Finite Group Databasebeta

← #13087403

Subgroups — #13087403

Order 1440.

346 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 78 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×#67151C211yesmaximal, characteristic
2C2×Dic3×D15C211yesmaximal, characteristic, complemented
3#10623908C211yesmaximal, characteristic, complemented
4C3×#6714C211yesmaximal, characteristic, complemented
5C3×C5×C4⋊Dic3C211yesmaximal, characteristic, complemented
6#10623814C211yesmaximal, characteristic, complemented
7#10623813C211yesmaximal, characteristic, complemented
8#91783—31nomaximal
9#91413—31nomaximal
10C22×C3⋊D15C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17#67151—22no—
18Dic3×D15—22no—
19#63890—51nomaximal, Hall
20C2×D60—31no—
21C2×C4×D15—31no—
22C2×#1202—31no—
23C2×D5×Dic3—31no—
24C3×C5×C4⋊C4—31no—
25#6714—31no—
26#6714—31no—
27#6714—31no—
28#6714—61no—
29#6714D311yescharacteristic, complemented
30#6712—31no—
31C5×C4⋊Dic3D311yescharacteristic, complemented
32C3×#685—31no—
33#6668—31no—
34#6667—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C3⋊D15—21no—
37C2×C3⋊D15—21no—
38C2×C3⋊D15—21no—
39C2×C3×D15—21no—
40C2×C3×D15—21no—
41C2×C3×D15D412yes—
42C32×C4×C5—21no—
43C3⋊Dic15—21no—
44C3×Dic15—21no—
45C3×C5×Dic3—21no—
46C3×C5×Dic3—21no—
47#4040—91noHall
48C2×#626—51no—
49C2×D3×Dic3—51no—
50#3752—51no—
51C3×(C2×D3)⋊C4—51no—
52C3×C4⋊Dic3D511yescharacteristic, complemented
53Dic3⋊Dic3—51no—
54#3722—51no—
55C22×D15—31no—
56C22×D15—31no—
57C22×D15—61no—
58C22×D15C2×D311yescharacteristic
59C22×C3×D5—31no—
60C2×C3×C4×C5—21no—
61C2×C3×C4×C5—31no—
62C2×C3×C4×C5—31no—
63C2×C3×C4×C5C2×D311yescharacteristic
64C2×C3×C4×C5C2×D311yescharacteristic
65C2×Dic15—31no—
66C2×Dic15—31no—
67C2×Dic15—61no—
68C2×Dic15C2×D311yescharacteristic
69D60—62no—
70C4×D15—62no—
71C2×C5×Dic3C2×D311yescharacteristic
72C2×C5×Dic3C2×D311yescharacteristic
73C2×C3×Dic5—31no—
74#1202—62no—
75D5×Dic3—62no—
76#894—151no—
77#886—151no—
78C2×C32×C5C2×D411yescharacteristic
79C2×C32×C5#4111yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C3⋊D15—41no—
82C3×D15—22no—
83C2×D20—91no—
84C2×C4×D5—91no—
85C5×C4⋊C4—31no—
86#685—31no—
87#685—91no—
88#685—91no—
89#683—91no—
90C22×C3⋊D3—51no—
91C22×C3×D3—51no—
92C2×C32×C4C2×D511yescharacteristic
93C2×C3⋊Dic3—51no—
94C2×C3×Dic3—51no—
95C2×C3×Dic3C2×D511yescharacteristic
96C2×C3×Dic3C2×D511yescharacteristic
97#626—102no—
98D3×Dic3—102no—
99C22×C3×C5—21no—
100C22×C3×C5C22×D311yescharacteristic
101C22×C3×C5C22×D311yescharacteristic
102C2×D15—21no—
103C2×D15—21no—
104C2×D15—62no—
105C2×D15—61no—
106C2×D15—61no—
107C2×D15—61no—
108C2×D15—61no—
109C2×D15—61no—
110C2×D15—61no—
111C2×D15—121no—
112C2×D15—121no—
113C2×D15D1212yes—
114C2×C3×D5—32no—
115C2×C3×D5—61no—
116C2×C3×D5—61no—
117C3×C4×C5—21no—
118C3×C4×C5—21no—
119C3×C4×C5—41no—
120C3×C4×C5—61no—
121C3×C4×C5—61no—
122Dic15—21no—
123Dic15—61no—
124Dic15—61no—
125Dic15—121no—
126C3×Dic5—61no—
127C5×Dic3—21no—
128C5×Dic3—21no—
129C2×#67—151no—
130C22×Dic3—151no—
131C2×D12—151no—
132C2×C4×D3—151no—
133C3×C4⋊C4—31no—
134C3×C22⋊C4—151no—
135(C2×D3)⋊C4—51no—
136(C2×D3)⋊C4—151no—
137(C2×D3)⋊C4—151no—
138(C2×D3)⋊C4—151no—
139(C2×D3)⋊C4—151no—
140(C2×D3)⋊C4—301no—
141C4⋊Dic3—151no—
142C4⋊Dic3D1511yescharacteristic, complemented
143Dic3⋊C4—151no—
144C32×C5#12311yescharacteristic, complemented, Hall
145C22×D5—31no—
146C22×D5—91no—
147C2×C4×C5—31no—
148C2×C4×C5—31no—
149C2×C4×C5D3211yescharacteristic
150C2×Dic5—31no—
151C2×Dic5—91no—
152D20—182no—
153C4×D5—182no—
154C22×C32C22×D511yescharacteristic
155C2×C3⋊D3—101no—
156C2×C3⋊D3—101no—
157C2×C3⋊D3—101no—
158C2×C3×D3—52no—
159C2×C3×D3—101no—
160C2×C3×D3—101no—
161C32×C4—21no—
162C3⋊Dic3—101no—
163C3×Dic3—21no—
164C3×Dic3—21no—
165C3×Dic3—101no—
166#123—451noSylow
167C2×C3×C5—21no—
168C2×C3×C5—21no—
169C2×C3×C5—21no—
170C2×C3×C5C2×D1211yescharacteristic
171C2×C3×C5#23511yescharacteristic
172C2×C3×C5D3×D411yescharacteristic
173C2×C3×C5#23711yescharacteristic
174C2×C3×C5#23711yescharacteristic
175C2×C3×C5#23911yescharacteristic
176D15—22no—
177D15—122no—
178D15—121no—
179D15—121no—
180C3×D5—62no—
181C23×C3—151no—
182C22×D3—51no—
183C22×D3—151no—
184C22×D3—151no—
185C22×D3—301no—
186C2×C3×C4—21no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—151no—
190C2×C3×C4D3×D511yescharacteristic
191C2×C3×C4C2×D1511yescharacteristic
192#67—302no—
193C2×Dic3—51no—
194C2×Dic3—151no—
195C2×Dic3—151no—
196C2×Dic3—301no—
197C2×Dic3—302no—
198C2×Dic3C2×D1511yescharacteristic
199C2×Dic3C2×D1511yescharacteristic
200D12—302no—
201C4×D3—302no—
202C22×C5C2×D3211yescharacteristic
203C2×D5—32no—
204C2×D5—61no—
205C2×D5—61no—
206C2×D5—181no—
207C2×D5—181no—
208C2×D5—181no—
209C4×C5—21no—
210C4×C5—61no—
211C4×C5—61no—
212Dic5—61no—
213Dic5—181no—
214C2×C32#70911yescharacteristic
215C2×C32D4×D511yescharacteristic
216C2×C32#71311yescharacteristic
217C3⋊D3—201no—
218C3×D3—102no—
219C2×D4—451no—
220C22×C4—451no—
221C4⋊C4—31no—
222C4⋊C4—451no—
223C22⋊C4—151no—
224C22⋊C4—451no—
225C22⋊C4—451no—
226C3×C5—21no—
227C3×C5#88611yescharacteristic, complemented
228C3×C5#89411yescharacteristic, complemented
229C22×C3—21no—
230C22×C3—152no—
231C22×C3—301no—
232C22×C3—301no—
233C22×C3C2×D3×D511yescharacteristic
234C22×C3C22×D1511yescharacteristic
235C2×D3—52no—
236C2×D3—101no—
237C2×D3—101no—
238C2×D3—302no—
239C2×D3—301no—
240C2×D3—301no—
241C2×D3—301no—
242C2×D3—301no—
243C2×D3—301no—
244C2×D3—301no—
245C2×D3—601no—
246C2×D3—601no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—41no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4—301no—
253Dic3—21no—
254Dic3—21no—
255Dic3—101no—
256Dic3—301no—
257Dic3—301no—
258Dic3—601no—
259C2×C5#379611yescharacteristic
260C2×C5D3×D1211yescharacteristic
261C2×C5#380411yescharacteristic
262D5—62no—
263D5—361no—
264C32#404011yescharacteristic, complemented, Sylow
265C23—151no—
266C23—451no—
267D4—902no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—151no—
271C2×C4—451no—
272C2×C4—902no—
273C2×C4D3×D1511yescharacteristic
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—302no—
278C2×C3#676611yescharacteristic
279C2×C3D5×D1211yescharacteristic
280C2×C3#677911yescharacteristic
281C2×C3#681711yescharacteristic
282C2×C3D4×D1511yescharacteristic
283C2×C3#682111yescharacteristic
284D3—102no—
285D3—602no—
286D3—601no—
287D3—601no—
288C5#6389011yescharacteristic, complemented, Sylow
289C22—152no—
290C22—301no—
291C22—301no—
292C22—901no—
293C22—901no—
294C22—901no—
295C22C2×D3×D1511yescharacteristic
296C4—21no—
297C4—61no—
298C4—61no—
299C4—301no—
300C4—901no—
301C3—21no—
302C3#9141311yescharacteristic, complemented
303C3#9178311yescharacteristic, complemented
304C2—302no—
305C2—1801no—
306C2#1062419411yescharacteristic
307C2D12×D1511yescharacteristic
308C2#1062420711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
prime radicalC2×#67151C2
hypercenterC2×C4D3×D15
hypocenterC32×C5#123
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×D15C2semidirect
3. #10623908C2semidirect
4. C3×#6714C2semidirect
5. C3×C5×C4⋊Dic3C2semidirect
6. #10623814C2semidirect
7. #10623813C2semidirect
29. #6714D3semidirect
31. C5×C4⋊Dic3D3semidirect
52. C3×C4⋊Dic3D5semidirect
142. C4⋊Dic3D15semidirect
144. C32×C5#123semidirect
227. C3×C5#886semidirect
228. C3×C5#894semidirect
264. C32#4040semidirect
288. C5#63890semidirect
302. C3#91413semidirect
303. C3#91783semidirect

Maximal subgroups

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

subgroupindex
C2×#671512
C2×Dic3×D152
#106239082
C3×#67142
C3×C5×C4⋊Dic32
#106238142
#106238132
#917833
#914133
#638905