Finite Group Databasebeta

← #67×Dic15 #13087558

Subgroups — #67×Dic15 #13087558

Order 1440.

410 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C22×C3×Dic15C211yesmaximal, characteristic, complemented
2C2×C3×C5×#67C211yesmaximal, characteristic
3C2×D3×Dic15C211yesmaximal, characteristic, complemented
4#10623913C211yesmaximal, characteristic, complemented
5Dic3⋊Dic15C211yesmaximal, characteristic, complemented
6(C2×D3)⋊Dic15C211yesmaximal, characteristic, complemented
7Dic3×Dic15C211yesmaximal, characteristic, complemented
8D4×Dic15—31nomaximal
9Dic5×#67—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3×Dic15—22no—
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C3×C5×#67C414yescomplemented
18C2×C3×C5×Dic3C2211yescharacteristic
19D3×Dic15—22no—
20Dic3×#67—51nomaximal, Hall
21C2×C3×C5×D4—31no—
22C22×Dic15—31no—
23C22×Dic15D311yescharacteristic, complemented
24C2×C5×#67D311yescharacteristic
25C22×C3×Dic5—31no—
26C2×D3×Dic5—31no—
27#6719—31no—
28#6719—31no—
29#6719—31no—
30#6719—61no—
31C4⋊Dic15—31no—
32C4×Dic15—31no—
33Dic3⋊Dic5—31no—
34(C2×D3)⋊Dic5—31no—
35Dic3×Dic5—31no—
36C22×C32×C5—21no—
37C22×C32×C5—21no—
38C22×C32×C5C2×C412yescomplemented
39C22×C32×C5C2311yescharacteristic
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3—21no—
42C2×C3×C5×D3C2×C412yescomplemented
43C3⋊Dic15—21no—
44C3⋊Dic15—21no—
45C3×Dic15—21no—
46C3×Dic15D412yescomplemented
47C3×C5×Dic3C2×C412yescomplemented
48D4×Dic5—91noHall
49C2×C3×#67D511yescharacteristic
50C22×C3×Dic3—51no—
51C2×D3×Dic3—51no—
52#3757—51no—
53Dic3⋊Dic3—51no—
54(C2×D3)⋊Dic3—51no—
55Dic32—51no—
56C23×C3×C5—21no—
57C23×C3×C5—31no—
58C23×C3×C5C2×D311yescharacteristic
59C23×C3×C5C2×D311yescharacteristic
60C22×C5×D3C2×D311yescharacteristic
61C3×C5×D4—34no—
62C2×C3×C4×C5—31no—
63C2×Dic15—22no—
64C2×Dic15—31no—
65C2×Dic15—31no—
66C2×Dic15—31no—
67C2×Dic15—31no—
68C2×Dic15—62no—
69C2×Dic15—61no—
70C2×Dic15—61no—
71C2×Dic15C2×D311yescharacteristic
72C2×Dic15C2×D311yescharacteristic
73C5×#67Dic314yescomplemented
74C2×C5×Dic3C2×D311yescharacteristic
75C2×C3×Dic5—31no—
76C2×C3×Dic5—31no—
77C2×C3×Dic5—62no—
78D3×Dic5—62no—
79D4×Dic3—151no—
80C4×#67—151no—
81C2×C32×C5—22no—
82C2×C32×C5C22×C411yescharacteristic
83C2×C32×C5C2×D411yescharacteristic
84C2×C32×C5#4111yescharacteristic
85C3×C5×D3—22no—
86C2×C5×D4—31no—
87C22×Dic5—31no—
88C22×Dic5—91no—
89C22⋊Dic5—91no—
90C22⋊Dic5—91no—
91C4⋊Dic5—91no—
92C4×Dic5—91no—
93C23×C32C2×D511yescharacteristic
94C22×C3×D3C2×D511yescharacteristic
95C2×C3⋊Dic3—51no—
96C2×C3⋊Dic3—51no—
97C3×#67Dic514yescomplemented
98C2×C3×Dic3—51no—
99C2×C3×Dic3—51no—
100C2×C3×Dic3—102no—
101C2×C3×Dic3C2×D511yescharacteristic
102D3×Dic3—102no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—22no—
108C22×C3×C5—21no—
109C22×C3×C5—32no—
110C22×C3×C5—41no—
111C22×C3×C5—41no—
112C22×C3×C5—61no—
113C22×C3×C5—61no—
114C22×C3×C5C4×D312yescomplemented
115C22×C3×C5C2×Dic312yescomplemented
116C22×C3×C5C22×D311yescharacteristic
117C22×C3×C5C22×D311yescharacteristic
118C2×C5×D3—21no—
119C2×C5×D3—21no—
120C2×C5×D3C2×Dic312yescomplemented
121C3×C4×C5—32no—
122Dic15—21no—
123Dic15—61no—
124Dic15—61no—
125Dic15—61no—
126Dic15—61no—
127Dic15—121no—
128Dic15—121no—
129Dic15#6712yesdirect product factor
130C3×Dic5—32no—
131C3×Dic5—61no—
132C5×Dic3C2×Dic312yescomplemented
133C2×C3×D4—31no—
134C22×C3×C4—151no—
135C2×#67D1511yescharacteristic
136C22×Dic3—51no—
137C22×Dic3—151no—
138C2×C4×D3—151no—
139#217—151no—
140#217—151no—
141#217—151no—
142#217—301no—
143(C2×D3)⋊C4—151no—
144C4⋊Dic3—151no—
145Dic3⋊C4—151no—
146C4×Dic3—151no—
147C4×Dic3—151no—
148C32×C5C4×D411yescharacteristic, complemented, Hall
149C23×C5—31no—
150C23×C5D3211yescharacteristic
151C5×D4—34no—
152C2×C4×C5—31no—
153C2×Dic5—31no—
154C2×Dic5—31no—
155C2×Dic5—62no—
156C2×Dic5—91no—
157C2×Dic5—91no—
158C2×Dic5—182no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32C2×Dic512yescomplemented
162C22×C32C22×D511yescharacteristic
163C2×C3×D3—21no—
164C2×C3×D3—21no—
165C2×C3×D3C2×Dic512yescomplemented
166C3⋊Dic3—101no—
167C3⋊Dic3—101no—
168C3×Dic3—52no—
169C3×Dic3—101no—
170C3×Dic3C2×Dic512yescomplemented
171C4×D4—451noSylow
172C2×C3×C5—22no—
173C2×C3×C5—22no—
174C2×C3×C5—21no—
175C2×C3×C5—21no—
176C2×C3×C5—21no—
177C2×C3×C5—42no—
178C2×C3×C5—62no—
179C2×C3×C5C2×C4×D311yescharacteristic
180C2×C3×C5#23511yescharacteristic
181C2×C3×C5D3×D411yescharacteristic
182C2×C3×C5#23711yescharacteristic
183C2×C3×C5C22×Dic311yescharacteristic
184C2×C3×C5C2×#6711yescharacteristic
185C5×D3—22no—
186C23×C3—21no—
187C23×C3—31no—
188C23×C3D3×D511yescharacteristic
189C23×C3C2×D1511yescharacteristic
190C22×D3C2×D1511yescharacteristic
191C3×D4—34no—
192C2×C3×C4—31no—
193C2×C3×C4—151no—
194C2×C3×C4—151no—
195C2×C3×C4—302no—
196#67Dic1514yesdirect product factor
197C2×Dic3—51no—
198C2×Dic3—51no—
199C2×Dic3—102no—
200C2×Dic3—151no—
201C2×Dic3—151no—
202C2×Dic3—151no—
203C2×Dic3—151no—
204C2×Dic3—301no—
205C2×Dic3—301no—
206C2×Dic3—302no—
207C2×Dic3C2×D1511yescharacteristic
208C4×D3—302no—
209C22×C5—21no—
210C22×C5—21no—
211C22×C5—32no—
212C22×C5—61no—
213C22×C5—61no—
214C22×C5D3×Dic312yescomplemented
215C22×C5C2×D3211yescharacteristic
216C4×C5—32no—
217Dic5—32no—
218Dic5—61no—
219Dic5—181no—
220Dic5—181no—
221C2×C32—22no—
222C2×C32D4×D511yescharacteristic
223C2×C32#71111yescharacteristic
224C2×C32C22×Dic511yescharacteristic
225C3×D3—22no—
226C2×D4—31no—
227C22×C4—151no—
228C22×C4—451no—
229C4⋊C4—451no—
230C22⋊C4—451no—
231C22⋊C4—451no—
232C42—451no—
233C3×C5—21no—
234C3×C5C4×#6711yescharacteristic, complemented
235C3×C5D4×Dic311yescharacteristic, complemented
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—21no—
242C22×C3—32no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3D3×Dic512yescomplemented
248C22×C3C2×Dic1512yescomplemented
249C22×C3C2×D3×D511yescharacteristic
250C22×C3C22×D1511yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3C2×Dic1512yescomplemented
254C3×C4—32no—
255C3×C4—152no—
256C3×C4—301no—
257Dic3—52no—
258Dic3—101no—
259Dic3—301no—
260Dic3—301no—
261Dic3—301no—
262Dic3—301no—
263Dic3—601no—
264Dic3—601no—
265Dic3C2×Dic1512yescomplemented
266C2×C5—22no—
267C2×C5—62no—
268C2×C5C2×D3×Dic311yescharacteristic
269C2×C5#380411yescharacteristic
270C2×C5D3×#6711yescharacteristic
271C32D4×Dic511yescharacteristic, complemented, Sylow
272C23—31no—
273C23D3×D1511yescharacteristic
274D4—34no—
275C2×C4—31no—
276C2×C4—151no—
277C2×C4—151no—
278C2×C4—302no—
279C2×C4—451no—
280C2×C4—451no—
281C2×C4—902no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—42no—
288C2×C3—62no—
289C2×C3C2×D3×Dic511yescharacteristic
290C2×C3#678211yescharacteristic
291C2×C3D5×#6711yescharacteristic
292C2×C3D4×D1511yescharacteristic
293C2×C3#681911yescharacteristic
294C2×C3C22×Dic1511yescharacteristic
295D3—22no—
296C5Dic3×#6711yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—32no—
300C22—61no—
301C22—61no—
302C22D3×Dic1512yescomplemented
303C22C2×D3×D1511yescharacteristic
304C4—32no—
305C4—152no—
306C4—301no—
307C4—901no—
308C4—901no—
309C3—21no—
310C3Dic5×#6711yescharacteristic, complemented
311C3D4×Dic1511yescharacteristic, complemented
312C2—22no—
313C2—62no—
314C2C2×D3×Dic1511yescharacteristic
315C2#1062421011yescharacteristic
316C2#67×D1511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutatorC2×C32×C5C22×C4
FittingC23×C32×C5C22
socle, cosocleC22×C32×C5C23
prime radicalC2×C3×C5×#67C2
hypercenterC23D3×D15
hypocenterC32×C5C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C3×Dic15C2semidirect
3. C2×D3×Dic15C2semidirect
4. #10623913C2semidirect
5. Dic3⋊Dic15C2semidirect
6. (C2×D3)⋊Dic15C2semidirect
7. Dic3×Dic15C2semidirect
17. C3×C5×#67C4semidirect
23. C22×Dic15D3semidirect
38. C22×C32×C5C2×C4semidirect
42. C2×C3×C5×D3C2×C4semidirect
46. C3×Dic15D4semidirect
47. C3×C5×Dic3C2×C4semidirect
73. C5×#67Dic3semidirect
97. C3×#67Dic5semidirect
114. C22×C3×C5C4×D3semidirect
115. C22×C3×C5C2×Dic3semidirect
120. C2×C5×D3C2×Dic3semidirect
129. Dic15#67direct (N × H)
132. C5×Dic3C2×Dic3semidirect
148. C32×C5C4×D4semidirect
161. C22×C32C2×Dic5semidirect
165. C2×C3×D3C2×Dic5semidirect
170. C3×Dic3C2×Dic5semidirect
196. #67Dic15direct (N × H)
214. C22×C5D3×Dic3semidirect
234. C3×C5C4×#67semidirect
235. C3×C5D4×Dic3semidirect
247. C22×C3D3×Dic5semidirect
248. C22×C3C2×Dic15semidirect
253. C2×D3C2×Dic15semidirect
265. Dic3C2×Dic15semidirect
271. C32D4×Dic5semidirect
296. C5Dic3×#67semidirect
302. C22D3×Dic15semidirect
310. C3Dic5×#67semidirect
311. C3D4×Dic15semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic152
C2×C3×C5×#672
C2×D3×Dic152
#106239132
Dic3⋊Dic152
(C2×D3)⋊Dic152
Dic3×Dic152
D4×Dic153
Dic5×#673
Dic3×#675