Finite Group Databasebeta

← #34111623

Subgroups — #34111623

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×#19097799C211yesmaximal, characteristic, complemented
2C2×D8×D157C211yesmaximal, characteristic, complemented
3#23757920C211yesmaximal, characteristic, complemented
4#23757907C211yesmaximal, characteristic, complemented
5#23757661C211yesmaximal, characteristic, complemented
6#23757503C211yesmaximal, characteristic, complemented
7#23757487C211yesmaximal, characteristic, complemented
8#23757467C211yesmaximal, characteristic, complemented
9#23757425C211yesmaximal, characteristic, complemented
10#23757416C211yesmaximal, characteristic, complemented
11#23757408C211yesmaximal, characteristic, complemented
12#23757406C211yesmaximal, characteristic, complemented
13#23757384C211yesmaximal, characteristic, complemented
14#23757342C211yesmaximal, characteristic, complemented
15#469×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic, complemented
17C22×D628C2211yescharacteristic, complemented
18C2×D4×D157—21no—
19C2×D4×D157C2211yescharacteristic
20C2×D4×D157C2211yescharacteristic, complemented
21#19097799—24no—
22C2×#14578943C2211yescharacteristic, complemented
23C2×#14578943C2211yescharacteristic, complemented
24C2×#14578942C2211yescharacteristic, complemented
25C2×D1256C2211yescharacteristic
26C2×D1256C2211yescharacteristic, complemented
27C2×#14578939C2211yescharacteristic
28C2×#14578939C2211yescharacteristic, complemented
29C2×#14578939C2211yescharacteristic, complemented
30D8×D157—24no—
31C2×C8×D157C2211yescharacteristic, complemented
32#19097767C2211yescharacteristic, complemented
33#19097758C2211yescharacteristic, complemented
34#19097753—21no—
35#19097751C2211yescharacteristic, complemented
36#19097746C2211yescharacteristic
37#19097744—21no—
38#19097742—21no—
39#19097741—21no—
40#19097739—21no—
41#19097739—21no—
42#19097733C2211yescharacteristic, complemented
43#19097733C2211yescharacteristic, complemented
44C4×C22⋊D157C2211yescharacteristic
45C22⋊C4×D157—21no—
46#19097700C2211yescharacteristic, complemented
47#19097694C2211yescharacteristic, complemented
48#19097691C2211yescharacteristic, complemented
49#19097691C2211yescharacteristic, complemented
50#19097691C2211yescharacteristic, complemented
51C22⋊#11929434C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097681C2211yescharacteristic, complemented
54C2×D8×C157C2211yescharacteristic, complemented
55C2×M16×C157C2211yescharacteristic, complemented
56#121×C157C2211yescharacteristic, complemented
57#121×C157C2211yescharacteristic, complemented
58#106×C157C2211yescharacteristic, complemented
59#102×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63C23×D157—21no—
64#14578954—22no—
65#14578954—41no—
66C2×Dic314C2311yescharacteristic
67C2×D628—22no—
68C2×D628—21no—
69C2×D628—21no—
70C2×D628—41no—
71C2×D628C2311yescharacteristic
72C2×D628C2311yescharacteristic
73C2×D628C2311yescharacteristic
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77C2×C22⋊D157—21no—
78C2×C22⋊D157—21no—
79C2×C22⋊D157D411yescharacteristic
80D4×D157—22no—
81D4×D157—22no—
82D4×D157—41no—
83D4×D157—41no—
84D4×D157—44no—
85C2×C4×D157—21no—
86C2×C4×D157—21no—
87C2×C4×D157—21no—
88C2×C4×D157C2311yescharacteristic
89#14578943—22no—
90#14578943—22no—
91#14578942—41no—
92D1256—22no—
93D1256D414yescomplemented
94#14578939—22no—
95#14578939—22no—
96#14578939—41no—
97C2×#11929434C2311yescharacteristic
98C2×#11929434C2311yescharacteristic
99C8×D157—22no—
100C4⋊Dic157—21no—
101#14578933D411yescharacteristic
102C22⋊Dic157—21no—
103C22⋊Dic157—21no—
104C22⋊Dic157D411yescharacteristic
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931—21no—
109#14578931D411yescharacteristic
110C4×Dic157C2311yescharacteristic
111C2×D4×C157—21no—
112C2×D4×C157—21no—
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115C22×C4×C157C2311yescharacteristic
116D8×C157—22no—
117M16×C157—22no—
118C2×C8×C157C2311yescharacteristic
119C2×C8×C157C2311yescharacteristic
120C4⋊C4×C157C2311yescharacteristic
121C4⋊C4×C157C2311yescharacteristic
122C22⋊C4×C157—21no—
123C22⋊C4×C157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—21no—
127C22×D157—42no—
128C22×D157—42no—
129C22×D157—42no—
130C22×D157—42no—
131C22×D157—42no—
132C22×D157—42no—
133C22×D157—81no—
134C22×D157—81no—
135C22×D157—81no—
136C22×D157C2×D411yescharacteristic
137Dic314—21no—
138Dic314—21no—
139D628—21no—
140D628—21no—
141D628—21no—
142D628—21no—
143D628—42no—
144D628—42no—
145D628C2×D412yes—
146D628C2×D412yescomplemented
147C22⋊D157—22no—
148C22⋊D157—42no—
149C22⋊D157—42no—
150C22⋊D157—42no—
151C22⋊D157—42no—
152C22⋊D157—81no—
153C2×Dic157—21no—
154C2×Dic157C2×D411yescharacteristic
155C2×Dic157C2×D411yescharacteristic
156C2×Dic157C2×D411yescharacteristic
157C4×D157—21no—
158C4×D157—21no—
159C4×D157—42no—
160C4×D157—42no—
161C4×D157—81no—
162#11929434—21no—
163#11929434—21no—
164C23×C157—21no—
165C23×C157—21no—
166C23×C157C2×D411yescharacteristic
167D4×C157—21no—
168D4×C157—21no—
169D4×C157—21no—
170D4×C157—21no—
171D4×C157—42no—
172D4×C157—81no—
173C2×C4×C157—21no—
174C2×C4×C157—21no—
175C2×C4×C157—21no—
176C2×C4×C157—21no—
177C2×C4×C157C2×D411yescharacteristic
178C2×C4×C157C2411yescharacteristic
179C8×C157—21no—
180C8×C157C2×D412yescomplemented
181C22×C157—21no—
182C22×C157—21no—
183C22×C157—21no—
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—22no—
194C2×D157—22no—
195C2×D157—41no—
196C2×D157—41no—
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—41no—
208C4×C157—41no—
209C4×C157C22×D411yescharacteristic
210C4×C157H(5,𝔽2)11yescharacteristic
211Dic157—21no—
212Dic157—21no—
213Dic157—21no—
214Dic157—41no—
215C2×C157—41no—
216C2×C157—81no—
217C2×C157—81no—
218C2×C157D4211yescharacteristic
219C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C157#57611yescharacteristic
221D157—22no—
222D157—42no—
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#469D15711yescharacteristic, complemented
233#465—1571no—
234#463—1571no—
235#459—1571no—
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×D8—1571no—
257C2×D8C2×D15711yescharacteristic, complemented
258C2×M16C2×D15711yescharacteristic, complemented
259C22×C8—1571no—
260#127—1571no—
261#124—1571no—
262#123—3141no—
263C22⋊Q8—1571no—
264#121—3141no—
265#121—3141no—
266#121—1571no—
267#121C2×D15711yescharacteristic, complemented
268#121C2×D15711yescharacteristic, complemented
269#120—3141no—
270#120—3141no—
271#120—3141no—
272C4×D4—1571no—
273C4×D4—1571no—
274C4×D4—1571no—
275C2×C22⋊C4—3141no—
276#106C2×D15711yescharacteristic, complemented
277C4⋊C8—1571no—
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—3142no—
296C2×D4—3141no—
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×D4—1571no—
313C2×D4C22×D15711yescharacteristic
314C2×D4C22×D15711yescharacteristic
315C22×C4—3141no—
316C22×C4—3141no—
317C22×C4—3141no—
318C22×C4—1571no—
319C22×C4C22×D15711yescharacteristic
320SD16—3142no—
321SD16—3142no—
322SD16—6281no—
323D8—22no—
324D8—3142no—
325D8—3142no—
326D8—3142no—
327D8—6281no—
328D8—1574no—
329M16—22no—
330C2×C8—3142no—
331C2×C8—1571no—
332C2×C8—1571no—
333C2×C8C22×D15711yescharacteristic
334C2×C8C22×D15711yescharacteristic
335C4⋊C4—3141no—
336C4⋊C4—1571no—
337C4⋊C4C22×D15711yescharacteristic
338C4⋊C4C22×D15711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—3141no—
342C22⋊C4—3141no—
343C22⋊C4—3141no—
344C22⋊C4—3141no—
345C22⋊C4—3141no—
346C22⋊C4—3141no—
347C22⋊C4—1571no—
348C22⋊C4—1571no—
349C42—1571no—
350C23—21no—
351C23—21no—
352C23—3141no—
353C23—3141no—
354C23—3141no—
355C23—6282no—
356C23—6282no—
357C23—6282no—
358C23—6282no—
359C23—6282no—
360C23—6282no—
361C23—1571no—
362C23—12561no—
363C23—12561no—
364C23—12561no—
365C23D4×D15711yescharacteristic
366Q8—3141no—
367Q8—3141no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—3142no—
375D4—3141no—
376D4—3141no—
377D4—3141no—
378D4—3141no—
379D4—6282no—
380D4—6282no—
381D4—6282no—
382D4—6282no—
383D4—6282no—
384D4—6282no—
385D4—1572no—
386D4—1572no—
387D4—12561no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—3141no—
393C2×C4—3141no—
394C2×C4—3141no—
395C2×C4—6282no—
396C2×C4—6282no—
397C2×C4—1571no—
398C2×C4—1571no—
399C2×C4—1571no—
400C2×C4—12561no—
401C2×C4D4×D15711yescharacteristic
402C2×C4C23×D15711yescharacteristic
403C8—21no—
404C8—3141no—
405C8—3141no—
406C8D4×D15712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—3141no—
417C22—3141no—
418C22—3142no—
419C22—3142no—
420C22—6281no—
421C22—6281no—
422C22—6281no—
423C22—6281no—
424C22—6281no—
425C22—6281no—
426C22—6281no—
427C22—1572no—
428C22—12564no—
429C22—12564no—
430C22—12564no—
431C22C2×D4×D15711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—3141no—
436C4—3141no—
437C4—3141no—
438C4—6281no—
439C4C2×D4×D15711yescharacteristic
440C4#1909781711yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—3142no—
445C2—6282no—
446C2—6282no—
447C2—12561no—
448C2#2375790711yescharacteristic
449C2AGL(1,ℤ/8)×D15711yescharacteristic
450C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#469×C157C2
socleC22×C157C22×D4
hypercenter#469D157
hypocenterC157#3264
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. C2×D8×D157C2semidirect
3. #23757920C2semidirect
4. #23757907C2semidirect
5. #23757661C2semidirect
6. #23757503C2semidirect
7. #23757487C2semidirect
8. #23757467C2semidirect
9. #23757425C2semidirect
10. #23757416C2semidirect
11. #23757408C2semidirect
12. #23757406C2semidirect
13. #23757384C2semidirect
14. #23757342C2semidirect
15. #469×C157C2semidirect
16. C2×#14578954C22semidirect
17. C22×D628C22semidirect
20. C2×D4×D157C22semidirect
22. C2×#14578943C22semidirect
23. C2×#14578943C22semidirect
24. C2×#14578942C22semidirect
26. C2×D1256C22semidirect
28. C2×#14578939C22semidirect
29. C2×#14578939C22semidirect
31. C2×C8×D157C22semidirect
32. #19097767C22semidirect
33. #19097758C22semidirect
35. #19097751C22semidirect
42. #19097733C22semidirect
43. #19097733C22semidirect
46. #19097700C22semidirect
47. #19097694C22semidirect
48. #19097691C22semidirect
49. #19097691C22semidirect
50. #19097691C22semidirect
51. C22⋊#11929434C22semidirect
52. #19097683C22semidirect
53. #19097681C22semidirect
54. C2×D8×C157C22semidirect
55. C2×M16×C157C22semidirect
56. #121×C157C22semidirect
57. #121×C157C22semidirect
58. #106×C157C22semidirect
59. #102×C157C22semidirect
60. #102×C157C22semidirect
93. D1256D4semidirect
146. D628C2×D4semidirect
180. C8×C157C2×D4semidirect
205. C2×D157AGL(1,ℤ/8)semidirect
225. C157#3264semidirect
232. #469D157semidirect
257. C2×D8C2×D157semidirect
258. C2×M16C2×D157semidirect
267. #121C2×D157semidirect
268. #121C2×D157semidirect
276. #106C2×D157semidirect
282. #102C2×D157semidirect
283. #102C2×D157semidirect
406. C8D4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
C2×D8×D1572
#237579202
#237579072
#237576612
#237575032
#237574872
#237574672
#237574252
#237574162
#237574082
#237574062
#237573842
#237573422
#469×C1572
#3264157