Finite Group Databasebeta

← #34112098

Subgroups — #34112098

Order 20096.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757924C211yesmaximal, characteristic, complemented
2#23757911C211yesmaximal, characteristic, complemented
3#23757634C211yesmaximal, characteristic, complemented
4#23757602C211yesmaximal, characteristic, complemented
5#23757591C211yesmaximal, characteristic, complemented
6#23757561C211yesmaximal, characteristic, complemented
7#23757526C211yesmaximal, characteristic, complemented
8#23757505C211yesmaximal, characteristic, complemented
9#23757491C211yesmaximal, characteristic, complemented
10#23757418C211yesmaximal, characteristic, complemented
11#23757408C211yesmaximal, characteristic, complemented
12#23757382C211yesmaximal, characteristic, complemented
13#23757342C211yesmaximal, characteristic, complemented
14#102×D157C211yesmaximal, characteristic, complemented
15#417×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—21no—
17C2×D4×D157C2211yescharacteristic
18C2×#14578941C2211yescharacteristic, complemented
19C2×D1256C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21C2×C8×D157C2211yescharacteristic
22#19097770C2211yescharacteristic, complemented
23#19097760C2211yescharacteristic
24#19097756—21no—
25#19097754—21no—
26#19097754—21no—
27#19097749—21no—
28#19097747C2211yescharacteristic
29#19097746C2211yescharacteristic
30#19097745C2211yescharacteristic
31#19097745C2211yescharacteristic, complemented
32#19097743—21no—
33#19097739—21no—
34D4×Dic157C2211yescharacteristic
35#19097733C2211yescharacteristic, complemented
36C4×C22⋊D157C2211yescharacteristic
37C4⋊C4×D157C2211yescharacteristic
38C2×#14578931—21no—
39#19097703C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41#19097702C2211yescharacteristic, complemented
42#19097700C2211yescharacteristic, complemented
43#19097695C2211yescharacteristic, complemented
44#19097693C2211yescharacteristic
45#19097693C2211yescharacteristic, complemented
46#19097692C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic, complemented
50#19097683C2211yescharacteristic
51#19097681C2211yescharacteristic, complemented
52C2×#41×C157C2211yescharacteristic
53C2×M16×C157C2211yescharacteristic, complemented
54C2×C4⋊C4×C157C2211yescharacteristic, complemented
55#103×C157C2211yescharacteristic, complemented
56#103×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59C23×D157—21no—
60#14578954—44no—
61C2×Dic314—21no—
62C2×D628—21no—
63C2×D628C2311yescharacteristic
64C2×D628C2311yescharacteristic
65C2×C22⋊D157—21no—
66C2×C22⋊D157—21no—
67C2×C22⋊D157—21no—
68C2×C22⋊D157D411yescharacteristic
69D4×D157—22no—
70D4×D157—41no—
71C22×Dic157—21no—
72C2×C4×D157—21no—
73C2×C4×D157—21no—
74C2×C4×D157C2311yescharacteristic
75#14578941—41no—
76D1256—41no—
77#14578939—22no—
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C8×D157—41no—
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933D411yescharacteristic
87C22⋊Dic157—21no—
88C22⋊Dic157D411yescharacteristic
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931D411yescharacteristic
97C4×Dic157C2311yescharacteristic
98#41×C157—22no—
99#41×C157—41no—
100C2×Q8×C157C2311yescharacteristic
101C2×D4×C157—21no—
102C2×D4×C157C2311yescharacteristic
103C22×C4×C157—21no—
104C22×C4×C157—21no—
105C22×C4×C157C2311yescharacteristic
106M16×C157—41no—
107C2×C8×C157C2311yescharacteristic
108C2×C8×C157C2311yescharacteristic
109C4⋊C4×C157—21no—
110C4⋊C4×C157C2311yescharacteristic
111C4⋊C4×C157C2311yescharacteristic
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—42no—
115C22×D157—42no—
116C22×D157—81no—
117C22×D157C2×D411yescharacteristic
118Dic314—42no—
119D628—21no—
120D628—21no—
121D628—21no—
122D628—21no—
123D628—42no—
124C22⋊D157—22no—
125C22⋊D157—42no—
126C22⋊D157—42no—
127C22⋊D157—81no—
128C2×Dic157—21no—
129C2×Dic157—21no—
130C2×Dic157—81no—
131C2×Dic157C2×D411yescharacteristic
132C2×Dic157C2×D411yescharacteristic
133C2×Dic157C2×D411yescharacteristic
134C4×D157—21no—
135C4×D157—21no—
136C4×D157—42no—
137C4×D157—42no—
138#11929434—21no—
139#11929434—21no—
140C23×C157—21no—
141C23×C157C2×D411yescharacteristic
142Q8×C157—21no—
143Q8×C157—21no—
144D4×C157—21no—
145D4×C157—42no—
146D4×C157#4112yescomplemented
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—21no—
151C2×C4×C157—21no—
152C2×C4×C157—42no—
153C2×C4×C157—42no—
154C2×C4×C157C2×D411yescharacteristic
155C2×C4×C157C2411yescharacteristic
156C8×C157—21no—
157C8×C157—21no—
158C22×C157—21no—
159C22×C157—22no—
160C22×C157—21no—
161C22×C157—21no—
162C22×C157—41no—
163C22×C157—41no—
164C22×C157C22×D411yescharacteristic
165C2×D157—21no—
166C2×D157—21no—
167C2×D157—41no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—41no—
172C2×D157—41no—
173C2×D157—84no—
174C2×D157AGL(1,ℤ/8)12yes—
175C4×C157—21no—
176C4×C157—41no—
177C4×C157—41no—
178C4×C157—41no—
179C4×C157C2×#4111yescharacteristic
180C4×C157H(5,𝔽2)11yescharacteristic
181Dic157—21no—
182Dic157—21no—
183Dic157—21no—
184Dic157—41no—
185Dic157—41no—
186C2×C157—42no—
187C2×C157—41no—
188C2×C157#54611yescharacteristic
189C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C157#57711yescharacteristic
191D157—22no—
192D157—81no—
193D157—81no—
194C157#330611yescharacteristic, complemented, Sylow
195#3306—1571nomaximal, Sylow
196#547—1571no—
197#546—1571no—
198#489—1571no—
199#483—1571no—
200#481—1571no—
201#476—1571no—
202#469—1571no—
203#465—1571no—
204#463—1571no—
205#449—1571no—
206#447—1571no—
207#442—1571no—
208#435—1571no—
209#417D15711yescharacteristic, complemented
210C2×#102—1571no—
211C2×#41—3141no—
212C2×#41C2×D15711yescharacteristic
213C22×D4—1571no—
214C2×SD16—1571no—
215C2×D8—1571no—
216C2×D8—1571no—
217C2×M16C2×D15711yescharacteristic, complemented
218C22×C8—1571no—
219C4⋊Q8—1571no—
220#124—1571no—
221#123—3141no—
222#123—3141no—
223#123—3141no—
224C22⋊Q8—3141no—
225C22⋊Q8—1571no—
226#121—3141no—
227#121—1571no—
228#121—1571no—
229#121—1571no—
230#120—3141no—
231C4×D4—1571no—
232C4×D4—1571no—
233C4×D4—1571no—
234C2×C4⋊C4—1571no—
235C2×C4⋊C4C2×D15711yescharacteristic, complemented
236C2×C22⋊C4—3141no—
237#106—1571no—
238#106—1571no—
239#106—1571no—
240C4⋊C8—1571no—
241#103—1571no—
242#103C2×D15711yescharacteristic, complemented
243#103C2×D15711yescharacteristic, complemented
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102—1571no—
248#102—1571no—
249#102C2×D15711yescharacteristic
250#102C2×D15711yescharacteristic
251C22⋊C8—1571no—
252C22⋊C8—1571no—
253#97—1571no—
254C24—3141no—
255#41—22no—
256#41—41no—
257#41—6284no—
258C2×Q8—3141no—
259C2×Q8C22×D15711yescharacteristic
260C2×D4—21no—
261C2×D4—3141no—
262C2×D4—3142no—
263C2×D4—3141no—
264C2×D4—3141no—
265C2×D4—3141no—
266C2×D4—6281no—
267C2×D4—1571no—
268C2×D4—1571no—
269C2×D4—1571no—
270C2×D4C22×D15711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3141no—
274C22×C4—3141no—
275C22×C4—3141no—
276C22×C4—1571no—
277C22×C4C22×D15711yescharacteristic
278SD16—6281no—
279D8—3142no—
280D8—6281no—
281M16—41no—
282C2×C8—6281no—
283C2×C8—1571no—
284C2×C8—1571no—
285C2×C8C22×D15711yescharacteristic
286C2×C8C22×D15711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3141no—
289C4⋊C4—3141no—
290C4⋊C4—3141no—
291C4⋊C4—1571no—
292C4⋊C4—1571no—
293C4⋊C4—1571no—
294C4⋊C4C22×D15711yescharacteristic
295C4⋊C4C22×D15711yescharacteristic
296C22⋊C4—3141no—
297C22⋊C4—3141no—
298C22⋊C4—3141no—
299C22⋊C4—3141no—
300C22⋊C4—3141no—
301C22⋊C4—3141no—
302C22⋊C4—3141no—
303C22⋊C4—3141no—
304C22⋊C4—1571no—
305C22⋊C4—1571no—
306C42—1571no—
307C23—21no—
308C23—3141no—
309C23—3141no—
310C23—6282no—
311C23—6282no—
312C23—1571no—
313C23—12561no—
314C23D4×D15711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—6282no—
318D4—21no—
319D4—42no—
320D4—3142no—
321D4—3141no—
322D4—3141no—
323D4—3141no—
324D4—3141no—
325D4—6282no—
326D4—6282no—
327D4—6282no—
328D4—12561no—
329D4#1457895412yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—3141no—
338C2×C4—3141no—
339C2×C4—3141no—
340C2×C4—3141no—
341C2×C4—6282no—
342C2×C4—6282no—
343C2×C4—1571no—
344C2×C4—1571no—
345C2×C4—1571no—
346C2×C4—12561no—
347C2×C4D4×D15711yescharacteristic
348C2×C4C23×D15711yescharacteristic
349C8—21no—
350C8—21no—
351C8—3141no—
352C8—3141no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—3141no—
360C22—3141no—
361C22—6281no—
362C22—6281no—
363C22—6281no—
364C22—6281no—
365C22—6281no—
366C22—6281no—
367C22—1572no—
368C22—12564no—
369C22C2×D4×D15711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—3141no—
375C4—3141no—
376C4—3141no—
377C4—6281no—
378C4—6281no—
379C4C2×#1457895411yescharacteristic
380C4#1909781611yescharacteristic
381C2—41no—
382C2—42no—
383C2—3142no—
384C2—12561no—
385C2—12561no—
386C2#2375791311yescharacteristic
387C2AGL(1,ℤ/8)×D15711yescharacteristic
388C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#417×C157C2
socleC22×C157C22×D4
hypercenter#417D157
hypocenterC157#3306
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #23757924C2semidirect
2. #23757911C2semidirect
3. #23757634C2semidirect
4. #23757602C2semidirect
5. #23757591C2semidirect
6. #23757561C2semidirect
7. #23757526C2semidirect
8. #23757505C2semidirect
9. #23757491C2semidirect
10. #23757418C2semidirect
11. #23757408C2semidirect
12. #23757382C2semidirect
13. #23757342C2semidirect
14. #102×D157C2semidirect
15. #417×C157C2semidirect
18. C2×#14578941C22semidirect
22. #19097770C22semidirect
31. #19097745C22semidirect
35. #19097733C22semidirect
39. #19097703C22semidirect
40. #19097702C22semidirect
41. #19097702C22semidirect
42. #19097700C22semidirect
43. #19097695C22semidirect
45. #19097693C22semidirect
49. C22⋊#11929434C22semidirect
51. #19097681C22semidirect
53. C2×M16×C157C22semidirect
54. C2×C4⋊C4×C157C22semidirect
55. #103×C157C22semidirect
56. #103×C157C22semidirect
146. D4×C157#41semidirect
194. C157#3306semidirect
209. #417D157semidirect
217. C2×M16C2×D157semidirect
235. C2×C4⋊C4C2×D157semidirect
242. #103C2×D157semidirect
243. #103C2×D157semidirect
329. D4#14578954semidirect

Maximal subgroups

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

subgroupindex
#237579242
#237579112
#237576342
#237576022
#237575912
#237575612
#237575262
#237575052
#237574912
#237574182
#237574082
#237573822
#237573422
#102×D1572
#417×C1572
#3306157