Finite Group Databasebeta

← #34109423

Subgroups — #34109423

Order 20096.

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

normal subgroups only

This group has 732 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×#19097819C211yesmaximal, characteristic, complemented
2C2×#19097799C211yesmaximal, characteristic, complemented
3C22×#14578941C211yesmaximal, characteristic, complemented
4#23757842C211yesmaximal, characteristic, complemented
5#23757421C214yesmaximal, complemented
6#23757413C214yesmaximal, complemented
7#23757267C211yesmaximal, characteristic, complemented
8#23757241C211yesmaximal, characteristic, complemented
9#419×C157C211yesmaximal, characteristic, complemented
10#19097819—24no—
11#19097819—24no—
12C2×#14578955—21no—
13C2×#14578955C2212yescomplemented
14C2×#14578954—22no—
15C2×#14578954C2211yescharacteristic
16C2×#14578954C2211yescharacteristic
17C2×Q8×D157—21no—
18C2×Q8×D157C2212yescomplemented
19C22×D628C2211yescharacteristic
20#19097799—24no—
21C2×#14578943C2212yes—
22C2×#14578941—24no—
23C2×#14578941C2214yes—
24C2×D1256C2212yes—
25C22×#11929434C2211yescharacteristic
26#19097768—21no—
27#19097759—21no—
28#19097746C2214yes—
29#19097741—22no—
30#19097694C2212yescomplemented
31#19097691C2212yescomplemented
32#19097683C2214yescomplemented
33C22×Q8×C157C2211yescharacteristic
34C2×M16×C157C2211yescharacteristic
35#117×C157C2211yescharacteristic
36#103×C157C2214yescomplemented
37C23×D157—21no—
38#14578955—28no—
39#14578955—24no—
40#14578955—24no—
41#14578954—216no—
42#14578954—22no—
43#14578954—22no—
44#14578954—22no—
45#14578954D414yescomplemented
46C2×Dic314—22no—
47C2×Dic314D411yescharacteristic
48C2×Dic314C2311yescharacteristic
49Q8×D157—28no—
50Q8×D157—24no—
51Q8×D157—24no—
52C2×D628—22no—
53C2×D628—24no—
54C2×D628—22no—
55C2×D628—22no—
56C2×D628—21no—
57C2×D628D411yescharacteristic
58C2×D628C2312yes—
59C2×D628C2311yescharacteristic
60C2×C22⋊D157—21no—
61C2×C22⋊D157D412yes—
62C2×C4×D157—22no—
63C2×C4×D157—22no—
64C2×C4×D157—21no—
65C2×C4×D157C2312yes—
66#14578943—24no—
67#14578941—28no—
68D1256—24no—
69C2×#11929434—22no—
70C2×#11929434C2312yes—
71#14578931—24no—
72C2×Q8×C157—21no—
73C2×Q8×C157—22no—
74C2×Q8×C157D414yescomplemented
75C2×Q8×C157C2312yes—
76C22×C4×C157—21no—
77C22×C4×C157C2311yescharacteristic
78M16×C157—22no—
79C2×C8×C157C2312yes—
80C4⋊C4×C157C2312yes—
81C22⋊C4×C157—21no—
82C42×C157—21no—
83C22×D157—21no—
84C22×D157—22no—
85C22×D157—42no—
86C22×D157—82no—
87C22×D157C2×D412yes—
88Dic314—28no—
89Dic314—22no—
90Dic314—21no—
91Dic314C2×D412yes—
92D628—28no—
93D628—22no—
94D628—21no—
95D628—22no—
96D628—22no—
97D628—42no—
98D628—84no—
99D628C2×D412yes—
100C22⋊D157—24no—
101C22⋊D157—24no—
102C2×Dic157—21no—
103C2×Dic157C2×D412yes—
104C4×D157—28no—
105C4×D157—28no—
106C4×D157—24no—
107C4×D157—22no—
108C4×D157—22no—
109#11929434—22no—
110C23×C157C2×D411yescharacteristic
111Q8×C157—24no—
112Q8×C157—22no—
113Q8×C157C2×D414yes—
114C2×C4×C157—24no—
115C2×C4×C157—22no—
116C2×C4×C157—22no—
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2×D412yes—
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2411yescharacteristic
121C8×C157—22no—
122C22×C157—22no—
123C22×C157#12012yes—
124C22×C157C22×D411yescharacteristic
125C2×D157—22no—
126C2×D157—22no—
127C2×D157—24no—
128C2×D157—42no—
129C2×D157—42no—
130C2×D157—44no—
131C2×D157—84no—
132C4×C157—24no—
133C4×C157—42no—
134C4×C157#12012yes—
135C4×C157C22×D411yescharacteristic
136C4×C157C22×D411yescharacteristic
137Dic157—22no—
138Dic157—22no—
139C2×C157—22no—
140C2×C157C2×#12011yescharacteristic
141C2×C157#57712yes—
142D157—42no—
143D157—42no—
144D157—82no—
145C157#301011yescharacteristic, complemented, Sylow
146#3010—1571nomaximal, Sylow
147C2×2−1+4—1571no—
148C2×AGL(1,ℤ/8)—1571no—
149C22×SD16—1571no—
150#535—1571no—
151#449—1574no—
152#448—1574no—
153#419D15711yescharacteristic, complemented
154#416—1571no—
155#408—1571no—
1562−1+4—3144no—
1572−1+4—3144no—
158C2×#41—3142no—
159C2×#41—3141no—
160C2×#41—1572no—
161C2×#41—1571no—
162C2×#41—1571no—
163C22×Q8—3141no—
164C22×Q8—1572no—
165C22×Q8C2×D15711yescharacteristic
166C22×D4—1571no—
167AGL(1,ℤ/8)—3144no—
168C2×SD16—3144no—
169C2×SD16—1574no—
170C2×SD16—1572no—
171C2×D8—1572no—
172C2×M16C2×D15711yescharacteristic
173C22×C8—1571no—
174#127—3141no—
175#124—3141no—
176#121—1574no—
177#120—3142no—
178#117C2×D15711yescharacteristic
179#103—1572no—
180#103C2×D15714yescomplemented
181#102—1572no—
182C22⋊C8—1574no—
183C24—3141no—
184#41—31416no—
185#41—3148no—
186#41—3144no—
187#41—3144no—
188#41—3142no—
189#41—3142no—
190#41—3142no—
191#41—1574no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—3142no—
195C2×Q8—3148no—
196C2×Q8—3144no—
197C2×Q8—3144no—
198C2×Q8—1571no—
199C2×Q8—1571no—
200C2×Q8D62814yescomplemented
201C2×Q8C22×D15712yes—
202C2×D4—3142no—
203C2×D4—3144no—
204C2×D4—3141no—
205C2×D4—3142no—
206C2×D4—3141no—
207C2×D4—3142no—
208C2×D4—1572no—
209C2×D4—1571no—
210C2×D4—1572no—
211C2×D4—1571no—
212C22×C4—21no—
213C22×C4—3142no—
214C22×C4—3142no—
215C22×C4—3141no—
216C22×C4—1572no—
217C22×C4C22×D15711yescharacteristic
218SD16—3144no—
219SD16—3148no—
220D8—3144no—
221M16—22no—
222C2×C8—3142no—
223C2×C8—1572no—
224C2×C8C22×D15712yes—
225C4⋊C4C22×D15712yes—
226C22⋊C4—21no—
227C22⋊C4—3144no—
228C42—21no—
229C23—3141no—
230C23—3142no—
231C23—6282no—
232C23—1572no—
233C23—12562no—
234C23D4×D15711yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—3148no—
238Q8—3142no—
239Q8—3141no—
240Q8—1572no—
241Q8C2×D62814yes—
242D4—3148no—
243D4—3144no—
244D4—3144no—
245D4—3142no—
246D4—3142no—
247D4—3141no—
248D4—3142no—
249D4—6282no—
250D4—1572no—
251D4—12564no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3148no—
256C2×C4—3148no—
257C2×C4—3141no—
258C2×C4—3144no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—1572no—
262C2×C4D4×D15711yescharacteristic
263C2×C4D4×D15712yes—
264C2×C4C2×D62812yes—
265C2×C4C23×D15711yescharacteristic
266C8—22no—
267C8—3142no—
268C22—22no—
269C22—3144no—
270C22—3142no—
271C22—3142no—
272C22—6282no—
273C22—6284no—
274C22—6282no—
275C22—12564no—
276C22#1909774112yes—
277C22C2×D4×D15711yescharacteristic
278C4—24no—
279C4—42no—
280C4—3142no—
281C4—3142no—
282C4#1909774112yes—
283C4C2×D4×D15711yescharacteristic
284C4C22×D62811yescharacteristic
285C2—22no—
286C2—6282no—
287C2—6282no—
288C2—12562no—
289C2C2×#1909774111yescharacteristic
290C2#2375808212yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#419×C157C2
socleC22×C157C22×D4
hypercenter#419D157
hypocenterC157#3010
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#19097819C2semidirect
2. C2×#19097799C2semidirect
3. C22×#14578941C2semidirect
4. #23757842C2semidirect
5. #23757421C2semidirect
6. #23757413C2semidirect
7. #23757267C2semidirect
8. #23757241C2semidirect
9. #419×C157C2semidirect
13. C2×#14578955C22semidirect
18. C2×Q8×D157C22semidirect
30. #19097694C22semidirect
31. #19097691C22semidirect
32. #19097683C22semidirect
36. #103×C157C22semidirect
45. #14578954D4semidirect
74. C2×Q8×C157D4semidirect
145. C157#3010semidirect
153. #419D157semidirect
180. #103C2×D157semidirect
200. C2×Q8D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190978192
C2×#190977992
C22×#145789412
#237578422
#237574212
#237574132
#237572672
#237572412
#419×C1572
#3010157