Finite Group Databasebeta

← #34111638

Subgroups — #34111638

Order 20096.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097799C211yesmaximal, characteristic, complemented
2C2×SD16×D157C211yesmaximal, characteristic, complemented
3#23757933C211yesmaximal, characteristic, complemented
4#23757907C211yesmaximal, characteristic, complemented
5#23757659C211yesmaximal, characteristic, complemented
6#23757510C211yesmaximal, characteristic, complemented
7#23757491C211yesmaximal, characteristic, complemented
8#23757472C211yesmaximal, characteristic, complemented
9#23757423C211yesmaximal, characteristic, complemented
10#23757416C211yesmaximal, characteristic, complemented
11#23757413C211yesmaximal, characteristic, complemented
12#23757408C211yesmaximal, characteristic, complemented
13#23757372C211yesmaximal, characteristic, complemented
14#23757342C211yesmaximal, characteristic, complemented
15#468×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic, complemented
17C2×Q8×D157C2211yescharacteristic, complemented
18C22×D628C2211yescharacteristic
19C2×D4×D157—21no—
20C2×D4×D157C2211yescharacteristic
21#19097799—24no—
22C2×#14578943C2211yescharacteristic
23C2×#14578943C2211yescharacteristic, complemented
24C2×#14578942C2211yescharacteristic, complemented
25C2×#14578941C2211yescharacteristic
26C2×#14578941C2211yescharacteristic, complemented
27SD16×D157—24no—
28C2×D1256C2211yescharacteristic
29C2×D1256C2211yescharacteristic, complemented
30C2×#14578939C2211yescharacteristic, complemented
31C2×C8×D157C2211yescharacteristic, complemented
32#19097767C2211yescharacteristic
33#19097760C2211yescharacteristic, complemented
34#19097757—21no—
35#19097749—21no—
36#19097746C2211yescharacteristic
37#19097746C2211yescharacteristic, complemented
38#19097744—21no—
39#19097744—21no—
40#19097741—21no—
41#19097739—21no—
42#19097736C2211yescharacteristic, complemented
43#19097733—21no—
44#19097733C2211yescharacteristic
45C4×C22⋊D157C2211yescharacteristic
46#19097700C2211yescharacteristic, complemented
47#19097694C2211yescharacteristic, complemented
48#19097691C2211yescharacteristic
49#19097691C2211yescharacteristic, complemented
50#19097691C2211yescharacteristic, complemented
51C22⋊#11929434C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097681C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C2×M16×C157C2211yescharacteristic, complemented
56C22⋊Q8×C157C2211yescharacteristic, complemented
57#121×C157C2211yescharacteristic
58#107×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63#14578954—22no—
64#14578954—41no—
65C2×Dic314—21no—
66C2×Dic314C2311yescharacteristic
67Q8×D157—22no—
68Q8×D157—41no—
69C2×D628—22no—
70C2×D628—21no—
71C2×D628—21no—
72C2×D628—21no—
73C2×D628—41no—
74C2×D628C2311yescharacteristic
75C2×D628C2311yescharacteristic
76C2×D628C2311yescharacteristic
77C2×C22⋊D157—21no—
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C2×C22⋊D157—21no—
81C2×C22⋊D157D411yescharacteristic
82D4×D157—22no—
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#14578943D414yescomplemented
91#14578942—22no—
92#14578941—22no—
93#14578941—41no—
94D1256—22no—
95D1256—22no—
96#14578939—41no—
97C2×#11929434C2311yescharacteristic
98C2×#11929434C2311yescharacteristic
99C8×D157—22no—
100#14578933—21no—
101#14578933D411yescharacteristic
102C22⋊Dic157D411yescharacteristic
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931D411yescharacteristic
109C4×Dic157—21no—
110C4×Dic157C2311yescharacteristic
111C2×Q8×C157C2311yescharacteristic
112C2×D4×C157—21no—
113C2×D4×C157C2311yescharacteristic
114C22×C4×C157C2311yescharacteristic
115SD16×C157—22no—
116M16×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—42no—
128C22×D157—42no—
129C22×D157—42no—
130C22×D157—42no—
131C22×D157—81no—
132C22×D157—81no—
133C22×D157C2×D411yescharacteristic
134Dic314—21no—
135Dic314—42no—
136Dic314C2×D412yescomplemented
137D628—21no—
138D628—21no—
139D628—21no—
140D628—21no—
141D628—21no—
142D628—42no—
143D628—42no—
144D628—42no—
145D628C2×D412yes—
146C22⋊D157—22no—
147C22⋊D157—42no—
148C22⋊D157—42no—
149C22⋊D157—42no—
150C22⋊D157—81no—
151C2×Dic157—21no—
152C2×Dic157C2×D411yescharacteristic
153C2×Dic157C2×D411yescharacteristic
154C2×Dic157C2×D411yescharacteristic
155C4×D157—21no—
156C4×D157—21no—
157C4×D157—42no—
158C4×D157—42no—
159C4×D157—42no—
160#11929434—21no—
161#11929434—21no—
162C23×C157—21no—
163C23×C157C2×D411yescharacteristic
164Q8×C157—21no—
165Q8×C157—21no—
166D4×C157—21no—
167D4×C157—21no—
168D4×C157—42no—
169C2×C4×C157—21no—
170C2×C4×C157—21no—
171C2×C4×C157—21no—
172C2×C4×C157—21no—
173C2×C4×C157—21no—
174C2×C4×C157C2×D411yescharacteristic
175C2×C4×C157C2411yescharacteristic
176C8×C157—21no—
177C8×C157C2×D412yescomplemented
178C22×C157—21no—
179C22×C157—21no—
180C22×C157—21no—
181C22×C157—41no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157C22×D411yescharacteristic
185C2×D157—21no—
186C2×D157—21no—
187C2×D157—22no—
188C2×D157—41no—
189C2×D157—41no—
190C2×D157—41no—
191C2×D157—41no—
192C2×D157—41no—
193C2×D157—41no—
194C2×D157—41no—
195C2×D157—41no—
196C2×D157—84no—
197C2×D157—84no—
198C2×D157AGL(1,ℤ/8)12yescomplemented
199C4×C157—21no—
200C4×C157—41no—
201C4×C157—41no—
202C4×C157—41no—
203C4×C157C22×D411yescharacteristic
204C4×C157H(5,𝔽2)11yescharacteristic
205Dic157—21no—
206Dic157—22no—
207Dic157—21no—
208Dic157—21no—
209C2×C157—41no—
210C2×C157—81no—
211C2×C157D4211yescharacteristic
212C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C157#57711yescharacteristic
214D157—22no—
215D157—42no—
216D157—81no—
217D157—81no—
218C157#326511yescharacteristic, complemented, Sylow
219#3265—1571nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1571no—
221C22×SD16—1571no—
222#548—1571no—
223D42—1571no—
224#496—1571no—
225#468D15711yescharacteristic, complemented
226#466—1571no—
227#463—1571no—
228#460—1571no—
229#450—1571no—
230#449—1571no—
231#448—1571no—
232#447—1571no—
233#440—1571no—
234#435—1571no—
235C2×#41—1571no—
236C22×Q8—1571no—
237C22×D4—3141no—
238C22×D4—1571no—
239C22×D4—1571no—
240AGL(1,ℤ/8)—3144no—
241C2×SD16—3144no—
242C2×SD16—1571no—
243C2×SD16—1571no—
244C2×SD16—1571no—
245C2×SD16—1571no—
246C2×SD16—1571no—
247C2×SD16C2×D15711yescharacteristic, complemented
248C2×D8—1571no—
249C2×D8—1571no—
250C2×D8—1571no—
251C2×M16C2×D15711yescharacteristic, complemented
252C22×C8—1571no—
253#127—1571no—
254#124—3141no—
255#124—1571no—
256C22⋊Q8—3141no—
257C22⋊Q8C2×D15711yescharacteristic, complemented
258#121—3141no—
259#121—3141no—
260#121—1571no—
261#121—1571no—
262#121C2×D15711yescharacteristic
263#120—3141no—
264#120—3141no—
265C4×Q8—1571no—
266C4×D4—3141no—
267C4×D4—1571no—
268C4×D4—1571no—
269#107C2×D15711yescharacteristic, complemented
270C4⋊C8—1571no—
271#103—1571no—
272#103C2×D15711yescharacteristic, complemented
273#102—1571no—
274#102—1571no—
275#102—1571no—
276#102C2×D15711yescharacteristic, complemented
277C22⋊C8—1571no—
278C22⋊C8—1571no—
279#97—1571no—
280C24—3141no—
281C24—3141no—
282#41—3142no—
283#41—6281no—
284C2×Q8—3141no—
285C2×Q8—3142no—
286C2×Q8—6281no—
287C2×Q8—1571no—
288C2×Q8C22×D15711yescharacteristic
289C2×D4—21no—
290C2×D4—3142no—
291C2×D4—3141no—
292C2×D4—3141no—
293C2×D4—3142no—
294C2×D4—3141no—
295C2×D4—3141no—
296C2×D4—3141no—
297C2×D4—3141no—
298C2×D4—3141no—
299C2×D4—6281no—
300C2×D4—6281no—
301C2×D4—6284no—
302C2×D4—1571no—
303C2×D4—1571no—
304C2×D4—1571no—
305C2×D4—1571no—
306C2×D4C22×D15711yescharacteristic
307C22×C4—3141no—
308C22×C4—3141no—
309C22×C4—3141no—
310C22×C4—1571no—
311C22×C4C22×D15711yescharacteristic
312SD16—22no—
313SD16—3142no—
314SD16—3142no—
315SD16—3142no—
316SD16—6281no—
317SD16—1574no—
318D8—3142no—
319D8—3142no—
320D8—6281no—
321M16—22no—
322C2×C8—3142no—
323C2×C8—1571no—
324C2×C8—1571no—
325C2×C8C22×D15711yescharacteristic
326C2×C8C22×D15711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—3141no—
329C4⋊C4—1571no—
330C4⋊C4C22×D15711yescharacteristic
331C4⋊C4C22×D15711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—3141no—
335C22⋊C4—3141no—
336C22⋊C4—3141no—
337C22⋊C4—3141no—
338C22⋊C4—3141no—
339C22⋊C4—1571no—
340C22⋊C4—1571no—
341C42—3141no—
342C42—1571no—
343C23—21no—
344C23—3141no—
345C23—3141no—
346C23—3141no—
347C23—6282no—
348C23—6282no—
349C23—6282no—
350C23—6282no—
351C23—1571no—
352C23—12561no—
353C23—12561no—
354C23D4×D15711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3141no—
358Q8—6282no—
359Q8—1572no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—3142no—
364D4—3141no—
365D4—3141no—
366D4—3141no—
367D4—3141no—
368D4—3141no—
369D4—6282no—
370D4—6282no—
371D4—6282no—
372D4—6282no—
373D4—6282no—
374D4—6282no—
375D4—1572no—
376D4—12561no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—3141no—
383C2×C4—3141no—
384C2×C4—3141no—
385C2×C4—6282no—
386C2×C4—6282no—
387C2×C4—6282no—
388C2×C4—1571no—
389C2×C4—1571no—
390C2×C4—1571no—
391C2×C4D4×D15711yescharacteristic
392C2×C4C23×D15711yescharacteristic
393C8—21no—
394C8—3141no—
395C8—3141no—
396C8D4×D15712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—3141no—
404C22—3141no—
405C22—3142no—
406C22—6281no—
407C22—6281no—
408C22—6281no—
409C22—6281no—
410C22—6281no—
411C22—6281no—
412C22—6281no—
413C22—6281no—
414C22—1572no—
415C22—12564no—
416C22—12564no—
417C22C2×D4×D15711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—3141no—
423C4—3141no—
424C4—3142no—
425C4—3141no—
426C4C2×D4×D15711yescharacteristic
427C4#1909781711yescharacteristic
428C2—41no—
429C2—81no—
430C2—3142no—
431C2—6282no—
432C2—12561no—
433C2—12561no—
434C2#2375790711yescharacteristic
435C2AGL(1,ℤ/8)×D15711yescharacteristic
436C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#468×C157C2
socleC22×C157C22×D4
hypercenter#468D157
hypocenterC157#3265
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. C2×SD16×D157C2semidirect
3. #23757933C2semidirect
4. #23757907C2semidirect
5. #23757659C2semidirect
6. #23757510C2semidirect
7. #23757491C2semidirect
8. #23757472C2semidirect
9. #23757423C2semidirect
10. #23757416C2semidirect
11. #23757413C2semidirect
12. #23757408C2semidirect
13. #23757372C2semidirect
14. #23757342C2semidirect
15. #468×C157C2semidirect
16. C2×#14578954C22semidirect
17. C2×Q8×D157C22semidirect
23. C2×#14578943C22semidirect
24. C2×#14578942C22semidirect
26. C2×#14578941C22semidirect
29. C2×D1256C22semidirect
30. C2×#14578939C22semidirect
31. C2×C8×D157C22semidirect
33. #19097760C22semidirect
37. #19097746C22semidirect
42. #19097736C22semidirect
46. #19097700C22semidirect
47. #19097694C22semidirect
49. #19097691C22semidirect
50. #19097691C22semidirect
51. C22⋊#11929434C22semidirect
52. #19097683C22semidirect
53. #19097681C22semidirect
54. C2×SD16×C157C22semidirect
55. C2×M16×C157C22semidirect
56. C22⋊Q8×C157C22semidirect
58. #107×C157C22semidirect
59. #103×C157C22semidirect
60. #102×C157C22semidirect
90. #14578943D4semidirect
136. Dic314C2×D4semidirect
177. C8×C157C2×D4semidirect
198. C2×D157AGL(1,ℤ/8)semidirect
218. C157#3265semidirect
225. #468D157semidirect
247. C2×SD16C2×D157semidirect
251. C2×M16C2×D157semidirect
257. C22⋊Q8C2×D157semidirect
269. #107C2×D157semidirect
272. #103C2×D157semidirect
276. #102C2×D157semidirect
396. 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×SD16×D1572
#237579332
#237579072
#237576592
#237575102
#237574912
#237574722
#237574232
#237574162
#237574132
#237574082
#237573722
#237573422
#468×C1572
#3265157