Finite Group Databasebeta

← #450×D157 #34109315

Subgroups — #450×D157 #34109315

Order 20096.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 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
2C2×SD16×D157C212yesmaximal, complemented
3C22⋊Q8×D157C211yesmaximal, characteristic, complemented
4#23757426C211yesmaximal, characteristic, complemented
5#23757425C212yesmaximal, complemented
6#23757424C211yesmaximal, characteristic, complemented
7#23757423C211yesmaximal, characteristic, complemented
8#23757422C212yesmaximal, complemented
9#102×D157C212yesmaximal, complemented
10C22⋊C8×D157C211yesmaximal, characteristic, complemented
11#450×C157C211yesmaximal, characteristic, complemented
12C24×D157—21no—
13C2×Q8×D157C2211yescharacteristic, 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
22C2×#14578943C2212yescomplemented
23C2×#14578942C2212yescomplemented
24C2×#14578941C2212yescomplemented
25SD16×D157—28no—
26C2×C8×D157C2212yescomplemented
27#19097751C2211yescharacteristic, complemented
28#19097750C2211yescharacteristic, complemented
29#19097749—21no—
30#19097748—21no—
31#19097747C2211yescharacteristic, complemented
32C4⋊C4×D157—21no—
33C4⋊C4×D157C2211yescharacteristic, complemented
34C22⋊C4×D157—21no—
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
52C2×Dic314—21no—
53C2×Dic314C2311yescharacteristic, complemented
54Q8×D157—24no—
55C2×D628—24no—
56C2×D628—21no—
57C2×D628C2312yes—
58C2×C22⋊D157—28no—
59C2×C22⋊D157—22no—
60C2×C22⋊D157—22no—
61D4×D157—28no—
62D4×D157—28no—
63D4×D157—24no—
64D4×D157—24no—
65D4×D157—24no—
66D4×D157D418yes—
67C22×Dic157D411yescharacteristic
68C2×C4×D157—24no—
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157D411yescharacteristic
72C2×C4×D157D411yescharacteristic
73C2×C4×D157C2311yescharacteristic
74C2×C4×D157C2311yescharacteristic
75#14578943—24no—
76#14578942—24no—
77#14578941—24no—
78C2×#11929434C2312yescomplemented
79C8×D157—24no—
80C4⋊Dic157—21no—
81C4⋊Dic157C2311yescharacteristic, complemented
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85C22⋊Dic157—21no—
86#14578931—21no—
87#14578931—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—28no—
102C22×D157—24no—
103C22×D157—22no—
104C22×D157—28no—
105C22×D157—44no—
106C22×D157—432no—
107C22×D157—42no—
108C22×D157—42no—
109C22×D157—44no—
110C22×D157—44no—
111C22×D157—44no—
112C22×D157SD1614yescomplemented
113C22×D157C2×D411yescharacteristic
114C22×D157C2×D411yescharacteristic
115Dic314—22no—
116Dic314—42no—
117D628—24no—
118D628—22no—
119D628C2×D414yes—
120C22⋊D157—28no—
121C22⋊D157—28no—
122C2×Dic157—22no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—22no—
128C4×D157—22no—
129C4×D157—22no—
130C4×D157—42no—
131C4×D157—42no—
132C4×D157C2×D414yes—
133#11929434—22no—
134C23×C157—24no—
135C23×C157—22no—
136C23×C157—42no—
137C23×C157—42no—
138C23×C157C2×D411yescharacteristic
139Q8×C157—22no—
140D4×C157—24no—
141D4×C157—22no—
142D4×C157C2×D414yes—
143C2×C4×C157—22no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C8×C157—22no—
149C22×C157—21no—
150C22×C157—21no—
151C22×C157—24no—
152C22×C157—48no—
153C22×C157—42no—
154C22×C157—42no—
155C22×C157C2×SD1612yescomplemented
156C22×C157C22×D411yescharacteristic
157C2×D157—21no—
158C2×D157—21no—
159C2×D157—28no—
160C2×D157—24no—
161C2×D157—416no—
162C2×D157—48no—
163C2×D157—42no—
164C2×D157—42no—
165C2×D157—416no—
166C2×D157#12012yes—
167C2×D157C2×SD1612yes—
168C2×D157C2×SD1612yescomplemented
169C2×D157AGL(1,ℤ/8)12yes—
170C4×C157—21no—
171C4×C157—41no—
172C4×C157—41no—
173C4×C157C22×D412yes—
174Dic157—21no—
175Dic157—41no—
176Dic157—41no—
177Dic157#12012yes—
178C2×C157—22no—
179C2×C157—44no—
180C2×C157C2×#12011yescharacteristic
181C2×C157C22×SD1611yescharacteristic
182C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
183D157—22no—
184D157—44no—
185D157#45014yesdirect product factor
186C157C2×#45011yescharacteristic, complemented, Sylow
187C2×#450—1571nomaximal, Sylow
188C23×D4—1571no—
189C22×SD16—1572no—
190C2×C22⋊Q8—1571no—
191#450—1571no—
192#450—1572no—
193#450—1571no—
194#450—1572no—
195#450—1571no—
196#450D15711yescharacteristic, direct product factor
197C2×#102—1572no—
198C2×C22⋊C8—1571no—
199C25—3141no—
200C22×Q8—1571no—
201C22×D4—3148no—
202C22×D4—3141no—
203C22×D4—3141no—
204C22×D4—3141no—
205C22×D4—1572no—
206C22×D4—1572no—
207C22×D4—1571no—
208C22×D4C2×D15711yescharacteristic
209C23×C4—1571no—
210C2×SD16—3148no—
211C2×SD16—1572no—
212C2×SD16—1572no—
213C2×SD16—1572no—
214C2×SD16C2×D15712yescomplemented
215C22×C8—1572no—
216C22⋊Q8—3141no—
217C22⋊Q8—3141no—
218C22⋊Q8—1571no—
219C22⋊Q8—1571no—
220C22⋊Q8—1571no—
221C22⋊Q8C2×D15711yescharacteristic, complemented
222C2×C4⋊C4—3141no—
223C2×C4⋊C4—1571no—
224C2×C22⋊C4—3141no—
225#102—1572no—
226#102—1572no—
227#102—1572no—
228#102C2×D15712yescomplemented
229C22⋊C8—1571no—
230C22⋊C8—1572no—
231C22⋊C8C2×D15711yescharacteristic, complemented
232C24—21no—
233C24—3148no—
234C24—3142no—
235C24—3142no—
236C24—3141no—
237C24—6284no—
238C24—6284no—
239C24—1571no—
240C2×Q8—3141no—
241C2×Q8—3144no—
242C2×Q8—1571no—
243C2×Q8C22×D15711yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—3144no—
247C2×D4—3148no—
248C2×D4—3148no—
249C2×D4—3144no—
250C2×D4—3148no—
251C2×D4—3144no—
252C2×D4—3144no—
253C2×D4—3142no—
254C2×D4—3142no—
255C2×D4—3141no—
256C2×D4—1578no—
257C2×D4—1572no—
258C2×D4C22×D15712yes—
259C22×C4—3144no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4—1571no—
263C22×C4—1571no—
264C22×C4—1571no—
265C22×C4—1571no—
266C22×C4—1571no—
267C22×C4C22×D15711yescharacteristic
268SD16—24no—
269SD16—3144no—
270SD16—3144no—
271SD16—3144no—
272C2×C8—3144no—
273C2×C8—1572no—
274C2×C8C22×D15712yescomplemented
275C4⋊C4—21no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—3141no—
279C4⋊C4—3141no—
280C4⋊C4—1571no—
281C4⋊C4C22×D15711yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—3141no—
284C22⋊C4—3141no—
285C22⋊C4—3141no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—3142no—
291C23—3142no—
292C23—3144no—
293C23—3148no—
294C23—3148no—
295C23—3142no—
296C23—6282no—
297C23—62832no—
298C23—6284no—
299C23—6284no—
300C23—6284no—
301C23—6284no—
302C23—6282no—
303C23—1574no—
304C23—1571no—
305C23—1571no—
306C23D4×D15711yescharacteristic
307Q8—22no—
308Q8—3142no—
309Q8—6282no—
310D4—24no—
311D4—22no—
312D4—3144no—
313D4—3148no—
314D4—3148no—
315D4—3142no—
316D4—1574no—
317D4D4×D15714yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—3142no—
322C2×C4—3142no—
323C2×C4—3142no—
324C2×C4—3142no—
325C2×C4—3141no—
326C2×C4—3141no—
327C2×C4—6282no—
328C2×C4—6282no—
329C2×C4—1571no—
330C2×C4—1571no—
331C2×C4—1574no—
332C2×C4D4×D15711yescharacteristic
333C2×C4C23×D15711yescharacteristic
334C8—22no—
335C8—3142no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—3148no—
343C22—3141no—
344C22—3141no—
345C22—3144no—
346C22—6288no—
347C22—62816no—
348C22—62816no—
349C22—6282no—
350C22—6282no—
351C22—1572no—
352C22—1572no—
353C22—1572no—
354C22—1572no—
355C22SD16×D15712yescomplemented
356C22C2×D4×D15711yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—3141no—
361C4—6281no—
362C4—6281no—
363C4—1572no—
364C4C2×D4×D15712yes—
365C2—22no—
366C2—44no—
367C2—3142no—
368C2—6284no—
369C2—1574no—
370C2#120×D15711yescharacteristic
371C2C2×SD16×D15711yescharacteristic
372C2AGL(1,ℤ/8)×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radicalC22×D4×D157C2
hypercenter#450D157
hypocenterC157C2×#450
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
2. C2×SD16×D157C2semidirect
3. C22⋊Q8×D157C2semidirect
4. #23757426C2semidirect
5. #23757425C2semidirect
6. #23757424C2semidirect
7. #23757423C2semidirect
8. #23757422C2semidirect
9. #102×D157C2semidirect
10. C22⋊C8×D157C2semidirect
11. #450×C157C2semidirect
13. C2×Q8×D157C22semidirect
22. C2×#14578943C22semidirect
23. C2×#14578942C22semidirect
24. C2×#14578941C22semidirect
26. C2×C8×D157C22semidirect
27. #19097751C22semidirect
28. #19097750C22semidirect
31. #19097747C22semidirect
33. C4⋊C4×D157C22semidirect
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
53. C2×Dic314C23semidirect
78. C2×#11929434C23semidirect
81. C4⋊Dic157C23semidirect
89. C2×Q8×C157C23semidirect
95. C2×C8×C157C23semidirect
97. C4⋊C4×C157C23semidirect
112. C22×D157SD16semidirect
155. C22×C157C2×SD16semidirect
168. C2×D157C2×SD16semidirect
185. D157#450direct (N × H)
186. C157C2×#450semidirect
196. #450D157direct (N × H)
214. C2×SD16C2×D157semidirect
221. C22⋊Q8C2×D157semidirect
228. #102C2×D157semidirect
231. C22⋊C8C2×D157semidirect
243. C2×Q8C22×D157semidirect
274. C2×C8C22×D157semidirect
281. C4⋊C4C22×D157semidirect
355. C22SD16×D157semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1572
C2×SD16×D1572
C22⋊Q8×D1572
#237574262
#237574252
#237574242
#237574232
#237574222
#102×D1572
C22⋊C8×D1572
#450×C1572
C2×#450157