Finite Group Databasebeta

← #34109402

Subgroups — #34109402

Order 20096.

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

normal subgroups only

This group has 784 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×#19097817C211yesmaximal, characteristic, complemented
2C2×#19097797C211yesmaximal, characteristic, complemented
3C22×#14578939C211yesmaximal, characteristic, complemented
4#23757416C214yesmaximal, complemented
5#23757406C214yesmaximal, complemented
6#23757277C211yesmaximal, characteristic, complemented
7#23757267C211yesmaximal, characteristic, complemented
8#23757240C211yesmaximal, characteristic, complemented
9#535×C157C211yesmaximal, characteristic, complemented
10#19097817—24no—
11#19097817—24no—
12C2×#14578955—21no—
13C2×#14578954C2211yescharacteristic
14C2×#14578954C2212yescomplemented
15C22×D628C2211yescharacteristic
16C22×D628C2212yescomplemented
17C2×D4×D157—21no—
18C2×D4×D157—22no—
19#19097797—24no—
20C2×#14578942C2212yescomplemented
21C2×#14578939—24no—
22C2×#14578939C2212yescomplemented
23C2×#14578939C2214yescomplemented
24C2×#14578938C2211yescharacteristic, complemented
25C22×#11929434C2211yescharacteristic, complemented
26#19097694C2212yescomplemented
27#19097691C2212yescomplemented
28#19097691C2214yescomplemented
29C22⋊#11929434C2214yescomplemented
30C2×#41×C157C2211yescharacteristic
31C22×D4×C157C2211yescharacteristic, complemented
32#127×C157—21no—
33#124×C157—21no—
34#121×C157C2214yescomplemented
35#120×C157—22no—
36#117×C157C2211yescharacteristic, complemented
37C23×D157—21no—
38C23×D157—22no—
39#14578955—28no—
40#14578954—24no—
41#14578954—24no—
42#14578954—22no—
43#14578954D414yescomplemented
44C2×Dic314C2311yescharacteristic
45C2×D628—24no—
46C2×D628—24no—
47C2×D628—21no—
48C2×D628—22no—
49C2×D628—22no—
50C2×D628D414yescomplemented
51C2×D628C2312yes—
52C2×D628C2311yescharacteristic
53C2×C22⋊D157—21no—
54C2×C22⋊D157—22no—
55D4×D157—216no—
56D4×D157—28no—
57C2×C4×D157—21no—
58C2×C4×D157—22no—
59#14578942—24no—
60#14578939—28no—
61#14578939—24no—
62#14578938—22no—
63C2×#11929434—22no—
64C2×#11929434C2312yes—
65C2×#11929434C2312yes—
66C24×C157—21no—
67#41×C157—22no—
68#41×C157—22no—
69C2×Q8×C157D411yescharacteristic
70C2×D4×C157—24no—
71C2×D4×C157—22no—
72C2×D4×C157—22no—
73C2×D4×C157—21no—
74C2×D4×C157D412yes—
75C2×D4×C157D411yescharacteristic
76C2×D4×C157C2312yes—
77C22×C4×C157C2312yes—
78C22×C4×C157C2311yescharacteristic
79C4⋊C4×C157C2312yes—
80C22⋊C4×C157—21no—
81C22⋊C4×C157—24no—
82C42×C157—21no—
83C22×D157—21no—
84C22×D157—22no—
85C22×D157—28no—
86C22×D157—24no—
87C22×D157—82no—
88C22×D157—84no—
89Dic314—21no—
90Dic314C2×D412yes—
91D628—28no—
92D628—24no—
93D628—22no—
94D628—21no—
95D628C2×D412yes—
96D628C2×D414yes—
97C22⋊D157—28no—
98C22⋊D157—24no—
99C2×Dic157—21no—
100C4×D157—28no—
101C4×D157—24no—
102#11929434—22no—
103#11929434—22no—
104C23×C157—22no—
105C23×C157—42no—
106C23×C157—82no—
107C23×C157C2×D412yes—
108C23×C157C2×D411yescharacteristic
109Q8×C157—22no—
110D4×C157—24no—
111D4×C157—22no—
112D4×C157—22no—
113D4×C157—22no—
114D4×C157—42no—
115D4×C157—84no—
116C2×C4×C157—22no—
117C2×C4×C157—22no—
118C2×C4×C157—22no—
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2×D412yes—
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2×D411yescharacteristic
123C2×C4×C157C2411yescharacteristic
124C22×C157—22no—
125C22×C157—22no—
126C22×C157—24no—
127C22×C157—42no—
128C22×C157—44no—
129C22×C157—84no—
130C22×C157#12012yes—
131C22×C157C22×D411yescharacteristic
132C2×D157—24no—
133C2×D157—22no—
134C2×D157—44no—
135C2×D157—42no—
136C2×D157—84no—
137C2×D157—88no—
138C4×C157—22no—
139C4×C157—42no—
140C4×C157#12012yes—
141C4×C157C22×D411yescharacteristic
142C4×C157C22×D411yescharacteristic
143Dic157—22no—
144C2×C157—22no—
145C2×C157—42no—
146C2×C157—82no—
147C2×C157C2×#12011yescharacteristic
148C2×C157#57612yes—
149D157—44no—
150D157—42no—
151C157#300811yescharacteristic, complemented, Sylow
152#3008—1571nomaximal, Sylow
153C2×H(5,𝔽2)—1571no—
154C2×AGL(1,ℤ/8)—1571no—
155C22×D8—1571no—
156#535D15711yescharacteristic, complemented
157#449—1574no—
158#447—1574no—
159#418—1571no—
160#416—1571no—
161#408—1571no—
162H(5,𝔽2)—3144no—
163H(5,𝔽2)—3144no—
164C2×#41—3141no—
165C2×#41—1572no—
166C2×#41—1571no—
167C2×#41C2×D15711yescharacteristic
168C22×D4—3142no—
169C22×D4—3141no—
170C22×D4—1572no—
171C22×D4—1571no—
172C22×D4C2×D15711yescharacteristic, complemented
173AGL(1,ℤ/8)—3144no—
174C2×SD16—1572no—
175C2×D8—3144no—
176C2×D8—1574no—
177C2×D8—1572no—
178C2×M16—1571no—
179C22×C8—1571no—
180#127—21no—
181#124—21no—
182#121C2×D15714yescomplemented
183#120—22no—
184#117C2×D15711yescharacteristic, complemented
185#103—1572no—
186#102—1572no—
187#102—1574no—
188C22⋊C8—1574no—
189C24—21no—
190C24—3142no—
191C24—3141no—
192#41—22no—
193#41—22no—
194#41—3144no—
195#41—3144no—
196#41—3148no—
197#41—3142no—
198#41—1574no—
199C2×Q8—1571no—
200C2×Q8C22⋊D15711yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—3144no—
206C2×D4—3144no—
207C2×D4—31416no—
208C2×D4—3148no—
209C2×D4—3142no—
210C2×D4—3141no—
211C2×D4—3142no—
212C2×D4—3142no—
213C2×D4—3141no—
214C2×D4—1574no—
215C2×D4—1571no—
216C2×D4—1572no—
217C2×D4C22⋊D15712yes—
218C2×D4C22⋊D15711yescharacteristic
219C2×D4C22×D15712yes—
220C22×C4—3142no—
221C22×C4—3141no—
222C22×C4C22×D15712yes—
223C22×C4C22×D15711yescharacteristic
224SD16—3144no—
225D8—3148no—
226D8—3144no—
227M16—3142no—
228C2×C8—3142no—
229C2×C8—1572no—
230C2×C8—1572no—
231C4⋊C4C22×D15712yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—3148no—
239C23—3142no—
240C23—3141no—
241C23—3144no—
242C23—12562no—
243C23—12564no—
244C23C2×C22⋊D15712yes—
245C23C2×C22⋊D15711yescharacteristic
246Q8—22no—
247Q8—3141no—
248Q8—1572no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—3148no—
256D4—3148no—
257D4—3144no—
258D4—3144no—
259D4—3142no—
260D4—3141no—
261D4—1572no—
262D4—1574no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—3148no—
267C2×C4—3141no—
268C2×C4—3144no—
269C2×C4D4×D15712yes—
270C2×C4D4×D15712yes—
271C2×C4C2×C22⋊D15712yes—
272C2×C4C2×C22⋊D15711yescharacteristic
273C2×C4C23×D15711yescharacteristic
274C8—3142no—
275C8—3142no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—3144no—
283C22—3142no—
284C22—6284no—
285C22—6282no—
286C22—12564no—
287C22—12568no—
288C22#1909773912yes—
289C22C22×C22⋊D15711yescharacteristic
290C4—22no—
291C4—42no—
292C4—3142no—
293C4#1909773912yes—
294C4C2×D4×D15711yescharacteristic
295C4C2×D4×D15711yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—6284no—
300C2—6282no—
301C2C2×#1909773911yescharacteristic
302C2#2375807612yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#535×C157C2
socleC22×C157C22×D4
hypercenter#535D157
hypocenterC157#3008
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19097817C2semidirect
2. C2×#19097797C2semidirect
3. C22×#14578939C2semidirect
4. #23757416C2semidirect
5. #23757406C2semidirect
6. #23757277C2semidirect
7. #23757267C2semidirect
8. #23757240C2semidirect
9. #535×C157C2semidirect
14. C2×#14578954C22semidirect
16. C22×D628C22semidirect
20. C2×#14578942C22semidirect
22. C2×#14578939C22semidirect
23. C2×#14578939C22semidirect
24. C2×#14578938C22semidirect
25. C22×#11929434C22semidirect
26. #19097694C22semidirect
27. #19097691C22semidirect
28. #19097691C22semidirect
29. C22⋊#11929434C22semidirect
31. C22×D4×C157C22semidirect
34. #121×C157C22semidirect
36. #117×C157C22semidirect
43. #14578954D4semidirect
50. C2×D628D4semidirect
151. C157#3008semidirect
156. #535D157semidirect
172. C22×D4C2×D157semidirect
182. #121C2×D157semidirect
184. #117C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978172
C2×#190977972
C22×#145789392
#237574162
#237574062
#237572772
#237572672
#237572402
#535×C1572
#3008157