Finite Group Databasebeta

← #34109376

Subgroups — #34109376

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×#19097801C211yesmaximal, characteristic, complemented
3C2×#19097796C211yesmaximal, characteristic, complemented
4#23757433C214yesmaximal, complemented
5#23757416C214yesmaximal, complemented
6#23757281C211yesmaximal, characteristic, complemented
7#23757277C211yesmaximal, characteristic, complemented
8#23757237C211yesmaximal, characteristic, complemented
9#525×C157C211yesmaximal, characteristic, complemented
10C2×#14578954—22no—
11C2×#14578954—24no—
12C2×#14578954—24no—
13C2×#14578954C2214yes—
14C22×Dic314C2211yescharacteristic
15C22×D628C2211yescharacteristic
16C22×C22⋊D157—21no—
17C22×C4×D157—21no—
18#19097801—24no—
19#19097796—24no—
20C2×#14578944C2212yescomplemented
21C2×#14578942C2212yescomplemented
22C2×#14578941C2212yescomplemented
23C2×#14578939C2212yescomplemented
24C2×#14578938C2211yescharacteristic
25C2×#14578938C2211yescharacteristic
26#19097694C2214yescomplemented
27#19097691C2214yescomplemented
28C22⋊#11929434C2214yescomplemented
29C2×#41×C157C2211yescharacteristic
30C23×C4×C157C2211yescharacteristic
31#123×C157—21no—
32C22⋊Q8×C157C2212yescomplemented
33#121×C157C2212yescomplemented
34#120×C157—21no—
35C4×D4×C157—22no—
36#117×C157C2211yescharacteristic
37C23×D157—21no—
38#14578954—216no—
39#14578954—28no—
40#14578954—28no—
41C2×Dic314—22no—
42C2×Dic314—21no—
43C2×Dic314D414yescomplemented
44C2×Dic314C2312yes—
45C2×D628—22no—
46C2×D628—21no—
47C2×D628D414yescomplemented
48C2×D628C2312yes—
49C2×C22⋊D157—28no—
50C2×C22⋊D157—24no—
51C22×Dic157—21no—
52C2×C4×D157—28no—
53C2×C4×D157—24no—
54#14578944—24no—
55#14578942—24no—
56#14578941—24no—
57#14578939—24no—
58#14578938—22no—
59#14578938—22no—
60C2×#11929434C2312yes—
61C2×#11929434C2312yes—
62C24×C157D411yescharacteristic
63#41×C157—22no—
64#41×C157—22no—
65C2×Q8×C157C2311yescharacteristic
66C2×D4×C157—22no—
67C2×D4×C157—21no—
68C2×D4×C157C2311yescharacteristic
69C22×C4×C157—22no—
70C22×C4×C157—22no—
71C22×C4×C157—21no—
72C22×C4×C157D412yes—
73C22×C4×C157D411yescharacteristic
74C22×C4×C157C2312yes—
75C22×C4×C157C2311yescharacteristic
76C4⋊C4×C157—22no—
77C4⋊C4×C157C2312yes—
78C22⋊C4×C157—21no—
79C22⋊C4×C157—22no—
80C22⋊C4×C157—22no—
81C42×C157—21no—
82C22×D157—24no—
83C22×D157—22no—
84C22×D157—44no—
85Dic314—24no—
86Dic314—22no—
87Dic314C2×D414yes—
88D628—24no—
89D628—22no—
90D628C2×D414yes—
91C22⋊D157—216no—
92C2×Dic157—24no—
93C2×Dic157—22no—
94C4×D157—216no—
95#11929434—22no—
96#11929434—22no—
97C23×C157—22no—
98C23×C157—21no—
99C23×C157—42no—
100C23×C157C2×D412yes—
101C23×C157C2×D411yescharacteristic
102Q8×C157—21no—
103Q8×C157—21no—
104D4×C157—21no—
105D4×C157—21no—
106D4×C157—44no—
107D4×C157—42no—
108C2×C4×C157—22no—
109C2×C4×C157—24no—
110C2×C4×C157—22no—
111C2×C4×C157—22no—
112C2×C4×C157—22no—
113C2×C4×C157—21no—
114C2×C4×C157—42no—
115C2×C4×C157C2×D412yes—
116C2×C4×C157C2×D412yes—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157C2411yescharacteristic
120C22×C157—22no—
121C22×C157—22no—
122C22×C157—22no—
123C22×C157—22no—
124C22×C157—44no—
125C22×C157—41no—
126C22×C157—42no—
127C22×C157#12012yes—
128C22×C157C22×D411yescharacteristic
129C2×D157—24no—
130C2×D157—48no—
131C2×D157—44no—
132C4×C157—22no—
133C4×C157—41no—
134C4×C157—42no—
135C4×C157#12012yes—
136C4×C157C22×D411yescharacteristic
137C4×C157C22×D411yescharacteristic
138Dic157—24no—
139C2×C157—22no—
140C2×C157—42no—
141C2×C157—81no—
142C2×C157C2×#12011yescharacteristic
143C2×C157#57512yes—
144D157—44no—
145C157#300511yescharacteristic, complemented, Sylow
146#3005—1571nomaximal, Sylow
147C22×#41—1571no—
148C2×#137—1571no—
149C2×AGL(1,ℤ/8)—1571no—
150#525D15711yescharacteristic, complemented
151#452—1574no—
152#449—1574no—
153#419—1571no—
154#418—1571no—
155#407—1571no—
156C2×#41—3144no—
157C2×#41—3144no—
158C2×#41—3142no—
159C2×#41—1574no—
160C2×#41C2×D15711yescharacteristic
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×M16—1571no—
173C2×M16—1571no—
174#123—21no—
175C22⋊Q8C2×D15712yescomplemented
176#121C2×D15712yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D15711yescharacteristic
180#103—1574no—
181#102—1574no—
182C22⋊C8—1574no—
183C24—3141no—
184C24C22⋊D15711yescharacteristic
185#41—22no—
186#41—22no—
187#41—31416no—
188#41—3148no—
189#41—3148no—
190C2×Q8—3142no—
191C2×Q8—3141no—
192C2×Q8—1574no—
193C2×Q8—1572no—
194C2×Q8C22×D15711yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—3148no—
198C2×D4—3144no—
199C2×D4—3142no—
200C2×D4—3141no—
201C2×D4—1574no—
202C2×D4—1572no—
203C2×D4C22×D15711yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—3148no—
208C22×C4—3141no—
209C22×C4—3144no—
210C22×C4C22⋊D15712yes—
211C22×C4C22⋊D15711yescharacteristic
212C22×C4C22×D15712yes—
213C22×C4C22×D15711yescharacteristic
214Q16—3144no—
215SD16—3144no—
216SD16—3144no—
217D8—3144no—
218M16—3142no—
219M16—3142no—
220C2×C8—1572no—
221C2×C8—1572no—
222C4⋊C4—22no—
223C4⋊C4C22×D15712yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—3144no—
232C23—3142no—
233C23—6284no—
234C23C2×C22⋊D15712yes—
235C23C2×C22⋊D15711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—3144no—
239Q8—3142no—
240Q8—1574no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—31416no—
246D4—3144no—
247D4—3142no—
248D4—1574no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—31416no—
257C2×C4—3144no—
258C2×C4—3142no—
259C2×C4D4×D15712yes—
260C2×C4D4×D15712yes—
261C2×C4C2×C22⋊D15712yes—
262C2×C4C2×C22⋊D15711yescharacteristic
263C2×C4C23×D15711yescharacteristic
264C8—3142no—
265C8—3142no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—3144no—
274C22—6288no—
275C22—6284no—
276C22#1909773912yes—
277C22C22×C22⋊D15711yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—3144no—
282C4#1909773912yes—
283C4C2×D4×D15711yescharacteristic
284C4C2×D4×D15711yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—6284no—
289C2C2×#1909773911yescharacteristic
290C2#2375807112yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#525×C157C2
socleC22×C157C22×D4
hypercenter#525D157
hypocenterC157#3005
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#14578954C2semidirect
2. C2×#19097801C2semidirect
3. C2×#19097796C2semidirect
4. #23757433C2semidirect
5. #23757416C2semidirect
6. #23757281C2semidirect
7. #23757277C2semidirect
8. #23757237C2semidirect
9. #525×C157C2semidirect
20. C2×#14578944C22semidirect
21. C2×#14578942C22semidirect
22. C2×#14578941C22semidirect
23. C2×#14578939C22semidirect
26. #19097694C22semidirect
27. #19097691C22semidirect
28. C22⋊#11929434C22semidirect
32. C22⋊Q8×C157C22semidirect
33. #121×C157C22semidirect
43. C2×Dic314D4semidirect
47. C2×D628D4semidirect
145. C157#3005semidirect
150. #525D157semidirect
175. C22⋊Q8C2×D157semidirect
176. #121C2×D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789542
C2×#190978012
C2×#190977962
#237574332
#237574162
#237572812
#237572772
#237572372
#525×C1572
#3005157