Finite Group Databasebeta

← C3×#10604881 #14280992

Subgroups — C3×#10604881 #14280992

Order 1920.

624 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C22×C3×C4⋊Dic5C211yesmaximal, characteristic
2C3×#65798C212yesmaximal
3C3×#64913C214yesmaximal, complemented
4#10604881C311yesmaximal, characteristic, direct product factor, Hall
5C23×C3×C4×C5C411yescharacteristic
6C23×C3×Dic5C411yescharacteristic
7C23×C3×Dic5C2211yescharacteristic
8C2×C3×#704—22no—
9C2×C3×C4⋊Dic5—22no—
10C2×C3×C4⋊Dic5—22no—
11C2×C3×C4⋊Dic5C412yes—
12C2×C3×C4⋊Dic5C2212yes—
13C3×C22⋊#170—22no—
14C3×C22⋊#170C2214yes—
15C3×#1463—51nomaximal, Hall
16C22×C4⋊Dic5C2×C311yescharacteristic
17#65798C2×C312yes—
18#64913C2×C314yescomplemented
19C24×C3×C5C2×C411yescharacteristic
20C22×C3×C4×C5—21no—
21C22×C3×C4×C5—21no—
22C22×C3×C4×C5—24no—
23C22×C3×C4×C5C2×C412yes—
24C22×C3×Dic5—21no—
25C22×C3×Dic5—21no—
26C22×C3×Dic5—42no—
27C22×C3×Dic5—42no—
28C22×C3×Dic5C2×C412yes—
29C22×C3×Dic5C2×C411yescharacteristic
30C22×C3×Dic5D412yes—
31C22×C3×Dic5D412yes—
32C22×C3×Dic5C2311yescharacteristic
33C3×#704—44no—
34C2×C3×#170—24no—
35C3×C4⋊Dic5—24no—
36C3×C4⋊Dic5—24no—
37C22×C3×C4⋊C4—51no—
38C3×#407—52no—
39C3×#324—54no—
40C23×C4×C5C3×C411yescharacteristic
41C23×Dic5C3×C411yescharacteristic
42C23×Dic5C22×C311yescharacteristic
43C2×#704—22no—
44C2×C4⋊Dic5—22no—
45C2×C4⋊Dic5—22no—
46C2×C4⋊Dic5C3×C412yes—
47C2×C4⋊Dic5C22×C312yes—
48C22⋊#170—22no—
49C22⋊#170C22×C314yes—
50#1463—51noSylow
51C23×C3×C5—22no—
52C23×C3×C5—22no—
53C23×C3×C5—22no—
54C23×C3×C5C22⋊C411yescharacteristic
55C23×C3×C5C22⋊C411yescharacteristic
56C23×C3×C5C22×C411yescharacteristic
57C2×C3×C4×C5—24no—
58C2×C3×C4×C5—24no—
59C2×C3×C4×C5—22no—
60C2×C3×C4×C5—22no—
61C2×C3×C4×C5M1614yescomplemented
62C2×C3×Dic5—24no—
63C2×C3×Dic5—22no—
64C2×C3×Dic5—41no—
65C2×C3×Dic5—42no—
66C2×C3×Dic5—41no—
67C2×C3×Dic5—42no—
68C2×C3×Dic5—42no—
69C2×C3×Dic5—42no—
70C2×C3×Dic5C22⋊C412yes—
71C2×C3×Dic5C2×D412yes—
72C3×#170—44no—
73C23×C3×C4—51no—
74C23×C3×C4—51no—
75C23×C3×C4AGL(1,𝔽5)11yescharacteristic
76C2×C3×M16—102no—
77C2×C3×C4⋊C4—52no—
78C2×C3×C4⋊C4—52no—
79C2×C3×C4⋊C4—102no—
80C2×C3×C4⋊C4—102no—
81C3×C22⋊C8—54no—
82C3×C22⋊C8—102no—
83C24×C5C2×C3×C411yescharacteristic
84C22×C4×C5—21no—
85C22×C4×C5—21no—
86C22×C4×C5—24no—
87C22×C4×C5C2×C3×C412yes—
88C22×Dic5—21no—
89C22×Dic5—21no—
90C22×Dic5—42no—
91C22×Dic5—42no—
92C22×Dic5C2×C3×C412yes—
93C22×Dic5C2×C3×C411yescharacteristic
94C22×Dic5C3×D412yes—
95C22×Dic5C3×D412yes—
96C22×Dic5C23×C311yescharacteristic
97#704—44no—
98C2×#170—24no—
99C4⋊Dic5—24no—
100C4⋊Dic5—24no—
101C22×C4⋊C4—51no—
102#407—52no—
103#324—54no—
104C22×C3×C5—28no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—21no—
108C22×C3×C5—21no—
109C22×C3×C5—21no—
110C22×C3×C5—21no—
111C22×C3×C5#9912yes—
112C22×C3×C5#10112yes—
113C22×C3×C5C2×C22⋊C411yescharacteristic
114C22×C3×C5C2×M1612yes—
115C3×C4×C5—24no—
116C3×Dic5—22no—
117C3×Dic5—22no—
118C3×Dic5—42no—
119C24×C3C2×AGL(1,𝔽5)11yescharacteristic
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—24no—
123C22×C3×C4—52no—
124C22×C3×C4—52no—
125C22×C3×C4—51no—
126C22×C3×C4—51no—
127C22×C3×C4—52no—
128C22×C3×C4—101no—
129C22×C3×C4—101no—
130C22×C3×C4—202no—
131C22×C3×C4—202no—
132C22×C3×C4C2×AGL(1,𝔽5)12yes—
133C3×M16—204no—
134C2×C3×C8—104no—
135C3×C4⋊C4—104no—
136C3×C4⋊C4—104no—
137C23×C5—22no—
138C23×C5—22no—
139C23×C5—22no—
140C23×C5C3×C22⋊C411yescharacteristic
141C23×C5C3×C22⋊C411yescharacteristic
142C23×C5C22×C3×C411yescharacteristic
143C2×C4×C5—24no—
144C2×C4×C5—24no—
145C2×C4×C5—22no—
146C2×C4×C5—22no—
147C2×C4×C5C3×M1614yescomplemented
148C2×Dic5—24no—
149C2×Dic5—22no—
150C2×Dic5—41no—
151C2×Dic5—42no—
152C2×Dic5—41no—
153C2×Dic5—42no—
154C2×Dic5—42no—
155C2×Dic5—42no—
156C2×Dic5C3×C22⋊C412yes—
157C2×Dic5C2×C3×D412yes—
158#170—44no—
159C23×C4—51no—
160C23×C4—51no—
161C23×C4C3×AGL(1,𝔽5)11yescharacteristic
162C2×M16—102no—
163C2×C4⋊C4—52no—
164C2×C4⋊C4—52no—
165C2×C4⋊C4—102no—
166C2×C4⋊C4—102no—
167C22⋊C8—54no—
168C22⋊C8—102no—
169C2×C3×C5—22no—
170C2×C3×C5—22no—
171C2×C3×C5—22no—
172C2×C3×C5#40711yescharacteristic
173C2×C3×C5C2×#9911yescharacteristic
174C2×C3×C5C2×#10111yescharacteristic
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
179C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
180C23×C3C22×AGL(1,𝔽5)11yescharacteristic
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—52no—
186C2×C3×C4—52no—
187C2×C3×C4—104no—
188C2×C3×C4—102no—
189C2×C3×C4—201no—
190C2×C3×C4—202no—
191C2×C3×C4—201no—
192C2×C3×C4—202no—
193C2×C3×C4—202no—
194C2×C3×C4—202no—
195C2×C3×C4#70414yescomplemented
196C3×C8—204no—
197C22×C5—28no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—21no—
202C22×C5—21no—
203C22×C5—21no—
204C22×C5C3×#9912yes—
205C22×C5C3×#10112yes—
206C22×C5C2×C3×C22⋊C411yescharacteristic
207C22×C5C2×C3×M1612yes—
208C4×C5—24no—
209Dic5—22no—
210Dic5—22no—
211Dic5—42no—
212C24C2×C3×AGL(1,𝔽5)11yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—52no—
217C22×C4—52no—
218C22×C4—51no—
219C22×C4—51no—
220C22×C4—52no—
221C22×C4—101no—
222C22×C4—101no—
223C22×C4—202no—
224C22×C4—202no—
225C22×C4C2×C3×AGL(1,𝔽5)12yes—
226M16—204no—
227C2×C8—104no—
228C4⋊C4—104no—
229C4⋊C4—104no—
230C3×C5#146311yescharacteristic, complemented, Hall
231C22×C3—28no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3#399912yes—
239C22×C3#400512yes—
240C22×C3C2×#70412yes—
241C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
242C3×C4—24no—
243C3×C4—102no—
244C3×C4—102no—
245C3×C4—202no—
246C2×C5—22no—
247C2×C5—22no—
248C2×C5—22no—
249C2×C5C3×#40711yescharacteristic
250C2×C5C2×C3×#9911yescharacteristic
251C2×C5C2×C3×#10111yescharacteristic
252C23—22no—
253C23—22no—
254C23—22no—
255C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
256C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
257C23C22×C3×AGL(1,𝔽5)11yescharacteristic
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—52no—
263C2×C4—52no—
264C2×C4—104no—
265C2×C4—102no—
266C2×C4—201no—
267C2×C4—202no—
268C2×C4—201no—
269C2×C4—202no—
270C2×C4—202no—
271C2×C4—202no—
272C2×C4C3×#70414yescomplemented
273C8—204no—
274C2×C3—22no—
275C2×C3—22no—
276C2×C3—22no—
277C2×C3C2×#399911yescharacteristic
278C2×C3C2×#400511yescharacteristic
279C2×C3#6579811yescharacteristic
280C5C3×#146311yescharacteristic, complemented, Sylow
281C22—28no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22C3×#399912yes—
289C22C3×#400512yes—
290C22C2×C3×#70412yes—
291C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
292C4—24no—
293C4—102no—
294C4—102no—
295C4—202no—
296C3#1060488111yescharacteristic, direct product factor, Sylow
297C2—22no—
298C2—22no—
299C2—22no—
300C2C2×C3×#399911yescharacteristic
301C2C2×C3×#400511yescharacteristic
302C2C3×#6579811yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC23×C3×C4×C5C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC24×C3×C5C2×C4
cosocleC22×Dic5C23×C3
hypercenterC23×C3×C4AGL(1,𝔽5)
hypocenterC5C3×#1463
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#64913C2semidirect
4. #10604881C3direct (N × H)
18. #64913C2×C3semidirect
61. C2×C3×C4×C5M16semidirect
147. C2×C4×C5C3×M16semidirect
195. C2×C3×C4#704semidirect
230. C3×C5#1463semidirect
272. C2×C4C3×#704semidirect
280. C5C3×#1463semidirect
296. C3#10604881direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Dic52
C3×#657982
C3×#649132
#106048813
C3×#14635