Finite Group Databasebeta

← C3×#91228 #13086276

Subgroups — C3×#91228 #13086276

Order 1440.

262 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 58 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×C32×D20C211yesmaximal, characteristic
2C3×#6763C212yesmaximal, complemented
3#91228C311yesmaximal, characteristic, direct product factor
4C3×#3999—31nomaximal
5C22×C32×D5C411yescharacteristic, complemented
6C22×C32×D5C2211yescharacteristic
7C2×C3×ΓL(1,𝔽16)—22no—
8C2×C32×C4×C5C411yescharacteristic, complemented
9C32×D20—22no—
10C3×#834—51nomaximal, Hall
11C2×C3×D20—21no—
12C2×C3×D20D311yescharacteristic
13C2×C3×D20C2×C311yescharacteristic
14#6763C2×C312yescomplemented
15C3×C22⋊AGL(1,𝔽5)—32no—
16C22×C32×C5C2×C411yescharacteristic
17C2×C32×D5—21no—
18C2×C32×D5—41no—
19C2×C32×D5—41no—
20C2×C32×D5D412yes—
21C3×ΓL(1,𝔽16)—42no—
22C32×C4×C5—21no—
23#3999—31noHall
24C2×C32×D4—51no—
25C3×#217—52no—
26C22×C3×D5—21no—
27C22×C3×D5—21no—
28C22×C3×D5Dic311yescharacteristic, complemented
29C22×C3×D5C3×C411yescharacteristic, complemented
30C22×C3×D5C2×D311yescharacteristic
31C22×C3×D5C22×C311yescharacteristic
32C2×ΓL(1,𝔽16)—22no—
33C2×C3×AGL(1,𝔽5)—62no—
34C2×C3×C4×C5—21no—
35C2×C3×C4×C5Dic311yescharacteristic, complemented
36C2×C3×C4×C5C3×C411yescharacteristic, complemented
37C3×D20—24no—
38C3×D20—22no—
39C3×D20—22no—
40C3×#99—151no—
41#834—51no—
42C2×C32×C5—21no—
43C2×C32×C5C22⋊C411yescharacteristic
44C32×D5—22no—
45C32×D5—41no—
46C2×D20C3×D311yescharacteristic
47C22⋊AGL(1,𝔽5)—32no—
48C23×C32—51no—
49C23×C32—51no—
50C32×D4—102no—
51C2×C32×C4AGL(1,𝔽5)11yescharacteristic, complemented
52C2×C3×Dic3—102no—
53C22×C3×C5—21no—
54C22×C3×C5C2×Dic311yescharacteristic
55C22×C3×C5C2×C3×C411yescharacteristic
56C2×C3×D5—22no—
57C2×C3×D5—21no—
58C2×C3×D5—21no—
59C2×C3×D5—22no—
60C2×C3×D5—41no—
61C2×C3×D5—41no—
62C2×C3×D5—42no—
63C2×C3×D5—42no—
64C2×C3×D5—41no—
65C2×C3×D5—41no—
66C2×C3×D5#6712yes—
67C2×C3×D5C3×D412yes—
68ΓL(1,𝔽16)—42no—
69C3×AGL(1,𝔽5)—122no—
70C3×C4×C5—22no—
71C3×C4×C5—21no—
72C3×C4×C5—21no—
73C2×C3×D4—51no—
74C2×C3×D4—51no—
75C2×C3×D4—101no—
76C3×C22⋊C4—152no—
77#217—52no—
78C32×C5#9911yescharacteristic, complemented, Hall
79C22×D5C3×Dic311yescharacteristic, complemented
80C22×D5C2×C3×D311yescharacteristic
81C2×AGL(1,𝔽5)—62no—
82C2×C4×C5C3×Dic311yescharacteristic, complemented
83D20—22no—
84C22×C32—52no—
85C22×C32—101no—
86C22×C32—201no—
87C22×C32—201no—
88C22×C32C2×AGL(1,𝔽5)11yescharacteristic
89C32×C4—21no—
90C3×Dic3—202no—
91#99—151noSylow
92C2×C3×C5—22no—
93C2×C3×C5—21no—
94C2×C3×C5—21no—
95C2×C3×C5—21no—
96C2×C3×C5#21711yescharacteristic
97C2×C3×C5C3×C22⋊C411yescharacteristic
98C3×D5—22no—
99C3×D5—22no—
100C3×D5—42no—
101C3×D5—42no—
102C3×D5—41no—
103C3×D5—41no—
104C23×C3—51no—
105C23×C3—51no—
106C23×C3—51no—
107C23×C3—51no—
108C23×C3—101no—
109C23×C3—101no—
110C3×D4—102no—
111C3×D4—102no—
112C3×D4—104no—
113C2×C3×C4—21no—
114C2×C3×C4—302no—
115C2×C3×C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
116C2×C3×C4ΓL(1,𝔽16)11yescharacteristic, complemented
117C2×Dic3—102no—
118C22×C5C2×C3×Dic311yescharacteristic
119C2×D5—21no—
120C2×D5—41no—
121C2×D5—41no—
122C2×D5C3×#6712yes—
123AGL(1,𝔽5)—122no—
124C4×C5—21no—
125C2×C32—21no—
126C2×C32—102no—
127C2×C32—201no—
128C2×C32C22⋊AGL(1,𝔽5)11yescharacteristic
129C2×D4—51no—
130C22⋊C4—152no—
131C3×C5—21no—
132C3×C5#83411yescharacteristic, complemented
133C3×C5C3×#9911yescharacteristic, complemented
134C22×C3—21no—
135C22×C3—52no—
136C22×C3—52no—
137C22×C3—102no—
138C22×C3—102no—
139C22×C3—101no—
140C22×C3—101no—
141C22×C3—201no—
142C22×C3—201no—
143C22×C3—201no—
144C22×C3—201no—
145C22×C3—202no—
146C22×C3—202no—
147C22×C3C2×C3×AGL(1,𝔽5)11yescharacteristic
148C22×C3C2×ΓL(1,𝔽16)11yescharacteristic
149C3×C4—22no—
150C3×C4—21no—
151C3×C4—21no—
152C3×C4—602no—
153Dic3—202no—
154C2×C5—21no—
155C2×C5C3×#21711yescharacteristic
156D5—22no—
157D5—41no—
158C32#399911yescharacteristic, complemented, Sylow
159C23—51no—
160C23—51no—
161D4—102no—
162C2×C4—302no—
163C2×C4C3×ΓL(1,𝔽16)11yescharacteristic, complemented
164C2×C3—22no—
165C2×C3—21no—
166C2×C3—21no—
167C2×C3—21no—
168C2×C3—102no—
169C2×C3—102no—
170C2×C3—202no—
171C2×C3—202no—
172C2×C3—201no—
173C2×C3—201no—
174C2×C3C3×C22⋊AGL(1,𝔽5)11yescharacteristic
175C2×C3#676311yescharacteristic
176C5C3×#83411yescharacteristic, complemented, Sylow
177C22—52no—
178C22—101no—
179C22—201no—
180C22—201no—
181C22C2×C3×ΓL(1,𝔽16)11yescharacteristic
182C4—21no—
183C4—602no—
184C3—21no—
185C3C3×#399911yescharacteristic, complemented
186C3#9122811yescharacteristic, direct product factor
187C2—21no—
188C2—102no—
189C2—201no—
190C2C3×#676311yescharacteristic
subgroupG/H
centerC2×C3#6763
commutatorC22×C3×C5C2×C3×C4
FrattiniC22C2×C3×ΓL(1,𝔽16)
FittingC2×C32×C4×C5C4
socleC2×C32×C5C22⋊C4
prime radicalC2×C32×D20C2
cosocleC22×C3×D5C22×C3
hypercenterC2×C3×C4ΓL(1,𝔽16)
hypocenterC3×C5C3×#99
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#6763C2semidirect
3. #91228C3direct (N × H)
5. C22×C32×D5C4semidirect
8. C2×C32×C4×C5C4semidirect
14. #6763C2×C3semidirect
28. C22×C3×D5Dic3semidirect
29. C22×C3×D5C3×C4semidirect
35. C2×C3×C4×C5Dic3semidirect
36. C2×C3×C4×C5C3×C4semidirect
51. C2×C32×C4AGL(1,𝔽5)semidirect
78. C32×C5#99semidirect
79. C22×D5C3×Dic3semidirect
82. C2×C4×C5C3×Dic3semidirect
115. C2×C3×C4C3×AGL(1,𝔽5)semidirect
116. C2×C3×C4ΓL(1,𝔽16)semidirect
132. C3×C5#834semidirect
133. C3×C5C3×#99semidirect
158. C32#3999semidirect
163. C2×C4C3×ΓL(1,𝔽16)semidirect
176. C5C3×#834semidirect
185. C3C3×#3999semidirect
186. C3#91228direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×D202
C3×#67632
#912283
C3×#39993
C3×#8345