Finite Group Databasebeta

← #34109365

Subgroups — #34109365

Order 20096.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×#19097799C211yesmaximal, characteristic, complemented
3C2×#19097797C211yesmaximal, characteristic, complemented
4#23757785C211yesmaximal, characteristic, complemented
5#23757425C214yesmaximal, complemented
6#23757408C214yesmaximal, complemented
7#23757277C211yesmaximal, characteristic, complemented
8#23757236C211yesmaximal, characteristic, complemented
9#418×C157C211yesmaximal, characteristic, complemented
10C24×D157—21no—
11C2×#14578954C2211yescharacteristic, complemented
12C22×D628C2211yescharacteristic
13C22×C22⋊D157—21no—
14C2×D4×D157—24no—
15C2×D4×D157—24no—
16C2×D4×D157—22no—
17C2×D4×D157C2214yes—
18C22×C4×D157C2211yescharacteristic
19#19097799—24no—
20#19097797—24no—
21C2×#14578943C2212yescomplemented
22C2×#14578942C2212yescomplemented
23C2×D1256C2212yescomplemented
24C2×#14578939C2212yescomplemented
25C2×#14578938C2211yescharacteristic, complemented
26#19097755—21no—
27#19097751C2212yescomplemented
28#19097746C2212yescomplemented
29#19097741—21no—
30C4×D628—22no—
31#19097691C2214yescomplemented
32#19097683C2214yescomplemented
33C22×D4×C157C2211yescharacteristic
34C2×M16×C157C2211yescharacteristic, complemented
35#117×C157C2211yescharacteristic, complemented
36#102×C157C2214yescomplemented
37C23×D157—28no—
38C23×D157—24no—
39C23×D157—21no—
40C23×D157—48no—
41C23×D157D411yescharacteristic
42#14578954—22no—
43#14578954—22no—
44C2×Dic314C2311yescharacteristic, complemented
45C2×D628—22no—
46C2×D628—21no—
47C2×D628—22no—
48C2×D628D414yescomplemented
49C2×D628C2312yes—
50C2×D628C2311yescharacteristic, complemented
51C2×C22⋊D157—28no—
52C2×C22⋊D157—24no—
53C2×C22⋊D157—21no—
54D4×D157—216no—
55D4×D157—28no—
56D4×D157—28no—
57C22×Dic157D411yescharacteristic
58C2×C4×D157—22no—
59C2×C4×D157—22no—
60C2×C4×D157—21no—
61C2×C4×D157D412yes—
62C2×C4×D157C2312yes—
63#14578943—24no—
64#14578942—24no—
65D1256—24no—
66#14578939—24no—
67#14578938—22no—
68C2×#11929434C2312yescomplemented
69C4⋊Dic157—22no—
70#14578931—22no—
71#14578931—22no—
72C24×C157—21no—
73C2×D4×C157—21no—
74C2×D4×C157—22no—
75C2×D4×C157D414yescomplemented
76C2×D4×C157C2312yes—
77C22×C4×C157C2311yescharacteristic
78M16×C157—22no—
79C2×C8×C157C2312yescomplemented
80C4⋊C4×C157C2312yescomplemented
81C22⋊C4×C157—21no—
82C42×C157—21no—
83C22×D157—22no—
84C22×D157—24no—
85C22×D157—216no—
86C22×D157—24no—
87C22×D157—22no—
88C22×D157—21no—
89C22×D157—432no—
90C22×D157—48no—
91C22×D157—44no—
92C22×D157—48no—
93C22×D157C2×D412yes—
94Dic314—21no—
95Dic314—21no—
96D628—24no—
97D628—22no—
98D628—21no—
99D628—21no—
100D628—44no—
101D628C2×D414yes—
102C22⋊D157—216no—
103C22⋊D157—42no—
104C2×Dic157—22no—
105C2×Dic157—21no—
106C2×Dic157C2×D412yes—
107C4×D157—24no—
108C4×D157—22no—
109C4×D157—22no—
110C4×D157—42no—
111#11929434—22no—
112C23×C157—24no—
113C23×C157—22no—
114C23×C157—44no—
115C23×C157C2×D411yescharacteristic
116D4×C157—24no—
117D4×C157—22no—
118D4×C157C2×D414yes—
119C2×C4×C157—22no—
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2×D412yes—
123C2×C4×C157C2411yescharacteristic
124C8×C157—22no—
125C22×C157—22no—
126C22×C157—24no—
127C22×C157—48no—
128C22×C157—44no—
129C22×C157#12012yes—
130C22×C157C22×D411yescharacteristic
131C2×D157—28no—
132C2×D157—22no—
133C2×D157—22no—
134C2×D157—24no—
135C2×D157—432no—
136C2×D157—48no—
137C2×D157—44no—
138C2×D157—41no—
139C2×D157—42no—
140C2×D157AGL(1,ℤ/8)14yescomplemented
141C4×C157—42no—
142C4×C157#12012yes—
143C4×C157C22×D411yescharacteristic
144C4×C157C22×D411yescharacteristic
145Dic157—22no—
146Dic157—41no—
147C2×C157—22no—
148C2×C157—44no—
149C2×C157C2×#12011yescharacteristic
150C2×C157C2×AGL(1,ℤ/8)12yes—
151D157—24no—
152D157—44no—
153D157—81no—
154C157#300311yescharacteristic, complemented, Sylow
155#3003—1571nomaximal, Sylow
156C23×D4—1571no—
157C2×AGL(1,ℤ/8)—1571no—
158C2×AGL(1,ℤ/8)—1571no—
159#525—1571no—
160#450—1574no—
161#447—1574no—
162#418—1571no—
163#418D15711yescharacteristic, complemented
164#407—1571no—
165C25—3141no—
166C2×#41—1571no—
167C22×D4—3144no—
168C22×D4—3144no—
169C22×D4—3142no—
170C22×D4—3141no—
171C22×D4—1574no—
172C22×D4—1571no—
173C22×D4C2×D15711yescharacteristic
174C23×C4—1571no—
175AGL(1,ℤ/8)—3144no—
176AGL(1,ℤ/8)—3144no—
177C2×SD16—1572no—
178C2×SD16—1572no—
179C2×D8—1572no—
180C2×D8—1572no—
181C2×M16—1571no—
182C2×M16C2×D15711yescharacteristic, complemented
183#123—3141no—
184C22⋊Q8—1572no—
185#121—1572no—
186#120—3141no—
187C4×D4—3142no—
188#117C2×D15711yescharacteristic, complemented
189#102—1574no—
190#102C2×D15714yescomplemented
191C22⋊C8—1574no—
192C24—21no—
193C24—3148no—
194C24—3144no—
195C24—3141no—
196C24—6288no—
197C24—1571no—
198#41—3142no—
199#41—3142no—
200C2×Q8—1571no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—3142no—
204C2×D4—31416no—
205C2×D4—3148no—
206C2×D4—3148no—
207C2×D4—3148no—
208C2×D4—3141no—
209C2×D4—3144no—
210C2×D4—3141no—
211C2×D4—3142no—
212C2×D4—1574no—
213C2×D4—1571no—
214C2×D4—1572no—
215C2×D4D62814yescomplemented
216C2×D4C22×D15712yes—
217C22×C4—3142no—
218C22×C4—3142no—
219C22×C4—3141no—
220C22×C4—1571no—
221C22×C4—1572no—
222C22×C4—1572no—
223C22×C4C22×D15711yescharacteristic
224SD16—3144no—
225SD16—3144no—
226D8—3144no—
227D8—3144no—
228M16—22no—
229M16—3142no—
230C2×C8—1572no—
231C2×C8C22×D15712yescomplemented
232C4⋊C4—3142no—
233C4⋊C4C22×D15712yescomplemented
234C22⋊C4—21no—
235C22⋊C4—3142no—
236C22⋊C4—3142no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—3144no—
242C23—31416no—
243C23—3142no—
244C23—3144no—
245C23—3142no—
246C23—3141no—
247C23—6284no—
248C23—62832no—
249C23—6288no—
250C23—6288no—
251C23—1572no—
252C23D4×D15711yescharacteristic
253Q8—3141no—
254Q8—3141no—
255D4—24no—
256D4—22no—
257D4—31416no—
258D4—3144no—
259D4—3141no—
260D4—3142no—
261D4—3141no—
262D4—6284no—
263D4—6282no—
264D4—1574no—
265D4C2×D62814yes—
266C2×C4—22no—
267C2×C4—3142no—
268C2×C4—3144no—
269C2×C4—3142no—
270C2×C4—3142no—
271C2×C4—3141no—
272C2×C4—6282no—
273C2×C4—1572no—
274C2×C4D4×D15711yescharacteristic
275C2×C4D4×D15712yes—
276C2×C4C2×D62812yes—
277C2×C4C23×D15711yescharacteristic
278C8—22no—
279C8—3142no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—3148no—
285C22—3142no—
286C22—3142no—
287C22—3144no—
288C22—6288no—
289C22—62832no—
290C22—6284no—
291C22—6281no—
292C22—6282no—
293C22—1574no—
294C22#1909774112yes—
295C22C2×D4×D15711yescharacteristic
296C4—42no—
297C4—3142no—
298C4—6281no—
299C4#1909774112yes—
300C4C2×D4×D15711yescharacteristic
301C4C22×D62811yescharacteristic
302C2—22no—
303C2—44no—
304C2—3144no—
305C2—6284no—
306C2—12561no—
307C2C2×#1909774111yescharacteristic
308C2AGL(1,ℤ/8)×D15712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#418×C157C2
socleC22×C157C22×D4
hypercenter#418D157
hypocenterC157#3003
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D157C2semidirect
2. C2×#19097799C2semidirect
3. C2×#19097797C2semidirect
4. #23757785C2semidirect
5. #23757425C2semidirect
6. #23757408C2semidirect
7. #23757277C2semidirect
8. #23757236C2semidirect
9. #418×C157C2semidirect
11. C2×#14578954C22semidirect
21. C2×#14578943C22semidirect
22. C2×#14578942C22semidirect
23. C2×D1256C22semidirect
24. C2×#14578939C22semidirect
25. C2×#14578938C22semidirect
27. #19097751C22semidirect
28. #19097746C22semidirect
31. #19097691C22semidirect
32. #19097683C22semidirect
34. C2×M16×C157C22semidirect
35. #117×C157C22semidirect
36. #102×C157C22semidirect
44. C2×Dic314C23semidirect
48. C2×D628D4semidirect
50. C2×D628C23semidirect
68. C2×#11929434C23semidirect
75. C2×D4×C157D4semidirect
79. C2×C8×C157C23semidirect
80. C4⋊C4×C157C23semidirect
140. C2×D157AGL(1,ℤ/8)semidirect
154. C157#3003semidirect
163. #418D157semidirect
182. C2×M16C2×D157semidirect
188. #117C2×D157semidirect
190. #102C2×D157semidirect
215. C2×D4D628semidirect
231. C2×C8C22×D157semidirect
233. C4⋊C4C22×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×#190977992
C2×#190977972
#237577852
#237574252
#237574082
#237572772
#237572362
#418×C1572
#3003157