Finite Group Databasebeta

← C2×#20245174 #26369310

Subgroups — C2×#20245174 #26369310

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093964C211yesmaximal, characteristic
2C2×#15689161C211yesmaximal, characteristic
3#20245174C218yesmaximal, direct product factor
4C2×#15688950C212yesmaximal, complemented
5C2×#15688948C212yesmaximal, complemented
6C22×#107×C47C211yesmaximal, characteristic
7C2×#13093983—21no—
8C2×#13093964C418yescomplemented
9C2×#13093964C2214yes—
10C22×C47⋊C16C2212yes—
11#15689161—22no—
12#15689161C2214yes—
13C2×#13093958C2211yescharacteristic
14C2×#13093958C2211yescharacteristic
15C2×#13093956—21no—
16#15688950C2218yescomplemented
17#15688948C2218yescomplemented
18C23×C8×C47C2211yescharacteristic
19C22×C4⋊C4×C47—21no—
20C2×#107×C47—21no—
21C2×#107×C47—22no—
22C2×#107×C47C2212yes—
23C2×#107×C47C2214yes—
24#13093983—24no—
25#13093983—42no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C22⋊Dic47—41no—
29C2×C4×Dic47—41no—
30#13093964C2×C4116yescomplemented
31C2×C47⋊C16C2×C418yescomplemented
32C2×C47⋊C16C2314yes—
33#13093958—21no—
34#13093958—21no—
35#13093958C2312yes—
36#13093958C2312yes—
37#13093956—24no—
38C23×C4×C47—41no—
39C23×C4×C47D411yescharacteristic
40C22×C8×C47C2×C414yes—
41C22×C8×C47D411yescharacteristic
42C22×C8×C47D412yes—
43C22×C8×C47Q814yes—
44C22×C8×C47C2311yescharacteristic
45C22×C8×C47C2312yes—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47—24no—
48C2×C4⋊C4×C47—41no—
49C2×C4⋊C4×C47—42no—
50#107×C47—24no—
51#107×C47—22no—
52#107×C47C2314yes—
53C24×C47D811yescharacteristic
54C22×C4×C47—44no—
55C22×C4×C47—42no—
56C22×C4×C47—84no—
57C22×C4×C47C4⋊C414yes—
58C22×C4×C47D811yescharacteristic
59C22×C4×C47D812yes—
60C22×C4×C47Q1614yes—
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D412yes—
63C22×Dic47—41no—
64C22×Dic47—41no—
65C2×C8×C47C4⋊C414yes—
66C2×C8×C47C4⋊C418yes—
67C2×C8×C47C22×C412yes—
68C2×C8×C47C22×C414yes—
69C2×C8×C47C2×D412yes—
70C2×C8×C47C2×D411yescharacteristic
71C2×C8×C47C2×Q812yes—
72C2×C8×C47C2×Q814yes—
73C2×C8×C47C2411yescharacteristic
74C4⋊C4×C47—24no—
75C4⋊C4×C47—44no—
76C4⋊C4×C47—42no—
77C22⋊Dic47—44no—
78C4⋊Dic47—22no—
79C4⋊Dic47—22no—
80C4⋊Dic47—41no—
81C4⋊Dic47—41no—
82C4×Dic47—44no—
83C47⋊C16C22×C418yescomplemented
84C23×C47—22no—
85C23×C47—22no—
86C23×C47#10714yes—
87C23×C47C2×D812yes—
88C23×C47C2×D811yescharacteristic
89C2×C4×C47—44no—
90C2×C4×C47—88no—
91C2×C4×C47—84no—
92C2×C4×C47#10718yes—
93C2×C4×C47#10714yes—
94C2×C4×C47C2×C4⋊C414yes—
95C2×C4×C47C2×C4⋊C412yes—
96C2×C4×C47C2×D812yes—
97C2×C4×C47C2×D811yescharacteristic
98C2×C4×C47C2×Q1614yes—
99C2×C4×C47C2×Q1612yes—
100C2×C4×C47C22×D411yescharacteristic
101C2×Dic47—42no—
102C2×Dic47—42no—
103C2×Dic47—82no—
104C2×Dic47—82no—
105C8×C47C2×C4⋊C412yes—
106C8×C47C2×C4⋊C414yes—
107C8×C47C23×C411yescharacteristic
108C8×C47C22×Q811yescharacteristic
109C2×#2159—471nomaximal, Sylow
110C22×C47—22no—
111C22×C47—22no—
112C22×C47—28no—
113C22×C47C2×#10714yes—
114C22×C47C2×#10712yes—
115C22×C47#50912yes—
116C22×C47#51012yes—
117C22×C47C22×D811yescharacteristic
118C4×C47—84no—
119C4×C47C2×#10714yes—
120C4×C47C2×#10712yes—
121C4×C47C22×C4⋊C411yescharacteristic
122C4×C47C22×Q1611yescharacteristic
123Dic47—82no—
124Dic47—82no—
125C22×M32—471no—
126C2×#427—471no—
127C22×#107D4711yescharacteristic
128#2159—478no—
129C2×#367—472no—
130C2×#366—472no—
131C2×C47—24no—
132C2×C47#215914yescomplemented
133C2×C47C22×#10711yescharacteristic
134C2×C47C2×#50911yescharacteristic
135C2×C47C2×#51011yescharacteristic
136C23×C8C2×D4711yescharacteristic
137C2×#117—941no—
138C22×C4⋊C4—21no—
139C2×M32—478no—
140C2×M32—474no—
141C22×C16—472no—
142#427—474no—
143#427—942no—
144C2×#107—21no—
145C2×#107—22no—
146C2×#107—471no—
147C2×#107—471no—
148C2×#107C2×D4712yes—
149C2×#107C2×D4714yes—
150C2×#106—941no—
151#367—478no—
152#366—478no—
153C47C2×#215911yescharacteristic, complemented, Sylow
154C23×C4—41no—
155C23×C4C22⋊D4711yescharacteristic
156C22×C8Dic9414yes—
157C22×C8C4×D4714yes—
158C22×C8C22⋊D4711yescharacteristic
159C22×C8C22⋊D4712yes—
160C22×C8C22×D4711yescharacteristic
161C22×C8C22×D4712yes—
162#117—944no—
163#117—1882no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—941no—
169C2×C4⋊C4—941no—
170C2×C22⋊C4—1881no—
171C2×C42—1881no—
172M32—4716no—
173C2×C16—478no—
174C2×C16—474no—
175#107—24no—
176#107—22no—
177#107—472no—
178#107—472no—
179#107—941no—
180#107—941no—
181#107C22×D4714yes—
182#106—944no—
183C24#1062533411yescharacteristic
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—84no—
187C22×C4—1881no—
188C22×C4—1881no—
189C22×C4Dic47⋊C414yes—
190C22×C4#1062533412yes—
191C22×C4#1062533411yescharacteristic
192C22×C4#1062533714yes—
193C22×C4C2×C22⋊D4712yes—
194C22×C4C2×C22⋊D4711yescharacteristic
195C2×C8Dic47⋊C414yes—
196C2×C8Dic47⋊C418yes—
197C2×C8C2×Dic9412yes—
198C2×C8C2×Dic9414yes—
199C2×C8C2×C4×D4712yes—
200C2×C8C2×C4×D4714yes—
201C2×C8C2×C22⋊D4712yes—
202C2×C8C2×C22⋊D4711yescharacteristic
203C2×C8C23×D4711yescharacteristic
204C4⋊C4—24no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—942no—
208C4⋊C4—942no—
209C4⋊C4—1881no—
210C4⋊C4—1881no—
211C22⋊C4—1884no—
212C42—1884no—
213C16—478no—
214C23—22no—
215C23—22no—
216C23#1309395714yes—
217C23C2×#1062533412yes—
218C23C2×#1062533411yescharacteristic
219C2×C4—44no—
220C2×C4—88no—
221C2×C4—84no—
222C2×C4—3762no—
223C2×C4—3762no—
224C2×C4—1882no—
225C2×C4—1882no—
226C2×C4#1309395718yes—
227C2×C4#1309395714yes—
228C2×C4C2×Dic47⋊C414yes—
229C2×C4C2×Dic47⋊C412yes—
230C2×C4C2×#1062533412yes—
231C2×C4C2×#1062533411yescharacteristic
232C2×C4C2×#1062533714yes—
233C2×C4C2×#1062533712yes—
234C2×C4C22×C22⋊D4711yescharacteristic
235C8C2×Dic47⋊C412yes—
236C8C2×Dic47⋊C414yes—
237C8C22×C4×D4711yescharacteristic
238C8C22×Dic9411yescharacteristic
239C22—22no—
240C22—28no—
241C22—22no—
242C22C2×#1309395714yes—
243C22C2×#1309395712yes—
244C22#1568955912yes—
245C22#1568956412yes—
246C22C22×#1062533411yescharacteristic
247C4—84no—
248C4—3762no—
249C4—3762no—
250C4C2×#1309395714yes—
251C4C2×#1309395712yes—
252C4C22×Dic47⋊C411yescharacteristic
253C4C22×#1062533711yescharacteristic
254C2—24no—
255C2#2024517414yesdirect product factor
256C2C22×#1309395711yescharacteristic
257C2C2×#1568955911yescharacteristic
258C2C2×#1568956411yescharacteristic
subgroupG/H
centerC23C2×#10625334
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC22×#107×C47C2
socleC23×C47C2×D8
prime radicalC24×C47D8
cosocleC2×C8×C47C24
hypercenterC22×#107D47
hypocenterC47C2×#2159
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
3. #20245174C2direct (N × H)
4. C2×#15688950C2semidirect
5. C2×#15688948C2semidirect
8. C2×#13093964C4semidirect
16. #15688950C22semidirect
17. #15688948C22semidirect
30. #13093964C2×C4semidirect
31. C2×C47⋊C16C2×C4semidirect
83. C47⋊C16C22×C4semidirect
132. C2×C47#2159semidirect
153. C47C2×#2159semidirect
255. C2#20245174direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939642
C2×#156891612
#202451742
C2×#156889502
C2×#156889482
C22×#107×C472
C2×#215947