Finite Group Databasebeta

← #34109500

Subgroups — #34109500

Order 20096.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#23758071C212yesmaximal, complemented
2#23757847C211yesmaximal, characteristic, complemented
3#23757456C214yesmaximal, complemented
4#23757446C214yesmaximal, complemented
5#23757290C212yesmaximal, complemented
6C2×#19097690C211yesmaximal, characteristic, complemented
7C2×2−1+4×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802—22no—
10#19097801C2214yescomplemented
11#19097797—22no—
12#19097796C2214yescomplemented
13#19097791—24no—
14C2×#14578938C2212yes—
15#19097769C2212yescomplemented
16#19097760C2212yescomplemented
17#19097754—22no—
18#19097747—22no—
19#19097730C2211yescharacteristic
20#19097698C2218yescomplemented
21#19097690C2214yescomplemented
222−1+4×C157—24no—
232−1+4×C157—22no—
242−1+4×C157C2214yes—
25C2×#41×C157—22no—
26C2×#41×C157—22no—
27C2×#41×C157C2212yes—
28C22×Q8×C157—22no—
29C22×Q8×C157C2211yescharacteristic
30#14578954—22no—
31#14578954—22no—
32C2×Dic314C2311yescharacteristic
33C2×D628C2311yescharacteristic
34C2×C22⋊D157—21no—
35C2×C4×D157—21no—
36#14578944—24no—
37#14578942—24no—
38#14578941—24no—
39#14578939—24no—
40#14578938—22no—
41#14578938C2314yes—
42C2×#11929434—22no—
43C4⋊Dic157—21no—
44#14578933—24no—
45C22⋊Dic157—21no—
46#14578931—24no—
47C4×Dic157C2312yes—
48#41×C157—28no—
49#41×C157—28no—
50#41×C157—28no—
51#41×C157—28no—
52#41×C157—22no—
53#41×C157—22no—
54#41×C157D414yes—
55#41×C157C2314yes—
56C2×Q8×C157—22no—
57C2×Q8×C157—28no—
58C2×Q8×C157—28no—
59C2×Q8×C157—22no—
60C2×Q8×C157—22no—
61C2×Q8×C157D412yes—
62C2×Q8×C157D412yes—
63C2×Q8×C157C2312yes—
64C2×D4×C157—22no—
65C2×D4×C157—22no—
66C2×D4×C157D412yes—
67C22×C4×C157—24no—
68C22×C4×C157—22no—
69C22×C4×C157D412yes—
70C22×C4×C157C2311yescharacteristic
71C22×D157—21no—
72Dic314—22no—
73D628—22no—
74C22⋊D157—41no—
75C22⋊D157—41no—
76C2×Dic157—21no—
77C2×Dic157—22no—
78C4×D157—42no—
79#11929434—24no—
80C23×C157—22no—
81C23×C157C2×D411yescharacteristic
82Q8×C157—28no—
83Q8×C157—22no—
84Q8×C157—24no—
85Q8×C157—24no—
86Q8×C157C2×D414yes—
87D4×C157—28no—
88D4×C157—24no—
89D4×C157—24no—
90D4×C157—22no—
91D4×C157C2×D414yes—
92C2×C4×C157—216no—
93C2×C4×C157—22no—
94C2×C4×C157—22no—
95C2×C4×C157—22no—
96C2×C4×C157—24no—
97C2×C4×C157—24no—
98C2×C4×C157C2×D414yes—
99C2×C4×C157C2×D411yescharacteristic
100C2×C4×C157C2×D412yes—
101C2×C4×C157C2×D412yes—
102C2×C4×C157C2411yescharacteristic
103C22×C157—24no—
104C22×C157—41no—
105C22×C157—44no—
106C22×C157#12011yescharacteristic
107C22×C157#12011yescharacteristic
108C22×C157C22×D411yescharacteristic
109C2×D157—42no—
110C2×D157—41no—
111C4×C157—24no—
112C4×C157—24no—
113C4×C157#12012yes—
114C4×C157C22×D412yes—
115Dic157—42no—
116Dic157—41no—
117C2×C157—21no—
118C2×C157—21no—
119C2×C157—21no—
120C2×C157—44no—
121C2×C157C2×#12011yescharacteristic
122D157—81no—
123C157#302011yescharacteristic, complemented, Sylow
124#3020—1571nomaximal, Sylow
125C2×2−1+4D15711yescharacteristic, complemented
126#575—1572no—
127#536—1571no—
128#456—1574no—
129#454—1574no—
130#421—1572no—
131C2×#101—1571no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D15714yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—1571no—
138C2×#41C2×D15712yes—
139C22×Q8—22no—
140C22×Q8C2×D15711yescharacteristic
141#137—3142no—
142#137—1574no—
143AGL(1,ℤ/8)—3142no—
144AGL(1,ℤ/8)—1574no—
145#135—3144no—
146C2×M16—1572no—
147C4⋊Q8—1572no—
148#124—1572no—
149#123—3142no—
150C22⋊Q8—3142no—
151#117—1571no—
152#104—1578no—
153#101—1574no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—3142no—
161#41—3142no—
162#41C22⋊D15714yes—
163#41C22×D15714yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—1571no—
170C2×Q8C22⋊D15712yes—
171C2×Q8C22⋊D15712yes—
172C2×Q8C22×D15712yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—3141no—
176C2×D4—1571no—
177C2×D4C22⋊D15712yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—3141no—
181C22×C4C22⋊D15712yes—
182C22×C4C22×D15711yescharacteristic
183Q16—3144no—
184SD16—3144no—
185SD16—3144no—
186D8—3144no—
187M16—3142no—
188M16—1574no—
189C2×C8—3142no—
190C4⋊C4—3141no—
191C4⋊C4—3144no—
192C22⋊C4—3141no—
193C22⋊C4—3144no—
194C42—1572no—
195C23—22no—
196C23—3141no—
197C23C2×C22⋊D15711yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—3142no—
203Q8C2×C22⋊D15714yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—3142no—
209D4—6281no—
210D4—6281no—
211D4C2×C22⋊D15714yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—3142no—
219C2×C4—3141no—
220C2×C4—6282no—
221C2×C4C2×C22⋊D15714yes—
222C2×C4C2×C22⋊D15711yescharacteristic
223C2×C4C2×C22⋊D15712yes—
224C2×C4C2×C22⋊D15712yes—
225C2×C4C23×D15711yescharacteristic
226C8—3144no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—6282no—
231C22—6281no—
232C22#1909774011yescharacteristic
233C22#1909774011yescharacteristic
234C22C22×C22⋊D15711yescharacteristic
235C4—24no—
236C4—24no—
237C4—6282no—
238C4—6281no—
239C4#1909774012yes—
240C4C22×C22⋊D15712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—12561no—
246C2C2×#1909774011yescharacteristic
subgroupG/H
centerC2C2×#19097740
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×2−1+4×C157C2
socleC2×C157C2×#120
hypercenterC2×2−1+4D157
hypocenterC157#3020
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #23758071C2semidirect
2. #23757847C2semidirect
3. #23757456C2semidirect
4. #23757446C2semidirect
5. #23757290C2semidirect
6. C2×#19097690C2semidirect
7. C2×2−1+4×C157C2semidirect
10. #19097801C22semidirect
12. #19097796C22semidirect
15. #19097769C22semidirect
16. #19097760C22semidirect
20. #19097698C22semidirect
21. #19097690C22semidirect
123. C157#3020semidirect
125. C2×2−1+4D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237578472
#237574562
#237574462
#237572902
C2×#190976902
C2×2−1+4×C1572
#3020157