Finite Group Databasebeta

← D15×#493 #14438765

Subgroups — D15×#493 #14438765

Order 1920.

1520 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1520 conjugacy classes of subgroups — too many to draw. Only its 231 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
1C2×D8×D15C214yesmaximal, complemented
2D15×#127C212yesmaximal, complemented
3C3×C5×#493C211yesmaximal, characteristic, complemented
4#11751252C214yesmaximal, complemented
5#11751174C212yesmaximal, complemented
6#11750796C211yesmaximal, characteristic, complemented
7C4×C8×D15C211yesmaximal, characteristic, complemented
8D5×#493—31nomaximal, Hall
9C2×D4×D15—24no—
10C2×D4×D15C2214yes—
11C2×C3×C5×D8C2214yescomplemented
12C3×C5×#127C2212yescomplemented
13#91829—24no—
14#91829C2214yescomplemented
15C2×#6715C2218yescomplemented
16D8×D15—216no—
17C2×D120C2214yescomplemented
18C2×C8×D15C2212yes—
19#91763C2212yescomplemented
20C42×D15C2211yescharacteristic
21C3×C4×C5×C8C2211yescharacteristic, complemented
22C8×Dic15C2212yescomplemented
23C4×C3⋊#168C2211yescharacteristic, complemented
24D3×#493—51nomaximal, Hall
25C2×D5×D8—34no—
26D5×#127—32no—
27C5×#493D311yescharacteristic, complemented
28#65440—34no—
29#65362—32no—
30#64984—31no—
31C4×C8×D5—31no—
32C23×D15—24no—
33C2×C3×C5×D4—24no—
34C2×C3×C5×D4C2314yes—
35C2×#1220—24no—
36C2×#1220—28no—
37D4×D15—216no—
38D4×D15—416no—
39C2×D60—24no—
40C2×D60C2314yes—
41C2×C4×D15D412yes—
42C2×C4×D15C2311yescharacteristic
43C3×C5×D8—28no—
44C2×C3×C5×C8C2312yes—
45C3×C42×C5C2311yescharacteristic
46#6715—216no—
47C4×Dic15D412yes—
48C4×Dic15C2311yescharacteristic
49C2×C3⋊#168C2312yes—
50D120—28no—
51C8×D15D418yescomplemented
52C2×D3×D8—54no—
53D3×#127—52no—
54C3×#493D511yescharacteristic, complemented
55#5151—54no—
56#5073—52no—
57#4695—51no—
58C4×D3×C8—51no—
59C2×D4×D5—34no—
60C2×D4×D5—64no—
61C2×C5×D8C2×D314yescomplemented
62C5×#127C2×D312yescomplemented
63#4086—34no—
64#4086—64no—
65C2×#686—38no—
66D5×D8—616no—
67C2×D40—34no—
68C2×C8×D5—32no—
69#4020—32no—
70C42×D5—31no—
71C4×C5×C8C2×D311yescharacteristic, complemented
72C8×Dic5—32no—
73C4×#168—31no—
74C2×#493—151noSylow
75C23×C3×C5—24no—
76C22×D15—24no—
77C22×D15—48no—
78C22×D15—416no—
79C22×D15C2×D411yescharacteristic
80C3×C5×D4—28no—
81C3×C5×D4—48no—
82C2×C3×C4×C5C2×D412yes—
83C2×C3×C4×C5C2411yescharacteristic
84#1220—48no—
85#1220—416no—
86C2×Dic15C2×D411yescharacteristic
87C2×Dic15C2×D412yes—
88D60—28no—
89D60—48no—
90C4×D15D818yescomplemented
91C4×D15C2×D414yes—
92C3×C5×C8C2×D414yescomplemented
93C3⋊#168C2×D414yescomplemented
94C2×D3×D4—54no—
95C2×D3×D4—104no—
96C2×C3×D8C2×D514yescomplemented
97C3×#127C2×D512yescomplemented
98#940—54no—
99#940—104no—
100C2×#213—58no—
101D3×D8—1016no—
102C2×D24—54no—
103C2×D3×C8—52no—
104#874—52no—
105C42×D3—51no—
106C3×C4×C8C2×D511yescharacteristic, complemented
107C8×Dic3—52no—
108C4×C3⋊C8—51no—
109C23×D5—64no—
110C2×C5×D4—24no—
111C2×C5×D4C22×D314yes—
112C2×C22⋊D5—64no—
113C2×C22⋊D5—68no—
114D4×D5—616no—
115D4×D5—1216no—
116C2×D20—34no—
117C2×D20—64no—
118C2×C4×D5—32no—
119C2×C4×D5—31no—
120C5×D8—28no—
121C2×C5×C8C22×D312yes—
122C42×C5C22×D311yescharacteristic
123#686—616no—
124C4×Dic5—32no—
125C4×Dic5—31no—
126C2×#168—32no—
127D40—68no—
128C8×D5—38no—
129C22×D8—154no—
130C2×#127—152no—
131#493—154no—
132#493—152no—
133#493—151no—
134#493D1511yescharacteristic, direct product factor
135C2×C4×C8—151no—
136C22×C3×C5—44no—
137C22×C3×C5—48no—
138C22×C3×C5C22×D411yescharacteristic
139C2×D15—44no—
140C2×D15—48no—
141C2×D15—816no—
142C2×D15#12712yes—
143C2×D15C2×D814yes—
144C3×C4×C5C2×D814yescomplemented
145C3×C4×C5C22×D412yes—
146Dic15#12712yes—
147Dic15C2×D814yescomplemented
148C23×D3—104no—
149C2×C3×D4—24no—
150C2×C3×D4C22×D514yes—
151C2×#67—108no—
152C2×#67—104no—
153D3×D4—1016no—
154D3×D4—2016no—
155C2×D12—54no—
156C2×D12—104no—
157C2×C4×D3—52no—
158C2×C4×D3—51no—
159C3×D8—28no—
160C2×C3×C8C22×D512yes—
161C3×C42C22×D511yescharacteristic
162#213—1016no—
163C4×Dic3—52no—
164C4×Dic3—51no—
165C2×C3⋊C8—52no—
166D24—108no—
167D3×C8—58no—
168C23×C5—24no—
169C22×D5—31no—
170C22×D5—64no—
171C22×D5—128no—
172C22×D5—1216no—
173C5×D4—28no—
174C5×D4—48no—
175C2×C4×C5D3×D412yes—
176C2×C4×C5C23×D311yescharacteristic
177C22⋊D5—128no—
178C22⋊D5—1216no—
179C2×Dic5—31no—
180C2×Dic5—32no—
181D20—68no—
182D20—128no—
183C4×D5—38no—
184C4×D5—34no—
185C5×C8D3×D414yescomplemented
186#168—34no—
187C22×D4—154no—
188C22×D4—304no—
189C2×D8—158no—
190C2×D8—154no—
191C2×D8—3016no—
192C2×D8C2×D1514yescomplemented
193C22×C8—152no—
194#127—154no—
195#127—152no—
196#127—304no—
197#127C2×D1512yescomplemented
198C2×C42—151no—
199C4×C8—152no—
200C4×C8—151no—
201C4×C8C2×D1511yescharacteristic, complemented
202C2×C3×C5—84no—
203C2×C3×C5C2×#12711yescharacteristic
204C2×C3×C5C22×D812yes—
205D15—84no—
206D15#49314yesdirect product factor
207C23×C3—24no—
208C22×D3—51no—
209C22×D3—104no—
210C22×D3—208no—
211C22×D3—2016no—
212C3×D4—28no—
213C3×D4—48no—
214C2×C3×C4D4×D512yes—
215C2×C3×C4C23×D511yescharacteristic
216#67—208no—
217#67—2016no—
218C2×Dic3—51no—
219C2×Dic3—52no—
220D12—108no—
221D12—208no—
222C4×D3—54no—
223C4×D3—58no—
224C3×C8D4×D514yescomplemented
225C3⋊C8—54no—
226C22×C5—44no—
227C22×C5—48no—
228C22×C5C2×D3×D411yescharacteristic
229C2×D5—32no—
230C2×D5—34no—
231C2×D5—124no—
232C2×D5—128no—
233C2×D5—2416no—
234C4×C5D3×D814yescomplemented
235C4×C5C2×D3×D412yes—
236Dic5—32no—
237Dic5—34no—
238C24—304no—
239C2×D4—24no—
240C2×D4—154no—
241C2×D4—3016no—
242C2×D4—308no—
243C2×D4—304no—
244C2×D4—304no—
245C2×D4—6016no—
246C2×D4C22×D1514yes—
247C22×C4—152no—
248C22×C4—151no—
249D8—28no—
250D8—3016no—
251D8—308no—
252C2×C8—152no—
253C2×C8—158no—
254C2×C8C22×D1512yes—
255C42—152no—
256C42—151no—
257C42C22×D1511yescharacteristic
258C3×C5C2×#49311yescharacteristic, complemented, Hall
259C22×C3—44no—
260C22×C3—48no—
261C22×C3C2×D4×D511yescharacteristic
262C2×D3—52no—
263C2×D3—54no—
264C2×D3—204no—
265C2×D3—208no—
266C2×D3—4016no—
267C3×C4D5×D814yescomplemented
268C3×C4C2×D4×D512yes—
269Dic3—52no—
270Dic3—54no—
271C2×C5—84no—
272C2×C5D3×#12711yescharacteristic
273C2×C5C2×D3×D812yes—
274D5—34no—
275D5—244no—
276C23—24no—
277C23—151no—
278C23—304no—
279C23—608no—
280C23—6016no—
281D4—28no—
282D4—48no—
283D4—308no—
284D4—608no—
285D4—6016no—
286D4—608no—
287C2×C4—151no—
288C2×C4—152no—
289C2×C4—158no—
290C2×C4—154no—
291C2×C4D4×D1512yes—
292C2×C4C23×D1511yescharacteristic
293C8—154no—
294C8D4×D1514yescomplemented
295C2×C3—84no—
296C2×C3D5×#12711yescharacteristic
297C2×C3C2×D5×D812yes—
298D3—54no—
299D3—404no—
300C5D3×#49311yescharacteristic, complemented, Sylow
301C22—44no—
302C22—48no—
303C22—152no—
304C22—154no—
305C22—604no—
306C22—608no—
307C22—12016no—
308C22C2×D4×D1511yescharacteristic
309C4—152no—
310C4—154no—
311C4D8×D1514yescomplemented
312C4C2×D4×D1512yes—
313C3D5×#49311yescharacteristic, complemented, Sylow
314C2—84no—
315C2—154no—
316C2—1204no—
317C2D15×#12711yescharacteristic
318C2C2×D8×D1512yes—
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#493C2
socleC22×C3×C5C22×D4
hypercenter#493D15
hypocenterC3×C5C2×#493
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×D8×D15C2semidirect
2. D15×#127C2semidirect
3. C3×C5×#493C2semidirect
4. #11751252C2semidirect
5. #11751174C2semidirect
6. #11750796C2semidirect
7. C4×C8×D15C2semidirect
11. C2×C3×C5×D8C22semidirect
12. C3×C5×#127C22semidirect
14. #91829C22semidirect
15. C2×#6715C22semidirect
17. C2×D120C22semidirect
19. #91763C22semidirect
21. C3×C4×C5×C8C22semidirect
22. C8×Dic15C22semidirect
23. C4×C3⋊#168C22semidirect
27. C5×#493D3semidirect
51. C8×D15D4semidirect
54. C3×#493D5semidirect
61. C2×C5×D8C2×D3semidirect
62. C5×#127C2×D3semidirect
71. C4×C5×C8C2×D3semidirect
90. C4×D15D8semidirect
92. C3×C5×C8C2×D4semidirect
93. C3⋊#168C2×D4semidirect
96. C2×C3×D8C2×D5semidirect
97. C3×#127C2×D5semidirect
106. C3×C4×C8C2×D5semidirect
134. #493D15direct (N × H)
144. C3×C4×C5C2×D8semidirect
147. Dic15C2×D8semidirect
185. C5×C8D3×D4semidirect
192. C2×D8C2×D15semidirect
197. #127C2×D15semidirect
201. C4×C8C2×D15semidirect
206. D15#493direct (N × H)
224. C3×C8D4×D5semidirect
234. C4×C5D3×D8semidirect
258. C3×C5C2×#493semidirect
267. C3×C4D5×D8semidirect
294. C8D4×D15semidirect
300. C5D3×#493semidirect
311. C4D8×D15semidirect
313. C3D5×#493semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D152
D15×#1272
C3×C5×#4932
#117512522
#117511742
#117507962
C4×C8×D152
D5×#4933
D3×#4935