Finite Group Databasebeta

← C22×#65035 #12965910

Subgroups — C22×#65035 #12965910

Order 1280.

5110 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4144C211yesmaximal, characteristic
2C22×#4082C211yesmaximal, characteristic, complemented
3C22×#4055C212yesmaximal, complemented
4C22×C5×#100C211yesmaximal, characteristic, complemented
5C22×#3957C212yesmaximal, complemented
6C2×#65035C2124yesmaximal, direct product factor
7C23×C5×D4C2211yescharacteristic
8C23×C22⋊D5—22no—
9C2×#4144—26no—
10C2×#4144—212no—
11C2×#4144C22112yes—
12C22×#711—22no—
13C22×D4×D5—22no—
14C22×#709C2212yes—
15C23×Dic10C2211yescharacteristic, complemented
16C22×C5×M16C2212yescomplemented
17C22×C22⋊Dic5—22no—
18C2×#4082—26no—
19C2×#4082C22112yescomplemented
20C22×C4×Dic5C2211yescharacteristic, complemented
21C2×#4055—212no—
22C2×#4055C22124yescomplemented
23C22×Dic20—22no—
24C22×#677—22no—
25C2×C5×#100C22112yescomplemented
26C2×#3957C22124yescomplemented
27#65035C22164yesdirect product factor
28C22×#455—51nomaximal, Sylow
29C25×C5D412yes—
30C24×D5—22no—
31C22×C5×D4—22no—
32C22×C5×D4—26no—
33C22×C5×D4D416yes—
34C22×C5×D4C2316yes—
35C23×C4×C5C2311yescharacteristic
36C22×C22⋊D5—24no—
37C22×C22⋊D5—224no—
38C22×C22⋊D5—22no—
39C22×C22⋊D5—224no—
40C23×Dic5—22no—
41C23×Dic5—21no—
42C23×Dic5—21no—
43#4144—248no—
44#4144—224no—
45#4144—248no—
46#4144C23116yes—
47C2×#711—224no—
48C2×#711—224no—
49C2×D4×D5—224no—
50C2×D4×D5—224no—
51C2×#709—26no—
52C2×#709—26no—
53C2×#709D4112yes—
54C2×#709C23112yes—
55C22×D20D412yes—
56C22×C4×D5—22no—
57C22×Dic10—22no—
58C22×Dic10—26no—
59C22×Dic10—23no—
60C22×Dic10D412yes—
61C22×Dic10C2316yescomplemented
62C2×C5×M16—26no—
63C2×C5×M16C23112yescomplemented
64C22×C5×C8—22no—
65C2×C22⋊Dic5—224no—
66#4082—224no—
67#4082C23116yescomplemented
68C2×C4×Dic5—23no—
69C2×C4×Dic5C2316yescomplemented
70#4055—248no—
71#4055C23132yescomplemented
72C2×Dic20—224no—
73C2×#677—224no—
74C5×#100C23116yescomplemented
75#3957C23132yescomplemented
76C22×H(5,𝔽2)—51no—
77C22×#137—52no—
78C22×#124—51no—
79C2×#455—524no—
80C22×#104—52no—
81C22×#100D511yescharacteristic, complemented
82C24×C5—22no—
83C24×C5—26no—
84C24×C5—48no—
85C24×C5C2×D411yescharacteristic
86C24×C5C2×D4112yes—
87C23×D5—22no—
88C23×D5—224no—
89C23×D5—88no—
90C2×C5×D4—212no—
91C2×C5×D4—212no—
92C2×C5×D4—212no—
93C2×C5×D4—212no—
94C2×C5×D4C2×D4112yes—
95C2×C5×D4C2414yes—
96C22×C4×C5C2×D413yes—
97C22×C4×C5C2×D412yes—
98C22×C4×C5C2×D416yes—
99C22×C4×C5C2413yes—
100C2×C22⋊D5—248no—
101C2×C22⋊D5—296no—
102C2×C22⋊D5—212no—
103C2×C22⋊D5—212no—
104C2×C22⋊D5—232no—
105C22×Dic5—224no—
106C22×Dic5—22no—
107C22×Dic5—26no—
108C22×Dic5—26no—
109C22×Dic5—44no—
110C22×Dic5—43no—
111C22×Dic5—41no—
112#711—296no—
113#711—232no—
114D4×D5—296no—
115D4×D5—232no—
116#709—224no—
117#709—212no—
118#709—212no—
119#709C2×D4124yes—
120#709C2418yes—
121C2×D20—26no—
122C2×D20C2×D4112yes—
123C2×C4×D5—212no—
124C2×C4×D5—212no—
125C2×Dic10—26no—
126C2×Dic10—212no—
127C2×Dic10—212no—
128C2×Dic10—26no—
129C2×Dic10—46no—
130C2×Dic10—46no—
131C2×Dic10C2×D4112yes—
132C2×Dic10C2414yescomplemented
133C5×M16—212no—
134C5×M16C2418yescomplemented
135C2×C5×C8—212no—
136C22⋊Dic5—232no—
137C4×Dic5—26no—
138C4×Dic5C2414yescomplemented
139Dic20—232no—
140#677—232no—
141C2×H(5,𝔽2)—512no—
142C2×H(5,𝔽2)—1012no—
143C2×H(5,𝔽2)—106no—
144C22×#41—52no—
145C22×#41—102no—
146C23×Q8—51no—
147C23×D4—102no—
148C23×D4—102no—
149C23×D4C2×D511yescharacteristic
150C2×#137—524no—
151C2×#137—1012no—
152C22×Q16—102no—
153C22×SD16—102no—
154C22×M16C2×D512yescomplemented
155C2×#124—512no—
156C2×#124—106no—
157C22×C22⋊C4—102no—
158C22×C42—51no—
159#455—564no—
160C2×#104—524no—
161C2×#100C2×D5112yescomplemented
162C23×C5—24no—
163C23×C5—212no—
164C23×C5—212no—
165C23×C5—424no—
166C23×C5—48no—
167C23×C5—812no—
168C23×C5#12013yes—
169C23×C5#12011yescharacteristic
170C23×C5C22×D413yes—
171C23×C5C22×D418yes—
172C22×D5—212no—
173C22×D5—232no—
174C22×D5—48no—
175C22×D5—848no—
176C5×D4—224no—
177C5×D4—28no—
178C2×C4×C5#12016yes—
179C2×C4×C5#12016yes—
180C2×C4×C5C22×D413yes—
181C2×C4×C5C22×D416yes—
182C2×C4×C5C22×D416yes—
183C2×C4×C5C2511yescharacteristic
184C22⋊D5—264no—
185C22⋊D5—224no—
186C22⋊D5—28no—
187C2×Dic5—232no—
188C2×Dic5—212no—
189C2×Dic5—24no—
190C2×Dic5—24no—
191C2×Dic5—412no—
192C2×Dic5—46no—
193C2×Dic5—46no—
194D20—212no—
195D20C22×D418yes—
196C4×D5—224no—
197C4×D5—28no—
198Dic10—212no—
199Dic10—28no—
200Dic10—412no—
201Dic10C22×D418yes—
202C5×C8—28no—
203C25—102no—
204C25D2012yes—
205H(5,𝔽2)—516no—
206H(5,𝔽2)—1048no—
207H(5,𝔽2)—1048no—
208H(5,𝔽2)—1024no—
209C2×#41—512no—
210C2×#41—512no—
211C2×#41—1024no—
212C2×#41—1024no—
213C2×#41—106no—
214C2×#41—106no—
215C22×Q8—52no—
216C22×Q8—56no—
217C22×Q8—103no—
218C22×Q8—102no—
219C22×Q8—106no—
220C22×D4—22no—
221C22×D4—26no—
222C22×D4—52no—
223C22×D4—1024no—
224C22×D4—1024no—
225C22×D4—104no—
226C22×D4—1024no—
227C22×D4—102no—
228C22×D4—1024no—
229C22×D4D2016yes—
230C22×D4C22×D516yes—
231C23×C4—102no—
232C23×C4—102no—
233C23×C4—101no—
234C23×C4—101no—
235C23×C4C22×D511yescharacteristic
236#137—532no—
237#137—1048no—
238C2×Q16—1024no—
239C2×SD16—1024no—
240C2×M16—26no—
241C2×M16C22×D5112yescomplemented
242C22×C8—22no—
243#124—516no—
244#124—1024no—
245C2×C22⋊C4—1024no—
246C2×C42—56no—
247C2×C42—103no—
248#104—532no—
249#100C22×D5116yescomplemented
250C22×C5—26no—
251C22×C5—26no—
252C22×C5—28no—
253C22×C5—48no—
254C22×C5—424no—
255C22×C5—812no—
256C22×C5#45514yescomplemented
257C22×C5C2×#12013yes—
258C22×C5C2×#12013yes—
259C22×C5C23×D411yescharacteristic
260C2×D5—28no—
261C2×D5—424no—
262C2×D5—832no—
263C4×C5C2×#12016yes—
264C4×C5C23×D412yes—
265Dic5—28no—
266Dic5—44no—
267Dic5—44no—
268C24—22no—
269C24—26no—
270C24—48no—
271C24—1024no—
272C24—102no—
273C24—408no—
274C24C2×D2011yescharacteristic
275C24C2×D20112yes—
276#41—524no—
277#41—58no—
278#41—1032no—
279#41—1096no—
280#41—1012no—
281#41—1012no—
282#41—1024no—
283C2×Q8—512no—
284C2×Q8—54no—
285C2×Q8—106no—
286C2×Q8—106no—
287C2×Q8—1012no—
288C2×Q8—1012no—
289C2×Q8—206no—
290C2×Q8—206no—
291C2×D4—212no—
292C2×D4—212no—
293C2×D4—212no—
294C2×D4—212no—
295C2×D4—512no—
296C2×D4—1032no—
297C2×D4—1096no—
298C2×D4—1048no—
299C2×D4—1096no—
300C2×D4—1012no—
301C2×D4—1032no—
302C2×D4—1012no—
303C2×D4—106no—
304C2×D4C2×D20112yes—
305C2×D4C23×D514yes—
306C22×C4—1024no—
307C22×C4—102no—
308C22×C4—1012no—
309C22×C4—1012no—
310C22×C4—106no—
311C22×C4—106no—
312C22×C4—204no—
313C22×C4—203no—
314C22×C4—201no—
315C22×C4C2×D2013yes—
316C22×C4D4×D516yes—
317C22×C4D4×D512yes—
318C22×C4C23×D513yes—
319Q16—1032no—
320SD16—1032no—
321M16—212no—
322M16C23×D518yescomplemented
323C2×C8—212no—
324C22⋊C4—1032no—
325C42—54no—
326C42—106no—
327C2×C5—21no—
328C2×C5—23no—
329C2×C5—48no—
330C2×C5C2×#45516yescomplemented
331C2×C5C22×#12011yescharacteristic
332D5—48no—
333C23—24no—
334C23—212no—
335C23—212no—
336C23—424no—
337C23—48no—
338C23—812no—
339C23—1032no—
340C23—1012no—
341C23—208no—
342C23—4048no—
343C23#402813yes—
344C23#402811yescharacteristic
345C23C22×D2013yes—
346C23C22×D2018yes—
347Q8—58no—
348Q8—1012no—
349Q8—108no—
350Q8—2012no—
351D4—224no—
352D4—28no—
353D4—58no—
354D4—1064no—
355D4—1024no—
356D4—108no—
357D4—1012no—
358C2×C4—1032no—
359C2×C4—1012no—
360C2×C4—1024no—
361C2×C4—108no—
362C2×C4—104no—
363C2×C4—104no—
364C2×C4—2012no—
365C2×C4—206no—
366C2×C4—206no—
367C2×C4#402816yes—
368C2×C4#402816yes—
369C2×C4C22×D2013yes—
370C2×C4C2×D4×D516yes—
371C2×C4C2×D4×D516yes—
372C2×C4C24×D511yescharacteristic
373C8—28no—
374C5C22×#45511yescharacteristic, complemented, Sylow
375C22—26no—
376C22—26no—
377C22—28no—
378C22—48no—
379C22—424no—
380C22—812no—
381C22—108no—
382C22—2024no—
383C22—4032no—
384C22#6503514yesdirect product factor
385C22C2×#402813yes—
386C22C2×#402813yes—
387C22C23×D2011yescharacteristic
388C4—108no—
389C4—204no—
390C4—204no—
391C4C2×#402816yes—
392C4C22×D4×D512yes—
393C2—21no—
394C2—23no—
395C2—48no—
396C2—208no—
397C2C2×#6503516yesdirect product factor
398C2C22×#402811yescharacteristic
subgroupG/H
centerC23#4028
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC22×C5×#100C2
socleC23×C5#120
prime radicalC22×#4144C2
hypercenterC22×#100D5
hypocenterC5C22×#455
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#100

Complements

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

NH ≅ G/Ntype
2. C22×#4082C2semidirect
3. C22×#4055C2semidirect
4. C22×C5×#100C2semidirect
5. C22×#3957C2semidirect
6. C2×#65035C2direct (N × H)
15. C23×Dic10C22semidirect
16. C22×C5×M16C22semidirect
19. C2×#4082C22semidirect
20. C22×C4×Dic5C22semidirect
22. C2×#4055C22semidirect
25. C2×C5×#100C22semidirect
26. C2×#3957C22semidirect
27. #65035C22direct (N × H)
61. C22×Dic10C23semidirect
63. C2×C5×M16C23semidirect
67. #4082C23semidirect
69. C2×C4×Dic5C23semidirect
71. #4055C23semidirect
74. C5×#100C23semidirect
75. #3957C23semidirect
81. C22×#100D5semidirect
132. C2×Dic10C24semidirect
134. C5×M16C24semidirect
138. C4×Dic5C24semidirect
154. C22×M16C2×D5semidirect
161. C2×#100C2×D5semidirect
241. C2×M16C22×D5semidirect
249. #100C22×D5semidirect
256. C22×C5#455semidirect
322. M16C23×D5semidirect
330. C2×C5C2×#455semidirect
374. C5C22×#455semidirect
384. C22#65035direct (N × H)
397. C2C2×#65035direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#41442
C22×#40822
C22×#40552
C22×C5×#1002
C22×#39572
C2×#650352
C22×#4555