Finite Group Databasebeta

← C3×#10604365 #14285456

Subgroups — C3×#10604365 #14285456

Order 1920.

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

normal subgroups only

This group has 420 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
1C3×#65851C211yesmaximal, characteristic
2C3×#65758C212yesmaximal
3C3×#64873C214yesmaximal, complemented
4#10604365C311yesmaximal, characteristic, direct product factor, Hall
5C2×C3×#704—22no—
6C3×C5×#117C411yescharacteristic
7C3×#4072C411yescharacteristic
8C2×C3×C4×Dic5C2211yescharacteristic
9C3×#4036C412yes—
10C3×#4034C2212yes—
11C3×#4024—22no—
12C3×C4×Dic10—22no—
13C3×#4004C2214yes—
14C3×C4⋊#170—22no—
15C3×#1561—51nomaximal, Hall
16#65851C2×C311yescharacteristic
17#65758C2×C312yes—
18#64873C2×C314yescomplemented
19C22×C3×C4×C5C2×C411yescharacteristic
20C22×C3×Dic5D412yes—
21C2×C3×Dic10—22no—
22C3×#704—44no—
23C2×C3×#170—24no—
24C3×C5×C4⋊C4C2×C412yes—
25C3×C5×C22⋊C4—21no—
26C3×C42×C5—21no—
27C3×C22⋊Dic5—21no—
28C3×C4⋊Dic5—22no—
29C3×C4⋊Dic5C2×C412yes—
30C3×#683—22no—
31C3×#683—22no—
32C3×C4×Dic5—21no—
33C3×C4×Dic5C2×C411yescharacteristic
34C3×C4×Dic5D412yes—
35C3×C4×Dic5C2311yescharacteristic
36C3×#533—51no—
37C3×#423—52no—
38C3×#332—54no—
39C2×#704—22no—
40C5×#117C3×C411yescharacteristic
41#4072C3×C411yescharacteristic
42C2×C4×Dic5C22×C311yescharacteristic
43#4036C3×C412yes—
44#4034C22×C312yes—
45#4024—22no—
46C4×Dic10—22no—
47#4004C22×C314yes—
48C4⋊#170—22no—
49#1561—51noSylow
50C23×C3×C5C22⋊C411yescharacteristic
51C2×C3×C4×C5—22no—
52C2×C3×C4×C5—22no—
53C2×C3×C4×C5C22⋊C411yescharacteristic
54C2×C3×C4×C5C22×C411yescharacteristic
55C2×C3×Dic5—22no—
56C2×C3×Dic5—22no—
57C2×C3×Dic5—22no—
58C2×C3×Dic5C22⋊C412yes—
59C2×C3×Dic5C2×D412yes—
60C3×Dic10—42no—
61C3×Dic10—42no—
62C3×#170—44no—
63C2×C3×M16—102no—
64C3×C4⋊Q8—52no—
65C3×#125—52no—
66C3×C22⋊Q8—102no—
67C3×C4×Q8—102no—
68C3×#117—51no—
69C3×#117AGL(1,𝔽5)11yescharacteristic
70C2×C3×C42—51no—
71C3×C4⋊C8—54no—
72C3×C4⋊C8—102no—
73C22×C4×C5C2×C3×C411yescharacteristic
74C22×Dic5C3×D412yes—
75C2×Dic10—22no—
76#704—44no—
77C2×#170—24no—
78C5×C4⋊C4C2×C3×C412yes—
79C5×C22⋊C4—21no—
80C42×C5—21no—
81C22⋊Dic5—21no—
82C4⋊Dic5—22no—
83C4⋊Dic5C2×C3×C412yes—
84#683—22no—
85#683—22no—
86C4×Dic5—21no—
87C4×Dic5C2×C3×C411yescharacteristic
88C4×Dic5C3×D412yes—
89C4×Dic5C23×C311yescharacteristic
90#533—51no—
91#423—52no—
92#332—54no—
93C22×C3×C5—21no—
94C22×C3×C5—22no—
95C22×C3×C5C2×C22⋊C411yescharacteristic
96C3×C4×C5—21no—
97C3×C4×C5—42no—
98C3×C4×C5#10112yes—
99C3×Dic5—22no—
100C3×Dic5—42no—
101C3×Dic5AGL(1,ℤ/8)12yes—
102C3×Dic5#13712yes—
103C2×C3×Q8—102no—
104C22×C3×C4—52no—
105C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
106C3×M16—204no—
107C2×C3×C8—104no—
108C3×C4⋊C4—52no—
109C3×C4⋊C4—102no—
110C3×C4⋊C4—102no—
111C3×C4⋊C4—102no—
112C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
113C3×C22⋊C4—21no—
114C3×C22⋊C4—101no—
115C3×C42—21no—
116C3×C42—52no—
117C3×C42—51no—
118C3×C42—51no—
119C3×C42—101no—
120C23×C5C3×C22⋊C411yescharacteristic
121C2×C4×C5—22no—
122C2×C4×C5—22no—
123C2×C4×C5C3×C22⋊C411yescharacteristic
124C2×C4×C5C22×C3×C411yescharacteristic
125C2×Dic5—22no—
126C2×Dic5—22no—
127C2×Dic5—22no—
128C2×Dic5C3×C22⋊C412yes—
129C2×Dic5C2×C3×D412yes—
130Dic10—42no—
131Dic10—42no—
132#170—44no—
133C2×M16—102no—
134C4⋊Q8—52no—
135#125—52no—
136C22⋊Q8—102no—
137C4×Q8—102no—
138#117—51no—
139#117C3×AGL(1,𝔽5)11yescharacteristic
140C2×C42—51no—
141C4⋊C8—54no—
142C4⋊C8—102no—
143C2×C3×C5—41no—
144C2×C3×C5C2×#10111yescharacteristic
145C2×C3×C5#41712yes—
146C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
147C3×Q8—202no—
148C3×Q8—202no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—52no—
152C2×C3×C4—52no—
153C2×C3×C4—102no—
154C2×C3×C4—102no—
155C2×C3×C4—102no—
156C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
157C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
158C3×C8—204no—
159C22×C5—21no—
160C22×C5—22no—
161C22×C5C2×C3×C22⋊C411yescharacteristic
162C4×C5—21no—
163C4×C5—42no—
164C4×C5C3×#10112yes—
165Dic5—22no—
166Dic5—42no—
167Dic5C3×AGL(1,ℤ/8)12yes—
168Dic5C3×#13712yes—
169C2×Q8—102no—
170C22×C4—52no—
171C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
172M16—204no—
173C2×C8—104no—
174C4⋊C4—52no—
175C4⋊C4—102no—
176C4⋊C4—102no—
177C4⋊C4—102no—
178C4⋊C4C2×C3×AGL(1,𝔽5)12yes—
179C22⋊C4—21no—
180C22⋊C4—101no—
181C42—21no—
182C42—52no—
183C42—51no—
184C42—51no—
185C42—101no—
186C3×C5#156111yescharacteristic, complemented, Hall
187C22×C3—22no—
188C22×C3—21no—
189C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
190C3×C4—21no—
191C3×C4—42no—
192C3×C4—52no—
193C3×C4—52no—
194C3×C4—102no—
195C3×C4—202no—
196C3×C4#400512yes—
197C2×C5—41no—
198C2×C5C2×C3×#10111yescharacteristic
199C2×C5C3×#41712yes—
200C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
201Q8—202no—
202Q8—202no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—52no—
206C2×C4—52no—
207C2×C4—102no—
208C2×C4—102no—
209C2×C4—102no—
210C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
211C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
212C8—204no—
213C2×C3—41no—
214C2×C3C2×#400511yescharacteristic
215C2×C3#6579312yes—
216C5C3×#156111yescharacteristic, complemented, Sylow
217C22—21no—
218C22—22no—
219C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
220C4—21no—
221C4—42no—
222C4—52no—
223C4—52no—
224C4—102no—
225C4—202no—
226C4C3×#400512yes—
227C3#1060436511yescharacteristic, direct product factor, Sylow
228C2—41no—
229C2C2×C3×#400511yescharacteristic
230C2C3×#6579312yes—
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#117C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×C5C22⋊C4
cosocleC4×Dic5C23×C3
hypercenterC3×#117AGL(1,𝔽5)
hypocenterC5C3×#1561
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
3. C3×#64873C2semidirect
4. #10604365C3direct (N × H)
18. #64873C2×C3semidirect
186. C3×C5#1561semidirect
216. C5C3×#1561semidirect
227. C3#10604365direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658512
C3×#657582
C3×#648732
#106043653
C3×#15615