Finite Group Databasebeta

← #34109381

Subgroups — #34109381

Order 20096.

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

normal subgroups only

This group has 758 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
1C22×#14578954C211yesmaximal, characteristic, complemented
2C2×#19097804C211yesmaximal, characteristic, complemented
3C2×#19097799C211yesmaximal, characteristic, complemented
4#23757784C211yesmaximal, characteristic, complemented
5#23757437C214yesmaximal, complemented
6#23757420C214yesmaximal, complemented
7#23757283C211yesmaximal, characteristic, complemented
8#23757279C211yesmaximal, characteristic, complemented
9#407×C157C211yesmaximal, characteristic, complemented
10C2×#14578954—22no—
11C2×#14578954—24no—
12C2×#14578954—24no—
13C2×#14578954C2214yes—
14C2×#14578954C2211yescharacteristic
15C22×Dic314C2211yescharacteristic
16C22×D628C2211yescharacteristic
17C22×C22⋊D157—21no—
18C22×C4×D157—21no—
19#19097804—24no—
20#19097799—24no—
21C2×Dic628C2212yescomplemented
22C2×#14578943C2212yescomplemented
23C2×#14578943C2212yescomplemented
24C2×D1256C2212yescomplemented
25#19097754—21no—
26#19097750C2212yescomplemented
27#19097745C2212yescomplemented
28#19097740—21no—
29C4×C22⋊D157—22no—
30#19097730C2211yescharacteristic
31#19097696C2214yescomplemented
32#19097693C2214yescomplemented
33C23×C4×C157C2211yescharacteristic
34C2×M16×C157C2211yescharacteristic
35C2×M16×C157C2211yescharacteristic
36C22⋊C8×C157C2214yescomplemented
37C23×D157—21no—
38#14578954—216no—
39#14578954—28no—
40#14578954—28no—
41#14578954—22no—
42#14578954—22no—
43C2×Dic314—22no—
44C2×Dic314—21no—
45C2×Dic314D414yescomplemented
46C2×Dic314C2311yescharacteristic
47C2×Dic314C2312yes—
48C2×D628—22no—
49C2×D628—21no—
50C2×D628D414yescomplemented
51C2×D628C2311yescharacteristic
52C2×D628C2312yes—
53C2×C22⋊D157—28no—
54C2×C22⋊D157—24no—
55C2×C22⋊D157—22no—
56C2×C22⋊D157—21no—
57C22×Dic157—21no—
58C2×C4×D157—28no—
59C2×C4×D157—24no—
60C2×C4×D157—21no—
61Dic628—24no—
62#14578943—24no—
63#14578943—24no—
64D1256—24no—
65C4⋊Dic157C2312yes—
66#14578933—22no—
67C22⋊Dic157—21no—
68C22⋊Dic157—22no—
69#14578931—22no—
70C4×Dic157—21no—
71C24×C157D411yescharacteristic
72C22×C4×C157—22no—
73C22×C4×C157—22no—
74C22×C4×C157D412yes—
75C22×C4×C157D411yescharacteristic
76C22×C4×C157C2312yes—
77C22×C4×C157C2311yescharacteristic
78M16×C157—22no—
79M16×C157—22no—
80C2×C8×C157C2312yes—
81C2×C8×C157C2312yes—
82C22×D157—24no—
83C22×D157—22no—
84C22×D157—21no—
85C22×D157—44no—
86Dic314—24no—
87Dic314—22no—
88Dic314—21no—
89Dic314—21no—
90Dic314C2×D414yes—
91D628—24no—
92D628—22no—
93D628—21no—
94D628—21no—
95D628C2×D414yes—
96C22⋊D157—216no—
97C22⋊D157—44no—
98C22⋊D157—42no—
99C2×Dic157—24no—
100C2×Dic157—22no—
101C2×Dic157—21no—
102C2×Dic157—22no—
103C4×D157—216no—
104C4×D157—42no—
105C23×C157—22no—
106C23×C157—42no—
107C23×C157C2×D412yes—
108C23×C157C2×D411yescharacteristic
109C2×C4×C157—22no—
110C2×C4×C157—24no—
111C2×C4×C157—22no—
112C2×C4×C157—22no—
113C2×C4×C157C2×D412yes—
114C2×C4×C157C2×D412yes—
115C2×C4×C157C2×D412yes—
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157C2411yescharacteristic
118C8×C157—22no—
119C8×C157—22no—
120C22×C157—22no—
121C22×C157—22no—
122C22×C157—22no—
123C22×C157—22no—
124C22×C157—44no—
125C22×C157#12012yes—
126C22×C157C22×D411yescharacteristic
127C2×D157—24no—
128C2×D157—48no—
129C2×D157—44no—
130C2×D157—41no—
131C2×D157—42no—
132C4×C157—22no—
133C4×C157#12012yes—
134C4×C157C22×D411yescharacteristic
135C4×C157C22×D411yescharacteristic
136Dic157—24no—
137Dic157—41no—
138Dic157—42no—
139C2×C157—22no—
140C2×C157—42no—
141C2×C157C2×#12011yescharacteristic
142C2×C157#57512yes—
143D157—44no—
144D157—81no—
145C157#300511yescharacteristic, complemented, Sylow
146#3005—1571nomaximal, Sylow
147C22×#41—1571no—
148C2×#137—1571no—
149C2×AGL(1,ℤ/8)—1571no—
150#525—1571no—
151#452—1574no—
152#449—1574no—
153#419—1571no—
154#418—1571no—
155#407D15711yescharacteristic, complemented
156C2×#41—3144no—
157C2×#41—3144no—
158C2×#41—3142no—
159C2×#41—1571no—
160C2×#41—1574no—
161C22×Q8—1571no—
162C22×D4—3141no—
163C22×D4—1571no—
164C23×C4—3141no—
165C23×C4C2×D15711yescharacteristic
166#137—3144no—
167AGL(1,ℤ/8)—3144no—
168C2×Q16—1572no—
169C2×SD16—1572no—
170C2×SD16—1572no—
171C2×D8—1572no—
172C2×M16C2×D15711yescharacteristic
173C2×M16C2×D15711yescharacteristic
174#123—3141no—
175C22⋊Q8—1572no—
176#121—1572no—
177#120—3141no—
178C4×D4—3142no—
179#117—1571no—
180#103—1574no—
181#102—1574no—
182C22⋊C8C2×D15714yescomplemented
183C24—3141no—
184C24D62811yescharacteristic
185#41—31416no—
186#41—3148no—
187#41—3148no—
188#41—3142no—
189#41—3142no—
190C2×Q8—3142no—
191C2×Q8—3141no—
192C2×Q8—1574no—
193C2×Q8—1571no—
194C2×Q8—1572no—
195C2×D4—3148no—
196C2×D4—3144no—
197C2×D4—3142no—
198C2×D4—3141no—
199C2×D4—3142no—
200C2×D4—3141no—
201C2×D4—1574no—
202C2×D4—1571no—
203C2×D4—1572no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—3148no—
207C22×C4—3141no—
208C22×C4—3144no—
209C22×C4—3141no—
210C22×C4D62812yes—
211C22×C4D62811yescharacteristic
212C22×C4C22×D15712yes—
213C22×C4C22×D15711yescharacteristic
214Q16—3144no—
215SD16—3144no—
216SD16—3144no—
217D8—3144no—
218M16—22no—
219M16—22no—
220C2×C8C22×D15712yes—
221C2×C8C22×D15712yes—
222C4⋊C4—3142no—
223C4⋊C4—1572no—
224C22⋊C4—3141no—
225C22⋊C4—3142no—
226C22⋊C4—3142no—
227C42—3141no—
228C23—22no—
229C23—42no—
230C23—3144no—
231C23—3142no—
232C23—3141no—
233C23—6284no—
234C23C2×D62812yes—
235C23C2×D62811yescharacteristic
236Q8—3144no—
237Q8—3142no—
238Q8—3141no—
239Q8—3141no—
240Q8—1574no—
241D4—31416no—
242D4—3144no—
243D4—3142no—
244D4—3141no—
245D4—3141no—
246D4—6284no—
247D4—6282no—
248D4—1574no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—31416no—
254C2×C4—3144no—
255C2×C4—3142no—
256C2×C4—3142no—
257C2×C4—3141no—
258C2×C4—6282no—
259C2×C4D4×D15712yes—
260C2×C4D4×D15712yes—
261C2×C4C2×D62812yes—
262C2×C4C2×D62811yescharacteristic
263C2×C4C23×D15711yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—3144no—
272C22—6288no—
273C22—6284no—
274C22—6281no—
275C22—6282no—
276C22#1909774112yes—
277C22C22×D62811yescharacteristic
278C4—22no—
279C4—3144no—
280C4—6281no—
281C4—6282no—
282C4#1909774112yes—
283C4C2×D4×D15711yescharacteristic
284C4C2×D4×D15711yescharacteristic
285C2—22no—
286C2—42no—
287C2—6284no—
288C2—12561no—
289C2C2×#1909774111yescharacteristic
290C2#2375807312yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#407×C157C2
socleC22×C157C22×D4
hypercenter#407D157
hypocenterC157#3005
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#14578954C2semidirect
2. C2×#19097804C2semidirect
3. C2×#19097799C2semidirect
4. #23757784C2semidirect
5. #23757437C2semidirect
6. #23757420C2semidirect
7. #23757283C2semidirect
8. #23757279C2semidirect
9. #407×C157C2semidirect
21. C2×Dic628C22semidirect
22. C2×#14578943C22semidirect
23. C2×#14578943C22semidirect
24. C2×D1256C22semidirect
26. #19097750C22semidirect
27. #19097745C22semidirect
31. #19097696C22semidirect
32. #19097693C22semidirect
36. C22⋊C8×C157C22semidirect
45. C2×Dic314D4semidirect
50. C2×D628D4semidirect
145. C157#3005semidirect
155. #407D157semidirect
182. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789542
C2×#190978042
C2×#190977992
#237577842
#237574372
#237574202
#237572832
#237572792
#407×C1572
#3005157