Finite Group Databasebeta

← #34110053

Subgroups — #34110053

Order 20096.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757785C211yesmaximal, characteristic
2#117×D157C211yesmaximal, characteristic, complemented
3#23757617C211yesmaximal, characteristic, complemented
4#23757616C211yesmaximal, characteristic, complemented
5#23757610C211yesmaximal, characteristic, complemented
6#23757605C211yesmaximal, characteristic, complemented
7#23757600C211yesmaximal, characteristic, complemented
8#23757599C211yesmaximal, characteristic, complemented
9#23757593C211yesmaximal, characteristic, complemented
10#23757588C211yesmaximal, characteristic, complemented
11#23757309C211yesmaximal, characteristic, complemented
12#23757270C211yesmaximal, characteristic, complemented
13#23757267C211yesmaximal, characteristic, complemented
14C2×#19097683C211yesmaximal, characteristic, complemented
15#427×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×C4×D157C2211yescharacteristic
18C22×#11929434C2211yescharacteristic
19#19097755—21no—
20#19097751C2211yescharacteristic
21#19097751C2211yescharacteristic
22#19097746C2211yescharacteristic
23#19097746C2211yescharacteristic
24#19097741—21no—
25C4×D628—21no—
26C4×D628—21no—
27#19097730C2211yescharacteristic
28#19097729C2211yescharacteristic
29#19097729C2211yescharacteristic
30#19097728—21no—
31#19097727—21no—
32C4⋊C4×D157C2211yescharacteristic
33C4⋊C4×D157C2211yescharacteristic
34C22⋊C4×D157—21no—
35C42×D157—21no—
36#19097704C2211yescharacteristic, complemented
37#19097704C2211yescharacteristic, complemented
38#19097702C2211yescharacteristic, complemented
39#19097702C2211yescharacteristic, complemented
40#19097696C2211yescharacteristic, complemented
41#19097696C2211yescharacteristic, complemented
42#19097694C2211yescharacteristic, complemented
43#19097694C2211yescharacteristic, complemented
44#19097693C2211yescharacteristic, complemented
45#19097693C2211yescharacteristic, complemented
46#19097691C2211yescharacteristic, complemented
47#19097691C2211yescharacteristic, complemented
48#19097683C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097683C2211yescharacteristic
52C22×C8×C157C2211yescharacteristic
53#117×C157C2211yescharacteristic
54#117×C157C2211yescharacteristic
55#107×C157C2211yescharacteristic, complemented
56#107×C157C2211yescharacteristic, complemented
57#106×C157C2211yescharacteristic, complemented
58#106×C157C2211yescharacteristic, complemented
59C23×D157D411yescharacteristic
60#14578954—22no—
61#14578954—22no—
62C2×Dic314C2311yescharacteristic
63C2×D628—21no—
64C2×D628—21no—
65C2×D628C2311yescharacteristic
66C2×C22⋊D157—21no—
67C22×Dic157D411yescharacteristic
68C2×C4×D157—21no—
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—22no—
72C2×C4×D157—22no—
73C2×C4×D157D411yescharacteristic
74C2×C4×D157D411yescharacteristic
75C2×C4×D157C2311yescharacteristic
76C2×C4×D157C2311yescharacteristic
77C2×#11929434—22no—
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C4⋊Dic157—21no—
81C4⋊Dic157—21no—
82C4⋊Dic157C2311yescharacteristic
83C4⋊Dic157C2311yescharacteristic
84#14578933—21no—
85#14578933—21no—
86C22⋊Dic157—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93C4×Dic157—21no—
94C4×Dic157—21no—
95C4×Dic157—21no—
96C22×C4×C157C2311yescharacteristic
97C2×C8×C157—22no—
98C2×C8×C157C2311yescharacteristic
99C2×C8×C157C2311yescharacteristic
100C4⋊C4×C157C2311yescharacteristic
101C4⋊C4×C157C2311yescharacteristic
102C4⋊C4×C157C2311yescharacteristic
103C4⋊C4×C157C2311yescharacteristic
104C22⋊C4×C157—21no—
105C22⋊C4×C157—21no—
106C42×C157—21no—
107C42×C157—21no—
108C22×D157—22no—
109C22×D157—22no—
110C22×D157—21no—
111C22×D157—41no—
112C22×D157C2×D411yescharacteristic
113C22×D157C2×D411yescharacteristic
114Dic314—21no—
115Dic314—21no—
116D628—21no—
117D628—21no—
118D628—42no—
119D628—42no—
120C22⋊D157—42no—
121C2×Dic157—22no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—22no—
128C4×D157—22no—
129C4×D157—21no—
130C4×D157—21no—
131C4×D157—21no—
132C4×D157—21no—
133C4×D157—42no—
134C4×D157—42no—
135C4×D157—42no—
136#11929434—21no—
137#11929434—21no—
138C23×C157C2×D411yescharacteristic
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157#4112yes—
145C2×C4×C157#4112yes—
146C2×C4×C157C2411yescharacteristic
147C8×C157—21no—
148C8×C157—21no—
149C22×C157—21no—
150C22×C157—21no—
151C22×C157#12312yes—
152C22×C157C22×D411yescharacteristic
153C2×D157—21no—
154C2×D157—21no—
155C2×D157—21no—
156C2×D157—21no—
157C2×D157—21no—
158C2×D157—44no—
159C2×D157—41no—
160C2×D157—41no—
161C2×D157—41no—
162C2×D157#13512yes—
163C4×C157—41no—
164C4×C157—41no—
165C4×C157—41no—
166C4×C157—41no—
167C4×C157#12312yes—
168C4×C157C2×#4111yescharacteristic
169C4×C157C2×#4111yescharacteristic
170Dic157—21no—
171Dic157—21no—
172Dic157—41no—
173Dic157—41no—
174Dic157—41no—
175C2×C157—22no—
176C2×C157C2×#12311yescharacteristic
177C2×C157C2×#13511yescharacteristic
178C2×C157#57511yescharacteristic
179D157—22no—
180D157—41no—
181D157—81no—
182C157#309111yescharacteristic, complemented, Sylow
183#3091—1571nomaximal, Sylow
184#525—1571no—
185C2×#117—1571no—
186#485—1571no—
187#485—1571no—
188#484—1571no—
189#483—1571no—
190#482—1571no—
191#482—1571no—
192#481—1571no—
193#480—1571no—
194#427—1571no—
195#427D15711yescharacteristic, complemented
196#416—1571no—
197#416—1571no—
198C2×C22⋊C8—1571no—
199C2×#41—1571no—
200C23×C4—1571no—
201C22×C8—1571no—
202C22×C8C2×D15711yescharacteristic
203#123—3141no—
204C22⋊Q8—1571no—
205C22⋊Q8—1571no—
206#121—1571no—
207#121—1571no—
208#120—3141no—
209C4×D4—3141no—
210C4×D4—3141no—
211#117—3141no—
212#117—3141no—
213#117—1571no—
214#117—1571no—
215#117—1571no—
216#117C2×D15711yescharacteristic
217#117C2×D15711yescharacteristic
218C2×C4⋊C4—1571no—
219C2×C4⋊C4—1571no—
220C2×C22⋊C4—3141no—
221C2×C42—3141no—
222#107—1571no—
223#107—1571no—
224#107C2×D15711yescharacteristic, complemented
225#107C2×D15711yescharacteristic, complemented
226#106—1571no—
227#106—1571no—
228#106C2×D15711yescharacteristic, complemented
229#106C2×D15711yescharacteristic, complemented
230#103—1571no—
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#102—1571no—
235#102—1571no—
236#102—1571no—
237#102—1571no—
238C22⋊C8—1571no—
239C22⋊C8—1571no—
240C22⋊C8—1571no—
241C22⋊C8—1571no—
242C24—1571no—
243#41—3142no—
244#41—3142no—
245C2×Q8—1571no—
246C2×D4—3141no—
247C2×D4—3141no—
248C2×D4—3141no—
249C2×D4—1571no—
250C22×C4—3141no—
251C22×C4—3141no—
252C22×C4—3142no—
253C22×C4—3142no—
254C22×C4—3141no—
255C22×C4—1571no—
256C22×C4—1571no—
257C22×C4—1571no—
258C22×C4—1571no—
259C22×C4—1571no—
260C22×C4C22×D15711yescharacteristic
261C2×C8—22no—
262C2×C8—3142no—
263C2×C8—1571no—
264C2×C8—1571no—
265C2×C8C22×D15711yescharacteristic
266C2×C8C22×D15711yescharacteristic
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—3141no—
271C4⋊C4—1571no—
272C4⋊C4—1571no—
273C4⋊C4C22×D15711yescharacteristic
274C4⋊C4C22×D15711yescharacteristic
275C4⋊C4C22×D15711yescharacteristic
276C4⋊C4C22×D15711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—3141no—
285C22⋊C4—3141no—
286C42—21no—
287C42—21no—
288C42—3141no—
289C42—3141no—
290C42—3141no—
291C23—3142no—
292C23—3142no—
293C23—3141no—
294C23—6281no—
295C23—1571no—
296C23—1571no—
297C23D4×D15711yescharacteristic
298Q8—3141no—
299Q8—3141no—
300D4—3141no—
301D4—3141no—
302D4—6282no—
303D4—6282no—
304D4—6282no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3142no—
310C2×C4—3142no—
311C2×C4—3142no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3141no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—6282no—
320C2×C4—6282no—
321C2×C4—6282no—
322C2×C4—1571no—
323C2×C4—1571no—
324C2×C4D4×D15711yescharacteristic
325C2×C4#1457895412yes—
326C2×C4#1457895512yes—
327C2×C4C23×D15711yescharacteristic
328C8—21no—
329C8—21no—
330C8—3141no—
331C8—3141no—
332C22—21no—
333C22—21no—
334C22—3141no—
335C22—3141no—
336C22—3141no—
337C22—3141no—
338C22—3141no—
339C22—6284no—
340C22—6281no—
341C22—6281no—
342C22—6281no—
343C22—1572no—
344C22#1909775612yes—
345C22C2×D4×D15711yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3141no—
351C4—3141no—
352C4—6281no—
353C4—6281no—
354C4—6281no—
355C4#1909775612yes—
356C4C2×#1457895411yescharacteristic
357C4C2×#1457895511yescharacteristic
358C2—22no—
359C2—3142no—
360C2—6281no—
361C2—12561no—
362C2C2×#1909775611yescharacteristic
363C2#135×D15711yescharacteristic
364C2#2375807011yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#427×C157C2
socleC22×C157C22×D4
prime radical#23757785C2
hypercenter#427D157
hypocenterC157#3091
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #117×D157C2semidirect
3. #23757617C2semidirect
4. #23757616C2semidirect
5. #23757610C2semidirect
6. #23757605C2semidirect
7. #23757600C2semidirect
8. #23757599C2semidirect
9. #23757593C2semidirect
10. #23757588C2semidirect
11. #23757309C2semidirect
12. #23757270C2semidirect
13. #23757267C2semidirect
14. C2×#19097683C2semidirect
15. #427×C157C2semidirect
36. #19097704C22semidirect
37. #19097704C22semidirect
38. #19097702C22semidirect
39. #19097702C22semidirect
40. #19097696C22semidirect
41. #19097696C22semidirect
42. #19097694C22semidirect
43. #19097694C22semidirect
44. #19097693C22semidirect
45. #19097693C22semidirect
46. #19097691C22semidirect
47. #19097691C22semidirect
55. #107×C157C22semidirect
56. #107×C157C22semidirect
57. #106×C157C22semidirect
58. #106×C157C22semidirect
182. C157#3091semidirect
195. #427D157semidirect
224. #107C2×D157semidirect
225. #107C2×D157semidirect
228. #106C2×D157semidirect
229. #106C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237577852
#117×D1572
#237576172
#237576162
#237576102
#237576052
#237576002
#237575992
#237575932
#237575882
#237573092
#237572702
#237572672
C2×#190976832
#427×C1572
#3091157