Finite Group Databasebeta

← C22×#64708 #12252236

Subgroups — C22×#64708 #12252236

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×#4050C211yesmaximal, characteristic
2C22×C5×#106C211yesmaximal, characteristic, complemented
3C22×#3944C211yesmaximal, characteristic, complemented
4C2×#64708C2112yesmaximal, direct product factor
5C22×#709—21no—
6C23×C5×C8C2211yescharacteristic
7C22×C5×C4⋊C4—21no—
8C22×Dic20C411yescharacteristic, complemented
9C2×#4050—23no—
10C2×#4050—23no—
11C2×#4050C416yescomplemented
12C2×#4050C2216yes—
13C22×D40C411yescharacteristic, complemented
14C22×#677—21no—
15C2×C5×#106—23no—
16C2×C5×#106C2216yescomplemented
17C2×#3944—23no—
18C2×#3944C2216yescomplemented
19C22×#672—21no—
20#64708C22116yesdirect product factor
21C22×#360—51nomaximal, Sylow
22C23×C4×C5—41no—
23C23×C4×C5D411yescharacteristic
24C22×C22⋊D5—41no—
25C2×#709—26no—
26C2×#709—26no—
27C2×#709—43no—
28C2×#709—43no—
29C22×D20—21no—
30C22×C4×D5—41no—
31C22×Dic10—21no—
32C22×C5×C8C2×C411yescharacteristic
33C22×C5×C8C2×C413yes—
34C22×C5×C8D411yescharacteristic
35C22×C5×C8D413yes—
36C22×C5×C8D413yes—
37C22×C5×C8C2313yes—
38C2×C5×C4⋊C4—26no—
39C2×C5×C4⋊C4—43no—
40C2×Dic20—23no—
41C2×Dic20C2×C416yescomplemented
42#4050—212no—
43#4050—26no—
44#4050—26no—
45#4050C2×C4112yescomplemented
46#4050C2314yes—
47C2×D40—23no—
48C2×D40C2×C416yescomplemented
49C2×#677—26no—
50C2×#677—26no—
51C5×#106—26no—
52C5×#106C2314yescomplemented
53#3944—26no—
54#3944C2314yescomplemented
55C2×#672—26no—
56C22×#135—51no—
57C22×M32—51no—
58C22×#106D511yescharacteristic, complemented
59C2×#360—512no—
60C24×C5D811yescharacteristic
61C23×D5—41no—
62C22×C4×C5—46no—
63C22×C4×C5—84no—
64C22×C4×C5C22⋊C411yescharacteristic
65C22×C4×C5C22⋊C413yes—
66C22×C4×C5D813yes—
67C22×C4×C5SD1611yescharacteristic
68C22×C4×C5SD1613yes—
69C22×C4×C5C2×D413yes—
70C2×C22⋊D5—46no—
71C2×C22⋊D5—46no—
72C22×Dic5—41no—
73#709—212no—
74#709—24no—
75#709—412no—
76#709—46no—
77#709—46no—
78C2×D20—26no—
79C2×D20—43no—
80C2×C4×D5—46no—
81C2×C4×D5—46no—
82C2×Dic10—26no—
83C2×Dic10—43no—
84C2×C5×C8C22⋊C413yes—
85C2×C5×C8C22⋊C413yes—
86C2×C5×C8C22⋊C416yes—
87C2×C5×C8C22×C413yes—
88C2×C5×C8C22×C413yes—
89C2×C5×C8C2×D413yes—
90C2×C5×C8C2×D413yes—
91C2×C5×C8C2×D413yes—
92C2×C5×C8C2411yescharacteristic
93C5×C4⋊C4—24no—
94C5×C4⋊C4—46no—
95Dic20—26no—
96Dic20C22×C414yescomplemented
97D40—26no—
98D40C22×C414yescomplemented
99#677—212no—
100#677—24no—
101#672—24no—
102C22×#41—101no—
103C2×#135—56no—
104C2×#135—56no—
105C2×#135—103no—
106C2×#135—103no—
107C22×Q16—51no—
108C22×SD16—101no—
109C22×D8—51no—
110C23×C8C2×D511yescharacteristic
111C22×C4⋊C4—21no—
112C2×M32—56no—
113C2×M32—103no—
114C22×C16—101no—
115C2×#106—23no—
116C2×#106C2×D516yescomplemented
117#360—516no—
118C23×C5—24no—
119C23×C5#10213yes—
120C23×C5#10211yescharacteristic
121C23×C5C2×D813yes—
122C22×D5—46no—
123C22×D5—84no—
124C2×C4×C5—44no—
125C2×C4×C5—812no—
126C2×C4×C5#10213yes—
127C2×C4×C5#10213yes—
128C2×C4×C5#10216yes—
129C2×C4×C5C2×C22⋊C413yes—
130C2×C4×C5C2×C22⋊C413yes—
131C2×C4×C5C2×D813yes—
132C2×C4×C5C2×SD1613yes—
133C2×C4×C5C2×SD1613yes—
134C2×C4×C5C22×D411yescharacteristic
135C22⋊D5—44no—
136C22⋊D5—412no—
137C2×Dic5—46no—
138D20—24no—
139D20—46no—
140C4×D5—44no—
141C4×D5—412no—
142Dic10—24no—
143Dic10—46no—
144C5×C8C2×C22⋊C413yes—
145C5×C8C2×C22⋊C413yes—
146C5×C8C23×C411yescharacteristic
147C5×C8C22×D411yescharacteristic
148C2×#41—106no—
149C2×#41—106no—
150C2×#41—203no—
151C2×#41—203no—
152C22×Q8—101no—
153C22×D4—101no—
154C22×D4—201no—
155C23×C4—41no—
156C23×C4—201no—
157C23×C4C22⋊D511yescharacteristic
158#135—512no—
159#135—54no—
160#135—1012no—
161#135—106no—
162#135—106no—
163C2×Q16—56no—
164C2×Q16—103no—
165C2×SD16—106no—
166C2×SD16—106no—
167C2×D8—56no—
168C2×D8—103no—
169C22×C8C4×D511yescharacteristic
170C22×C8C4×D513yes—
171C22×C8D2011yescharacteristic
172C22×C8D2013yes—
173C22×C8C22⋊D513yes—
174C22×C8C22×D513yes—
175C2×C4⋊C4—26no—
176C2×C4⋊C4—43no—
177M32—54no—
178M32—106no—
179C2×C16—106no—
180#106—26no—
181#106C22×D514yescomplemented
182C22×C5—26no—
183C22×C5—26no—
184C22×C5#36014yescomplemented
185C22×C5C2×#10213yes—
186C22×C5C2×#10213yes—
187C22×C5C22×D811yescharacteristic
188C2×D5—44no—
189C2×D5—812no—
190C4×C5—84no—
191C4×C5C2×#10213yes—
192C4×C5C2×#10213yes—
193C4×C5C22×C22⋊C411yescharacteristic
194C4×C5C22×SD1611yescharacteristic
195Dic5—44no—
196C24—201no—
197C24#68611yescharacteristic
198#41—104no—
199#41—1012no—
200#41—2012no—
201#41—206no—
202#41—206no—
203C2×Q8—106no—
204C2×Q8—203no—
205C2×D4—106no—
206C2×D4—206no—
207C2×D4—206no—
208C2×D4—203no—
209C22×C4—46no—
210C22×C4—84no—
211C22×C4—201no—
212C22×C4—206no—
213C22×C4—206no—
214C22×C4#68513yes—
215C22×C4#68511yescharacteristic
216C22×C4#68613yes—
217C22×C4#68813yes—
218C22×C4#68811yescharacteristic
219C22×C4C2×C22⋊D513yes—
220Q16—54no—
221Q16—106no—
222SD16—104no—
223SD16—1012no—
224D8—54no—
225D8—106no—
226C2×C8#68513yes—
227C2×C8#68513yes—
228C2×C8#68516yes—
229C2×C8C2×C4×D513yes—
230C2×C8C2×C4×D513yes—
231C2×C8C2×D2013yes—
232C2×C8C2×D2013yes—
233C2×C8C2×C22⋊D513yes—
234C2×C8C23×D511yescharacteristic
235C4⋊C4—24no—
236C4⋊C4—46no—
237C16—104no—
238C2×C5—21no—
239C2×C5—23no—
240C2×C5C2×#36016yescomplemented
241C2×C5C22×#10211yescharacteristic
242D5—84no—
243C23—24no—
244C23—206no—
245C23—404no—
246C23#394113yes—
247C23#394111yescharacteristic
248C23C2×#68613yes—
249Q8—104no—
250Q8—206no—
251D4—104no—
252D4—204no—
253D4—2012no—
254D4—206no—
255C2×C4—44no—
256C2×C4—812no—
257C2×C4—206no—
258C2×C4—204no—
259C2×C4—2012no—
260C2×C4#394113yes—
261C2×C4#394113yes—
262C2×C4#394116yes—
263C2×C4C2×#68513yes—
264C2×C4C2×#68513yes—
265C2×C4C2×#68613yes—
266C2×C4C2×#68813yes—
267C2×C4C2×#68813yes—
268C2×C4C22×C22⋊D511yescharacteristic
269C8C2×#68513yes—
270C8C2×#68513yes—
271C8C22×C4×D511yescharacteristic
272C8C22×D2011yescharacteristic
273C5C22×#36011yescharacteristic, complemented, Sylow
274C22—26no—
275C22—26no—
276C22—204no—
277C22—4012no—
278C22#6470814yesdirect product factor
279C22C2×#394113yes—
280C22C2×#394113yes—
281C22C22×#68611yescharacteristic
282C4—84no—
283C4—204no—
284C4C2×#394113yes—
285C4C2×#394113yes—
286C4C22×#68511yescharacteristic
287C4C22×#68811yescharacteristic
288C2—21no—
289C2—23no—
290C2—404no—
291C2C2×#6470816yesdirect product factor
292C2C22×#394111yescharacteristic
subgroupG/H
centerC23#3941
commutatorC5×C8C23×C4
FrattiniC2×C8C23×D5
FittingC22×C5×#106C2
socleC23×C5#102
prime radicalC22×#4050C2
cosocleC2×C5×C8C24
hypercenterC22×#106D5
hypocenterC5C22×#360
LCSG ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
2. C22×C5×#106C2semidirect
3. C22×#3944C2semidirect
4. C2×#64708C2direct (N × H)
8. C22×Dic20C4semidirect
11. C2×#4050C4semidirect
13. C22×D40C4semidirect
16. C2×C5×#106C22semidirect
18. C2×#3944C22semidirect
20. #64708C22direct (N × H)
41. C2×Dic20C2×C4semidirect
45. #4050C2×C4semidirect
48. C2×D40C2×C4semidirect
52. C5×#106C23semidirect
54. #3944C23semidirect
58. C22×#106D5semidirect
96. Dic20C22×C4semidirect
98. D40C22×C4semidirect
116. C2×#106C2×D5semidirect
181. #106C22×D5semidirect
184. C22×C5#360semidirect
240. C2×C5C2×#360semidirect
273. C5C22×#360semidirect
278. C22#64708direct (N × H)
291. C2C2×#64708direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40502
C22×C5×#1062
C22×#39442
C2×#647082
C22×#3605