Finite Group Databasebeta

← C2×#20276501 #27537124

Subgroups — C2×#20276501 #27537124

Order 13568.

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×#13133157C211yesmaximal, characteristic
2C2×#16810661C211yesmaximal, characteristic
3#20276501C218yesmaximal, direct product factor
4C2×#16810450C212yesmaximal, complemented
5C2×#16810448C212yesmaximal, complemented
6C22×#107×C53C211yesmaximal, characteristic
7C2×#13133176—21no—
8C2×#13133157C418yescomplemented
9C2×#13133157C2214yes—
10C22×#11719908C2212yes—
11#16810661—22no—
12#16810661C2214yes—
13C2×#13133151C2211yescharacteristic
14C2×#13133151C2211yescharacteristic
15C2×#13133149—21no—
16#16810450C2218yescomplemented
17#16810448C2218yescomplemented
18C23×C8×C53C2211yescharacteristic
19C22×C4⋊C4×C53—21no—
20C2×#107×C53—21no—
21C2×#107×C53—22no—
22C2×#107×C53C2212yes—
23C2×#107×C53C2214yes—
24#13133176—24no—
25#13133176—42no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C22⋊Dic53—41no—
29C2×C4×Dic53—41no—
30#13133157C2×C4116yescomplemented
31C2×#11719908C2×C418yescomplemented
32C2×#11719908C2314yes—
33#13133151—21no—
34#13133151—21no—
35#13133151C2312yes—
36#13133151C2312yes—
37#13133149—24no—
38C23×C4×C53—41no—
39C23×C4×C53D411yescharacteristic
40C22×C8×C53C2×C414yes—
41C22×C8×C53D411yescharacteristic
42C22×C8×C53D412yes—
43C22×C8×C53Q814yes—
44C22×C8×C53C2311yescharacteristic
45C22×C8×C53C2312yes—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53—24no—
48C2×C4⋊C4×C53—41no—
49C2×C4⋊C4×C53—42no—
50#107×C53—24no—
51#107×C53—22no—
52#107×C53C2314yes—
53C24×C53D811yescharacteristic
54C22×C4×C53—44no—
55C22×C4×C53—42no—
56C22×C4×C53—84no—
57C22×C4×C53C4⋊C414yes—
58C22×C4×C53D811yescharacteristic
59C22×C4×C53D812yes—
60C22×C4×C53Q1614yes—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D412yes—
63C22×Dic53—41no—
64C22×Dic53—41no—
65C2×C8×C53C4⋊C414yes—
66C2×C8×C53C4⋊C418yes—
67C2×C8×C53C22×C412yes—
68C2×C8×C53C22×C414yes—
69C2×C8×C53C2×D412yes—
70C2×C8×C53C2×D411yescharacteristic
71C2×C8×C53C2×Q812yes—
72C2×C8×C53C2×Q814yes—
73C2×C8×C53C2411yescharacteristic
74C4⋊C4×C53—24no—
75C4⋊C4×C53—44no—
76C4⋊C4×C53—42no—
77C22⋊Dic53—44no—
78C4⋊Dic53—22no—
79C4⋊Dic53—22no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4×Dic53—44no—
83#11719908C22×C418yescomplemented
84C23×C53—22no—
85C23×C53—22no—
86C23×C53#10714yes—
87C23×C53C2×D812yes—
88C23×C53C2×D811yescharacteristic
89C2×C4×C53—44no—
90C2×C4×C53—88no—
91C2×C4×C53—84no—
92C2×C4×C53#10718yes—
93C2×C4×C53#10714yes—
94C2×C4×C53C2×C4⋊C414yes—
95C2×C4×C53C2×C4⋊C412yes—
96C2×C4×C53C2×D812yes—
97C2×C4×C53C2×D811yescharacteristic
98C2×C4×C53C2×Q1614yes—
99C2×C4×C53C2×Q1612yes—
100C2×C4×C53C22×D411yescharacteristic
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—82no—
104C2×Dic53—82no—
105C8×C53C2×C4⋊C412yes—
106C8×C53C2×C4⋊C414yes—
107C8×C53C23×C411yescharacteristic
108C8×C53C22×Q811yescharacteristic
109C2×#2159—531nomaximal, Sylow
110C22×C53—22no—
111C22×C53—22no—
112C22×C53—28no—
113C22×C53C2×#10714yes—
114C22×C53C2×#10712yes—
115C22×C53#50912yes—
116C22×C53#51012yes—
117C22×C53C22×D811yescharacteristic
118C4×C53—84no—
119C4×C53C2×#10714yes—
120C4×C53C2×#10712yes—
121C4×C53C22×C4⋊C411yescharacteristic
122C4×C53C22×Q1611yescharacteristic
123Dic53—82no—
124Dic53—82no—
125C22×M32—531no—
126C2×#427—531no—
127C22×#107D5311yescharacteristic
128#2159—538no—
129C2×#367—532no—
130C2×#366—532no—
131C2×C53—24no—
132C2×C53#215914yescomplemented
133C2×C53C22×#10711yescharacteristic
134C2×C53C2×#50911yescharacteristic
135C2×C53C2×#51011yescharacteristic
136C23×C8C2×D5311yescharacteristic
137C2×#117—1061no—
138C22×C4⋊C4—21no—
139C2×M32—538no—
140C2×M32—534no—
141C22×C16—532no—
142#427—534no—
143#427—1062no—
144C2×#107—21no—
145C2×#107—22no—
146C2×#107—531no—
147C2×#107—531no—
148C2×#107C2×D5312yes—
149C2×#107C2×D5314yes—
150C2×#106—1061no—
151#367—538no—
152#366—538no—
153C53C2×#215911yescharacteristic, complemented, Sylow
154C23×C4—41no—
155C23×C4C22⋊D5311yescharacteristic
156C22×C8Dic10614yes—
157C22×C8C4×D5314yes—
158C22×C8C22⋊D5311yescharacteristic
159C22×C8C22⋊D5312yes—
160C22×C8C22×D5311yescharacteristic
161C22×C8C22×D5312yes—
162#117—1064no—
163#117—2122no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—1061no—
169C2×C4⋊C4—1061no—
170C2×C22⋊C4—2121no—
171C2×C42—2121no—
172M32—5316no—
173C2×C16—538no—
174C2×C16—534no—
175#107—24no—
176#107—22no—
177#107—532no—
178#107—532no—
179#107—1061no—
180#107—1061no—
181#107C22×D5314yes—
182#106—1064no—
183C24#1171992211yescharacteristic
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—84no—
187C22×C4—2121no—
188C22×C4—2121no—
189C22×C4#1171991914yes—
190C22×C4#1171992212yes—
191C22×C4#1171992211yescharacteristic
192C22×C4#1171992514yes—
193C22×C4C2×C22⋊D5312yes—
194C22×C4C2×C22⋊D5311yescharacteristic
195C2×C8#1171991914yes—
196C2×C8#1171991918yes—
197C2×C8C2×Dic10612yes—
198C2×C8C2×Dic10614yes—
199C2×C8C2×C4×D5312yes—
200C2×C8C2×C4×D5314yes—
201C2×C8C2×C22⋊D5312yes—
202C2×C8C2×C22⋊D5311yescharacteristic
203C2×C8C23×D5311yescharacteristic
204C4⋊C4—24no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—1062no—
208C4⋊C4—1062no—
209C4⋊C4—2121no—
210C4⋊C4—2121no—
211C22⋊C4—2124no—
212C42—2124no—
213C16—538no—
214C23—22no—
215C23—22no—
216C23#1313315014yes—
217C23C2×#1171992212yes—
218C23C2×#1171992211yescharacteristic
219C2×C4—44no—
220C2×C4—88no—
221C2×C4—84no—
222C2×C4—4242no—
223C2×C4—4242no—
224C2×C4—2122no—
225C2×C4—2122no—
226C2×C4#1313315018yes—
227C2×C4#1313315014yes—
228C2×C4C2×#1171991914yes—
229C2×C4C2×#1171991912yes—
230C2×C4C2×#1171992212yes—
231C2×C4C2×#1171992211yescharacteristic
232C2×C4C2×#1171992514yes—
233C2×C4C2×#1171992512yes—
234C2×C4C22×C22⋊D5311yescharacteristic
235C8C2×#1171991912yes—
236C8C2×#1171991914yes—
237C8C22×C4×D5311yescharacteristic
238C8C22×Dic10611yescharacteristic
239C22—28no—
240C22—22no—
241C22—22no—
242C22C2×#1313315014yes—
243C22C2×#1313315012yes—
244C22#1681105912yes—
245C22#1681106412yes—
246C22C22×#1171992211yescharacteristic
247C4—84no—
248C4—4242no—
249C4—4242no—
250C4C2×#1313315014yes—
251C4C2×#1313315012yes—
252C4C22×#1171991911yescharacteristic
253C4C22×#1171992511yescharacteristic
254C2—24no—
255C2#2027650114yesdirect product factor
256C2C22×#1313315011yescharacteristic
257C2C2×#1681105911yescharacteristic
258C2C2×#1681106411yescharacteristic
subgroupG/H
centerC23C2×#11719922
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC22×#107×C53C2
socleC23×C53C2×D8
prime radicalC24×C53D8
cosocleC2×C8×C53C24
hypercenterC22×#107D53
hypocenterC53C2×#2159
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
3. #20276501C2direct (N × H)
4. C2×#16810450C2semidirect
5. C2×#16810448C2semidirect
8. C2×#13133157C4semidirect
16. #16810450C22semidirect
17. #16810448C22semidirect
30. #13133157C2×C4semidirect
31. C2×#11719908C2×C4semidirect
83. #11719908C22×C4semidirect
132. C2×C53#2159semidirect
153. C53C2×#2159semidirect
255. C2#20276501direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331572
C2×#168106612
#202765012
C2×#168104502
C2×#168104482
C22×#107×C532
C2×#215953