Finite Group Databasebeta

← #34110278

Subgroups — #34110278

Order 20096.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757810C211yesmaximal, characteristic, complemented
2C4×D4×D157C211yesmaximal, characteristic, complemented
3#23757602C211yesmaximal, characteristic, complemented
4#23757588C211yesmaximal, characteristic, complemented
5#23757576C211yesmaximal, characteristic, complemented
6#23757575C211yesmaximal, characteristic, complemented
7#23757491C211yesmaximal, characteristic, complemented
8#23757490C211yesmaximal, characteristic, complemented
9#23757422C211yesmaximal, characteristic, complemented
10#23757408C211yesmaximal, characteristic
11#23757386C211yesmaximal, characteristic, complemented
12#23757383C211yesmaximal, characteristic, complemented
13#23757371C211yesmaximal, characteristic, complemented
14#23757328C211yesmaximal, characteristic, complemented
15#440×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C22×C4×D157—21no—
18C2×#14578941C2211yescharacteristic
19C2×D1256C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21#19097766—21no—
22#19097762C2211yescharacteristic, complemented
23#19097760C2211yescharacteristic, complemented
24#19097756—21no—
25#19097747C2211yescharacteristic
26#19097746C2211yescharacteristic
27C4×D628—21no—
28C4×D628C2211yescharacteristic
29C4×D628C2211yescharacteristic, complemented
30D4×Dic157C2211yescharacteristic
31#19097733C2211yescharacteristic
32C4×C22⋊D157—21no—
33#19097731—21no—
34#19097728—21no—
35C4⋊C4×D157C2211yescharacteristic
36C22⋊C4×D157—21no—
37C42×D157C2211yescharacteristic
38#19097704C2211yescharacteristic, complemented
39#19097703C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097700C2211yescharacteristic, complemented
43#19097695C2211yescharacteristic, complemented
44#19097693C2211yescharacteristic
45#19097693C2211yescharacteristic, complemented
46#19097692C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic, complemented
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097682C2211yescharacteristic, complemented
52C2×SD16×C157C2211yescharacteristic
53C4×Q8×C157C2211yescharacteristic, complemented
54C4×D4×C157C2211yescharacteristic
55#107×C157C2211yescharacteristic, complemented
56#103×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C2×D628—21no—
61C2×D628C2311yescharacteristic
62C2×D628C2311yescharacteristic
63C2×C22⋊D157—21no—
64D4×D157—22no—
65D4×D157—22no—
66C22×Dic157—21no—
67C2×C4×D157—21no—
68C2×C4×D157—21no—
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—44no—
72C2×C4×D157D411yescharacteristic
73C2×C4×D157D411yescharacteristic
74C2×C4×D157C2311yescharacteristic
75#14578941—22no—
76D1256—22no—
77#14578939—22no—
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C4⋊Dic157—21no—
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
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—
93#14578931—21no—
94C4×Dic157D411yescharacteristic
95C4×Dic157D411yescharacteristic
96C4×Dic157C2311yescharacteristic
97C2×Q8×C157C2311yescharacteristic
98C2×D4×C157C2311yescharacteristic
99C22×C4×C157—21no—
100SD16×C157—22no—
101C2×C8×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C4⋊C4×C157—21no—
104C4⋊C4×C157C2311yescharacteristic
105C4⋊C4×C157C2311yescharacteristic
106C22⋊C4×C157—21no—
107C42×C157—21no—
108C42×C157C2311yescharacteristic
109C22×D157—24no—
110C22×D157—21no—
111C22×D157—21no—
112C22×D157—42no—
113C22×D157—42no—
114C22×D157C2×D411yescharacteristic
115D628—21no—
116D628—21no—
117D628—21no—
118D628—42no—
119D628#4112yes—
120C22⋊D157—24no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—42no—
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—22no—
128C4×D157—21no—
129C4×D157—21no—
130C4×D157—41no—
131C4×D157—42no—
132C4×D157—42no—
133C4×D157—42no—
134C4×D157—42no—
135#11929434—21no—
136#11929434—21no—
137C23×C157—21no—
138Q8×C157—21no—
139Q8×C157—21no—
140D4×C157—21no—
141D4×C157#4112yes—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—42no—
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2411yescharacteristic
149C8×C157—21no—
150C8×C157—21no—
151C22×C157—22no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157C22×D411yescharacteristic
155C2×D157—21no—
156C2×D157—21no—
157C2×D157—22no—
158C2×D157—48no—
159C2×D157—41no—
160C2×D157—41no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157AGL(1,ℤ/8)12yes—
165C4×C157—21no—
166C4×C157—21no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157—41no—
170C4×C157C2×#4111yescharacteristic
171C4×C157C2×#4111yescharacteristic
172Dic157—21no—
173Dic157—21no—
174Dic157—21no—
175Dic157—41no—
176Dic157—41no—
177C2×C157—42no—
178C2×C157#52511yescharacteristic
179C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C157#57511yescharacteristic
181D157—22no—
182D157—42no—
183D157—81no—
184C157#311411yescharacteristic, complemented, Sylow
185#3114—1571nomaximal, Sylow
186#529—1571no—
187C2×C4×D4—1571no—
188#483—1571no—
189#480—1571no—
190#478—1571no—
191#478—1571no—
192#463—1571no—
193#463—1571no—
194#450—1571no—
195#447—1571no—
196#442—1571no—
197#442—1571no—
198#440—1571no—
199#440D15711yescharacteristic, complemented
200#432—1571no—
201C22×D4—1571no—
202C23×C4—3141no—
203C2×SD16—1571no—
204C2×SD16C2×D15711yescharacteristic
205C2×D8—1571no—
206C2×D8—1571no—
207#126—3141no—
208#125—1571no—
209#124—1571no—
210#123—3141no—
211C22⋊Q8—1571no—
212#121—1571no—
213C4×Q8C2×D15711yescharacteristic, complemented
214C4×D4—3141no—
215C4×D4—3141no—
216C4×D4—3141no—
217C4×D4—1571no—
218C4×D4—1571no—
219C4×D4—1571no—
220C4×D4—1571no—
221C4×D4C2×D15711yescharacteristic
222#117—3141no—
223C2×C4⋊C4—1571no—
224C2×C22⋊C4—3141no—
225C2×C42—1571no—
226#107—1571no—
227#107C2×D15711yescharacteristic, complemented
228#106—1571no—
229#106—1571no—
230C4⋊C8—1571no—
231C4⋊C8—1571no—
232#103—1571no—
233#103C2×D15711yescharacteristic, complemented
234#102—1571no—
235#102—1571no—
236#102—1571no—
237#102—1571no—
238#102—1571no—
239#102C2×D15711yescharacteristic
240C22⋊C8—1571no—
241C22⋊C8—1571no—
242#97—1571no—
243#97C2×D15711yescharacteristic, complemented
244C24—3141no—
245C2×Q8C22×D15711yescharacteristic
246C2×D4—3141no—
247C2×D4—3142no—
248C2×D4—3142no—
249C2×D4—3141no—
250C2×D4—1571no—
251C2×D4—1571no—
252C2×D4C22×D15711yescharacteristic
253C22×C4—21no—
254C22×C4—3141no—
255C22×C4—3141no—
256C22×C4—3141no—
257C22×C4—3141no—
258C22×C4—3141no—
259C22×C4—6284no—
260C22×C4—1571no—
261C22×C4—1571no—
262C22×C4—1571no—
263SD16—22no—
264SD16—3142no—
265D8—3142no—
266D8—3142no—
267C2×C8—1571no—
268C2×C8—1571no—
269C2×C8C22×D15711yescharacteristic
270C2×C8C22×D15711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—3141no—
273C4⋊C4—3141no—
274C4⋊C4—3141no—
275C4⋊C4—3141no—
276C4⋊C4—1571no—
277C4⋊C4—1571no—
278C4⋊C4C22×D15711yescharacteristic
279C4⋊C4C22×D15711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—3141no—
285C22⋊C4—3141no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C42—21no—
290C42—1571no—
291C42—1571no—
292C42—1571no—
293C42C22×D15711yescharacteristic
294C23—21no—
295C23—3144no—
296C23—3141no—
297C23—3141no—
298C23—6282no—
299C23—6282no—
300C23—1571no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—3144no—
305D4—3141no—
306D4—3141no—
307D4—3141no—
308D4—6282no—
309D4—1572no—
310D4#1457895412yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—3142no—
316C2×C4—3141no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—3141no—
320C2×C4—6281no—
321C2×C4—6282no—
322C2×C4—6282no—
323C2×C4—6282no—
324C2×C4—6282no—
325C2×C4—6282no—
326C2×C4—1571no—
327C2×C4—1571no—
328C2×C4—1571no—
329C2×C4D4×D15711yescharacteristic
330C2×C4D4×D15711yescharacteristic
331C2×C4C23×D15711yescharacteristic
332C8—21no—
333C8—21no—
334C8—3141no—
335C8—3141no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—3141no—
340C22—3141no—
341C22—3142no—
342C22—6288no—
343C22—6281no—
344C22—6281no—
345C22—6281no—
346C22—6281no—
347C22—6281no—
348C22—1572no—
349C22C2×D4×D15711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3141no—
356C4—3141no—
357C4—3141no—
358C4—6281no—
359C4—6281no—
360C4#41×D15711yescharacteristic
361C4C2×#1457895411yescharacteristic
362C2—42no—
363C2—3142no—
364C2—6282no—
365C2—12561no—
366C2#2375778211yescharacteristic
367C2AGL(1,ℤ/8)×D15711yescharacteristic
368C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#440×C157C2
socleC22×C157C22×D4
prime radical#23757408C2
hypercenter#440D157
hypocenterC157#3114
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #23757810C2semidirect
2. C4×D4×D157C2semidirect
3. #23757602C2semidirect
4. #23757588C2semidirect
5. #23757576C2semidirect
6. #23757575C2semidirect
7. #23757491C2semidirect
8. #23757490C2semidirect
9. #23757422C2semidirect
11. #23757386C2semidirect
12. #23757383C2semidirect
13. #23757371C2semidirect
14. #23757328C2semidirect
15. #440×C157C2semidirect
22. #19097762C22semidirect
23. #19097760C22semidirect
29. C4×D628C22semidirect
38. #19097704C22semidirect
39. #19097703C22semidirect
40. #19097702C22semidirect
41. #19097700C22semidirect
42. #19097700C22semidirect
43. #19097695C22semidirect
45. #19097693C22semidirect
48. #19097691C22semidirect
51. #19097682C22semidirect
53. C4×Q8×C157C22semidirect
55. #107×C157C22semidirect
56. #103×C157C22semidirect
58. #97×C157C22semidirect
184. C157#3114semidirect
199. #440D157semidirect
213. C4×Q8C2×D157semidirect
227. #107C2×D157semidirect
233. #103C2×D157semidirect
243. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578102
C4×D4×D1572
#237576022
#237575882
#237575762
#237575752
#237574912
#237574902
#237574222
#237574082
#237573862
#237573832
#237573712
#237573282
#440×C1572
#3114157