Finite Group Databasebeta

← C2×#11717026 #13106077

Subgroups — C2×#11717026 #13106077

Order 1600.

944 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 208 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
1C22×C5×D20C211yesmaximal, characteristic, complemented
2C22×C4×C5×D5C211yesmaximal, characteristic, complemented
3C22×#6037C212yesmaximal, complemented
4C2×C4⋊#1039C211yesmaximal, characteristic, complemented
5C2×#87778C212yesmaximal, complemented
6#11717026C218yesmaximal, direct product factor
7C23×C5×D5C2212yes—
8C23×C5×D5C2211yescharacteristic
9C22×C4×C52C2211yescharacteristic
10C22×#1039C2212yes—
11C22×C5×Dic5C2211yescharacteristic
12C2×C5×D20—24no—
13C2×C5×D20C2214yescomplemented
14C2×C4×C5×D5—24no—
15C2×C4×C5×D5C2214yescomplemented
16C2×#6037—28no—
17C2×#6037C2218yescomplemented
18C4⋊#1039C2214yescomplemented
19#87778C2218yescomplemented
20C2×#4084—51nomaximal
21C2×#4076—51nomaximal
22C23×C52C2311yescharacteristic
23C22×C5×D5—22no—
24C22×C5×D5—22no—
25C22×C5×D5—24no—
26C22×C5×D5—24no—
27C22×C5×D5—24no—
28C22×C5×D5D414yescomplemented
29C22×C5×D5C2314yes—
30C22×C5×D5C2312yes—
31C2×C4×C52D414yescomplemented
32C2×C4×C52C2312yes—
33C2×#1039—24no—
34C2×#1039C2314yes—
35C2×C5×Dic5—22no—
36C2×C5×Dic5C2312yes—
37C5×D20—28no—
38C4×C5×D5—28no—
39#6037—216no—
40C22×C5×D4—51no—
41C23×C4×C5—51no—
42C22×C22⋊D5—52no—
43C22×C22⋊D5—52no—
44C22×D20D511yescharacteristic, complemented
45C22×C4×D5D511yescharacteristic, complemented
46C2×C22⋊Dic5—52no—
47#4084—58no—
48#4076—58no—
49C2×#685—52no—
50C2×C4⋊Dic5—51no—
51C2×C4⋊Dic5—51no—
52C2×C4⋊Dic5—102no—
53C22×C52C2×D412yes—
54C22×C52C2×D412yes—
55C22×C52#4112yes—
56C22×C52C2411yescharacteristic
57C2×C5×D5—28no—
58C2×C5×D5—22no—
59C2×C5×D5—24no—
60C2×C5×D5—24no—
61C2×C5×D5—22no—
62C2×C5×D5—24no—
63C2×C5×D5—48no—
64C2×C5×D5C2×D414yescomplemented
65C4×C52C2×D414yescomplemented
66#1039—24no—
67C5×Dic5—22no—
68C24×C5—51no—
69C24×C5—52no—
70C23×D5C2×D512yes—
71C23×D5C2×D511yescharacteristic
72C2×C5×D4—54no—
73C2×C5×D4—104no—
74C22×C4×C5—22no—
75C22×C4×C5—51no—
76C22×C4×C5—54no—
77C22×C4×C5—104no—
78C22×C4×C5C2×D511yescharacteristic
79C22×C4×C5C2×D511yescharacteristic
80C2×C22⋊D5—58no—
81C2×C22⋊D5—58no—
82C2×C22⋊D5—108no—
83C2×C22⋊D5—108no—
84C22×Dic5—52no—
85C22×Dic5—52no—
86C22×Dic5—104no—
87C22×Dic5C2×D511yescharacteristic
88C2×D20—24no—
89C2×D20C2×D514yescomplemented
90C2×C4×D5—24no—
91C2×C4×D5C2×D514yescomplemented
92C22⋊Dic5—58no—
93#685—58no—
94C4⋊Dic5—54no—
95C4⋊Dic5—54no—
96C4⋊Dic5—108no—
97C2×#121—251noSylow
98C2×C52#12114yescomplemented
99C2×C52C22×D411yescharacteristic
100C2×C52C22×D411yescharacteristic
101C2×C52C2×#4111yescharacteristic
102C5×D5—24no—
103C5×D5—44no—
104C23×C5—22no—
105C23×C5—54no—
106C23×C5—52no—
107C23×C5—54no—
108C23×C5—102no—
109C23×C5—102no—
110C23×C5—104no—
111C23×C5—104no—
112C23×C5—104no—
113C23×C5C22×D511yescharacteristic
114C23×C5C22×D511yescharacteristic
115C22×D5—22no—
116C22×D5—22no—
117C22×D5—24no—
118C22×D5—24no—
119C22×D5—24no—
120C22×D5D2014yescomplemented
121C22×D5C22×D514yes—
122C22×D5C22×D512yes—
123C5×D4—108no—
124C2×C4×C5—28no—
125C2×C4×C5—24no—
126C2×C4×C5—52no—
127C2×C4×C5—102no—
128C2×C4×C5—108no—
129C2×C4×C5C22⋊D514yescomplemented
130C2×C4×C5C22⋊D514yescomplemented
131C2×C4×C5C22×D512yes—
132C2×C4×C5C22×D512yes—
133C22⋊D5—1016no—
134C22⋊D5—1016no—
135C2×Dic5—22no—
136C2×Dic5—54no—
137C2×Dic5—54no—
138C2×Dic5—108no—
139C2×Dic5—104no—
140C2×Dic5—104no—
141C2×Dic5—208no—
142C2×Dic5C22×D512yes—
143D20—28no—
144C4×D5—28no—
145C22×D4—51no—
146C22×D4—252no—
147C23×C4—51no—
148#121—258no—
149C2×C4⋊C4—251no—
150C2×C22⋊C4—252no—
151C52C2×#12111yescharacteristic, complemented, Sylow
152C22×C5—24no—
153C22×C5—24no—
154C22×C5—24no—
155C22×C5—22no—
156C22×C5—54no—
157C22×C5—102no—
158C22×C5—102no—
159C22×C5—108no—
160C22×C5—104no—
161C22×C5—104no—
162C22×C5—104no—
163C22×C5—208no—
164C22×C5C2×D2012yes—
165C22×C5#70912yes—
166C22×C5D4×D512yes—
167C22×C5#71112yes—
168C22×C5C2×C22⋊D512yes—
169C22×C5C2×C22⋊D512yes—
170C22×C5C23×D511yescharacteristic
171C22×C5C23×D511yescharacteristic
172C2×D5—22no—
173C2×D5—22no—
174C2×D5—28no—
175C2×D5—24no—
176C2×D5—24no—
177C2×D5—24no—
178C2×D5—48no—
179C2×D5C2×D2014yescomplemented
180C4×C5—28no—
181C4×C5—102no—
182C4×C5C2×C22⋊D514yescomplemented
183C4×C5C2×C22⋊D514yescomplemented
184Dic5—22no—
185Dic5—104no—
186Dic5—104no—
187Dic5—208no—
188C24—51no—
189C24—52no—
190C2×D4—54no—
191C2×D4—104no—
192C2×D4—258no—
193C2×D4—508no—
194C22×C4—51no—
195C22×C4—54no—
196C22×C4—104no—
197C22×C4—252no—
198C22×C4D5211yescharacteristic
199C4⋊C4—254no—
200C22⋊C4—258no—
201C2×C5—22no—
202C2×C5—22no—
203C2×C5—22no—
204C2×C5—28no—
205C2×C5—104no—
206C2×C5—204no—
207C2×C5#407614yescomplemented
208C2×C5#408414yescomplemented
209C2×C5C22×D2011yescharacteristic
210C2×C5C2×#70911yescharacteristic
211C2×C5C2×D4×D511yescharacteristic
212C2×C5C2×#71111yescharacteristic
213C2×C5C22×C22⋊D511yescharacteristic
214C2×C5C22×C22⋊D511yescharacteristic
215D5—24no—
216D5—44no—
217C23—54no—
218C23—52no—
219C23—54no—
220C23—102no—
221C23—102no—
222C23—104no—
223C23—104no—
224C23—104no—
225C23C2×D5211yescharacteristic
226D4—108no—
227D4—5016no—
228C2×C4—52no—
229C2×C4—102no—
230C2×C4—108no—
231C2×C4—254no—
232C2×C4—504no—
233C2×C4#603714yescomplemented
234C2×C4C2×D5212yes—
235C5—22no—
236C5C2×#407611yescharacteristic, complemented
237C5C2×#408411yescharacteristic, complemented
238C22—54no—
239C22—102no—
240C22—102no—
241C22—108no—
242C22—104no—
243C22—104no—
244C22—104no—
245C22—208no—
246C22#8787012yes—
247C22D5×D2012yes—
248C22C2×#603712yes—
249C22C22×D5211yescharacteristic
250C4—102no—
251C4—504no—
252C4C2×#603714yescomplemented
253C2—104no—
254C2—204no—
255C2#1171702614yesdirect product factor
256C2C2×#8787011yescharacteristic
257C2C2×D5×D2011yescharacteristic
258C2C22×#603711yescharacteristic
subgroupG/H
centerC23C2×D52
commutator, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC22×C4×C52C22
socleC23×C52C23
hypercenterC22×C4D52
hypocenterC52C2×#121
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C5×D20C2semidirect
2. C22×C4×C5×D5C2semidirect
3. C22×#6037C2semidirect
4. C2×C4⋊#1039C2semidirect
5. C2×#87778C2semidirect
6. #11717026C2direct (N × H)
13. C2×C5×D20C22semidirect
15. C2×C4×C5×D5C22semidirect
17. C2×#6037C22semidirect
18. C4⋊#1039C22semidirect
19. #87778C22semidirect
28. C22×C5×D5D4semidirect
31. C2×C4×C52D4semidirect
44. C22×D20D5semidirect
45. C22×C4×D5D5semidirect
64. C2×C5×D5C2×D4semidirect
65. C4×C52C2×D4semidirect
89. C2×D20C2×D5semidirect
91. C2×C4×D5C2×D5semidirect
98. C2×C52#121semidirect
120. C22×D5D20semidirect
129. C2×C4×C5C22⋊D5semidirect
130. C2×C4×C5C22⋊D5semidirect
151. C52C2×#121semidirect
179. C2×D5C2×D20semidirect
182. C4×C5C2×C22⋊D5semidirect
183. C4×C5C2×C22⋊D5semidirect
207. C2×C5#4076semidirect
208. C2×C5#4084semidirect
233. C2×C4#6037semidirect
236. C5C2×#4076semidirect
237. C5C2×#4084semidirect
252. C4C2×#6037semidirect
255. C2#11717026direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×D202
C22×C4×C5×D52
C22×#60372
C2×C4⋊#10392
C2×#877782
#117170262
C2×#40845
C2×#40765