Finite Group Databasebeta

← C3×#91298 #13086497

Subgroups — C3×#91298 #13086497

Order 1440.

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

normal subgroups only

This group has 326 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×#235C211yesmaximal, characteristic, complemented
2C2×C32×D20C211yesmaximal, characteristic, complemented
3C3×#6710C211yesmaximal, characteristic, complemented
4C3×#6655C212yesmaximal, complemented
5C3×#6652C212yesmaximal, complemented
6C3×#4095—31nomaximal
7#91298C311yesmaximal, characteristic, direct product factor
8C22×C32×D5—21no—
9C2×C32×C4×C5C2211yescharacteristic
10C3×C5×#67—21no—
11C3×C5×D12C2211yescharacteristic, complemented
12C3×C4×C5×D3—21no—
13C3×C5×Dic6C2211yescharacteristic, complemented
14C32×D20—21no—
15C32×D20C2212yes—
16C3×C3⋊#168C2212yescomplemented
17C3×#932—51nomaximal, Hall
18C3×C5×#41—31no—
19C5×#235C2×C311yescharacteristic, complemented
20C2×C3×D20—21no—
21C2×C3×D20D311yescharacteristic, complemented
22C2×C3×D20C2×C311yescharacteristic, complemented
23#6710C2×C311yescharacteristic, complemented
24C3×#688—32no—
25C3×#686—32no—
26C3×#681—31no—
27#6655C2×C312yescomplemented
28#6652C2×C312yescomplemented
29C22×C32×C5D411yescharacteristic
30C2×C3×C5×D3—21no—
31C2×C32×D5—22no—
32C2×C32×D5—42no—
33C32×C4×C5D411yescharacteristic
34C32×C4×C5C2311yescharacteristic
35C3×C5×Dic3—21no—
36#4095—31noHall
37C2×C32×D4—51no—
38C3×#235D511yescharacteristic, complemented
39C3×#214—52no—
40C3×#213—52no—
41C3×#208—51no—
42C22×C3×D5—22no—
43C22×C3×D5—21no—
44C22×C3×D5—21no—
45C3×C5×Q8—31no—
46C3×C5×D4—31no—
47C3×C5×D4—61no—
48C2×C3×C4×C5—21no—
49C2×C3×C4×C5—61no—
50C2×C3×C4×C5C2×D311yescharacteristic
51C2×C3×C4×C5C22×C311yescharacteristic
52C5×#67—21no—
53C5×D12C22×C311yescharacteristic, complemented
54C4×C5×D3—21no—
55C5×Dic6C22×C311yescharacteristic, complemented
56C3×D20—22no—
57C3×D20—22no—
58C3×D20—21no—
59C3×D20—21no—
60C3×D20C2×D312yes—
61C3×D20C22×C312yes—
62C3⋊#168C22×C312yescomplemented
63C3×#168—32no—
64C3×AGL(1,ℤ/8)—151no—
65#932—51no—
66C2×C32×C5—21no—
67C2×C32×C5C2×D411yescharacteristic
68C3×C5×D3—41no—
69C32×D5—42no—
70C5×#41—31no—
71C2×D20C3×D311yescharacteristic, complemented
72#688—32no—
73#686—32no—
74#681—31no—
75C23×C32—101no—
76C32×D4—52no—
77C32×D4—101no—
78C2×C32×C4C2×D511yescharacteristic
79C3×#67—21no—
80C3×D12C2×D511yescharacteristic, complemented
81C3×C4×D3—21no—
82C3×Dic6C2×D511yescharacteristic, complemented
83C3×C3⋊C8—52no—
84C22×C3×C5—21no—
85C22×C3×C5—61no—
86C22×C3×C5#6711yescharacteristic
87C22×C3×C5C3×D411yescharacteristic
88C2×C5×D3—21no—
89C2×C3×D5—24no—
90C2×C3×D5—22no—
91C2×C3×D5—22no—
92C2×C3×D5—44no—
93C2×C3×D5—42no—
94C2×C3×D5—42no—
95C3×C4×C5—21no—
96C3×C4×C5—21no—
97C3×C4×C5—61no—
98C3×C4×C5#6711yescharacteristic
99C3×C4×C5C3×D411yescharacteristic
100C3×C4×C5C22×D311yescharacteristic
101C3×C4×C5C23×C311yescharacteristic
102C5×Dic3—21no—
103C3×#41—31no—
104C2×C3×D4—51no—
105C2×C3×D4—51no—
106C2×C3×D4—101no—
107#235C3×D511yescharacteristic, complemented
108C3×SD16—152no—
109C3×D8—152no—
110C3×M16—151no—
111#214—52no—
112#213—52no—
113#208—51no—
114C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
115C22×D5—21no—
116C5×Q8—31no—
117C5×D4—31no—
118C5×D4—61no—
119C2×C4×C5—61no—
120C2×C4×C5C2×C3×D311yescharacteristic
121D20—21no—
122D20C2×C3×D312yes—
123#168—32no—
124C22×C32—102no—
125C22×C32—202no—
126C22×C32C22⋊D511yescharacteristic
127C2×C3×D3—21no—
128C32×C4C22⋊D511yescharacteristic
129C32×C4C22×D511yescharacteristic
130C3×Dic3—21no—
131AGL(1,ℤ/8)—151noSylow
132C2×C3×C5—22no—
133C2×C3×C5—21no—
134C2×C3×C5—21no—
135C2×C3×C5—21no—
136C2×C3×C5—121no—
137C2×C3×C5C2×#6711yescharacteristic
138C2×C3×C5C2×C3×D411yescharacteristic
139C3×D5—44no—
140C3×D5—42no—
141C3×D5—42no—
142C5×D3—41no—
143C23×C3—102no—
144C23×C3—101no—
145C23×C3—101no—
146C3×Q8—31no—
147C3×D4—31no—
148C3×D4—52no—
149C3×D4—52no—
150C3×D4—61no—
151C3×D4—102no—
152C3×D4—102no—
153C3×D4—101no—
154C3×D4—101no—
155C2×C3×C4—21no—
156C2×C3×C4—61no—
157C2×C3×C4D3×D511yescharacteristic
158C2×C3×C4C2×C3×D511yescharacteristic
159#67—21no—
160D12C2×C3×D511yescharacteristic, complemented
161C4×D3—21no—
162Dic6C2×C3×D511yescharacteristic, complemented
163C3×C8—152no—
164C3⋊C8—52no—
165C22×C5—61no—
166C22×C5C3×#6711yescharacteristic
167C2×D5—22no—
168C2×D5—42no—
169C4×C5—61no—
170C4×C5C3×#6711yescharacteristic
171C4×C5C22×C3×D311yescharacteristic
172C2×C32—21no—
173C2×C32—202no—
174C2×C32C2×C22⋊D511yescharacteristic
175C3×D3—41no—
176#41—31no—
177C2×D4—51no—
178SD16—152no—
179D8—152no—
180M16—151no—
181C3×C5—21no—
182C3×C5#93211yescharacteristic, complemented
183C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
184C22×C3—21no—
185C22×C3—61no—
186C22×C3—104no—
187C22×C3—102no—
188C22×C3—102no—
189C22×C3—204no—
190C22×C3—202no—
191C22×C3—202no—
192C22×C3#120111yescharacteristic
193C22×C3C3×C22⋊D511yescharacteristic
194C2×D3—21no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—61no—
198C3×C4#120111yescharacteristic
199C3×C4C3×C22⋊D511yescharacteristic
200C3×C4C2×D3×D511yescharacteristic
201C3×C4C22×C3×D511yescharacteristic
202Dic3—21no—
203C2×C5—21no—
204C2×C5—121no—
205C2×C5C2×C3×#6711yescharacteristic
206D5—42no—
207C32#409511yescharacteristic, complemented, Sylow
208C23—101no—
209Q8—31no—
210D4—31no—
211D4—52no—
212D4—61no—
213D4—101no—
214C2×C4—61no—
215C2×C4C3×D3×D511yescharacteristic
216C8—152no—
217C2×C3—22no—
218C2×C3—21no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3—121no—
222C2×C3—204no—
223C2×C3—202no—
224C2×C3—202no—
225C2×C3C2×#120111yescharacteristic
226C2×C3C2×C3×C22⋊D511yescharacteristic
227D3—41no—
228C5C3×#93211yescharacteristic, complemented, Sylow
229C22—61no—
230C22—102no—
231C22—202no—
232C22C3×#120111yescharacteristic
233C4—61no—
234C4C3×#120111yescharacteristic
235C4C2×C3×D3×D511yescharacteristic
236C3—21no—
237C3#9129811yescharacteristic, direct product factor
238C3C3×#409511yescharacteristic, complemented
239C2—21no—
240C2—121no—
241C2—202no—
242C2C2×C3×#120111yescharacteristic
subgroupG/H
centerC2×C3C2×#1201
commutator, cosocleC3×C4×C5C23×C3
FrattiniC4C2×C3×D3×D5
FittingC2×C32×C4×C5C22
socleC2×C32×C5C2×D4
hypercenterC2×C3×C4D3×D5
hypocenterC3×C5C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×C5×#235C2semidirect
2. C2×C32×D20C2semidirect
3. C3×#6710C2semidirect
4. C3×#6655C2semidirect
5. C3×#6652C2semidirect
7. #91298C3direct (N × H)
11. C3×C5×D12C22semidirect
13. C3×C5×Dic6C22semidirect
16. C3×C3⋊#168C22semidirect
19. C5×#235C2×C3semidirect
21. C2×C3×D20D3semidirect
22. C2×C3×D20C2×C3semidirect
23. #6710C2×C3semidirect
27. #6655C2×C3semidirect
28. #6652C2×C3semidirect
38. C3×#235D5semidirect
53. C5×D12C22×C3semidirect
55. C5×Dic6C22×C3semidirect
62. C3⋊#168C22×C3semidirect
71. C2×D20C3×D3semidirect
80. C3×D12C2×D5semidirect
82. C3×Dic6C2×D5semidirect
107. #235C3×D5semidirect
114. C32×C5AGL(1,ℤ/8)semidirect
160. D12C2×C3×D5semidirect
162. Dic6C2×C3×D5semidirect
182. C3×C5#932semidirect
183. C3×C5C3×AGL(1,ℤ/8)semidirect
207. C32#4095semidirect
228. C5C3×#932semidirect
237. C3#91298direct (N × H)
238. C3C3×#4095semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#2352
C2×C32×D202
C3×#67102
C3×#66552
C3×#66522
C3×#40953
#912983
C3×#9325