Finite Group Databasebeta

← C3×#91299 #13086498

Subgroups — C3×#91299 #13086498

Order 1440.

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

normal subgroups only

This group has 312 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
1C2×C3×C5×D12C211yesmaximal, characteristic, complemented
2C32×#709C211yesmaximal, characteristic, complemented
3C3×#6710C211yesmaximal, characteristic, complemented
4C3×#6656C212yesmaximal, complemented
5C3×#6652C212yesmaximal, complemented
6C3×#4078—31nomaximal
7#91299C311yesmaximal, characteristic, direct product factor
8C22×C3×C5×D3—21no—
9C2×C32×C4×C5C2211yescharacteristic
10C3×C5×D12—21no—
11C3×C5×D12C2212yes—
12C32×C22⋊D5—21no—
13C32×D20C2211yescharacteristic, complemented
14C32×C4×D5—21no—
15C32×Dic10C2211yescharacteristic, complemented
16C3×C3⋊#168C2212yescomplemented
17C3×#949—51nomaximal, Hall
18C2×C3×C5×D4—31no—
19C2×C5×D12C2×C311yescharacteristic, complemented
20C3×#709—21no—
21C3×#709D311yescharacteristic, complemented
22C3×#709C2×C311yescharacteristic, complemented
23#6710C2×C311yescharacteristic, complemented
24C3×#687—32no—
25C3×#686—32no—
26C3×#681—31no—
27#6656C2×C312yescomplemented
28#6652C2×C312yescomplemented
29C22×C32×C5D411yescharacteristic
30C2×C3×C5×D3—22no—
31C2×C3×C5×D3—42no—
32C2×C32×D5—21no—
33C32×C4×C5D411yescharacteristic
34C32×C4×C5C2311yescharacteristic
35C32×Dic5—21no—
36#4078—31noHall
37C32×#41—51no—
38C2×C3×D12D511yescharacteristic, complemented
39C3×#215—52no—
40C3×#213—52no—
41C3×#208—51no—
42C23×C3×C5—61no—
43C22×C5×D3—21no—
44C3×C5×D4—32no—
45C3×C5×D4—61no—
46C2×C3×C4×C5—21no—
47C2×C3×C4×C5C2×D311yescharacteristic
48C2×C3×C4×C5C22×C311yescharacteristic
49C5×D12—21no—
50C5×D12C22×C312yes—
51C3×C22⋊D5—22no—
52C3×C22⋊D5—21no—
53C3×C22⋊D5—21no—
54C3×D20—21no—
55C3×D20C2×D311yescharacteristic, complemented
56C3×D20C22×C311yescharacteristic, complemented
57C3×C4×D5—22no—
58C3×C4×D5—21no—
59C3×C4×D5—21no—
60C3×Dic10—21no—
61C3×Dic10C2×D311yescharacteristic, complemented
62C3×Dic10C22×C311yescharacteristic, complemented
63C3⋊#168C22×C312yescomplemented
64C3×#168—32no—
65C3×AGL(1,ℤ/8)—151no—
66#949—51no—
67C2×C32×C5—21no—
68C2×C32×C5C2×D411yescharacteristic
69C3×C5×D3—42no—
70C32×D5—41no—
71C2×C5×D4—31no—
72#709C3×D311yescharacteristic, complemented
73#687—32no—
74#686—32no—
75#681—31no—
76C22×C3×D3—21no—
77C32×Q8—51no—
78C32×D4—51no—
79C32×D4—101no—
80C2×C32×C4—101no—
81C2×C32×C4C2×D511yescharacteristic
82C3×D12—21no—
83C3×D12C2×D512yes—
84C3×C3⋊C8—52no—
85C22×C3×C5—21no—
86C22×C3×C5—62no—
87C22×C3×C5—122no—
88C22×C3×C5#6711yescharacteristic
89C22×C3×C5C3×D411yescharacteristic
90C2×C5×D3—22no—
91C2×C5×D3—42no—
92C2×C3×D5—22no—
93C2×C3×D5—21no—
94C2×C3×D5—21no—
95C3×C4×C5—21no—
96C3×C4×C5—21no—
97C3×C4×C5#6711yescharacteristic
98C3×C4×C5C3×D411yescharacteristic
99C3×C4×C5C22×D311yescharacteristic
100C3×C4×C5C23×C311yescharacteristic
101C3×Dic5—22no—
102C3×Dic5—21no—
103C3×Dic5—21no—
104C3×#41—51no—
105C3×#41—51no—
106C3×#41—101no—
107C2×C3×D4—31no—
108C2×D12C3×D511yescharacteristic, complemented
109C3×SD16—152no—
110C3×D8—152no—
111C3×M16—151no—
112#215—52no—
113#213—52no—
114#208—51no—
115C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
116C23×C5—61no—
117C5×D4—32no—
118C5×D4—61no—
119C2×C4×C5C2×C3×D311yescharacteristic
120C22⋊D5—21no—
121D20C2×C3×D311yescharacteristic, complemented
122C4×D5—21no—
123Dic10C2×C3×D311yescharacteristic, complemented
124#168—32no—
125C22×C32—101no—
126C22×C32C22⋊D511yescharacteristic
127C2×C3×D3—22no—
128C2×C3×D3—42no—
129C32×C4—101no—
130C32×C4C22⋊D511yescharacteristic
131C32×C4C22×D511yescharacteristic
132AGL(1,ℤ/8)—151noSylow
133C2×C3×C5—21no—
134C2×C3×C5—21no—
135C2×C3×C5—21no—
136C2×C3×C5—41no—
137C2×C3×C5—122no—
138C2×C3×C5C2×#6711yescharacteristic
139C2×C3×C5C2×C3×D411yescharacteristic
140C3×D5—42no—
141C3×D5—41no—
142C3×D5—41no—
143C5×D3—42no—
144C23×C3—61no—
145C22×D3—21no—
146C3×Q8—51no—
147C3×Q8—51no—
148C3×Q8—101no—
149C3×D4—32no—
150C3×D4—51no—
151C3×D4—51no—
152C3×D4—61no—
153C3×D4—101no—
154C3×D4—102no—
155C3×D4—101no—
156C3×D4—101no—
157C2×C3×C4—21no—
158C2×C3×C4—102no—
159C2×C3×C4—101no—
160C2×C3×C4—101no—
161C2×C3×C4D3×D511yescharacteristic
162C2×C3×C4C2×C3×D511yescharacteristic
163D12—21no—
164D12C2×C3×D512yes—
165C3×C8—152no—
166C3⋊C8—52no—
167C22×C5—62no—
168C22×C5—122no—
169C22×C5C3×#6711yescharacteristic
170C2×D5—21no—
171C4×C5C3×#6711yescharacteristic
172C4×C5C22×C3×D311yescharacteristic
173Dic5—21no—
174C2×C32—21no—
175C2×C32—201no—
176C2×C32C2×C22⋊D511yescharacteristic
177C3×D3—42no—
178#41—51no—
179C2×D4—31no—
180SD16—152no—
181D8—152no—
182M16—151no—
183C3×C5—21no—
184C3×C5#94911yescharacteristic, complemented
185C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
186C22×C3—21no—
187C22×C3—62no—
188C22×C3—102no—
189C22×C3—101no—
190C22×C3—101no—
191C22×C3—122no—
192C22×C3#120111yescharacteristic
193C22×C3C3×C22⋊D511yescharacteristic
194C2×D3—22no—
195C2×D3—42no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—102no—
199C3×C4—101no—
200C3×C4—101no—
201C3×C4#120111yescharacteristic
202C3×C4C3×C22⋊D511yescharacteristic
203C3×C4C2×D3×D511yescharacteristic
204C3×C4C22×C3×D511yescharacteristic
205C2×C5—21no—
206C2×C5—122no—
207C2×C5C2×C3×#6711yescharacteristic
208D5—41no—
209C32#407811yescharacteristic, complemented, Sylow
210C23—61no—
211Q8—51no—
212D4—32no—
213D4—51no—
214D4—61no—
215D4—101no—
216C2×C4—101no—
217C2×C4C3×D3×D511yescharacteristic
218C8—152no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3—21no—
222C2×C3—41no—
223C2×C3—122no—
224C2×C3—202no—
225C2×C3—201no—
226C2×C3—201no—
227C2×C3C2×#120111yescharacteristic
228C2×C3C2×C3×C22⋊D511yescharacteristic
229D3—42no—
230C5C3×#94911yescharacteristic, complemented, Sylow
231C22—62no—
232C22—101no—
233C22—122no—
234C22C3×#120111yescharacteristic
235C4—101no—
236C4C3×#120111yescharacteristic
237C4C2×C3×D3×D511yescharacteristic
238C3—21no—
239C3#9129911yescharacteristic, direct product factor
240C3C3×#407811yescharacteristic, complemented
241C2—21no—
242C2—122no—
243C2—201no—
244C2C2×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. C2×C3×C5×D12C2semidirect
2. C32×#709C2semidirect
3. C3×#6710C2semidirect
4. C3×#6656C2semidirect
5. C3×#6652C2semidirect
7. #91299C3direct (N × H)
13. C32×D20C22semidirect
15. C32×Dic10C22semidirect
16. C3×C3⋊#168C22semidirect
19. C2×C5×D12C2×C3semidirect
21. C3×#709D3semidirect
22. C3×#709C2×C3semidirect
23. #6710C2×C3semidirect
27. #6656C2×C3semidirect
28. #6652C2×C3semidirect
38. C2×C3×D12D5semidirect
55. C3×D20C2×D3semidirect
56. C3×D20C22×C3semidirect
61. C3×Dic10C2×D3semidirect
62. C3×Dic10C22×C3semidirect
63. C3⋊#168C22×C3semidirect
72. #709C3×D3semidirect
108. C2×D12C3×D5semidirect
115. C32×C5AGL(1,ℤ/8)semidirect
121. D20C2×C3×D3semidirect
123. Dic10C2×C3×D3semidirect
184. C3×C5#949semidirect
185. C3×C5C3×AGL(1,ℤ/8)semidirect
209. C32#4078semidirect
230. C5C3×#949semidirect
239. C3#91299direct (N × H)
240. C3C3×#4078semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×D122
C32×#7092
C3×#67102
C3×#66562
C3×#66522
C3×#40783
#912993
C3×#9495