Finite Group Databasebeta

← #34109514

Subgroups — #34109514

Order 20096.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#23758109C211yesmaximal, characteristic
2C2×#19097802C212yesmaximal, complemented
3#23757455C214yesmaximal, complemented
4#23757450C214yesmaximal, complemented
5#23757291C212yesmaximal, complemented
6#23757257C211yesmaximal, characteristic, complemented
7#536×C157C211yesmaximal, characteristic, complemented
8#19097819C2212yes—
9#19097818—22no—
10#19097817—22no—
11#19097816—22no—
12#19097816C2212yes—
13C2×#14578954—22no—
14C2×#14578954C2212yes—
15#41×D157—24no—
16#19097802—24no—
17#19097802C2218yescomplemented
18C2×#14578944—22no—
19C2×#14578941—22no—
20C2×#14578938C2212yescomplemented
21#19097699C2218yescomplemented
22#19097690C2212yescomplemented
23#19097688C2212yescomplemented
24C2×#41×C157C2211yescharacteristic
25C22×Q8×C157C2211yescharacteristic, complemented
26C4⋊Q8×C157C2212yescomplemented
27#124×C157C2212yescomplemented
28#123×C157—22no—
29C22⋊Q8×C157—22no—
30#117×C157C2211yescharacteristic, complemented
31#14578955—24no—
32#14578954—24no—
33#14578954—24no—
34#14578954—22no—
35#14578954—24no—
36#14578954—22no—
37#14578954D414yes—
38#14578954C2314yes—
39#14578953—28no—
40#14578953—24no—
41C2×Dic314—22no—
42C2×Dic314D412yes—
43Q8×D157—24no—
44C2×D628—22no—
45C2×D628D412yes—
46C2×C22⋊D157—24no—
47C2×C22⋊D157—22no—
48D4×D157—28no—
49D4×D157—24no—
50C2×C4×D157—24no—
51C2×C4×D157—22no—
52#14578944—28no—
53#14578941—28no—
54#14578938—22no—
55#14578938C2314yes—
56C2×#11929434—22no—
57#41×C157—22no—
58#41×C157—22no—
59C2×Q8×C157—21no—
60C2×Q8×C157—22no—
61C2×Q8×C157—22no—
62C2×Q8×C157C2312yes—
63C2×Q8×C157C2311yescharacteristic
64C2×D4×C157D412yes—
65C2×D4×C157C2311yescharacteristic
66C22×C4×C157—21no—
67C22×C4×C157D412yes—
68C22×C4×C157C2311yescharacteristic
69C4⋊C4×C157—21no—
70C4⋊C4×C157—24no—
71C22⋊C4×C157—21no—
72C22⋊C4×C157—24no—
73C42×C157C2312yes—
74C22×D157—24no—
75C22×D157—22no—
76Dic314—24no—
77Dic314—22no—
78Dic314C2×D414yes—
79D628—24no—
80D628—22no—
81D628C2×D414yes—
82C22⋊D157—28no—
83C22⋊D157—24no—
84C22⋊D157—24no—
85C2×Dic157—24no—
86C2×Dic157—22no—
87C4×D157—28no—
88C4×D157—24no—
89C4×D157—24no—
90#11929434—24no—
91C23×C157C2×D412yes—
92C23×C157C2×D411yescharacteristic
93Q8×C157—22no—
94Q8×C157—24no—
95Q8×C157—41no—
96Q8×C157—41no—
97D4×C157—24no—
98D4×C157—22no—
99C2×C4×C157—22no—
100C2×C4×C157—22no—
101C2×C4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157—42no—
104C2×C4×C157C2×D412yes—
105C2×C4×C157C2×D412yes—
106C2×C4×C157C2×D412yes—
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2411yescharacteristic
109C22×C157—22no—
110C22×C157—41no—
111C22×C157—42no—
112C22×C157#12012yes—
113C22×C157C22×D411yescharacteristic
114C2×D157—24no—
115C2×D157—82no—
116C2×D157—84no—
117C4×C157—22no—
118C4×C157—42no—
119C4×C157—42no—
120C4×C157#12011yescharacteristic
121C4×C157#12011yescharacteristic
122C4×C157C22×D412yes—
123Dic157—24no—
124C2×C157—21no—
125C2×C157—22no—
126C2×C157—42no—
127C2×C157C2×#12011yescharacteristic
128D157—44no—
129C157#302211yescharacteristic, complemented, Sylow
130#3022—1571nomaximal, Sylow
131#585—1571no—
132C2×#137—1572no—
133#536D15711yescharacteristic, complemented
134#456—1574no—
135#455—1574no—
136#421—1572no—
137#413—1571no—
1382−1+4—3142no—
1392−1+4—1572no—
140H(5,𝔽2)—3142no—
141H(5,𝔽2)—3142no—
142H(5,𝔽2)—1572no—
143C2×#41—3144no—
144C2×#41—3142no—
145C2×#41—1572no—
146C2×#41C2×D15711yescharacteristic
147C22×Q8C2×D15711yescharacteristic, complemented
148#137—3144no—
149#137—1578no—
150C2×Q16—3142no—
151C2×SD16—3142no—
152C2×M16—1572no—
153C4⋊Q8C2×D15712yescomplemented
154#124C2×D15712yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D15711yescharacteristic, complemented
158#104—1578no—
159#101—1572no—
160#100—1572no—
161#41—22no—
162#41—22no—
163#41—3144no—
164#41—3144no—
165#41—3148no—
166#41—3144no—
167#41—3144no—
168#41—3142no—
169#41—3142no—
170#41—3144no—
171#41—1574no—
172#41—1574no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—3142no—
177C2×Q8—3144no—
178C2×Q8—1572no—
179C2×Q8C22×D15712yes—
180C2×Q8C22×D15711yescharacteristic
181C2×D4—3142no—
182C2×D4—3148no—
183C2×D4—3144no—
184C2×D4—3144no—
185C2×D4—3142no—
186C2×D4—1572no—
187C2×D4C22⋊D15712yes—
188C2×D4C22×D15711yescharacteristic
189C22×C4—21no—
190C22×C4—3144no—
191C22×C4—3142no—
192C22×C4C22⋊D15712yes—
193C22×C4C22×D15711yescharacteristic
194Q16—3148no—
195SD16—3148no—
196M16—3142no—
197M16—1574no—
198C2×C8—3142no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D15712yes—
204C23—3144no—
205C23—3142no—
206C23C2×C22⋊D15712yes—
207C23C2×C22⋊D15711yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—3144no—
213Q8—3142no—
214Q8—1574no—
215D4—24no—
216D4—22no—
217D4—3144no—
218D4—3148no—
219D4—3144no—
220D4—3144no—
221D4—3142no—
222D4—1574no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—3148no—
229C2×C4—3144no—
230C2×C4—3142no—
231C2×C4—3144no—
232C2×C4—3144no—
233C2×C4D4×D15712yes—
234C2×C4D4×D15712yes—
235C2×C4C2×C22⋊D15712yes—
236C2×C4C2×C22⋊D15711yescharacteristic
237C2×C4C23×D15711yescharacteristic
238C8—3144no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—3144no—
243C22—12562no—
244C22—12564no—
245C22#1909773912yes—
246C22C22×C22⋊D15711yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—3144no—
251C4#1909773911yescharacteristic
252C4#1909773911yescharacteristic
253C4C2×D4×D15712yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—6284no—
258C2C2×#1909773911yescharacteristic
subgroupG/H
centerC2C2×#19097739
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#536×C157C2
socleC2×C157C2×#120
prime radical#23758109C2
hypercenter#536D157
hypocenterC157#3022
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#19097802C2semidirect
3. #23757455C2semidirect
4. #23757450C2semidirect
5. #23757291C2semidirect
6. #23757257C2semidirect
7. #536×C157C2semidirect
17. #19097802C22semidirect
20. C2×#14578938C22semidirect
21. #19097699C22semidirect
22. #19097690C22semidirect
23. #19097688C22semidirect
25. C22×Q8×C157C22semidirect
26. C4⋊Q8×C157C22semidirect
27. #124×C157C22semidirect
30. #117×C157C22semidirect
129. C157#3022semidirect
133. #536D157semidirect
147. C22×Q8C2×D157semidirect
153. C4⋊Q8C2×D157semidirect
154. #124C2×D157semidirect
157. #117C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
C2×#190978022
#237574552
#237574502
#237572912
#237572572
#536×C1572
#3022157