Finite Group Databasebeta

← #34101165

Subgroups — #34101165

Order 20096.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19097802C211yesmaximal, characteristic, complemented
2#23757840C211yesmaximal, characteristic
3C2×#19097699C211yesmaximal, characteristic, complemented
4#23757282C211yesmaximal, characteristic, complemented
5C2×#19097688C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7C2×#124×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802—22no—
10#19097802—22no—
11C2×#14578944—21no—
12C2×#14578941—21no—
13C2×#14578938C2211yescharacteristic
14C2×#14578938C2211yescharacteristic
15#19097767—21no—
16#19097758—21no—
17#19097742—22no—
18#19097740—22no—
19#19097730C2211yescharacteristic
20#19097699—22no—
21#19097695—22no—
22#19097688—22no—
23C22×Q8×C157C2211yescharacteristic
24C22×D4×C157C2211yescharacteristic
25#124×C157—22no—
26#124×C157—22no—
27C2×C22⋊C4×C157—22no—
28C2×C42×C157C2211yescharacteristic
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32#14578954—21no—
33C2×Dic314D411yescharacteristic
34C2×D628D411yescharacteristic
35C2×C22⋊D157—24no—
36C2×C22⋊D157—21no—
37C2×C4×D157—21no—
38#14578944—24no—
39#14578941—24no—
40#14578938—21no—
41#14578938—21no—
42#14578938—21no—
43#14578938D412yescomplemented
44C2×#11929434—21no—
45C2×#11929434—21no—
46C4⋊Dic157D412yes—
47C22⋊Dic157—24no—
48C22⋊Dic157—21no—
49C4×Dic157—21no—
50C24×C157D412yes—
51C2×Q8×C157—21no—
52C2×Q8×C157—21no—
53C2×Q8×C157—21no—
54C2×Q8×C157—21no—
55C2×Q8×C157—21no—
56C2×Q8×C157D412yescomplemented
57C2×D4×C157—21no—
58C2×D4×C157—21no—
59C2×D4×C157—21no—
60C2×D4×C157—21no—
61C2×D4×C157—21no—
62C2×D4×C157—21no—
63C22×C4×C157—21no—
64C22×C4×C157—21no—
65C22×C4×C157C2311yescharacteristic
66C22⋊C4×C157—28no—
67C42×C157—21no—
68C42×C157D412yescomplemented
69C22×D157—21no—
70Dic314—21no—
71Dic314—21no—
72D628—21no—
73D628—21no—
74C22⋊D157—41no—
75C22⋊D157—41no—
76C22⋊D157—84no—
77C2×Dic157—21no—
78C2×Dic157—22no—
79C4×D157—41no—
80C4×D157—41no—
81#11929434—22no—
82#11929434—41no—
83C23×C157—22no—
84C23×C157—22no—
85C23×C157—22no—
86C23×C157—44no—
87C23×C157C2×D411yescharacteristic
88Q8×C157—22no—
89Q8×C157—22no—
90Q8×C157—41no—
91Q8×C157—41no—
92D4×C157—41no—
93D4×C157—41no—
94D4×C157—41no—
95D4×C157—41no—
96C2×C4×C157—22no—
97C2×C4×C157—22no—
98C2×C4×C157—42no—
99C2×C4×C157—41no—
100C2×C4×C157—41no—
101C2×C4×C157C2×D411yescharacteristic
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×D411yescharacteristic
105C2×C4×C157C2×D411yescharacteristic
106C2×C4×C157#4111yescharacteristic
107C22×C157—22no—
108C22×C157—44no—
109C22×C157—42no—
110C22×C157—44no—
111C22×C157—82no—
112C22×C157#12011yescharacteristic
113C22×C157#12011yescharacteristic
114C22×C157#12111yescharacteristic
115C2×D157—41no—
116C2×D157—42no—
117C4×C157—42no—
118C4×C157—42no—
119C4×C157#12011yescharacteristic
120C4×C157#12111yescharacteristic
121C4×C157#12111yescharacteristic
122C4×C157#12711yescharacteristic
123Dic157—41no—
124Dic157—42no—
125C2×C157—21no—
126C2×C157—21no—
127C2×C157—82no—
128C2×C157#39211yescharacteristic
129C2×C157#45512yes—
130D157—81no—
131C157#200411yescharacteristic, complemented, Sylow
132#2004—1571nomaximal, Sylow
133C2×#137—1571no—
134#535—1571no—
135C2×#124D15711yescharacteristic, complemented
136C2×#104—1571no—
137#419—1571no—
138C2×#100—1571no—
139#339—1571no—
140C2×#41—1571no—
141C22×Q8C2×D15711yescharacteristic
142C22×D4C2×D15711yescharacteristic
143#137—3142no—
144#137—3142no—
145C2×Q16—3141no—
146C2×SD16—3141no—
147C2×M16—1571no—
148C2×M16—1571no—
149#127—3141no—
150#124—22no—
151#124—22no—
152#124—3141no—
153#121—3142no—
154#120—3142no—
155#117—1571no—
156C2×C22⋊C4—22no—
157C2×C42C2×D15711yescharacteristic
158#104—3142no—
159#103—3142no—
160#100—3142no—
161C24C22⋊D15712yes—
162#41—3141no—
163#41—3141no—
164#41—3141no—
165#41—3141no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—1571no—
172C2×Q8C22⋊D15712yescomplemented
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—3141no—
180C2×D4—3144no—
181C2×D4—1571no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—3141no—
185C22×C4C22×D15711yescharacteristic
186Q16—3144no—
187SD16—3144no—
188M16—3141no—
189M16—3141no—
190M16—3141no—
191M16—1572no—
192C2×C8—3141no—
193C2×C8—3141no—
194C4⋊C4—1572no—
195C22⋊C4—28no—
196C22⋊C4—3144no—
197C22⋊C4—3141no—
198C42—21no—
199C42—3141no—
200C42C22⋊D15712yescomplemented
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—3141no—
206C23C2×C22⋊D15711yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—3141no—
212Q8—3141no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—3141no—
218D4—3141no—
219D4—6281no—
220D4—6281no—
221D4—12564no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—3142no—
228C2×C4—3141no—
229C2×C4—6281no—
230C2×C4—6281no—
231C2×C4D4×D15711yescharacteristic
232C2×C4D4×D15711yescharacteristic
233C2×C4D4×D15711yescharacteristic
234C2×C4C2×C22⋊D15711yescharacteristic
235C2×C4C2×C22⋊D15711yescharacteristic
236C2×C4#1457895311yescharacteristic
237C8—3142no—
238C8—6281no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—6281no—
245C22—6282no—
246C22#1909773911yescharacteristic
247C22#1909774011yescharacteristic
248C22#1909774311yescharacteristic
249C4—42no—
250C4—42no—
251C4—6281no—
252C4—6282no—
253C4#1909773911yescharacteristic
254C4#1909774211yescharacteristic
255C4#1909774311yescharacteristic
256C4#1909776711yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—12561no—
261C2#2375718111yescharacteristic
262C2#2375745012yes—
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#124×C157C2
socleC22×C157#120
prime radical#23757840C2
hypercenterC2×#124D157
hypocenterC157#2004
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#19097802C2semidirect
3. C2×#19097699C2semidirect
4. #23757282C2semidirect
5. C2×#19097688C2semidirect
6. #23757033C2semidirect
7. C2×#124×C157C2semidirect
43. #14578938D4semidirect
56. C2×Q8×C157D4semidirect
68. C42×C157D4semidirect
131. C157#2004semidirect
135. C2×#124D157semidirect
172. C2×Q8C22⋊D157semidirect
200. C42C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978022
#237578402
C2×#190976992
#237572822
C2×#190976882
#237570332
C2×#124×C1572
#2004157