Finite Group Databasebeta

← C3×#91235 #13086283

Subgroups — C3×#91235 #13086283

Order 1440.

358 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C22×C3×ΓL(1,𝔽16)C212yesmaximal
2C2×C32×C4×D5C211yesmaximal, characteristic
3#91235C311yesmaximal, characteristic, direct product factor
4C3×(C2×AGL(1,𝔽5))⋊C4—31nomaximal
5C22×C32×D5C2211yescharacteristic
6C2×C3×ΓL(1,𝔽16)—22no—
7C2×C3×ΓL(1,𝔽16)—22no—
8C2×C3×ΓL(1,𝔽16)C414yescomplemented
9C2×C32×C4×C5C411yescharacteristic, complemented
10C2×C32×Dic5C411yescharacteristic, complemented
11C32×C4×D5—22no—
12C3×C3⋊#95—51nomaximal, Hall
13C22×ΓL(1,𝔽16)C2×C312yes—
14C22×C3×AGL(1,𝔽5)—32no—
15C2×C3×C4×D5—21no—
16C2×C3×C4×D5D311yescharacteristic
17C2×C3×C4×D5C2×C311yescharacteristic
18C22×C32×C5C2×C411yescharacteristic
19C2×C32×D5C2×C412yes—
20C2×C32×D5D411yescharacteristic
21C2×C32×D5D412yes—
22C2×C32×D5Q811yescharacteristic
23C3×ΓL(1,𝔽16)—24no—
24C32×C4×C5—21no—
25C32×Dic5—21no—
26(C2×AGL(1,𝔽5))⋊C4—31noHall
27C22×C32×C4—51no—
28C22×C3×Dic3—52no—
29C22×C3×D5—21no—
30C22×C3×D5C2×D311yescharacteristic
31C22×C3×D5C22×C311yescharacteristic
32C2×ΓL(1,𝔽16)—22no—
33C2×ΓL(1,𝔽16)—22no—
34C2×ΓL(1,𝔽16)C3×C414yescomplemented
35C2×C3×AGL(1,𝔽5)—34no—
36C2×C3×AGL(1,𝔽5)—62no—
37C2×C3×AGL(1,𝔽5)—62no—
38C2×C3×C4×C5—21no—
39C2×C3×C4×C5Dic311yescharacteristic, complemented
40C2×C3×C4×C5C3×C411yescharacteristic, complemented
41C2×C3×Dic5—21no—
42C2×C3×Dic5Dic311yescharacteristic, complemented
43C2×C3×Dic5C3×C411yescharacteristic, complemented
44C3×C4×D5—24no—
45C3×C4×D5—22no—
46C3×C4×D5—22no—
47C3×#95—151no—
48C3⋊#95—51no—
49C2×C32×C5C4211yescharacteristic
50C2×C32×C5C22⋊C411yescharacteristic
51C2×C32×C5C4⋊C411yescharacteristic
52C32×D5C22⋊C412yes—
53C32×D5C4⋊C412yes—
54C22×AGL(1,𝔽5)—32no—
55C2×C4×D5C3×D311yescharacteristic
56C23×C32—51no—
57C2×C32×C4—51no—
58C2×C32×C4—102no—
59C2×C32×C4AGL(1,𝔽5)11yescharacteristic, complemented
60C2×C3×Dic3—54no—
61C2×C3×Dic3—102no—
62C2×C3×Dic3—102no—
63C22×C3×C5—21no—
64C22×C3×C5C2×Dic311yescharacteristic
65C22×C3×C5C2×C3×C411yescharacteristic
66C2×C3×D5—21no—
67C2×C3×D5—22no—
68C2×C3×D5—21no—
69C2×C3×D5—22no—
70C2×C3×D5Dic611yescharacteristic
71C2×C3×D5C4×D312yes—
72C2×C3×D5D1211yescharacteristic
73C2×C3×D5#6712yes—
74C2×C3×D5C2×C3×C412yes—
75C2×C3×D5C3×D411yescharacteristic
76C2×C3×D5C3×D412yes—
77C2×C3×D5C3×Q811yescharacteristic
78ΓL(1,𝔽16)—24no—
79C3×AGL(1,𝔽5)—64no—
80C3×C4×C5—22no—
81C3×C4×C5—21no—
82C3×C4×C5—21no—
83C3×Dic5—22no—
84C3×Dic5—21no—
85C3×Dic5—21no—
86C22×C3×C4—51no—
87C22×C3×C4—51no—
88C22×C3×C4—101no—
89C22×C3×C4—152no—
90C22×Dic3—52no—
91C32×C5#9511yescharacteristic, complemented, Hall
92C22×D5C2×C3×D311yescharacteristic
93C2×AGL(1,𝔽5)—34no—
94C2×AGL(1,𝔽5)—62no—
95C2×AGL(1,𝔽5)—62no—
96C2×C4×C5C3×Dic311yescharacteristic, complemented
97C2×Dic5C3×Dic311yescharacteristic, complemented
98C4×D5—22no—
99C22×C32—52no—
100C22×C32—51no—
101C22×C32—51no—
102C22×C32—52no—
103C22×C32C2×AGL(1,𝔽5)11yescharacteristic
104C32×C4—21no—
105C32×C4—101no—
106C3×Dic3—104no—
107#95—151noSylow
108C2×C3×C5—21no—
109C2×C3×C5—21no—
110C2×C3×C5—21no—
111C2×C3×C5C4×Dic311yescharacteristic
112C2×C3×C5C4⋊Dic311yescharacteristic
113C2×C3×C5#21711yescharacteristic
114C2×C3×C5C3×C4211yescharacteristic
115C2×C3×C5C3×C22⋊C411yescharacteristic
116C2×C3×C5C3×C4⋊C411yescharacteristic
117C3×D5—22no—
118C3×D5—22no—
119C3×D5Dic3⋊C412yes—
120C3×D5(C2×D3)⋊C412yes—
121C3×D5C3×C22⋊C412yes—
122C3×D5C3×C4⋊C412yes—
123C23×C3—51no—
124C23×C3—51no—
125C23×C3—101no—
126C2×C3×C4—21no—
127C2×C3×C4—51no—
128C2×C3×C4—51no—
129C2×C3×C4—101no—
130C2×C3×C4—102no—
131C2×C3×C4—102no—
132C2×C3×C4—104no—
133C2×C3×C4—154no—
134C2×C3×C4—302no—
135C2×C3×C4—302no—
136C2×C3×C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
137C2×C3×C4ΓL(1,𝔽16)11yescharacteristic, complemented
138C2×Dic3—54no—
139C2×Dic3—102no—
140C2×Dic3—102no—
141C22×C5C2×C3×Dic311yescharacteristic
142C2×D5C3×Dic611yescharacteristic
143C2×D5C3×C4×D312yes—
144C2×D5C3×D1211yescharacteristic
145C2×D5C3×#6712yes—
146AGL(1,𝔽5)—64no—
147C4×C5—21no—
148Dic5—21no—
149C2×C32—52no—
150C2×C32—52no—
151C2×C32C4×AGL(1,𝔽5)11yescharacteristic
152C2×C32C4⋊AGL(1,𝔽5)11yescharacteristic
153C2×C32C22⋊AGL(1,𝔽5)11yescharacteristic
154C22×C4—51no—
155C22×C4—152no—
156C3×C5—21no—
157C3×C5C3⋊#9511yescharacteristic, complemented
158C3×C5C3×#9511yescharacteristic, complemented
159C22×C3—21no—
160C22×C3—52no—
161C22×C3—51no—
162C22×C3—51no—
163C22×C3—52no—
164C22×C3—52no—
165C22×C3—51no—
166C22×C3—51no—
167C22×C3—52no—
168C22×C3—102no—
169C22×C3—102no—
170C22×C3—101no—
171C22×C3—101no—
172C22×C3C2×C3×AGL(1,𝔽5)11yescharacteristic
173C22×C3C2×ΓL(1,𝔽16)11yescharacteristic
174C3×C4—22no—
175C3×C4—21no—
176C3×C4—21no—
177C3×C4—102no—
178C3×C4—101no—
179C3×C4—101no—
180C3×C4—304no—
181Dic3—104no—
182C2×C5C3×C4×Dic311yescharacteristic
183C2×C5C3×C4⋊Dic311yescharacteristic
184C2×C5C3×#21711yescharacteristic
185D5C3×Dic3⋊C412yes—
186D5C3×(C2×D3)⋊C412yes—
187C32(C2×AGL(1,𝔽5))⋊C411yescharacteristic, complemented, Sylow
188C23—51no—
189C2×C4—51no—
190C2×C4—102no—
191C2×C4—154no—
192C2×C4—302no—
193C2×C4—302no—
194C2×C4C3×ΓL(1,𝔽16)11yescharacteristic, complemented
195C2×C3—21no—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3—52no—
199C2×C3—52no—
200C2×C3—52no—
201C2×C3—52no—
202C2×C3—102no—
203C2×C3—102no—
204C2×C3C3×C4×AGL(1,𝔽5)11yescharacteristic
205C2×C3C3×C4⋊AGL(1,𝔽5)11yescharacteristic
206C2×C3C3×C22⋊AGL(1,𝔽5)11yescharacteristic
207C2×C3C4×ΓL(1,𝔽16)11yescharacteristic
208C2×C3C4⋊ΓL(1,𝔽16)11yescharacteristic
209C2×C3#676311yescharacteristic
210C5C3×C3⋊#9511yescharacteristic, complemented, Sylow
211C22—52no—
212C22—51no—
213C22—51no—
214C22—52no—
215C22C2×C3×ΓL(1,𝔽16)11yescharacteristic
216C4—21no—
217C4—101no—
218C4—304no—
219C3—21no—
220C3C3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic, complemented
221C3#9123511yescharacteristic, direct product factor
222C2—52no—
223C2—52no—
224C2C3×C4×ΓL(1,𝔽16)11yescharacteristic
225C2C3×C4⋊ΓL(1,𝔽16)11yescharacteristic
226C2C3×#676311yescharacteristic
subgroupG/H
centerC22×C3C2×ΓL(1,𝔽16)
commutatorC2×C3×C5C3×C42
FrattiniC22C2×C3×ΓL(1,𝔽16)
FittingC2×C32×C4×C5C4
socleC22×C32×C5C2×C4
prime radicalC22×C32×D5C22
cosocleC22×C3×D5C22×C3
hypercenterC2×C3×C4ΓL(1,𝔽16)
hypocenterC3×C5C3×#95
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
3. #91235C3direct (N × H)
8. C2×C3×ΓL(1,𝔽16)C4semidirect
9. C2×C32×C4×C5C4semidirect
10. C2×C32×Dic5C4semidirect
34. C2×ΓL(1,𝔽16)C3×C4semidirect
39. C2×C3×C4×C5Dic3semidirect
40. C2×C3×C4×C5C3×C4semidirect
42. C2×C3×Dic5Dic3semidirect
43. C2×C3×Dic5C3×C4semidirect
59. C2×C32×C4AGL(1,𝔽5)semidirect
91. C32×C5#95semidirect
96. C2×C4×C5C3×Dic3semidirect
97. C2×Dic5C3×Dic3semidirect
136. C2×C3×C4C3×AGL(1,𝔽5)semidirect
137. C2×C3×C4ΓL(1,𝔽16)semidirect
157. C3×C5C3⋊#95semidirect
158. C3×C5C3×#95semidirect
187. C32(C2×AGL(1,𝔽5))⋊C4semidirect
194. C2×C4C3×ΓL(1,𝔽16)semidirect
210. C5C3×C3⋊#95semidirect
220. C3C3×(C2×AGL(1,𝔽5))⋊C4semidirect
221. C3#91235direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×ΓL(1,𝔽16)2
C2×C32×C4×D52
#912353
C3×(C2×AGL(1,𝔽5))⋊C43
C3×C3⋊#955