Finite Group Databasebeta

← #34111613

Subgroups — #34111613

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×#19097798C211yesmaximal, characteristic, complemented
2C22×D1256C211yesmaximal, characteristic, complemented
3#23757914C211yesmaximal, characteristic, complemented
4#23757907C211yesmaximal, characteristic, complemented
5#23757659C211yesmaximal, characteristic, complemented
6#23757527C211yesmaximal, characteristic, complemented
7#23757491C211yesmaximal, characteristic, complemented
8#23757467C211yesmaximal, characteristic, complemented
9#23757423C211yesmaximal, characteristic, complemented
10#23757421C211yesmaximal, characteristic, complemented
11#23757408C211yesmaximal, characteristic, complemented
12#23757406C211yesmaximal, characteristic, complemented
13#23757384C211yesmaximal, characteristic, complemented
14#23757344C211yesmaximal, characteristic, complemented
15#465×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic, complemented
17C22×D628C2211yescharacteristic
18C22×D628C2211yescharacteristic, complemented
19C2×D4×D157—21no—
20C2×D4×D157C2211yescharacteristic, complemented
21#19097798—24no—
22C2×#14578941C2211yescharacteristic, complemented
23C2×#14578941C2211yescharacteristic, complemented
24C2×D1256—24no—
25C2×D1256C2211yescharacteristic
26C2×D1256C2211yescharacteristic
27C2×D1256C2211yescharacteristic, complemented
28C2×D1256C2211yescharacteristic, complemented
29C2×#14578939C2211yescharacteristic, complemented
30C2×#14578939C2211yescharacteristic, complemented
31C2×#14578937C2211yescharacteristic, complemented
32#19097767C2211yescharacteristic, complemented
33#19097760C2211yescharacteristic, complemented
34#19097756—21no—
35#19097746—21no—
36#19097746C2211yescharacteristic, complemented
37#19097746C2211yescharacteristic, complemented
38#19097744—21no—
39#19097741—21no—
40#19097741—21no—
41#19097739—21no—
42#19097733C2211yescharacteristic, complemented
43#19097733C2211yescharacteristic, complemented
44C4×C22⋊D157C2211yescharacteristic
45C22⋊C4×D157—21no—
46#19097700C2211yescharacteristic, complemented
47#19097691C2211yescharacteristic, complemented
48#19097691C2211yescharacteristic, complemented
49#19097691C2211yescharacteristic, complemented
50#19097691C2211yescharacteristic, complemented
51C22⋊#11929434C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097681C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C22×C8×C157C2211yescharacteristic, complemented
56C22⋊Q8×C157C2211yescharacteristic, complemented
57#121×C157C2211yescharacteristic
58#106×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63C23×D157—21no—
64#14578955—22no—
65#14578955—41no—
66C2×D628—22no—
67C2×D628—22no—
68C2×D628—21no—
69C2×D628—21no—
70C2×D628—21no—
71C2×D628—21no—
72C2×D628—21no—
73C2×D628—41no—
74C2×D628—41no—
75C2×D628C2311yescharacteristic
76C2×D628C2311yescharacteristic
77C2×D628C2311yescharacteristic
78C2×D628C2311yescharacteristic
79C2×C22⋊D157—21no—
80C2×C22⋊D157—21no—
81C2×C22⋊D157—21no—
82C2×C22⋊D157D411yescharacteristic
83D4×D157—22no—
84D4×D157—41no—
85D4×D157—44no—
86C2×C4×D157—21no—
87C2×C4×D157—21no—
88C2×C4×D157—21no—
89C2×C4×D157C2311yescharacteristic
90#14578941—22no—
91#14578941—41no—
92D1256—22no—
93D1256—22no—
94D1256—22no—
95D1256D414yescomplemented
96#14578939—22no—
97#14578939—41no—
98#14578937—22no—
99C2×#11929434C2311yescharacteristic
100C2×#11929434C2311yescharacteristic
101#14578933D411yescharacteristic
102C22⋊Dic157D411yescharacteristic
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931—21no—
109#14578931D411yescharacteristic
110C4×Dic157C2311yescharacteristic
111C2×Q8×C157C2311yescharacteristic
112C2×D4×C157—21no—
113C2×D4×C157C2311yescharacteristic
114C22×C4×C157C2311yescharacteristic
115SD16×C157—22no—
116C2×C8×C157—22no—
117C2×C8×C157C2311yescharacteristic
118C2×C8×C157C2311yescharacteristic
119C4⋊C4×C157—21no—
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—21no—
128C22×D157—42no—
129C22×D157—42no—
130C22×D157—42no—
131C22×D157—42no—
132C22×D157—42no—
133C22×D157—42no—
134C22×D157—81no—
135C22×D157—81no—
136C22×D157—81no—
137C22×D157C2×D411yescharacteristic
138D628—21no—
139D628—21no—
140D628—21no—
141D628—21no—
142D628—21no—
143D628—21no—
144D628—42no—
145D628—42no—
146D628—42no—
147D628—42no—
148D628—81no—
149D628C2×D412yes—
150D628C2×D412yescomplemented
151C22⋊D157—22no—
152C22⋊D157—42no—
153C22⋊D157—42no—
154C22⋊D157—81no—
155C2×Dic157C2×D411yescharacteristic
156C2×Dic157C2×D411yescharacteristic
157C2×Dic157C2×D411yescharacteristic
158C4×D157—21no—
159C4×D157—21no—
160C4×D157—42no—
161C4×D157—42no—
162C4×D157—81no—
163#11929434—21no—
164#11929434—21no—
165C23×C157—21no—
166C23×C157C2×D411yescharacteristic
167Q8×C157—21no—
168Q8×C157—21no—
169D4×C157—21no—
170D4×C157—21no—
171D4×C157—42no—
172C2×C4×C157—21no—
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—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157AGL(1,ℤ/8)12yescomplemented
187C22×C157C22×D411yescharacteristic
188C2×D157—21no—
189C2×D157—22no—
190C2×D157—22no—
191C2×D157—21no—
192C2×D157—21no—
193C2×D157—41no—
194C2×D157—41no—
195C2×D157—41no—
196C2×D157—41no—
197C2×D157—41no—
198C2×D157—41no—
199C2×D157—41no—
200C2×D157—41no—
201C2×D157—41no—
202C2×D157—41no—
203C2×D157—84no—
204C2×D157—84no—
205C2×D157—84no—
206C4×C157—21no—
207C4×C157—41no—
208C4×C157—41no—
209C4×C157—41no—
210C4×C157C22×D411yescharacteristic
211C4×C157H(5,𝔽2)11yescharacteristic
212Dic157—21no—
213Dic157—21no—
214Dic157—21no—
215C2×C157—22no—
216C2×C157—81no—
217C2×C157D4211yescharacteristic
218C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C157#57611yescharacteristic
220D157—41no—
221D157—42no—
222D157—42no—
223D157—81no—
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#465D15711yescharacteristic, complemented
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×SD16C2×D15711yescharacteristic, complemented
251C2×D8—3144no—
252C2×D8—1571no—
253C2×D8—1571no—
254C2×D8—1571no—
255C2×D8—1571no—
256C2×D8—1571no—
257C2×D8—1571no—
258C2×M16—1571no—
259C22×C8C2×D15711yescharacteristic, complemented
260#127—1571no—
261#124—1571no—
262#123—3141no—
263C22⋊Q8C2×D15711yescharacteristic, complemented
264#121—3141no—
265#121—3141no—
266#121—1571no—
267#121—1571no—
268#121C2×D15711yescharacteristic
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#103C2×D15711yescharacteristic, complemented
279#102—1571no—
280#102—1571no—
281#102—1571no—
282#102—1571no—
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×Q8C22×D15711yescharacteristic
293C2×D4—21no—
294C2×D4—3142no—
295C2×D4—3142no—
296C2×D4—3141no—
297C2×D4—3141no—
298C2×D4—3141no—
299C2×D4—3142no—
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×D4—1571no—
314C2×D4C22×D15711yescharacteristic
315C22×C4—3141no—
316C22×C4—3141no—
317C22×C4—3141no—
318C22×C4—1571no—
319C22×C4C22×D15711yescharacteristic
320SD16—22no—
321SD16—3142no—
322SD16—6281no—
323D8—3142no—
324D8—3142no—
325D8—3142no—
326D8—3142no—
327D8—6281no—
328D8—1574no—
329M16—3142no—
330C2×C8—22no—
331C2×C8—1571no—
332C2×C8—1571no—
333C2×C8C22×D15711yescharacteristic
334C2×C8C22×D15711yescharacteristic
335C4⋊C4—21no—
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—3141no—
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—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—42no—
371D4—3142no—
372D4—3141no—
373D4—3141no—
374D4—3141no—
375D4—3141no—
376D4—3141no—
377D4—3141no—
378D4—6282no—
379D4—6282no—
380D4—6282no—
381D4—6282no—
382D4—6282no—
383D4—6282no—
384D4—1572no—
385D4—1572no—
386D4—12561no—
387D4—12561no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
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—41no—
410C22—41no—
411C22—41no—
412C22—3141no—
413C22—3142no—
414C22—3142no—
415C22—3141no—
416C22—3141no—
417C22—6281no—
418C22—6281no—
419C22—6281no—
420C22—6281no—
421C22—6281no—
422C22—6281no—
423C22—6281no—
424C22—6281no—
425C22—6281no—
426C22—6281no—
427C22—12564no—
428C22—12564no—
429C22—12564no—
430C22#1909779812yescomplemented
431C22C2×D4×D15711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—3141no—
437C4—3141no—
438C4—3141no—
439C4C2×D4×D15711yescharacteristic
440C4#1909781711yescharacteristic
441C2—22no—
442C2—81no—
443C2—6281no—
444C2—6282no—
445C2—6282no—
446C2—12561no—
447C2—12561no—
448C2#2375790711yescharacteristic
449C2C2×#1909779811yescharacteristic
450C2#2375807511yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#465×C157C2
socleC22×C157C22×D4
hypercenter#465D157
hypocenterC157#3264
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. C22×D1256C2semidirect
3. #23757914C2semidirect
4. #23757907C2semidirect
5. #23757659C2semidirect
6. #23757527C2semidirect
7. #23757491C2semidirect
8. #23757467C2semidirect
9. #23757423C2semidirect
10. #23757421C2semidirect
11. #23757408C2semidirect
12. #23757406C2semidirect
13. #23757384C2semidirect
14. #23757344C2semidirect
15. #465×C157C2semidirect
16. C2×#14578955C22semidirect
18. C22×D628C22semidirect
20. C2×D4×D157C22semidirect
22. C2×#14578941C22semidirect
23. C2×#14578941C22semidirect
27. C2×D1256C22semidirect
28. C2×D1256C22semidirect
29. C2×#14578939C22semidirect
30. C2×#14578939C22semidirect
31. C2×#14578937C22semidirect
32. #19097767C22semidirect
33. #19097760C22semidirect
36. #19097746C22semidirect
37. #19097746C22semidirect
42. #19097733C22semidirect
43. #19097733C22semidirect
46. #19097700C22semidirect
47. #19097691C22semidirect
48. #19097691C22semidirect
49. #19097691C22semidirect
50. #19097691C22semidirect
51. C22⋊#11929434C22semidirect
52. #19097683C22semidirect
53. #19097681C22semidirect
54. C2×SD16×C157C22semidirect
55. C22×C8×C157C22semidirect
56. C22⋊Q8×C157C22semidirect
58. #106×C157C22semidirect
59. #103×C157C22semidirect
60. #102×C157C22semidirect
95. D1256D4semidirect
150. D628C2×D4semidirect
180. C8×C157C2×D4semidirect
186. C22×C157AGL(1,ℤ/8)semidirect
225. C157#3264semidirect
233. #465D157semidirect
250. C2×SD16C2×D157semidirect
259. C22×C8C2×D157semidirect
263. C22⋊Q8C2×D157semidirect
276. #106C2×D157semidirect
278. #103C2×D157semidirect
283. #102C2×D157semidirect
406. C8D4×D157semidirect
430. C22#19097798semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
C22×D12562
#237579142
#237579072
#237576592
#237575272
#237574912
#237574672
#237574232
#237574212
#237574082
#237574062
#237573842
#237573442
#465×C1572
#3264157