Finite Group Databasebeta

← #34111742

Subgroups — #34111742

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×#19097798C211yesmaximal, characteristic, complemented
2C2×#19097795C211yesmaximal, characteristic, complemented
3#23757918C211yesmaximal, characteristic, complemented
4#23757916C211yesmaximal, characteristic, complemented
5#23757655C211yesmaximal, characteristic, complemented
6#23757522C211yesmaximal, characteristic, complemented
7#23757518C211yesmaximal, characteristic, complemented
8#23757489C211yesmaximal, characteristic, complemented
9#23757438C211yesmaximal, characteristic, complemented
10#23757432C211yesmaximal, characteristic, complemented
11#23757408C211yesmaximal, characteristic, complemented
12#23757405C211yesmaximal, characteristic, complemented
13C4×#14578939C211yesmaximal, characteristic, complemented
14#23757350C211yesmaximal, characteristic, complemented
15#463×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic, complemented
19C2×D4×D157C2211yescharacteristic, complemented
20#19097798—24no—
21#19097795—24no—
22C2×#14578944C2211yescharacteristic, complemented
23C2×#14578943C2211yescharacteristic, complemented
24C2×#14578942C2211yescharacteristic, complemented
25C2×#14578941C2211yescharacteristic, complemented
26C2×D1256C2211yescharacteristic
27C2×#14578939C2211yescharacteristic
28C2×#14578939C2211yescharacteristic
29C2×#14578937C2211yescharacteristic, complemented
30C2×#14578937C2211yescharacteristic, complemented
31#19097759C2211yescharacteristic, complemented
32#19097755—21no—
33#19097753—21no—
34#19097751C2211yescharacteristic, complemented
35#19097747C2211yescharacteristic, complemented
36#19097746C2211yescharacteristic
37#19097745—21no—
38#19097744—21no—
39#19097742C2211yescharacteristic
40#19097741—21no—
41#19097741—21no—
42C4×D628C2211yescharacteristic
43C4×D628C2211yescharacteristic, complemented
44C2×#14578931—21no—
45C22⋊C4×D157—21no—
46#19097704C2211yescharacteristic, complemented
47#19097696C2211yescharacteristic, complemented
48#19097693C2211yescharacteristic, complemented
49#19097692C2211yescharacteristic, complemented
50#19097691C2211yescharacteristic, complemented
51#19097683C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53C4×#11929434C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C2×D8×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic, complemented
57C4×D4×C157C2211yescharacteristic, complemented
58C4⋊C8×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63#14578955—22no—
64#14578955—41no—
65#14578953—22no—
66#14578953—41no—
67C2×Dic314C2311yescharacteristic
68C2×D628—21no—
69C2×D628—21no—
70C2×D628D411yescharacteristic
71C2×D628C2311yescharacteristic
72C2×D628C2311yescharacteristic
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77D4×D157—22no—
78D4×D157—22no—
79D4×D157—41no—
80D4×D157—41no—
81C22×Dic157—21no—
82C2×C4×D157—21no—
83C2×C4×D157—21no—
84C2×C4×D157C2311yescharacteristic
85C2×C4×D157C2311yescharacteristic
86#14578944—41no—
87#14578943—22no—
88#14578942—22no—
89#14578941—22no—
90D1256—22no—
91#14578939—41no—
92#14578939D414yescomplemented
93#14578937—22no—
94#14578937—22no—
95C2×#11929434C2311yescharacteristic
96C2×#11929434C2311yescharacteristic
97C4⋊Dic157—21no—
98C4⋊Dic157D411yescharacteristic
99C4⋊Dic157C2311yescharacteristic
100#14578933—21no—
101C22⋊Dic157—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931D411yescharacteristic
109#14578931D411yescharacteristic
110C2×Q8×C157C2311yescharacteristic
111C2×D4×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C22×C4×C157—21no—
114SD16×C157—22no—
115D8×C157—22no—
116C2×C8×C157C2311yescharacteristic
117C2×C8×C157C2311yescharacteristic
118C4⋊C4×C157C2311yescharacteristic
119C22⋊C4×C157—21no—
120C22⋊C4×C157—21no—
121C22⋊C4×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—42no—
126C22×D157—42no—
127C22×D157—42no—
128C22×D157—42no—
129C22×D157—81no—
130C22×D157—81no—
131C22×D157C2×D411yescharacteristic
132C22×D157C2×D411yescharacteristic
133Dic314—21no—
134Dic314—21no—
135D628—21no—
136D628—21no—
137D628—21no—
138D628—41no—
139D628—42no—
140D628—81no—
141D628C2×D412yes—
142C22⋊D157—42no—
143C22⋊D157—42no—
144C22⋊D157—42no—
145C22⋊D157—81no—
146C2×Dic157—21no—
147C2×Dic157—21no—
148C2×Dic157—42no—
149C2×Dic157C2×D411yescharacteristic
150C2×Dic157C2×D411yescharacteristic
151C4×D157—21no—
152C4×D157—21no—
153C4×D157—21no—
154C4×D157—21no—
155C4×D157—42no—
156C4×D157—81no—
157#11929434—21no—
158#11929434C2×D412yes—
159C23×C157—21no—
160C23×C157—21no—
161Q8×C157—21no—
162Q8×C157—21no—
163D4×C157—21no—
164D4×C157—21no—
165D4×C157—21no—
166D4×C157C2×D412yes—
167C2×C4×C157—21no—
168C2×C4×C157—21no—
169C2×C4×C157—81no—
170C2×C4×C157C2×D411yescharacteristic
171C2×C4×C157C2×D411yescharacteristic
172C2×C4×C157C2411yescharacteristic
173C8×C157—21no—
174C8×C157—21no—
175C22×C157—22no—
176C22×C157—41no—
177C22×C157—41no—
178C22×C157—41no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157C22×D411yescharacteristic
182C2×D157—21no—
183C2×D157—21no—
184C2×D157—22no—
185C2×D157—21no—
186C2×D157—21no—
187C2×D157—21no—
188C2×D157—21no—
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—21no—
198C4×C157—41no—
199C4×C157—41no—
200C4×C157C22×D411yescharacteristic
201C4×C157H(5,𝔽2)11yescharacteristic
202Dic157—21no—
203Dic157—21no—
204Dic157—41no—
205Dic157—41no—
206C2×C157—42no—
207C2×C157—81no—
208C2×C157D4211yescharacteristic
209C2×C157#57511yescharacteristic
210C2×C157#57611yescharacteristic
211D157—41no—
212D157—41no—
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#468—1571no—
224#463—1571no—
225#463D15711yescharacteristic, complemented
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×D8C2×D15711yescharacteristic
247C2×M16—1571no—
248C2×M16—1571no—
249#124—1571no—
250#124C2×D15711yescharacteristic, complemented
251#123—3141no—
252#123—3141no—
253C22⋊Q8—1571no—
254C22⋊Q8—1571no—
255#121—3141no—
256#121—3141no—
257#121—1571no—
258#121—1571no—
259#120—3141no—
260#120—3141no—
261C4×D4—1571no—
262C4×D4—1571no—
263C4×D4C2×D15711yescharacteristic, complemented
264C2×C22⋊C4—3141no—
265C2×C22⋊C4—3141no—
266#107—1571no—
267C4⋊C8C2×D15711yescharacteristic, complemented
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—3141no—
286C2×D4—3141no—
287C2×D4—3142no—
288C2×D4—3142no—
289C2×D4—3141no—
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×D4C22×D15711yescharacteristic
299C2×D4C22×D15711yescharacteristic
300C22×C4—21no—
301C22×C4—3141no—
302C22×C4—3141no—
303C22×C4—3141no—
304C22×C4—1571no—
305C22×C4—1571no—
306Q16—6281no—
307SD16—22no—
308SD16—3142no—
309SD16—3142no—
310SD16—3142no—
311D8—22no—
312D8—3142no—
313D8—6281no—
314D8—1574no—
315M16—3142no—
316M16—3142no—
317C2×C8—1571no—
318C2×C8—1571no—
319C2×C8C22×D15711yescharacteristic
320C2×C8C22×D15711yescharacteristic
321C4⋊C4—3141no—
322C4⋊C4—3141no—
323C4⋊C4—1571no—
324C4⋊C4—1571no—
325C4⋊C4C22×D15711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
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—
338C42C22×D15711yescharacteristic
339C23—21no—
340C23—21no—
341C23—3141no—
342C23—3141no—
343C23—6282no—
344C23—6282no—
345C23—6282no—
346C23—6282no—
347C23—1571no—
348C23—1571no—
349C23—12561no—
350C23—12561no—
351Q8—21no—
352Q8—21no—
353Q8—3141no—
354Q8—3141no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—3141no—
359D4—3141no—
360D4—3141no—
361D4—6281no—
362D4—6282no—
363D4—6282no—
364D4—6282no—
365D4—6282no—
366D4—1572no—
367D4—12561no—
368D4—12561no—
369D4D4×D15712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—3141no—
374C2×C4—3141no—
375C2×C4—3141no—
376C2×C4—3141no—
377C2×C4—3141no—
378C2×C4—3141no—
379C2×C4—6282no—
380C2×C4—6282no—
381C2×C4—1571no—
382C2×C4—1571no—
383C2×C4—12561no—
384C2×C4D4×D15711yescharacteristic
385C2×C4D4×D15711yescharacteristic
386C2×C4C23×D15711yescharacteristic
387C8—21no—
388C8—21no—
389C8—3141no—
390C8—1572no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3141no—
398C22—3141no—
399C22—3142no—
400C22—3141no—
401C22—3141no—
402C22—3141no—
403C22—3141no—
404C22—6281no—
405C22—6281no—
406C22—6281no—
407C22—6281no—
408C22—6281no—
409C22—12564no—
410C22—12564no—
411C22C2×D4×D15711yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—3141no—
417C4—3141no—
418C4—6281no—
419C4—6281no—
420C4C2×D4×D15711yescharacteristic
421C4#1909781711yescharacteristic
422C2—42no—
423C2—81no—
424C2—6281no—
425C2—6281no—
426C2—6282no—
427C2—12561no—
428C2#2375790711yescharacteristic
429C2#2375807011yescharacteristic
430C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#463×C157C2
socleC22×C157C22×D4
hypercenter#463D157
hypocenterC157#3273
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. C2×#19097795C2semidirect
3. #23757918C2semidirect
4. #23757916C2semidirect
5. #23757655C2semidirect
6. #23757522C2semidirect
7. #23757518C2semidirect
8. #23757489C2semidirect
9. #23757438C2semidirect
10. #23757432C2semidirect
11. #23757408C2semidirect
12. #23757405C2semidirect
13. C4×#14578939C2semidirect
14. #23757350C2semidirect
15. #463×C157C2semidirect
18. C2×D4×D157C22semidirect
19. C2×D4×D157C22semidirect
22. C2×#14578944C22semidirect
23. C2×#14578943C22semidirect
24. C2×#14578942C22semidirect
25. C2×#14578941C22semidirect
29. C2×#14578937C22semidirect
30. C2×#14578937C22semidirect
31. #19097759C22semidirect
34. #19097751C22semidirect
35. #19097747C22semidirect
43. C4×D628C22semidirect
46. #19097704C22semidirect
47. #19097696C22semidirect
48. #19097693C22semidirect
49. #19097692C22semidirect
50. #19097691C22semidirect
51. #19097683C22semidirect
52. #19097683C22semidirect
53. C4×#11929434C22semidirect
54. C2×SD16×C157C22semidirect
56. #124×C157C22semidirect
57. C4×D4×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #103×C157C22semidirect
60. #102×C157C22semidirect
92. #14578939D4semidirect
215. C157#3273semidirect
225. #463D157semidirect
242. C2×SD16C2×D157semidirect
250. #124C2×D157semidirect
263. C4×D4C2×D157semidirect
267. C4⋊C8C2×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×#190977982
C2×#190977952
#237579182
#237579162
#237576552
#237575222
#237575182
#237574892
#237574382
#237574322
#237574082
#237574052
C4×#145789392
#237573502
#463×C1572
#3273157