Finite Group Databasebeta

← #34111618

Subgroups — #34111618

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
4D4×D628C211yesmaximal, characteristic, complemented
5#23757662C211yesmaximal, characteristic, complemented
6#23757527C211yesmaximal, characteristic, complemented
7#23757503C211yesmaximal, characteristic, complemented
8#23757488C211yesmaximal, characteristic, complemented
9#23757468C211yesmaximal, characteristic, complemented
10#23757425C211yesmaximal, characteristic, complemented
11#23757419C211yesmaximal, characteristic, complemented
12#23757408C211yesmaximal, characteristic, complemented
13#23757408C211yesmaximal, characteristic, complemented
14#23757346C211yesmaximal, characteristic, complemented
15#442×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic, complemented
17C22×D628—21no—
18C2×D4×D157C2211yescharacteristic
19C2×D4×D157C2211yescharacteristic, complemented
20C2×D4×D157C2211yescharacteristic, complemented
21#19097795—24no—
22C2×#14578943C2211yescharacteristic, complemented
23C2×#14578943C2211yescharacteristic, complemented
24C2×#14578942C2211yescharacteristic, complemented
25C2×D1256C2211yescharacteristic, complemented
26C2×D1256C2211yescharacteristic, complemented
27C2×#14578939C2211yescharacteristic
28C2×#14578939C2211yescharacteristic, complemented
29C2×#14578939C2211yescharacteristic, complemented
30D8×D157—24no—
31C2×#14578937C2211yescharacteristic, complemented
32C2×C8×D157C2211yescharacteristic, complemented
33#19097768C2211yescharacteristic, complemented
34#19097759C2211yescharacteristic, complemented
35#19097755—21no—
36#19097751C2211yescharacteristic, complemented
37#19097746C2211yescharacteristic
38#19097746C2211yescharacteristic, complemented
39#19097746C2211yescharacteristic, complemented
40#19097745—21no—
41#19097745—21no—
42#19097741—21no—
43#19097741—21no—
44#19097741—21no—
45C4×D628C2211yescharacteristic
46C2×#14578931—21no—
47C4⋊#11929434C2211yescharacteristic, complemented
48#19097694C2211yescharacteristic, complemented
49#19097691C2211yescharacteristic, complemented
50#19097691C2211yescharacteristic, complemented
51#19097691C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097683C2211yescharacteristic, complemented
54C2×D8×C157C2211yescharacteristic
55C4×D4×C157C2211yescharacteristic, complemented
56C4×D4×C157C2211yescharacteristic, complemented
57#106×C157C2211yescharacteristic, complemented
58#102×C157C2211yescharacteristic, complemented
59#102×C157C2211yescharacteristic, complemented
60#97×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63C23×D157—21no—
64#14578953—22no—
65#14578953—41no—
66C2×Dic314C2311yescharacteristic
67C2×D628—21no—
68C2×D628—21no—
69C2×D628—21no—
70C2×D628—21no—
71C2×D628—21no—
72C2×D628—44no—
73C2×D628D411yescharacteristic
74C2×D628C2311yescharacteristic
75C2×D628C2311yescharacteristic
76C2×D628C2311yescharacteristic
77C2×C22⋊D157—21no—
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C2×C22⋊D157—21no—
81D4×D157—22no—
82D4×D157—22no—
83D4×D157—22no—
84D4×D157—41no—
85D4×D157—41no—
86D4×D157—41no—
87C22×Dic157—21no—
88C2×C4×D157C2311yescharacteristic
89C2×C4×D157C2311yescharacteristic
90#14578943—22no—
91#14578943—41no—
92#14578942—22no—
93D1256—22no—
94D1256—41no—
95#14578939—22no—
96#14578939—22no—
97#14578939—22no—
98#14578937—22no—
99C2×#11929434C2311yescharacteristic
100C2×#11929434C2311yescharacteristic
101C8×D157—22no—
102C4⋊Dic157—21no—
103C4⋊Dic157D411yescharacteristic
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931—21no—
109#14578931—21no—
110#14578931D411yescharacteristic
111#14578931D411yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C2×D4×C157C2311yescharacteristic
114C22×C4×C157—21no—
115C22×C4×C157—21no—
116D8×C157D414yescomplemented
117C2×C8×C157C2311yescharacteristic
118C2×C8×C157C2311yescharacteristic
119C4⋊C4×C157C2311yescharacteristic
120C4⋊C4×C157C2311yescharacteristic
121C22⋊C4×C157—21no—
122C22⋊C4×C157—21no—
123C42×C157C2311yescharacteristic
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
137C22×D157C2×D411yescharacteristic
138Dic314—21no—
139Dic314—21no—
140D628—22no—
141D628—21no—
142D628—21no—
143D628—21no—
144D628—21no—
145D628—21no—
146D628—21no—
147D628—42no—
148D628—42no—
149D628—42no—
150D628—81no—
151D628—81no—
152C22⋊D157—42no—
153C22⋊D157—42no—
154C22⋊D157—42no—
155C22⋊D157—42no—
156C2×Dic157—21no—
157C2×Dic157—42no—
158C2×Dic157C2×D411yescharacteristic
159C2×Dic157C2×D411yescharacteristic
160C4×D157—21no—
161C4×D157—21no—
162C4×D157—21no—
163C4×D157—21no—
164#11929434—21no—
165#11929434—21no—
166C23×C157—21no—
167C23×C157—21no—
168D4×C157—21no—
169D4×C157—21no—
170D4×C157C2×D412yes—
171D4×C157C2×D412yescomplemented
172C2×C4×C157—21no—
173C2×C4×C157—21no—
174C2×C4×C157—42no—
175C2×C4×C157—81no—
176C2×C4×C157C2×D411yescharacteristic
177C2×C4×C157C2×D411yescharacteristic
178C2×C4×C157C2411yescharacteristic
179C8×C157—21no—
180C8×C157C2×D412yescomplemented
181C22×C157—22no—
182C22×C157—22no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157—41no—
187C22×C157C22×D411yescharacteristic
188C2×D157—21no—
189C2×D157—21no—
190C2×D157—21no—
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—84no—
203C2×D157—84no—
204C2×D157—84no—
205C2×D157AGL(1,ℤ/8)12yescomplemented
206C4×C157—21no—
207C4×C157—21no—
208C4×C157—41no—
209C4×C157—41no—
210C4×C157C22×D411yescharacteristic
211C4×C157H(5,𝔽2)11yescharacteristic
212Dic157—21no—
213Dic157—21no—
214Dic157—41no—
215C2×C157—42no—
216C2×C157—42no—
217C2×C157D4211yescharacteristic
218C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C157#57611yescharacteristic
220D157—22no—
221D157—41no—
222D157—81no—
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#465—1571no—
234#463—1571no—
235#459—1571no—
236#450—1571no—
237#449—1571no—
238#447—1571no—
239#447—1571no—
240#442D15711yescharacteristic, complemented
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
258C2×M16—1571no—
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#121—1571no—
268#121—1571no—
269#120—3141no—
270#120—3141no—
271#120—3141no—
272C4×D4—1571no—
273C4×D4C2×D15711yescharacteristic, complemented
274C4×D4C2×D15711yescharacteristic, complemented
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#97C2×D15711yescharacteristic, complemented
287C24—3141no—
288C24—3141no—
289C24—3141no—
290#41—3142no—
291#41—6281no—
292C2×Q8—1571no—
293C2×D4—3141no—
294C2×D4—3141no—
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—6284no—
306C2×D4—6281no—
307C2×D4—6281no—
308C2×D4—6281no—
309C2×D4—1571no—
310C2×D4—1571no—
311C2×D4—1571no—
312C2×D4—1571no—
313C2×D4C22×D15711yescharacteristic
314C2×D4C22×D15711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—3141no—
318C22×C4—1571no—
319C22×C4—1571no—
320SD16—3142no—
321SD16—3142no—
322SD16—6281no—
323D8—3142no—
324D8—3142no—
325D8—3142no—
326D8—3142no—
327D8—6281no—
328D8D62814yescomplemented
329M16—3142no—
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—
349C42C22×D15711yescharacteristic
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—1571no—
363C23—12561no—
364C23—12561no—
365C23—12561no—
366Q8—3141no—
367Q8—3141no—
368D4—21no—
369D4—21no—
370D4—3142no—
371D4—3141no—
372D4—3141no—
373D4—3141no—
374D4—3141no—
375D4—3141no—
376D4—3141no—
377D4—6282no—
378D4—6282no—
379D4—6282no—
380D4—6282no—
381D4—6282no—
382D4—6282no—
383D4—6282no—
384D4—12561no—
385D4—12561no—
386D4C2×D62812yes—
387D4C2×D62812yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—3141no—
393C2×C4—3141no—
394C2×C4—3141no—
395C2×C4—3141no—
396C2×C4—3141no—
397C2×C4—6282no—
398C2×C4—1571no—
399C2×C4—1571no—
400C2×C4D4×D15711yescharacteristic
401C2×C4D4×D15711yescharacteristic
402C2×C4C23×D15711yescharacteristic
403C8—21no—
404C8—3141no—
405C8—3141no—
406C8C2×D62812yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—3141no—
414C22—3141no—
415C22—3141no—
416C22—3141no—
417C22—3141no—
418C22—6281no—
419C22—6281no—
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—21no—
434C4—41no—
435C4—41no—
436C4—3141no—
437C4—3141no—
438C4—6281no—
439C4C22×D62811yescharacteristic
440C4#1909781711yescharacteristic
441C2—42no—
442C2—42no—
443C2—3142no—
444C2—6281no—
445C2—12561no—
446C2—12561no—
447C2—12561no—
448C2D4×D62811yescharacteristic
449C2AGL(1,ℤ/8)×D15711yescharacteristic
450C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#442×C157C2
socleC22×C157C22×D4
hypercenter#442D157
hypocenterC157#3264
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. C2×D8×D157C2semidirect
3. #23757918C2semidirect
4. D4×D628C2semidirect
5. #23757662C2semidirect
6. #23757527C2semidirect
7. #23757503C2semidirect
8. #23757488C2semidirect
9. #23757468C2semidirect
10. #23757425C2semidirect
11. #23757419C2semidirect
12. #23757408C2semidirect
13. #23757408C2semidirect
14. #23757346C2semidirect
15. #442×C157C2semidirect
16. C2×#14578953C22semidirect
19. C2×D4×D157C22semidirect
20. C2×D4×D157C22semidirect
22. C2×#14578943C22semidirect
23. C2×#14578943C22semidirect
24. C2×#14578942C22semidirect
25. C2×D1256C22semidirect
26. C2×D1256C22semidirect
28. C2×#14578939C22semidirect
29. C2×#14578939C22semidirect
31. C2×#14578937C22semidirect
32. C2×C8×D157C22semidirect
33. #19097768C22semidirect
34. #19097759C22semidirect
36. #19097751C22semidirect
38. #19097746C22semidirect
39. #19097746C22semidirect
47. C4⋊#11929434C22semidirect
48. #19097694C22semidirect
49. #19097691C22semidirect
50. #19097691C22semidirect
51. #19097691C22semidirect
52. #19097683C22semidirect
53. #19097683C22semidirect
55. C4×D4×C157C22semidirect
56. C4×D4×C157C22semidirect
57. #106×C157C22semidirect
58. #102×C157C22semidirect
59. #102×C157C22semidirect
60. #97×C157C22semidirect
116. D8×C157D4semidirect
171. D4×C157C2×D4semidirect
180. C8×C157C2×D4semidirect
205. C2×D157AGL(1,ℤ/8)semidirect
225. C157#3264semidirect
240. #442D157semidirect
273. C4×D4C2×D157semidirect
274. C4×D4C2×D157semidirect
276. #106C2×D157semidirect
282. #102C2×D157semidirect
283. #102C2×D157semidirect
286. #97C2×D157semidirect
328. D8D628semidirect
387. D4C2×D628semidirect
406. C8C2×D628semidirect

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
D4×D6282
#237576622
#237575272
#237575032
#237574882
#237574682
#237574252
#237574192
#237574082
#237573462
#442×C1572
#3264157