Finite Group Databasebeta

← #34101162

Subgroups — #34101162

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
1#23757837C211yesmaximal, characteristic
2C2×#19097760C211yesmaximal, characteristic, complemented
3C2×#19097698C211yesmaximal, characteristic, complemented
4#23757282C211yesmaximal, characteristic, complemented
5C2×#19097687C211yesmaximal, characteristic, complemented
6#23757034C211yesmaximal, characteristic, complemented
7C2×#137×C157C211yesmaximal, characteristic, complemented
8C22×D628C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10#19097767—21no—
11#19097760—22no—
12#19097760—21no—
13#19097760—22no—
14#19097745—22no—
15#19097739—22no—
16#19097730C2211yescharacteristic
17C2×#14578931—22no—
18C2×C4×Dic157C2211yescharacteristic
19#19097698—22no—
20#19097695—22no—
21#19097687—22no—
22C2×#41×C157C2211yescharacteristic
23C22×Q8×C157C2211yescharacteristic
24#137×C157—22no—
25#137×C157—22no—
26C2×Q16×C157—21no—
27C2×SD16×C157—21no—
28C2×M16×C157C2211yescharacteristic
29C23×D157D412yes—
30C2×D628—21no—
31C2×D628—21no—
32C2×D628—21no—
33C2×D628—21no—
34C2×D628—21no—
35C2×D628—21no—
36C2×C22⋊D157—24no—
37C22×Dic157—21no—
38#14578938—21no—
39#14578938—21no—
40C2×#11929434—21no—
41C4⋊Dic157D412yes—
42C22⋊Dic157—21no—
43#14578931—28no—
44#14578931—24no—
45C4×Dic157—21no—
46C4×Dic157—21no—
47C4×Dic157D412yescomplemented
48#41×C157—21no—
49#41×C157—21no—
50#41×C157—21no—
51#41×C157—21no—
52C2×Q8×C157—21no—
53C2×Q8×C157—21no—
54C2×Q8×C157—21no—
55C2×Q8×C157—21no—
56C2×Q8×C157—21no—
57C2×Q8×C157D411yescharacteristic
58C2×Q8×C157D412yescomplemented
59C2×D4×C157—21no—
60C2×D4×C157D411yescharacteristic
61C22×C4×C157—21no—
62C22×C4×C157—21no—
63C22×C4×C157C2311yescharacteristic
64Q16×C157—24no—
65SD16×C157—24no—
66M16×C157—21no—
67M16×C157D412yescomplemented
68C2×C8×C157—21no—
69C22×D157—22no—
70C22×D157—22no—
71C22×D157—22no—
72C22×D157—44no—
73D628—41no—
74D628—41no—
75D628—41no—
76D628—41no—
77C22⋊D157—84no—
78C2×Dic157—22no—
79C2×Dic157—22no—
80C2×Dic157—42no—
81#11929434—41no—
82C23×C157—21no—
83C23×C157C2×D411yescharacteristic
84Q8×C157—22no—
85Q8×C157—21no—
86Q8×C157—22no—
87Q8×C157—21no—
88Q8×C157—41no—
89Q8×C157—41no—
90D4×C157—21no—
91D4×C157—21no—
92D4×C157—41no—
93D4×C157—41no—
94C2×C4×C157—21no—
95C2×C4×C157—22no—
96C2×C4×C157—41no—
97C2×C4×C157—41no—
98C2×C4×C157—41no—
99C2×C4×C157—41no—
100C2×C4×C157C2×D411yescharacteristic
101C2×C4×C157C2×D411yescharacteristic
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×D411yescharacteristic
105C2×C4×C157#4111yescharacteristic
106C8×C157—22no—
107C22×C157—22no—
108C22×C157—41no—
109C22×C157—42no—
110C22×C157#12011yescharacteristic
111C22×C157#12011yescharacteristic
112C22×C157#12111yescharacteristic
113C2×D157—44no—
114C2×D157—42no—
115C2×D157—44no—
116C2×D157—82no—
117C4×C157—41no—
118C4×C157—42no—
119C4×C157#12011yescharacteristic
120C4×C157#12111yescharacteristic
121C4×C157#12111yescharacteristic
122C4×C157#12711yescharacteristic
123Dic157—42no—
124Dic157—42no—
125C2×C157—21no—
126C2×C157—21no—
127C2×C157—81no—
128C2×C157#39211yescharacteristic
129C2×C157#45512yes—
130D157—82no—
131C157#200411yescharacteristic, complemented, Sylow
132#2004—1571nomaximal, Sylow
133C2×#137D15711yescharacteristic, complemented
134#535—1571no—
135C2×#124—1571no—
136C2×#104—1571no—
137#419—1571no—
138C2×#100—1571no—
139#339—1571no—
140C2×#41C2×D15711yescharacteristic
141C22×Q8C2×D15711yescharacteristic
142C22×D4—1571no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—1571no—
148C2×M16C2×D15711yescharacteristic
149#127—3141no—
150#124—3141no—
151#124—3142no—
152#124—3142no—
153#121—3142no—
154#120—3142no—
155#117—1571no—
156C2×C22⋊C4—3142no—
157C2×C42—1571no—
158#104—3142no—
159#103—3142no—
160#100—3142no—
161C24—1572no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D15711yescharacteristic
172C2×Q8C22⋊D15712yescomplemented
173C2×D4—21no—
174C2×D4—3144no—
175C2×D4—3141no—
176C2×D4—3141no—
177C2×D4—3141no—
178C2×D4—3141no—
179C2×D4—3141no—
180C2×D4—3141no—
181C2×D4C22⋊D15711yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—3141no—
185C22×C4C22×D15711yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—3141no—
190M16—3141no—
191M16C22⋊D15712yescomplemented
192C2×C8—21no—
193C2×C8—3141no—
194C4⋊C4—1572no—
195C22⋊C4—3148no—
196C22⋊C4—3144no—
197C22⋊C4—3141no—
198C42—3141no—
199C42—3141no—
200C42—1572no—
201C23—21no—
202C23—3142no—
203C23—3142no—
204C23—3142no—
205C23—6284no—
206C23D4×D15711yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—6281no—
218D4—6281no—
219D4—6281no—
220D4—6281no—
221D4—12564no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3142no—
229C2×C4—3142no—
230C2×C4—6282no—
231C2×C4D4×D15711yescharacteristic
232C2×C4D4×D15711yescharacteristic
233C2×C4C2×C22⋊D15711yescharacteristic
234C2×C4C2×C22⋊D15711yescharacteristic
235C2×C4C2×C22⋊D15711yescharacteristic
236C2×C4#1457895311yescharacteristic
237C8—22no—
238C8—6281no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—6284no—
243C22—6282no—
244C22—6284no—
245C22—12562no—
246C22#1909773911yescharacteristic
247C22#1909773911yescharacteristic
248C22#1909774211yescharacteristic
249C4—41no—
250C4—42no—
251C4—6282no—
252C4—6282no—
253C4#1909774011yescharacteristic
254C4#1909774311yescharacteristic
255C4#1909774311yescharacteristic
256C4#1909776711yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—12562no—
261C2#2375718111yescharacteristic
262C2#2375744912yes—
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#137×C157C2
socleC22×C157#120
prime radical#23757837C2
hypercenterC2×#137D157
hypocenterC157#2004
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#19097760C2semidirect
3. C2×#19097698C2semidirect
4. #23757282C2semidirect
5. C2×#19097687C2semidirect
6. #23757034C2semidirect
7. C2×#137×C157C2semidirect
47. C4×Dic157D4semidirect
58. C2×Q8×C157D4semidirect
67. M16×C157D4semidirect
131. C157#2004semidirect
133. C2×#137D157semidirect
172. C2×Q8C22⋊D157semidirect
191. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578372
C2×#190977602
C2×#190976982
#237572822
C2×#190976872
#237570342
C2×#137×C1572
#2004157