Finite Group Databasebeta

← #34111744

Subgroups — #34111744

Order 20096.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097799C211yesmaximal, characteristic, complemented
2C2×#19097798C211yesmaximal, characteristic, complemented
3#23757920C211yesmaximal, characteristic, complemented
4#23757914C211yesmaximal, characteristic, complemented
5#23757654C211yesmaximal, characteristic, complemented
6#23757520C211yesmaximal, characteristic, complemented
7#23757491C211yesmaximal, characteristic, complemented
8#23757487C211yesmaximal, characteristic, complemented
9#23757438C211yesmaximal, characteristic, complemented
10#23757433C211yesmaximal, characteristic, complemented
11#23757408C211yesmaximal, characteristic, complemented
12#23757406C211yesmaximal, characteristic, complemented
13#23757355C211yesmaximal, characteristic, complemented
14#23757349C211yesmaximal, characteristic, complemented
15#468×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578954C2211yescharacteristic
18C22×D628C2211yescharacteristic, complemented
19C2×D4×D157C2211yescharacteristic, complemented
20#19097799—24no—
21#19097798—24no—
22C2×#14578944C2211yescharacteristic, complemented
23C2×#14578943C2211yescharacteristic, complemented
24C2×#14578943C2211yescharacteristic, complemented
25C2×#14578941C2211yescharacteristic, complemented
26C2×D1256C2211yescharacteristic
27C2×D1256C2211yescharacteristic
28C2×#14578939C2211yescharacteristic
29C2×#14578939C2211yescharacteristic
30C2×#14578937C2211yescharacteristic, complemented
31#19097760C2211yescharacteristic, complemented
32#19097758C2211yescharacteristic, complemented
33#19097756—21no—
34#19097753—21no—
35#19097751C2211yescharacteristic, complemented
36#19097746—21no—
37#19097746C2211yescharacteristic
38#19097742—21no—
39#19097741—21no—
40#19097739—21no—
41#19097733C2211yescharacteristic, complemented
42#19097733C2211yescharacteristic, complemented
43C4×C22⋊D157C2211yescharacteristic
44C22⋊C4×D157—21no—
45C22⋊C4×D157—21no—
46#19097700C2211yescharacteristic, complemented
47#19097694C2211yescharacteristic, complemented
48#19097691C2211yescharacteristic, complemented
49#19097691C2211yescharacteristic, complemented
50#19097691C2211yescharacteristic, complemented
51C22⋊#11929434C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53C8×Dic157C2211yescharacteristic, 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#14578955—22no—
64#14578955—41no—
65#14578954—22no—
66#14578954—41no—
67C2×Dic314C2311yescharacteristic
68C2×D628—22no—
69C2×D628—21no—
70C2×D628—21no—
71C2×D628—21no—
72C2×D628—41no—
73C2×D628C2311yescharacteristic
74C2×D628C2311yescharacteristic
75C2×D628C2311yescharacteristic
76C2×C22⋊D157—21no—
77C2×C22⋊D157—21no—
78C2×C22⋊D157D411yescharacteristic
79D4×D157—22no—
80D4×D157—41no—
81C2×C4×D157—21no—
82C2×C4×D157—21no—
83C2×C4×D157—21no—
84C2×C4×D157—21no—
85C2×C4×D157C2311yescharacteristic
86#14578944—41no—
87#14578943—22no—
88#14578943—22no—
89#14578941—22no—
90D1256—22no—
91D1256D414yescomplemented
92#14578939—22no—
93#14578939—41no—
94#14578937—22no—
95C2×#11929434C2311yescharacteristic
96C2×#11929434C2311yescharacteristic
97C4⋊Dic157—21no—
98#14578933D411yescharacteristic
99C22⋊Dic157—21no—
100C22⋊Dic157—21no—
101C22⋊Dic157D411yescharacteristic
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931D411yescharacteristic
109C4×Dic157C2311yescharacteristic
110C2×Q8×C157C2311yescharacteristic
111C2×D4×C157—21no—
112C2×D4×C157C2311yescharacteristic
113C22×C4×C157C2311yescharacteristic
114SD16×C157—22no—
115M16×C157—22no—
116C2×C8×C157C2311yescharacteristic
117C2×C8×C157C2311yescharacteristic
118C4⋊C4×C157—21no—
119C4⋊C4×C157C2311yescharacteristic
120C4⋊C4×C157C2311yescharacteristic
121C22⋊C4×C157—21no—
122C22⋊C4×C157—21no—
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—42no—
127C22×D157—42no—
128C22×D157—42no—
129C22×D157—42no—
130C22×D157—81no—
131C22×D157—81no—
132C22×D157C2×D411yescharacteristic
133Dic314—21no—
134Dic314—21no—
135D628—21no—
136D628—21no—
137D628—21no—
138D628—21no—
139D628—42no—
140D628—42no—
141D628—81no—
142D628C2×D412yes—
143D628C2×D412yes—
144C22⋊D157—41no—
145C22⋊D157—42no—
146C22⋊D157—42no—
147C2×Dic157—21no—
148C2×Dic157C2×D411yescharacteristic
149C2×Dic157C2×D411yescharacteristic
150C2×Dic157C2×D411yescharacteristic
151C4×D157—21no—
152C4×D157—21no—
153C4×D157—42no—
154C4×D157—42no—
155C4×D157—81no—
156C4×D157—81no—
157#11929434—21no—
158#11929434—21no—
159C23×C157—21no—
160C23×C157C2×D411yescharacteristic
161Q8×C157—21no—
162Q8×C157—21no—
163D4×C157—21no—
164D4×C157—21no—
165D4×C157—81no—
166C2×C4×C157—21no—
167C2×C4×C157—21no—
168C2×C4×C157—21no—
169C2×C4×C157—21no—
170C2×C4×C157—21no—
171C2×C4×C157C2×D411yescharacteristic
172C2×C4×C157C2411yescharacteristic
173C8×C157—21no—
174C8×C157C2×D412yes—
175C22×C157—21no—
176C22×C157—21no—
177C22×C157—21no—
178C22×C157—41no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157C22×D411yescharacteristic
182C2×D157—21no—
183C2×D157—22no—
184C2×D157—22no—
185C2×D157—21no—
186C2×D157—21no—
187C2×D157—41no—
188C2×D157—41no—
189C2×D157—41no—
190C2×D157—41no—
191C2×D157—41no—
192C2×D157—41no—
193C2×D157—41no—
194C2×D157—84no—
195C2×D157—84no—
196C4×C157—21no—
197C4×C157—41no—
198C4×C157—41no—
199C4×C157—41no—
200C4×C157C22×D411yescharacteristic
201C4×C157H(5,𝔽2)11yescharacteristic
202Dic157—21no—
203Dic157—21no—
204Dic157—21no—
205Dic157—41no—
206C2×C157—41no—
207C2×C157—81no—
208C2×C157D4211yescharacteristic
209C2×C157#57511yescharacteristic
210C2×C157#57611yescharacteristic
211D157—41no—
212D157—42no—
213D157—42no—
214D157—81no—
215C157#327311yescharacteristic, complemented, Sylow
216#3273—1571nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1571no—
218C2×AGL(1,ℤ/8)—1571no—
219#546—1571no—
220#546—1571no—
221#495—1571no—
222#468—1571no—
223#468D15711yescharacteristic, complemented
224#463—1571no—
225#463—1571no—
226#452—1571no—
227#452—1571no—
228#447—1571no—
229#447—1571no—
230C4×D8—1571no—
231#436—1571no—
232C2×#41—1571no—
233C2×#41—1571no—
234C22×D4—1571no—
235C22×D4—1571no—
236AGL(1,ℤ/8)—3144no—
237AGL(1,ℤ/8)—3144no—
238C2×Q16—1571no—
239C2×SD16—1571no—
240C2×SD16—1571no—
241C2×SD16—1571no—
242C2×SD16C2×D15711yescharacteristic, complemented
243C2×D8—1571no—
244C2×D8—1571no—
245C2×D8—1571no—
246C2×D8—1571no—
247C2×M16—1571no—
248C2×M16C2×D15711yescharacteristic, complemented
249#124—1571no—
250#124—1571no—
251#123—3141no—
252#123—3141no—
253C22⋊Q8—1571no—
254C22⋊Q8C2×D15711yescharacteristic, complemented
255#121—3141no—
256#121—3141no—
257#121—1571no—
258#121C2×D15711yescharacteristic
259#120—3141no—
260#120—3141no—
261C4×D4—1571no—
262C4×D4—1571no—
263C4×D4—1571no—
264C2×C22⋊C4—3141no—
265C2×C22⋊C4—3141no—
266#107C2×D15711yescharacteristic, complemented
267C4⋊C8—1571no—
268#103—1571no—
269#103C2×D15711yescharacteristic, complemented
270#102—1571no—
271#102—1571no—
272#102—1571no—
273#102C2×D15711yescharacteristic, complemented
274C22⋊C8—1571no—
275C22⋊C8—1571no—
276C4×C8—1571no—
277C24—3141no—
278C24—3141no—
279#41—3142no—
280#41—3142no—
281#41—6281no—
282#41—6281no—
283C2×Q8—1571no—
284C2×Q8C22×D15711yescharacteristic
285C2×D4—21no—
286C2×D4—3142no—
287C2×D4—3141no—
288C2×D4—3141no—
289C2×D4—3142no—
290C2×D4—3141no—
291C2×D4—3141no—
292C2×D4—3141no—
293C2×D4—6281no—
294C2×D4—6281no—
295C2×D4—1571no—
296C2×D4—1571no—
297C2×D4—1571no—
298C2×D4—1571no—
299C2×D4C22×D15711yescharacteristic
300C22×C4—3141no—
301C22×C4—3141no—
302C22×C4—3141no—
303C22×C4—3141no—
304C22×C4—1571no—
305C22×C4C22×D15711yescharacteristic
306Q16—6281no—
307SD16—22no—
308SD16—3142no—
309SD16—3142no—
310SD16—3142no—
311D8—3142no—
312D8—3142no—
313D8—6281no—
314D8—1574no—
315M16—22no—
316M16—3142no—
317C2×C8—1571no—
318C2×C8—1571no—
319C2×C8C22×D15711yescharacteristic
320C2×C8C22×D15711yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—3141no—
323C4⋊C4—1571no—
324C4⋊C4C22×D15711yescharacteristic
325C4⋊C4C22×D15711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—3141no—
329C22⋊C4—3141no—
330C22⋊C4—3141no—
331C22⋊C4—3141no—
332C22⋊C4—3141no—
333C22⋊C4—3141no—
334C22⋊C4—3141no—
335C22⋊C4—3141no—
336C22⋊C4—1571no—
337C22⋊C4—1571no—
338C42—1571no—
339C23—21no—
340C23—3141no—
341C23—3141no—
342C23—3141no—
343C23—6282no—
344C23—6282no—
345C23—6282no—
346C23—6282no—
347C23—1571no—
348C23—12561no—
349C23—12561no—
350C23D4×D15711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3141no—
354Q8—3141no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—3141no—
359D4—3141no—
360D4—3141no—
361D4—3141no—
362D4—6282no—
363D4—6282no—
364D4—6281no—
365D4—6282no—
366D4—6282no—
367D4—1572no—
368D4—1572no—
369D4—12561no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—3141no—
376C2×C4—3141no—
377C2×C4—3141no—
378C2×C4—6282no—
379C2×C4—6282no—
380C2×C4—1571no—
381C2×C4—1571no—
382C2×C4—1571no—
383C2×C4—12561no—
384C2×C4—12561no—
385C2×C4D4×D15711yescharacteristic
386C2×C4C23×D15711yescharacteristic
387C8—21no—
388C8—3141no—
389C8—3141no—
390C8D4×D15712yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3141no—
398C22—3142no—
399C22—3142no—
400C22—3141no—
401C22—3141no—
402C22—6281no—
403C22—6281no—
404C22—6281no—
405C22—6281no—
406C22—6281no—
407C22—6281no—
408C22—6281no—
409C22—12564no—
410C22—12564no—
411C22C2×D4×D15711yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—3141no—
417C4—3141no—
418C4—3141no—
419C4—6281no—
420C4C2×D4×D15711yescharacteristic
421C4#1909781711yescharacteristic
422C2—41no—
423C2—81no—
424C2—6281no—
425C2—6282no—
426C2—6282no—
427C2—12561no—
428C2#2375790711yescharacteristic
429C2#2375807211yescharacteristic
430C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#468×C157C2
socleC22×C157C22×D4
hypercenter#468D157
hypocenterC157#3273
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×#19097798C2semidirect
3. #23757920C2semidirect
4. #23757914C2semidirect
5. #23757654C2semidirect
6. #23757520C2semidirect
7. #23757491C2semidirect
8. #23757487C2semidirect
9. #23757438C2semidirect
10. #23757433C2semidirect
11. #23757408C2semidirect
12. #23757406C2semidirect
13. #23757355C2semidirect
14. #23757349C2semidirect
15. #468×C157C2semidirect
18. C22×D628C22semidirect
19. C2×D4×D157C22semidirect
22. C2×#14578944C22semidirect
23. C2×#14578943C22semidirect
24. C2×#14578943C22semidirect
25. C2×#14578941C22semidirect
30. C2×#14578937C22semidirect
31. #19097760C22semidirect
32. #19097758C22semidirect
35. #19097751C22semidirect
41. #19097733C22semidirect
42. #19097733C22semidirect
46. #19097700C22semidirect
47. #19097694C22semidirect
48. #19097691C22semidirect
49. #19097691C22semidirect
50. #19097691C22semidirect
51. C22⋊#11929434C22semidirect
52. #19097683C22semidirect
53. C8×Dic157C22semidirect
54. C2×SD16×C157C22semidirect
55. C2×M16×C157C22semidirect
56. C22⋊Q8×C157C22semidirect
58. #107×C157C22semidirect
59. #103×C157C22semidirect
60. #102×C157C22semidirect
91. D1256D4semidirect
215. C157#3273semidirect
223. #468D157semidirect
242. C2×SD16C2×D157semidirect
248. C2×M16C2×D157semidirect
254. C22⋊Q8C2×D157semidirect
266. #107C2×D157semidirect
269. #103C2×D157semidirect
273. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
C2×#190977982
#237579202
#237579142
#237576542
#237575202
#237574912
#237574872
#237574382
#237574332
#237574082
#237574062
#237573552
#237573492
#468×C1572
#3273157