Finite Group Databasebeta

← #34109424

Subgroups — #34109424

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×#19097818C211yesmaximal, characteristic, complemented
2C2×#19097799C211yesmaximal, characteristic, complemented
3C22×#14578943C211yesmaximal, characteristic, complemented
4#23757837C211yesmaximal, characteristic, complemented
5#23757420C214yesmaximal, complemented
6#23757414C214yesmaximal, complemented
7#23757283C211yesmaximal, characteristic, complemented
8#23757270C211yesmaximal, characteristic, complemented
9#408×C157C211yesmaximal, characteristic, complemented
10#19097818—24no—
11#19097818—24no—
12C2×#14578954C2211yescharacteristic
13C2×#14578954C2212yescomplemented
14C2×#14578953—21no—
15C2×#14578953—22no—
16C22×Dic314C2211yescharacteristic
17C22×Dic314C2212yescomplemented
18C2×Q8×D157—21no—
19C22×D628C2211yescharacteristic
20#19097799—24no—
21C2×#14578943—24no—
22C2×#14578943C2212yes—
23C2×#14578943C2214yes—
24C2×D1256C2212yes—
25#19097767—21no—
26#19097760—21no—
27#19097745C2214yes—
28#19097739—22no—
29#19097730C2211yescharacteristic
30#19097696C2212yescomplemented
31#19097696C2214yescomplemented
32#19097693C2212yescomplemented
33C2×#41×C157C2211yescharacteristic
34C2×M16×C157C2211yescharacteristic
35C22×C8×C157C2211yescharacteristic
36C22⋊C8×C157C2214yescomplemented
37C23×D157—21no—
38#14578954—24no—
39#14578954—24no—
40#14578954—22no—
41#14578954D414yescomplemented
42#14578953—216no—
43#14578953—28no—
44C2×Dic314—24no—
45C2×Dic314—24no—
46C2×Dic314—21no—
47C2×Dic314—22no—
48C2×Dic314—22no—
49C2×Dic314D414yescomplemented
50C2×Dic314C2312yes—
51C2×Dic314C2311yescharacteristic
52Q8×D157—28no—
53C2×D628—22no—
54C2×D628—22no—
55C2×D628—21no—
56C2×D628C2312yes—
57C2×D628C2311yescharacteristic
58C2×C22⋊D157—21no—
59C2×C22⋊D157—22no—
60C2×C22⋊D157—24no—
61C22×Dic157—21no—
62C22×Dic157—22no—
63C2×C4×D157—21no—
64C2×C4×D157—22no—
65#14578943—28no—
66#14578943—24no—
67D1256—24no—
68C4⋊Dic157C2312yes—
69C22⋊Dic157—21no—
70#14578931—24no—
71C4×Dic157—21no—
72#41×C157—22no—
73#41×C157—22no—
74C2×Q8×C157D411yescharacteristic
75C2×D4×C157D412yes—
76C2×D4×C157D411yescharacteristic
77C22×C4×C157C2312yes—
78C22×C4×C157C2311yescharacteristic
79M16×C157—22no—
80C2×C8×C157—22no—
81C2×C8×C157C2312yes—
82C2×C8×C157C2312yes—
83C22×D157—21no—
84C22×D157—22no—
85C22×D157—42no—
86C22×D157—82no—
87Dic314—28no—
88Dic314—24no—
89Dic314—21no—
90Dic314—22no—
91Dic314C2×D412yes—
92Dic314C2×D414yes—
93D628—21no—
94D628—22no—
95D628—22no—
96D628—42no—
97D628C2×D412yes—
98C22⋊D157—28no—
99C22⋊D157—24no—
100C22⋊D157—84no—
101C2×Dic157—28no—
102C2×Dic157—24no—
103C2×Dic157—21no—
104C2×Dic157—22no—
105C2×Dic157—22no—
106C4×D157—28no—
107C4×D157—24no—
108C23×C157C2×D412yes—
109C23×C157C2×D411yescharacteristic
110Q8×C157—22no—
111D4×C157—24no—
112D4×C157—22no—
113C2×C4×C157—22no—
114C2×C4×C157—22no—
115C2×C4×C157C2×D412yes—
116C2×C4×C157C2×D412yes—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157C2411yescharacteristic
120C8×C157—22no—
121C8×C157—22no—
122C22×C157—22no—
123C22×C157—22no—
124C22×C157—24no—
125C22×C157#12012yes—
126C22×C157C22×D411yescharacteristic
127C2×D157—22no—
128C2×D157—42no—
129C2×D157—42no—
130C2×D157—44no—
131C2×D157—84no—
132C4×C157—22no—
133C4×C157#12012yes—
134C4×C157C22×D411yescharacteristic
135C4×C157C22×D411yescharacteristic
136Dic157—24no—
137Dic157—22no—
138Dic157—42no—
139C2×C157—22no—
140C2×C157—42no—
141C2×C157C2×#12011yescharacteristic
142C2×C157#57712yes—
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#419—1571no—
154#416—1571no—
155#408D15711yescharacteristic, complemented
1562−1+4—3144no—
1572−1+4—3144no—
158C2×#41—3142no—
159C2×#41—3141no—
160C2×#41—1572no—
161C2×#41—1571no—
162C2×#41C2×D15711yescharacteristic
163C22×Q8—3141no—
164C22×Q8—1572no—
165C22×Q8—1571no—
166C22×D4—1571no—
167AGL(1,ℤ/8)—3144no—
168C2×SD16—3144no—
169C2×SD16—1574no—
170C2×SD16—1572no—
171C2×D8—1572no—
172C2×M16C2×D15711yescharacteristic
173C22×C8C2×D15711yescharacteristic
174#127—3141no—
175#124—3141no—
176#121—1574no—
177#120—3142no—
178#117—1571no—
179#103—1572no—
180#103—1574no—
181#102—1572no—
182C22⋊C8C2×D15714yescomplemented
183C24—3141no—
184#41—22no—
185#41—22no—
186#41—3144no—
187#41—3144no—
188#41—31416no—
189#41—3148no—
190#41—3142no—
191#41—1574no—
192C2×Q8—3144no—
193C2×Q8—3144no—
194C2×Q8—3148no—
195C2×Q8—3142no—
196C2×Q8—3142no—
197C2×Q8—3141no—
198C2×Q8—1574no—
199C2×Q8—1571no—
200C2×Q8—1572no—
201C2×Q8D62811yescharacteristic
202C2×D4—3142no—
203C2×D4—3141no—
204C2×D4—3144no—
205C2×D4—3142no—
206C2×D4—3142no—
207C2×D4—3141no—
208C2×D4—1572no—
209C2×D4—1571no—
210C2×D4D62812yes—
211C2×D4D62811yescharacteristic
212C22×C4—3142no—
213C22×C4—3141no—
214C22×C4—3142no—
215C22×C4—3141no—
216C22×C4C22×D15712yes—
217C22×C4C22×D15711yescharacteristic
218SD16—3148no—
219SD16—3144no—
220D8—3144no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D15712yes—
224C2×C8C22×D15712yes—
225C4⋊C4—1572no—
226C22⋊C4—3141no—
227C22⋊C4—3144no—
228C42—3141no—
229C23—3141no—
230C23—3142no—
231C23—6282no—
232C23—12562no—
233C23C2×D62812yes—
234C23C2×D62811yescharacteristic
235Q8—22no—
236Q8—3148no—
237Q8—3144no—
238Q8—3141no—
239Q8—3142no—
240Q8—1572no—
241Q8—1574no—
242D4—24no—
243D4—22no—
244D4—3148no—
245D4—3144no—
246D4—3141no—
247D4—3142no—
248D4—3142no—
249D4—6282no—
250D4—1572no—
251D4—12564no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—3148no—
255C2×C4—3148no—
256C2×C4—3144no—
257C2×C4—3142no—
258C2×C4—3141no—
259C2×C4—3144no—
260C2×C4—3142no—
261C2×C4D4×D15712yes—
262C2×C4D4×D15712yes—
263C2×C4C2×D62812yes—
264C2×C4C2×D62811yescharacteristic
265C2×C4C23×D15711yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—3142no—
272C22—6282no—
273C22—6284no—
274C22—6282no—
275C22—12564no—
276C22#1909774112yes—
277C22C22×D62811yescharacteristic
278C4—22no—
279C4—3144no—
280C4—3142no—
281C4—6282no—
282C4#1909774112yes—
283C4C2×D4×D15711yescharacteristic
284C4C2×D4×D15711yescharacteristic
285C2—22no—
286C2—42no—
287C2—6282no—
288C2—12562no—
289C2C2×#1909774111yescharacteristic
290C2#2375808512yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#408×C157C2
socleC22×C157C22×D4
hypercenter#408D157
hypocenterC157#3010
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#19097818C2semidirect
2. C2×#19097799C2semidirect
3. C22×#14578943C2semidirect
4. #23757837C2semidirect
5. #23757420C2semidirect
6. #23757414C2semidirect
7. #23757283C2semidirect
8. #23757270C2semidirect
9. #408×C157C2semidirect
13. C2×#14578954C22semidirect
17. C22×Dic314C22semidirect
30. #19097696C22semidirect
31. #19097696C22semidirect
32. #19097693C22semidirect
36. C22⋊C8×C157C22semidirect
41. #14578954D4semidirect
49. C2×Dic314D4semidirect
145. C157#3010semidirect
155. #408D157semidirect
182. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978182
C2×#190977992
C22×#145789432
#237578372
#237574202
#237574142
#237572832
#237572702
#408×C1572
#3010157