Finite Group Databasebeta

← #13085468

Subgroups — #13085468

Order 1440.

242 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 48 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×C3×#1202C211yesmaximal, characteristic
2#10624098C211yesmaximal, characteristic, complemented
3C3×#6763C211yesmaximal, characteristic, complemented
4#91228—31nomaximal
5#91173—31nomaximal
6C22×C3×D15C411yescharacteristic, complemented
7C22×C32×D5C2211yescharacteristic
8C2×C3⋊ΓL(1,𝔽16)—21no—
9C2×C3×ΓL(1,𝔽16)—21no—
10C2×C3×C5×Dic3C411yescharacteristic, complemented
11C3×#1202—21no—
12C3×#1202—21no—
13#63580—51nomaximal, Hall
14C2×C3×D20—31no—
15C2×#1202D311yescharacteristic
16#6763—31no—
17#6763—31no—
18#6763—61no—
19#6763D311yescharacteristic, complemented
20C3×C22⋊AGL(1,𝔽5)—31no—
21C22×C32×C5C2×C411yescharacteristic
22C2×C3×D15—21no—
23C2×C3×D15—41no—
24C2×C32×D5—41no—
25C2×C32×D5D411yescharacteristic
26C2×C32×D5D411yescharacteristic
27C3⋊ΓL(1,𝔽16)—41no—
28C3×ΓL(1,𝔽16)—41no—
29C3×C5×Dic3—21no—
30#3999—91noHall
31C2×C3×#67—51no—
32#3757—51no—
33C3×#217—51no—
34C22×D15Dic311yescharacteristic, complemented
35C22×C3×D5—21no—
36C22×C3×D5—31no—
37C22×C3×D5C2×D311yescharacteristic
38C22×C3×D5C2×D311yescharacteristic
39C2×ΓL(1,𝔽16)—21no—
40C2×ΓL(1,𝔽16)—62no—
41C2×ΓL(1,𝔽16)—61no—
42C2×ΓL(1,𝔽16)—61no—
43C2×C3×AGL(1,𝔽5)—61no—
44C2×C3×C4×C5—31no—
45C2×C5×Dic3Dic311yescharacteristic, complemented
46C3×D20—61no—
47C3×D20—61no—
48#1202—21no—
49#1202—21no—
50#834—151no—
51#806—151no—
52C2×C32×C5—21no—
53C2×C32×C5C22⋊C411yescharacteristic
54C3×D15—41no—
55C32×D5—21no—
56C32×D5—21no—
57C2×D20—31no—
58C22⋊AGL(1,𝔽5)—31no—
59C22⋊AGL(1,𝔽5)—91no—
60C23×C32—51no—
61C22×C3×D3—51no—
62C2×C3⋊Dic3—101no—
63C3×#67—101no—
64C3×#67—101no—
65C2×C3×Dic3—101no—
66C2×C3×Dic3AGL(1,𝔽5)11yescharacteristic, complemented
67C22×C3×C5—21no—
68C22×C3×C5C4×D311yescharacteristic
69C22×C3×C5C2×Dic311yescharacteristic
70C2×D15—21no—
71C2×D15—41no—
72C2×C3×D5—21no—
73C2×C3×D5—21no—
74C2×C3×D5—41no—
75C2×C3×D5—41no—
76C2×C3×D5—42no—
77C2×C3×D5—61no—
78C2×C3×D5—121no—
79C2×C3×D5D1211yescharacteristic
80C2×C3×D5#6711yescharacteristic
81C2×C3×D5#6711yescharacteristic
82C2×C3×D5#6711yescharacteristic
83ΓL(1,𝔽16)—41no—
84ΓL(1,𝔽16)—122no—
85ΓL(1,𝔽16)—121no—
86ΓL(1,𝔽16)—121no—
87C3×AGL(1,𝔽5)—121no—
88C3×C4×C5—61no—
89C5×Dic3—21no—
90C2×C3×D4—151no—
91C2×#67—51no—
92C3×C22⋊C4—151no—
93#217—51no—
94#217—151no—
95#217—151no—
96#217—301no—
97C32×C5#9911yescharacteristic, complemented, Hall
98C22×D5—31no—
99C22×D5D3211yescharacteristic
100C2×AGL(1,𝔽5)—61no—
101C2×AGL(1,𝔽5)—181no—
102C2×C4×C5—31no—
103D20—61no—
104D20—61no—
105C22×C32—51no—
106C22×C32—51no—
107C22×C32—201no—
108C22×C32C2×AGL(1,𝔽5)11yescharacteristic
109C2×C3×D3—101no—
110C2×C3×D3—201no—
111C3⋊Dic3—201no—
112C3×Dic3—21no—
113C3×Dic3—201no—
114#99—451noSylow
115C2×C3×C5—21no—
116C2×C3×C5—21no—
117C2×C3×C5—21no—
118C2×C3×C5—41no—
119C2×C3×C5(C2×D3)⋊C411yescharacteristic
120C2×C3×C5#21711yescharacteristic
121D15—41no—
122C3×D5—22no—
123C3×D5—21no—
124C3×D5—21no—
125C3×D5—21no—
126C3×D5—21no—
127C3×D5—41no—
128C3×D5—121no—
129C23×C3—51no—
130C23×C3—51no—
131C23×C3—101no—
132C23×C3—151no—
133C22×D3—51no—
134C3×D4—301no—
135C3×D4—301no—
136C2×C3×C4—31no—
137C2×C3×C4—301no—
138#67—101no—
139#67—101no—
140C2×Dic3—101no—
141C2×Dic3—301no—
142C2×Dic3—301no—
143C2×Dic3—302no—
144C2×Dic3ΓL(1,𝔽16)11yescharacteristic, complemented
145C22×C5D3×Dic311yescharacteristic
146C2×D5—41no—
147C2×D5—61no—
148C2×D5—121no—
149C2×D5#62511yescharacteristic
150C2×D5#62611yescharacteristic
151AGL(1,𝔽5)—121no—
152AGL(1,𝔽5)—361no—
153C4×C5—61no—
154C2×C32—21no—
155C2×C32—101no—
156C2×C32—101no—
157C2×C32C22⋊AGL(1,𝔽5)11yescharacteristic
158C3×D3—201no—
159C2×D4—151no—
160C22⋊C4—151no—
161C22⋊C4—451no—
162C3×C5—21no—
163C3×C5#80611yescharacteristic, complemented
164C3×C5#83411yescharacteristic, complemented
165C22×C3—21no—
166C22×C3—51no—
167C22×C3—51no—
168C22×C3—51no—
169C22×C3—51no—
170C22×C3—101no—
171C22×C3—101no—
172C22×C3—201no—
173C22×C3—201no—
174C22×C3—202no—
175C22×C3—301no—
176C22×C3—601no—
177C22×C3D3×AGL(1,𝔽5)11yescharacteristic
178C22×C3C2×ΓL(1,𝔽16)11yescharacteristic
179C2×D3—101no—
180C2×D3—201no—
181C3×C4—61no—
182C3×C4—601no—
183Dic3—21no—
184Dic3—201no—
185Dic3—602no—
186Dic3—601no—
187Dic3—601no—
188C2×C5—21no—
189C2×C5(C2×D3)⋊Dic311yescharacteristic
190D5—21no—
191D5—21no—
192D5—121no—
193C32#399911yescharacteristic, complemented, Sylow
194C23—51no—
195C23—151no—
196D4—301no—
197D4—301no—
198C2×C4—31no—
199C2×C4—301no—
200C2×C4—901no—
201C2×C3—21no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—41no—
205C2×C3—102no—
206C2×C3—101no—
207C2×C3—101no—
208C2×C3—101no—
209C2×C3—101no—
210C2×C3—201no—
211C2×C3—601no—
212C2×C3(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
213C2×C3#676311yescharacteristic
214D3—201no—
215C5#6358011yescharacteristic, complemented, Sylow
216C22—51no—
217C22—51no—
218C22—201no—
219C22—301no—
220C22—601no—
221C22D3×ΓL(1,𝔽16)11yescharacteristic
222C4—61no—
223C4—601no—
224C4—1801no—
225C3—21no—
226C3#9117311yescharacteristic, complemented
227C3#9122811yescharacteristic, complemented
228C2—21no—
229C2—101no—
230C2—101no—
231C2—601no—
232C2(C2×D3)⋊ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
centerC2(C2×D3)⋊ΓL(1,𝔽16)
commutator, FittingC22×C32×C5C2×C4
Frattini, hypercenterC22D3×ΓL(1,𝔽16)
socleC2×C32×C5C22⋊C4
prime radicalC2×C3×#1202C2
cosocleC22×C32×D5C22
hypocenterC32×C5#99
LCSG ▷ C22×C32×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #10624098C2semidirect
3. C3×#6763C2semidirect
6. C22×C3×D15C4semidirect
10. C2×C3×C5×Dic3C4semidirect
19. #6763D3semidirect
34. C22×D15Dic3semidirect
45. C2×C5×Dic3Dic3semidirect
66. C2×C3×Dic3AGL(1,𝔽5)semidirect
97. C32×C5#99semidirect
144. C2×Dic3ΓL(1,𝔽16)semidirect
163. C3×C5#806semidirect
164. C3×C5#834semidirect
193. C32#3999semidirect
215. C5#63580semidirect
226. C3#91173semidirect
227. C3#91228semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#12022
#106240982
C3×#67632
#912283
#911733
#635805