Finite Group Databasebeta

← #34111829

Subgroups — #34111829

Order 20096.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14578943C211yesmaximal, characteristic, complemented
2C2×SD16×D157C211yesmaximal, characteristic, complemented
3#23757936C211yesmaximal, characteristic, complemented
4#23757907C211yesmaximal, characteristic
5#23757644C211yesmaximal, characteristic, complemented
6#23757503C211yesmaximal, characteristic, complemented
7#23757477C211yesmaximal, characteristic, complemented
8#23757472C211yesmaximal, characteristic, complemented
9#23757425C211yesmaximal, characteristic, complemented
10#23757423C211yesmaximal, characteristic, complemented
11#23757413C211yesmaximal, characteristic, complemented
12#23757411C211yesmaximal, characteristic, complemented
13#23757360C211yesmaximal, characteristic, complemented
14C8×C22⋊D157C211yesmaximal, characteristic, complemented
15#465×C157C211yesmaximal, characteristic, complemented
16C22×Dic314C2211yescharacteristic, complemented
17C2×Q8×D157C2211yescharacteristic, complemented
18C22×D628C2211yescharacteristic
19C2×D4×D157—21no—
20C2×D4×D157C2211yescharacteristic
21C2×#14578943—24no—
22C2×#14578943C2211yescharacteristic
23C2×#14578943C2211yescharacteristic
24C2×#14578943C2211yescharacteristic, complemented
25C2×#14578943C2211yescharacteristic, complemented
26C2×#14578942C2211yescharacteristic, complemented
27C2×#14578942C2211yescharacteristic, complemented
28C2×#14578941C2211yescharacteristic
29C2×#14578941C2211yescharacteristic, complemented
30SD16×D157—24no—
31C2×C8×D157C2211yescharacteristic, complemented
32#19097770—21no—
33#19097769C2211yescharacteristic, complemented
34#19097767C2211yescharacteristic
35#19097751C2211yescharacteristic, complemented
36#19097749—21no—
37#19097748—21no—
38#19097746C2211yescharacteristic
39#19097744—21no—
40#19097741—21no—
41#19097739—21no—
42#19097736C2211yescharacteristic, complemented
43#19097733C2211yescharacteristic
44C4×C22⋊D157C2211yescharacteristic
45#19097731—21no—
46#19097700C2211yescharacteristic, complemented
47#19097694C2211yescharacteristic, complemented
48#19097694C2211yescharacteristic, complemented
49#19097691C2211yescharacteristic
50#19097691C2211yescharacteristic, complemented
51C22⋊#11929434C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53C8×Dic157C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C22×C8×C157C2211yescharacteristic, complemented
56C22⋊Q8×C157C2211yescharacteristic, complemented
57#121×C157C2211yescharacteristic
58#106×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63C2×Dic314—21no—
64C2×Dic314—22no—
65C2×Dic314—21no—
66C2×Dic314—41no—
67C2×Dic314C2311yescharacteristic
68C2×Dic314C2311yescharacteristic
69Q8×D157—22no—
70Q8×D157—41no—
71C2×D628—22no—
72C2×D628—21no—
73C2×D628—21no—
74C2×D628—41no—
75C2×D628C2311yescharacteristic
76C2×D628C2311yescharacteristic
77C2×C22⋊D157—21no—
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C2×C22⋊D157D411yescharacteristic
81D4×D157—22no—
82D4×D157—41no—
83D4×D157—44no—
84C22×Dic157—21no—
85C2×C4×D157—21no—
86C2×C4×D157—21no—
87C2×C4×D157C2311yescharacteristic
88#14578943—22no—
89#14578943—22no—
90#14578943—22no—
91#14578943D414yescomplemented
92#14578942—22no—
93#14578942—41no—
94#14578941—22no—
95#14578941—41no—
96C2×#11929434C2311yescharacteristic
97C2×#11929434C2311yescharacteristic
98C8×D157—22no—
99C4⋊Dic157—21no—
100#14578933—21no—
101#14578933—21no—
102#14578933—21no—
103#14578933D411yescharacteristic
104C22⋊Dic157D411yescharacteristic
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—
116C2×C8×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—42no—
127C22×D157—42no—
128C22×D157—42no—
129C22×D157—42no—
130C22×D157—81no—
131C22×D157—81no—
132C22×D157C2×D411yescharacteristic
133Dic314—21no—
134Dic314—21no—
135Dic314—21no—
136Dic314—42no—
137Dic314—42no—
138Dic314C2×D412yescomplemented
139D628—21no—
140D628—21no—
141D628—21no—
142D628—42no—
143D628—42no—
144D628C2×D412yes—
145C22⋊D157—42no—
146C22⋊D157—42no—
147C22⋊D157—81no—
148C22⋊D157SD1614yescomplemented
149C2×Dic157—21no—
150C2×Dic157—21no—
151C2×Dic157—42no—
152C2×Dic157C2×D411yescharacteristic
153C2×Dic157C2×D411yescharacteristic
154C2×Dic157C2×D411yescharacteristic
155C4×D157—21no—
156C4×D157—21no—
157C4×D157—42no—
158C4×D157—42no—
159#11929434—21no—
160#11929434—21no—
161C23×C157—21no—
162C23×C157C2×D411yescharacteristic
163Q8×C157—21no—
164Q8×C157—21no—
165D4×C157—21no—
166D4×C157—21no—
167D4×C157—42no—
168C2×C4×C157—21no—
169C2×C4×C157—21no—
170C2×C4×C157—21no—
171C2×C4×C157—21no—
172C2×C4×C157—21no—
173C2×C4×C157C2×D411yescharacteristic
174C2×C4×C157C2411yescharacteristic
175C8×C157—21no—
176C8×C157C2×D412yescomplemented
177C22×C157—21no—
178C22×C157—21no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157—41no—
182C22×C157C2×SD1612yescomplemented
183C22×C157C22×D411yescharacteristic
184C2×D157—21no—
185C2×D157—21no—
186C2×D157—22no—
187C2×D157—41no—
188C2×D157—41no—
189C2×D157—41no—
190C2×D157—41no—
191C2×D157—41no—
192C2×D157—84no—
193C2×D157—84no—
194C2×D157C2×SD1612yescomplemented
195C4×C157—21no—
196C4×C157—41no—
197C4×C157—41no—
198C4×C157—41no—
199C4×C157C22×D411yescharacteristic
200C4×C157H(5,𝔽2)11yescharacteristic
201Dic157—21no—
202Dic157—22no—
203Dic157—21no—
204Dic157—41no—
205Dic157C2×SD1612yescomplemented
206C2×C157—22no—
207C2×C157—81no—
208C2×C157D4211yescharacteristic
209C2×C157C22×SD1611yescharacteristic
210C2×C157#57711yescharacteristic
211D157—22no—
212D157—42no—
213D157—81no—
214C157D4×SD1611yescharacteristic, complemented, Sylow
215D4×SD16—1571nomaximal, Sylow
216C22×SD16—1571no—
217C22×SD16—1571no—
218D4×Q8—1571no—
219D42—1571no—
220#492—1571no—
221#465—1571no—
222#465D15711yescharacteristic, complemented
223#461—1571no—
224#460—1571no—
225#450—1571no—
226#450—1571no—
227#448—1571no—
228#448—1571no—
229C4×SD16—1571no—
230C8×D4—1571no—
231C22×Q8—1571no—
232C22×Q8—1571no—
233C22×D4—3141no—
234C22×D4—1571no—
235C22×D4—1571no—
236C2×SD16—3144no—
237C2×SD16—3144no—
238C2×SD16—1571no—
239C2×SD16—1571no—
240C2×SD16—1571no—
241C2×SD16—1571no—
242C2×SD16—1571no—
243C2×SD16—1571no—
244C2×SD16—1571no—
245C2×SD16—1571no—
246C2×SD16C2×D15711yescharacteristic, complemented
247C22×C8—1571no—
248C22×C8C2×D15711yescharacteristic, complemented
249C4⋊Q8—3141no—
250C4⋊Q8—1571no—
251#127—1571no—
252C22⋊Q8—3141no—
253C22⋊Q8—3141no—
254C22⋊Q8—1571no—
255C22⋊Q8C2×D15711yescharacteristic, complemented
256#121—3141no—
257#121—1571no—
258#121C2×D15711yescharacteristic
259#120—3141no—
260#120—3141no—
261C4×Q8—1571no—
262C4×D4—3141no—
263C4×D4—1571no—
264C4×D4—1571no—
265#106C2×D15711yescharacteristic, complemented
266C4⋊C8—1571no—
267#103—1571no—
268#103—1571no—
269#103C2×D15711yescharacteristic, complemented
270#102—1571no—
271#102—1571no—
272#102C2×D15711yescharacteristic, complemented
273C22⋊C8—1571no—
274C22⋊C8—1571no—
275C4×C8—1571no—
276C24—3141no—
277C24—3141no—
278C2×Q8—3141no—
279C2×Q8—3142no—
280C2×Q8—3141no—
281C2×Q8—3142no—
282C2×Q8—6281no—
283C2×Q8—6281no—
284C2×Q8—1571no—
285C2×Q8—1571no—
286C2×Q8C22×D15711yescharacteristic
287C2×D4—21no—
288C2×D4—3142no—
289C2×D4—3141no—
290C2×D4—3142no—
291C2×D4—3141no—
292C2×D4—3141no—
293C2×D4—3141no—
294C2×D4—3141no—
295C2×D4—6281no—
296C2×D4—6281no—
297C2×D4—6284no—
298C2×D4—1571no—
299C2×D4—1571no—
300C2×D4—1571no—
301C2×D4C22×D15711yescharacteristic
302C22×C4—3141no—
303C22×C4—3141no—
304C22×C4—3141no—
305C22×C4—1571no—
306C22×C4C22×D15711yescharacteristic
307SD16—22no—
308SD16—3142no—
309SD16—3142no—
310SD16—3142no—
311SD16—3142no—
312SD16—3142no—
313SD16—6281no—
314SD16—6281no—
315SD16—1574no—
316C2×C8—22no—
317C2×C8—3142no—
318C2×C8—1571no—
319C2×C8—1571no—
320C2×C8C22×D15711yescharacteristic
321C2×C8C22×D15711yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—3141no—
324C4⋊C4—3141no—
325C4⋊C4—3141no—
326C4⋊C4—3141no—
327C4⋊C4—1571no—
328C4⋊C4C22×D15711yescharacteristic
329C4⋊C4C22×D15711yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—3141no—
333C22⋊C4—3141no—
334C22⋊C4—3141no—
335C22⋊C4—1571no—
336C22⋊C4—1571no—
337C42—3141no—
338C42—1571no—
339C23—21no—
340C23—3141no—
341C23—3141no—
342C23—6282no—
343C23—6282no—
344C23—6282no—
345C23—6282no—
346C23—1571no—
347C23—12561no—
348C23—12561no—
349C23D4×D15711yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—3141no—
353Q8—3141no—
354Q8—3141no—
355Q8—6282no—
356Q8—6282no—
357Q8—1572no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—3141no—
362D4—3141no—
363D4—3141no—
364D4—6282no—
365D4—6282no—
366D4—6282no—
367D4—6282no—
368D4—1574no—
369D4—1572no—
370D4—12561no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—3141no—
377C2×C4—3141no—
378C2×C4—3141no—
379C2×C4—3141no—
380C2×C4—6282no—
381C2×C4—6282no—
382C2×C4—6282no—
383C2×C4—1571no—
384C2×C4—1571no—
385C2×C4—1571no—
386C2×C4D4×D15711yescharacteristic
387C2×C4C23×D15711yescharacteristic
388C8—21no—
389C8—3141no—
390C8—3141no—
391C8D4×D15712yescomplemented
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3141no—
398C22—3141no—
399C22—3142no—
400C22—6281no—
401C22—6281no—
402C22—6281no—
403C22—6281no—
404C22—6281no—
405C22—1572no—
406C22—12564no—
407C22—12564no—
408C22SD16×D15712yescomplemented
409C22C2×D4×D15711yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—3141no—
415C4—3141no—
416C4—3142no—
417C4—6281no—
418C4—1572no—
419C4C2×D4×D15711yescharacteristic
420C4#1909781711yescharacteristic
421C2—22no—
422C2—81no—
423C2—3142no—
424C2—6282no—
425C2—12561no—
426C2#2375790711yescharacteristic
427C2C2×SD16×D15711yescharacteristic
428C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#465×C157C2
socleC22×C157C22×D4
prime radical#23757907C2
hypercenter#465D157
hypocenterC157D4×SD16
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C22×#14578943C2semidirect
2. C2×SD16×D157C2semidirect
3. #23757936C2semidirect
5. #23757644C2semidirect
6. #23757503C2semidirect
7. #23757477C2semidirect
8. #23757472C2semidirect
9. #23757425C2semidirect
10. #23757423C2semidirect
11. #23757413C2semidirect
12. #23757411C2semidirect
13. #23757360C2semidirect
14. C8×C22⋊D157C2semidirect
15. #465×C157C2semidirect
16. C22×Dic314C22semidirect
17. C2×Q8×D157C22semidirect
24. C2×#14578943C22semidirect
25. C2×#14578943C22semidirect
26. C2×#14578942C22semidirect
27. C2×#14578942C22semidirect
29. C2×#14578941C22semidirect
31. C2×C8×D157C22semidirect
33. #19097769C22semidirect
35. #19097751C22semidirect
42. #19097736C22semidirect
46. #19097700C22semidirect
47. #19097694C22semidirect
48. #19097694C22semidirect
50. #19097691C22semidirect
51. C22⋊#11929434C22semidirect
52. #19097683C22semidirect
53. C8×Dic157C22semidirect
54. C2×SD16×C157C22semidirect
55. C22×C8×C157C22semidirect
56. C22⋊Q8×C157C22semidirect
58. #106×C157C22semidirect
59. #103×C157C22semidirect
60. #102×C157C22semidirect
91. #14578943D4semidirect
138. Dic314C2×D4semidirect
148. C22⋊D157SD16semidirect
176. C8×C157C2×D4semidirect
182. C22×C157C2×SD16semidirect
194. C2×D157C2×SD16semidirect
205. Dic157C2×SD16semidirect
214. C157D4×SD16semidirect
222. #465D157semidirect
246. C2×SD16C2×D157semidirect
248. C22×C8C2×D157semidirect
255. C22⋊Q8C2×D157semidirect
265. #106C2×D157semidirect
269. #103C2×D157semidirect
272. #102C2×D157semidirect
391. C8D4×D157semidirect
408. C22SD16×D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789432
C2×SD16×D1572
#237579362
#237579072
#237576442
#237575032
#237574772
#237574722
#237574252
#237574232
#237574132
#237574112
#237573602
C8×C22⋊D1572
#465×C1572
D4×SD16157