Finite Group Databasebeta

← D15×#213 #13087490

Subgroups — D15×#213 #13087490

Order 1440.

326 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C3×D4×D15C211yesmaximal, characteristic, complemented
2D12×D15C211yesmaximal, characteristic, complemented
3#10623909C211yesmaximal, characteristic, complemented
4C3×C5×#213C211yesmaximal, characteristic, complemented
5#10623799C211yesmaximal, characteristic, complemented
6#10623798C211yesmaximal, characteristic, complemented
7C3⋊C8×D15C211yesmaximal, characteristic, complemented
8D8×D15—31nomaximal
9D5×#213—31nomaximal
10C22×C3×D15—21no—
11C2×D3×D15—21no—
12C32×C5×D4C2211yescharacteristic, complemented
13C4⋊(C3⋊D15)C2211yescharacteristic, complemented
14C3×#1220—21no—
15C3×D60C2211yescharacteristic, complemented
16C3×C4×D15C2211yescharacteristic
17C3×C5×D12C2211yescharacteristic, complemented
18#67152—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21D3×#213—51nomaximal, Hall
22D4×D15—31no—
23D4×D15D311yescharacteristic, complemented
24C3×D4×D5—31no—
25D5×D12—31no—
26C3×C5×D8—31no—
27#6715—31no—
28#6715—31no—
29#6715—31no—
30#6715—61no—
31D120—31no—
32C8×D15—31no—
33C5×#213D311yescharacteristic, complemented
34#6653—31no—
35#6652—31no—
36D5×C3⋊C8—31no—
37C22×C32×C5—21no—
38C2×C3⋊D15—21no—
39C2×C3×D15—21no—
40C2×C3×D15—42no—
41C2×C3×D15D411yescharacteristic
42C2×C3×C5×D3—21no—
43D3×D15—42no—
44C32×C4×C5C2311yescharacteristic
45C3×Dic15D411yescharacteristic
46D5×D8—91noHall
47C3×D3×D4—51no—
48D3×D12—51no—
49#3753—51no—
50C3×#213D511yescharacteristic, complemented
51#3714—51no—
52#3713—51no—
53D3×C3⋊C8—51no—
54C22×D15—21no—
55C22×D15—61no—
56C22×C3×D5—61no—
57C2×D3×D5—61no—
58C3×C5×D4—21no—
59C3×C5×D4—31no—
60C3×C5×D4C2×D311yescharacteristic, complemented
61C3×C5×D4C2×D311yescharacteristic, complemented
62#1220—21no—
63#1220—61no—
64D60—31no—
65D60—31no—
66D60—61no—
67D60C2×D311yescharacteristic, complemented
68C4×D15C2×D311yescharacteristic
69C5×D12C2×D311yescharacteristic, complemented
70C3×C22⋊D5—61no—
71C3×D20—31no—
72C3×C4×D5—31no—
73#1203—61no—
74C3×C5×C8—31no—
75C3⋊#168—31no—
76C3⋊#168—31no—
77C3⋊#168—61no—
78C5×C3⋊C8C2×D311yescharacteristic, complemented
79C2×#213—151no—
80D3×D8—151no—
81C2×C32×C5—41no—
82C2×C32×C5C2×D411yescharacteristic
83C3⋊D15—41no—
84C3×D15—41no—
85C3×D15D812yescomplemented
86C3×C5×D3—41no—
87D4×D5—31no—
88D4×D5—91no—
89C5×D8—31no—
90#686—91no—
91#686—91no—
92D40—91no—
93C8×D5—91no—
94C22×C3×D3—101no—
95C2×D32—101no—
96C32×D4C2×D511yescharacteristic, complemented
97C4⋊(C3⋊D3)—51no—
98C3×#67—101no—
99C3×D12—51no—
100C3×D12C2×D511yescharacteristic, complemented
101C3×C4×D3—51no—
102#626—101no—
103C3⋊(C3⋊C8)—51no—
104C3×C3⋊C8C2×D511yescharacteristic, complemented
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—41no—
108C22×C3×C5—61no—
109C2×D15—21no—
110C2×D15—42no—
111C2×D15—61no—
112C2×D15—61no—
113C2×D15—121no—
114C2×D15—122no—
115C2×D15#6711yescharacteristic
116C2×C5×D3—21no—
117C2×C3×D5—31no—
118C2×C3×D5—61no—
119C2×C3×D5—122no—
120D3×D5—122no—
121C3×C4×C5—21no—
122C3×C4×C5C22×D311yescharacteristic
123C3×C4×C5C22×D311yescharacteristic
124Dic15#6711yescharacteristic
125C3×Dic5—31no—
126C2×C3×D4—151no—
127D3×D4—51no—
128D3×D4—151no—
129C2×D12—151no—
130C3×D8—31no—
131#213—151no—
132#213—151no—
133#213—151no—
134#213—151no—
135#213—151no—
136#213—301no—
137#213D1511yescharacteristic, direct product factor
138C2×C3⋊C8—151no—
139D24—151no—
140D3×C8—151no—
141C32×C5C2×D811yescharacteristic, complemented, Hall
142C22×D5—61no—
143C22×D5—181no—
144C5×D4—31no—
145C5×D4D3211yescharacteristic, complemented
146C22⋊D5—61no—
147C22⋊D5—181no—
148D20—31no—
149D20—91no—
150C4×D5—31no—
151C5×C8—31no—
152#168—91no—
153C22×C32—21no—
154C2×C3⋊D3—101no—
155C2×C3×D3—21no—
156C2×C3×D3—51no—
157C2×C3×D3—101no—
158C2×C3×D3—202no—
159D32—202no—
160C32×C4C22×D511yescharacteristic
161C3×Dic3—51no—
162C2×D8—451noSylow
163C2×C3×C5—21no—
164C2×C3×C5—41no—
165C2×C3×C5—41no—
166C2×C3×C5—81no—
167C2×C3×C5—121no—
168C2×C3×C5D3×D411yescharacteristic
169C2×C3×C5C2×#6711yescharacteristic
170D15—41no—
171D15—121no—
172D15—121no—
173D15—241no—
174D15#21312yesdirect product factor
175C3×D5—32no—
176C3×D5—121no—
177C5×D3—41no—
178C23×C3—301no—
179C22×D3—101no—
180C22×D3—301no—
181C22×D3—301no—
182C3×D4—21no—
183C3×D4—31no—
184C3×D4—151no—
185C3×D4—301no—
186C3×D4D3×D511yescharacteristic, complemented
187C3×D4C2×D1511yescharacteristic, complemented
188C2×C3×C4—151no—
189#67—101no—
190#67—301no—
191D12—51no—
192D12—151no—
193D12—151no—
194D12—301no—
195D12—301no—
196D12C2×D1511yescharacteristic, complemented
197C4×D3—51no—
198C3×C8—31no—
199C3⋊C8—151no—
200C3⋊C8—151no—
201C3⋊C8—301no—
202C3⋊C8C2×D1511yescharacteristic, complemented
203C22×C5—21no—
204C22×C5—61no—
205C2×D5—31no—
206C2×D5—61no—
207C2×D5—122no—
208C2×D5—181no—
209C2×D5—362no—
210C4×C5C2×D3211yescharacteristic
211Dic5—31no—
212C2×C32—41no—
213C2×C32D4×D511yescharacteristic
214C3⋊D3—201no—
215C3×D3—41no—
216C3×D3—52no—
217C3×D3—201no—
218C2×D4—151no—
219C2×D4—451no—
220D8—31no—
221D8—451no—
222D8—451no—
223D8—451no—
224C2×C8—451no—
225C3×C5—21no—
226C3×C5D3×D811yescharacteristic, complemented
227C3×C5C2×#21311yescharacteristic, complemented
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—41no—
231C22×C3—61no—
232C22×C3—151no—
233C22×C3—301no—
234C22×C3—602no—
235C2×D3—21no—
236C2×D3—51no—
237C2×D3—101no—
238C2×D3—202no—
239C2×D3—301no—
240C2×D3—301no—
241C2×D3—601no—
242C2×D3—602no—
243C2×D3—602no—
244C3×C4—21no—
245C3×C4—151no—
246C3×C4C2×D3×D511yescharacteristic
247C3×C4C22×D1511yescharacteristic
248Dic3—51no—
249C2×C5—41no—
250C2×C5—121no—
251C2×C5D3×#6711yescharacteristic
252D5—32no—
253D5—121no—
254D5—361no—
255C32D5×D811yescharacteristic, complemented, Sylow
256C23—301no—
257C23—901no—
258D4—31no—
259D4—151no—
260D4—301no—
261D4—451no—
262D4—901no—
263D4D3×D1511yescharacteristic, complemented
264C2×C4—151no—
265C8—31no—
266C8—451no—
267C2×C3—21no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—81no—
271C2×C3—121no—
272C2×C3—152no—
273C2×C3—601no—
274C2×C3D5×#6711yescharacteristic
275C2×C3D4×D1511yescharacteristic
276D3—41no—
277D3—52no—
278D3—201no—
279D3—601no—
280D3—601no—
281D3—1201no—
282C5D3×#21311yescharacteristic, complemented, Sylow
283C22—21no—
284C22—61no—
285C22—151no—
286C22—301no—
287C22—602no—
288C22—901no—
289C22—1802no—
290C4—151no—
291C4C2×D3×D1511yescharacteristic
292C3—21no—
293C3D5×#21311yescharacteristic, complemented
294C3D8×D1511yescharacteristic, complemented
295C2—41no—
296C2—121no—
297C2—152no—
298C2—601no—
299C2—1801no—
300C2#67×D1511yescharacteristic
subgroupG/H
centerC2#67×D15
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterD4D3×D15
hypocenterC32×C5C2×D8
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×D15C2semidirect
2. D12×D15C2semidirect
3. #10623909C2semidirect
4. C3×C5×#213C2semidirect
5. #10623799C2semidirect
6. #10623798C2semidirect
7. C3⋊C8×D15C2semidirect
12. C32×C5×D4C22semidirect
13. C4⋊(C3⋊D15)C22semidirect
15. C3×D60C22semidirect
17. C3×C5×D12C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
23. D4×D15D3semidirect
33. C5×#213D3semidirect
50. C3×#213D5semidirect
60. C3×C5×D4C2×D3semidirect
61. C3×C5×D4C2×D3semidirect
67. D60C2×D3semidirect
69. C5×D12C2×D3semidirect
78. C5×C3⋊C8C2×D3semidirect
85. C3×D15D8semidirect
96. C32×D4C2×D5semidirect
100. C3×D12C2×D5semidirect
104. C3×C3⋊C8C2×D5semidirect
137. #213D15direct (N × H)
141. C32×C5C2×D8semidirect
145. C5×D4D32semidirect
174. D15#213direct (N × H)
186. C3×D4D3×D5semidirect
187. C3×D4C2×D15semidirect
196. D12C2×D15semidirect
202. C3⋊C8C2×D15semidirect
226. C3×C5D3×D8semidirect
227. C3×C5C2×#213semidirect
255. C32D5×D8semidirect
263. D4D3×D15semidirect
282. C5D3×#213semidirect
293. C3D5×#213semidirect
294. C3D8×D15semidirect

Maximal subgroups

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

subgroupindex
C3×D4×D152
D12×D152
#106239092
C3×C5×#2132
#106237992
#106237982
C3⋊C8×D152
D8×D153
D5×#2133
D3×#2135