Finite Group Databasebeta

← C3×#91776 #13087984

Subgroups — C3×#91776 #13087984

Order 1440.

306 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 130 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×Dic30C211yesmaximal, characteristic
2C32×C5×C4⋊C4C211yesmaximal, characteristic
3C3×#6712C212yesmaximal
4C3×C4×Dic15C211yesmaximal, characteristic
5C3×C4×Dic15C212yesmaximal
6#91776C311yesmaximal, characteristic, direct product factor
7C3×#4033—31nomaximal
8C2×C32×C4×C5C2212yes—
9C2×C32×C4×C5C2211yescharacteristic
10C2×C3×Dic15C2211yescharacteristic
11C2×C3×Dic15C2212yes—
12C2×C3×Dic15C2211yescharacteristic
13C3×Dic30C414yescomplemented
14C3×#887—51nomaximal, Hall
15C2×Dic30C2×C311yescharacteristic
16C2×C3×Dic10—31no—
17C3×C5×C4⋊C4—21no—
18C3×C5×C4⋊C4D311yescharacteristic
19C3×C5×C4⋊C4C2×C311yescharacteristic
20#6712C2×C312yes—
21C4×Dic15C2×C311yescharacteristic
22C4×Dic15C2×C312yes—
23C3×#683—32no—
24C3×C4×Dic5—32no—
25C3×C4×Dic5—31no—
26C22×C32×C5C2311yescharacteristic
27C32×C4×C5—22no—
28C32×C4×C5C2×C412yes—
29C3×Dic15—21no—
30C3×Dic15C2×C414yes—
31C3×Dic15Q812yescomplemented
32#4033—31noHall
33C2×C3×Dic6—51no—
34C32×C4⋊C4D511yescharacteristic
35C3×Dic3⋊C4—52no—
36C3×C4×Dic3—52no—
37C3×C4×Dic3—51no—
38C2×C3×C4×C5—21no—
39C2×C3×C4×C5—22no—
40C2×C3×C4×C5C2×D312yes—
41C2×C3×C4×C5C2×D311yescharacteristic
42C2×C3×C4×C5C22×C312yes—
43C2×C3×C4×C5C22×C311yescharacteristic
44C2×Dic15C22×C311yescharacteristic
45C2×Dic15C22×C312yes—
46C2×Dic15C22×C311yescharacteristic
47Dic30C3×C414yescomplemented
48C2×C3×Dic5—31no—
49C2×C3×Dic5—32no—
50C2×C3×Dic5—31no—
51C3×Dic10—34no—
52C3×C4×Q8—151no—
53#887—51no—
54C2×C32×C5C22×C411yescharacteristic
55C2×C32×C5C2×Q811yescharacteristic
56C2×C32×C5#4111yescharacteristic
57C2×Dic10—31no—
58C5×C4⋊C4C3×D311yescharacteristic
59#683—32no—
60C4×Dic5—32no—
61C4×Dic5—31no—
62C2×C32×C4C2×D512yes—
63C2×C32×C4C2×D511yescharacteristic
64C2×C3×Dic3—51no—
65C2×C3×Dic3—52no—
66C2×C3×Dic3—51no—
67C3×Dic6—54no—
68C22×C3×C5—21no—
69C22×C3×C5C22×D311yescharacteristic
70C22×C3×C5C23×C311yescharacteristic
71C3×C4×C5—22no—
72C3×C4×C5—22no—
73C3×C4×C5—22no—
74C3×C4×C5—42no—
75C3×C4×C5C4×D312yes—
76C3×C4×C5C2×C3×C412yes—
77Dic15—21no—
78Dic15C2×C3×C414yes—
79Dic15C3×Q812yescomplemented
80C3×Dic5—34no—
81C3×Dic5—32no—
82C3×Dic5—61no—
83C2×C3×Q8—151no—
84C2×Dic6—51no—
85C3×C4⋊C4—21no—
86C3×C4⋊C4—152no—
87C3×C4⋊C4C3×D511yescharacteristic
88C3×C4⋊C4D1511yescharacteristic
89C3×C42—152no—
90C3×C42—151no—
91Dic3⋊C4—52no—
92C4×Dic3—52no—
93C4×Dic3—51no—
94C32×C5C4×Q811yescharacteristic, complemented, Hall
95C2×C4×C5C2×C3×D312yes—
96C2×C4×C5C2×C3×D311yescharacteristic
97C2×Dic5—31no—
98C2×Dic5—32no—
99C2×Dic5—31no—
100Dic10—34no—
101C22×C32C22×D511yescharacteristic
102C32×C4—22no—
103C32×C4C4×D512yes—
104C3×Dic3—54no—
105C3×Dic3—52no—
106C3×Dic3—101no—
107C4×Q8—151noSylow
108C2×C3×C5—21no—
109C2×C3×C5—21no—
110C2×C3×C5—21no—
111C2×C3×C5C2×C4×D311yescharacteristic
112C2×C3×C5#23711yescharacteristic
113C2×C3×C5D3×Q811yescharacteristic
114C2×C3×C5C22×C3×C411yescharacteristic
115C2×C3×C5C2×C3×Q811yescharacteristic
116C2×C3×C5C3×#4111yescharacteristic
117C3×Q8—154no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
120C2×C3×C4—151no—
121C2×C3×C4—152no—
122C2×C3×C4—151no—
123C2×C3×C4C2×C3×D512yes—
124C2×C3×C4C2×C3×D511yescharacteristic
125C2×C3×C4C2×D1512yes—
126C2×C3×C4C2×D1511yescharacteristic
127C2×Dic3—51no—
128C2×Dic3—52no—
129C2×Dic3—51no—
130Dic6—54no—
131C22×C5C22×C3×D311yescharacteristic
132C4×C5—22no—
133C4×C5C3×C4×D312yes—
134Dic5—34no—
135Dic5—32no—
136Dic5—61no—
137C2×C32C2×C4×D511yescharacteristic
138C2×C32#71111yescharacteristic
139C2×C32Q8×D511yescharacteristic
140C2×Q8—151no—
141C4⋊C4—152no—
142C4⋊C4C3×D1511yescharacteristic
143C42—152no—
144C42—151no—
145C3×C5—21no—
146C3×C5#88711yescharacteristic, complemented
147C3×C5C3×C4×Q811yescharacteristic, complemented
148C22×C3—21no—
149C22×C3C22×C3×D511yescharacteristic
150C22×C3C22×D1511yescharacteristic
151C3×C4—22no—
152C3×C4—22no—
153C3×C4—22no—
154C3×C4—42no—
155C3×C4—154no—
156C3×C4—152no—
157C3×C4—301no—
158C3×C4C3×C4×D512yes—
159C3×C4C4×D1512yes—
160Dic3—54no—
161Dic3—52no—
162Dic3—101no—
163C2×C5C2×C3×C4×D311yescharacteristic
164C2×C5C3×#23711yescharacteristic
165C2×C5C3×D3×Q811yescharacteristic
166C32#403311yescharacteristic, complemented, Sylow
167Q8—154no—
168C2×C4—151no—
169C2×C4—152no—
170C2×C4—151no—
171C2×C4C2×C3×D1512yes—
172C2×C4C2×C3×D1511yescharacteristic
173C2×C3—21no—
174C2×C3—21no—
175C2×C3—21no—
176C2×C3C2×C3×C4×D511yescharacteristic
177C2×C3C3×#71111yescharacteristic
178C2×C3C3×Q8×D511yescharacteristic
179C2×C3C2×C4×D1511yescharacteristic
180C2×C3#681911yescharacteristic
181C2×C3Q8×D1511yescharacteristic
182C5C3×#88711yescharacteristic, complemented, Sylow
183C22C22×C3×D1511yescharacteristic
184C4—22no—
185C4—154no—
186C4—152no—
187C4—301no—
188C4C3×C4×D1512yes—
189C3—21no—
190C3C3×#403311yescharacteristic, complemented
191C3#9177611yescharacteristic, direct product factor
192C2C2×C3×C4×D1511yescharacteristic
193C2C3×#681911yescharacteristic
194C2C3×Q8×D1511yescharacteristic
subgroupG/H
centerC22×C3C22×D15
commutatorC2×C3×C5C22×C3×C4
FrattiniC22C22×C3×D15
FittingC32×C5×C4⋊C4C2
socle, prime radicalC22×C32×C5C23
cosocleC22×C3×C5C23×C3
hypercenterC3×C4⋊C4D15
hypocenterC3×C5C3×C4×Q8
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #91776C3direct (N × H)
13. C3×Dic30C4semidirect
31. C3×Dic15Q8semidirect
47. Dic30C3×C4semidirect
79. Dic15C3×Q8semidirect
94. C32×C5C4×Q8semidirect
146. C3×C5#887semidirect
147. C3×C5C3×C4×Q8semidirect
166. C32#4033semidirect
182. C5C3×#887semidirect
190. C3C3×#4033semidirect
191. C3#91776direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Dic302
C32×C5×C4⋊C42
C3×#67122
C3×C4×Dic152
#917763
C3×#40333
C3×#8875