Finite Group Databasebeta

← #13087554

Subgroups — #13087554

Order 1440.

366 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 90 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×#67184C211yesmaximal, characteristic
2C22×C3×Dic15C211yesmaximal, characteristic, complemented
3C3×C5×#217C211yesmaximal, characteristic, complemented
4Dic3⋊Dic15C212yesmaximal, complemented
5#10623813C212yesmaximal, complemented
6#91775—31nomaximal
7#91546—31nomaximal
8C23×C32×C5C2211yescharacteristic
9C22×C3⋊D15C2211yescharacteristic
10#67184—22no—
11C2×C3⋊Dic15C2211yescharacteristic
12C2×C3×Dic15—22no—
13C2×C3×Dic15C2212yes—
14C2×C3×C5×Dic3C2212yes—
15#63966—51nomaximal, Hall
16C2×#1220—31no—
17C2×#1220—31no—
18C2×#1220—61no—
19C22×Dic15D311yescharacteristic, complemented
20C22×C3×Dic5—31no—
21C3×C5×C22⋊C4—31no—
22#6714—32no—
23C4⋊Dic15—32no—
24C5×#217D311yescharacteristic, complemented
25Dic3⋊Dic5—32no—
26#6667—32no—
27C22×C32×C5—22no—
28C22×C32×C5D412yes—
29C22×C32×C5C2311yescharacteristic
30C2×C3⋊D15—21no—
31C2×C3⋊D15—22no—
32C3⋊Dic15—21no—
33C3×Dic15—22no—
34C3×C5×Dic3—22no—
35#4032—91noHall
36C2×#638—51no—
37C22×C3×Dic3—51no—
38C3×#217D511yescharacteristic, complemented
39Dic3⋊Dic3—52no—
40#3722—52no—
41C23×C3×C5—21no—
42C23×C3×C5C2×D311yescharacteristic
43C23×C3×C5C2×D311yescharacteristic
44C22×D15—31no—
45C22×D15—31no—
46C22×D15—61no—
47C2×C3×C4×C5—32no—
48#1220—64no—
49#1220—62no—
50#1220—62no—
51C2×Dic15—22no—
52C2×Dic15—31no—
53C2×Dic15—31no—
54C2×Dic15—61no—
55C2×Dic15C2×D312yes—
56C2×C5×Dic3C2×D312yes—
57C2×C3×Dic5—32no—
58C2×C3×Dic5—62no—
59#929—151no—
60#886—151no—
61C2×C32×C5—22no—
62C2×C32×C5C2×D411yescharacteristic
63C2×C32×C5#4112yes—
64C3⋊D15—41no—
65C2×C22⋊D5—91no—
66C22×Dic5—31no—
67C5×C22⋊C4—31no—
68#685—92no—
69C4⋊Dic5—92no—
70C23×C32C2×D511yescharacteristic
71C22×C3⋊D3—51no—
72#638—102no—
73C2×C3⋊Dic3—51no—
74C2×C3×Dic3—52no—
75C2×C3×Dic3—102no—
76C2×C3×Dic3C2×D512yes—
77C22×C3×C5—22no—
78C22×C3×C5—22no—
79C22×C3×C5—22no—
80C22×C3×C5—21no—
81C22×C3×C5—42no—
82C22×C3×C5D1212yes—
83C22×C3×C5#6712yes—
84C22×C3×C5C22×D311yescharacteristic
85C22×C3×C5C22×D311yescharacteristic
86C2×D15—62no—
87C2×D15—61no—
88C2×D15—62no—
89C2×D15—62no—
90C2×D15—61no—
91C2×D15—122no—
92C3×C4×C5—62no—
93Dic15—22no—
94Dic15—62no—
95Dic15—61no—
96Dic15—61no—
97C3×Dic5—62no—
98C5×Dic3—22no—
99C22×C3×C4—151no—
100C2×#67—151no—
101C2×#67—151no—
102C2×#67—301no—
103C22×Dic3—51no—
104C3×C22⋊C4—31no—
105#217D1511yescharacteristic, complemented
106(C2×D3)⋊C4—152no—
107(C2×D3)⋊C4—152no—
108C4⋊Dic3—152no—
109Dic3⋊C4—152no—
110C32×C5#12311yescharacteristic, complemented, Hall
111C23×C5D3211yescharacteristic
112C22×D5—91no—
113C2×C4×C5—32no—
114C22⋊D5—182no—
115C2×Dic5—32no—
116C2×Dic5—62no—
117C2×Dic5—91no—
118C22×C32—22no—
119C22×C32D2012yes—
120C22×C32C22×D511yescharacteristic
121C2×C3⋊D3—101no—
122C2×C3⋊D3—102no—
123C3⋊Dic3—101no—
124C3×Dic3—22no—
125C3×Dic3—102no—
126#123—451noSylow
127C2×C3×C5—22no—
128C2×C3×C5—22no—
129C2×C3×C5—22no—
130C2×C3×C5—21no—
131C2×C3×C5—42no—
132C2×C3×C5C2×D1211yescharacteristic
133C2×C3×C5#23512yes—
134C2×C3×C5#23712yes—
135C2×C3×C5C2×#6711yescharacteristic
136D15—122no—
137D15—121no—
138D15—121no—
139C23×C3—21no—
140C23×C3D3×D511yescharacteristic
141C23×C3C2×D1511yescharacteristic
142C22×D3—151no—
143C22×D3—151no—
144C22×D3—301no—
145C2×C3×C4—32no—
146C2×C3×C4—152no—
147C2×C3×C4—302no—
148#67—304no—
149#67—302no—
150#67—302no—
151C2×Dic3—52no—
152C2×Dic3—102no—
153C2×Dic3—151no—
154C2×Dic3—151no—
155C2×Dic3—301no—
156C2×Dic3C2×D1512yes—
157C22×C5—22no—
158C22×C5#62612yes—
159C22×C5C2×D3211yescharacteristic
160C2×D5—182no—
161C2×D5—181no—
162C4×C5—62no—
163Dic5—62no—
164Dic5—181no—
165C2×C32—22no—
166C2×C32C2×D2011yescharacteristic
167C2×C32#71112yes—
168C3⋊D3—201no—
169C2×D4—451no—
170C22×C4—151no—
171C4⋊C4—452no—
172C22⋊C4—31no—
173C22⋊C4—452no—
174C3×C5—21no—
175C3×C5#88611yescharacteristic, complemented
176C3×C5#92911yescharacteristic, complemented
177C22×C3—22no—
178C22×C3—22no—
179C22×C3—22no—
180C22×C3—21no—
181C22×C3—42no—
182C22×C3#120212yes—
183C22×C3D6012yes—
184C22×C3C2×D3×D511yescharacteristic
185C22×C3C22×D1511yescharacteristic
186C2×D3—302no—
187C2×D3—302no—
188C2×D3—301no—
189C2×D3—301no—
190C2×D3—302no—
191C2×D3—602no—
192C3×C4—62no—
193C3×C4—302no—
194Dic3—22no—
195Dic3—102no—
196Dic3—302no—
197Dic3—301no—
198Dic3—301no—
199C2×C5—22no—
200C2×C5#380412yes—
201C2×C5C2×#62611yescharacteristic
202D5—361no—
203C32#403211yescharacteristic, complemented, Sylow
204C23—451no—
205C23D3×D1511yescharacteristic
206D4—902no—
207C2×C4—32no—
208C2×C4—152no—
209C2×C4—302no—
210C2×C4—451no—
211C2×C3—22no—
212C2×C3—22no—
213C2×C3—22no—
214C2×C3—21no—
215C2×C3—42no—
216C2×C3#678212yes—
217C2×C3C2×#120211yescharacteristic
218C2×C3C2×D6011yescharacteristic
219C2×C3#681912yes—
220D3—602no—
221D3—601no—
222D3—601no—
223C5#6396611yescharacteristic, complemented, Sylow
224C22—22no—
225C22—902no—
226C22—901no—
227C22#6715112yes—
228C22C2×D3×D1511yescharacteristic
229C4—62no—
230C4—302no—
231C4—901no—
232C3—21no—
233C3#9154611yescharacteristic, complemented
234C3#9177511yescharacteristic, complemented
235C2—22no—
236C2—1801no—
237C2#1062421012yes—
238C2C2×#6715111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC23×C32×C5C22
prime radicalC2×#67184C2
hypercenterC23D3×D15
hypocenterC32×C5#123
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×Dic15C2semidirect
3. C3×C5×#217C2semidirect
4. Dic3⋊Dic15C2semidirect
5. #10623813C2semidirect
19. C22×Dic15D3semidirect
24. C5×#217D3semidirect
38. C3×#217D5semidirect
105. #217D15semidirect
110. C32×C5#123semidirect
175. C3×C5#886semidirect
176. C3×C5#929semidirect
203. C32#4032semidirect
223. C5#63966semidirect
233. C3#91546semidirect
234. C3#91775semidirect

Maximal subgroups

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

subgroupindex
C2×#671842
C22×C3×Dic152
C3×C5×#2172
Dic3⋊Dic152
#106238132
#917753
#915463
#639665