Finite Group Databasebeta

← #67⋊Dic15 #13087568

Subgroups — #67⋊Dic15 #13087568

Order 1440.

430 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 124 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
4C3×#6719C211yesmaximal, characteristic, complemented
5Dic3⋊Dic15C211yesmaximal, characteristic, complemented
6(C2×D3)⋊Dic15C211yesmaximal, characteristic, complemented
7Dic3×Dic15C211yesmaximal, characteristic, complemented
8D4×Dic15—31nomaximal
9#67⋊Dic5—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C3⋊Dic15—22no—
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3⋊Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C3×C5×#67C414yescomplemented
18C2×C3×C5×Dic3C2211yescharacteristic
19D3×Dic15—22no—
20#67⋊Dic3—51nomaximal, Hall
21C2×C3×C5×D4—31no—
22C22×Dic15—31no—
23C22×Dic15—31no—
24C22×Dic15—31no—
25C22×Dic15—61no—
26C2×C5×#67D311yescharacteristic
27C2×D3×Dic5—31no—
28#6719—31no—
29#6719D311yescharacteristic, complemented
30C4⋊Dic15—31no—
31C4×Dic15—31no—
32C3×C22⋊Dic5—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⋊Dic15D412yescomplemented
45C3×Dic15—21no—
46C3×Dic15—21no—
47C3×C5×Dic3C2×C412yescomplemented
48D4×Dic5—91noHall
49C22×C3⋊Dic3—51no—
50C2×C3×#67D511yescharacteristic
51C2×D3×Dic3—51no—
52C3×#217—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—31no—
64C2×Dic15—31no—
65C2×Dic15—31no—
66C2×Dic15—31no—
67C2×Dic15—62no—
68C2×Dic15—62no—
69C2×Dic15—62no—
70C2×Dic15—61no—
71C2×Dic15—61no—
72C2×Dic15—64no—
73C2×Dic15C2×D311yescharacteristic
74C2×Dic15C2×D311yescharacteristic
75C5×#67Dic314yescomplemented
76C2×C5×Dic3C2×D311yescharacteristic
77C2×C3×Dic5—31no—
78C2×C3×Dic5—31no—
79D3×Dic5—62no—
80D4×Dic3—151no—
81#67⋊C4—151no—
82C2×C32×C5—22no—
83C2×C32×C5C22×C411yescharacteristic
84C2×C32×C5C2×D411yescharacteristic
85C2×C32×C5#4111yescharacteristic
86C3×C5×D3—22no—
87C2×C5×D4—31no—
88C22×Dic5—91no—
89C22×Dic5—91no—
90C22⋊Dic5—31no—
91C22⋊Dic5—91no—
92C4⋊Dic5—91no—
93C4×Dic5—91no—
94C23×C32C2×D511yescharacteristic
95C22×C3×D3C2×D511yescharacteristic
96C2×C3⋊Dic3—51no—
97C2×C3⋊Dic3—51no—
98C2×C3⋊Dic3—102no—
99C3×#67Dic514yescomplemented
100C2×C3×Dic3—51no—
101C2×C3×Dic3—51no—
102C2×C3×Dic3C2×D511yescharacteristic
103D3×Dic3—102no—
104C22×C3×C5—22no—
105C22×C3×C5—22no—
106C22×C3×C5—21no—
107C22×C3×C5—21no—
108C22×C3×C5—21no—
109C22×C3×C5—21no—
110C22×C3×C5—22no—
111C22×C3×C5—21no—
112C22×C3×C5—32no—
113C22×C3×C5—61no—
114C22×C3×C5—61no—
115C22×C3×C5C4×D312yescomplemented
116C22×C3×C5C2×Dic312yescomplemented
117C22×C3×C5C22×D311yescharacteristic
118C22×C3×C5C22×D311yescharacteristic
119C2×C5×D3—21no—
120C2×C5×D3—21no—
121C2×C5×D3C2×Dic312yescomplemented
122C3×C4×C5—32no—
123Dic15—21no—
124Dic15—21no—
125Dic15—32no—
126Dic15—32no—
127Dic15—62no—
128Dic15—62no—
129Dic15—61no—
130Dic15—61no—
131C3×Dic5—61no—
132C3×Dic5—61no—
133C5×Dic3C2×Dic312yescomplemented
134C2×C3×D4—31no—
135C2×#67D1511yescharacteristic
136C22×Dic3—151no—
137C22×Dic3—151no—
138C22×Dic3—151no—
139C22×Dic3—301no—
140C2×C4×D3—151no—
141C3×C22⋊C4—151no—
142#217—51no—
143#217—151no—
144(C2×D3)⋊C4—151no—
145C4⋊Dic3—151no—
146Dic3⋊C4—151no—
147C4×Dic3—151no—
148C4×Dic3—151no—
149C32×C5C4×D411yescharacteristic, complemented, Hall
150C23×C5—31no—
151C23×C5D3211yescharacteristic
152C5×D4—34no—
153C2×C4×C5—31no—
154C2×Dic5—31no—
155C2×Dic5—31no—
156C2×Dic5—91no—
157C2×Dic5—91no—
158C2×Dic5—182no—
159C2×Dic5—182no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32C2×Dic512yescomplemented
163C22×C32C22×D511yescharacteristic
164C2×C3×D3—21no—
165C2×C3×D3—21no—
166C2×C3×D3C2×Dic512yescomplemented
167C3⋊Dic3—52no—
168C3⋊Dic3—101no—
169C3×Dic3—101no—
170C3×Dic3—101no—
171C3×Dic3C2×Dic512yescomplemented
172C4×D4—451noSylow
173C2×C3×C5—24no—
174C2×C3×C5—22no—
175C2×C3×C5—22no—
176C2×C3×C5—21no—
177C2×C3×C5—21no—
178C2×C3×C5—21no—
179C2×C3×C5—62no—
180C2×C3×C5C2×C4×D311yescharacteristic
181C2×C3×C5D3×D411yescharacteristic
182C2×C3×C5D3×D411yescharacteristic
183C2×C3×C5#23711yescharacteristic
184C2×C3×C5#23711yescharacteristic
185C2×C3×C5C22×Dic311yescharacteristic
186C5×D3—22no—
187C23×C3—21no—
188C23×C3—31no—
189C23×C3D3×D511yescharacteristic
190C23×C3C2×D1511yescharacteristic
191C22×D3C2×D1511yescharacteristic
192C3×D4—34no—
193C2×C3×C4—31no—
194C2×C3×C4—151no—
195C2×C3×C4—151no—
196#67Dic1514yescomplemented
197C2×Dic3—51no—
198C2×Dic3—51no—
199C2×Dic3—151no—
200C2×Dic3—151no—
201C2×Dic3—151no—
202C2×Dic3—151no—
203C2×Dic3—301no—
204C2×Dic3—301no—
205C2×Dic3—302no—
206C2×Dic3—302no—
207C2×Dic3—302no—
208C2×Dic3—304no—
209C2×Dic3C2×D1511yescharacteristic
210C4×D3—302no—
211C22×C5—21no—
212C22×C5—21no—
213C22×C5—32no—
214C22×C5—61no—
215C22×C5—61no—
216C22×C5D3×Dic312yescomplemented
217C22×C5C2×D3211yescharacteristic
218C4×C5—32no—
219Dic5—61no—
220Dic5—61no—
221Dic5—92no—
222Dic5—181no—
223C2×C32—22no—
224C2×C32D4×D511yescharacteristic
225C2×C32#71111yescharacteristic
226C2×C32C22×Dic511yescharacteristic
227C3×D3—22no—
228C2×D4—31no—
229C22×C4—451no—
230C22×C4—451no—
231C4⋊C4—451no—
232C22⋊C4—151no—
233C22⋊C4—451no—
234C42—451no—
235C3×C5—21no—
236C3×C5#67⋊C411yescharacteristic, complemented
237C3×C5D4×Dic311yescharacteristic, complemented
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—32no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3D3×Dic512yescomplemented
250C22×C3C2×Dic1512yescomplemented
251C22×C3C2×D3×D511yescharacteristic
252C22×C3C22×D1511yescharacteristic
253C2×D3—21no—
254C2×D3—21no—
255C2×D3C2×Dic1512yescomplemented
256C3×C4—32no—
257C3×C4—301no—
258C3×C4—301no—
259Dic3—101no—
260Dic3—101no—
261Dic3—152no—
262Dic3—152no—
263Dic3—302no—
264Dic3—302no—
265Dic3—301no—
266Dic3—301no—
267Dic3C2×Dic1512yescomplemented
268C2×C5—22no—
269C2×C5—62no—
270C2×C5C2×D3×Dic311yescharacteristic
271C2×C5#380511yescharacteristic
272C2×C5#381111yescharacteristic
273C32D4×Dic511yescharacteristic, complemented, Sylow
274C23—31no—
275C23D3×D1511yescharacteristic
276D4—34no—
277C2×C4—31no—
278C2×C4—151no—
279C2×C4—151no—
280C2×C4—451no—
281C2×C4—451no—
282C2×C4—902no—
283C2×C4—902no—
284C2×C3—24no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—62no—
291C2×C3#678011yescharacteristic
292C2×C3C2×D3×Dic511yescharacteristic
293C2×C3#679011yescharacteristic
294C2×C3D4×D1511yescharacteristic
295C2×C3#681911yescharacteristic
296C2×C3C22×Dic1511yescharacteristic
297D3—22no—
298C5#67⋊Dic311yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—32no—
302C22—61no—
303C22—61no—
304C22D3×Dic1512yescomplemented
305C22C2×D3×D1511yescharacteristic
306C4—32no—
307C4—301no—
308C4—301no—
309C4—452no—
310C4—901no—
311C3—21no—
312C3#67⋊Dic511yescharacteristic, complemented
313C3D4×Dic1511yescharacteristic, complemented
314C2—22no—
315C2—62no—
316C2#1062420811yescharacteristic
317C2C2×D3×Dic1511yescharacteristic
318C2#1062421811yescharacteristic
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. C3×#6719C2semidirect
5. Dic3⋊Dic15C2semidirect
6. (C2×D3)⋊Dic15C2semidirect
7. Dic3×Dic15C2semidirect
17. C3×C5×#67C4semidirect
29. #6719D3semidirect
38. C22×C32×C5C2×C4semidirect
42. C2×C3×C5×D3C2×C4semidirect
44. C3⋊Dic15D4semidirect
47. C3×C5×Dic3C2×C4semidirect
75. C5×#67Dic3semidirect
99. C3×#67Dic5semidirect
115. C22×C3×C5C4×D3semidirect
116. C22×C3×C5C2×Dic3semidirect
121. C2×C5×D3C2×Dic3semidirect
133. C5×Dic3C2×Dic3semidirect
149. C32×C5C4×D4semidirect
162. C22×C32C2×Dic5semidirect
166. C2×C3×D3C2×Dic5semidirect
171. C3×Dic3C2×Dic5semidirect
196. #67Dic15semidirect
216. C22×C5D3×Dic3semidirect
236. C3×C5#67⋊C4semidirect
237. C3×C5D4×Dic3semidirect
249. C22×C3D3×Dic5semidirect
250. C22×C3C2×Dic15semidirect
255. C2×D3C2×Dic15semidirect
267. Dic3C2×Dic15semidirect
273. C32D4×Dic5semidirect
298. C5#67⋊Dic3semidirect
304. C22D3×Dic15semidirect
312. C3#67⋊Dic5semidirect
313. 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
C3×#67192
Dic3⋊Dic152
(C2×D3)⋊Dic152
Dic3×Dic152
D4×Dic153
#67⋊Dic53
#67⋊Dic35