Finite Group Databasebeta

← C3×#10604663 #14305998

Subgroups — C3×#10604663 #14305998

Order 1920.

720 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 156 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×#66059C211yesmaximal, characteristic
2C2×C3×#4010C214yesmaximal, complemented
3C3×C42×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C3×#65681C211yesmaximal, characteristic, complemented
5#10604663C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×#713C2212yes—
7C2×C3×C4×AGL(1,𝔽5)—22no—
8C2×C3×C4×AGL(1,𝔽5)C2212yes—
9C2×C3×#700C2212yes—
10C3×C5×C4⋊Q8C411yescharacteristic, complemented
11C3×#4093C412yescomplemented
12C3×#4041—22no—
13C3×#4039—22no—
14C3×#4020C411yescharacteristic, complemented
15C3×C42×D5C2211yescharacteristic
16C3×#4010—28no—
17C3×C4⋊#170—22no—
18C3×#1962—51nomaximal, Hall
19#66059C2×C311yescharacteristic
20C2×#4010C2×C314yescomplemented
21C42×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
22#65681C2×C311yescharacteristic, complemented
23C22×C3×AGL(1,𝔽5)—22no—
24C2×C3×C5×Q8C2×C412yes—
25C3×#713—24no—
26C3×#713—24no—
27C2×C3×D20—22no—
28C2×C3×D20—22no—
29C2×C3×D20C2×C412yes—
30C2×C3×C4×D5—22no—
31C2×C3×C4×D5D412yes—
32C2×C3×C4×D5C2311yescharacteristic
33C2×C3×#170—22no—
34C3×C4×AGL(1,𝔽5)—28no—
35C3×C4×AGL(1,𝔽5)—22no—
36C3×C4×AGL(1,𝔽5)D414yescomplemented
37C3×#700—24no—
38C3×C5×C4⋊C4—22no—
39C3×C42×C5C2×C411yescharacteristic
40C3×#685—24no—
41C3×C4×Dic5C2×C411yescharacteristic
42C3×C4×Dic5D412yes—
43C3×#532—51no—
44C3×#423—51no—
45C2×C3×#104—54no—
46C3×C43—51no—
47C2×#713C22×C312yes—
48C2×C4×AGL(1,𝔽5)—22no—
49C2×C4×AGL(1,𝔽5)C22×C312yes—
50C2×#700C22×C312yes—
51C5×C4⋊Q8C3×C411yescharacteristic, complemented
52#4093C3×C412yescomplemented
53#4041—22no—
54#4039—22no—
55#4020C3×C411yescharacteristic, complemented
56C42×D5C22×C311yescharacteristic
57#4010—28no—
58C4⋊#170—22no—
59#1962—51noSylow
60C22×C3×D5—22no—
61C22×C3×D5C2×D411yescharacteristic
62C2×C3×AGL(1,𝔽5)—28no—
63C2×C3×AGL(1,𝔽5)—24no—
64C3×C5×Q8—24no—
65C2×C3×C4×C5—22no—
66C2×C3×C4×C5C22⋊C412yes—
67C2×C3×C4×C5C22×C411yescharacteristic
68C2×C3×Dic5C22⋊C412yes—
69C2×C3×Dic5C2×D411yescharacteristic
70C3×D20—24no—
71C3×D20—44no—
72C3×D20—44no—
73C3×C4×D5—24no—
74C3×C4×D5—42no—
75C3×C4×D5—42no—
76C3×C4×D5C2×D412yes—
77C3×C4×D5#4112yes—
78C3×#170—42no—
79C2×C3×#41—52no—
80C2×C3×M16—52no—
81C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
82C3×#127—51no—
83C3×#124—52no—
84C3×#121—102no—
85C3×C4×D4—102no—
86C2×C3×C42—51no—
87C2×C3×C42—52no—
88C2×C3×C42—102no—
89C3×C4⋊C8—102no—
90C3×#104—108no—
91C22×AGL(1,𝔽5)—22no—
92C2×C5×Q8C2×C3×C412yes—
93#713—24no—
94#713—24no—
95C2×D20—22no—
96C2×D20—22no—
97C2×D20C2×C3×C412yes—
98C2×C4×D5—22no—
99C2×C4×D5C3×D412yes—
100C2×C4×D5C23×C311yescharacteristic
101C2×#170—22no—
102C4×AGL(1,𝔽5)—28no—
103C4×AGL(1,𝔽5)—22no—
104C4×AGL(1,𝔽5)C3×D414yescomplemented
105#700—24no—
106C5×C4⋊C4—22no—
107C42×C5C2×C3×C411yescharacteristic
108#685—24no—
109C4×Dic5C2×C3×C411yescharacteristic
110C4×Dic5C3×D412yes—
111#532—51no—
112#423—51no—
113C2×#104—54no—
114C43—51no—
115C22×C3×C5C2×C22⋊C411yescharacteristic
116C2×C3×D5—22no—
117C2×C3×D5—42no—
118C2×C3×D5—44no—
119C2×C3×D5#12112yes—
120C3×AGL(1,𝔽5)—28no—
121C3×C4×C5—42no—
122C3×C4×C5#10414yescomplemented
123C3×C4×C5C4×D412yes—
124C3×Dic5—22no—
125C3×Dic5#12411yescharacteristic
126C3×Dic5#12711yescharacteristic
127C3×#41—104no—
128C3×#41—104no—
129C2×C3×Q8C2×AGL(1,𝔽5)12yes—
130C2×C3×D4—52no—
131C2×C3×D4—102no—
132C2×C3×D4—102no—
133C22×C3×C4—51no—
134C22×C3×C4—52no—
135C22×C3×C4—102no—
136C22×C3×C4—102no—
137C3×M16—104no—
138C2×C3×C8—102no—
139C3×C4⋊C4—22no—
140C3×C22⋊C4—104no—
141C3×C42—52no—
142C3×C42—51no—
143C3×C42—54no—
144C3×C42—102no—
145C3×C42—108no—
146C3×C42C2×AGL(1,𝔽5)11yescharacteristic
147C22×D5—22no—
148C22×D5C2×C3×D411yescharacteristic
149C2×AGL(1,𝔽5)—28no—
150C2×AGL(1,𝔽5)—24no—
151C5×Q8—24no—
152C2×C4×C5—22no—
153C2×C4×C5C3×C22⋊C412yes—
154C2×C4×C5C22×C3×C411yescharacteristic
155C2×Dic5C3×C22⋊C412yes—
156C2×Dic5C2×C3×D411yescharacteristic
157D20—24no—
158D20—44no—
159D20—44no—
160C4×D5—24no—
161C4×D5—42no—
162C4×D5—42no—
163C4×D5C2×C3×D412yes—
164C4×D5C3×#4112yes—
165#170—42no—
166C2×#41—52no—
167C2×M16—52no—
168C4⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
169#127—51no—
170#124—52no—
171#121—102no—
172C4×D4—102no—
173C2×C42—51no—
174C2×C42—52no—
175C2×C42—102no—
176C4⋊C8—102no—
177#104—108no—
178C2×C3×C5#39011yescharacteristic
179C2×C3×C5C2×#10412yes—
180C3×D5—22no—
181C3×D5—82no—
182C23×C3—51no—
183C23×C3—102no—
184C3×Q8—24no—
185C3×D4—104no—
186C3×D4—204no—
187C3×D4—204no—
188C2×C3×C4—22no—
189C2×C3×C4—51no—
190C2×C3×C4—52no—
191C2×C3×C4—52no—
192C2×C3×C4—52no—
193C2×C3×C4—104no—
194C2×C3×C4—108no—
195C2×C3×C4—104no—
196C2×C3×C4—202no—
197C2×C3×C4—202no—
198C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
199C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
200C3×C8—202no—
201C22×C5C2×C3×C22⋊C411yescharacteristic
202C2×D5—22no—
203C2×D5—42no—
204C2×D5—44no—
205C2×D5C3×#12112yes—
206AGL(1,𝔽5)—28no—
207C4×C5—42no—
208C4×C5C3×#10414yescomplemented
209C4×C5C3×C4×D412yes—
210Dic5—22no—
211Dic5C3×#12411yescharacteristic
212Dic5C3×#12711yescharacteristic
213#41—104no—
214#41—104no—
215C2×Q8C2×C3×AGL(1,𝔽5)12yes—
216C2×D4—52no—
217C2×D4—102no—
218C2×D4—102no—
219C22×C4—51no—
220C22×C4—52no—
221C22×C4—102no—
222C22×C4—102no—
223M16—104no—
224C2×C8—102no—
225C4⋊C4—22no—
226C22⋊C4—104no—
227C42—52no—
228C42—51no—
229C42—54no—
230C42—102no—
231C42—108no—
232C42C2×C3×AGL(1,𝔽5)11yescharacteristic
233C3×C5#196211yescharacteristic, complemented, Hall
234C22×C3—52no—
235C22×C3—102no—
236C22×C3—202no—
237C22×C3—204no—
238C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
239C3×C4—42no—
240C3×C4—51no—
241C3×C4—51no—
242C3×C4—108no—
243C3×C4—102no—
244C3×C4#401014yescomplemented
245C3×C4D4×AGL(1,𝔽5)12yes—
246C2×C5C3×#39011yescharacteristic
247C2×C5C2×C3×#10412yes—
248D5—22no—
249D5—82no—
250C23—51no—
251C23—102no—
252Q8—24no—
253D4—104no—
254D4—204no—
255D4—204no—
256C2×C4—22no—
257C2×C4—51no—
258C2×C4—52no—
259C2×C4—52no—
260C2×C4—52no—
261C2×C4—104no—
262C2×C4—108no—
263C2×C4—104no—
264C2×C4—202no—
265C2×C4—202no—
266C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
267C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
268C8—202no—
269C2×C3—102no—
270C2×C3—402no—
271C2×C3#6578011yescharacteristic
272C2×C3C2×#401012yes—
273C5C3×#196211yescharacteristic, complemented, Sylow
274C22—52no—
275C22—102no—
276C22—202no—
277C22—204no—
278C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
279C4—42no—
280C4—51no—
281C4—51no—
282C4—108no—
283C4—102no—
284C4C3×#401014yescomplemented
285C4C3×D4×AGL(1,𝔽5)12yes—
286C3#1060466311yescharacteristic, direct product factor, Sylow
287C2—102no—
288C2—402no—
289C2C3×#6578011yescharacteristic
290C2C2×C3×#401012yes—
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C4⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66059C2
cosocleC2×C4×D5C23×C3
hypercenterC3×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1962
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4010C2semidirect
3. C3×C42×AGL(1,𝔽5)C2semidirect
4. C3×#65681C2semidirect
5. #10604663C3direct (N × H)
10. C3×C5×C4⋊Q8C4semidirect
11. C3×#4093C4semidirect
14. C3×#4020C4semidirect
20. C2×#4010C2×C3semidirect
21. C42×AGL(1,𝔽5)C2×C3semidirect
22. #65681C2×C3semidirect
36. C3×C4×AGL(1,𝔽5)D4semidirect
51. C5×C4⋊Q8C3×C4semidirect
52. #4093C3×C4semidirect
55. #4020C3×C4semidirect
81. C3×C4⋊Q8AGL(1,𝔽5)semidirect
104. C4×AGL(1,𝔽5)C3×D4semidirect
122. C3×C4×C5#104semidirect
168. C4⋊Q8C3×AGL(1,𝔽5)semidirect
208. C4×C5C3×#104semidirect
233. C3×C5#1962semidirect
244. C3×C4#4010semidirect
273. C5C3×#1962semidirect
284. C4C3×#4010semidirect
286. C3#10604663direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660592
C2×C3×#40102
C3×C42×AGL(1,𝔽5)2
C3×#656812
#106046633
C3×#19625