Finite Group Databasebeta

← #20281038

Subgroups — #20281038

Order 6784.

802 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133199C211yesmaximal, characteristic, complemented
2C2×#13133192C211yesmaximal, characteristic, complemented
3C2×#13133187C211yesmaximal, characteristic, complemented
4C2×#13133187C211yesmaximal, characteristic, complemented
5#16810550C211yesmaximal, characteristic, complemented
6#16810545C211yesmaximal, characteristic, complemented
7#16810541C211yesmaximal, characteristic, complemented
8#16810539C211yesmaximal, characteristic, complemented
9#16810533C211yesmaximal, characteristic, complemented
10#16810532C211yesmaximal, characteristic, complemented
11#16810525C211yesmaximal, characteristic, complemented
12#16810525C211yesmaximal, characteristic, complemented
13#16810501C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C22×D212C2211yescharacteristic
21C22×D212C2211yescharacteristic, complemented
22C22×C22⋊D53C2211yescharacteristic
23C22×C22⋊D53C2211yescharacteristic
24C22×C4×D53C2211yescharacteristic, complemented
25#13133199—24no—
26#13133192—24no—
27#13133187—24no—
28#13133187—24no—
29C2×C4⋊Dic53C2211yescharacteristic, complemented
30C2×C4⋊Dic53C2211yescharacteristic, complemented
31C2×#11719919C2211yescharacteristic, complemented
32C2×C22⋊Dic53C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic, complemented
42C2×#11719921C2211yescharacteristic, complemented
43C2×#11719921C2211yescharacteristic, complemented
44C2×#11719921C2211yescharacteristic, complemented
45C2×#11719921C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47#13133124C2211yescharacteristic, complemented
48C2×C4⋊C4×C53C2211yescharacteristic, complemented
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C42×C53C2211yescharacteristic, complemented
53#95×C53C2211yescharacteristic, complemented
54#95×C53C2211yescharacteristic, complemented
55C24×C53C2311yescharacteristic
56C23×D53—22no—
57C23×D53—22no—
58C23×D53—22no—
59C23×D53—41no—
60C23×D53—41no—
61C23×D53—41no—
62C23×D53—41no—
63C23×D53C2311yescharacteristic
64C23×D53C2311yescharacteristic
65C23×D53C2311yescharacteristic
66C23×D53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
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—
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—22no—
86C2×D212—22no—
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212D414yescomplemented
98C2×C4×D53—22no—
99C2×C4×D53—22no—
100C2×C4×D53—41no—
101C4⋊C4×C53—22no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53D414yescomplemented
105C42×C53—41no—
106C22⋊Dic53—22no—
107#11719921—22no—
108#11719921—22no—
109#11719921—22no—
110#11719921—22no—
111#11719921—22no—
112#11719921—22no—
113#11719921—22no—
114#11719921—41no—
115#11719921—41no—
116#11719921—41no—
117#11719921—41no—
118#11719921—41no—
119#11719921—41no—
120C4⋊Dic53—22no—
121C4⋊Dic53—41no—
122#11719919—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53C2×D412yes—
130C23×C53C2411yescharacteristic
131C22×D53—21no—
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—21no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—21no—
156C22×D53—44no—
157C22×D53—44no—
158C22×D53—44no—
159C22×D53—44no—
160C22×D53—44no—
161C22×D53—44no—
162C22×D53—44no—
163C22×D53C2×D412yes—
164C22×D53C2×D412yes—
165C22×D53#4112yes—
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—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53C2×D412yes—
182C2×C4×C53C2×D412yes—
183C2×C4×C53C2×D412yes—
184C22⋊D53—44no—
185C22⋊D53—44no—
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—
198D212—44no—
199D212—44no—
200D212—44no—
201D212—44no—
202D212—44no—
203C4×D53—44no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53C22×D411yescharacteristic
212C22×C53C22×D411yescharacteristic
213C22×C53C2×#4111yescharacteristic
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—22no—
225C2×D53—22no—
226C2×D53—22no—
227C2×D53—416no—
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—
240C2×D53—41no—
241C2×D53—41no—
242C2×D53—41no—
243C2×D53—41no—
244C2×D53—41no—
245C2×D53—41no—
246C2×D53—41no—
247C4×C53—22no—
248C4×C53—22no—
249C4×C53—22no—
250C4×C53—41no—
251C4×C53—41no—
252C4×C53—41no—
253Dic53—41no—
254Dic53—41no—
255Dic53—41no—
256Dic53—41no—
257#2678—531nomaximal, Sylow
258C2×C53—42no—
259C2×C53#53411yescharacteristic
260C2×C53#53511yescharacteristic
261C2×C53#53811yescharacteristic
262C2×C53D4211yescharacteristic
263C2×C53#54611yescharacteristic
264C2×C53#54611yescharacteristic
265C2×C53#56011yescharacteristic
266D53—42no—
267D53—42no—
268D53—42no—
269D53—81no—
270C2×#123—531no—
271C2×#121—531no—
272C2×#120—531no—
273C2×#120—531no—
274#397—531no—
275#396—531no—
276#394—531no—
277#394—531no—
278#392—531no—
279#392—531no—
280#390—531no—
281#390—531no—
282#388D5311yescharacteristic, complemented
283#386—531no—
284#379—531no—
285C53#267811yescharacteristic, complemented, Sylow
286C25—531no—
287C22×D4—531no—
288C22×D4—531no—
289C22×D4—531no—
290C22×D4—531no—
291C22×D4—531no—
292C22×D4—531no—
293C22×D4—531no—
294C23×C4—531no—
295#123—1064no—
296#121—1064no—
297#120—1064no—
298#120—1064no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—531no—
302C2×C4⋊C4C2×D5311yescharacteristic, complemented
303C2×C22⋊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
320C2×C42C2×D5311yescharacteristic, complemented
321#95—531no—
322#95—531no—
323#95C2×D5311yescharacteristic, complemented
324#95C2×D5311yescharacteristic, complemented
325C24—531no—
326C24—531no—
327C24—531no—
328C24—531no—
329C24—1062no—
330C24—1062no—
331C24—1062no—
332C24—2121no—
333C24—2121no—
334C24—2121no—
335C24—2121no—
336C24C22×D5311yescharacteristic
337C2×D4—534no—
338C2×D4—1062no—
339C2×D4—1062no—
340C2×D4—1062no—
341C2×D4—1062no—
342C2×D4—1062no—
343C2×D4—1062no—
344C2×D4—1062no—
345C2×D4—1062no—
346C2×D4—1062no—
347C2×D4—1062no—
348C2×D4—2121no—
349C2×D4—2121no—
350C2×D4—2121no—
351C2×D4—2121no—
352C2×D4—2121no—
353C2×D4—2121no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4—2121no—
358C22×C4—531no—
359C22×C4—531no—
360C22×C4—531no—
361C22×C4—531no—
362C22×C4—1062no—
363C22×C4—1062no—
364C22×C4—2121no—
365C22×C4C22×D5311yescharacteristic
366C22×C4C22×D5311yescharacteristic
367C22×C4C22×D5311yescharacteristic
368C22×C4C22×D5311yescharacteristic
369C22×C4C22×D5311yescharacteristic
370C22×C4C22×D5311yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—1062no—
373C4⋊C4—1062no—
374C4⋊C4—2121no—
375C22⋊C4—22no—
376C22⋊C4—41no—
377C22⋊C4—1062no—
378C22⋊C4—1062no—
379C22⋊C4—1062no—
380C22⋊C4—1062no—
381C22⋊C4—1062no—
382C22⋊C4—1062no—
383C22⋊C4—1062no—
384C22⋊C4—1062no—
385C22⋊C4—2121no—
386C22⋊C4—2121no—
387C22⋊C4—2121no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C22⋊C4D21214yescomplemented
392C42—41no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—532no—
400C23—532no—
401C23—532no—
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—1061no—
415C23—1061no—
416C23—1061no—
417C23—1061no—
418C23—1061no—
419C23—1061no—
420C23—1061no—
421C23—1061no—
422C23—1061no—
423C23—1061no—
424C23—1061no—
425C23—1061no—
426C23—1061no—
427C23—2124no—
428C23—2124no—
429C23—2124no—
430C23—2124no—
431C23—2124no—
432C23—2124no—
433C23—2124no—
434C23C2×D21212yes—
435C23C23×D5311yescharacteristic
436D4—2124no—
437D4—2124no—
438D4—2124no—
439D4—2124no—
440D4—2124no—
441D4—2124no—
442D4—2124no—
443C2×C4—21no—
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—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
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—2124no—
471C2×C4C2×D21212yes—
472C2×C4C2×D21212yes—
473C2×C4D4×D5312yes—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—1062no—
482C22—1062no—
483C22—1062no—
484C22—1062no—
485C22—1062no—
486C22—1062no—
487C22—1062no—
488C22—1062no—
489C22—1062no—
490C22—21216no—
491C22—2121no—
492C22—2121no—
493C22—2121no—
494C22—2121no—
495C22—2121no—
496C22—2121no—
497C22—2121no—
498C22—2121no—
499C22—2121no—
500C22—2121no—
501C22—2121no—
502C22—2121no—
503C22—2121no—
504C22—2121no—
505C22—2121no—
506C22—2121no—
507C22—2121no—
508C22—2121no—
509C22—2121no—
510C22C2×D4×D5311yescharacteristic
511C22C22×D21211yescharacteristic
512C22#41×D5311yescharacteristic
513C22#1313326211yescharacteristic
514C22#1313326311yescharacteristic
515C22#1313326311yescharacteristic
516C22#1313326311yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—2121no—
524C4—2121no—
525C4—2121no—
526C4—2121no—
527C2—42no—
528C2—4241no—
529C2—2122no—
530C2—2122no—
531C2—2122no—
532C2#1681118111yescharacteristic
533C2#1681119211yescharacteristic
534C2#1681121211yescharacteristic
535C2D4×D21211yescharacteristic
536C2#1681126611yescharacteristic
537C2#1681126811yescharacteristic
538C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#388C2
hypercenter#388D53
hypocenterC53#2678
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133199C2semidirect
2. C2×#13133192C2semidirect
3. C2×#13133187C2semidirect
4. C2×#13133187C2semidirect
5. #16810550C2semidirect
6. #16810545C2semidirect
7. #16810541C2semidirect
8. #16810539C2semidirect
9. #16810533C2semidirect
10. #16810532C2semidirect
11. #16810525C2semidirect
12. #16810525C2semidirect
13. #16810501C2semidirect
14. #16810478C2semidirect
15. C53×#388C2semidirect
21. C22×D212C22semidirect
24. C22×C4×D53C22semidirect
29. C2×C4⋊Dic53C22semidirect
30. C2×C4⋊Dic53C22semidirect
31. C2×#11719919C22semidirect
41. C2×#11719921C22semidirect
42. C2×#11719921C22semidirect
43. C2×#11719921C22semidirect
44. C2×#11719921C22semidirect
45. C2×#11719921C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
48. C2×C4⋊C4×C53C22semidirect
52. C2×C42×C53C22semidirect
53. #95×C53C22semidirect
54. #95×C53C22semidirect
97. C2×D212D4semidirect
104. C22⋊C4×C53D4semidirect
282. #388D53semidirect
285. C53#2678semidirect
302. C2×C4⋊C4C2×D53semidirect
320. C2×C42C2×D53semidirect
323. #95C2×D53semidirect
324. #95C2×D53semidirect
391. C22⋊C4D212semidirect