Finite Group Databasebeta

← C3×#10604219 #14298635

Subgroups — C3×#10604219 #14298635

Order 1920.

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

normal subgroups only

This group has 592 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×#65820C211yesmaximal, characteristic
2C2×C3×#4011C212yesmaximal
3C3×#64866C214yesmaximal, complemented
4#10604219C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C22⋊D5C2211yescharacteristic
6C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
7C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
8C2×C3×C22⋊Dic5C411yescharacteristic, complemented
9C3×C4×C22⋊D5—22no—
10C3×#4027—22no—
11C3×D5×C22⋊C4C2212yes—
12C3×#4023C412yescomplemented
13C3×#4011—22no—
14C3×#4011—22no—
15C3×#1828—51nomaximal, Hall
16#65820C2×C311yescharacteristic
17C2×#4011C2×C312yes—
18#64866C2×C314yescomplemented
19C24×C3×C5C2×C411yescharacteristic
20C23×C3×D5C2311yescharacteristic
21C22×C3×AGL(1,𝔽5)—24no—
22C22×C3×C4×C5—21no—
23C2×C3×C22⋊D5—24no—
24C2×C3×C22⋊D5D412yes—
25C2×C3×C22⋊D5D412yes—
26C22×C3×Dic5—21no—
27C22×C3×Dic5C2×C411yescharacteristic
28C2×C3×C4×D5—22no—
29C3×C22⋊AGL(1,𝔽5)—24no—
30C3×C22⋊AGL(1,𝔽5)—24no—
31C3×C5×C22⋊C4—21no—
32C3×C5×C22⋊C4C2×C412yes—
33C3×C22⋊Dic5—21no—
34C3×C22⋊Dic5C2×C412yes—
35C3×#685—22no—
36C3×#683—22no—
37C3×C4×Dic5—22no—
38C3×#518—51no—
39C2×C3×#99—52no—
40C3×#342—54no—
41C22×C22⋊D5C22×C311yescharacteristic
42C2×C22⋊AGL(1,𝔽5)C22×C314yes—
43C2×C5×C22⋊C4C3×C411yescharacteristic, complemented
44C2×C22⋊Dic5C3×C411yescharacteristic, complemented
45C4×C22⋊D5—22no—
46#4027—22no—
47D5×C22⋊C4C22×C312yes—
48#4023C3×C412yescomplemented
49#4011—22no—
50#4011—22no—
51#1828—51noSylow
52C23×C3×C5—22no—
53C23×C3×C5—81no—
54C23×C3×C5C22⋊C411yescharacteristic
55C23×C3×C5C22⋊C411yescharacteristic
56C23×C3×C5C22×C411yescharacteristic
57C22×C3×D5—81no—
58C22×C3×D5C2×D412yes—
59C22×C3×D5#4114yes—
60C2×C3×AGL(1,𝔽5)—44no—
61C2×C3×AGL(1,𝔽5)—84no—
62C2×C3×C4×C5—22no—
63C2×C3×C4×C5—81no—
64C3×C22⋊D5—24no—
65C3×C22⋊D5—24no—
66C2×C3×Dic5—22no—
67C2×C3×Dic5—22no—
68C2×C3×Dic5—81no—
69C2×C3×Dic5C22⋊C412yes—
70C3×C4×D5—82no—
71C22×C3×D4—51no—
72C3×C4×D4—102no—
73C3×C4×D4—102no—
74C3×#117—52no—
75C2×C3×C22⋊C4—51no—
76C2×C3×C22⋊C4—52no—
77C2×C3×C22⋊C4—54no—
78C2×C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
79C3×#99—102no—
80C3×#99—102no—
81C24×C5C2×C3×C411yescharacteristic
82C23×D5C23×C311yescharacteristic
83C22×AGL(1,𝔽5)—24no—
84C22×C4×C5—21no—
85C2×C22⋊D5—24no—
86C2×C22⋊D5C3×D412yes—
87C2×C22⋊D5C3×D412yes—
88C22×Dic5—21no—
89C22×Dic5C2×C3×C411yescharacteristic
90C2×C4×D5—22no—
91C22⋊AGL(1,𝔽5)—24no—
92C22⋊AGL(1,𝔽5)—24no—
93C5×C22⋊C4—21no—
94C5×C22⋊C4C2×C3×C412yes—
95C22⋊Dic5—21no—
96C22⋊Dic5C2×C3×C412yes—
97#685—22no—
98#683—22no—
99C4×Dic5—22no—
100#518—51no—
101C2×#99—52no—
102#342—54no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—41no—
106C22×C3×C5—41no—
107C22×C3×C5—41no—
108C22×C3×C5—82no—
109C22×C3×C5C2×C22⋊C411yescharacteristic
110C22×C3×C5#11712yes—
111C2×C3×D5—42no—
112C2×C3×D5—44no—
113C2×C3×D5#12314yes—
114C3×AGL(1,𝔽5)—84no—
115C3×C4×C5—42no—
116C3×Dic5—22no—
117C3×Dic5—42no—
118C24×C3—51no—
119C24×C3C2×AGL(1,𝔽5)11yescharacteristic
120C2×C3×D4—52no—
121C2×C3×D4—52no—
122C2×C3×D4—104no—
123C22×C3×C4—21no—
124C22×C3×C4—51no—
125C22×C3×C4—102no—
126C22×C3×C4—101no—
127C22×C3×C4—104no—
128C3×C4⋊C4—102no—
129C3×C22⋊C4—21no—
130C3×C22⋊C4—52no—
131C3×C22⋊C4—102no—
132C3×C22⋊C4—101no—
133C3×C22⋊C4—104no—
134C3×C22⋊C4—104no—
135C3×C22⋊C4C2×AGL(1,𝔽5)12yes—
136C3×C42—102no—
137C23×C5—22no—
138C23×C5—81no—
139C23×C5C3×C22⋊C411yescharacteristic
140C23×C5C3×C22⋊C411yescharacteristic
141C23×C5C22×C3×C411yescharacteristic
142C22×D5—81no—
143C22×D5C2×C3×D412yes—
144C22×D5C3×#4114yes—
145C2×AGL(1,𝔽5)—44no—
146C2×AGL(1,𝔽5)—84no—
147C2×C4×C5—22no—
148C2×C4×C5—81no—
149C22⋊D5—24no—
150C22⋊D5—24no—
151C2×Dic5—22no—
152C2×Dic5—22no—
153C2×Dic5—81no—
154C2×Dic5C3×C22⋊C412yes—
155C4×D5—82no—
156C22×D4—51no—
157C4×D4—102no—
158C4×D4—102no—
159#117—52no—
160C2×C22⋊C4—51no—
161C2×C22⋊C4—52no—
162C2×C22⋊C4—54no—
163C2×C22⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
164#99—102no—
165#99—102no—
166C2×C3×C5—21no—
167C2×C3×C5—22no—
168C2×C3×C5—41no—
169C2×C3×C5—41no—
170C2×C3×C5#38111yescharacteristic
171C3×D5—24no—
172C23×C3—22no—
173C23×C3—54no—
174C23×C3—52no—
175C23×C3—81no—
176C23×C3—401no—
177C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
178C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
179C23×C3C22×AGL(1,𝔽5)11yescharacteristic
180C3×D4—104no—
181C3×D4—104no—
182C2×C3×C4—22no—
183C2×C3×C4—52no—
184C2×C3×C4—81no—
185C2×C3×C4—102no—
186C2×C3×C4—102no—
187C2×C3×C4—204no—
188C2×C3×C4—402no—
189C2×C3×C4—401no—
190C2×C3×C4—404no—
191C22×C5—21no—
192C22×C5—21no—
193C22×C5—41no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5—82no—
197C22×C5C2×C3×C22⋊C411yescharacteristic
198C22×C5C3×#11712yes—
199C2×D5—42no—
200C2×D5—44no—
201C2×D5C3×#12314yes—
202AGL(1,𝔽5)—84no—
203C4×C5—42no—
204Dic5—22no—
205Dic5—42no—
206C24—51no—
207C24C2×C3×AGL(1,𝔽5)11yescharacteristic
208C2×D4—52no—
209C2×D4—52no—
210C2×D4—104no—
211C22×C4—21no—
212C22×C4—51no—
213C22×C4—102no—
214C22×C4—101no—
215C22×C4—104no—
216C4⋊C4—102no—
217C22⋊C4—21no—
218C22⋊C4—52no—
219C22⋊C4—102no—
220C22⋊C4—101no—
221C22⋊C4—104no—
222C22⋊C4—104no—
223C22⋊C4C2×C3×AGL(1,𝔽5)12yes—
224C42—102no—
225C3×C5#182811yescharacteristic, complemented, Hall
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—54no—
232C22×C3—82no—
233C22×C3—202no—
234C22×C3—204no—
235C22×C3#413012yes—
236C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
237C3×C4—42no—
238C3×C4—102no—
239C3×C4—202no—
240C3×C4—404no—
241C2×C5—21no—
242C2×C5—22no—
243C2×C5—41no—
244C2×C5—41no—
245C2×C5C3×#38111yescharacteristic
246D5—24no—
247C23—22no—
248C23—54no—
249C23—52no—
250C23—81no—
251C23—401no—
252C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
253C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
254C23C22×C3×AGL(1,𝔽5)11yescharacteristic
255D4—104no—
256D4—104no—
257C2×C4—22no—
258C2×C4—52no—
259C2×C4—81no—
260C2×C4—102no—
261C2×C4—102no—
262C2×C4—204no—
263C2×C4—402no—
264C2×C4—401no—
265C2×C4—404no—
266C2×C3—21no—
267C2×C3—22no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—104no—
271C2×C3#6576411yescharacteristic
272C5C3×#182811yescharacteristic, complemented, Sylow
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—54no—
279C22—82no—
280C22—202no—
281C22—204no—
282C22C3×#413012yes—
283C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
284C4—42no—
285C4—102no—
286C4—202no—
287C4—404no—
288C3#1060421911yescharacteristic, direct product factor, Sylow
289C2—21no—
290C2—22no—
291C2—41no—
292C2—41no—
293C2—104no—
294C2C3×#6576411yescharacteristic
subgroupG/H
centerC2×C3#65764
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C22⋊C4C4
socleC2×C3×C5#381
prime radicalC22×C3×C22⋊D5C22
cosocleC23×D5C23×C3
hypercenterC2×C3×C22⋊C4AGL(1,𝔽5)
hypocenterC5C3×#1828
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#64866C2semidirect
4. #10604219C3direct (N × H)
7. C2×C3×C5×C22⋊C4C4semidirect
8. C2×C3×C22⋊Dic5C4semidirect
12. C3×#4023C4semidirect
18. #64866C2×C3semidirect
43. C2×C5×C22⋊C4C3×C4semidirect
44. C2×C22⋊Dic5C3×C4semidirect
48. #4023C3×C4semidirect
78. C2×C3×C22⋊C4AGL(1,𝔽5)semidirect
163. C2×C22⋊C4C3×AGL(1,𝔽5)semidirect
225. C3×C5#1828semidirect
272. C5C3×#1828semidirect
288. C3#10604219direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658202
C2×C3×#40112
C3×#648662
#106042193
C3×#18285