Finite Group Databasebeta

← C3×#10604927 #14285228

Subgroups — C3×#10604927 #14285228

Order 1920.

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

normal subgroups only

This group has 664 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×#4086C211yesmaximal, characteristic
2C3×#65757C212yesmaximal, complemented
3C3×#64925C214yesmaximal, complemented
4#10604927C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5×D4C411yescharacteristic
6C22×C3×C22⋊D5—22no—
7C22×C3×D20C411yescharacteristic
8C3×#4086—22no—
9C3×#4086C412yes—
10C3×#4086C2212yes—
11C2×C3×C4×Dic5C2211yescharacteristic
12C3×C22⋊#170—22no—
13C3×#4004C2214yescomplemented
14C3×#4002—22no—
15C3×#1555—51nomaximal, Hall
16C2×#4086C2×C311yescharacteristic
17#65757C2×C312yescomplemented
18#64925C2×C314yescomplemented
19C24×C3×C5—21no—
20C23×C3×D5—21no—
21C2×C3×C5×D4—21no—
22C2×C3×C5×D4—42no—
23C2×C3×C5×D4C2×C412yes—
24C22×C3×C4×C5C2×C411yescharacteristic
25C2×C3×C22⋊D5—24no—
26C2×C3×C22⋊D5—24no—
27C2×C3×C22⋊D5—48no—
28C22×C3×Dic5D412yes—
29C2×C3×D20—21no—
30C2×C3×D20—42no—
31C2×C3×D20C2×C412yes—
32C2×C3×#170—24no—
33C3×C4×Dic5C2×C411yescharacteristic
34C3×C4×Dic5D412yes—
35C3×C4×Dic5C2311yescharacteristic
36C2×C3×#127—51no—
37C3×#432—52no—
38C3×#331—54no—
39C22×C5×D4C3×C411yescharacteristic
40C22×C22⋊D5—22no—
41C22×D20C3×C411yescharacteristic
42#4086—22no—
43#4086C3×C412yes—
44#4086C22×C312yes—
45C2×C4×Dic5C22×C311yescharacteristic
46C22⋊#170—22no—
47#4004C22×C314yescomplemented
48#4002—22no—
49#1555—51noSylow
50C23×C3×C5—22no—
51C23×C3×C5—44no—
52C23×C3×C5—42no—
53C23×C3×C5C22⋊C411yescharacteristic
54C22×C3×D5—22no—
55C22×C3×D5—44no—
56C22×C3×D5—42no—
57C3×C5×D4—42no—
58C3×C5×D4—42no—
59C2×C3×C4×C5—22no—
60C2×C3×C4×C5C22⋊C411yescharacteristic
61C2×C3×C4×C5C22×C411yescharacteristic
62C3×C22⋊D5—48no—
63C3×C22⋊D5—48no—
64C2×C3×Dic5—24no—
65C2×C3×Dic5C22⋊C412yes—
66C2×C3×Dic5C2×D412yes—
67C3×D20—42no—
68C3×D20—42no—
69C3×#170—44no—
70C22×C3×D4—51no—
71C22×C3×D4—102no—
72C22×C3×D4AGL(1,𝔽5)11yescharacteristic
73C3×#127—52no—
74C3×#127—52no—
75C3×#127—102no—
76C2×C3×C42—51no—
77C3×C4⋊C8—54no—
78C3×C22⋊C8—102no—
79C3×#97—102no—
80C24×C5—21no—
81C23×D5—21no—
82C2×C5×D4—21no—
83C2×C5×D4—42no—
84C2×C5×D4C2×C3×C412yes—
85C22×C4×C5C2×C3×C411yescharacteristic
86C2×C22⋊D5—24no—
87C2×C22⋊D5—24no—
88C2×C22⋊D5—48no—
89C22×Dic5C3×D412yes—
90C2×D20—21no—
91C2×D20—42no—
92C2×D20C2×C3×C412yes—
93C2×#170—24no—
94C4×Dic5C2×C3×C411yescharacteristic
95C4×Dic5C3×D412yes—
96C4×Dic5C23×C311yescharacteristic
97C2×#127—51no—
98#432—52no—
99#331—54no—
100C22×C3×C5—22no—
101C22×C3×C5—42no—
102C22×C3×C5—44no—
103C22×C3×C5—84no—
104C22×C3×C5#10012yes—
105C22×C3×C5C2×C22⋊C411yescharacteristic
106C2×C3×D5—42no—
107C2×C3×D5—44no—
108C2×C3×D5—84no—
109C3×C4×C5—21no—
110C3×C4×C5—21no—
111C3×Dic5—22no—
112C3×Dic5AGL(1,ℤ/8)14yes—
113C24×C3—21no—
114C24×C3—101no—
115C2×C3×D4—21no—
116C2×C3×D4—42no—
117C2×C3×D4—52no—
118C2×C3×D4—104no—
119C2×C3×D4—104no—
120C2×C3×D4—101no—
121C2×C3×D4—202no—
122C2×C3×D4—208no—
123C2×C3×D4C2×AGL(1,𝔽5)12yes—
124C22×C3×C4—52no—
125C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
126C2×C3×C8—104no—
127C3×C42—52no—
128C3×C42—51no—
129C3×C42—51no—
130C23×C5—22no—
131C23×C5—44no—
132C23×C5—42no—
133C23×C5C3×C22⋊C411yescharacteristic
134C22×D5—22no—
135C22×D5—44no—
136C22×D5—42no—
137C5×D4—42no—
138C5×D4—42no—
139C2×C4×C5—22no—
140C2×C4×C5C3×C22⋊C411yescharacteristic
141C2×C4×C5C22×C3×C411yescharacteristic
142C22⋊D5—48no—
143C22⋊D5—48no—
144C2×Dic5—24no—
145C2×Dic5C3×C22⋊C412yes—
146C2×Dic5C2×C3×D412yes—
147D20—42no—
148D20—42no—
149#170—44no—
150C22×D4—51no—
151C22×D4—102no—
152C22×D4C3×AGL(1,𝔽5)11yescharacteristic
153#127—52no—
154#127—52no—
155#127—102no—
156C2×C42—51no—
157C4⋊C8—54no—
158C22⋊C8—102no—
159#97—102no—
160C2×C3×C5—22no—
161C2×C3×C5—82no—
162C2×C3×C5C2×#10011yescharacteristic
163C2×C3×C5#41812yes—
164C3×D5—82no—
165C23×C3—22no—
166C23×C3—44no—
167C23×C3—42no—
168C23×C3—102no—
169C23×C3—204no—
170C23×C3—202no—
171C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
172C3×D4—42no—
173C3×D4—42no—
174C3×D4—202no—
175C3×D4—202no—
176C3×D4—208no—
177C3×D4—208no—
178C2×C3×C4—22no—
179C2×C3×C4—52no—
180C2×C3×C4—52no—
181C2×C3×C4—104no—
182C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
183C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
184C3×C8—204no—
185C22×C5—22no—
186C22×C5—42no—
187C22×C5—44no—
188C22×C5—84no—
189C22×C5C3×#10012yes—
190C22×C5C2×C3×C22⋊C411yescharacteristic
191C2×D5—42no—
192C2×D5—44no—
193C2×D5—84no—
194C4×C5—21no—
195C4×C5—21no—
196Dic5—22no—
197Dic5C3×AGL(1,ℤ/8)14yes—
198C24—21no—
199C24—101no—
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—52no—
203C2×D4—104no—
204C2×D4—104no—
205C2×D4—101no—
206C2×D4—202no—
207C2×D4—208no—
208C2×D4C2×C3×AGL(1,𝔽5)12yes—
209C22×C4—52no—
210C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
211C2×C8—104no—
212C42—52no—
213C42—51no—
214C42—51no—
215C3×C5#155511yescharacteristic, complemented, Hall
216C22×C3—22no—
217C22×C3—42no—
218C22×C3—44no—
219C22×C3—84no—
220C22×C3—202no—
221C22×C3—204no—
222C22×C3—404no—
223C22×C3#401312yes—
224C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—54no—
228C3×C4—102no—
229C2×C5—22no—
230C2×C5—82no—
231C2×C5C2×C3×#10011yescharacteristic
232C2×C5C3×#41812yes—
233D5—82no—
234C23—22no—
235C23—44no—
236C23—42no—
237C23—102no—
238C23—204no—
239C23—202no—
240C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
241D4—42no—
242D4—42no—
243D4—202no—
244D4—202no—
245D4—208no—
246D4—208no—
247C2×C4—22no—
248C2×C4—52no—
249C2×C4—52no—
250C2×C4—104no—
251C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
252C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
253C8—204no—
254C2×C3—22no—
255C2×C3—82no—
256C2×C3—402no—
257C2×C3#6576712yes—
258C2×C3C2×#401311yescharacteristic
259C5C3×#155511yescharacteristic, complemented, Sylow
260C22—22no—
261C22—44no—
262C22—42no—
263C22—84no—
264C22—204no—
265C22—202no—
266C22—404no—
267C22C3×#401312yes—
268C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
269C4—21no—
270C4—21no—
271C4—54no—
272C4—102no—
273C3#1060492711yescharacteristic, direct product factor, Sylow
274C2—22no—
275C2—82no—
276C2—402no—
277C2C3×#6576712yes—
278C2C2×C3×#401311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#4086C2
cosocleC4×Dic5C23×C3
hypercenterC22×C3×D4AGL(1,𝔽5)
hypocenterC5C3×#1555
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#65757C2semidirect
3. C3×#64925C2semidirect
4. #10604927C3direct (N × H)
13. C3×#4004C22semidirect
17. #65757C2×C3semidirect
18. #64925C2×C3semidirect
47. #4004C22×C3semidirect
215. C3×C5#1555semidirect
259. C5C3×#1555semidirect
273. C3#10604927direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40862
C3×#657572
C3×#649252
#106049273
C3×#15555