Finite Group Databasebeta

← #34109930

Subgroups — #34109930

Order 20096.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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#23757826C211yesmaximal, characteristic, complemented
2C2×D4×Dic157C211yesmaximal, characteristic
3#23757575C212yesmaximal, complemented
4#23757571C212yesmaximal, complemented
5#23757561C212yesmaximal, complemented
6#23757551C212yesmaximal, complemented
7#23757316C211yesmaximal, characteristic, complemented
8#23757315C211yesmaximal, characteristic, complemented
9#23757296C211yesmaximal, characteristic, complemented
10#23757278C211yesmaximal, characteristic
11#418×C157C211yesmaximal, characteristic
12C23×Dic157—21no—
13C2×#14578938C2211yescharacteristic
14#19097770C2211yescharacteristic, complemented
15#19097770C2211yescharacteristic, complemented
16#19097762C2211yescharacteristic, complemented
17#19097762C2211yescharacteristic, complemented
18#19097748—21no—
19#19097748—21no—
20C4×Dic314—21no—
21C4×Dic314—21no—
22D4×Dic157—21no—
23D4×Dic157—21no—
24D4×Dic157C2212yes—
25D4×Dic157C2212yes—
26#19097730C2211yescharacteristic
27C2×C4⋊Dic157C2211yescharacteristic
28C2×C22⋊Dic157—21no—
29C2×C4×Dic157C2211yescharacteristic
30#19097705C2212yescomplemented
31#19097704C2212yescomplemented
32#19097703C2212yescomplemented
33#19097702C2212yescomplemented
34#19097700C2212yescomplemented
35#19097700C2212yescomplemented
36#19097692C2212yes—
37#19097692C2212yes—
38C22×D4×C157C2211yescharacteristic
39C2×M16×C157C2211yescharacteristic
40#117×C157C2211yescharacteristic
41#102×C157C2212yes—
42#102×C157C2212yes—
43C2×Dic314—21no—
44C2×Dic314—21no—
45C22×Dic157—22no—
46C22×Dic157—22no—
47C22×Dic157—21no—
48C22×Dic157—44no—
49C22×Dic157D411yescharacteristic
50C22×Dic157D411yescharacteristic
51#14578938—22no—
52C2×#11929434C2312yes—
53C4⋊Dic157—21no—
54C4⋊Dic157—21no—
55C4⋊Dic157—21no—
56C4⋊Dic157C2311yescharacteristic
57C4⋊Dic157C2311yescharacteristic
58C4⋊Dic157C2312yes—
59#14578933—21no—
60#14578933—21no—
61#14578933—21no—
62#14578933—21no—
63C22⋊Dic157—22no—
64C22⋊Dic157—22no—
65C22⋊Dic157—21no—
66C4×Dic157—21no—
67C4×Dic157D411yescharacteristic
68C4×Dic157D411yescharacteristic
69C4×Dic157C2311yescharacteristic
70C4×Dic157C2311yescharacteristic
71C24×C157—21no—
72C2×D4×C157—21no—
73C2×D4×C157—22no—
74C2×D4×C157Q814yescomplemented
75C2×D4×C157C2312yes—
76C22×C4×C157C2311yescharacteristic
77M16×C157—22no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157C2311yescharacteristic
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×C157—21no—
82C42×C157—21no—
83Dic314—42no—
84Dic314—42no—
85C2×Dic157—22no—
86C2×Dic157—22no—
87C2×Dic157—21no—
88C2×Dic157—21no—
89C2×Dic157—41no—
90C2×Dic157—44no—
91C2×Dic157—42no—
92C2×Dic157—44no—
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97#11929434—22no—
98C23×C157—24no—
99C23×C157—22no—
100C23×C157—42no—
101C23×C157—42no—
102C23×C157C2×D411yescharacteristic
103D4×C157—24no—
104D4×C157—22no—
105D4×C157C2×Q814yes—
106C2×C4×C157—21no—
107C2×C4×C157—21no—
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×Q812yes—
110C2×C4×C157#4112yes—
111C2×C4×C157C2411yescharacteristic
112C8×C157—22no—
113C22×C157—21no—
114C22×C157—21no—
115C22×C157—24no—
116C22×C157—48no—
117C22×C157—42no—
118C22×C157—42no—
119C22×C157C22⋊Q812yes—
120C22×C157C22×D411yescharacteristic
121C4×C157—41no—
122C4×C157—41no—
123C4×C157C22⋊Q812yes—
124C4×C157C22×Q811yescharacteristic
125C4×C157C2×#4111yescharacteristic
126Dic157—21no—
127Dic157—21no—
128Dic157—21no—
129Dic157—42no—
130Dic157—41no—
131Dic157—41no—
132Dic157AGL(1,ℤ/8)12yes—
133C2×C157—22no—
134C2×C157—44no—
135C2×C157C2×C22⋊Q811yescharacteristic
136C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C157#57511yescharacteristic
138C157#307711yescharacteristic, complemented, Sylow
139#3077—1571nomaximal, Sylow
140#533—1571no—
141C2×C4×D4—1571no—
142#478—1572no—
143#478—1572no—
144#476—1572no—
145#474—1572no—
146#428—1571no—
147#428—1571no—
148#423—1571no—
149#418—1571no—
150#418D15711yescharacteristic
151C22×D4C2×D15711yescharacteristic
152C23×C4—3141no—
153C2×M16—1571no—
154C2×M16C2×D15711yescharacteristic
155C4⋊Q8—1571no—
156C4⋊Q8—1571no—
157#125—1571no—
158#125—1571no—
159C22⋊Q8—3141no—
160C22⋊Q8—3141no—
161C4×Q8—3141no—
162C4×Q8—3141no—
163C4×D4—3141no—
164C4×D4—3141no—
165C4×D4—1572no—
166C4×D4—1572no—
167#117—1571no—
168#117C2×D15711yescharacteristic
169C2×C4⋊C4—1571no—
170C2×C22⋊C4—3141no—
171C2×C42—1571no—
172#107—1572no—
173#107—1572no—
174#106—1572no—
175#106—1572no—
176C4⋊C8—1572no—
177C4⋊C8—1572no—
178#102—1572no—
179#102—1572no—
180#102C2×D15712yes—
181#102C2×D15712yes—
182C24—21no—
183C2×Q8—3141no—
184C2×Q8—3141no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic31414yescomplemented
188C2×D4C22×D15712yes—
189C22×C4—3141no—
190C22×C4—3142no—
191C22×C4—3142no—
192C22×C4—6284no—
193C22×C4—1571no—
194C22×C4—1571no—
195C22×C4C22×D15711yescharacteristic
196M16—22no—
197M16—3142no—
198C2×C8—1572no—
199C2×C8C22×D15712yes—
200C4⋊C4—3141no—
201C4⋊C4—3141no—
202C4⋊C4—3141no—
203C4⋊C4—3141no—
204C4⋊C4—3141no—
205C4⋊C4—3141no—
206C4⋊C4—3141no—
207C4⋊C4—1571no—
208C4⋊C4—1571no—
209C4⋊C4—1572no—
210C4⋊C4C22×D15711yescharacteristic
211C4⋊C4C22×D15711yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—3141no—
214C22⋊C4—3142no—
215C22⋊C4—3142no—
216C42—21no—
217C42—3141no—
218C42—1571no—
219C42—1571no—
220C42—1571no—
221C42—1571no—
222C23—24no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23D4×D15711yescharacteristic
227Q8—6282no—
228Q8—6282no—
229D4—24no—
230D4—22no—
231D4C2×Dic31414yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—3142no—
235C2×C4—3141no—
236C2×C4—3141no—
237C2×C4—3142no—
238C2×C4—6281no—
239C2×C4—6284no—
240C2×C4—6282no—
241C2×C4—6284no—
242C2×C4—1571no—
243C2×C4—1571no—
244C2×C4—1571no—
245C2×C4—1571no—
246C2×C4D4×D15711yescharacteristic
247C2×C4C2×Dic31412yes—
248C2×C4#1457895312yes—
249C2×C4C23×D15711yescharacteristic
250C8—22no—
251C8—3142no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#1909774812yes—
259C22C2×D4×D15711yescharacteristic
260C4—41no—
261C4—41no—
262C4—3141no—
263C4—3141no—
264C4—3141no—
265C4—6282no—
266C4—6281no—
267C4—6281no—
268C4—1572no—
269C4#1909774812yes—
270C4C22×Dic31411yescharacteristic
271C4C2×#1457895311yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#1909774811yescharacteristic
275C2AGL(1,ℤ/8)×D15711yescharacteristic
276C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#418×C157C2
socleC22×C157C22×D4
prime radicalC22×D4×C157C22
hypercenter#418D157
hypocenterC157#3077
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #23757826C2semidirect
3. #23757575C2semidirect
4. #23757571C2semidirect
5. #23757561C2semidirect
6. #23757551C2semidirect
7. #23757316C2semidirect
8. #23757315C2semidirect
9. #23757296C2semidirect
14. #19097770C22semidirect
15. #19097770C22semidirect
16. #19097762C22semidirect
17. #19097762C22semidirect
30. #19097705C22semidirect
31. #19097704C22semidirect
32. #19097703C22semidirect
33. #19097702C22semidirect
34. #19097700C22semidirect
35. #19097700C22semidirect
74. C2×D4×C157Q8semidirect
138. C157#3077semidirect
187. C2×D4Dic314semidirect

Maximal subgroups

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

subgroupindex
#237578262
C2×D4×Dic1572
#237575752
#237575712
#237575612
#237575512
#237573162
#237573152
#237572962
#237572782
#418×C1572
#3077157