Finite Group Databasebeta

← C3×#91786 #13087994

Subgroups — C3×#91786 #13087994

Order 1440.

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

normal subgroups only

This group has 322 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
1C2×C3×C4×D15C211yesmaximal, characteristic
2C2×C3×Dic30C211yesmaximal, characteristic, complemented
3C32×C5×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#6714C212yesmaximal
5C3×C4⋊Dic15C212yesmaximal, complemented
6#91786C311yesmaximal, characteristic, direct product factor
7C3×#4043—31nomaximal
8C22×C3×D15C2211yescharacteristic
9C2×C32×C4×C5C2212yes—
10C2×C32×C4×C5C2211yescharacteristic
11C2×C3×Dic15C2212yes—
12C2×C3×Dic15C2211yescharacteristic
13C3×C4×D15—22no—
14C3×Dic30—22no—
15C3×#897—51nomaximal, Hall
16C2×C4×D15C2×C311yescharacteristic
17C2×Dic30C2×C311yescharacteristic, complemented
18C2×C3×C4×D5—31no—
19C2×C3×Dic10—31no—
20C3×C5×C4⋊C4—21no—
21C3×C5×C4⋊C4D311yescharacteristic, complemented
22C3×C5×C4⋊C4C2×C311yescharacteristic, complemented
23#6714C2×C312yes—
24C4⋊Dic15C2×C312yescomplemented
25C3×#685—32no—
26C3×C4⋊Dic5—32no—
27C22×C32×C5C2311yescharacteristic
28C2×C3×D15—21no—
29C2×C3×D15—21no—
30C2×C3×D15Q812yescomplemented
31C32×C4×C5—22no—
32C32×C4×C5D412yes—
33C3×Dic15—22no—
34C3×Dic15—21no—
35#4043—31noHall
36C2×C3×C4×D3—51no—
37C2×C3×Dic6—51no—
38C32×C4⋊C4D511yescharacteristic, complemented
39C3×(C2×D3)⋊C4—52no—
40C3×C4⋊Dic3—52no—
41C22×D15C22×C311yescharacteristic
42C22×C3×D5—31no—
43C2×C3×C4×C5—21no—
44C2×C3×C4×C5—22no—
45C2×C3×C4×C5C2×D312yes—
46C2×C3×C4×C5C2×D311yescharacteristic
47C2×C3×C4×C5C22×C312yes—
48C2×C3×C4×C5C22×C311yescharacteristic
49C2×Dic15C22×C311yescharacteristic
50C2×Dic15C22×C312yes—
51C4×D15—22no—
52Dic30—22no—
53C2×C3×Dic5—31no—
54C2×C3×Dic5—32no—
55C3×C4×D5—62no—
56C3×Dic10—62no—
57C3×C22⋊Q8—151no—
58#897—51no—
59C2×C32×C5C2×D411yescharacteristic
60C2×C32×C5C2×Q811yescharacteristic
61C2×C32×C5#4111yescharacteristic
62C3×D15—22no—
63C2×C4×D5—31no—
64C2×Dic10—31no—
65C5×C4⋊C4C3×D311yescharacteristic, complemented
66#685—32no—
67C4⋊Dic5—32no—
68C22×C3×D3—51no—
69C2×C32×C4C2×D512yes—
70C2×C32×C4C2×D511yescharacteristic
71C2×C3×Dic3—51no—
72C2×C3×Dic3—52no—
73C3×C4×D3—102no—
74C3×Dic6—102no—
75C22×C3×C5—21no—
76C22×C3×C5C22×D311yescharacteristic
77C22×C3×C5C23×C311yescharacteristic
78C2×D15—21no—
79C2×D15—21no—
80C2×D15C3×Q812yescomplemented
81C2×C3×D5—32no—
82C2×C3×D5—61no—
83C2×C3×D5—61no—
84C3×C4×C5—22no—
85C3×C4×C5—22no—
86C3×C4×C5—22no—
87C3×C4×C5—42no—
88C3×C4×C5D1212yes—
89C3×C4×C5C3×D412yes—
90Dic15—22no—
91Dic15—21no—
92C3×Dic5—62no—
93C3×Dic5—61no—
94C2×C3×Q8—151no—
95C22×C3×C4—151no—
96C2×C4×D3—51no—
97C2×Dic6—51no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4—152no—
100C3×C4⋊C4C3×D511yescharacteristic, complemented
101C3×C4⋊C4D1511yescharacteristic, complemented
102C3×C22⋊C4—152no—
103(C2×D3)⋊C4—52no—
104C4⋊Dic3—52no—
105C32×C5C22⋊Q811yescharacteristic, complemented, Hall
106C22×D5—31no—
107C2×C4×C5C2×C3×D312yes—
108C2×C4×C5C2×C3×D311yescharacteristic
109C2×Dic5—31no—
110C2×Dic5—32no—
111C4×D5—62no—
112Dic10—62no—
113C22×C32C22×D511yescharacteristic
114C2×C3×D3—52no—
115C2×C3×D3—101no—
116C2×C3×D3—101no—
117C32×C4—22no—
118C32×C4D2012yes—
119C3×Dic3—102no—
120C3×Dic3—101no—
121C22⋊Q8—151noSylow
122C2×C3×C5—21no—
123C2×C3×C5—21no—
124C2×C3×C5—21no—
125C2×C3×C5C2×D1211yescharacteristic
126C2×C3×C5#23711yescharacteristic
127C2×C3×C5D3×Q811yescharacteristic
128C2×C3×C5C2×C3×D411yescharacteristic
129C2×C3×C5C2×C3×Q811yescharacteristic
130C2×C3×C5C3×#4111yescharacteristic
131D15—22no—
132C3×D5—62no—
133C23×C3—151no—
134C22×D3—51no—
135C3×Q8—302no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—152no—
139C2×C3×C4—151no—
140C2×C3×C4—302no—
141C2×C3×C4C2×C3×D512yes—
142C2×C3×C4C2×C3×D511yescharacteristic
143C2×C3×C4C2×D1512yes—
144C2×C3×C4C2×D1511yescharacteristic
145C2×Dic3—51no—
146C2×Dic3—52no—
147C4×D3—102no—
148Dic6—102no—
149C22×C5C22×C3×D311yescharacteristic
150C2×D5—32no—
151C2×D5—61no—
152C2×D5—61no—
153C4×C5—22no—
154C4×C5C3×D1212yes—
155Dic5—62no—
156Dic5—61no—
157C2×C32C2×D2011yescharacteristic
158C2×C32#71111yescharacteristic
159C2×C32Q8×D511yescharacteristic
160C3×D3—102no—
161C2×Q8—151no—
162C22×C4—151no—
163C4⋊C4—152no—
164C4⋊C4C3×D1511yescharacteristic, complemented
165C22⋊C4—152no—
166C3×C5—21no—
167C3×C5#89711yescharacteristic, complemented
168C3×C5C3×C22⋊Q811yescharacteristic, complemented
169C22×C3—21no—
170C22×C3—152no—
171C22×C3—301no—
172C22×C3—301no—
173C22×C3C22×C3×D511yescharacteristic
174C22×C3C22×D1511yescharacteristic
175C2×D3—52no—
176C2×D3—101no—
177C2×D3—101no—
178C3×C4—22no—
179C3×C4—22no—
180C3×C4—22no—
181C3×C4—42no—
182C3×C4—301no—
183C3×C4—302no—
184C3×C4C3×D2012yes—
185C3×C4D6012yes—
186Dic3—102no—
187Dic3—101no—
188C2×C5C2×C3×D1211yescharacteristic
189C2×C5C3×#23711yescharacteristic
190C2×C5C3×D3×Q811yescharacteristic
191D5—62no—
192C32#404311yescharacteristic, complemented, Sylow
193C23—151no—
194Q8—302no—
195C2×C4—152no—
196C2×C4—151no—
197C2×C4—302no—
198C2×C4C2×C3×D1512yes—
199C2×C4C2×C3×D1511yescharacteristic
200C2×C3—21no—
201C2×C3—21no—
202C2×C3—21no—
203C2×C3—302no—
204C2×C3C2×C3×D2011yescharacteristic
205C2×C3C3×#71111yescharacteristic
206C2×C3C3×Q8×D511yescharacteristic
207C2×C3C2×D6011yescharacteristic
208C2×C3#681911yescharacteristic
209C2×C3Q8×D1511yescharacteristic
210D3—102no—
211C5C3×#89711yescharacteristic, complemented, Sylow
212C22—152no—
213C22—301no—
214C22—301no—
215C22C22×C3×D1511yescharacteristic
216C4—22no—
217C4—301no—
218C4—302no—
219C4C3×D6012yes—
220C3—21no—
221C3C3×#404311yescharacteristic, complemented
222C3#9178611yescharacteristic, direct product factor
223C2—302no—
224C2C2×C3×D6011yescharacteristic
225C2C3×#681911yescharacteristic
226C2C3×Q8×D1511yescharacteristic
subgroupG/H
centerC22×C3C22×D15
commutator, cosocleC22×C3×C5C23×C3
FrattiniC22C22×C3×D15
FittingC32×C5×C4⋊C4C2
socleC22×C32×C5C23
prime radicalC22×C3×D15C22
hypercenterC3×C4⋊C4D15
hypocenterC3×C5C3×C22⋊Q8
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic30C2semidirect
3. C32×C5×C4⋊C4C2semidirect
5. C3×C4⋊Dic15C2semidirect
6. #91786C3direct (N × H)
17. C2×Dic30C2×C3semidirect
21. C3×C5×C4⋊C4D3semidirect
22. C3×C5×C4⋊C4C2×C3semidirect
24. C4⋊Dic15C2×C3semidirect
30. C2×C3×D15Q8semidirect
38. C32×C4⋊C4D5semidirect
65. C5×C4⋊C4C3×D3semidirect
80. C2×D15C3×Q8semidirect
100. C3×C4⋊C4C3×D5semidirect
101. C3×C4⋊C4D15semidirect
105. C32×C5C22⋊Q8semidirect
164. C4⋊C4C3×D15semidirect
167. C3×C5#897semidirect
168. C3×C5C3×C22⋊Q8semidirect
192. C32#4043semidirect
211. C5C3×#897semidirect
221. C3C3×#4043semidirect
222. C3#91786direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D152
C2×C3×Dic302
C32×C5×C4⋊C42
C3×#67142
C3×C4⋊Dic152
#917863
C3×#40433
C3×#8975