Finite Group Databasebeta

← #34111611

Subgroups — #34111611

Order 20096.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097795C211yesmaximal, characteristic, complemented
2C2×D8×D157C211yesmaximal, characteristic, complemented
3#23757918C211yesmaximal, characteristic, complemented
4#23757908C211yesmaximal, characteristic, complemented
5#23757660C211yesmaximal, characteristic, complemented
6#23757525C211yesmaximal, characteristic, complemented
7#23757501C211yesmaximal, characteristic, complemented
8#23757489C211yesmaximal, characteristic, complemented
9#23757425C211yesmaximal, characteristic, complemented
10#23757415C211yesmaximal, characteristic, complemented
11#23757408C211yesmaximal, characteristic, complemented
12#23757405C211yesmaximal, characteristic, complemented
13#23757383C211yesmaximal, characteristic, complemented
14#23757343C211yesmaximal, characteristic, complemented
15#459×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic, complemented
17C2×D4×D157—21no—
18C2×D4×D157C2211yescharacteristic
19C2×D4×D157C2211yescharacteristic, complemented
20C2×D4×D157C2211yescharacteristic, complemented
21#19097795—24no—
22C2×#14578943C2211yescharacteristic, complemented
23C2×#14578942C2211yescharacteristic, complemented
24C2×#14578942C2211yescharacteristic, complemented
25C2×D1256C2211yescharacteristic
26C2×#14578939C2211yescharacteristic
27C2×#14578939C2211yescharacteristic, complemented
28C2×#14578939C2211yescharacteristic, complemented
29D8×D157—24no—
30C2×#14578937C2211yescharacteristic, complemented
31C2×C8×D157C2211yescharacteristic, complemented
32#19097759C2211yescharacteristic, complemented
33#19097755—21no—
34#19097751C2211yescharacteristic, complemented
35#19097746C2211yescharacteristic, complemented
36#19097745—21no—
37#19097742—21no—
38#19097742C2211yescharacteristic
39#19097742C2211yescharacteristic, complemented
40#19097741—21no—
41#19097739—21no—
42#19097739—21no—
43C4×D628C2211yescharacteristic
44C4×D628C2211yescharacteristic, complemented
45C2×#14578931—21no—
46#19097702C2211yescharacteristic, complemented
47#19097696C2211yescharacteristic, complemented
48#19097693C2211yescharacteristic, complemented
49#19097692C2211yescharacteristic, complemented
50#19097691C2211yescharacteristic, complemented
51#19097683C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097682C2211yescharacteristic, complemented
54C2×D8×C157C2211yescharacteristic, complemented
55C2×D8×C157C2211yescharacteristic, complemented
56#127×C157C2211yescharacteristic, complemented
57C4×D4×C157C2211yescharacteristic, complemented
58C4⋊C8×C157C2211yescharacteristic, complemented
59#102×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63C23×D157—21no—
64#14578953—22no—
65#14578953—41no—
66C2×Dic314C2311yescharacteristic
67C2×D628—21no—
68C2×D628D411yescharacteristic
69C2×D628C2311yescharacteristic
70C2×D628C2311yescharacteristic
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77D4×D157—22no—
78D4×D157—22no—
79D4×D157—22no—
80D4×D157—41no—
81D4×D157—41no—
82D4×D157—41no—
83D4×D157—44no—
84C22×Dic157—21no—
85C2×C4×D157—21no—
86C2×C4×D157C2311yescharacteristic
87C2×C4×D157C2311yescharacteristic
88#14578943—22no—
89#14578942—22no—
90#14578942—41no—
91D1256—22no—
92#14578939—22no—
93#14578939—41no—
94#14578939D414yescomplemented
95#14578937—22no—
96C2×#11929434C2311yescharacteristic
97C2×#11929434C2311yescharacteristic
98C8×D157—22no—
99C4⋊Dic157—21no—
100C4⋊Dic157D411yescharacteristic
101C4⋊Dic157C2311yescharacteristic
102C22⋊Dic157—21no—
103C22⋊Dic157—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931—21no—
109#14578931D411yescharacteristic
110#14578931D411yescharacteristic
111C2×D4×C157—21no—
112C2×D4×C157—21no—
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115C2×D4×C157C2311yescharacteristic
116C22×C4×C157—21no—
117D8×C157—22no—
118D8×C157—22no—
119C2×C8×C157C2311yescharacteristic
120C2×C8×C157C2311yescharacteristic
121C4⋊C4×C157C2311yescharacteristic
122C22⋊C4×C157—21no—
123C42×C157C2311yescharacteristic
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—42no—
127C22×D157—42no—
128C22×D157—42no—
129C22×D157—42no—
130C22×D157—42no—
131C22×D157—42no—
132C22×D157—81no—
133C22×D157—81no—
134C22×D157—81no—
135C22×D157C2×D411yescharacteristic
136C22×D157C2×D411yescharacteristic
137Dic314—21no—
138Dic314—21no—
139D628—22no—
140D628—21no—
141D628—21no—
142D628—21no—
143D628—81no—
144D628C2×D412yes—
145C22⋊D157—42no—
146C22⋊D157—42no—
147C22⋊D157—42no—
148C22⋊D157—42no—
149C22⋊D157—42no—
150C22⋊D157—42no—
151C2×Dic157—21no—
152C2×Dic157—21no—
153C2×Dic157—42no—
154C2×Dic157C2×D411yescharacteristic
155C2×Dic157C2×D411yescharacteristic
156C4×D157—21no—
157C4×D157—21no—
158C4×D157—21no—
159C4×D157—21no—
160C4×D157—42no—
161#11929434—21no—
162#11929434C2×D412yescomplemented
163C23×C157—21no—
164C23×C157—21no—
165C23×C157—21no—
166D4×C157—21no—
167D4×C157—21no—
168D4×C157—21no—
169D4×C157—21no—
170D4×C157—21no—
171D4×C157—42no—
172D4×C157—81no—
173D4×C157C2×D412yescomplemented
174C2×C4×C157—21no—
175C2×C4×C157—81no—
176C2×C4×C157C2×D411yescharacteristic
177C2×C4×C157C2×D411yescharacteristic
178C2×C4×C157C2411yescharacteristic
179C8×C157—21no—
180C8×C157—21no—
181C22×C157—22no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157—41no—
187C22×C157—41no—
188C22×C157—41no—
189C22×C157—41no—
190C22×C157C22×D411yescharacteristic
191C2×D157—21no—
192C2×D157—21no—
193C2×D157—21no—
194C2×D157—22no—
195C2×D157—21no—
196C2×D157—21no—
197C2×D157—41no—
198C2×D157—41no—
199C2×D157—41no—
200C2×D157—41no—
201C2×D157—41no—
202C2×D157—84no—
203C2×D157—84no—
204C2×D157—84no—
205C2×D157AGL(1,ℤ/8)12yescomplemented
206C4×C157—21no—
207C4×C157—21no—
208C4×C157—41no—
209C4×C157C22×D411yescharacteristic
210C4×C157H(5,𝔽2)11yescharacteristic
211Dic157—21no—
212Dic157—21no—
213Dic157—41no—
214Dic157—41no—
215C2×C157—42no—
216C2×C157—81no—
217C2×C157—81no—
218C2×C157D4211yescharacteristic
219C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C157#57611yescharacteristic
221D157—22no—
222D157—41no—
223D157—42no—
224D157—81no—
225C157#326411yescharacteristic, complemented, Sylow
226#3264—1571nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1571no—
228C22×D8—1571no—
229#546—1571no—
230D42—1571no—
231#496—1571no—
232#469—1571no—
233#465—1571no—
234#463—1571no—
235#459D15711yescharacteristic, complemented
236#450—1571no—
237#449—1571no—
238#447—1571no—
239#447—1571no—
240#442—1571no—
241#435—1571no—
242C2×#41—1571no—
243C22×D4—3141no—
244C22×D4—1571no—
245C22×D4—1571no—
246C22×D4—1571no—
247AGL(1,ℤ/8)—3144no—
248C2×SD16—1571no—
249C2×SD16—1571no—
250C2×SD16—1571no—
251C2×D8—3144no—
252C2×D8—1571no—
253C2×D8—1571no—
254C2×D8—1571no—
255C2×D8—1571no—
256C2×D8C2×D15711yescharacteristic, complemented
257C2×D8C2×D15711yescharacteristic, complemented
258C2×M16—1571no—
259C22×C8—1571no—
260#127C2×D15711yescharacteristic, complemented
261#124—1571no—
262#123—3141no—
263C22⋊Q8—1571no—
264#121—3141no—
265#121—3141no—
266#121—1571no—
267#121—1571no—
268#121—1571no—
269#120—3141no—
270#120—3141no—
271#120—3141no—
272C4×D4—1571no—
273C4×D4—1571no—
274C4×D4C2×D15711yescharacteristic, complemented
275C2×C22⋊C4—3141no—
276#106—1571no—
277C4⋊C8C2×D15711yescharacteristic, complemented
278#103—1571no—
279#102—1571no—
280#102—1571no—
281#102—1571no—
282#102C2×D15711yescharacteristic, complemented
283#102C2×D15711yescharacteristic, complemented
284C22⋊C8—1571no—
285C22⋊C8—1571no—
286#97—1571no—
287C24—3141no—
288C24—3141no—
289C24—3141no—
290#41—3142no—
291#41—6281no—
292C2×Q8—1571no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3141no—
296C2×D4—3142no—
297C2×D4—3142no—
298C2×D4—3142no—
299C2×D4—3141no—
300C2×D4—3141no—
301C2×D4—3141no—
302C2×D4—3141no—
303C2×D4—3141no—
304C2×D4—3141no—
305C2×D4—6281no—
306C2×D4—6281no—
307C2×D4—6281no—
308C2×D4—6284no—
309C2×D4—1571no—
310C2×D4—1571no—
311C2×D4—1571no—
312C2×D4C22×D15711yescharacteristic
313C2×D4C22×D15711yescharacteristic
314C2×D4C22×D15711yescharacteristic
315C22×C4—21no—
316C22×C4—3141no—
317C22×C4—3141no—
318C22×C4—1571no—
319C22×C4—1571no—
320SD16—3142no—
321SD16—3142no—
322SD16—6281no—
323D8—22no—
324D8—22no—
325D8—3142no—
326D8—3142no—
327D8—6281no—
328D8—1574no—
329M16—3142no—
330C2×C8—3142no—
331C2×C8—1571no—
332C2×C8—1571no—
333C2×C8C22×D15711yescharacteristic
334C2×C8C22×D15711yescharacteristic
335C4⋊C4—3141no—
336C4⋊C4—1571no—
337C4⋊C4—1571no—
338C4⋊C4C22×D15711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—3141no—
341C22⋊C4—3141no—
342C22⋊C4—3141no—
343C22⋊C4—3141no—
344C22⋊C4—3141no—
345C22⋊C4—3141no—
346C22⋊C4—3141no—
347C22⋊C4—1571no—
348C22⋊C4—1571no—
349C42C22×D15711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—3141no—
354C23—3141no—
355C23—6282no—
356C23—6282no—
357C23—6282no—
358C23—6282no—
359C23—6282no—
360C23—6282no—
361C23—1571no—
362C23—1571no—
363C23—12561no—
364C23—12561no—
365C23—12561no—
366Q8—3141no—
367Q8—3141no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—3142no—
376D4—3141no—
377D4—3141no—
378D4—3141no—
379D4—6282no—
380D4—6282no—
381D4—6282no—
382D4—6282no—
383D4—6282no—
384D4—6282no—
385D4—1572no—
386D4—12561no—
387D4D4×D15712yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—3141no—
391C2×C4—3141no—
392C2×C4—3141no—
393C2×C4—3141no—
394C2×C4—3141no—
395C2×C4—3141no—
396C2×C4—6282no—
397C2×C4—6282no—
398C2×C4—1571no—
399C2×C4—1571no—
400C2×C4D4×D15711yescharacteristic
401C2×C4D4×D15711yescharacteristic
402C2×C4C23×D15711yescharacteristic
403C8—21no—
404C8—21no—
405C8—3141no—
406C8—1572no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—3141no—
417C22—3141no—
418C22—3141no—
419C22—3142no—
420C22—3141no—
421C22—3141no—
422C22—6281no—
423C22—6281no—
424C22—6281no—
425C22—6281no—
426C22—6281no—
427C22—1572no—
428C22—12564no—
429C22—12564no—
430C22—12564no—
431C22C2×D4×D15711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—3141no—
436C4—3141no—
437C4—6281no—
438C4—6281no—
439C4C2×D4×D15711yescharacteristic
440C4#1909781711yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—3142no—
445C2—6281no—
446C2—6282no—
447C2—12561no—
448C2#2375790711yescharacteristic
449C2AGL(1,ℤ/8)×D15711yescharacteristic
450C2#2375807411yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#459×C157C2
socleC22×C157C22×D4
hypercenter#459D157
hypocenterC157#3264
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. C2×D8×D157C2semidirect
3. #23757918C2semidirect
4. #23757908C2semidirect
5. #23757660C2semidirect
6. #23757525C2semidirect
7. #23757501C2semidirect
8. #23757489C2semidirect
9. #23757425C2semidirect
10. #23757415C2semidirect
11. #23757408C2semidirect
12. #23757405C2semidirect
13. #23757383C2semidirect
14. #23757343C2semidirect
15. #459×C157C2semidirect
16. C2×#14578953C22semidirect
19. C2×D4×D157C22semidirect
20. C2×D4×D157C22semidirect
22. C2×#14578943C22semidirect
23. C2×#14578942C22semidirect
24. C2×#14578942C22semidirect
27. C2×#14578939C22semidirect
28. C2×#14578939C22semidirect
30. C2×#14578937C22semidirect
31. C2×C8×D157C22semidirect
32. #19097759C22semidirect
34. #19097751C22semidirect
35. #19097746C22semidirect
39. #19097742C22semidirect
44. C4×D628C22semidirect
46. #19097702C22semidirect
47. #19097696C22semidirect
48. #19097693C22semidirect
49. #19097692C22semidirect
50. #19097691C22semidirect
51. #19097683C22semidirect
52. #19097683C22semidirect
53. #19097682C22semidirect
54. C2×D8×C157C22semidirect
55. C2×D8×C157C22semidirect
56. #127×C157C22semidirect
57. C4×D4×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #102×C157C22semidirect
60. #102×C157C22semidirect
94. #14578939D4semidirect
162. #11929434C2×D4semidirect
173. D4×C157C2×D4semidirect
205. C2×D157AGL(1,ℤ/8)semidirect
225. C157#3264semidirect
235. #459D157semidirect
256. C2×D8C2×D157semidirect
257. C2×D8C2×D157semidirect
260. #127C2×D157semidirect
274. C4×D4C2×D157semidirect
277. C4⋊C8C2×D157semidirect
282. #102C2×D157semidirect
283. #102C2×D157semidirect
387. D4D4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
C2×D8×D1572
#237579182
#237579082
#237576602
#237575252
#237575012
#237574892
#237574252
#237574152
#237574082
#237574052
#237573832
#237573432
#459×C1572
#3264157