Finite Group Databasebeta

← C22×#64786 #12252237

Subgroups — C22×#64786 #12252237

Order 1280.

1214 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#135C211yesmaximal, characteristic
2C22×#3949C211yesmaximal, characteristic, complemented
3C22×#3944C211yesmaximal, characteristic, complemented
4C2×#64786C2112yesmaximal, direct product factor
5C22×C5×#41—21no—
6C2×C5×#135—23no—
7C2×C5×#135—23no—
8C2×C5×#135C416yescomplemented
9C2×C5×#135C2216yes—
10C22×C5×Q16C411yescharacteristic, complemented
11C22×C5×SD16—21no—
12C22×C5×D8C411yescharacteristic, complemented
13C23×C5×C8C2211yescharacteristic
14C22×C4⋊Dic5—21no—
15C2×#3949—23no—
16C2×#3949C2216yescomplemented
17C2×#3944—23no—
18C2×#3944C2216yescomplemented
19C22×#672—21no—
20#64786C22116yesdirect product factor
21C22×#360—51nomaximal, Sylow
22C2×C5×#41—26no—
23C2×C5×#41—26no—
24C2×C5×#41—43no—
25C2×C5×#41—43no—
26C22×C5×Q8—21no—
27C22×C5×D4—21no—
28C22×C5×D4—41no—
29C23×C4×C5—41no—
30C23×C4×C5D411yescharacteristic
31C23×Dic5—41no—
32C5×#135—212no—
33C5×#135—26no—
34C5×#135—26no—
35C5×#135C2×C4112yescomplemented
36C5×#135C2314yes—
37C2×C5×Q16—23no—
38C2×C5×Q16C2×C416yescomplemented
39C2×C5×SD16—26no—
40C2×C5×SD16—26no—
41C2×C5×D8—23no—
42C2×C5×D8C2×C416yescomplemented
43C22×C5×C8C2×C411yescharacteristic
44C22×C5×C8C2×C413yes—
45C22×C5×C8D413yes—
46C22×C5×C8D411yescharacteristic
47C22×C5×C8D413yes—
48C22×C5×C8C2313yes—
49C2×C4⋊Dic5—26no—
50C2×C4⋊Dic5—43no—
51#3949—26no—
52#3949C2314yescomplemented
53#3944—26no—
54#3944C2314yescomplemented
55C2×#672—26no—
56C22×#135D511yescharacteristic
57C22×M32—51no—
58C22×#106—51no—
59C2×#360—512no—
60C24×C5—41no—
61C24×C5D811yescharacteristic
62C5×#41—212no—
63C5×#41—24no—
64C5×#41—412no—
65C5×#41—46no—
66C5×#41—46no—
67C2×C5×Q8—26no—
68C2×C5×Q8—43no—
69C2×C5×D4—26no—
70C2×C5×D4—46no—
71C2×C5×D4—46no—
72C2×C5×D4—43no—
73C22×C4×C5—41no—
74C22×C4×C5—46no—
75C22×C4×C5—46no—
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C22⋊C413yes—
78C22×C4×C5D813yes—
79C22×C4×C5SD1611yescharacteristic
80C22×C4×C5SD1613yes—
81C22×C4×C5C2×D413yes—
82C22×Dic5—46no—
83C22×Dic5—84no—
84C5×Q16—26no—
85C5×Q16C22×C414yescomplemented
86C5×SD16—212no—
87C5×SD16—24no—
88C5×D8—26no—
89C5×D8C22×C414yescomplemented
90C2×C5×C8C22⋊C413yes—
91C2×C5×C8C22⋊C413yes—
92C2×C5×C8C22⋊C416yes—
93C2×C5×C8C22×C413yes—
94C2×C5×C8C22×C413yes—
95C2×C5×C8C2×D413yes—
96C2×C5×C8C2×D413yes—
97C2×C5×C8C2×D413yes—
98C2×C5×C8C2411yescharacteristic
99C4⋊Dic5—24no—
100C4⋊Dic5—46no—
101#672—24no—
102C22×#41—21no—
103C2×#135—23no—
104C2×#135—23no—
105C2×#135Dic516yescomplemented
106C2×#135C2×D516yes—
107C22×Q16Dic511yescharacteristic, complemented
108C22×SD16—21no—
109C22×D8Dic511yescharacteristic, complemented
110C23×C8C2×D511yescharacteristic
111C22×C4⋊C4—101no—
112C2×M32—56no—
113C2×M32—103no—
114C22×C16—101no—
115C2×#106—56no—
116C2×#106—103no—
117#360—516no—
118C23×C5—24no—
119C23×C5—46no—
120C23×C5—84no—
121C23×C5#10213yes—
122C23×C5#10211yescharacteristic
123C23×C5C2×D813yes—
124C5×Q8—24no—
125C5×Q8—46no—
126C5×D4—24no—
127C5×D4—44no—
128C5×D4—412no—
129C5×D4—46no—
130C2×C4×C5—46no—
131C2×C4×C5—44no—
132C2×C4×C5—412no—
133C2×C4×C5#10213yes—
134C2×C4×C5#10213yes—
135C2×C4×C5#10216yes—
136C2×C4×C5C2×C22⋊C413yes—
137C2×C4×C5C2×C22⋊C413yes—
138C2×C4×C5C2×D813yes—
139C2×C4×C5C2×SD1613yes—
140C2×C4×C5C2×SD1613yes—
141C2×C4×C5C22×D411yescharacteristic
142C2×Dic5—44no—
143C2×Dic5—812no—
144C5×C8C2×C22⋊C413yes—
145C5×C8C2×C22⋊C413yes—
146C5×C8C23×C411yescharacteristic
147C5×C8C22×D411yescharacteristic
148C2×#41—26no—
149C2×#41—26no—
150C2×#41—43no—
151C2×#41—43no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—41no—
155C23×C4—41no—
156C23×C4—201no—
157C23×C4C22⋊D511yescharacteristic
158#135—212no—
159#135—26no—
160#135—26no—
161#135C2×Dic5112yescomplemented
162#135C22×D514yes—
163C2×Q16—23no—
164C2×Q16C2×Dic516yescomplemented
165C2×SD16—26no—
166C2×SD16—26no—
167C2×D8—23no—
168C2×D8C2×Dic516yescomplemented
169C22×C8C2×Dic511yescharacteristic
170C22×C8C2×Dic513yes—
171C22×C8C22⋊D513yes—
172C22×C8C22⋊D511yescharacteristic
173C22×C8C22⋊D513yes—
174C22×C8C22×D513yes—
175C2×C4⋊C4—106no—
176C2×C4⋊C4—203no—
177M32—54no—
178M32—106no—
179C2×C16—106no—
180#106—54no—
181#106—106no—
182C22×C5—26no—
183C22×C5—26no—
184C22×C5—44no—
185C22×C5—812no—
186C22×C5#36014yescomplemented
187C22×C5C2×#10213yes—
188C22×C5C2×#10213yes—
189C22×C5C22×D811yescharacteristic
190C4×C5—44no—
191C4×C5C2×#10213yes—
192C4×C5C2×#10213yes—
193C4×C5C22×C22⋊C411yescharacteristic
194C4×C5C22×SD1611yescharacteristic
195Dic5—84no—
196C24—41no—
197C24#68611yescharacteristic
198#41—212no—
199#41—24no—
200#41—412no—
201#41—46no—
202#41—46no—
203C2×Q8—26no—
204C2×Q8—43no—
205C2×D4—26no—
206C2×D4—46no—
207C2×D4—46no—
208C2×D4—43no—
209C22×C4—41no—
210C22×C4—46no—
211C22×C4—46no—
212C22×C4—206no—
213C22×C4—404no—
214C22×C4#68613yes—
215C22×C4#68713yes—
216C22×C4#68711yescharacteristic
217C22×C4C22⋊Dic513yes—
218C22×C4C22⋊Dic511yescharacteristic
219C22×C4C2×C22⋊D513yes—
220Q16—26no—
221Q16C22×Dic514yescomplemented
222SD16—212no—
223SD16—24no—
224D8—26no—
225D8C22×Dic514yescomplemented
226C2×C8C22⋊Dic513yes—
227C2×C8C22⋊Dic513yes—
228C2×C8C22⋊Dic516yes—
229C2×C8C22×Dic513yes—
230C2×C8C22×Dic513yes—
231C2×C8C2×C22⋊D513yes—
232C2×C8C2×C22⋊D513yes—
233C2×C8C2×C22⋊D513yes—
234C2×C8C23×D511yescharacteristic
235C4⋊C4—104no—
236C4⋊C4—206no—
237C16—104no—
238C2×C5—21no—
239C2×C5—23no—
240C2×C5—84no—
241C2×C5C2×#36016yescomplemented
242C2×C5C22×#10211yescharacteristic
243C23—24no—
244C23—46no—
245C23—84no—
246C23#396413yes—
247C23#396411yescharacteristic
248C23C2×#68613yes—
249Q8—24no—
250Q8—46no—
251D4—24no—
252D4—44no—
253D4—412no—
254D4—46no—
255C2×C4—46no—
256C2×C4—44no—
257C2×C4—412no—
258C2×C4—204no—
259C2×C4—4012no—
260C2×C4#396413yes—
261C2×C4#396413yes—
262C2×C4#396416yes—
263C2×C4C2×#68613yes—
264C2×C4C2×#68713yes—
265C2×C4C2×#68713yes—
266C2×C4C2×C22⋊Dic513yes—
267C2×C4C2×C22⋊Dic513yes—
268C2×C4C22×C22⋊D511yescharacteristic
269C8C2×C22⋊Dic513yes—
270C8C2×C22⋊Dic513yes—
271C8C23×Dic511yescharacteristic
272C8C22×C22⋊D511yescharacteristic
273C5C22×#36011yescharacteristic, complemented, Sylow
274C22—26no—
275C22—26no—
276C22—44no—
277C22—812no—
278C22#6478614yesdirect product factor
279C22C2×#396413yes—
280C22C2×#396413yes—
281C22C22×#68611yescharacteristic
282C4—44no—
283C4—404no—
284C4C2×#396413yes—
285C4C2×#396413yes—
286C4C22×#68711yescharacteristic
287C4C22×C22⋊Dic511yescharacteristic
288C2—21no—
289C2—23no—
290C2—84no—
291C2C2×#6478616yesdirect product factor
292C2C22×#396411yescharacteristic
subgroupG/H
centerC23#3964
commutatorC5×C8C23×C4
FrattiniC2×C8C23×D5
Fitting, prime radicalC22×C5×#135C2
socleC23×C5#102
cosocleC2×C5×C8C24
hypercenterC22×#135D5
hypocenterC5C22×#360
LCSG ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
2. C22×#3949C2semidirect
3. C22×#3944C2semidirect
4. C2×#64786C2direct (N × H)
8. C2×C5×#135C4semidirect
10. C22×C5×Q16C4semidirect
12. C22×C5×D8C4semidirect
16. C2×#3949C22semidirect
18. C2×#3944C22semidirect
20. #64786C22direct (N × H)
35. C5×#135C2×C4semidirect
38. C2×C5×Q16C2×C4semidirect
42. C2×C5×D8C2×C4semidirect
52. #3949C23semidirect
54. #3944C23semidirect
85. C5×Q16C22×C4semidirect
89. C5×D8C22×C4semidirect
105. C2×#135Dic5semidirect
107. C22×Q16Dic5semidirect
109. C22×D8Dic5semidirect
161. #135C2×Dic5semidirect
164. C2×Q16C2×Dic5semidirect
168. C2×D8C2×Dic5semidirect
186. C22×C5#360semidirect
221. Q16C22×Dic5semidirect
225. D8C22×Dic5semidirect
241. C2×C5C2×#360semidirect
273. C5C22×#360semidirect
278. C22#64786direct (N × H)
291. C2C2×#64786direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#1352
C22×#39492
C22×#39442
C2×#647862
C22×#3605