Finite Group Databasebeta

← C5×D3×#214 #13087200

Subgroups — C5×D3×#214 #13087200

Order 1440.

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

normal subgroups only

This group has 292 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
1C3×C5×D3×D4C211yesmaximal, characteristic
2C5×D3×Dic6C211yesmaximal, characteristic, complemented
3C5×#3754C211yesmaximal, characteristic, complemented
4C3×C5×#214C211yesmaximal, characteristic, complemented
5C5×#3716C211yesmaximal, characteristic, complemented
6C5×#3715C211yesmaximal, characteristic, complemented
7C5×D3×C3⋊C8C211yesmaximal, characteristic, complemented
8C2×C5×#214—31nomaximal
9C5×D3×SD16—31nomaximal
10C22×C3×C5×D3—21no—
11C32×C5×D4C2211yescharacteristic
12C5×#634C2211yescharacteristic, complemented
13C3×C5×#67—21no—
14C3×C5×D12C2211yescharacteristic
15C3×C4×C5×D3C2211yescharacteristic
16C3×C5×Dic6C2211yescharacteristic, complemented
17C5×#627—21no—
18C5×D3×Dic3—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21D3×#214C511yesmaximal, characteristic, direct product factor, Hall
22C2×C3×C5×D4—31no—
23C5×D3×Q8—31no—
24C5×D3×D4D311yescharacteristic
25C2×C5×Dic6—31no—
26C3×C5×SD16—31no—
27C5×#215—31no—
28C5×#214—31no—
29C5×#214—31no—
30C5×#214—31no—
31C5×#214—31no—
32C5×#214—61no—
33C5×#214D311yescharacteristic, direct product factor
34C2×C5×C3⋊C8—31no—
35C5×#204—31no—
36C5×D3×C8—31no—
37C22×C32×C5—21no—
38C2×C3×C5×D3—21no—
39C2×C3×C5×D3—42no—
40C2×C3×C5×D3D411yescharacteristic
41C32×C4×C5C2311yescharacteristic
42C5×C3⋊Dic3—21no—
43C3×C5×Dic3—21no—
44C3×C5×Dic3D411yescharacteristic
45C2×C5×SD16—91noHall
46C3×D3×D4C2×C511yescharacteristic
47D3×Dic6C2×C511yescharacteristic, complemented
48#3754C2×C511yescharacteristic, complemented
49C3×#214C2×C511yescharacteristic, complemented
50#3716C2×C511yescharacteristic, complemented
51#3715C2×C511yescharacteristic, complemented
52D3×C3⋊C8C2×C511yescharacteristic, complemented
53C23×C3×C5—61no—
54C22×C5×D3—21no—
55C3×C5×Q8—31no—
56C3×C5×D4—21no—
57C3×C5×D4—31no—
58C3×C5×D4—61no—
59C3×C5×D4C2×D311yescharacteristic
60C3×C5×D4C2×D311yescharacteristic
61C2×C3×C4×C5—31no—
62C5×#67—21no—
63C2×C5×Dic3—61no—
64C5×D12C2×D311yescharacteristic
65C4×C5×D3—61no—
66C4×C5×D3C2×D311yescharacteristic
67C5×Dic6—31no—
68C5×Dic6—31no—
69C5×Dic6—61no—
70C5×Dic6—61no—
71C5×Dic6—61no—
72C5×Dic6C2×D311yescharacteristic, complemented
73C3×C5×C8—31no—
74C5×C3⋊C8—31no—
75C5×C3⋊C8—31no—
76C5×C3⋊C8—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78C2×#214—31no—
79D3×SD16—31no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C3×C5×D3—41no—
83C3×C5×D3SD1612yescomplemented
84C2×C5×Q8—91no—
85C2×C5×D4—31no—
86C5×SD16—31no—
87C5×SD16—91no—
88C5×SD16—91no—
89C5×SD16—91no—
90C2×C5×C8—91no—
91C22×C3×D3—21no—
92C32×D4C22×C511yescharacteristic
93#634C22×C511yescharacteristic, complemented
94C3×#67—21no—
95C3×D12C22×C511yescharacteristic
96C3×C4×D3C22×C511yescharacteristic
97C3×Dic6C22×C511yescharacteristic, complemented
98#627—21no—
99D3×Dic3—21no—
100C3⋊(C3⋊C8)C22×C511yescharacteristic, complemented
101C3×C3⋊C8C22×C511yescharacteristic, complemented
102C22×C3×C5—21no—
103C22×C3×C5—21no—
104C22×C3×C5—31no—
105C22×C3×C5—41no—
106C22×C3×C5—61no—
107C22×C3×C5—122no—
108C2×C5×D3—21no—
109C2×C5×D3—42no—
110C2×C5×D3#6711yescharacteristic
111C3×C4×C5—21no—
112C3×C4×C5—31no—
113C3×C4×C5—61no—
114C3×C4×C5C22×D311yescharacteristic
115C3×C4×C5C22×D311yescharacteristic
116C5×Dic3—21no—
117C5×Dic3—61no—
118C5×Dic3—61no—
119C5×Dic3—121no—
120C5×Dic3#6711yescharacteristic
121C2×C3×D4—31no—
122D3×Q8—31no—
123D3×D4C5×D311yescharacteristic
124C2×Dic6—31no—
125C3×SD16—31no—
126#215—31no—
127#214—31no—
128#214—31no—
129#214—31no—
130#214—31no—
131#214—61no—
132#214C5×D311yescharacteristic, direct product factor
133C2×C3⋊C8—31no—
134#204—31no—
135D3×C8—31no—
136C32×C5C2×SD1611yescharacteristic, complemented, Hall
137C23×C5—61no—
138C5×Q8—31no—
139C5×Q8—91no—
140C5×Q8—181no—
141C5×D4—31no—
142C5×D4—61no—
143C5×D4D3211yescharacteristic
144C2×C4×C5—31no—
145C2×C4×C5—181no—
146C5×C8—31no—
147C5×C8—91no—
148C22×C32—21no—
149C2×C3×D3—21no—
150C2×C3×D3—42no—
151C2×C3×D3C5×D411yescharacteristic
152C32×C4C23×C511yescharacteristic
153C3⋊Dic3—21no—
154C3×Dic3—21no—
155C3×Dic3C5×D411yescharacteristic
156C2×SD16—91noSylow
157C2×C3×C5—21no—
158C2×C3×C5—32no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5—81no—
162C2×C3×C5—121no—
163C2×C3×C5D3×D411yescharacteristic
164C2×C3×C5C2×#6711yescharacteristic
165C5×D3—41no—
166C5×D3#21412yesdirect product factor
167C23×C3—61no—
168C22×D3—21no—
169C3×Q8—31no—
170C3×D4—21no—
171C3×D4—31no—
172C3×D4—61no—
173C3×D4C2×C5×D311yescharacteristic
174C3×D4C2×C5×D311yescharacteristic
175C2×C3×C4—31no—
176#67—21no—
177C2×Dic3—61no—
178D12C2×C5×D311yescharacteristic
179C4×D3—61no—
180C4×D3C2×C5×D311yescharacteristic
181Dic6—31no—
182Dic6—31no—
183Dic6—61no—
184Dic6—61no—
185Dic6—61no—
186Dic6C2×C5×D311yescharacteristic, complemented
187C3×C8—31no—
188C3⋊C8—31no—
189C3⋊C8—31no—
190C3⋊C8—61no—
191C3⋊C8C2×C5×D311yescharacteristic, complemented
192C22×C5—21no—
193C22×C5—31no—
194C22×C5—61no—
195C22×C5—122no—
196C4×C5—31no—
197C4×C5—61no—
198C4×C5—181no—
199C4×C5C2×D3211yescharacteristic
200C2×C32—41no—
201C2×C32C2×C5×D411yescharacteristic
202C3×D3—41no—
203C3×D3C5×SD1612yescomplemented
204C2×Q8—91no—
205C2×D4—31no—
206SD16—31no—
207SD16—91no—
208SD16—91no—
209SD16—91no—
210C2×C8—91no—
211C3×C5—21no—
212C3×C5D3×SD1611yescharacteristic, complemented
213C3×C5C2×#21411yescharacteristic, complemented
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—31no—
217C22×C3—41no—
218C22×C3—61no—
219C22×C3—122no—
220C2×D3—21no—
221C2×D3—42no—
222C2×D3C5×#6711yescharacteristic
223C3×C4—21no—
224C3×C4—31no—
225C3×C4—61no—
226C3×C4C22×C5×D311yescharacteristic
227C3×C4C22×C5×D311yescharacteristic
228Dic3—21no—
229Dic3—61no—
230Dic3—61no—
231Dic3—121no—
232Dic3C5×#6711yescharacteristic
233C2×C5—32no—
234C2×C5—41no—
235C2×C5—121no—
236C2×C5D3×#6711yescharacteristic
237C32C2×C5×SD1611yescharacteristic, complemented, Sylow
238C23—61no—
239Q8—31no—
240Q8—91no—
241Q8—181no—
242D4—31no—
243D4—61no—
244D4C5×D3211yescharacteristic
245C2×C4—31no—
246C2×C4—181no—
247C8—31no—
248C8—91no—
249C2×C3—21no—
250C2×C3—32no—
251C2×C3—41no—
252C2×C3—41no—
253C2×C3—81no—
254C2×C3—121no—
255C2×C3C5×D3×D411yescharacteristic
256C2×C3C2×C5×#6711yescharacteristic
257D3—41no—
258D3C5×#21412yesdirect product factor
259C5D3×#21411yescharacteristic, direct product factor, Sylow
260C22—21no—
261C22—31no—
262C22—61no—
263C22—122no—
264C4—31no—
265C4—61no—
266C4—181no—
267C4C2×C5×D3211yescharacteristic
268C3—21no—
269C3C5×D3×SD1611yescharacteristic, complemented
270C3C2×C5×#21411yescharacteristic, complemented
271C2—32no—
272C2—41no—
273C2—121no—
274C2C5×D3×#6711yescharacteristic
subgroupG/H
centerC2×C5D3×#67
commutator, cosocleC32×C4C23×C5
FrattiniC4C2×C5×D32
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
prime radicalC3×C5×D3×D4C2
hypercenterC5×D4D32
hypocenterC32C2×C5×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×D4

Complements

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

NH ≅ G/Ntype
2. C5×D3×Dic6C2semidirect
3. C5×#3754C2semidirect
4. C3×C5×#214C2semidirect
5. C5×#3716C2semidirect
6. C5×#3715C2semidirect
7. C5×D3×C3⋊C8C2semidirect
12. C5×#634C22semidirect
16. C3×C5×Dic6C22semidirect
19. C5×C3⋊(C3⋊C8)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
21. D3×#214C5direct (N × H)
33. C5×#214D3direct (N × H)
47. D3×Dic6C2×C5semidirect
48. #3754C2×C5semidirect
49. C3×#214C2×C5semidirect
50. #3716C2×C5semidirect
51. #3715C2×C5semidirect
52. D3×C3⋊C8C2×C5semidirect
72. C5×Dic6C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
83. C3×C5×D3SD16semidirect
93. #634C22×C5semidirect
97. C3×Dic6C22×C5semidirect
100. C3⋊(C3⋊C8)C22×C5semidirect
101. C3×C3⋊C8C22×C5semidirect
132. #214C5×D3direct (N × H)
136. C32×C5C2×SD16semidirect
166. C5×D3#214direct (N × H)
186. Dic6C2×C5×D3semidirect
191. C3⋊C8C2×C5×D3semidirect
203. C3×D3C5×SD16semidirect
212. C3×C5D3×SD16semidirect
213. C3×C5C2×#214semidirect
237. C32C2×C5×SD16semidirect
258. D3C5×#214direct (N × H)
259. C5D3×#214direct (N × H)
269. C3C5×D3×SD16semidirect
270. C3C2×C5×#214semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D3×D42
C5×D3×Dic62
C5×#37542
C3×C5×#2142
C5×#37162
C5×#37152
C5×D3×C3⋊C82
C2×C5×#2143
C5×D3×SD163
D3×#2145