Finite Group Databasebeta

← #34111822

Subgroups — #34111822

Order 20096.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19097790C211yesmaximal, characteristic, complemented
2C2×D8×D157C211yesmaximal, characteristic, complemented
3#23757918C212yesmaximal, complemented
4#23757657C211yesmaximal, characteristic, complemented
5#23757515C211yesmaximal, characteristic, complemented
6#23757510C211yesmaximal, characteristic, complemented
7#23757488C212yesmaximal, complemented
8#23757425C212yesmaximal, complemented
9#23757419C212yesmaximal, complemented
10C8×D628C211yesmaximal, characteristic, complemented
11C4×D8×C157C211yesmaximal, characteristic, complemented
12C2×#14578953C2212yescomplemented
13C2×D4×D157C2212yes—
14#19097790—24no—
15C2×Dic628C2211yescharacteristic, complemented
16C2×#14578943C2212yescomplemented
17C2×#14578942C2212yescomplemented
18C2×D1256C2211yescharacteristic
19C2×#14578939C2212yes—
20D8×D157—24no—
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic, complemented
23#19097759C2212yescomplemented
24#19097755—22no—
25#19097751C2212yes—
26#19097746C2212yescomplemented
27#19097745—22no—
28#19097741—22no—
29C4×D628C2211yescharacteristic
30C2×#14578931—22no—
31C4⋊#11929434C2211yescharacteristic, complemented
32#19097694C2212yescomplemented
33#19097691C2212yescomplemented
34#19097683C2211yescharacteristic
35#19097683C2211yescharacteristic, complemented
36C2×D8×C157C2211yescharacteristic
37C4×D4×C157C2212yescomplemented
38#107×C157C2211yescharacteristic, complemented
39#102×C157C2212yescomplemented
40C4×C8×C157C2211yescharacteristic, complemented
41C23×D157—22no—
42#14578953—24no—
43#14578953—42no—
44C2×Dic314C2312yes—
45C2×D628—22no—
46C2×D628D411yescharacteristic
47C2×D628C2312yes—
48C2×C22⋊D157—22no—
49C2×C22⋊D157—22no—
50D4×D157—24no—
51D4×D157—42no—
52C22×Dic157—22no—
53C2×C4×D157C2311yescharacteristic
54C2×C4×D157C2311yescharacteristic
55Dic628—22no—
56#14578943—42no—
57#14578942—24no—
58D1256—22no—
59#14578939—24no—
60C2×#11929434C2311yescharacteristic
61C2×#11929434C2311yescharacteristic
62C8×D157—22no—
63C8×D157—22no—
64C4⋊Dic157—22no—
65C4⋊Dic157D411yescharacteristic
66#14578931—22no—
67#14578931—22no—
68#14578931—22no—
69#14578931—22no—
70#14578931D411yescharacteristic
71#14578931D411yescharacteristic
72C2×D4×C157C2312yes—
73C22×C4×C157—22no—
74D8×C157D414yescomplemented
75C2×C8×C157C2311yescharacteristic
76C2×C8×C157C2311yescharacteristic
77C4⋊C4×C157C2312yes—
78C22⋊C4×C157—22no—
79C42×C157C2311yescharacteristic
80C22×D157—22no—
81C22×D157—44no—
82C22×D157—44no—
83C22×D157—82no—
84C22×D157C2×D411yescharacteristic
85C22×D157C2×D411yescharacteristic
86Dic314—22no—
87Dic314—22no—
88D628—22no—
89D628—22no—
90D628—22no—
91D628—82no—
92C22⋊D157—44no—
93C22⋊D157—44no—
94C2×Dic157—22no—
95C2×Dic157—44no—
96C2×Dic157C2×D411yescharacteristic
97C2×Dic157C2×D411yescharacteristic
98C4×D157—21no—
99C4×D157—21no—
100C4×D157—21no—
101C4×D157—21no—
102#11929434—21no—
103#11929434—21no—
104C23×C157—22no—
105D4×C157—22no—
106D4×C157C2×D414yes—
107C2×C4×C157—22no—
108C2×C4×C157—82no—
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2411yescharacteristic
112C8×C157—21no—
113C8×C157C2×D412yes—
114C22×C157—24no—
115C22×C157—42no—
116C22×C157—42no—
117C22×C157C22×D411yescharacteristic
118C2×D157—21no—
119C2×D157—21no—
120C2×D157—21no—
121C2×D157—21no—
122C2×D157—21no—
123C2×D157—42no—
124C2×D157—42no—
125C2×D157—42no—
126C2×D157—88no—
127C2×D157#13512yescomplemented
128C4×C157—21no—
129C4×C157—21no—
130C4×C157—42no—
131C4×C157C22×D411yescharacteristic
132C4×C157H(5,𝔽2)11yescharacteristic
133Dic157—21no—
134Dic157—21no—
135Dic157—42no—
136C2×C157—44no—
137C2×C157D4211yescharacteristic
138C2×C157C2×#13511yescharacteristic
139C2×C157#57611yescharacteristic
140D157—22no—
141D157—41no—
142D157—82no—
143C157#328011yescharacteristic, complemented, Sylow
144#3280—1571nomaximal, Sylow
145C2×#135—1571no—
146C22×D8—1571no—
147#546—1572no—
148#495—1571no—
149#467—1571no—
150#466—1571no—
151#463—1572no—
152#450—1572no—
153#449—1572no—
154C4×D8D15711yescharacteristic, complemented
155C8×D4—1571no—
156C2×#41—1572no—
157C22×D4—1572no—
158#135—3144no—
159C2×Q16—1571no—
160C2×SD16—1572no—
161C2×SD16—1572no—
162C2×D8—3144no—
163C2×D8—1572no—
164C2×D8—1571no—
165C2×D8C2×D15711yescharacteristic
166C22×C8—1571no—
167C22×C8—1571no—
168#124—1572no—
169#123—3142no—
170C22⋊Q8—1572no—
171#121—3142no—
172#121—1572no—
173#120—3142no—
174C4×D4—1571no—
175C4×D4C2×D15712yescomplemented
176C2×C22⋊C4—3142no—
177#107C2×D15711yescharacteristic, complemented
178C4⋊C8—1571no—
179#103—1572no—
180#102—1572no—
181#102C2×D15712yescomplemented
182C22⋊C8—1571no—
183C22⋊C8—1571no—
184C4×C8C2×D15711yescharacteristic, complemented
185C24—3142no—
186#41—3144no—
187#41—6282no—
188C2×Q8—1572no—
189C2×D4—3142no—
190C2×D4—3144no—
191C2×D4—3142no—
192C2×D4—3142no—
193C2×D4—6282no—
194C2×D4—1571no—
195C2×D4—1572no—
196C2×D4C22×D15712yes—
197C22×C4—22no—
198C22×C4—3142no—
199C22×C4—1571no—
200C22×C4—1571no—
201Q16—3142no—
202SD16—3144no—
203SD16—6282no—
204D8—3144no—
205D8—3142no—
206D8D62814yescomplemented
207C2×C8—3142no—
208C2×C8—3142no—
209C2×C8—1571no—
210C2×C8—1571no—
211C2×C8C22×D15711yescharacteristic
212C2×C8C22×D15711yescharacteristic
213C4⋊C4—3142no—
214C4⋊C4—1571no—
215C4⋊C4C22×D15712yes—
216C22⋊C4—22no—
217C22⋊C4—3142no—
218C22⋊C4—3142no—
219C22⋊C4—3142no—
220C22⋊C4—3142no—
221C22⋊C4—1571no—
222C22⋊C4—1571no—
223C42C22×D15711yescharacteristic
224C23—22no—
225C23—3142no—
226C23—6284no—
227C23—6284no—
228C23—1571no—
229C23—1571no—
230C23—12562no—
231Q8—3142no—
232Q8—3142no—
233D4—22no—
234D4—3142no—
235D4—3142no—
236D4—3142no—
237D4—6284no—
238D4—6284no—
239D4—12562no—
240D4C2×D62814yes—
241C2×C4—22no—
242C2×C4—82no—
243C2×C4—3141no—
244C2×C4—3141no—
245C2×C4—3141no—
246C2×C4—3141no—
247C2×C4—3142no—
248C2×C4—6284no—
249C2×C4—1571no—
250C2×C4—1571no—
251C2×C4D4×D15711yescharacteristic
252C2×C4D4×D15711yescharacteristic
253C2×C4C23×D15711yescharacteristic
254C8—21no—
255C8—3141no—
256C8—3141no—
257C8C2×D62812yes—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—3141no—
262C22—3141no—
263C22—3141no—
264C22—3141no—
265C22—3141no—
266C22—6282no—
267C22—6282no—
268C22—6282no—
269C22—1572no—
270C22—12568no—
271C22C2×D4×D15711yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—3141no—
276C4—3141no—
277C4—6282no—
278C4C22×D62811yescharacteristic
279C4#1909781711yescharacteristic
280C2—44no—
281C2—3142no—
282C2—6281no—
283C2—12562no—
284C2D4×D62811yescharacteristic
285C2#135×D15711yescharacteristic
286C2#2375807411yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC4×D8×C157C2
socleC22×C157C22×D4
hypercenterC4×D8D157
hypocenterC157#3280
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19097790C2semidirect
2. C2×D8×D157C2semidirect
3. #23757918C2semidirect
4. #23757657C2semidirect
5. #23757515C2semidirect
6. #23757510C2semidirect
7. #23757488C2semidirect
8. #23757425C2semidirect
9. #23757419C2semidirect
10. C8×D628C2semidirect
11. C4×D8×C157C2semidirect
12. C2×#14578953C22semidirect
15. C2×Dic628C22semidirect
16. C2×#14578943C22semidirect
17. C2×#14578942C22semidirect
22. C2×C8×D157C22semidirect
23. #19097759C22semidirect
26. #19097746C22semidirect
31. C4⋊#11929434C22semidirect
32. #19097694C22semidirect
33. #19097691C22semidirect
35. #19097683C22semidirect
37. C4×D4×C157C22semidirect
38. #107×C157C22semidirect
39. #102×C157C22semidirect
40. C4×C8×C157C22semidirect
74. D8×C157D4semidirect
127. C2×D157#135semidirect
143. C157#3280semidirect
154. C4×D8D157semidirect
175. C4×D4C2×D157semidirect
177. #107C2×D157semidirect
181. #102C2×D157semidirect
184. C4×C8C2×D157semidirect
206. D8D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190977902
C2×D8×D1572
#237579182
#237576572
#237575152
#237575102
#237574882
#237574252
#237574192
C8×D6282
C4×D8×C1572
#3280157