Finite Group Databasebeta

← C5×#63977 #13087245

Subgroups — C5×#63977 #13087245

Order 1440.

574 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C23×C3×C5×D3C211yesmaximal, characteristic, complemented
2C2×C5×#638C211yesmaximal, characteristic, complemented
3C2×C5×#626C212yesmaximal, complemented
4C3×C5×#217C211yesmaximal, characteristic, complemented
5C5×(C2×D3)⋊Dic3C212yesmaximal, complemented
6C5×#953—31nomaximal
7C5×#882—31nomaximal
8C23×C32×C5C2211yescharacteristic
9C22×C5×C3⋊D3C2211yescharacteristic, complemented
10C22×C3×C5×D3—22no—
11C22×C3×C5×D3—24no—
12C22×C3×C5×D3C2212yes—
13C5×#638—22no—
14C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
15C2×C3×C5×Dic3C2212yescomplemented
16C5×#626—24no—
17#63977C511yesmaximal, characteristic, direct product factor, Hall
18C24×C3×C5—31no—
19C23×C5×D3D311yescharacteristic, complemented
20C2×C5×#67—31no—
21C2×C5×#67—31no—
22C2×C5×#67—32no—
23C2×C5×#67—61no—
24C2×C5×D12—32no—
25C3×C5×C22⋊C4—31no—
26C5×#217—32no—
27C5×#217D311yescharacteristic, complemented
28C5×(C2×D3)⋊C4—32no—
29C22×C32×C5—22no—
30C22×C32×C5D412yescomplemented
31C22×C32×C5C2311yescharacteristic
32C2×C5×C3⋊D3—21no—
33C2×C5×C3⋊D3—22no—
34C2×C3×C5×D3—28no—
35C2×C3×C5×D3—22no—
36C2×C3×C5×D3—22no—
37C2×C3×C5×D3D414yescomplemented
38C5×C3⋊Dic3—21no—
39C3×C5×Dic3—22no—
40C5×#120—91noHall
41C23×C3×D3C2×C511yescharacteristic, complemented
42C2×#638C2×C511yescharacteristic, complemented
43C2×#626C2×C512yescomplemented
44C3×#217C2×C511yescharacteristic, complemented
45(C2×D3)⋊Dic3C2×C512yescomplemented
46C23×C3×C5—21no—
47C23×C3×C5—32no—
48C23×C3×C5—64no—
49C23×C3×C5—62no—
50C23×C3×C5C2×D311yescharacteristic
51C23×C3×C5C2×D311yescharacteristic
52C22×C5×D3—22no—
53C22×C5×D3—24no—
54C22×C5×D3—31no—
55C22×C5×D3—31no—
56C22×C5×D3—61no—
57C22×C5×D3C2×D312yes—
58C2×C3×C4×C5—32no—
59C5×#67—64no—
60C5×#67—64no—
61C5×#67—62no—
62C5×#67—62no—
63C2×C5×Dic3—31no—
64C2×C5×Dic3—31no—
65C2×C5×Dic3—61no—
66C2×C5×Dic3C2×D312yescomplemented
67C5×D12—64no—
68#953—31no—
69#882—31no—
70C2×C32×C5—22no—
71C2×C32×C5C2×D411yescharacteristic
72C2×C32×C5C2×D412yes—
73C5×C3⋊D3—41no—
74C3×C5×D3—24no—
75C24×C5—31no—
76C2×C5×D4—92no—
77C2×C5×D4—91no—
78C5×C22⋊C4—31no—
79C5×C22⋊C4—92no—
80C23×C32C22×C511yescharacteristic
81C22×C3⋊D3C22×C511yescharacteristic, complemented
82C22×C3×D3—22no—
83C22×C3×D3—24no—
84C22×C3×D3C22×C512yes—
85#638—22no—
86C2×C3⋊Dic3C22×C511yescharacteristic, complemented
87C2×C3×Dic3C22×C512yescomplemented
88#626—24no—
89C22×C3×C5—22no—
90C22×C3×C5—22no—
91C22×C3×C5—22no—
92C22×C3×C5—21no—
93C22×C3×C5—34no—
94C22×C3×C5—42no—
95C22×C3×C5—68no—
96C22×C3×C5—62no—
97C22×C3×C5—62no—
98C22×C3×C5D1212yescomplemented
99C22×C3×C5#6712yescomplemented
100C22×C3×C5C22×D311yescharacteristic
101C22×C3×C5C22×D311yescharacteristic
102C2×C5×D3—28no—
103C2×C5×D3—22no—
104C2×C5×D3—22no—
105C2×C5×D3—62no—
106C2×C5×D3—61no—
107C2×C5×D3—62no—
108C2×C5×D3—62no—
109C2×C5×D3—61no—
110C2×C5×D3—122no—
111C2×C5×D3#6714yescomplemented
112C3×C4×C5—62no—
113C5×Dic3—22no—
114C5×Dic3—62no—
115C5×Dic3—61no—
116C5×Dic3—61no—
117C24×C3—31no—
118C23×D3C5×D311yescharacteristic, complemented
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—32no—
122C2×#67—61no—
123C2×D12—32no—
124C3×C22⋊C4—31no—
125#217—32no—
126#217C5×D311yescharacteristic, complemented
127(C2×D3)⋊C4—32no—
128C32×C5#12011yescharacteristic, complemented, Hall
129C23×C5—32no—
130C23×C5—64no—
131C23×C5—62no—
132C23×C5—91no—
133C23×C5D3211yescharacteristic
134C5×D4—184no—
135C5×D4—182no—
136C2×C4×C5—32no—
137C2×C4×C5—91no—
138C22×C32—22no—
139C22×C32C5×D412yescomplemented
140C22×C32C23×C511yescharacteristic
141C2×C3⋊D3—21no—
142C2×C3⋊D3—22no—
143C2×C3×D3—28no—
144C2×C3×D3—22no—
145C2×C3×D3—22no—
146C2×C3×D3C5×D414yescomplemented
147C3⋊Dic3—21no—
148C3×Dic3—22no—
149#120—91noSylow
150C2×C3×C5—22no—
151C2×C3×C5—22no—
152C2×C3×C5—21no—
153C2×C3×C5—22no—
154C2×C3×C5—42no—
155C2×C3×C5—64no—
156C2×C3×C5C2×D1211yescharacteristic
157C2×C3×C5D3×D412yes—
158C2×C3×C5C2×#6711yescharacteristic
159C2×C3×C5C2×#6712yes—
160C5×D3—24no—
161C5×D3—122no—
162C5×D3—121no—
163C5×D3—121no—
164C23×C3—21no—
165C23×C3—32no—
166C23×C3—64no—
167C23×C3—62no—
168C23×C3C2×C5×D311yescharacteristic
169C23×C3C2×C5×D311yescharacteristic
170C22×D3—22no—
171C22×D3—24no—
172C22×D3—31no—
173C22×D3—31no—
174C22×D3—61no—
175C22×D3C2×C5×D312yes—
176C2×C3×C4—32no—
177#67—64no—
178#67—64no—
179#67—62no—
180#67—62no—
181C2×Dic3—31no—
182C2×Dic3—31no—
183C2×Dic3—61no—
184C2×Dic3C2×C5×D312yescomplemented
185D12—64no—
186C22×C5—22no—
187C22×C5—34no—
188C22×C5—68no—
189C22×C5—62no—
190C22×C5—62no—
191C22×C5—181no—
192C22×C5—182no—
193C22×C5#62612yescomplemented
194C22×C5C2×D3211yescharacteristic
195C4×C5—62no—
196C4×C5—181no—
197C2×C32—22no—
198C2×C32C2×C5×D411yescharacteristic
199C2×C32C2×C5×D412yes—
200C3⋊D3—41no—
201C3×D3—24no—
202C24—31no—
203C2×D4—92no—
204C2×D4—91no—
205C22⋊C4—31no—
206C22⋊C4—92no—
207C3×C5—21no—
208C3×C5#88211yescharacteristic, complemented
209C3×C5#95311yescharacteristic, complemented
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—21no—
214C22×C3—34no—
215C22×C3—42no—
216C22×C3—68no—
217C22×C3—62no—
218C22×C3—62no—
219C22×C3C5×D1212yescomplemented
220C22×C3C5×#6712yescomplemented
221C22×C3C22×C5×D311yescharacteristic
222C22×C3C22×C5×D311yescharacteristic
223C2×D3—28no—
224C2×D3—22no—
225C2×D3—22no—
226C2×D3—62no—
227C2×D3—61no—
228C2×D3—62no—
229C2×D3—62no—
230C2×D3—61no—
231C2×D3—122no—
232C2×D3C5×#6714yescomplemented
233C3×C4—62no—
234Dic3—22no—
235Dic3—62no—
236Dic3—61no—
237Dic3—61no—
238C2×C5—22no—
239C2×C5—64no—
240C2×C5—361no—
241C2×C5C2×#62611yescharacteristic
242C2×C5D3×#6712yes—
243C32C5×#12011yescharacteristic, complemented, Sylow
244C23—32no—
245C23—64no—
246C23—62no—
247C23—91no—
248C23C5×D3211yescharacteristic
249D4—184no—
250D4—182no—
251C2×C4—32no—
252C2×C4—91no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—21no—
256C2×C3—22no—
257C2×C3—42no—
258C2×C3—64no—
259C2×C3C2×C5×D1211yescharacteristic
260C2×C3C5×D3×D412yes—
261C2×C3C2×C5×#6711yescharacteristic
262C2×C3C2×C5×#6712yes—
263D3—24no—
264D3—122no—
265D3—121no—
266D3—121no—
267C5#6397711yescharacteristic, direct product factor, Sylow
268C22—22no—
269C22—34no—
270C22—68no—
271C22—62no—
272C22—62no—
273C22—181no—
274C22—182no—
275C22C5×#62612yescomplemented
276C22C2×C5×D3211yescharacteristic
277C4—62no—
278C4—181no—
279C3—21no—
280C3C5×#88211yescharacteristic, complemented
281C3C5×#95311yescharacteristic, complemented
282C2—22no—
283C2—64no—
284C2—361no—
285C2C2×C5×#62611yescharacteristic
286C2C5×D3×#6712yes—
subgroupG/H
centerC22×C5C2×D32
commutator, cosocleC22×C32C23×C5
FrattiniC22C2×C5×D32
FittingC23×C32×C5C22
socleC22×C32×C5C23
hypercenterC23×C5D32
hypocenterC32C5×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C5 ◁ C23×C5

Complements

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

NH ≅ G/Ntype
1. C23×C3×C5×D3C2semidirect
2. C2×C5×#638C2semidirect
3. C2×C5×#626C2semidirect
4. C3×C5×#217C2semidirect
5. C5×(C2×D3)⋊Dic3C2semidirect
9. C22×C5×C3⋊D3C22semidirect
14. C2×C5×C3⋊Dic3C22semidirect
15. C2×C3×C5×Dic3C22semidirect
17. #63977C5direct (N × H)
19. C23×C5×D3D3semidirect
27. C5×#217D3semidirect
30. C22×C32×C5D4semidirect
37. C2×C3×C5×D3D4semidirect
41. C23×C3×D3C2×C5semidirect
42. C2×#638C2×C5semidirect
43. C2×#626C2×C5semidirect
44. C3×#217C2×C5semidirect
45. (C2×D3)⋊Dic3C2×C5semidirect
66. C2×C5×Dic3C2×D3semidirect
81. C22×C3⋊D3C22×C5semidirect
86. C2×C3⋊Dic3C22×C5semidirect
87. C2×C3×Dic3C22×C5semidirect
98. C22×C3×C5D12semidirect
99. C22×C3×C5#67semidirect
111. C2×C5×D3#67semidirect
118. C23×D3C5×D3semidirect
126. #217C5×D3semidirect
128. C32×C5#120semidirect
139. C22×C32C5×D4semidirect
146. C2×C3×D3C5×D4semidirect
184. C2×Dic3C2×C5×D3semidirect
193. C22×C5#626semidirect
208. C3×C5#882semidirect
209. C3×C5#953semidirect
219. C22×C3C5×D12semidirect
220. C22×C3C5×#67semidirect
232. C2×D3C5×#67semidirect
243. C32C5×#120semidirect
267. C5#63977direct (N × H)
275. C22C5×#626semidirect
280. C3C5×#882semidirect
281. C3C5×#953semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C5×D32
C2×C5×#6382
C2×C5×#6262
C3×C5×#2172
C5×(C2×D3)⋊Dic32
C5×#9533
C5×#8823
#639775