Finite Group Databasebeta

← #34110230

Subgroups — #34110230

Order 20096.

444 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 123 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#23757825C211yesmaximal, characteristic, complemented
2C2×D4×Dic157C211yesmaximal, characteristic
3#23757609C211yesmaximal, characteristic, complemented
4#23757606C211yesmaximal, characteristic, complemented
5#23757551C212yesmaximal, complemented
6#23757476C212yesmaximal, complemented
7#23757424C211yesmaximal, characteristic
8#23757374C212yesmaximal, complemented
9#23757370C212yesmaximal, complemented
10#23757327C211yesmaximal, characteristic, complemented
11#450×C157C211yesmaximal, characteristic
12C23×Dic157—21no—
13C2×#14578942C2212yes—
14#19097770C2211yescharacteristic, complemented
15#19097769C2211yescharacteristic, complemented
16#19097761—21no—
17#19097750C2211yescharacteristic
18#19097748—21no—
19Q8×Dic157C2211yescharacteristic, complemented
20#19097736—21no—
21#19097736C2211yescharacteristic, complemented
22D4×Dic157—21no—
23D4×Dic157—21no—
24D4×Dic157C2212yes—
25D4×Dic157C2212yes—
26#19097727—21no—
27C2×C4⋊Dic157C2211yescharacteristic
28C2×C22⋊Dic157—21no—
29C2×C4×Dic157C2211yescharacteristic
30#19097705C2212yescomplemented
31#19097704C2212yescomplemented
32#19097700C2212yescomplemented
33#19097696C2212yescomplemented
34#19097695C2212yescomplemented
35#19097692C2212yes—
36C22⋊#11929434C2211yescharacteristic
37#19097681C2212yescomplemented
38C22×D4×C157C2211yescharacteristic
39C2×SD16×C157C2212yes—
40C22⋊Q8×C157C2211yescharacteristic
41#102×C157C2212yes—
42C22⋊C8×C157C2211yescharacteristic
43C2×Dic314—21no—
44C2×Dic314C2311yescharacteristic
45C22×Dic157—22no—
46C22×Dic157—22no—
47C22×Dic157—21no—
48C22×Dic157—44no—
49C22×Dic157D411yescharacteristic
50C22×Dic157D411yescharacteristic
51#14578942—24no—
52C2×#11929434C2312yes—
53C4⋊Dic157—21no—
54C4⋊Dic157—21no—
55C4⋊Dic157C2311yescharacteristic
56C4⋊Dic157C2312yes—
57#14578933—21no—
58#14578933—21no—
59#14578933—21no—
60#14578933—21no—
61#14578933—21no—
62C22⋊Dic157—22no—
63C22⋊Dic157—22no—
64C22⋊Dic157—21no—
65C4×Dic157—21no—
66C4×Dic157—21no—
67C4×Dic157D411yescharacteristic
68C4×Dic157D411yescharacteristic
69C4×Dic157C2311yescharacteristic
70C4×Dic157C2311yescharacteristic
71C24×C157—21no—
72C2×Q8×C157C2311yescharacteristic
73C2×D4×C157—24no—
74C2×D4×C157—21no—
75C2×D4×C157C2312yes—
76C22×C4×C157C2311yescharacteristic
77SD16×C157—24no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×C157—21no—
82Dic314—22no—
83Dic314—42no—
84C2×Dic157—22no—
85C2×Dic157—22no—
86C2×Dic157—22no—
87C2×Dic157—21no—
88C2×Dic157—21no—
89C2×Dic157—44no—
90C2×Dic157—42no—
91C2×Dic157—44no—
92C2×Dic157C2×D411yescharacteristic
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96#11929434—22no—
97C23×C157—24no—
98C23×C157—22no—
99C23×C157—42no—
100C23×C157—42no—
101C23×C157C2×D411yescharacteristic
102Q8×C157—22no—
103D4×C157—24no—
104D4×C157—22no—
105D4×C157#4114yes—
106C2×C4×C157—22no—
107C2×C4×C157—21no—
108C2×C4×C157—21no—
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2411yescharacteristic
111C8×C157—22no—
112C22×C157—21no—
113C22×C157—21no—
114C22×C157—24no—
115C22×C157—48no—
116C22×C157—42no—
117C22×C157—42no—
118C22×C157#13712yes—
119C22×C157C22×D411yescharacteristic
120C4×C157—21no—
121C4×C157—41no—
122C4×C157—41no—
123C4×C157C2×#4112yes—
124Dic157—21no—
125Dic157—21no—
126Dic157—21no—
127Dic157—41no—
128Dic157—42no—
129Dic157—41no—
130Dic157AGL(1,ℤ/8)12yes—
131C2×C157—22no—
132C2×C157—44no—
133C2×C157#52511yescharacteristic
134C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C157C2×#13711yescharacteristic
136C157#311011yescharacteristic, complemented, Sylow
137#3110—1571nomaximal, Sylow
138#533—1571no—
139C2×C4×D4—1571no—
140#484—1571no—
141#484—1571no—
142#474—1572no—
143#461—1572no—
144#450—1571no—
145#450D15711yescharacteristic
146#440—1572no—
147#440—1572no—
148#432—1571no—
149C22×D4C2×D15711yescharacteristic
150C23×C4—3141no—
151C2×SD16—1572no—
152C2×SD16C2×D15712yes—
153C4⋊Q8—1571no—
154C4⋊Q8—1571no—
155#125—3141no—
156C22⋊Q8—3141no—
157C22⋊Q8—1571no—
158C22⋊Q8C2×D15711yescharacteristic
159C4×Q8—3141no—
160C4×Q8—1571no—
161C4×Q8—1571no—
162C4×D4—3141no—
163C4×D4—3141no—
164C4×D4—1572no—
165C4×D4—1572no—
166#117—3141no—
167C2×C4⋊C4—1571no—
168C2×C22⋊C4—3141no—
169C2×C42—1571no—
170#107—1572no—
171#107—1572no—
172C4⋊C8—1572no—
173#103—1572no—
174#103—1572no—
175#102—1572no—
176#102C2×D15712yes—
177C22⋊C8—1571no—
178C22⋊C8C2×D15711yescharacteristic
179#97—1572no—
180C24—21no—
181C2×Q8—3141no—
182C2×Q8—1571no—
183C2×Q8C22×D15711yescharacteristic
184C2×D4—24no—
185C2×D4—21no—
186C2×D4C22×D15712yes—
187C22×C4—3141no—
188C22×C4—3142no—
189C22×C4—3142no—
190C22×C4—6284no—
191C22×C4—1571no—
192C22×C4—1571no—
193C22×C4C22×D15711yescharacteristic
194SD16—24no—
195SD16—3144no—
196C2×C8—1572no—
197C2×C8C22×D15712yes—
198C4⋊C4—21no—
199C4⋊C4—3141no—
200C4⋊C4—3141no—
201C4⋊C4—3141no—
202C4⋊C4—3141no—
203C4⋊C4—3141no—
204C4⋊C4—3141no—
205C4⋊C4—3141no—
206C4⋊C4—1571no—
207C4⋊C4—1572no—
208C4⋊C4C22×D15711yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—3141no—
211C22⋊C4—3142no—
212C22⋊C4—3142no—
213C42—3141no—
214C42—3141no—
215C42—1571no—
216C42—1571no—
217C42—1571no—
218C42—1571no—
219C23—24no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23D4×D15711yescharacteristic
224Q8—22no—
225Q8—3142no—
226Q8—6282no—
227D4—24no—
228D4—22no—
229D4#1457895314yes—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—3142no—
234C2×C4—3142no—
235C2×C4—3141no—
236C2×C4—3141no—
237C2×C4—3142no—
238C2×C4—6284no—
239C2×C4—6282no—
240C2×C4—6284no—
241C2×C4—1571no—
242C2×C4—1571no—
243C2×C4—1571no—
244C2×C4—1571no—
245C2×C4D4×D15711yescharacteristic
246C2×C4C23×D15711yescharacteristic
247C8—22no—
248C8—3142no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22#1909780012yes—
256C22C2×D4×D15711yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—3141no—
261C4—3141no—
262C4—3141no—
263C4—6281no—
264C4—6282no—
265C4—6281no—
266C4—1572no—
267C4C2×#1457895312yes—
268C2—22no—
269C2—44no—
270C2#2375777911yescharacteristic
271C2AGL(1,ℤ/8)×D15711yescharacteristic
272C2C2×#1909780011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radicalC22×D4×C157C22
hypercenter#450D157
hypocenterC157#3110
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23757825C2semidirect
3. #23757609C2semidirect
4. #23757606C2semidirect
5. #23757551C2semidirect
6. #23757476C2semidirect
8. #23757374C2semidirect
9. #23757370C2semidirect
10. #23757327C2semidirect
14. #19097770C22semidirect
15. #19097769C22semidirect
19. Q8×Dic157C22semidirect
21. #19097736C22semidirect
30. #19097705C22semidirect
31. #19097704C22semidirect
32. #19097700C22semidirect
33. #19097696C22semidirect
34. #19097695C22semidirect
37. #19097681C22semidirect
136. C157#3110semidirect

Maximal subgroups

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

subgroupindex
#237578252
C2×D4×Dic1572
#237576092
#237576062
#237575512
#237574762
#237574242
#237573742
#237573702
#237573272
#450×C1572
#3110157