Finite Group Databasebeta

← C3×#91172 #13085405

Subgroups — C3×#91172 #13085405

Order 1440.

242 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 52 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×#1201C211yesmaximal, characteristic
2C3×#6763C211yesmaximal, characteristic, complemented
3C32×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C3×#4011—31nomaximal
5#91172C311yesmaximal, characteristic, direct product factor
6C22×C3×C5×D3C411yescharacteristic, complemented
7C22×C32×D5C2211yescharacteristic
8C2×C3×ΓL(1,𝔽16)—21no—
9C2×C32×AGL(1,𝔽5)—21no—
10C2×C3×Dic15C411yescharacteristic, complemented
11C3×#1201—21no—
12C3×#1201—21no—
13C3×#806—51nomaximal, Hall
14C2×C3×C22⋊D5—31no—
15C2×#1201C2×C311yescharacteristic
16#6763C2×C311yescharacteristic, complemented
17C3×C22⋊AGL(1,𝔽5)—21no—
18C3×C22⋊AGL(1,𝔽5)—31no—
19C3×C22⋊AGL(1,𝔽5)D311yescharacteristic, complemented
20C3×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
21C22×C32×C5C2×C411yescharacteristic
22C2×C3×C5×D3—21no—
23C2×C3×C5×D3—41no—
24C2×C32×D5—41no—
25C2×C32×D5D411yescharacteristic
26C2×C32×D5D411yescharacteristic
27C3×ΓL(1,𝔽16)—41no—
28C32×AGL(1,𝔽5)—41no—
29C3×Dic15—21no—
30#4011—31noHall
31C2×C3×#67—51no—
32C32×C22⋊C4—51no—
33C3×#217—51no—
34C23×C3×C5—31no—
35C22×C5×D3C3×C411yescharacteristic, complemented
36C22×C3×D5—21no—
37C22×C3×D5C2×D311yescharacteristic
38C22×C3×D5C22×C311yescharacteristic
39C2×ΓL(1,𝔽16)—21no—
40C2×C3×AGL(1,𝔽5)—22no—
41C2×C3×AGL(1,𝔽5)—21no—
42C2×C3×AGL(1,𝔽5)—21no—
43C2×C3×AGL(1,𝔽5)—61no—
44C2×Dic15C3×C411yescharacteristic, complemented
45C3×C22⋊D5—61no—
46C3×C22⋊D5—61no—
47C2×C3×Dic5—31no—
48#1201—21no—
49#1201—21no—
50C3×#99—151no—
51#806—51no—
52C2×C32×C5—21no—
53C2×C32×C5C22⋊C411yescharacteristic
54C3×C5×D3—41no—
55C32×D5—21no—
56C32×D5—21no—
57C2×C22⋊D5—31no—
58C22⋊AGL(1,𝔽5)—31no—
59C22⋊AGL(1,𝔽5)C3×D311yescharacteristic, complemented
60C23×C32—51no—
61C22×C3×D3AGL(1,𝔽5)11yescharacteristic, complemented
62C2×C32×C4—101no—
63C3×#67—101no—
64C3×#67—101no—
65C2×C3×Dic3—51no—
66C2×C3×Dic3—101no—
67C22×C3×C5—21no—
68C22×C3×C5—61no—
69C22×C3×C5—121no—
70C22×C3×C5C4×D311yescharacteristic
71C22×C3×C5C2×C3×C411yescharacteristic
72C2×C5×D3—21no—
73C2×C5×D3—41no—
74C2×C3×D5—21no—
75C2×C3×D5—21no—
76C2×C3×D5—41no—
77C2×C3×D5—41no—
78C2×C3×D5—42no—
79C2×C3×D5D1211yescharacteristic
80C2×C3×D5#6711yescharacteristic
81C2×C3×D5C3×D411yescharacteristic
82C2×C3×D5C3×D411yescharacteristic
83ΓL(1,𝔽16)—41no—
84C3×AGL(1,𝔽5)—42no—
85C3×AGL(1,𝔽5)—41no—
86C3×AGL(1,𝔽5)—41no—
87C3×AGL(1,𝔽5)—121no—
88Dic15—21no—
89C3×Dic5—61no—
90C2×C3×D4—151no—
91C2×#67—51no—
92C3×C22⋊C4—51no—
93C3×C22⋊C4—51no—
94C3×C22⋊C4—101no—
95C3×C22⋊C4—151no—
96#217—51no—
97C32×C5#9911yescharacteristic, complemented, Hall
98C23×C5—31no—
99C22×D5C2×C3×D311yescharacteristic
100C2×AGL(1,𝔽5)—21no—
101C2×AGL(1,𝔽5)—61no—
102C22⋊D5—61no—
103C22⋊D5—61no—
104C2×Dic5—31no—
105C22×C32—51no—
106C22×C32—51no—
107C22×C32—201no—
108C22×C32C2×AGL(1,𝔽5)11yescharacteristic
109C2×C3×D3—21no—
110C2×C3×D3—41no—
111C32×C4—201no—
112C3×Dic3—101no—
113C3×Dic3—201no—
114#99—151noSylow
115C2×C3×C5—21no—
116C2×C3×C5—21no—
117C2×C3×C5—21no—
118C2×C3×C5—41no—
119C2×C3×C5—121no—
120C2×C3×C5(C2×D3)⋊C411yescharacteristic
121C2×C3×C5C3×C22⋊C411yescharacteristic
122C3×D5—22no—
123C3×D5—21no—
124C3×D5—21no—
125C3×D5—21no—
126C3×D5—21no—
127C3×D5—41no—
128C5×D3—41no—
129C23×C3—31no—
130C23×C3—51no—
131C23×C3—51no—
132C23×C3—101no—
133C22×D3C3×AGL(1,𝔽5)11yescharacteristic, complemented
134C3×D4—301no—
135C3×D4—301no—
136C2×C3×C4—101no—
137C2×C3×C4—101no—
138C2×C3×C4—102no—
139C2×C3×C4—151no—
140C2×C3×C4—301no—
141#67—101no—
142#67—101no—
143C2×Dic3—51no—
144C2×Dic3—101no—
145C22×C5—61no—
146C22×C5—121no—
147C22×C5C3×C4×D311yescharacteristic
148C2×D5—41no—
149C2×D5C3×D1211yescharacteristic
150C2×D5C3×#6711yescharacteristic
151AGL(1,𝔽5)—41no—
152AGL(1,𝔽5)—121no—
153Dic5—61no—
154C2×C32—21no—
155C2×C32—101no—
156C2×C32—101no—
157C2×C32C22⋊AGL(1,𝔽5)11yescharacteristic
158C3×D3—41no—
159C2×D4—151no—
160C22⋊C4—51no—
161C22⋊C4—151no—
162C3×C5—21no—
163C3×C5#80611yescharacteristic, complemented
164C3×C5C3×#9911yescharacteristic, complemented
165C22×C3—21no—
166C22×C3—51no—
167C22×C3—51no—
168C22×C3—51no—
169C22×C3—51no—
170C22×C3—61no—
171C22×C3—101no—
172C22×C3—101no—
173C22×C3—121no—
174C22×C3—201no—
175C22×C3—201no—
176C22×C3—202no—
177C22×C3D3×AGL(1,𝔽5)11yescharacteristic
178C22×C3C2×C3×AGL(1,𝔽5)11yescharacteristic
179C2×D3—21no—
180C2×D3—41no—
181C3×C4—202no—
182C3×C4—201no—
183C3×C4—201no—
184C3×C4—301no—
185C3×C4—601no—
186Dic3—101no—
187Dic3—201no—
188C2×C5—21no—
189C2×C5—121no—
190C2×C5C3×(C2×D3)⋊C411yescharacteristic
191D5—21no—
192D5—21no—
193C32#401111yescharacteristic, complemented, Sylow
194C23—31no—
195C23—51no—
196D4—301no—
197D4—301no—
198C2×C4—101no—
199C2×C4—151no—
200C2×C4—301no—
201C2×C3—21no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—41no—
205C2×C3—102no—
206C2×C3—101no—
207C2×C3—101no—
208C2×C3—101no—
209C2×C3—101no—
210C2×C3—121no—
211C2×C3—201no—
212C2×C3(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
213C2×C3C3×C22⋊AGL(1,𝔽5)11yescharacteristic
214D3—41no—
215C5C3×#80611yescharacteristic, complemented, Sylow
216C22—51no—
217C22—51no—
218C22—61no—
219C22—121no—
220C22—201no—
221C22C3×D3×AGL(1,𝔽5)11yescharacteristic
222C4—201no—
223C4—301no—
224C4—601no—
225C3—21no—
226C3#9117211yescharacteristic, direct product factor
227C3C3×#401111yescharacteristic, complemented
228C2—21no—
229C2—101no—
230C2—101no—
231C2—121no—
232C2C3×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3(C2×D3)⋊AGL(1,𝔽5)
commutatorC22×C3×C5C2×C3×C4
FrattiniC22C3×D3×AGL(1,𝔽5)
FittingC22×C32×C5C2×C4
socleC2×C32×C5C22⋊C4
prime radicalC2×C3×#1201C2
cosocleC22×C3×D5C22×C3
hypercenterC22×C3D3×AGL(1,𝔽5)
hypocenterC3×C5C3×#99
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×#6763C2semidirect
3. C32×C22⋊AGL(1,𝔽5)C2semidirect
5. #91172C3direct (N × H)
6. C22×C3×C5×D3C4semidirect
10. C2×C3×Dic15C4semidirect
16. #6763C2×C3semidirect
19. C3×C22⋊AGL(1,𝔽5)D3semidirect
20. C3×C22⋊AGL(1,𝔽5)C2×C3semidirect
35. C22×C5×D3C3×C4semidirect
44. C2×Dic15C3×C4semidirect
59. C22⋊AGL(1,𝔽5)C3×D3semidirect
61. C22×C3×D3AGL(1,𝔽5)semidirect
97. C32×C5#99semidirect
133. C22×D3C3×AGL(1,𝔽5)semidirect
163. C3×C5#806semidirect
164. C3×C5C3×#99semidirect
193. C32#4011semidirect
215. C5C3×#806semidirect
226. C3#91172direct (N × H)
227. C3C3×#4011semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#12012
C3×#67632
C32×C22⋊AGL(1,𝔽5)2
C3×#40113
#911723
C3×#8065