Finite Group Databasebeta

← D5×#64162 #72229926

Subgroups — D5×#64162 #72229926

Order 2880.

3972 conjugacy classes of subgroups; only the 476 normal subgroups are shown.

normal subgroups only

This group has 3972 conjugacy classes of subgroups — too many to draw. Only its 476 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
1C24×C32×D5C211yesmaximal, characteristic, complemented
2C2×D5×#638C213yesmaximal, complemented
3#13088222C211yesmaximal, characteristic, complemented
4C5×#64162C211yesmaximal, characteristic, complemented
5#13087043C213yesmaximal, complemented
6#13087041C213yesmaximal, complemented
7D5×#3757C213yesmaximal, complemented
8D5×#953—34nomaximal
9C24×C32×C5C2211yescharacteristic, complemented
10C23×C32×D5—212no—
11C23×C32×D5C2213yes—
12C23×C32×D5C2213yescomplemented
13C22×D5×C3⋊D3C2211yescharacteristic, complemented
14C2×#67184C2213yescomplemented
15C2×C5×#638C2213yescomplemented
16D5×#638—212no—
17C2×#67139C2213yescomplemented
18C2×#67138C2213yescomplemented
19C2×D5×C3⋊Dic3C2213yescomplemented
20#10623913C2213yescomplemented
21C5×#3757C2213yescomplemented
22#10623770C2216yescomplemented
23C2×#64162—51nomaximal, Hall
24C24×C3×D5D314yescomplemented
25C2×D5×#67—312no—
26#91842—34no—
27C5×#953—34no—
28#91564—312no—
29#91563—312no—
30D5×#217—312no—
31C23×C32×C5—26no—
32C23×C32×C5C2313yes—
33C22×C3⋊D15C2311yescharacteristic, complemented
34C22×C5×C3⋊D3C2311yescharacteristic, complemented
35C22×C32×D5—232no—
36C22×C32×D5—26no—
37C22×C32×D5—212no—
38C22×C32×D5—212no—
39C22×C32×D5D4112yescomplemented
40C22×C32×D5C2313yes—
41C22×C32×D5C2311yescharacteristic
42C2×D5×C3⋊D3—26no—
43#67184—26no—
44C2×C3⋊Dic15C2313yescomplemented
45C5×#638—26no—
46C2×C5×C3⋊Dic3C2313yescomplemented
47#67139—26no—
48#67138—26no—
49D5×C3⋊Dic3—26no—
50D5×#120—91noHall
51C25×C32—51no—
52C22×#638—53no—
53#64162—53no—
54#64162—53no—
55#64162—51no—
56#64162D511yescharacteristic, direct product factor
57C2×#3757—53no—
58C24×C3×C5C2×D314yescomplemented
59C23×C3×D5—248no—
60C23×C3×D5C2×D3112yes—
61C23×C3×D5C2×D3112yescomplemented
62C22×D3×D5—34no—
63C2×#1220—312no—
64C2×C5×#67—312no—
65D5×#67—648no—
66C2×#1202—312no—
67C2×#1201—312no—
68C2×D5×Dic3—312no—
69#6719—312no—
70C5×#217—312no—
71#6665—324no—
72C2×#953—154no—
73C22×C32×C5—28no—
74C22×C32×C5—26no—
75C22×C32×C5C2×D416yescomplemented
76C22×C32×C5C2411yescharacteristic
77C2×C3⋊D15—23no—
78C2×C5×C3⋊D3—23no—
79C2×C32×D5—224no—
80C2×C32×D5—26no—
81C2×C32×D5—224no—
82C2×C32×D5C2×D416yes—
83C2×C32×D5C2×D416yescomplemented
84D5×C3⋊D3—44no—
85C3⋊Dic15—23no—
86C5×C3⋊Dic3—23no—
87C24×D5C3⋊D311yescharacteristic, complemented
88C2×D4×D5—93no—
89C5×#120—91no—
90#4099—91no—
91#4083—93no—
92#4028—93no—
93D5×C22⋊C4—93no—
94C24×C32—53no—
95C24×C32—53no—
96C24×C32—1012no—
97C24×C32C2×D511yescharacteristic, complemented
98C23×C3⋊D3—51no—
99C2×#638—53no—
100C2×#638—53no—
101C2×#638—53no—
102C2×#638—1012no—
103C2×#638C2×D513yescomplemented
104C22×C3⋊Dic3—53no—
105#3757—53no—
106#3757—56no—
107#3757C2×D513yescomplemented
108C23×C3×C5—224no—
109C23×C3×C5C22×D3112yes—
110C22×D15—34no—
111C22×C5×D3—34no—
112C22×C3×D5—2128no—
113C22×C3×D5—224no—
114C22×C3×D5—248no—
115C22×C3×D5—248no—
116C22×C3×D5#67148yescomplemented
117C22×C3×D5C22×D3112yes—
118C22×C3×D5C22×D314yes—
119C2×D3×D5—624no—
120#1220—624no—
121C2×Dic15—312no—
122C5×#67—624no—
123C2×C5×Dic3—312no—
124#1202—624no—
125#1201—624no—
126D5×Dic3—624no—
127C25×C3—54no—
128C22×#67—1512no—
129#953—34no—
130#953—1512no—
131#953—1512no—
132#953—154no—
133C2×#217—1512no—
134C2×C32×C5—26no—
135C2×C32×C5C22×D413yes—
136C3⋊D15—41no—
137C5×C3⋊D3—41no—
138C32×D5—26no—
139C32×D5#12014yescomplemented
140C24×C5C2×C3⋊D311yescharacteristic, complemented
141C23×D5—212no—
142C23×D5—91no—
143C23×D5C2×C3⋊D313yes—
144C23×D5C2×C3⋊D313yescomplemented
145C2×C5×D4—93no—
146C2×C22⋊D5—93no—
147C2×C22⋊D5—93no—
148D4×D5—1812no—
149C2×D20—93no—
150C2×C4×D5—93no—
151C5×C22⋊C4—93no—
152C22⋊Dic5—93no—
153#685—96no—
154C23×C32—26no—
155C23×C32—51no—
156C23×C32—512no—
157C23×C32—53no—
158C23×C32—1012no—
159C23×C32—1012no—
160C23×C32—1032no—
161C23×C32—106no—
162C23×C32C22×D513yes—
163C22×C3⋊D3—51no—
164C22×C3⋊D3—106no—
165C22×C3⋊D3C22×D511yescharacteristic, complemented
166#638—26no—
167#638—106no—
168#638—106no—
169#638—106no—
170C2×C3⋊Dic3—53no—
171C2×C3⋊Dic3—106no—
172C2×C3⋊Dic3C22×D513yescomplemented
173C2×#120—451noSylow
174C22×C3×C5—232no—
175C22×C3×C5—224no—
176C22×C3×C5C2×#67124yescomplemented
177C22×C3×C5C23×D314yes—
178C2×D15—612no—
179C2×C5×D3—612no—
180C2×C3×D5—296no—
181C2×C3×D5—224no—
182C2×C3×D5—296no—
183C2×C3×D5C2×#67124yes—
184C2×C3×D5C2×#67124yescomplemented
185D3×D5—1216no—
186Dic15—612no—
187C5×Dic3—612no—
188C24×C3—512no—
189C24×C3—512no—
190C24×C3—1048no—
191C24×C3D3×D514yescomplemented
192C23×D3—154no—
193C2×#67—312no—
194C2×#67—1512no—
195C2×#67—1512no—
196C2×#67—1512no—
197C2×#67—3048no—
198C22×Dic3—1512no—
199#217—312no—
200#217—1524no—
201#217—1512no—
202C32×C5C2×#12011yescharacteristic, complemented, Hall
203C23×C5—26no—
204C23×C5—91no—
205C23×C5C22×C3⋊D313yes—
206C22×D5—232no—
207C22×D5—26no—
208C22×D5—212no—
209C22×D5—212no—
210C22×D5—91no—
211C22×D5—186no—
212C22×D5#638112yescomplemented
213C22×D5C22×C3⋊D313yes—
214C22×D5C22×C3⋊D311yescharacteristic
215C5×D4—186no—
216C2×C4×C5—93no—
217C22⋊D5—186no—
218C22⋊D5—186no—
219C2×Dic5—93no—
220D20—186no—
221C4×D5—186no—
222C22×C32—28no—
223C22×C32—26no—
224C22×C32—56no—
225C22×C32—56no—
226C22×C32—1024no—
227C22×C32—1024no—
228C22×C32—106no—
229C22×C32D4×D516yescomplemented
230C22×C32C23×D511yescharacteristic
231C2×C3⋊D3—23no—
232C2×C3⋊D3—103no—
233C2×C3⋊D3—204no—
234C3⋊Dic3—23no—
235C3⋊Dic3—103no—
236C25—51no—
237C22×D4—453no—
238#120—91no—
239#120—453no—
240#120—453no—
241#120—451no—
242C2×C22⋊C4—453no—
243C2×C3×C5—224no—
244C2×C3×C5C22×#67112yes—
245D15—124no—
246C3×D5—224no—
247C3×D5#953116yescomplemented
248C5×D3—124no—
249C23×C3—224no—
250C23×C3—54no—
251C23×C3—548no—
252C23×C3—512no—
253C23×C3—1048no—
254C23×C3—1048no—
255C23×C3—10128no—
256C23×C3—1024no—
257C23×C3C2×D3×D5112yes—
258C22×D3—34no—
259C22×D3—154no—
260C22×D3—3024no—
261#67—624no—
262#67—3024no—
263#67—3024no—
264#67—3024no—
265C2×Dic3—312no—
266C2×Dic3—1512no—
267C2×Dic3—3024no—
268C22×C5—28no—
269C22×C5—26no—
270C22×C5—183no—
271C22×C5C2×#63816yescomplemented
272C22×C5C23×C3⋊D311yescharacteristic
273C2×D5—224no—
274C2×D5—26no—
275C2×D5—224no—
276C2×D5—183no—
277C2×D5—364no—
278C2×D5C2×#63816yes—
279C2×D5C2×#63816yescomplemented
280C4×C5—183no—
281Dic5—183no—
282C2×C32—26no—
283C2×C32—54no—
284C2×C32—106no—
285C2×C32C2×D4×D513yes—
286C3⋊D3—41no—
287C3⋊D3—201no—
288C24—53no—
289C24—53no—
290C24—1012no—
291C24—451no—
292C24D5×C3⋊D311yescharacteristic, complemented
293C2×D4—93no—
294C2×D4—453no—
295C2×D4—453no—
296C2×D4—453no—
297C2×D4—9012no—
298C22×C4—453no—
299C22⋊C4—93no—
300C22⋊C4—456no—
301C22⋊C4—453no—
302C3×C5C2×#95314yescomplemented
303C22×C3—232no—
304C22×C3—224no—
305C22×C3—524no—
306C22×C3—524no—
307C22×C3—1096no—
308C22×C3—1096no—
309C22×C3—1024no—
310C22×C3D5×#67124yescomplemented
311C22×C3C22×D3×D514yes—
312C2×D3—612no—
313C2×D3—3012no—
314C2×D3—6016no—
315Dic3—612no—
316Dic3—3012no—
317C2×C5—26no—
318C2×C5—361no—
319C2×C5C22×#63813yes—
320D5—26no—
321D5—361no—
322D5#6416214yesdirect product factor
323C32D5×#12011yescharacteristic, complemented, Sylow
324C23—26no—
325C23—51no—
326C23—53no—
327C23—512no—
328C23—91no—
329C23—1012no—
330C23—1012no—
331C23—1032no—
332C23—106no—
333C23—451no—
334C23—906no—
335C23C2×D5×C3⋊D313yes—
336D4—186no—
337D4—906no—
338D4—906no—
339D4—906no—
340C2×C4—93no—
341C2×C4—453no—
342C2×C4—906no—
343C2×C3—224no—
344C2×C3—516no—
345C2×C3—1024no—
346C2×C3C2×D5×#67112yes—
347D3—124no—
348D3—604no—
349C5C2×#6416211yescharacteristic, complemented, Sylow
350C22—28no—
351C22—26no—
352C22—56no—
353C22—56no—
354C22—1024no—
355C22—1024no—
356C22—106no—
357C22—183no—
358C22—903no—
359C22—1804no—
360C22D5×#63816yescomplemented
361C22C22×D5×C3⋊D311yescharacteristic
362C4—183no—
363C4—903no—
364C3D5×#95314yescomplemented
365C2—26no—
366C2—54no—
367C2—106no—
368C2—361no—
369C2—1801no—
370C2C2×D5×#63813yes—
subgroupG/H
center, FrattiniC22C22×D5×C3⋊D3
commutator, socle, cosocleC22×C32×C5C24
FittingC24×C32×C5C22
hypercenterC24D5×C3⋊D3
hypocenterC32×C5C2×#120
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×C32×D5C2semidirect
2. C2×D5×#638C2semidirect
3. #13088222C2semidirect
4. C5×#64162C2semidirect
5. #13087043C2semidirect
6. #13087041C2semidirect
7. D5×#3757C2semidirect
9. C24×C32×C5C22semidirect
12. C23×C32×D5C22semidirect
13. C22×D5×C3⋊D3C22semidirect
14. C2×#67184C22semidirect
15. C2×C5×#638C22semidirect
17. C2×#67139C22semidirect
18. C2×#67138C22semidirect
19. C2×D5×C3⋊Dic3C22semidirect
20. #10623913C22semidirect
21. C5×#3757C22semidirect
22. #10623770C22semidirect
24. C24×C3×D5D3semidirect
33. C22×C3⋊D15C23semidirect
34. C22×C5×C3⋊D3C23semidirect
39. C22×C32×D5D4semidirect
44. C2×C3⋊Dic15C23semidirect
46. C2×C5×C3⋊Dic3C23semidirect
56. #64162D5direct (N × H)
58. C24×C3×C5C2×D3semidirect
61. C23×C3×D5C2×D3semidirect
75. C22×C32×C5C2×D4semidirect
83. C2×C32×D5C2×D4semidirect
87. C24×D5C3⋊D3semidirect
97. C24×C32C2×D5semidirect
103. C2×#638C2×D5semidirect
107. #3757C2×D5semidirect
116. C22×C3×D5#67semidirect
139. C32×D5#120semidirect
140. C24×C5C2×C3⋊D3semidirect
144. C23×D5C2×C3⋊D3semidirect
165. C22×C3⋊D3C22×D5semidirect
172. C2×C3⋊Dic3C22×D5semidirect
176. C22×C3×C5C2×#67semidirect
184. C2×C3×D5C2×#67semidirect
191. C24×C3D3×D5semidirect
202. C32×C5C2×#120semidirect
212. C22×D5#638semidirect
229. C22×C32D4×D5semidirect
247. C3×D5#953semidirect
271. C22×C5C2×#638semidirect
279. C2×D5C2×#638semidirect
292. C24D5×C3⋊D3semidirect
302. C3×C5C2×#953semidirect
310. C22×C3D5×#67semidirect
322. D5#64162direct (N × H)
323. C32D5×#120semidirect
349. C5C2×#64162semidirect
360. C22D5×#638semidirect
364. C3D5×#953semidirect

Maximal subgroups

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

subgroupindex
C24×C32×D52
C2×D5×#6382
#130882222
C5×#641622
#130870432
#130870412
D5×#37572
D5×#9533
C2×#641625