Finite Group Databasebeta

← C3×#10604972 #14285342

Subgroups — C3×#10604972 #14285342

Order 1920.

456 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 124 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×#4090C211yesmaximal, characteristic
2C3×#65757C212yesmaximal
3C3×#64931C214yesmaximal, complemented
4#10604972C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5×Q8C411yescharacteristic
6C22×C3×Dic10C411yescharacteristic
7C3×#4090—22no—
8C3×#4090C412yes—
9C3×#4090C2212yes—
10C2×C3×#683—22no—
11C2×C3×C4×Dic5C2211yescharacteristic
12C3×C22⋊#170—22no—
13C3×#4004C2214yes—
14C3×#4002—22no—
15C3×#1558—51nomaximal, Hall
16C2×#4090C2×C311yescharacteristic
17#65757C2×C312yes—
18#64931C2×C314yescomplemented
19C2×C3×C5×Q8—21no—
20C2×C3×C5×Q8—42no—
21C2×C3×C5×Q8C2×C412yes—
22C22×C3×C4×C5—21no—
23C22×C3×C4×C5C2×C411yescharacteristic
24C22×C3×Dic5—21no—
25C22×C3×Dic5D412yes—
26C2×C3×Dic10—21no—
27C2×C3×Dic10—42no—
28C2×C3×Dic10C2×C412yes—
29C2×C3×#170—24no—
30C3×#683—24no—
31C3×#683—24no—
32C3×C4×Dic5C2×C411yescharacteristic
33C3×C4×Dic5D412yes—
34C3×C4×Dic5C2311yescharacteristic
35C2×C3×C4⋊Q8—51no—
36C3×#432—52no—
37C3×#333—54no—
38C22×C5×Q8C3×C411yescharacteristic
39C22×Dic10C3×C411yescharacteristic
40#4090—22no—
41#4090C3×C412yes—
42#4090C22×C312yes—
43C2×#683—22no—
44C2×C4×Dic5C22×C311yescharacteristic
45C22⋊#170—22no—
46#4004C22×C314yes—
47#4002—22no—
48#1558—51noSylow
49C23×C3×C5C22⋊C411yescharacteristic
50C3×C5×Q8—42no—
51C3×C5×Q8—42no—
52C2×C3×C4×C5—22no—
53C2×C3×C4×C5—22no—
54C2×C3×C4×C5—42no—
55C2×C3×C4×C5C22⋊C411yescharacteristic
56C2×C3×C4×C5C22×C411yescharacteristic
57C2×C3×Dic5—24no—
58C2×C3×Dic5—22no—
59C2×C3×Dic5—42no—
60C2×C3×Dic5C22⋊C412yes—
61C2×C3×Dic5C2×D412yes—
62C3×Dic10—42no—
63C3×Dic10—42no—
64C3×#170—44no—
65C22×C3×Q8—51no—
66C22×C3×Q8AGL(1,𝔽5)11yescharacteristic
67C3×C4⋊Q8—52no—
68C3×C4⋊Q8—52no—
69C3×C4⋊Q8—102no—
70C2×C3×C4⋊C4—102no—
71C2×C3×C42—51no—
72C3×C4⋊C8—54no—
73C3×C22⋊C8—102no—
74C3×#97—102no—
75C2×C5×Q8—21no—
76C2×C5×Q8—42no—
77C2×C5×Q8C2×C3×C412yes—
78C22×C4×C5—21no—
79C22×C4×C5C2×C3×C411yescharacteristic
80C22×Dic5—21no—
81C22×Dic5C3×D412yes—
82C2×Dic10—21no—
83C2×Dic10—42no—
84C2×Dic10C2×C3×C412yes—
85C2×#170—24no—
86#683—24no—
87#683—24no—
88C4×Dic5C2×C3×C411yescharacteristic
89C4×Dic5C3×D412yes—
90C4×Dic5C23×C311yescharacteristic
91C2×C4⋊Q8—51no—
92#432—52no—
93#333—54no—
94C22×C3×C5—22no—
95C22×C3×C5#10012yes—
96C22×C3×C5C2×C22⋊C411yescharacteristic
97C3×C4×C5—21no—
98C3×C4×C5—21no—
99C3×C4×C5—42no—
100C3×Dic5—22no—
101C3×Dic5—42no—
102C3×Dic5#13714yes—
103C2×C3×Q8—21no—
104C2×C3×Q8—42no—
105C2×C3×Q8—52no—
106C2×C3×Q8—101no—
107C2×C3×Q8—202no—
108C2×C3×Q8C2×AGL(1,𝔽5)12yes—
109C22×C3×C4—21no—
110C22×C3×C4—52no—
111C22×C3×C4—101no—
112C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
113C2×C3×C8—104no—
114C3×C4⋊C4—104no—
115C3×C4⋊C4—104no—
116C3×C42—52no—
117C3×C42—51no—
118C3×C42—51no—
119C23×C5C3×C22⋊C411yescharacteristic
120C5×Q8—42no—
121C5×Q8—42no—
122C2×C4×C5—22no—
123C2×C4×C5—22no—
124C2×C4×C5—42no—
125C2×C4×C5C3×C22⋊C411yescharacteristic
126C2×C4×C5C22×C3×C411yescharacteristic
127C2×Dic5—24no—
128C2×Dic5—22no—
129C2×Dic5—42no—
130C2×Dic5C3×C22⋊C412yes—
131C2×Dic5C2×C3×D412yes—
132Dic10—42no—
133Dic10—42no—
134#170—44no—
135C22×Q8—51no—
136C22×Q8C3×AGL(1,𝔽5)11yescharacteristic
137C4⋊Q8—52no—
138C4⋊Q8—52no—
139C4⋊Q8—102no—
140C2×C4⋊C4—102no—
141C2×C42—51no—
142C4⋊C8—54no—
143C22⋊C8—102no—
144#97—102no—
145C2×C3×C5—22no—
146C2×C3×C5C2×#10011yescharacteristic
147C2×C3×C5#41912yes—
148C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
149C3×Q8—42no—
150C3×Q8—42no—
151C3×Q8—202no—
152C3×Q8—202no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—52no—
157C2×C3×C4—52no—
158C2×C3×C4—104no—
159C2×C3×C4—102no—
160C2×C3×C4—202no—
161C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
162C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
163C3×C8—204no—
164C22×C5—22no—
165C22×C5C3×#10012yes—
166C22×C5C2×C3×C22⋊C411yescharacteristic
167C4×C5—21no—
168C4×C5—21no—
169C4×C5—42no—
170Dic5—22no—
171Dic5—42no—
172Dic5C3×#13714yes—
173C2×Q8—21no—
174C2×Q8—42no—
175C2×Q8—52no—
176C2×Q8—101no—
177C2×Q8—202no—
178C2×Q8C2×C3×AGL(1,𝔽5)12yes—
179C22×C4—21no—
180C22×C4—52no—
181C22×C4—101no—
182C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
183C2×C8—104no—
184C4⋊C4—104no—
185C4⋊C4—104no—
186C42—52no—
187C42—51no—
188C42—51no—
189C3×C5#155811yescharacteristic, complemented, Hall
190C22×C3—22no—
191C22×C3#401312yes—
192C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
193C3×C4—21no—
194C3×C4—21no—
195C3×C4—42no—
196C3×C4—54no—
197C3×C4—102no—
198C3×C4—202no—
199C2×C5—22no—
200C2×C5C2×C3×#10011yescharacteristic
201C2×C5C3×#41912yes—
202C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
203Q8—42no—
204Q8—42no—
205Q8—202no—
206Q8—202no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—52no—
211C2×C4—52no—
212C2×C4—104no—
213C2×C4—102no—
214C2×C4—202no—
215C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
216C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
217C8—204no—
218C2×C3—22no—
219C2×C3#6578212yes—
220C2×C3C2×#401311yescharacteristic
221C5C3×#155811yescharacteristic, complemented, Sylow
222C22—22no—
223C22C3×#401312yes—
224C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
225C4—21no—
226C4—21no—
227C4—42no—
228C4—54no—
229C4—102no—
230C4—202no—
231C3#1060497211yescharacteristic, direct product factor, Sylow
232C2—22no—
233C2C3×#6578212yes—
234C2C2×C3×#401311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×C5C22⋊C4
cosocleC4×Dic5C23×C3
hypercenterC22×C3×Q8AGL(1,𝔽5)
hypocenterC5C3×#1558
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#64931C2semidirect
4. #10604972C3direct (N × H)
18. #64931C2×C3semidirect
189. C3×C5#1558semidirect
221. C5C3×#1558semidirect
231. C3#10604972direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40902
C3×#657572
C3×#649312
#106049723
C3×#15585