Finite Group Databasebeta

← #34109364

Subgroups — #34109364

Order 20096.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D157C211yesmaximal, characteristic, complemented
2C2×#19097798C212yesmaximal, complemented
3#23757786C211yesmaximal, characteristic, complemented
4#23757423C211yesmaximal, characteristic, complemented
5#23757422C212yesmaximal, complemented
6#23757409C211yesmaximal, characteristic, complemented
7#23757408C212yesmaximal, complemented
8#23757407C211yesmaximal, characteristic, complemented
9#23757276C212yesmaximal, complemented
10#23757235C211yesmaximal, characteristic, complemented
11#450×C157C211yesmaximal, characteristic, complemented
12C24×D157—21no—
13C2×#14578955C2211yescharacteristic, complemented
14C22×D628C2211yescharacteristic
15C22×C22⋊D157—21no—
16C2×D4×D157—28no—
17C2×D4×D157—21no—
18C2×D4×D157—21no—
19C2×D4×D157C2212yes—
20C2×D4×D157C2212yes—
21C22×C4×D157C2211yescharacteristic
22#19097798—28no—
23C2×#14578941C2212yescomplemented
24C2×D1256C2212yescomplemented
25C2×#14578939C2212yescomplemented
26C2×#14578937C2212yescomplemented
27#19097756—21no—
28#19097747C2211yescharacteristic, complemented
29#19097746C2211yescharacteristic, complemented
30#19097745C2211yescharacteristic, complemented
31#19097741—21no—
32#19097733—21no—
33#19097731—21no—
34#19097729C2211yescharacteristic, complemented
35#19097693C2212yescomplemented
36#19097692C2212yescomplemented
37#19097691C2212yescomplemented
38C22⋊#11929434C2211yescharacteristic, complemented
39#19097683C2212yescomplemented
40C22×D4×C157C2211yescharacteristic
41C2×SD16×C157C2212yescomplemented
42C22⋊Q8×C157C2211yescharacteristic, complemented
43#102×C157C2212yescomplemented
44C22⋊C8×C157C2211yescharacteristic, complemented
45C23×D157—28no—
46C23×D157—22no—
47C23×D157—22no—
48C23×D157—21no—
49C23×D157—44no—
50C23×D157—44no—
51C23×D157D411yescharacteristic
52#14578955—24no—
53C2×D628—24no—
54C2×D628—21no—
55C2×D628—21no—
56C2×D628—21no—
57C2×D628C2312yes—
58C2×D628C2311yescharacteristic, complemented
59C2×C22⋊D157—28no—
60C2×C22⋊D157—22no—
61C2×C22⋊D157—22no—
62C2×C22⋊D157—21no—
63D4×D157—28no—
64D4×D157—28no—
65D4×D157—24no—
66D4×D157—24no—
67D4×D157—24no—
68D4×D157D418yescomplemented
69C22×Dic157D411yescharacteristic
70C2×C4×D157—24no—
71C2×C4×D157—21no—
72C2×C4×D157D411yescharacteristic
73C2×C4×D157D411yescharacteristic
74C2×C4×D157C2311yescharacteristic
75C2×C4×D157C2311yescharacteristic
76#14578941—24no—
77D1256—24no—
78#14578939—24no—
79#14578937—24no—
80C2×#11929434C2312yescomplemented
81C4⋊Dic157C2311yescharacteristic, complemented
82#14578933—21no—
83#14578931—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87C4×Dic157—21no—
88C24×C157—21no—
89C2×Q8×C157C2311yescharacteristic, complemented
90C2×D4×C157—24no—
91C2×D4×C157—21no—
92C2×D4×C157C2312yes—
93C22×C4×C157C2311yescharacteristic
94SD16×C157—24no—
95C2×C8×C157C2312yescomplemented
96C4⋊C4×C157—21no—
97C4⋊C4×C157C2311yescharacteristic, complemented
98C22⋊C4×C157—21no—
99C22×D157—22no—
100C22×D157—22no—
101C22×D157—22no—
102C22×D157—28no—
103C22×D157—24no—
104C22×D157—22no—
105C22×D157—28no—
106C22×D157—21no—
107C22×D157—432no—
108C22×D157—44no—
109C22×D157—44no—
110C22×D157—44no—
111C22×D157—42no—
112C22×D157—42no—
113C22×D157—44no—
114C22×D157C2×D411yescharacteristic
115C22×D157C2×D411yescharacteristic
116D628—24no—
117D628—22no—
118D628—22no—
119D628—42no—
120D628—42no—
121D628C2×D414yes—
122C22⋊D157—28no—
123C22⋊D157—28no—
124C22⋊D157—42no—
125C2×Dic157—22no—
126C2×Dic157—21no—
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C4×D157—22no—
130C4×D157—22no—
131C4×D157—22no—
132C4×D157—42no—
133C4×D157C2×D414yes—
134#11929434—22no—
135C23×C157—24no—
136C23×C157—22no—
137C23×C157—42no—
138C23×C157—42no—
139C23×C157C2×D411yescharacteristic
140Q8×C157—22no—
141D4×C157—24no—
142D4×C157—22no—
143D4×C157C2×D414yes—
144C2×C4×C157—22no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2411yescharacteristic
149C8×C157—22no—
150C22×C157—21no—
151C22×C157—21no—
152C22×C157—24no—
153C22×C157—48no—
154C22×C157—42no—
155C22×C157—42no—
156C22×C157AGL(1,ℤ/8)12yescomplemented
157C22×C157C22×D411yescharacteristic
158C2×D157—28no—
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—21no—
162C2×D157—21no—
163C2×D157—24no—
164C2×D157—416no—
165C2×D157—416no—
166C2×D157—48no—
167C2×D157—42no—
168C2×D157—42no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—41no—
172C2×D157#12012yes—
173C2×D157AGL(1,ℤ/8)12yescomplemented
174C4×C157—21no—
175C4×C157—41no—
176C4×C157—41no—
177C4×C157C22×D412yes—
178Dic157—21no—
179Dic157—41no—
180Dic157#12012yes—
181C2×C157—22no—
182C2×C157—44no—
183C2×C157C2×#12011yescharacteristic
184C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
186D157—22no—
187D157—22no—
188D157—44no—
189D157—81no—
190C157#300311yescharacteristic, complemented, Sylow
191#3003—1571nomaximal, Sylow
192C23×D4—1571no—
193C2×AGL(1,ℤ/8)—1572no—
194#525—1571no—
195#450—1572no—
196#450—1571no—
197#450D15711yescharacteristic, complemented
198#447—1571no—
199#447—1572no—
200#447—1571no—
201#418—1572no—
202#407—1571no—
203C25—3141no—
204C2×#41—1571no—
205C22×D4—3148no—
206C22×D4—3141no—
207C22×D4—3141no—
208C22×D4—3141no—
209C22×D4—1572no—
210C22×D4—1572no—
211C22×D4—1571no—
212C22×D4C2×D15711yescharacteristic
213C23×C4—1571no—
214AGL(1,ℤ/8)—3148no—
215C2×SD16—1572no—
216C2×SD16C2×D15712yescomplemented
217C2×D8—1572no—
218C2×D8—1572no—
219C2×M16—1572no—
220#123—3141no—
221C22⋊Q8—1571no—
222C22⋊Q8C2×D15711yescharacteristic, complemented
223#121—1571no—
224#121—1571no—
225#120—3141no—
226C4×D4—3141no—
227C4×D4—3141no—
228#117—1571no—
229#102—1572no—
230#102—1572no—
231#102—1572no—
232#102C2×D15712yescomplemented
233C22⋊C8—1571no—
234C22⋊C8—1572no—
235C22⋊C8C2×D15711yescharacteristic, complemented
236C24—21no—
237C24—3148no—
238C24—3142no—
239C24—3142no—
240C24—3141no—
241C24—6284no—
242C24—6284no—
243C24—1571no—
244#41—3144no—
245C2×Q8C22×D15711yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—3144no—
249C2×D4—3141no—
250C2×D4—3141no—
251C2×D4—3148no—
252C2×D4—3148no—
253C2×D4—3144no—
254C2×D4—3148no—
255C2×D4—3144no—
256C2×D4—3141no—
257C2×D4—3144no—
258C2×D4—3142no—
259C2×D4—3142no—
260C2×D4—3141no—
261C2×D4—1578no—
262C2×D4—1571no—
263C2×D4—1572no—
264C2×D4C22×D15712yes—
265C22×C4—3144no—
266C22×C4—3141no—
267C22×C4—1571no—
268C22×C4—1571no—
269C22×C4—1571no—
270C22×C4—1571no—
271C22×C4—1571no—
272C22×C4C22×D15711yescharacteristic
273SD16—24no—
274SD16—3144no—
275D8—3144no—
276D8—3144no—
277M16—3144no—
278C2×C8—1572no—
279C2×C8C22×D15712yescomplemented
280C4⋊C4—21no—
281C4⋊C4—3141no—
282C4⋊C4—1571no—
283C4⋊C4C22×D15711yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—3141no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C42—3141no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—3142no—
295C23—3142no—
296C23—3144no—
297C23—3148no—
298C23—3148no—
299C23—3142no—
300C23—3142no—
301C23—3141no—
302C23—6282no—
303C23—62832no—
304C23—6284no—
305C23—6284no—
306C23—6284no—
307C23—6284no—
308C23—6282no—
309C23—1571no—
310C23—1571no—
311C23D4×D15711yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—3144no—
316D4—3148no—
317D4—3148no—
318D4—3142no—
319D4—3142no—
320D4—6282no—
321D4—6282no—
322D4—6282no—
323D4—1574no—
324D4D4×D15714yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—3142no—
329C2×C4—3142no—
330C2×C4—3142no—
331C2×C4—3142no—
332C2×C4—3141no—
333C2×C4—6282no—
334C2×C4—1571no—
335C2×C4—1571no—
336C2×C4—1574no—
337C2×C4D4×D15711yescharacteristic
338C2×C4C23×D15711yescharacteristic
339C8—22no—
340C8—3142no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—3148no—
348C22—3141no—
349C22—3141no—
350C22—3141no—
351C22—3141no—
352C22—3144no—
353C22—6288no—
354C22—62816no—
355C22—62816no—
356C22—6282no—
357C22—6282no—
358C22—6281no—
359C22—6281no—
360C22—6281no—
361C22—1572no—
362C22—1572no—
363C22#1909779812yescomplemented
364C22C2×D4×D15711yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3141no—
369C4—6281no—
370C4—1572no—
371C4C2×D4×D15712yes—
372C2—22no—
373C2—44no—
374C2—3142no—
375C2—3142no—
376C2—6284no—
377C2—12561no—
378C2#120×D15711yescharacteristic
379C2AGL(1,ℤ/8)×D15711yescharacteristic
380C2C2×#1909779811yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
hypercenter#450D157
hypocenterC157#3003
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×D4×D157C2semidirect
2. C2×#19097798C2semidirect
3. #23757786C2semidirect
4. #23757423C2semidirect
5. #23757422C2semidirect
6. #23757409C2semidirect
7. #23757408C2semidirect
8. #23757407C2semidirect
9. #23757276C2semidirect
10. #23757235C2semidirect
11. #450×C157C2semidirect
13. C2×#14578955C22semidirect
23. C2×#14578941C22semidirect
24. C2×D1256C22semidirect
25. C2×#14578939C22semidirect
26. C2×#14578937C22semidirect
28. #19097747C22semidirect
29. #19097746C22semidirect
30. #19097745C22semidirect
34. #19097729C22semidirect
35. #19097693C22semidirect
36. #19097692C22semidirect
37. #19097691C22semidirect
38. C22⋊#11929434C22semidirect
39. #19097683C22semidirect
41. C2×SD16×C157C22semidirect
42. C22⋊Q8×C157C22semidirect
43. #102×C157C22semidirect
44. C22⋊C8×C157C22semidirect
58. C2×D628C23semidirect
68. D4×D157D4semidirect
80. C2×#11929434C23semidirect
81. C4⋊Dic157C23semidirect
89. C2×Q8×C157C23semidirect
95. C2×C8×C157C23semidirect
97. C4⋊C4×C157C23semidirect
156. C22×C157AGL(1,ℤ/8)semidirect
173. C2×D157AGL(1,ℤ/8)semidirect
190. C157#3003semidirect
197. #450D157semidirect
216. C2×SD16C2×D157semidirect
222. C22⋊Q8C2×D157semidirect
232. #102C2×D157semidirect
235. C22⋊C8C2×D157semidirect
245. C2×Q8C22×D157semidirect
279. C2×C8C22×D157semidirect
283. C4⋊C4C22×D157semidirect
363. C22#19097798semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1572
C2×#190977982
#237577862
#237574232
#237574222
#237574092
#237574082
#237574072
#237572762
#237572352
#450×C1572
#3003157