Finite Group Databasebeta

← #34109816

Subgroups — #34109816

Order 20096.

504 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19097799C211yesmaximal, characteristic, complemented
2#23757851C211yesmaximal, characteristic, complemented
3#23757837C211yesmaximal, characteristic, complemented
4#23757526C212yesmaximal, complemented
5#23757519C212yesmaximal, complemented
6#23757509C212yesmaximal, complemented
7#23757502C212yesmaximal, complemented
8#23757315C211yesmaximal, characteristic, complemented
9#23757270C211yesmaximal, characteristic, complemented
10C2×#19097693C211yesmaximal, characteristic, complemented
11C2×#131×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×D628C2211yescharacteristic
14#19097799—24no—
15C2×#14578943C2212yes—
16C2×D1256C2212yes—
17#19097767—21no—
18#19097760—21no—
19#19097750C2212yescomplemented
20#19097747—21no—
21#19097745C2212yes—
22#19097745C2212yes—
23#19097745C2212yescomplemented
24#19097743—22no—
25#19097742—21no—
26#19097739—22no—
27#19097730C2211yescharacteristic
28C2×C4⋊Dic157C2211yescharacteristic
29#19097705C2212yescomplemented
30#19097703C2212yescomplemented
31#19097696C2211yescharacteristic, complemented
32#19097696C2211yescharacteristic, complemented
33#19097693C2212yes—
34#19097693C2212yes—
35#19097693C2211yescharacteristic, complemented
36#19097693C2211yescharacteristic, complemented
37C2×#41×C157C2211yescharacteristic
38#131×C157—24no—
39C2×M16×C157C2211yescharacteristic
40C2×M16×C157C2212yescomplemented
41C22×C8×C157C2211yescharacteristic
42C22×C8×C157C2212yescomplemented
43C23×D157—21no—
44#14578954—22no—
45#14578954—22no—
46C2×Dic314C2311yescharacteristic
47C2×D628—22no—
48C2×D628—21no—
49C2×D628—22no—
50C2×D628C2312yes—
51C2×D628C2311yescharacteristic
52C2×C22⋊D157—21no—
53C2×C22⋊D157—22no—
54C2×C22⋊D157—22no—
55C2×C22⋊D157—22no—
56C22×Dic157—21no—
57C2×C4×D157—21no—
58#14578943—24no—
59D1256—24no—
60C4⋊Dic157—21no—
61C4⋊Dic157C2312yes—
62C4⋊Dic157C2311yescharacteristic
63C4⋊Dic157C2311yescharacteristic
64#14578933—22no—
65C22⋊Dic157—21no—
66C22⋊Dic157—22no—
67#14578931—22no—
68#14578931—22no—
69#14578931—22no—
70C4×Dic157—21no—
71#41×C157—22no—
72#41×C157—22no—
73C2×Q8×C157D411yescharacteristic
74C2×D4×C157D412yes—
75C2×D4×C157D411yescharacteristic
76C22×C4×C157C2312yes—
77C22×C4×C157C2311yescharacteristic
78M16×C157—24no—
79M16×C157D414yescomplemented
80C2×C8×C157—22no—
81C2×C8×C157—24no—
82C2×C8×C157C2311yescharacteristic
83C2×C8×C157C2311yescharacteristic
84C2×C8×C157C2312yes—
85C22×D157—21no—
86C22×D157—22no—
87C22×D157—42no—
88C22×D157—81no—
89C22×D157—81no—
90Dic314—21no—
91Dic314—21no—
92D628—22no—
93D628—21no—
94D628—22no—
95D628—21no—
96D628—42no—
97C22⋊D157—42no—
98C22⋊D157—82no—
99C22⋊D157—82no—
100C22⋊D157—82no—
101C2×Dic157—21no—
102C2×Dic157—22no—
103C2×Dic157—21no—
104C2×Dic157—21no—
105C2×Dic157—81no—
106C4×D157—42no—
107C23×C157C2×D412yes—
108C23×C157C2×D411yescharacteristic
109Q8×C157—22no—
110D4×C157—24no—
111D4×C157—22no—
112C2×C4×C157—22no—
113C2×C4×C157—22no—
114C2×C4×C157C2×D412yes—
115C2×C4×C157C2×D412yes—
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157#4112yes—
118C2×C4×C157C2411yescharacteristic
119C8×C157—21no—
120C8×C157—21no—
121C8×C157C2×D414yes—
122C22×C157—21no—
123C22×C157—21no—
124C22×C157—22no—
125C22×C157—22no—
126C22×C157—22no—
127C22×C157#12112yes—
128C22×C157C22×D411yescharacteristic
129C2×D157—41no—
130C2×D157—42no—
131C2×D157—41no—
132C2×D157—42no—
133C2×D157—42no—
134C2×D157—41no—
135C2×D157—84no—
136C4×C157—22no—
137C4×C157#12112yes—
138C4×C157C22×D411yescharacteristic
139C4×C157C2×#4111yescharacteristic
140Dic157—41no—
141Dic157—42no—
142Dic157—41no—
143Dic157—41no—
144C2×C157—22no—
145C2×C157—42no—
146C2×C157C2×#12111yescharacteristic
147C2×C157#57611yescharacteristic
148C2×C157#57711yescharacteristic
149D157—81no—
150D157—82no—
151C157#306211yescharacteristic, complemented, Sylow
152#3062—1571nomaximal, Sylow
153C2×AGL(1,ℤ/8)—1571no—
154C2×#131D15711yescharacteristic, complemented
155#537—1571no—
156#535—1571no—
157#469—1572no—
158#468—1572no—
159#466—1572no—
160#465—1572no—
161#428—1571no—
162#416—1571no—
163C2×#102—1571no—
164C2×#41—1571no—
165C2×#41C2×D15711yescharacteristic
166C22×D4—1571no—
167AGL(1,ℤ/8)—3144no—
168C2×SD16—1572no—
169C2×D8—1572no—
170#131—24no—
171C2×M16C2×D15711yescharacteristic
172C2×M16C2×D15712yescomplemented
173C22×C8C2×D15711yescharacteristic
174C22×C8C2×D15712yescomplemented
175#127—3141no—
176#124—3141no—
177C22⋊Q8—3141no—
178C22⋊Q8—1572no—
179#121—3141no—
180#121—3142no—
181#121—1572no—
182#121—1572no—
183#121—1572no—
184#120—3142no—
185#117—1571no—
186C2×C4⋊C4—1571no—
187#107—1572no—
188#106—1572no—
189#103—1571no—
190#103—1571no—
191#102—1572no—
192#102—1572no—
193#102—1571no—
194#102—1571no—
195C24—3141no—
196#41—22no—
197#41—22no—
198#41—3142no—
199#41—3142no—
200C2×Q8—1571no—
201C2×Q8D62811yescharacteristic
202C2×D4—3141no—
203C2×D4—3142no—
204C2×D4—3142no—
205C2×D4—3142no—
206C2×D4—3142no—
207C2×D4—3141no—
208C2×D4—3142no—
209C2×D4—1571no—
210C2×D4—1572no—
211C2×D4D62812yes—
212C2×D4D62811yescharacteristic
213C22×C4—3141no—
214C22×C4—3141no—
215C22×C4C22×D15712yes—
216C22×C4C22×D15711yescharacteristic
217SD16—3144no—
218D8—3144no—
219M16—24no—
220M16C22⋊D15714yescomplemented
221C2×C8—22no—
222C2×C8—24no—
223C2×C8C22×D15711yescharacteristic
224C2×C8C22×D15711yescharacteristic
225C2×C8C22×D15712yes—
226C4⋊C4—3141no—
227C4⋊C4—3142no—
228C4⋊C4—1572no—
229C4⋊C4—1571no—
230C4⋊C4—1571no—
231C22⋊C4—3141no—
232C22⋊C4—3142no—
233C22⋊C4—3142no—
234C22⋊C4—3142no—
235C22⋊C4—3142no—
236C42—3141no—
237C23—3141no—
238C23—3142no—
239C23—6282no—
240C23—12561no—
241C23—12561no—
242C23C2×D62812yes—
243C23C2×D62811yescharacteristic
244Q8—22no—
245Q8—3141no—
246Q8—3141no—
247D4—24no—
248D4—22no—
249D4—3141no—
250D4—3142no—
251D4—3142no—
252D4—3141no—
253D4—6282no—
254D4—6282no—
255D4—12562no—
256D4—12562no—
257D4—12562no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—3141no—
261C2×C4—3141no—
262C2×C4—3142no—
263C2×C4—3141no—
264C2×C4—6282no—
265C2×C4—12561no—
266C2×C4C2×C22⋊D15712yes—
267C2×C4C2×D62812yes—
268C2×C4C2×D62811yescharacteristic
269C2×C4#1457895412yes—
270C2×C4C23×D15711yescharacteristic
271C8—21no—
272C8—21no—
273C8C2×C22⋊D15714yes—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—6282no—
280C22—6281no—
281C22—6281no—
282C22—6282no—
283C22—6281no—
284C22—6282no—
285C22—12564no—
286C22#1909774512yes—
287C22C22×D62811yescharacteristic
288C4—22no—
289C4—6281no—
290C4—6281no—
291C4—6282no—
292C4—6281no—
293C4#1909774512yes—
294C4C22×C22⋊D15711yescharacteristic
295C4C2×#1457895411yescharacteristic
296C2—22no—
297C2—42no—
298C2—12562no—
299C2—12561no—
300C2C2×#1909774511yescharacteristic
301C2#2375807811yescharacteristic
302C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#131×C157C2
socleC22×C157C22×D4
hypercenterC2×#131D157
hypocenterC157#3062
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. #23757851C2semidirect
3. #23757837C2semidirect
4. #23757526C2semidirect
5. #23757519C2semidirect
6. #23757509C2semidirect
7. #23757502C2semidirect
8. #23757315C2semidirect
9. #23757270C2semidirect
10. C2×#19097693C2semidirect
11. C2×#131×C157C2semidirect
19. #19097750C22semidirect
23. #19097745C22semidirect
29. #19097705C22semidirect
30. #19097703C22semidirect
31. #19097696C22semidirect
32. #19097696C22semidirect
35. #19097693C22semidirect
36. #19097693C22semidirect
40. C2×M16×C157C22semidirect
42. C22×C8×C157C22semidirect
79. M16×C157D4semidirect
151. C157#3062semidirect
154. C2×#131D157semidirect
172. C2×M16C2×D157semidirect
174. C22×C8C2×D157semidirect
220. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
#237578512
#237578372
#237575262
#237575192
#237575092
#237575022
#237573152
#237572702
C2×#190976932
C2×#131×C1572
#3062157