Finite Group Databasebeta

← #449×D157 #34109333

Subgroups — #449×D157 #34109333

Order 20096.

812 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D157C211yesmaximal, characteristic, complemented
2C2×SD16×D157C211yesmaximal, characteristic, complemented
3C2×D8×D157C211yesmaximal, characteristic, complemented
4#121×D157C211yesmaximal, characteristic, complemented
5#23757421C211yesmaximal, characteristic, complemented
6#23757420C211yesmaximal, characteristic, complemented
7#23757419C211yesmaximal, characteristic, complemented
8#23757418C211yesmaximal, characteristic, complemented
9#23757417C211yesmaximal, characteristic, complemented
10#23757416C211yesmaximal, characteristic, complemented
11#23757415C211yesmaximal, characteristic, complemented
12#103×D157C211yesmaximal, characteristic, complemented
13#102×D157C211yesmaximal, characteristic, complemented
14C22⋊C8×D157C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic, complemented
17C2×#14578954—21no—
18C2×#14578954C2211yescharacteristic, complemented
19C2×#14578953—21no—
20C2×#14578953C2211yescharacteristic, complemented
21#41×D157—24no—
22#41×D157—24no—
23C2×Q8×D157C2211yescharacteristic
24C2×D4×D157—21no—
25C2×D4×D157—21no—
26C2×D4×D157C2211yescharacteristic
27C2×D4×D157C2211yescharacteristic
28C22×C4×D157—21no—
29C22×C4×D157C2211yescharacteristic
30C2×#14578943C2211yescharacteristic, complemented
31C2×#14578942C2211yescharacteristic, complemented
32C2×#14578941C2211yescharacteristic, complemented
33SD16×D157—24no—
34C2×D1256C2211yescharacteristic, complemented
35C2×#14578939C2211yescharacteristic, complemented
36C2×#14578939C2211yescharacteristic, complemented
37D8×D157—24no—
38C2×C8×D157C2211yescharacteristic
39C2×C8×D157C2211yescharacteristic
40#19097746C2211yescharacteristic, complemented
41#19097745C2211yescharacteristic, complemented
42#19097744—21no—
43#19097743—21no—
44#19097742C2211yescharacteristic, complemented
45C4⋊C4×D157C2211yescharacteristic
46C22⋊C4×D157—21no—
47#19097696C2211yescharacteristic, complemented
48#19097695C2211yescharacteristic, complemented
49#19097694C2211yescharacteristic, complemented
50#19097693C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic, complemented
52#19097691C2211yescharacteristic, complemented
53C22⋊#11929434C2211yescharacteristic, complemented
54#19097683C2211yescharacteristic, complemented
55#19097683C2211yescharacteristic, complemented
56C2×#41×C157C2211yescharacteristic, complemented
57C2×SD16×C157C2211yescharacteristic, complemented
58C2×D8×C157C2211yescharacteristic, complemented
59#121×C157C2211yescharacteristic, complemented
60#103×C157C2211yescharacteristic, complemented
61#102×C157C2211yescharacteristic, complemented
62C22⋊C8×C157C2211yescharacteristic, complemented
63C23×D157—21no—
64C23×D157—21no—
65C23×D157D411yescharacteristic
66#14578955—22no—
67#14578955—22no—
68#14578954—28no—
69#14578954—22no—
70#14578954—22no—
71#14578953—28no—
72#14578953—22no—
73#14578953—22no—
74C2×Dic314—21no—
75C2×Dic314C2311yescharacteristic
76Q8×D157—22no—
77Q8×D157D414yescomplemented
78C2×D628—21no—
79C2×D628—21no—
80C2×D628C2311yescharacteristic
81C2×D628C2311yescharacteristic
82C2×C22⋊D157—21no—
83C2×C22⋊D157—21no—
84C2×C22⋊D157—21no—
85C2×C22⋊D157—21no—
86C2×C22⋊D157—21no—
87C2×C22⋊D157—21no—
88D4×D157—28no—
89D4×D157—22no—
90D4×D157—22no—
91D4×D157—22no—
92D4×D157—44no—
93D4×D157D414yescomplemented
94C22×Dic157—21no—
95C22×Dic157D411yescharacteristic
96C2×C4×D157—28no—
97C2×C4×D157—21no—
98C2×C4×D157—21no—
99C2×C4×D157—22no—
100C2×C4×D157—22no—
101C2×C4×D157—21no—
102C2×C4×D157—21no—
103C2×C4×D157D411yescharacteristic
104C2×C4×D157D411yescharacteristic
105C2×C4×D157C2311yescharacteristic
106C2×C4×D157C2311yescharacteristic
107#14578943—22no—
108#14578942—22no—
109#14578941—22no—
110D1256—22no—
111#14578939—22no—
112#14578939—22no—
113C2×#11929434C2311yescharacteristic
114C2×#11929434C2311yescharacteristic
115C8×D157—22no—
116C8×D157—22no—
117C4⋊Dic157C2311yescharacteristic
118#14578933—21no—
119C22⋊Dic157—21no—
120#14578931—21no—
121#14578931—21no—
122#41×C157—22no—
123#41×C157—22no—
124C2×Q8×C157C2311yescharacteristic
125C2×D4×C157—21no—
126C2×D4×C157—21no—
127C2×D4×C157C2311yescharacteristic
128C2×D4×C157C2311yescharacteristic
129C22×C4×C157—21no—
130C22×C4×C157C2311yescharacteristic
131SD16×C157—22no—
132D8×C157—22no—
133C2×C8×C157C2311yescharacteristic
134C2×C8×C157C2311yescharacteristic
135C4⋊C4×C157C2311yescharacteristic
136C22⋊C4×C157—21no—
137C22×D157—21no—
138C22×D157—22no—
139C22×D157—24no—
140C22×D157—22no—
141C22×D157—22no—
142C22×D157—21no—
143C22×D157—42no—
144C22×D157—42no—
145C22×D157—42no—
146C22×D157—42no—
147C22×D157—42no—
148C22×D157C2×D411yescharacteristic
149C22×D157C2×D411yescharacteristic
150Dic314—21no—
151Dic314—24no—
152Dic314C2×D412yescomplemented
153D628—24no—
154D628—21no—
155D628—21no—
156D628—21no—
157D628—42no—
158D628C2×D412yescomplemented
159C22⋊D157—24no—
160C22⋊D157—24no—
161C22⋊D157—24no—
162C22⋊D157—42no—
163C22⋊D157—42no—
164C22⋊D157—42no—
165C2×Dic157—24no—
166C2×Dic157—21no—
167C2×Dic157—22no—
168C2×Dic157—21no—
169C2×Dic157C2×D411yescharacteristic
170C2×Dic157C2×D411yescharacteristic
171C4×D157—24no—
172C4×D157—24no—
173C4×D157—22no—
174C4×D157—24no—
175C4×D157—21no—
176C4×D157—21no—
177C4×D157—22no—
178C4×D157—42no—
179C4×D157C2×D412yes—
180C4×D157C2×D412yes—
181#11929434—21no—
182#11929434—21no—
183C23×C157—21no—
184C23×C157—21no—
185C23×C157C2×D411yescharacteristic
186Q8×C157—21no—
187Q8×C157C2×D412yescomplemented
188D4×C157—24no—
189D4×C157—21no—
190D4×C157—21no—
191D4×C157—21no—
192D4×C157—42no—
193D4×C157C2×D412yescomplemented
194C2×C4×C157—24no—
195C2×C4×C157—21no—
196C2×C4×C157—21no—
197C2×C4×C157—21no—
198C2×C4×C157—21no—
199C2×C4×C157C2×D411yescharacteristic
200C2×C4×C157C2411yescharacteristic
201C8×C157—21no—
202C8×C157—21no—
203C22×C157—21no—
204C22×C157—22no—
205C22×C157—21no—
206C22×C157—21no—
207C22×C157—41no—
208C22×C157—41no—
209C22×C157—41no—
210C22×C157—41no—
211C22×C157—41no—
212C22×C157C22×D411yescharacteristic
213C2×D157—21no—
214C2×D157—22no—
215C2×D157—21no—
216C2×D157—21no—
217C2×D157—41no—
218C2×D157—41no—
219C2×D157—44no—
220C2×D157—48no—
221C2×D157—41no—
222C2×D157—41no—
223C2×D157—41no—
224C2×D157—84no—
225C2×D157#12012yes—
226C2×D157#13512yes—
227C2×D157AGL(1,ℤ/8)12yes—
228C4×C157—21no—
229C4×C157—22no—
230C4×C157—41no—
231C4×C157C22×D411yescharacteristic
232C4×C157C22×D411yescharacteristic
233Dic157—21no—
234Dic157—22no—
235Dic157—41no—
236Dic157#12012yes—
237C2×C157—41no—
238C2×C157—42no—
239C2×C157—81no—
240C2×C157C2×#12011yescharacteristic
241C2×C157C2×#13511yescharacteristic
242C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
243D157—41no—
244D157—42no—
245D157—81no—
246D157#44914yesdirect product factor
247C157C2×#44911yescharacteristic, complemented, Sylow
248C2×#449—1571nomaximal, Sylow
249C22×#41—1571no—
250C22×SD16—1571no—
251C22×D8—1571no—
252C2×#121—1571no—
253#449—1571no—
254#449—1571no—
255#449—1571no—
256#449—1571no—
257#449—1571no—
258#449—1571no—
259#449—1571no—
260#449D15711yescharacteristic, direct product factor
261C2×#103—1571no—
262C2×#102—1571no—
263C2×C22⋊C8—1571no—
264C2×#41—3141no—
265C2×#41—3144no—
266C2×#41—3144no—
267C2×#41—3141no—
268C2×#41—1571no—
269C2×#41—1571no—
270C2×#41—1571no—
271C2×#41C2×D15711yescharacteristic, complemented
272C22×Q8—1571no—
273C22×D4—3141no—
274C22×D4—3141no—
275C22×D4—1571no—
276C22×D4—1571no—
277C23×C4—3141no—
278C23×C4—1571no—
279C2×SD16—3144no—
280C2×SD16—1571no—
281C2×SD16—1571no—
282C2×SD16—1571no—
283C2×SD16C2×D15711yescharacteristic, complemented
284C2×D8—3144no—
285C2×D8—1571no—
286C2×D8—1571no—
287C2×D8—1571no—
288C2×D8C2×D15711yescharacteristic, complemented
289C22×C8—1571no—
290C22×C8—1571no—
291#121—3141no—
292#121—3141no—
293#121—1571no—
294#121—1571no—
295#121—1571no—
296#121C2×D15711yescharacteristic, complemented
297C2×C4⋊C4—1571no—
298C2×C22⋊C4—3141no—
299#103—1571no—
300#103—1571no—
301#103—1571no—
302#103C2×D15711yescharacteristic, complemented
303#102—1571no—
304#102—1571no—
305#102—1571no—
306#102C2×D15711yescharacteristic, complemented
307C22⋊C8—1571no—
308C22⋊C8—1571no—
309C22⋊C8—1571no—
310C22⋊C8C2×D15711yescharacteristic, complemented
311C24—3141no—
312C24—3141no—
313C24—1571no—
314#41—22no—
315#41—22no—
316#41—3148no—
317#41—3142no—
318#41—3142no—
319#41—3148no—
320#41—3142no—
321#41—3142no—
322#41—3142no—
323#41—3142no—
324C2×Q8—3141no—
325C2×Q8—3142no—
326C2×Q8—1574no—
327C2×Q8—1571no—
328C2×Q8C22×D15711yescharacteristic
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—3141no—
332C2×D4—3141no—
333C2×D4—3148no—
334C2×D4—3141no—
335C2×D4—3141no—
336C2×D4—3141no—
337C2×D4—3141no—
338C2×D4—3142no—
339C2×D4—3142no—
340C2×D4—3142no—
341C2×D4—3141no—
342C2×D4—3141no—
343C2×D4—6284no—
344C2×D4—1574no—
345C2×D4—1571no—
346C2×D4—1571no—
347C2×D4C22×D15711yescharacteristic
348C2×D4C22×D15711yescharacteristic
349C22×C4—21no—
350C22×C4—3148no—
351C22×C4—3141no—
352C22×C4—3141no—
353C22×C4—3142no—
354C22×C4—3142no—
355C22×C4—3141no—
356C22×C4—3141no—
357C22×C4—3141no—
358C22×C4—1571no—
359C22×C4—1571no—
360C22×C4—1571no—
361C22×C4—1571no—
362C22×C4—1571no—
363C22×C4C22×D15711yescharacteristic
364SD16—22no—
365SD16—3142no—
366SD16—3142no—
367SD16—3142no—
368D8—22no—
369D8—3142no—
370D8—3142no—
371D8—3142no—
372C2×C8—3142no—
373C2×C8—3142no—
374C2×C8—1571no—
375C2×C8—1571no—
376C2×C8C22×D15711yescharacteristic
377C2×C8C22×D15711yescharacteristic
378C4⋊C4—3141no—
379C4⋊C4—1571no—
380C4⋊C4C22×D15711yescharacteristic
381C22⋊C4—21no—
382C22⋊C4—3141no—
383C22⋊C4—3141no—
384C22⋊C4—3141no—
385C23—21no—
386C23—21no—
387C23—3142no—
388C23—3144no—
389C23—3142no—
390C23—3142no—
391C23—3141no—
392C23—3141no—
393C23—6282no—
394C23—6282no—
395C23—6282no—
396C23—6282no—
397C23—6282no—
398C23—1571no—
399C23—1571no—
400C23D4×D15711yescharacteristic
401Q8—21no—
402Q8—3144no—
403Q8—3141no—
404Q8—1572no—
405Q8D4×D15712yescomplemented
406D4—24no—
407D4—21no—
408D4—21no—
409D4—21no—
410D4—42no—
411D4—3144no—
412D4—3144no—
413D4—3144no—
414D4—3144no—
415D4—3141no—
416D4—3141no—
417D4—3141no—
418D4—6282no—
419D4—6282no—
420D4—6282no—
421D4—6282no—
422D4—1572no—
423D4D4×D15712yescomplemented
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—3144no—
430C2×C4—3144no—
431C2×C4—3142no—
432C2×C4—3144no—
433C2×C4—3142no—
434C2×C4—3144no—
435C2×C4—3141no—
436C2×C4—3142no—
437C2×C4—3141no—
438C2×C4—3141no—
439C2×C4—3141no—
440C2×C4—6282no—
441C2×C4—1571no—
442C2×C4—1571no—
443C2×C4—1572no—
444C2×C4—1572no—
445C2×C4D4×D15711yescharacteristic
446C2×C4C23×D15711yescharacteristic
447C8—21no—
448C8—21no—
449C8—3141no—
450C8—3141no—
451C22—21no—
452C22—22no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—3141no—
461C22—3142no—
462C22—3141no—
463C22—3141no—
464C22—6284no—
465C22—6288no—
466C22—6281no—
467C22—6281no—
468C22—6281no—
469C22—6281no—
470C22—6281no—
471C22—1572no—
472C22—1572no—
473C22—1572no—
474C22—12564no—
475C22C2×D4×D15711yescharacteristic
476C4—21no—
477C4—22no—
478C4—41no—
479C4—3141no—
480C4—3142no—
481C4—6281no—
482C4—1572no—
483C4C2×D4×D15711yescharacteristic
484C4C2×D4×D15711yescharacteristic
485C2—41no—
486C2—42no—
487C2—81no—
488C2—6281no—
489C2—6282no—
490C2—1574no—
491C2—12561no—
492C2#120×D15711yescharacteristic
493C2#135×D15711yescharacteristic
494C2AGL(1,ℤ/8)×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157C2×#449
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#41×D157C2semidirect
2. C2×SD16×D157C2semidirect
3. C2×D8×D157C2semidirect
4. #121×D157C2semidirect
5. #23757421C2semidirect
6. #23757420C2semidirect
7. #23757419C2semidirect
8. #23757418C2semidirect
9. #23757417C2semidirect
10. #23757416C2semidirect
11. #23757415C2semidirect
12. #103×D157C2semidirect
13. #102×D157C2semidirect
14. C22⋊C8×D157C2semidirect
15. #449×C157C2semidirect
16. C2×#14578955C22semidirect
18. C2×#14578954C22semidirect
20. C2×#14578953C22semidirect
30. C2×#14578943C22semidirect
31. C2×#14578942C22semidirect
32. C2×#14578941C22semidirect
34. C2×D1256C22semidirect
35. C2×#14578939C22semidirect
36. C2×#14578939C22semidirect
40. #19097746C22semidirect
41. #19097745C22semidirect
44. #19097742C22semidirect
47. #19097696C22semidirect
48. #19097695C22semidirect
49. #19097694C22semidirect
50. #19097693C22semidirect
51. #19097692C22semidirect
52. #19097691C22semidirect
53. C22⋊#11929434C22semidirect
54. #19097683C22semidirect
55. #19097683C22semidirect
56. C2×#41×C157C22semidirect
57. C2×SD16×C157C22semidirect
58. C2×D8×C157C22semidirect
59. #121×C157C22semidirect
60. #103×C157C22semidirect
61. #102×C157C22semidirect
62. C22⋊C8×C157C22semidirect
77. Q8×D157D4semidirect
93. D4×D157D4semidirect
152. Dic314C2×D4semidirect
158. D628C2×D4semidirect
187. Q8×C157C2×D4semidirect
193. D4×C157C2×D4semidirect
246. D157#449direct (N × H)
247. C157C2×#449semidirect
260. #449D157direct (N × H)
271. C2×#41C2×D157semidirect
283. C2×SD16C2×D157semidirect
288. C2×D8C2×D157semidirect
296. #121C2×D157semidirect
302. #103C2×D157semidirect
306. #102C2×D157semidirect
310. C22⋊C8C2×D157semidirect
405. Q8D4×D157semidirect
423. D4D4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1572
C2×SD16×D1572
C2×D8×D1572
#121×D1572
#237574212
#237574202
#237574192
#237574182
#237574172
#237574162
#237574152
#103×D1572
#102×D1572
C22⋊C8×D1572
#449×C1572
C2×#449157