Finite Group Databasebeta

← C3×#65782 #11752315

Subgroups — C3×#65782 #11752315

Order 960.

300 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 100 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×Q8×D5C211yesmaximal, characteristic
2C3×#4130C211yesmaximal, characteristic
3C3×D5⋊M16C211yesmaximal, characteristic
4C3×#4009C214yesmaximal, complemented
5#65782C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×C5×Q8C411yescharacteristic, complemented
7C3×Q8×D5—21no—
8C3×Q8×D5—21no—
9C3×Q8×D5C412yescomplemented
10C3×Q8×D5C2212yes—
11C2×C3×C4×D5—21no—
12C2×C3×C4×D5C2211yescharacteristic
13C2×C3×Dic10—21no—
14C2×C3×Dic10C411yescharacteristic, complemented
15C3×C22⋊AGL(1,𝔽5)—21no—
16C3×#704—21no—
17C3×C4⋊AGL(1,𝔽5)C2212yes—
18C3×C4×AGL(1,𝔽5)—21no—
19C3×#700—21no—
20C3×D5⋊C8C2212yes—
21C3×#419—51nomaximal, Hall
22C2×Q8×D5C2×C311yescharacteristic
23#4130C2×C311yescharacteristic
24D5⋊M16C2×C311yescharacteristic
25#4009C2×C314yescomplemented
26C22×C3×D5D411yescharacteristic
27C2×C3×AGL(1,𝔽5)—22no—
28C3×C5×Q8—21no—
29C3×C5×Q8C2×C412yes—
30C2×C3×C4×C5—21no—
31C2×C3×C4×C5C2×C411yescharacteristic
32C2×C3×Dic5—21no—
33C2×C3×Dic5D411yescharacteristic
34C3×C4×D5—22no—
35C3×C4×D5—22no—
36C3×C4×D5C2×C411yescharacteristic
37C3×C4×D5D411yescharacteristic
38C3×C4×D5D411yescharacteristic
39C3×C4×D5C2311yescharacteristic
40C3×Dic10—21no—
41C3×Dic10—22no—
42C3×Dic10—22no—
43C3×Dic10C2×C412yes—
44C3×#170—22no—
45C22×C3×Q8—51no—
46C2×C3×M16—51no—
47C3×#117—51no—
48C3×#103—54no—
49C2×C5×Q8C3×C411yescharacteristic, complemented
50Q8×D5—21no—
51Q8×D5—21no—
52Q8×D5C3×C412yescomplemented
53Q8×D5C22×C312yes—
54C2×C4×D5—21no—
55C2×C4×D5C22×C311yescharacteristic
56C2×Dic10—21no—
57C2×Dic10C3×C411yescharacteristic, complemented
58C22⋊AGL(1,𝔽5)—21no—
59#704—21no—
60C4⋊AGL(1,𝔽5)C22×C312yes—
61C4×AGL(1,𝔽5)—21no—
62#700—21no—
63D5⋊C8C22×C312yes—
64#419—51noSylow
65C22×C3×C5C22⋊C411yescharacteristic
66C2×C3×D5—22no—
67C2×C3×D5C22⋊C411yescharacteristic
68C2×C3×D5C2×D411yescharacteristic
69C3×AGL(1,𝔽5)—42no—
70C3×C4×C5—22no—
71C3×C4×C5C22⋊C411yescharacteristic
72C3×C4×C5C22×C411yescharacteristic
73C3×Dic5—22no—
74C3×Dic5C22⋊C411yescharacteristic
75C3×Dic5C2×D411yescharacteristic
76C2×C3×Q8—52no—
77C2×C3×Q8—52no—
78C2×C3×Q8—51no—
79C2×C3×Q8—101no—
80C2×C3×Q8—101no—
81C2×C3×Q8—101no—
82C2×C3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
83C22×C3×C4—51no—
84C22×C3×C4—101no—
85C3×M16—101no—
86C3×M16—101no—
87C2×C3×C8—52no—
88C3×C4⋊C4—52no—
89C3×C22⋊C4—101no—
90C3×C42—101no—
91C22×D5C3×D411yescharacteristic
92C2×AGL(1,𝔽5)—22no—
93C5×Q8—21no—
94C5×Q8C2×C3×C412yes—
95C2×C4×C5—21no—
96C2×C4×C5C2×C3×C411yescharacteristic
97C2×Dic5—21no—
98C2×Dic5C3×D411yescharacteristic
99C4×D5—22no—
100C4×D5—22no—
101C4×D5C2×C3×C411yescharacteristic
102C4×D5C3×D411yescharacteristic
103C4×D5C3×D411yescharacteristic
104C4×D5C23×C311yescharacteristic
105Dic10—21no—
106Dic10—22no—
107Dic10—22no—
108Dic10C2×C3×C412yes—
109#170—22no—
110C22×Q8—51no—
111C2×M16—51no—
112#117—51no—
113#103—54no—
114C2×C3×C5—21no—
115C2×C3×C5C2×C22⋊C411yescharacteristic
116C3×D5—21no—
117C3×D5#13712yes—
118C23×C3—51no—
119C3×Q8—21no—
120C3×Q8—52no—
121C3×Q8—102no—
122C3×Q8—102no—
123C3×Q8—101no—
124C3×Q8C2×AGL(1,𝔽5)12yes—
125C2×C3×C4—21no—
126C2×C3×C4—51no—
127C2×C3×C4—51no—
128C2×C3×C4—51no—
129C2×C3×C4—51no—
130C2×C3×C4—51no—
131C2×C3×C4—102no—
132C2×C3×C4—102no—
133C2×C3×C4—101no—
134C2×C3×C4—102no—
135C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
136C3×C8—102no—
137C22×C5C3×C22⋊C411yescharacteristic
138C2×D5—22no—
139C2×D5C3×C22⋊C411yescharacteristic
140C2×D5C2×C3×D411yescharacteristic
141AGL(1,𝔽5)—42no—
142C4×C5—22no—
143C4×C5C3×C22⋊C411yescharacteristic
144C4×C5C22×C3×C411yescharacteristic
145Dic5—22no—
146Dic5C3×C22⋊C411yescharacteristic
147Dic5C2×C3×D411yescharacteristic
148C2×Q8—52no—
149C2×Q8—52no—
150C2×Q8—51no—
151C2×Q8—101no—
152C2×Q8—101no—
153C2×Q8—101no—
154C2×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
155C22×C4—51no—
156C22×C4—101no—
157M16—101no—
158M16—101no—
159C2×C8—52no—
160C4⋊C4—52no—
161C22⋊C4—101no—
162C42—101no—
163C3×C5#41911yescharacteristic, complemented, Hall
164C22×C3—51no—
165C22×C3—51no—
166C22×C3—102no—
167C22×C3C22⋊AGL(1,𝔽5)11yescharacteristic
168C3×C4—22no—
169C3×C4—51no—
170C3×C4—51no—
171C3×C4—102no—
172C3×C4—202no—
173C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
174C3×C4C22×AGL(1,𝔽5)11yescharacteristic
175C2×C5—21no—
176C2×C5C2×C3×C22⋊C411yescharacteristic
177D5—21no—
178D5C3×#13712yes—
179C23—51no—
180Q8—21no—
181Q8—52no—
182Q8—102no—
183Q8—102no—
184Q8—101no—
185Q8C2×C3×AGL(1,𝔽5)12yes—
186C2×C4—21no—
187C2×C4—51no—
188C2×C4—51no—
189C2×C4—51no—
190C2×C4—51no—
191C2×C4—51no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—101no—
195C2×C4—102no—
196C2×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
197C8—102no—
198C2×C3—21no—
199C2×C3—52no—
200C2×C3—101no—
201C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
202C5C3×#41911yescharacteristic, complemented, Sylow
203C22—51no—
204C22—51no—
205C22—102no—
206C22C3×C22⋊AGL(1,𝔽5)11yescharacteristic
207C4—22no—
208C4—51no—
209C4—51no—
210C4—102no—
211C4—202no—
212C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
213C4C22×C3×AGL(1,𝔽5)11yescharacteristic
214C3#6578211yescharacteristic, direct product factor, Sylow
215C2—21no—
216C2—52no—
217C2—101no—
218C2C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3C2×C22⋊AGL(1,𝔽5)
commutatorC4×C5C22×C3×C4
FrattiniC4C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C4
socleC2×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5D4
cosocleC4×D5C23×C3
hypercenterC2×C3×Q8AGL(1,𝔽5)
hypocenterC5C3×#419
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
4. C3×#4009C2semidirect
5. #65782C3direct (N × H)
6. C2×C3×C5×Q8C4semidirect
9. C3×Q8×D5C4semidirect
14. C2×C3×Dic10C4semidirect
25. #4009C2×C3semidirect
49. C2×C5×Q8C3×C4semidirect
52. Q8×D5C3×C4semidirect
57. C2×Dic10C3×C4semidirect
82. C2×C3×Q8AGL(1,𝔽5)semidirect
154. C2×Q8C3×AGL(1,𝔽5)semidirect
163. C3×C5#419semidirect
202. C5C3×#419semidirect
214. C3#65782direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D52
C3×#41302
C3×D5⋊M162
C3×#40092
#657823
C3×#4195