C23—1061no—
428C23—1061no—
429C23—1061no—
430C23—1061no—
431C23—1061no—
432C23D4×D5312yes—
433C23#1171994712yes—
434C23C2×C22⋊D5312yes—
435C23C2×C22⋊D5312yes—
436C23C23×D5311yescharacteristic
437D4—44no—
438D4—44no—
439D4—44no—
440D4—2124no—
441D4—2124no—
442D4—2124no—
443D4—2124no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—532no—
453C2×C4—1061no—
454C2×C4—1061no—
455C2×C4—1061no—
456C2×C4—1061no—
457C2×C4—1061no—
458C2×C4—1061no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—1061no—
463C2×C4—1061no—
464C2×C4—1061no—
465C2×C4—1061no—
466C2×C4—1061no—
467C2×C4—1061no—
468C2×C4—1061no—
469C2×C4—1061no—
470C2×C4—1061no—
471C2×C4—1061no—
472C2×C4—1061no—
473C2×C4—2124no—
474C2×C4—2124no—
475C2×C4C2×C22⋊D5312yes—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—416no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—1062no—
506C22—1062no—
507C22—1062no—
508C22—2121no—
509C22—2121no—
510C22—2121no—
511C22—2121no—
512C22C2×D4×D5311yescharacteristic
513C22C22×C22⋊D5311yescharacteristic
514C22C2×#1171994711yescharacteristic
515C22#1313326211yescharacteristic
516C22#1313326211yescharacteristic
517C22#1313326211yescharacteristic
518C22#1313326211yescharacteristic
519C4—22no—
520C4—41no—
521C4—41no—
522C4—1062no—
523C4—2121no—
524C4—2121no—
525C4—2121no—
526C4—2121no—
527C4—2121no—
528C4—2121no—
529C2—42no—
530C2—42no—
531C2—42no—
532C2—42no—
533C2—2122no—
534C2#1681118211yescharacteristic
535C2#1681118311yescharacteristic
536C2#1681120811yescharacteristic
537C2D4×C22⋊D5311yescharacteristic
538C2#1681126011yescharacteristic
539C2#1681126711yescharacteristic
540C2#1681136211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#120×C53C2
hypercenterC2×#120D53
hypocenterC53#2668
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. #16810559C2semidirect
5. #16810551C2semidirect
6. #16810548C2semidirect
7. #16810538C2semidirect
8. #16810537C2semidirect
9. #16810531C2semidirect
10. #16810503C2semidirect
11. #16810503C2semidirect
12. #16810499C2semidirect
13. #16810499C2semidirect
14. #16810479C2semidirect
15. C2×#120×C53C2semidirect
19. C22×C22⋊D53C22semidirect
20. C23×Dic53C22semidirect
21. C22×C4×D53C22semidirect
25. C2×C4⋊Dic53C22semidirect
26. C2×#11719919C22semidirect
27. C2×#11719919C22semidirect
37. C2×C22⋊Dic53C22semidirect
38. C2×C22⋊Dic53C22semidirect
39. C2×#11719921C22semidirect
40. C2×#11719921C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
75. C2×D4×C53D4semidirect
90. C2×C22⋊D53D4semidirect
274. C2×#120D53semidirect
286. C53#2668semidirect
358. C2×D4C22⋊D53semidirect