Finite Group Databasebeta

← #34111629

Subgroups — #34111629

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×#19097797C211yesmaximal, characteristic, complemented
2C2×SD16×D157C211yesmaximal, characteristic, complemented
3#23757933C211yesmaximal, characteristic, complemented
4D4×C22⋊D157C211yesmaximal, characteristic, complemented
5#23757659C211yesmaximal, characteristic, complemented
6#23757521C211yesmaximal, characteristic, complemented
7#23757508C211yesmaximal, characteristic, complemented
8#23757490C211yesmaximal, characteristic, complemented
9#23757472C211yesmaximal, characteristic, complemented
10#23757420C211yesmaximal, characteristic, complemented
11#23757410C211yesmaximal, characteristic, complemented
12#23757408C211yesmaximal, characteristic, complemented
13#23757370C211yesmaximal, characteristic, complemented
14#23757347C211yesmaximal, characteristic, complemented
15#450×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic, complemented
17C2×Q8×D157C2211yescharacteristic, complemented
18C22×C22⋊D157—21no—
19C2×D4×D157C2211yescharacteristic
20#19097797—24no—
21C2×#14578943C2211yescharacteristic, complemented
22C2×#14578942C2211yescharacteristic
23C2×#14578942C2211yescharacteristic, complemented
24C2×#14578941C2211yescharacteristic, complemented
25SD16×D157—24no—
26C2×D1256C2211yescharacteristic, complemented
27C2×#14578939C2211yescharacteristic
28C2×#14578939C2211yescharacteristic, complemented
29C2×#14578938C2211yescharacteristic, complemented
30C2×C8×D157C2211yescharacteristic, complemented
31#19097767C2211yescharacteristic
32#19097760C2211yescharacteristic, complemented
33#19097757—21no—
34#19097749—21no—
35#19097746C2211yescharacteristic, complemented
36#19097745C2211yescharacteristic
37#19097744—21no—
38#19097743—21no—
39#19097742C2211yescharacteristic
40#19097740—21no—
41#19097739—21no—
42#19097736C2211yescharacteristic, complemented
43D4×Dic157C2211yescharacteristic
44#19097733—21no—
45C4×C22⋊D157C2211yescharacteristic
46#19097704C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097695C2211yescharacteristic, complemented
50#19097693C2211yescharacteristic, complemented
51#19097691C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097681C2211yescharacteristic, complemented
54C22×D4×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56C2×SD16×C157C2211yescharacteristic, complemented
57C22⋊Q8×C157C2211yescharacteristic, complemented
58#102×C157C2211yescharacteristic
59#102×C157C2211yescharacteristic, complemented
60C22⋊C8×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578954—22no—
63#14578954—41no—
64C2×Dic314—21no—
65C2×Dic314C2311yescharacteristic
66Q8×D157—22no—
67Q8×D157—41no—
68C2×D628—21no—
69C2×D628C2311yescharacteristic
70C2×D628C2311yescharacteristic
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77C2×C22⋊D157—44no—
78C2×C22⋊D157D411yescharacteristic
79D4×D157—22no—
80D4×D157—41no—
81C22×Dic157—21no—
82C2×C4×D157—21no—
83C2×C4×D157—21no—
84C2×C4×D157C2311yescharacteristic
85#14578943—22no—
86#14578942—22no—
87#14578942D414yescomplemented
88#14578941—22no—
89D1256—41no—
90#14578939—22no—
91#14578939—22no—
92#14578938—22no—
93C2×#11929434C2311yescharacteristic
94C2×#11929434C2311yescharacteristic
95C8×D157—22no—
96C4⋊Dic157C2311yescharacteristic
97#14578933—21no—
98#14578933D411yescharacteristic
99C22⋊Dic157—21no—
100C22⋊Dic157—21no—
101C22⋊Dic157D411yescharacteristic
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931D411yescharacteristic
107C4×Dic157—21no—
108C4×Dic157C2311yescharacteristic
109C24×C157—21no—
110C2×Q8×C157C2311yescharacteristic
111C2×D4×C157—22no—
112C2×D4×C157—21no—
113C2×D4×C157—41no—
114C2×D4×C157C2311yescharacteristic
115C2×D4×C157C2311yescharacteristic
116C22×C4×C157C2311yescharacteristic
117SD16×C157—22no—
118SD16×C157—41no—
119C2×C8×C157C2311yescharacteristic
120C2×C8×C157C2311yescharacteristic
121C4⋊C4×C157—21no—
122C4⋊C4×C157C2311yescharacteristic
123C22⋊C4×C157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—42no—
127C22×D157—42no—
128C22×D157—81no—
129C22×D157C2×D411yescharacteristic
130Dic314—21no—
131Dic314—42no—
132Dic314C2×D412yescomplemented
133D628—21no—
134D628—21no—
135D628—21no—
136D628—21no—
137D628—42no—
138C22⋊D157—22no—
139C22⋊D157—42no—
140C22⋊D157—42no—
141C22⋊D157—42no—
142C22⋊D157—42no—
143C22⋊D157—42no—
144C22⋊D157—81no—
145C2×Dic157—21no—
146C2×Dic157—21no—
147C2×Dic157—42no—
148C2×Dic157C2×D411yescharacteristic
149C2×Dic157C2×D411yescharacteristic
150C2×Dic157C2×D411yescharacteristic
151C4×D157—21no—
152C4×D157—21no—
153C4×D157—42no—
154C4×D157—42no—
155#11929434—21no—
156#11929434C2×D412yescomplemented
157C23×C157—21no—
158C23×C157—21no—
159C23×C157—42no—
160C23×C157—42no—
161C23×C157—81no—
162C23×C157C2×D411yescharacteristic
163Q8×C157—21no—
164Q8×C157—21no—
165D4×C157—21no—
166D4×C157—21no—
167D4×C157—21no—
168D4×C157—42no—
169D4×C157C2×D412yes—
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×C157—21no—
178C22×C157—21no—
179C22×C157—22no—
180C22×C157—21no—
181C22×C157—21no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157—41no—
187C22×C157—84no—
188C22×C157C22×D411yescharacteristic
189C2×D157—21no—
190C2×D157—21no—
191C2×D157—41no—
192C2×D157—41no—
193C2×D157—41no—
194C2×D157—41no—
195C2×D157—41no—
196C2×D157—41no—
197C2×D157—84no—
198C2×D157AGL(1,ℤ/8)12yescomplemented
199C4×C157—21no—
200C4×C157—41no—
201C4×C157—41no—
202C4×C157C22×D411yescharacteristic
203C4×C157H(5,𝔽2)11yescharacteristic
204Dic157—21no—
205Dic157—22no—
206Dic157—21no—
207Dic157—21no—
208Dic157—41no—
209C2×C157—42no—
210C2×C157—41no—
211C2×C157—81no—
212C2×C157D4211yescharacteristic
213C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C157#57711yescharacteristic
215D157—22no—
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#468—1571no—
226#466—1571no—
227#463—1571no—
228#460—1571no—
229#450D15711yescharacteristic, complemented
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×D4C2×D15711yescharacteristic
240AGL(1,ℤ/8)—3144no—
241C2×SD16—3144no—
242C2×SD16—1571no—
243C2×SD16—1571no—
244C2×SD16—1571no—
245C2×SD16—1571no—
246C2×SD16C2×D15711yescharacteristic
247C2×SD16C2×D15711yescharacteristic, complemented
248C2×D8—1571no—
249C2×D8—1571no—
250C2×D8—1571no—
251C2×M16—1571no—
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#121—1571no—
263#120—3141no—
264#120—3141no—
265C4×Q8—1571no—
266C4×D4—3141no—
267C4×D4—1571no—
268C4×D4—1571no—
269#107—1571no—
270C4⋊C8—1571no—
271#103—1571no—
272#103—1571no—
273#102—1571no—
274#102—1571no—
275#102C2×D15711yescharacteristic
276#102C2×D15711yescharacteristic, complemented
277C22⋊C8—1571no—
278C22⋊C8C2×D15711yescharacteristic, complemented
279#97—1571no—
280C24—21no—
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—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—3141no—
293C2×D4—3142no—
294C2×D4—3141no—
295C2×D4—3141no—
296C2×D4—3141no—
297C2×D4—3141no—
298C2×D4—3141no—
299C2×D4—3141no—
300C2×D4—6281no—
301C2×D4—6284no—
302C2×D4—1571no—
303C2×D4—1571no—
304C2×D4—1571no—
305C2×D4C22×D15711yescharacteristic
306C2×D4C22×D15711yescharacteristic
307C22×C4—3141no—
308C22×C4—3141no—
309C22×C4—3141no—
310C22×C4—1571no—
311C22×C4C22×D15711yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—3142no—
315SD16—3142no—
316SD16—3142no—
317SD16—1574no—
318D8—3142no—
319D8—3142no—
320D8—6281no—
321M16—3142no—
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⋊C4—1571no—
331C4⋊C4C22×D15711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—3141no—
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—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—3141no—
349C23—3141no—
350C23—6282no—
351C23—6282no—
352C23—1571no—
353C23—12561no—
354C23D4×D15711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3141no—
358Q8—6282no—
359Q8—1572no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—3142no—
365D4—3141no—
366D4—3141no—
367D4—3141no—
368D4—3141no—
369D4—6282no—
370D4—6282no—
371D4—6282no—
372D4—6282no—
373D4—6282no—
374D4—6282no—
375D4—12561no—
376D4D4×D15712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—3141no—
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—21no—
395C8—3141no—
396C8—1572no—
397C22—21no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—3141no—
408C22—3141no—
409C22—6281no—
410C22—6281no—
411C22—6281no—
412C22—6281no—
413C22—6281no—
414C22—6281no—
415C22—1572no—
416C22—12564no—
417C22C2×D4×D15711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—3141no—
422C4—3142no—
423C4—3141no—
424C4—3141no—
425C4—6281no—
426C4C2×D4×D15711yescharacteristic
427C4#1909781611yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—3142no—
432C2—12561no—
433C2—12561no—
434C2#2375790511yescharacteristic
435C2AGL(1,ℤ/8)×D15711yescharacteristic
436C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
hypercenter#450D157
hypocenterC157#3265
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19097797C2semidirect
2. C2×SD16×D157C2semidirect
3. #23757933C2semidirect
4. D4×C22⋊D157C2semidirect
5. #23757659C2semidirect
6. #23757521C2semidirect
7. #23757508C2semidirect
8. #23757490C2semidirect
9. #23757472C2semidirect
10. #23757420C2semidirect
11. #23757410C2semidirect
12. #23757408C2semidirect
13. #23757370C2semidirect
14. #23757347C2semidirect
15. #450×C157C2semidirect
16. C2×#14578954C22semidirect
17. C2×Q8×D157C22semidirect
21. C2×#14578943C22semidirect
23. C2×#14578942C22semidirect
24. C2×#14578941C22semidirect
26. C2×D1256C22semidirect
28. C2×#14578939C22semidirect
29. C2×#14578938C22semidirect
30. C2×C8×D157C22semidirect
32. #19097760C22semidirect
35. #19097746C22semidirect
42. #19097736C22semidirect
46. #19097704C22semidirect
47. #19097700C22semidirect
48. #19097696C22semidirect
49. #19097695C22semidirect
50. #19097693C22semidirect
51. #19097691C22semidirect
52. #19097683C22semidirect
53. #19097681C22semidirect
56. C2×SD16×C157C22semidirect
57. C22⋊Q8×C157C22semidirect
59. #102×C157C22semidirect
60. C22⋊C8×C157C22semidirect
87. #14578942D4semidirect
132. Dic314C2×D4semidirect
156. #11929434C2×D4semidirect
198. C2×D157AGL(1,ℤ/8)semidirect
218. C157#3265semidirect
229. #450D157semidirect
247. C2×SD16C2×D157semidirect
257. C22⋊Q8C2×D157semidirect
276. #102C2×D157semidirect
278. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977972
C2×SD16×D1572
#237579332
D4×C22⋊D1572
#237576592
#237575212
#237575082
#237574902
#237574722
#237574202
#237574102
#237574082
#237573702
#237573472
#450×C1572
#3265157