Finite Group Databasebeta

← C3×#91818 #13088026

Subgroups — C3×#91818 #13088026

Order 1440.

366 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 110 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×C32×C4×C5C211yesmaximal, characteristic, complemented
2C2×C3×#1220C211yesmaximal, characteristic
3C3×#6719C211yesmaximal, characteristic, complemented
4C3×#6714C212yesmaximal, complemented
5C3×#6712C212yesmaximal, complemented
6#91818C311yesmaximal, characteristic, direct product factor
7C3×#4075—31nomaximal
8C23×C32×C5C2211yescharacteristic
9C22×C3×D15C2211yescharacteristic
10C2×C32×C4×C5—22no—
11C2×C32×C4×C5C2212yes—
12C3×#1220—22no—
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×Dic15C2212yes—
15C3×#929—51nomaximal, Hall
16C22×C3×C4×C5—21no—
17C22×C3×C4×C5D311yescharacteristic, complemented
18C22×C3×C4×C5C2×C311yescharacteristic, complemented
19C2×#1220C2×C311yescharacteristic
20C2×C3×C22⋊D5—31no—
21#6719C2×C311yescharacteristic, complemented
22#6714C2×C312yescomplemented
23#6712C2×C312yescomplemented
24C3×C22⋊Dic5—31no—
25C3×#685—32no—
26C3×#683—32no—
27C22×C32×C5—22no—
28C22×C32×C5D412yes—
29C22×C32×C5C2311yescharacteristic
30C2×C3×D15—21no—
31C2×C3×D15—22no—
32C32×C4×C5—22no—
33C3×Dic15—21no—
34C3×Dic15—22no—
35#4075—31noHall
36C22×C32×C4D511yescharacteristic, complemented
37C2×C3×#67—51no—
38C3×#217—51no—
39C3×(C2×D3)⋊C4—52no—
40C3×Dic3⋊C4—52no—
41C23×C3×C5—21no—
42C23×C3×C5C2×D311yescharacteristic
43C23×C3×C5C22×C311yescharacteristic
44C22×D15C22×C311yescharacteristic
45C22×C3×D5—31no—
46C2×C3×C4×C5—24no—
47C2×C3×C4×C5—22no—
48C2×C3×C4×C5—22no—
49C2×C3×C4×C5—22no—
50C2×C3×C4×C5C2×D312yes—
51C2×C3×C4×C5C22×C312yes—
52#1220—22no—
53C2×Dic15C22×C311yescharacteristic
54C2×Dic15C22×C312yes—
55C3×C22⋊D5—62no—
56C2×C3×Dic5—31no—
57C2×C3×Dic5—32no—
58C3×#123—151no—
59#929—51no—
60C2×C32×C5—22no—
61C2×C32×C5C2×D411yescharacteristic
62C2×C32×C5#4112yes—
63C3×D15—41no—
64C22×C4×C5C3×D311yescharacteristic, complemented
65C2×C22⋊D5—31no—
66C22⋊Dic5—31no—
67#685—32no—
68#683—32no—
69C23×C32C2×D511yescharacteristic
70C22×C3×D3—51no—
71C2×C32×C4—22no—
72C2×C32×C4C2×D512yes—
73C3×#67—102no—
74C2×C3×Dic3—51no—
75C2×C3×Dic3—52no—
76C22×C3×C5—24no—
77C22×C3×C5—22no—
78C22×C3×C5—22no—
79C22×C3×C5—22no—
80C22×C3×C5—21no—
81C22×C3×C5#6712yes—
82C22×C3×C5C3×D412yes—
83C22×C3×C5C22×D311yescharacteristic
84C22×C3×C5C23×C311yescharacteristic
85C2×D15—21no—
86C2×D15—22no—
87C2×C3×D5—62no—
88C2×C3×D5—61no—
89C3×C4×C5—24no—
90C3×C4×C5—22no—
91C3×C4×C5—22no—
92Dic15—21no—
93Dic15—22no—
94C3×Dic5—62no—
95C3×Dic5—61no—
96C2×C3×D4—151no—
97C22×C3×C4—21no—
98C22×C3×C4C3×D511yescharacteristic, complemented
99C22×C3×C4D1511yescharacteristic, complemented
100C2×#67—51no—
101C3×C4⋊C4—152no—
102C3×C22⋊C4—151no—
103C3×C22⋊C4—152no—
104#217—51no—
105(C2×D3)⋊C4—52no—
106Dic3⋊C4—52no—
107C32×C5#12311yescharacteristic, complemented, Hall
108C23×C5C2×C3×D311yescharacteristic
109C22×D5—31no—
110C2×C4×C5—22no—
111C2×C4×C5C2×C3×D312yes—
112C22⋊D5—62no—
113C2×Dic5—31no—
114C2×Dic5—32no—
115C22×C32—22no—
116C22×C32C22⋊D512yes—
117C22×C32C22×D511yescharacteristic
118C2×C3×D3—102no—
119C2×C3×D3—101no—
120C32×C4—22no—
121C3×Dic3—102no—
122C3×Dic3—101no—
123#123—151noSylow
124C2×C3×C5—24no—
125C2×C3×C5—22no—
126C2×C3×C5—22no—
127C2×C3×C5—22no—
128C2×C3×C5—21no—
129C2×C3×C5#23512yes—
130C2×C3×C5C2×#6711yescharacteristic
131C2×C3×C5C2×C3×D411yescharacteristic
132C2×C3×C5C3×#4112yes—
133D15—41no—
134C3×D5—121no—
135C23×C3—21no—
136C23×C3—151no—
137C23×C3C2×C3×D511yescharacteristic
138C23×C3C2×D1511yescharacteristic
139C22×D3—51no—
140C3×D4—302no—
141C2×C3×C4—24no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—152no—
146C2×C3×C4—151no—
147C2×C3×C4C2×C3×D512yes—
148C2×C3×C4C2×D1512yes—
149#67—102no—
150C2×Dic3—51no—
151C2×Dic3—52no—
152C22×C5—22no—
153C22×C5C3×#6712yes—
154C22×C5C22×C3×D311yescharacteristic
155C2×D5—62no—
156C2×D5—61no—
157C4×C5—22no—
158Dic5—62no—
159Dic5—61no—
160C2×C32—22no—
161C2×C32#70912yes—
162C2×C32C2×C22⋊D511yescharacteristic
163C3×D3—201no—
164C2×D4—151no—
165C22×C4C3×D1511yescharacteristic, complemented
166C4⋊C4—152no—
167C22⋊C4—151no—
168C22⋊C4—152no—
169C3×C5—21no—
170C3×C5#92911yescharacteristic, complemented
171C3×C5C3×#12311yescharacteristic, complemented
172C22×C3—24no—
173C22×C3—22no—
174C22×C3—22no—
175C22×C3—22no—
176C22×C3—21no—
177C22×C3—302no—
178C22×C3—301no—
179C22×C3C3×C22⋊D512yes—
180C22×C3#122012yes—
181C22×C3C22×C3×D511yescharacteristic
182C22×C3C22×D1511yescharacteristic
183C2×D3—102no—
184C2×D3—101no—
185C3×C4—24no—
186C3×C4—22no—
187C3×C4—22no—
188C3×C4—302no—
189C3×C4—301no—
190Dic3—102no—
191Dic3—101no—
192C2×C5—22no—
193C2×C5C3×#23512yes—
194C2×C5C2×C3×#6711yescharacteristic
195D5—121no—
196C32#407511yescharacteristic, complemented, Sylow
197C23—151no—
198C23C2×C3×D1511yescharacteristic
199D4—302no—
200C2×C4—22no—
201C2×C4—152no—
202C2×C4—151no—
203C2×C4C2×C3×D1512yes—
204C2×C3—24no—
205C2×C3—22no—
206C2×C3—22no—
207C2×C3—22no—
208C2×C3—21no—
209C2×C3—601no—
210C2×C3C3×#70912yes—
211C2×C3C2×C3×C22⋊D511yescharacteristic
212C2×C3#681712yes—
213C2×C3C2×#122011yescharacteristic
214D3—201no—
215C5C3×#92911yescharacteristic, complemented, Sylow
216C22—22no—
217C22—302no—
218C22—301no—
219C22C3×#122012yes—
220C22C22×C3×D1511yescharacteristic
221C4—22no—
222C4—302no—
223C4—301no—
224C3—21no—
225C3C3×#407511yescharacteristic, complemented
226C3#9181811yescharacteristic, direct product factor
227C2—22no—
228C2—601no—
229C2C3×#681712yes—
230C2C2×C3×#122011yescharacteristic
subgroupG/H
centerC22×C3C22×D15
commutator, cosocleC22×C3×C5C23×C3
FrattiniC22C22×C3×D15
FittingC22×C32×C4×C5C2
socleC22×C32×C5C23
prime radicalC2×C3×#1220C2
hypercenterC22×C3×C4D15
hypocenterC3×C5C3×#123
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C32×C4×C5C2semidirect
3. C3×#6719C2semidirect
4. C3×#6714C2semidirect
5. C3×#6712C2semidirect
6. #91818C3direct (N × H)
17. C22×C3×C4×C5D3semidirect
18. C22×C3×C4×C5C2×C3semidirect
21. #6719C2×C3semidirect
22. #6714C2×C3semidirect
23. #6712C2×C3semidirect
36. C22×C32×C4D5semidirect
64. C22×C4×C5C3×D3semidirect
98. C22×C3×C4C3×D5semidirect
99. C22×C3×C4D15semidirect
107. C32×C5#123semidirect
165. C22×C4C3×D15semidirect
170. C3×C5#929semidirect
171. C3×C5C3×#123semidirect
196. C32#4075semidirect
215. C5C3×#929semidirect
225. C3C3×#4075semidirect
226. C3#91818direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C4×C52
C2×C3×#12202
C3×#67192
C3×#67142
C3×#67122
#918183
C3×#40753
C3×#9295