Finite Group Databasebeta

← #34111615

Subgroups — #34111615

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
2C2×D8×D157C211yesmaximal, characteristic, complemented
3#23757916C211yesmaximal, characteristic, complemented
4#23757908C211yesmaximal, characteristic, complemented
5#23757523C211yesmaximal, characteristic, complemented
6#23757499C211yesmaximal, characteristic, complemented
7#23757485C211yesmaximal, characteristic, complemented
8#23757465C211yesmaximal, characteristic, complemented
9#23757422C211yesmaximal, characteristic, complemented
10#23757417C211yesmaximal, characteristic, complemented
11#23757405C211yesmaximal, characteristic, complemented
12#23757405C211yesmaximal, characteristic, complemented
13#23757386C211yesmaximal, characteristic, complemented
14#23757346C211yesmaximal, characteristic, complemented
15#496×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic, complemented
17C2×D4×D157—21no—
18C2×D4×D157C2211yescharacteristic
19C2×D4×D157C2211yescharacteristic, complemented
20C2×D4×D157C2211yescharacteristic, complemented
21#19097798—24no—
22C2×#14578941C2211yescharacteristic, complemented
23C2×D1256C2211yescharacteristic
24C2×#14578939C2211yescharacteristic
25C2×#14578939C2211yescharacteristic, complemented
26C2×#14578939C2211yescharacteristic, complemented
27D8×D157—24no—
28C2×#14578937C2211yescharacteristic, complemented
29C2×C8×D157C2211yescharacteristic, complemented
30#19097753—21no—
31#19097747C2211yescharacteristic, complemented
32#19097744—21no—
33#19097742—21no—
34#19097742C2211yescharacteristic
35#19097742C2211yescharacteristic, complemented
36#19097742C2211yescharacteristic, complemented
37#19097741—21no—
38#19097739—21no—
39#19097739—21no—
40C4×D628C2211yescharacteristic
41C4×D628C2211yescharacteristic, complemented
42C4×D628C2211yescharacteristic, complemented
43C22⋊C4×D157—21no—
44#19097703C2211yescharacteristic, complemented
45C4⋊#11929434C2211yescharacteristic, complemented
46#19097695C2211yescharacteristic, complemented
47#19097693C2211yescharacteristic, complemented
48#19097693C2211yescharacteristic, complemented
49#19097692C2211yescharacteristic, complemented
50#19097692C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097683C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C2×SD16×C157C2211yescharacteristic, complemented
56C2×D8×C157C2211yescharacteristic, complemented
57C2×D8×C157C2211yescharacteristic, complemented
58#127×C157C2211yescharacteristic, complemented
59#124×C157C2211yescharacteristic, complemented
60#97×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63C23×D157—21no—
64#14578955—22no—
65#14578955—41no—
66C2×D628—21no—
67C2×D628D411yescharacteristic
68C2×D628C2311yescharacteristic
69C2×D628C2311yescharacteristic
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76D4×D157—22no—
77D4×D157—22no—
78D4×D157—22no—
79D4×D157—41no—
80D4×D157—41no—
81D4×D157—41no—
82D4×D157—44no—
83C2×C4×D157—21no—
84C2×C4×D157—21no—
85C2×C4×D157—21no—
86C2×C4×D157C2311yescharacteristic
87C2×C4×D157C2311yescharacteristic
88#14578941—22no—
89D1256D414yescomplemented
90#14578939—22no—
91#14578939—22no—
92#14578939—22no—
93#14578937—22no—
94C2×#11929434C2311yescharacteristic
95C2×#11929434C2311yescharacteristic
96C8×D157—22no—
97C4⋊Dic157D411yescharacteristic
98C4⋊Dic157C2311yescharacteristic
99C4⋊Dic157C2311yescharacteristic
100#14578933—21no—
101C22⋊Dic157—21no—
102C22⋊Dic157—21no—
103C22⋊Dic157—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931D411yescharacteristic
108#14578931D411yescharacteristic
109C2×Q8×C157C2311yescharacteristic
110C2×D4×C157—21no—
111C2×D4×C157—21no—
112C2×D4×C157C2311yescharacteristic
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115SD16×C157—22no—
116SD16×C157—41no—
117D8×C157—22no—
118D8×C157—41no—
119C2×C8×C157C2311yescharacteristic
120C2×C8×C157C2311yescharacteristic
121C22⋊C4×C157—21no—
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
137D628—22no—
138D628—21no—
139D628—21no—
140D628—42no—
141D628C2×D412yes—
142D628C2×D412yescomplemented
143C22⋊D157—42no—
144C22⋊D157—42no—
145C22⋊D157—42no—
146C22⋊D157—42no—
147C22⋊D157—42no—
148C22⋊D157—81no—
149C2×Dic157—21no—
150C2×Dic157—21no—
151C2×Dic157C2×D411yescharacteristic
152C2×Dic157C2×D411yescharacteristic
153C4×D157—21no—
154C4×D157—21no—
155C4×D157—21no—
156C4×D157—21no—
157C4×D157—42no—
158C4×D157—42no—
159C4×D157—81no—
160#11929434—21no—
161#11929434—21no—
162C23×C157—21no—
163C23×C157—21no—
164C23×C157—21no—
165Q8×C157—21no—
166Q8×C157—21no—
167D4×C157—21no—
168D4×C157—21no—
169D4×C157—21no—
170D4×C157—21no—
171D4×C157—21no—
172D4×C157—21no—
173D4×C157—42no—
174D4×C157—81no—
175C2×C4×C157—21no—
176C2×C4×C157C2×D411yescharacteristic
177C2×C4×C157C2×D411yescharacteristic
178C2×C4×C157C2411yescharacteristic
179C8×C157—21no—
180C8×C157C2×D412yescomplemented
181C22×C157—41no—
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—22no—
196C2×D157—21no—
197C2×D157—21no—
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—81no—
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—42no—
225C157#326411yescharacteristic, complemented, Sylow
226#3264—1571nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1571no—
228C22×D8—1571no—
229#546—1571no—
230D42—1571no—
231#496D15711yescharacteristic, complemented
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#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×SD16C2×D15711yescharacteristic, complemented
250C2×SD16C2×D15711yescharacteristic, complemented
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#124C2×D15711yescharacteristic, complemented
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×D4—1571no—
275C2×C22⋊C4—3141no—
276#106—1571no—
277C4⋊C8—1571no—
278#103—1571no—
279#102—1571no—
280#102—1571no—
281#102—1571no—
282#102—1571no—
283#102—1571no—
284C22⋊C8—1571no—
285C22⋊C8—1571no—
286#97C2×D15711yescharacteristic, complemented
287C24—3141no—
288C24—3141no—
289C24—3141no—
290#41—3142no—
291#41—6281no—
292C2×Q8C22×D15711yescharacteristic
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—3141no—
316C22×C4—3141no—
317C22×C4—3141no—
318C22×C4—1571no—
319C22×C4—1571no—
320SD16—22no—
321SD16—41no—
322SD16—3142no—
323D8—22no—
324D8—41no—
325D8—3142no—
326D8—3142no—
327D8—3142no—
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⋊C4—1571no—
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—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—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—3142no—
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—3141no—
390C2×C4—3141no—
391C2×C4—3141no—
392C2×C4—3141no—
393C2×C4—3141no—
394C2×C4—3141no—
395C2×C4—6282no—
396C2×C4—6282no—
397C2×C4—1571no—
398C2×C4—1571no—
399C2×C4—12561no—
400C2×C4D4×D15711yescharacteristic
401C2×C4D4×D15711yescharacteristic
402C2×C4C23×D15711yescharacteristic
403C8—21no—
404C8—3141no—
405C8—3141no—
406C8D4×D15712yescomplemented
407C22—41no—
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—3142no—
421C22—3141no—
422C22—3141no—
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#1909781611yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—3142no—
445C2—6281no—
446C2—6282no—
447C2—6282no—
448C2#2375790811yescharacteristic
449C2AGL(1,ℤ/8)×D15711yescharacteristic
450C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#496×C157C2
socleC22×C157C22×D4
hypercenter#496D157
hypocenterC157#3264
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. C2×D8×D157C2semidirect
3. #23757916C2semidirect
4. #23757908C2semidirect
5. #23757523C2semidirect
6. #23757499C2semidirect
7. #23757485C2semidirect
8. #23757465C2semidirect
9. #23757422C2semidirect
10. #23757417C2semidirect
11. #23757405C2semidirect
12. #23757405C2semidirect
13. #23757386C2semidirect
14. #23757346C2semidirect
15. #496×C157C2semidirect
16. C2×#14578955C22semidirect
19. C2×D4×D157C22semidirect
20. C2×D4×D157C22semidirect
22. C2×#14578941C22semidirect
25. C2×#14578939C22semidirect
26. C2×#14578939C22semidirect
28. C2×#14578937C22semidirect
29. C2×C8×D157C22semidirect
31. #19097747C22semidirect
35. #19097742C22semidirect
36. #19097742C22semidirect
41. C4×D628C22semidirect
42. C4×D628C22semidirect
44. #19097703C22semidirect
45. C4⋊#11929434C22semidirect
46. #19097695C22semidirect
47. #19097693C22semidirect
48. #19097693C22semidirect
49. #19097692C22semidirect
50. #19097692C22semidirect
51. #19097692C22semidirect
52. #19097683C22semidirect
53. #19097683C22semidirect
54. C2×SD16×C157C22semidirect
55. C2×SD16×C157C22semidirect
56. C2×D8×C157C22semidirect
57. C2×D8×C157C22semidirect
58. #127×C157C22semidirect
59. #124×C157C22semidirect
60. #97×C157C22semidirect
89. D1256D4semidirect
142. D628C2×D4semidirect
180. C8×C157C2×D4semidirect
205. C2×D157AGL(1,ℤ/8)semidirect
225. C157#3264semidirect
231. #496D157semidirect
249. C2×SD16C2×D157semidirect
250. C2×SD16C2×D157semidirect
256. C2×D8C2×D157semidirect
257. C2×D8C2×D157semidirect
260. #127C2×D157semidirect
261. #124C2×D157semidirect
286. #97C2×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×#190977982
C2×D8×D1572
#237579162
#237579082
#237575232
#237574992
#237574852
#237574652
#237574222
#237574172
#237574052
#237573862
#237573462
#496×C1572
#3264157