Finite Group Databasebeta

← #21460604

Subgroups — #21460604

Order 7808.

1412 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16840599C211yesmaximal, characteristic, complemented
22−1+4×D61C211yesmaximal, characteristic, complemented
3#16840578C213yesmaximal, complemented
4#16840577C211yesmaximal, characteristic, complemented
5#16840575C213yesmaximal, complemented
6#16840574C211yesmaximal, characteristic, complemented
7#16840571C213yesmaximal, complemented
8#16840570C211yesmaximal, characteristic, complemented
9#16840569C213yesmaximal, complemented
10#16840561C213yesmaximal, complemented
11#16840559C213yesmaximal, complemented
12C2×#14519809C213yesmaximal, complemented
13#137×D61C213yesmaximal, complemented
14#16840527C211yesmaximal, characteristic, complemented
15C61×#578C211yesmaximal, characteristic, complemented
16#14519825—21no—
17#14519825—21no—
18#14519825C2213yescomplemented
19#14519825C2213yescomplemented
20#14519824—23no—
21#14519824C2213yescomplemented
22#14519824C2211yescharacteristic, complemented
23#14519823—23no—
24#14519823C2213yescomplemented
25#14519823C2211yescharacteristic, complemented
26C2×#11758226—21no—
27C2×#11758226C2213yes—
28#41×D61—23no—
29#41×D61—23no—
30#41×D61C2213yes—
31#41×D61C2211yescharacteristic
32#41×D61C2213yescomplemented
33C2×Q8×D61—21no—
34C2×Q8×D61C2213yescomplemented
35#14519810C2213yescomplemented
36#14519809C2216yescomplemented
37#14519809C2213yescomplemented
38#14519809C2216yescomplemented
39#14519809C2216yescomplemented
40#14519809C2213yescomplemented
41#14519808C2213yescomplemented
42#14519808C2213yescomplemented
43#14519807C2213yescomplemented
44#14519806C2213yescomplemented
45#14519806C2213yescomplemented
46#14519805C2213yescomplemented
47#14519804C2213yescomplemented
48#14519804C2213yescomplemented
49#14519802C2213yescomplemented
50#14519801C2211yescharacteristic, complemented
51#14519801C2213yescomplemented
52#14519800C2216yescomplemented
53#14519799C2213yescomplemented
54#14519798C2216yescomplemented
55#14519797C2213yescomplemented
56#14519797C2213yescomplemented
57C2×#11758202C2213yescomplemented
58Q16×D61C2216yescomplemented
59C2×#11758190C2213yescomplemented
60C2×#11758201C2213yescomplemented
61SD16×D61C2216yescomplemented
62#14519785C2213yescomplemented
63#14519784C2211yescharacteristic, complemented
64#14519783C2213yescomplemented
65C2×#11758189C2213yescomplemented
66M16×D61C2213yescomplemented
672−1+4×C61C2211yescharacteristic, complemented
682−1+4×C61C2211yescharacteristic, complemented
69#137×C61C2213yescomplemented
70#137×C61C2213yescomplemented
71#135×C61C2213yescomplemented
72C2×Q16×C61C2213yescomplemented
73#131×C61C2211yescharacteristic, complemented
74#41×C61—23no—
75#41×C61—23no—
76#41×C61C2313yes—
77#41×C61C2313yes—
78#41×C61C2311yescharacteristic
79C2×Q8×C61—21no—
80C2×Q8×C61—21no—
81C2×Q8×C61C2313yes—
82C2×Q8×C61C2313yes—
83#11758226—21no—
84#11758226—23no—
85#11758226—23no—
86#11758226—23no—
87#11758226—23no—
88#11758226D411yescharacteristic
89#11758226C2313yes—
90#11758226C2316yes—
91#11758226C2313yes—
92Q8×D61—21no—
93Q8×D61—23no—
94Q8×D61—23no—
95Q8×D61—23no—
96Q8×D61—23no—
97Q8×D61D411yescharacteristic
98Q8×D61C2313yes—
99Q8×D61C2316yes—
100Q8×D61C2313yes—
101#11758224—23no—
102#11758224—26no—
103#11758224—23no—
104#11758224—23no—
105#11758224D413yes—
106#11758224C2311yescharacteristic
107#11758224C2313yes—
108D4×D61—23no—
109D4×D61—23no—
110D4×D61—23no—
111D4×D61—26no—
112D4×D61D413yes—
113D4×D61C2313yes—
114D4×D61C2311yescharacteristic
115#11758222—26no—
116#11758222—26no—
117#11758222—23no—
118#11758222—23no—
119#11758222—23no—
120#11758222—23no—
121#11758222C2313yes—
122#11758222C2313yes—
123#11758222C2313yes—
124C2×D244—23no—
125C2×D244—23no—
126C2×D244C2313yes—
127C2×C4×D61—23no—
128C2×C4×D61—23no—
129C2×C4×D61—23no—
130C2×C4×D61—23no—
131C2×C4×D61C2313yes—
132C2×Dic122—23no—
133C2×Dic122—23no—
134C2×Dic122C2313yes—
135Q16×C61C2313yes—
136Q16×C61C2316yes—
137SD16×C61C2313yes—
138SD16×C61C2313yes—
139D8×C61C2311yescharacteristic
140M16×C61C2313yes—
141C2×C8×C61C2313yes—
142#11758202C2313yes—
143#11758202C2316yes—
144#11758202C2313yes—
145#11758201C2313yes—
146#11758201C2316yes—
147#11758201C2313yes—
148#11758200C2313yes—
149#11758200C2311yescharacteristic
150#11758199C2311yescharacteristic
151#11758199C2313yes—
152#11758194C2313yes—
153C2×#92418C2313yes—
154Dic244C2313yes—
155D488C2313yes—
156#11758190C2313yes—
157#11758190C2311yescharacteristic
158#11758190C2316yes—
159#11758189C2311yescharacteristic
160#11758189C2313yes—
161#11758189C2313yes—
162#11758189C2313yes—
163C8×D61C2316yes—
164C22×D61—23no—
165C22×D61—21no—
166C22×D61—23no—
167C22×D61C2×D413yes—
168Q8×C61—23no—
169Q8×C61—23no—
170Q8×C61C2×D411yescharacteristic
171Q8×C61C2413yes—
172Q8×C61C2413yes—
173D4×C61—23no—
174D4×C61—23no—
175D4×C61C2×D413yes—
176D4×C61C2411yescharacteristic
177D4×C61C2411yescharacteristic
178C2×C4×C61—23no—
179C2×C4×C61—23no—
180C2×C4×C61—23no—
181C2×C4×C61—23no—
182C2×C4×C61C2413yes—
183C22⋊D61—23no—
184C22⋊D61—23no—
185C22⋊D61—21no—
186C22⋊D61—23no—
187C22⋊D61—23no—
188C22⋊D61—21no—
189C22⋊D61C2×D416yes—
190C2×Dic61—23no—
191C2×Dic61—23no—
192C2×Dic61—21no—
193C2×Dic61C2×D413yes—
194D244—23no—
195D244—26no—
196D244—23no—
197D244—23no—
198D244C2×D413yes—
199D244C2413yes—
200D244C2411yescharacteristic
201C4×D61—23no—
202C4×D61—23no—
203C4×D61—26no—
204C4×D61—23no—
205C4×D61—23no—
206C4×D61—23no—
207C4×D61—26no—
208C4×D61—23no—
209C4×D61C2×D413yes—
210C4×D61C2×D413yes—
211C4×D61C2413yes—
212C4×D61C2411yescharacteristic
213Dic122—23no—
214Dic122—26no—
215Dic122—23no—
216Dic122—23no—
217Dic122C2×D413yes—
218Dic122C2413yes—
219Dic122C2411yescharacteristic
220C8×C61C2413yes—
221C8×C61C2411yescharacteristic
222#92418C2413yes—
223#92418C2411yescharacteristic
224C22×C61—21no—
225C22×C61—21no—
226C22×C61C22×D413yes—
227C2×D61—21no—
228C2×D61—23no—
229C2×D61—43no—
230C2×D61—42no—
231C2×D61—83no—
232C2×D61—83no—
233C2×D61C22×D411yescharacteristic
234C2×D61C22×D413yes—
235C4×C61—23no—
236C4×C61—23no—
237C4×C61C22×D413yes—
238C4×C61C2511yescharacteristic
239Dic61—23no—
240Dic61—21no—
241Dic61C22×D413yes—
242Dic61C22×D411yescharacteristic
243#3586—611nomaximal, Sylow
244C2×C61—23no—
245C2×C61—41no—
246C2×C61—41no—
247C2×C61C23×D411yescharacteristic
248D61—21no—
249D61—23no—
250D61—41no—
251D61—43no—
252#585—611no—
253C2×2−1+4—611no—
254#578—611no—
255#578—613no—
256#578—613no—
257#578D6111yescharacteristic, complemented
258#577—611no—
259#577—613no—
260#577—613no—
261#577—611no—
262#575—613no—
263#575—613no—
264C2×#137—613no—
265C2×#137—613no—
266#568—611no—
267C61#358611yescharacteristic, complemented, Sylow
2682−1+4—613no—
2692−1+4—611no—
2702−1+4—613no—
2712−1+4—613no—
2722−1+4—1221no—
2732−1+4—1221no—
2742−1+4—1223no—
2752−1+4C2×D6111yescharacteristic, complemented
2762−1+4C2×D6111yescharacteristic, complemented
277H(5,𝔽2)—613no—
278H(5,𝔽2)—611no—
279H(5,𝔽2)—1223no—
280C2×#41—611no—
281C2×#41—613no—
282C2×#41—613no—
283C2×#41—613no—
284C2×#41—1223no—
285C2×#41—1221no—
286C2×#41—1223no—
287C22×Q8—613no—
288C22×Q8—1221no—
289#137—613no—
290#137—613no—
291#137—613no—
292#137—616no—
293#137—613no—
294#137—616no—
295#137—616no—
296#137—613no—
297#137—613no—
298#137—613no—
299#137—613no—
300#137C2×D6113yescomplemented
301#137C2×D6113yescomplemented
302AGL(1,ℤ/8)—613no—
303AGL(1,ℤ/8)—613no—
304AGL(1,ℤ/8)—613no—
305AGL(1,ℤ/8)—611no—
306AGL(1,ℤ/8)—613no—
307AGL(1,ℤ/8)—613no—
308#135—613no—
309#135—613no—
310#135—616no—
311#135—616no—
312#135—613no—
313#135C2×D6113yescomplemented
314C2×Q16—613no—
315C2×Q16—616no—
316C2×Q16C2×D6113yescomplemented
317C2×SD16—613no—
318C2×SD16—616no—
319C2×SD16—613no—
320#131—611no—
321#131—613no—
322#131—613no—
323#131C2×D6111yescharacteristic, complemented
324C2×M16—613no—
325C2×M16—613no—
326#41—23no—
327#41—23no—
328#41—611no—
329#41—613no—
330#41—613no—
331#41—613no—
332#41—616no—
333#41—613no—
334#41—613no—
335#41—611no—
336#41—613no—
337#41—613no—
338#41—1223no—
339#41—1223no—
340#41—1226no—
341#41—1226no—
342#41—1223no—
343#41—1223no—
344#41—1223no—
345#41—1223no—
346#41—1223no—
347#41—1223no—
348#41—1226no—
349#41—1223no—
350#41—1223no—
351#41—1221no—
352#41—1223no—
353#41C22×D6113yes—
354#41C22×D6113yes—
355#41C22×D6111yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—611no—
359C2×Q8—616no—
360C2×Q8—613no—
361C2×Q8—613no—
362C2×Q8—613no—
363C2×Q8—1223no—
364C2×Q8—1223no—
365C2×Q8—1223no—
366C2×Q8—1223no—
367C2×Q8—1223no—
368C2×Q8—1223no—
369C2×Q8—1221no—
370C2×Q8C22×D6113yes—
371C2×Q8C22×D6113yes—
372C2×D4—613no—
373C2×D4—613no—
374C2×D4—611no—
375C2×D4—613no—
376C2×D4—1223no—
377C2×D4—1223no—
378C2×D4—1226no—
379C2×D4—1223no—
380C2×D4—1223no—
381C2×D4—1223no—
382C22×C4—613no—
383C22×C4—1223no—
384C22×C4—1223no—
385C22×C4—1223no—
386C22×C4—1223no—
387Q16—616no—
388Q16—613no—
389Q16—613no—
390Q16—613no—
391Q16C22×D6116yes—
392Q16C22×D6113yes—
393SD16—616no—
394SD16—613no—
395SD16—613no—
396SD16—613no—
397SD16—611no—
398SD16—616no—
399SD16—613no—
400SD16—611no—
401SD16C22×D6113yes—
402SD16C22×D6113yes—
403D8—611no—
404D8—613no—
405D8—613no—
406D8C22×D6111yescharacteristic
407M16—613no—
408M16—613no—
409M16—611no—
410M16—613no—
411M16—613no—
412M16C22×D6113yes—
413C2×C8—613no—
414C2×C8—616no—
415C2×C8C22×D6113yes—
416C23—613no—
417C23—1223no—
418C23—1221no—
419C23—1223no—
420Q8—23no—
421Q8—23no—
422Q8—613no—
423Q8—613no—
424Q8—611no—
425Q8—1223no—
426Q8—1226no—
427Q8—1223no—
428Q8—1223no—
429Q8D4×D6111yescharacteristic
430Q8C23×D6113yes—
431Q8C23×D6113yes—
432D4—23no—
433D4—23no—
434D4—616no—
435D4—613no—
436D4—611no—
437D4—613no—
438D4—1223no—
439D4—1226no—
440D4—1223no—
441D4—1223no—
442D4—1221no—
443D4—1221no—
444D4—1223no—
445D4—1223no—
446D4—1223no—
447D4—1223no—
448D4D4×D6113yes—
449D4C23×D6111yescharacteristic
450D4C23×D6111yescharacteristic
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—613no—
456C2×C4—613no—
457C2×C4—613no—
458C2×C4—613no—
459C2×C4—611no—
460C2×C4—1223no—
461C2×C4—1226no—
462C2×C4—1223no—
463C2×C4—1223no—
464C2×C4—1223no—
465C2×C4—1226no—
466C2×C4—1221no—
467C2×C4—1223no—
468C2×C4—1223no—
469C2×C4—1223no—
470C2×C4—1223no—
471C2×C4C23×D6113yes—
472C8—613no—
473C8—611no—
474C8C23×D6113yes—
475C8C23×D6111yescharacteristic
476C22—21no—
477C22—21no—
478C22—611no—
479C22—613no—
480C22—1221no—
481C22—1223no—
482C22—4883no—
483C22—4883no—
484C22—2443no—
485C22—2442no—
486C22C2×D4×D6113yes—
487C4—23no—
488C4—23no—
489C4—611no—
490C4—613no—
491C4—1221no—
492C4—1223no—
493C4C2×D4×D6113yes—
494C4C24×D6111yescharacteristic
495C2—23no—
496C2—41no—
497C2—41no—
498C2—1223no—
499C2—1221no—
500C2—2443no—
501C2—2441no—
502C2C22×D4×D6111yescharacteristic
subgroupG/H
centerC2C22×D4×D61
commutatorC4×C61C25
FrattiniC4C24×D61
FittingC61×#578C2
socleC2×C61C23×D4
hypercenter#578D61
hypocenterC61#3586
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ #41 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #16840599C2semidirect
2. 2−1+4×D61C2semidirect
3. #16840578C2semidirect
4. #16840577C2semidirect
5. #16840575C2semidirect
6. #16840574C2semidirect
7. #16840571C2semidirect
8. #16840570C2semidirect
9. #16840569C2semidirect
10. #16840561C2semidirect
11. #16840559C2semidirect
12. C2×#14519809C2semidirect
13. #137×D61C2semidirect
14. #16840527C2semidirect
15. C61×#578C2semidirect
18. #14519825C22semidirect
19. #14519825C22semidirect
21. #14519824C22semidirect
22. #14519824C22semidirect
24. #14519823C22semidirect
25. #14519823C22semidirect
32. #41×D61C22semidirect
34. C2×Q8×D61C22semidirect
35. #14519810C22semidirect
36. #14519809C22semidirect
37. #14519809C22semidirect
38. #14519809C22semidirect
39. #14519809C22semidirect
40. #14519809C22semidirect
41. #14519808C22semidirect
42. #14519808C22semidirect
43. #14519807C22semidirect
44. #14519806C22semidirect
45. #14519806C22semidirect
46. #14519805C22semidirect
47. #14519804C22semidirect
48. #14519804C22semidirect
49. #14519802C22semidirect
50. #14519801C22semidirect
51. #14519801C22semidirect
52. #14519800C22semidirect
53. #14519799C22semidirect
54. #14519798C22semidirect
55. #14519797C22semidirect
56. #14519797C22semidirect
57. C2×#11758202C22semidirect
58. Q16×D61C22semidirect
59. C2×#11758190C22semidirect
60. C2×#11758201C22semidirect
61. SD16×D61C22semidirect
62. #14519785C22semidirect
63. #14519784C22semidirect
64. #14519783C22semidirect
65. C2×#11758189C22semidirect
66. M16×D61C22semidirect
67. 2−1+4×C61C22semidirect
68. 2−1+4×C61C22semidirect
69. #137×C61C22semidirect
70. #137×C61C22semidirect
71. #135×C61C22semidirect
72. C2×Q16×C61C22semidirect
73. #131×C61C22semidirect
257. #578D61semidirect
267. C61#3586semidirect
275. 2−1+4C2×D61semidirect
276. 2−1+4C2×D61semidirect
300. #137C2×D61semidirect
301. #137C2×D61semidirect
313. #135C2×D61semidirect
316. C2×Q16C2×D61semidirect
323. #131C2×D61semidirect