Finite Group Databasebeta

← #34114921

Subgroups — #34114921

Order 20096.

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#23758110C211yesmaximal, characteristic, complemented
22−1+4×D157C211yesmaximal, characteristic, complemented
3#23758089C213yesmaximal, complemented
4#23758088C211yesmaximal, characteristic, complemented
5#23758086C213yesmaximal, complemented
6#23758085C211yesmaximal, characteristic, complemented
7#23758082C213yesmaximal, complemented
8#23758081C211yesmaximal, characteristic, complemented
9#23758080C213yesmaximal, complemented
10#23758072C213yesmaximal, complemented
11#23758070C213yesmaximal, complemented
12C2×#19097803C213yesmaximal, complemented
13#137×D157C213yesmaximal, complemented
14#23758038C211yesmaximal, characteristic, complemented
15#578×C157C211yesmaximal, characteristic, complemented
16#19097819—21no—
17#19097819—21no—
18#19097819C2213yescomplemented
19#19097819C2213yescomplemented
20#19097818—23no—
21#19097818C2211yescharacteristic, complemented
22#19097818C2213yescomplemented
23#19097817—23no—
24#19097817C2211yescharacteristic, complemented
25#19097817C2213yescomplemented
26C2×#14578955—21no—
27C2×#14578955C2213yes—
28#41×D157—23no—
29#41×D157—23no—
30#41×D157C2213yes—
31#41×D157C2211yescharacteristic
32#41×D157C2213yescomplemented
33C2×Q8×D157—21no—
34C2×Q8×D157C2213yescomplemented
35#19097804C2213yescomplemented
36#19097803C2216yescomplemented
37#19097803C2216yescomplemented
38#19097803C2213yescomplemented
39#19097803C2216yescomplemented
40#19097803C2213yescomplemented
41#19097802C2213yescomplemented
42#19097802C2213yescomplemented
43#19097801C2213yescomplemented
44#19097800C2213yescomplemented
45#19097800C2213yescomplemented
46#19097799C2213yescomplemented
47#19097798C2213yescomplemented
48#19097798C2213yescomplemented
49#19097796C2213yescomplemented
50#19097795C2213yescomplemented
51#19097795C2211yescharacteristic, complemented
52#19097794C2216yescomplemented
53#19097793C2213yescomplemented
54#19097792C2216yescomplemented
55#19097791C2213yescomplemented
56#19097791C2213yescomplemented
57C2×#14578944C2213yescomplemented
58Q16×D157C2216yescomplemented
59C2×#14578943C2213yescomplemented
60C2×#14578941C2213yescomplemented
61SD16×D157C2216yescomplemented
62#19097779C2213yescomplemented
63#19097778C2211yescharacteristic, complemented
64#19097777C2213yescomplemented
65C2×#14578937C2213yescomplemented
66M16×D157C2213yescomplemented
672−1+4×C157C2211yescharacteristic, complemented
682−1+4×C157C2211yescharacteristic, complemented
69#137×C157C2213yescomplemented
70#137×C157C2213yescomplemented
71#135×C157C2213yescomplemented
72C2×Q16×C157C2213yescomplemented
73#131×C157C2211yescharacteristic, complemented
74#14578955—23no—
75#14578955—23no—
76#14578955—23no—
77#14578955—23no—
78#14578955—21no—
79#14578955D411yescharacteristic
80#14578955C2316yes—
81#14578955C2313yes—
82#14578955C2313yes—
83#14578954—23no—
84#14578954—23no—
85#14578954—26no—
86#14578954—26no—
87#14578954—23no—
88#14578954—23no—
89#14578954C2313yes—
90#14578954C2313yes—
91#14578954C2313yes—
92#14578953—26no—
93#14578953—23no—
94#14578953—23no—
95#14578953—23no—
96#14578953D413yes—
97#14578953C2313yes—
98#14578953C2311yescharacteristic
99C2×Dic314—23no—
100C2×Dic314—23no—
101C2×Dic314C2313yes—
102Q8×D157—23no—
103Q8×D157—23no—
104Q8×D157—23no—
105Q8×D157—23no—
106Q8×D157—21no—
107Q8×D157D411yescharacteristic
108Q8×D157C2316yes—
109Q8×D157C2313yes—
110Q8×D157C2313yes—
111C2×D628—23no—
112C2×D628—23no—
113C2×D628C2313yes—
114D4×D157—23no—
115D4×D157—23no—
116D4×D157—26no—
117D4×D157—23no—
118D4×D157D413yes—
119D4×D157C2313yes—
120D4×D157C2311yescharacteristic
121C2×C4×D157—23no—
122C2×C4×D157—23no—
123C2×C4×D157—23no—
124C2×C4×D157—23no—
125C2×C4×D157C2313yes—
126Dic628C2313yes—
127#14578944C2313yes—
128#14578944C2316yes—
129#14578944C2313yes—
130#14578943C2316yes—
131#14578943C2313yes—
132#14578943C2311yescharacteristic
133#14578942C2313yes—
134#14578942C2311yescharacteristic
135#14578941C2316yes—
136#14578941C2313yes—
137#14578941C2313yes—
138D1256C2313yes—
139#14578939C2311yescharacteristic
140#14578939C2313yes—
141#14578938C2313yes—
142#14578937C2313yes—
143#14578937C2311yescharacteristic
144#14578937C2313yes—
145#14578937C2313yes—
146C2×#11929434C2313yes—
147C8×D157C2316yes—
148#41×C157—23no—
149#41×C157—23no—
150#41×C157C2313yes—
151#41×C157C2313yes—
152#41×C157C2311yescharacteristic
153C2×Q8×C157—21no—
154C2×Q8×C157—21no—
155C2×Q8×C157C2313yes—
156C2×Q8×C157C2313yes—
157Q16×C157C2313yes—
158Q16×C157C2316yes—
159SD16×C157C2313yes—
160SD16×C157C2313yes—
161D8×C157C2311yescharacteristic
162M16×C157C2313yes—
163C2×C8×C157C2313yes—
164C22×D157—23no—
165C22×D157—21no—
166C22×D157—23no—
167C22×D157C2×D413yes—
168Dic314—23no—
169Dic314—23no—
170Dic314—26no—
171Dic314—23no—
172Dic314C2×D413yes—
173Dic314C2411yescharacteristic
174Dic314C2413yes—
175D628—23no—
176D628—26no—
177D628—23no—
178D628—23no—
179D628C2×D413yes—
180D628C2413yes—
181D628C2411yescharacteristic
182C22⋊D157—23no—
183C22⋊D157—23no—
184C22⋊D157—21no—
185C22⋊D157—23no—
186C22⋊D157—23no—
187C22⋊D157—21no—
188C22⋊D157C2×D416yes—
189C2×Dic157—23no—
190C2×Dic157—21no—
191C2×Dic157—23no—
192C2×Dic157C2×D413yes—
193C4×D157—23no—
194C4×D157—23no—
195C4×D157—26no—
196C4×D157—23no—
197C4×D157—23no—
198C4×D157—23no—
199C4×D157—26no—
200C4×D157—23no—
201C4×D157C2×D413yes—
202C4×D157C2×D413yes—
203C4×D157C2411yescharacteristic
204C4×D157C2413yes—
205#11929434C2413yes—
206#11929434C2411yescharacteristic
207Q8×C157—23no—
208Q8×C157—23no—
209Q8×C157C2×D411yescharacteristic
210Q8×C157C2413yes—
211Q8×C157C2413yes—
212D4×C157—23no—
213D4×C157—23no—
214D4×C157C2×D413yes—
215D4×C157C2411yescharacteristic
216D4×C157C2411yescharacteristic
217C2×C4×C157—23no—
218C2×C4×C157—23no—
219C2×C4×C157—23no—
220C2×C4×C157—23no—
221C2×C4×C157C2413yes—
222C8×C157C2413yes—
223C8×C157C2411yescharacteristic
224C22×C157—21no—
225C22×C157—21no—
226C22×C157C22×D413yes—
227C2×D157—21no—
228C2×D157—23no—
229C2×D157—43no—
230C2×D157—42no—
231C2×D157—83no—
232C2×D157—83no—
233C2×D157C22×D411yescharacteristic
234C2×D157C22×D413yes—
235C4×C157—23no—
236C4×C157—23no—
237C4×C157C22×D413yes—
238C4×C157C2511yescharacteristic
239Dic157—21no—
240Dic157—23no—
241Dic157C22×D411yescharacteristic
242Dic157C22×D413yes—
243C2×C157—23no—
244C2×C157—41no—
245C2×C157—41no—
246C2×C157C23×D411yescharacteristic
247D157—21no—
248D157—23no—
249D157—41no—
250D157—43no—
251C157#358611yescharacteristic, complemented, Sylow
252#3586—1571nomaximal, Sylow
253#585—1571no—
254C2×2−1+4—1571no—
255#578—1571no—
256#578—1573no—
257#578—1573no—
258#578D15711yescharacteristic, complemented
259#577—1571no—
260#577—1573no—
261#577—1573no—
262#577—1571no—
263#575—1573no—
264#575—1573no—
265C2×#137—1573no—
266C2×#137—1573no—
267#568—1571no—
2682−1+4—3141no—
2692−1+4—3141no—
2702−1+4—3143no—
2712−1+4—1573no—
2722−1+4—1571no—
2732−1+4—1573no—
2742−1+4—1573no—
2752−1+4C2×D15711yescharacteristic, complemented
2762−1+4C2×D15711yescharacteristic, complemented
277H(5,𝔽2)—3143no—
278H(5,𝔽2)—1573no—
279H(5,𝔽2)—1571no—
280C2×#41—3141no—
281C2×#41—3143no—
282C2×#41—3143no—
283C2×#41—1573no—
284C2×#41—1573no—
285C2×#41—1573no—
286C2×#41—1571no—
287C22×Q8—3141no—
288C22×Q8—1573no—
289#137—1573no—
290#137—1573no—
291#137—1573no—
292#137—1576no—
293#137—1573no—
294#137—1576no—
295#137—1576no—
296#137—1573no—
297#137—1573no—
298#137—1573no—
299#137—1573no—
300#137C2×D15713yescomplemented
301#137C2×D15713yescomplemented
302AGL(1,ℤ/8)—1573no—
303AGL(1,ℤ/8)—1573no—
304AGL(1,ℤ/8)—1573no—
305AGL(1,ℤ/8)—1571no—
306AGL(1,ℤ/8)—1573no—
307AGL(1,ℤ/8)—1573no—
308#135—1573no—
309#135—1573no—
310#135—1576no—
311#135—1576no—
312#135—1573no—
313#135C2×D15713yescomplemented
314C2×Q16—1573no—
315C2×Q16—1576no—
316C2×Q16C2×D15713yescomplemented
317C2×SD16—1573no—
318C2×SD16—1576no—
319C2×SD16—1573no—
320#131—1571no—
321#131—1573no—
322#131—1573no—
323#131C2×D15711yescharacteristic, complemented
324C2×M16—1573no—
325C2×M16—1573no—
326#41—23no—
327#41—23no—
328#41—3143no—
329#41—3146no—
330#41—3146no—
331#41—3143no—
332#41—3143no—
333#41—3143no—
334#41—3146no—
335#41—3141no—
336#41—3143no—
337#41—3143no—
338#41—3143no—
339#41—3143no—
340#41—3143no—
341#41—3143no—
342#41—3143no—
343#41—1573no—
344#41—1576no—
345#41—1573no—
346#41—1573no—
347#41—1571no—
348#41—1573no—
349#41—1573no—
350#41—1571no—
351#41—1573no—
352#41—1573no—
353#41C22×D15713yes—
354#41C22×D15713yes—
355#41C22×D15711yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—3143no—
359C2×Q8—3143no—
360C2×Q8—3143no—
361C2×Q8—3143no—
362C2×Q8—3141no—
363C2×Q8—3143no—
364C2×Q8—3143no—
365C2×Q8—1576no—
366C2×Q8—1573no—
367C2×Q8—1573no—
368C2×Q8—1573no—
369C2×Q8—1571no—
370C2×Q8C22×D15713yes—
371C2×Q8C22×D15713yes—
372C2×D4—3143no—
373C2×D4—3143no—
374C2×D4—3143no—
375C2×D4—3143no—
376C2×D4—3146no—
377C2×D4—3143no—
378C2×D4—1573no—
379C2×D4—1571no—
380C2×D4—1573no—
381C2×D4—1573no—
382C22×C4—3143no—
383C22×C4—3143no—
384C22×C4—3143no—
385C22×C4—3143no—
386C22×C4—1573no—
387Q16—1576no—
388Q16—1573no—
389Q16—1573no—
390Q16—1573no—
391Q16C22×D15716yes—
392Q16C22×D15713yes—
393SD16—1573no—
394SD16—1576no—
395SD16—1573no—
396SD16—1573no—
397SD16—1571no—
398SD16—1576no—
399SD16—1573no—
400SD16—1571no—
401SD16C22×D15713yes—
402SD16C22×D15713yes—
403D8—1571no—
404D8—1573no—
405D8—1573no—
406D8C22×D15711yescharacteristic
407M16—1573no—
408M16—1573no—
409M16—1571no—
410M16—1573no—
411M16—1573no—
412M16C22×D15713yes—
413C2×C8—1573no—
414C2×C8—1576no—
415C2×C8C22×D15713yes—
416C23—3141no—
417C23—3143no—
418C23—3143no—
419C23—1573no—
420Q8—23no—
421Q8—23no—
422Q8—3143no—
423Q8—3146no—
424Q8—3143no—
425Q8—3143no—
426Q8—1573no—
427Q8—1571no—
428Q8—1573no—
429Q8D4×D15711yescharacteristic
430Q8C23×D15713yes—
431Q8C23×D15713yes—
432D4—23no—
433D4—23no—
434D4—3143no—
435D4—3146no—
436D4—3143no—
437D4—3141no—
438D4—3143no—
439D4—3143no—
440D4—3143no—
441D4—3143no—
442D4—3141no—
443D4—3143no—
444D4—1576no—
445D4—1573no—
446D4—1571no—
447D4—1573no—
448D4D4×D15713yes—
449D4C23×D15711yescharacteristic
450D4C23×D15711yescharacteristic
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—3143no—
456C2×C4—3143no—
457C2×C4—3146no—
458C2×C4—3143no—
459C2×C4—3146no—
460C2×C4—3143no—
461C2×C4—3143no—
462C2×C4—3141no—
463C2×C4—3143no—
464C2×C4—3143no—
465C2×C4—3143no—
466C2×C4—1573no—
467C2×C4—1573no—
468C2×C4—1573no—
469C2×C4—1573no—
470C2×C4—1571no—
471C2×C4C23×D15713yes—
472C8—1571no—
473C8—1573no—
474C8C23×D15713yes—
475C8C23×D15711yescharacteristic
476C22—21no—
477C22—21no—
478C22—3141no—
479C22—3143no—
480C22—6283no—
481C22—6282no—
482C22—1571no—
483C22—1573no—
484C22—12563no—
485C22—12563no—
486C22C2×D4×D15713yes—
487C4—23no—
488C4—23no—
489C4—3141no—
490C4—3143no—
491C4—1571no—
492C4—1573no—
493C4C2×D4×D15713yes—
494C4C24×D15711yescharacteristic
495C2—23no—
496C2—41no—
497C2—41no—
498C2—3141no—
499C2—3143no—
500C2—6281no—
501C2—6283no—
502C2C22×D4×D15711yescharacteristic
subgroupG/H
centerC2C22×D4×D157
commutator, cosocleC4×C157C25
FrattiniC4C24×D157
Fitting#578×C157C2
socleC2×C157C23×D4
hypercenter#578D157
hypocenterC157#3586
LCSG ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ #41 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #23758110C2semidirect
2. 2−1+4×D157C2semidirect
3. #23758089C2semidirect
4. #23758088C2semidirect
5. #23758086C2semidirect
6. #23758085C2semidirect
7. #23758082C2semidirect
8. #23758081C2semidirect
9. #23758080C2semidirect
10. #23758072C2semidirect
11. #23758070C2semidirect
12. C2×#19097803C2semidirect
13. #137×D157C2semidirect
14. #23758038C2semidirect
15. #578×C157C2semidirect
18. #19097819C22semidirect
19. #19097819C22semidirect
21. #19097818C22semidirect
22. #19097818C22semidirect
24. #19097817C22semidirect
25. #19097817C22semidirect
32. #41×D157C22semidirect
34. C2×Q8×D157C22semidirect
35. #19097804C22semidirect
36. #19097803C22semidirect
37. #19097803C22semidirect
38. #19097803C22semidirect
39. #19097803C22semidirect
40. #19097803C22semidirect
41. #19097802C22semidirect
42. #19097802C22semidirect
43. #19097801C22semidirect
44. #19097800C22semidirect
45. #19097800C22semidirect
46. #19097799C22semidirect
47. #19097798C22semidirect
48. #19097798C22semidirect
49. #19097796C22semidirect
50. #19097795C22semidirect
51. #19097795C22semidirect
52. #19097794C22semidirect
53. #19097793C22semidirect
54. #19097792C22semidirect
55. #19097791C22semidirect
56. #19097791C22semidirect
57. C2×#14578944C22semidirect
58. Q16×D157C22semidirect
59. C2×#14578943C22semidirect
60. C2×#14578941C22semidirect
61. SD16×D157C22semidirect
62. #19097779C22semidirect
63. #19097778C22semidirect
64. #19097777C22semidirect
65. C2×#14578937C22semidirect
66. M16×D157C22semidirect
67. 2−1+4×C157C22semidirect
68. 2−1+4×C157C22semidirect
69. #137×C157C22semidirect
70. #137×C157C22semidirect
71. #135×C157C22semidirect
72. C2×Q16×C157C22semidirect
73. #131×C157C22semidirect
251. C157#3586semidirect
258. #578D157semidirect
275. 2−1+4C2×D157semidirect
276. 2−1+4C2×D157semidirect
300. #137C2×D157semidirect
301. #137C2×D157semidirect
313. #135C2×D157semidirect
316. C2×Q16C2×D157semidirect
323. #131C2×D157semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237581102
2−1+4×D1572
#237580892
#237580882
#237580862
#237580852
#237580822
#237580812
#237580802
#237580722
#237580702
C2×#190978032
#137×D1572
#237580382
#578×C1572
#3586157