Finite Group Databasebeta

← #34110274

Subgroups — #34110274

Order 20096.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#23757807C211yesmaximal, characteristic, complemented
2#23757603C211yesmaximal, characteristic, complemented
3#23757586C211yesmaximal, characteristic, complemented
4#23757573C212yesmaximal, complemented
5#23757488C212yesmaximal, complemented
6#23757424C211yesmaximal, characteristic, complemented
7#23757407C211yesmaximal, characteristic
8#23757383C212yesmaximal, complemented
9#23757373C212yesmaximal, complemented
10#23757330C211yesmaximal, characteristic, complemented
11C2×C4×D4×C157C211yesmaximal, characteristic, complemented
12C2×#14578942C2212yes—
13C2×#14578939C2212yes—
14#19097765—21no—
15#19097763C2211yescharacteristic, complemented
16#19097759C2211yescharacteristic, complemented
17#19097754—21no—
18#19097750C2211yescharacteristic
19#19097745C2211yescharacteristic
20C4×Dic314C2211yescharacteristic, complemented
21C4×D628C2211yescharacteristic, complemented
22C4×C22⋊D157—21no—
23#19097728—21no—
24#19097704C2212yescomplemented
25#19097702C2212yescomplemented
26C4⋊#11929434C2212yescomplemented
27#19097694C2212yescomplemented
28#19097692C2212yes—
29#19097692C2212yes—
30#19097691C2212yescomplemented
31C22⋊#11929434C2211yescharacteristic
32C22⋊#11929434C2211yescharacteristic
33#19097682C2212yescomplemented
34C22×D4×C157C2211yescharacteristic
35C23×C4×C157—21no—
36C4×D4×C157—21no—
37C4×D4×C157—21no—
38C4×D4×C157C2212yes—
39C4×D4×C157C2212yes—
40C2×C4⋊C4×C157C2211yescharacteristic
41C2×C22⋊C4×C157—21no—
42C2×C42×C157C2211yescharacteristic
43C2×Dic314C2311yescharacteristic
44C2×D628C2311yescharacteristic
45C2×C22⋊D157—21no—
46C2×C4×D157—21no—
47#14578942—24no—
48#14578939—24no—
49C2×#11929434C2312yes—
50C2×#11929434C2312yes—
51C4⋊Dic157C2311yescharacteristic
52C4⋊Dic157C2311yescharacteristic
53#14578933—21no—
54#14578933—21no—
55#14578933—21no—
56#14578933—21no—
57C22⋊Dic157—21no—
58#14578931—21no—
59#14578931—21no—
60#14578931—21no—
61#14578931—21no—
62C4×Dic157—21no—
63C24×C157—21no—
64C2×D4×C157—24no—
65C2×D4×C157—21no—
66C2×D4×C157C2312yes—
67C22×C4×C157—21no—
68C22×C4×C157—22no—
69C22×C4×C157—22no—
70C22×C4×C157—44no—
71C22×C4×C157D411yescharacteristic
72C22×C4×C157D411yescharacteristic
73C22×C4×C157C2311yescharacteristic
74C4⋊C4×C157—21no—
75C4⋊C4×C157C2312yes—
76C22⋊C4×C157—22no—
77C22⋊C4×C157—22no—
78C42×C157D411yescharacteristic
79C42×C157D411yescharacteristic
80C42×C157C2311yescharacteristic
81C42×C157C2311yescharacteristic
82C22×D157—21no—
83Dic314—22no—
84D628—22no—
85C22⋊D157—42no—
86C2×Dic157—21no—
87C2×Dic157—21no—
88C2×Dic157—21no—
89C4×D157—42no—
90#11929434—22no—
91#11929434—22no—
92C23×C157—24no—
93C23×C157—22no—
94C23×C157—42no—
95C23×C157—42no—
96C23×C157C2×D411yescharacteristic
97D4×C157—24no—
98D4×C157—22no—
99D4×C157#4114yes—
100C2×C4×C157—22no—
101C2×C4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157—41no—
104C2×C4×C157—44no—
105C2×C4×C157—44no—
106C2×C4×C157—42no—
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2×D411yescharacteristic
112C2×C4×C157C2411yescharacteristic
113C22×C157—21no—
114C22×C157—21no—
115C22×C157—24no—
116C22×C157—48no—
117C22×C157—42no—
118C22×C157—42no—
119C22×C157AGL(1,ℤ/8)12yes—
120C22×C157C22×D411yescharacteristic
121C2×D157—41no—
122C2×D157—41no—
123C2×D157—41no—
124C4×C157—21no—
125C4×C157—21no—
126C4×C157—21no—
127C4×C157—21no—
128C4×C157—21no—
129C4×C157—42no—
130C4×C157C2×#4112yes—
131Dic157—41no—
132Dic157—41no—
133Dic157—41no—
134C2×C157—22no—
135C2×C157—44no—
136C2×C157#52511yescharacteristic
137C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C157#57511yescharacteristic
139D157—81no—
140C157#311411yescharacteristic, complemented, Sylow
141#3114—1571nomaximal, Sylow
142#529—1571no—
143C2×C4×D4D15711yescharacteristic, complemented
144#483—1571no—
145#480—1571no—
146#478—1572no—
147#463—1572no—
148#450—1571no—
149#447—1571no—
150#442—1572no—
151#440—1572no—
152#432—1571no—
153C22×D4C2×D15711yescharacteristic
154C23×C4—21no—
155C2×SD16—1572no—
156C2×D8—1572no—
157#126—3141no—
158#125—1571no—
159#124—1571no—
160#123—3141no—
161C22⋊Q8—1571no—
162#121—1571no—
163C4×Q8—1571no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—3141no—
167C4×D4—1571no—
168C4×D4C2×D15712yes—
169C4×D4C2×D15712yes—
170#117—3141no—
171C2×C4⋊C4C2×D15711yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D15711yescharacteristic
174#107—1572no—
175#106—1572no—
176C4⋊C8—1572no—
177#103—1572no—
178#102—1572no—
179#102—1572no—
180#102—1572no—
181C22⋊C8—1571no—
182C22⋊C8—1571no—
183#97—1572no—
184C24—21no—
185C2×Q8—1571no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—3141no—
189C2×D4—1571no—
190C2×D4C22×D15712yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—3141no—
196C22×C4C22⋊D15711yescharacteristic
197C22×C4C22⋊D15711yescharacteristic
198C22×C4C22×D15711yescharacteristic
199SD16—3144no—
200D8—3144no—
201C2×C8—1572no—
202C2×C8—1572no—
203C4⋊C4—21no—
204C4⋊C4—3141no—
205C4⋊C4—3141no—
206C4⋊C4—3141no—
207C4⋊C4—3141no—
208C4⋊C4—1571no—
209C4⋊C4—1571no—
210C4⋊C4C22×D15712yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—3141no—
214C22⋊C4—3141no—
215C22⋊C4—3141no—
216C22⋊C4—3141no—
217C22⋊C4—3141no—
218C42—3141no—
219C42C22⋊D15711yescharacteristic
220C42C22⋊D15711yescharacteristic
221C42C22×D15711yescharacteristic
222C42C22×D15711yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—3141no—
228C23C2×C22⋊D15711yescharacteristic
229Q8—3142no—
230D4—24no—
231D4—22no—
232D4—3142no—
233D4—6282no—
234D4#1457895414yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—3141no—
243C2×C4—3141no—
244C2×C4—3141no—
245C2×C4—6282no—
246C2×C4C2×C22⋊D15711yescharacteristic
247C2×C4C2×C22⋊D15711yescharacteristic
248C2×C4C2×C22⋊D15711yescharacteristic
249C2×C4C2×C22⋊D15711yescharacteristic
250C2×C4C2×C22⋊D15711yescharacteristic
251C2×C4C23×D15711yescharacteristic
252C8—3142no—
253C8—3142no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—6281no—
261C22—6281no—
262C22—6281no—
263C22#1909779712yes—
264C22C22×C22⋊D15711yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—6281no—
272C4—6281no—
273C4—6281no—
274C4C2×#1457895412yes—
275C2—22no—
276C2—44no—
277C2—12561no—
278C2#2375778411yescharacteristic
279C2C2×#1909779711yescharacteristic
280C2#2375807111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×C4×D4×C157C2
socleC22×C157C22×D4
prime radical#23757407C2
hypercenterC2×C4×D4D157
hypocenterC157#3114
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #23757807C2semidirect
2. #23757603C2semidirect
3. #23757586C2semidirect
4. #23757573C2semidirect
5. #23757488C2semidirect
6. #23757424C2semidirect
8. #23757383C2semidirect
9. #23757373C2semidirect
10. #23757330C2semidirect
11. C2×C4×D4×C157C2semidirect
15. #19097763C22semidirect
16. #19097759C22semidirect
20. C4×Dic314C22semidirect
21. C4×D628C22semidirect
24. #19097704C22semidirect
25. #19097702C22semidirect
26. C4⋊#11929434C22semidirect
27. #19097694C22semidirect
30. #19097691C22semidirect
33. #19097682C22semidirect
140. C157#3114semidirect
143. C2×C4×D4D157semidirect

Maximal subgroups

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

subgroupindex
#237578072
#237576032
#237575862
#237575732
#237574882
#237574242
#237574072
#237573832
#237573732
#237573302
C2×C4×D4×C1572
#3114157