Finite Group Databasebeta

← #13085902

Subgroups — #13085902

Order 1440.

334 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 88 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×D3×Dic15C211yesmaximal, characteristic
2#10623831C211yesmaximal, characteristic, complemented
3Dic3×Dic15C211yesmaximal, characteristic, complemented
4C5×(C2×D3)⋊Dic3C211yesmaximal, characteristic
5Dic5×C3⋊Dic3C211yesmaximal, characteristic, complemented
6C3×#6669C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic5C211yesmaximal, characteristic
8#91532—31nomaximal
9#91400—31nomaximal
10C22×C3×C5×D3C2211yescharacteristic
11C2×C3⋊Dic15C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17D3×Dic15C414yescomplemented
18#63878—51nomaximal, Hall
19C22×Dic15—31no—
20C2×D3×Dic5—31no—
21C4×Dic15—31no—
22C5×#217—31no—
23C5×(C2×D3)⋊C4—31no—
24C3×C22⋊Dic5—31no—
25C3×#683—31no—
26#6669—31no—
27#6669D311yescharacteristic, complemented
28#6668—31no—
29(C2×D3)⋊Dic5D311yescharacteristic
30Dic3×Dic5—31no—
31Dic3×Dic5—31no—
32Dic3×Dic5—31no—
33Dic3×Dic5—61no—
34C22×C32×C5C2311yescharacteristic
35C2×C3×C5×D3—21no—
36C2×C3×C5×D3—21no—
37C2×C3×C5×D3C2×C412yes—
38C3⋊Dic15C2×C412yes—
39C5×C3⋊Dic3—21no—
40C3×Dic15—21no—
41C3×Dic15C2×C412yes—
42C3×C5×Dic3—21no—
43C32×Dic5—21no—
44#4023—91noHall
45C2×D3×Dic3—51no—
46C4×C3⋊Dic3—51no—
47C3×(C2×D3)⋊C4—51no—
48C3×Dic3⋊C4—51no—
49Dic3⋊Dic3—51no—
50(C2×D3)⋊Dic3D511yescharacteristic
51Dic32—51no—
52C23×C3×C5—31no—
53C22×C5×D3C2×D311yescharacteristic
54C2×C3×C4×C5—31no—
55C2×Dic15—31no—
56C2×Dic15—31no—
57C2×Dic15—34no—
58C2×Dic15—61no—
59C2×Dic15C2×D311yescharacteristic
60C2×Dic15C2×D311yescharacteristic
61C2×C5×Dic3—31no—
62C2×C5×Dic3—31no—
63C2×C5×Dic3—61no—
64C2×C5×Dic3C2×D311yescharacteristic
65C2×C3×Dic5—21no—
66C2×C3×Dic5—31no—
67C2×C3×Dic5—31no—
68C2×C3×Dic5C2×D311yescharacteristic
69C2×C3×Dic5C2×D311yescharacteristic
70D3×Dic5—34no—
71#892—151no—
72#877—151no—
73C2×C32×C5C22×C411yescharacteristic
74C2×C32×C5#4111yescharacteristic
75C2×C32×C5#4111yescharacteristic
76C3×C5×D3—22no—
77C22×Dic5—91no—
78C5×C22⋊C4—91no—
79C22⋊Dic5—31no—
80#683—31no—
81#683—91no—
82C4×Dic5—91no—
83C4×Dic5—91no—
84C22×C3×D3C2×D511yescharacteristic
85C2×C32×C4—51no—
86C2×C3⋊Dic3—51no—
87C2×C3⋊Dic3C2×D511yescharacteristic
88C2×C3×Dic3—51no—
89C2×C3×Dic3—51no—
90C2×C3×Dic3C2×D511yescharacteristic
91D3×Dic3—54no—
92C22×C3×C5—21no—
93C22×C3×C5—32no—
94C22×C3×C5—61no—
95C22×C3×C5—61no—
96C22×C3×C5C22×D311yescharacteristic
97C22×C3×C5C22×D311yescharacteristic
98C2×C5×D3—21no—
99C2×C5×D3—21no—
100C2×C5×D3C4×D312yes—
101C3×C4×C5—61no—
102Dic15—21no—
103Dic15—32no—
104Dic15—32no—
105Dic15—62no—
106Dic15C4×D312yes—
107C3×Dic5—22no—
108C3×Dic5—21no—
109C3×Dic5—21no—
110C3×Dic5—32no—
111C3×Dic5—61no—
112C5×Dic3—21no—
113C5×Dic3—62no—
114C5×Dic3—61no—
115C5×Dic3—61no—
116C22×Dic3—151no—
117C2×C4×D3—151no—
118C3×C4⋊C4—151no—
119C3×C22⋊C4—151no—
120#217—31no—
121(C2×D3)⋊C4—31no—
122(C2×D3)⋊C4—51no—
123C4⋊Dic3—151no—
124Dic3⋊C4—51no—
125Dic3⋊C4—151no—
126C4×Dic3—151no—
127C4×Dic3—151no—
128C4×Dic3—151no—
129C4×Dic3—151no—
130C4×Dic3—301no—
131C32×C5#11711yescharacteristic, complemented, Hall
132C23×C5—31no—
133C2×C4×C5—31no—
134C2×C4×C5—91no—
135C2×Dic5—31no—
136C2×Dic5—31no—
137C2×Dic5—94no—
138C2×Dic5—91no—
139C2×Dic5D3211yescharacteristic
140C22×C32C22×D511yescharacteristic
141C2×C3×D3—21no—
142C2×C3×D3—21no—
143C2×C3×D3C4×D512yes—
144C32×C4—101no—
145C3⋊Dic3—21no—
146C3⋊Dic3—52no—
147C3×Dic3—21no—
148C3×Dic3—52no—
149C3×Dic3—101no—
150#117—451noSylow
151C2×C3×C5—21no—
152C2×C3×C5—21no—
153C2×C3×C5—21no—
154C2×C3×C5—62no—
155C2×C3×C5C2×C4×D311yescharacteristic
156C2×C3×C5C2×C4×D311yescharacteristic
157C2×C3×C5#23711yescharacteristic
158C2×C3×C5#23711yescharacteristic
159C2×C3×C5#23711yescharacteristic
160C2×C3×C5#23911yescharacteristic
161C5×D3—22no—
162C23×C3—31no—
163C22×D3D3×D511yescharacteristic
164C2×C3×C4—31no—
165C2×C3×C4—51no—
166C2×C3×C4—51no—
167C2×C3×C4—101no—
168C2×C3×C4—151no—
169C2×C3×C4—151no—
170C2×Dic3—31no—
171C2×Dic3—31no—
172C2×Dic3—51no—
173C2×Dic3—51no—
174C2×Dic3—61no—
175C2×Dic3—154no—
176C2×Dic3—151no—
177C2×Dic3—151no—
178C2×Dic3—301no—
179C2×Dic3D3×D511yescharacteristic
180C4×D3—154no—
181C22×C5—32no—
182C22×C5—61no—
183C22×C5—61no—
184C22×C5C2×D3211yescharacteristic
185C4×C5—61no—
186C4×C5—181no—
187Dic5—21no—
188Dic5—32no—
189Dic5—61no—
190Dic5—92no—
191C2×C32C2×C4×D511yescharacteristic
192C2×C32#71111yescharacteristic
193C2×C32#71111yescharacteristic
194C3×D3—22no—
195C22×C4—451no—
196C4⋊C4—151no—
197C4⋊C4—451no—
198C22⋊C4—91no—
199C22⋊C4—151no—
200C42—451no—
201C42—451no—
202C3×C5—21no—
203C3×C5#87711yescharacteristic, complemented
204C3×C5#89211yescharacteristic, complemented
205C22×C3—21no—
206C22×C3—32no—
207C22×C3—61no—
208C22×C3—61no—
209C22×C3C2×D3×D511yescharacteristic
210C22×C3C2×D3×D511yescharacteristic
211C2×D3—21no—
212C2×D3—21no—
213C2×D3#120012yes—
214C3×C4—61no—
215C3×C4—102no—
216C3×C4—101no—
217C3×C4—101no—
218C3×C4—152no—
219C3×C4—301no—
220Dic3—21no—
221Dic3—52no—
222Dic3—62no—
223Dic3—61no—
224Dic3—61no—
225Dic3—101no—
226Dic3—152no—
227Dic3—152no—
228Dic3—302no—
229C2×C5—62no—
230C2×C5#379611yescharacteristic
231C2×C5C4×D3211yescharacteristic
232C2×C5#380511yescharacteristic
233C32#402311yescharacteristic, complemented, Sylow
234C23—31no—
235C2×C4—31no—
236C2×C4—51no—
237C2×C4—91no—
238C2×C4—151no—
239C2×C4—151no—
240C2×C4—454no—
241C2×C4—451no—
242C2×C3—21no—
243C2×C3—21no—
244C2×C3—21no—
245C2×C3—62no—
246C2×C3#676811yescharacteristic
247C2×C3C4×D3×D511yescharacteristic
248C2×C3#678011yescharacteristic
249C2×C3#678011yescharacteristic
250C2×C3#678011yescharacteristic
251C2×C3C2×#120011yescharacteristic
252D3—22no—
253C5#6387811yescharacteristic, complemented, Sylow
254C22—32no—
255C22—61no—
256C22—61no—
257C22D32×D511yescharacteristic
258C4—61no—
259C4—101no—
260C4—152no—
261C4—181no—
262C4—301no—
263C4—452no—
264C3—21no—
265C3#9140011yescharacteristic, complemented
266C3#9153211yescharacteristic, complemented
267C2—62no—
268C2D3×#120011yescharacteristic
269C2#1062404611yescharacteristic
270C2#1062405611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutatorC2×C32×C5C22×C4
Fitting, socle, cosocleC22×C32×C5C23
prime radicalC22×C3×C5×D3C22
hypocenterC32×C5#117
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #10623831C2semidirect
3. Dic3×Dic15C2semidirect
5. Dic5×C3⋊Dic3C2semidirect
6. C3×#6669C2semidirect
17. D3×Dic15C4semidirect
27. #6669D3semidirect
131. C32×C5#117semidirect
203. C3×C5#877semidirect
204. C3×C5#892semidirect
233. C32#4023semidirect
253. C5#63878semidirect
265. C3#91400semidirect
266. C3#91532semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic152
#106238312
Dic3×Dic152
C5×(C2×D3)⋊Dic32
Dic5×C3⋊Dic32
C3×#66692
C3×(C2×D3)⋊Dic52
#915323
#914003
#638785