Finite Group Databasebeta

← C3×#10604611 #14305996

Subgroups — C3×#10604611 #14305996

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×#66050C211yesmaximal, characteristic
2C2×C3×#4008C214yesmaximal, complemented
3C3×C42×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C3×#65681C211yesmaximal, characteristic, complemented
5#10604611C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×#711C2212yes—
7C2×C3×C4×AGL(1,𝔽5)—22no—
8C2×C3×C4×AGL(1,𝔽5)C2212yes—
9C2×C3×#700C2212yes—
10C3×C5×#127C411yescharacteristic, complemented
11C3×#4084—22no—
12C3×#4082C412yescomplemented
13C3×D4×Dic5—22no—
14C3×C42×D5C2211yescharacteristic
15C3×#4015C411yescharacteristic, complemented
16C3×#4008—28no—
17C3×C4⋊#170—22no—
18C3×#1962—51nomaximal, Hall
19#66050C2×C311yescharacteristic
20C2×#4008C2×C314yescomplemented
21C42×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
22#65681C2×C311yescharacteristic, complemented
23C22×C3×AGL(1,𝔽5)—22no—
24C2×C3×C5×D4—22no—
25C2×C3×C5×D4C2×C412yes—
26C2×C3×C22⋊D5—22no—
27C22×C3×Dic5—22no—
28C3×#711—24no—
29C3×#711—24no—
30C2×C3×C4×D5D412yes—
31C2×C3×C4×D5C2311yescharacteristic
32C2×C3×Dic10C2×C412yes—
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×C42×C5C2×C411yescharacteristic
39C3×C22⋊Dic5—24no—
40C3×C4⋊Dic5—22no—
41C3×C4×Dic5C2×C411yescharacteristic
42C3×C4×Dic5D412yes—
43C3×#532—51no—
44C3×#423—51no—
45C2×C3×#104—54no—
46C3×C43—51no—
47C2×#711C22×C312yes—
48C2×C4×AGL(1,𝔽5)—22no—
49C2×C4×AGL(1,𝔽5)C22×C312yes—
50C2×#700C22×C312yes—
51C5×#127C3×C411yescharacteristic, complemented
52#4084—22no—
53#4082C3×C412yescomplemented
54D4×Dic5—22no—
55C42×D5C22×C311yescharacteristic
56#4015C3×C411yescharacteristic, complemented
57#4008—28no—
58C4⋊#170—22no—
59#1962—51noSylow
60C23×C3×C5—22no—
61C22×C3×D5C2×D411yescharacteristic
62C2×C3×AGL(1,𝔽5)—28no—
63C2×C3×AGL(1,𝔽5)—24no—
64C3×C5×D4—24no—
65C3×C5×D4—44no—
66C2×C3×C4×C5C22⋊C412yes—
67C2×C3×C4×C5C22×C411yescharacteristic
68C3×C22⋊D5—44no—
69C2×C3×Dic5—22no—
70C2×C3×Dic5—42no—
71C2×C3×Dic5—42no—
72C2×C3×Dic5C22⋊C412yes—
73C2×C3×Dic5C2×D411yescharacteristic
74C3×C4×D5—24no—
75C3×C4×D5C2×D412yes—
76C3×C4×D5#4112yes—
77C3×Dic10—24no—
78C3×#170—42no—
79C2×C3×#41—52no—
80C2×C3×M16—52no—
81C3×C4⋊Q8—51no—
82C3×#127AGL(1,𝔽5)11yescharacteristic, complemented
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×D4—22no—
93C2×C5×D4C2×C3×C412yes—
94C2×C22⋊D5—22no—
95C22×Dic5—22no—
96#711—24no—
97#711—24no—
98C2×C4×D5C3×D412yes—
99C2×C4×D5C23×C311yescharacteristic
100C2×Dic10C2×C3×C412yes—
101C2×#170—22no—
102C4×AGL(1,𝔽5)—28no—
103C4×AGL(1,𝔽5)—22no—
104C4×AGL(1,𝔽5)C3×D414yescomplemented
105#700—24no—
106C42×C5C2×C3×C411yescharacteristic
107C22⋊Dic5—24no—
108C4⋊Dic5—22no—
109C4×Dic5C2×C3×C411yescharacteristic
110C4×Dic5C3×D412yes—
111#532—51no—
112#423—51no—
113C2×#104—54no—
114C43—51no—
115C22×C3×C5—42no—
116C22×C3×C5—44no—
117C22×C3×C5C2×C22⋊C411yescharacteristic
118C2×C3×D5—22no—
119C2×C3×D5#12112yes—
120C3×AGL(1,𝔽5)—28no—
121C3×C4×C5#10414yescomplemented
122C3×C4×C5C4×D412yes—
123C3×Dic5—22no—
124C3×Dic5—42no—
125C3×Dic5#12411yescharacteristic
126C3×Dic5#12711yescharacteristic
127C3×#41—104no—
128C3×#41—104no—
129C2×C3×Q8—52no—
130C2×C3×D4—22no—
131C2×C3×D4—102no—
132C2×C3×D4C2×AGL(1,𝔽5)12yes—
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—102no—
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
147C23×C5—22no—
148C22×D5C2×C3×D411yescharacteristic
149C2×AGL(1,𝔽5)—28no—
150C2×AGL(1,𝔽5)—24no—
151C5×D4—24no—
152C5×D4—44no—
153C2×C4×C5C3×C22⋊C412yes—
154C2×C4×C5C22×C3×C411yescharacteristic
155C22⋊D5—44no—
156C2×Dic5—22no—
157C2×Dic5—42no—
158C2×Dic5—42no—
159C2×Dic5C3×C22⋊C412yes—
160C2×Dic5C2×C3×D411yescharacteristic
161C4×D5—24no—
162C4×D5C2×C3×D412yes—
163C4×D5C3×#4112yes—
164Dic10—24no—
165#170—42no—
166C2×#41—52no—
167C2×M16—52no—
168C4⋊Q8—51no—
169#127C3×AGL(1,𝔽5)11yescharacteristic, complemented
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—82no—
179C2×C3×C5#39011yescharacteristic
180C2×C3×C5C2×#10412yes—
181C3×D5—22no—
182C23×C3—22no—
183C23×C3—51no—
184C3×Q8—104no—
185C3×D4—24no—
186C3×D4—44no—
187C3×D4—204no—
188C2×C3×C4—51no—
189C2×C3×C4—52no—
190C2×C3×C4—52no—
191C2×C3×C4—52no—
192C2×C3×C4—104no—
193C2×C3×C4—108no—
194C2×C3×C4—104no—
195C2×C3×C4—102no—
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×C5—42no—
202C22×C5—44no—
203C22×C5C2×C3×C22⋊C411yescharacteristic
204C2×D5—22no—
205C2×D5C3×#12112yes—
206AGL(1,𝔽5)—28no—
207C4×C5C3×#10414yescomplemented
208C4×C5C3×C4×D412yes—
209Dic5—22no—
210Dic5—42no—
211Dic5C3×#12411yescharacteristic
212Dic5C3×#12711yescharacteristic
213#41—104no—
214#41—104no—
215C2×Q8—52no—
216C2×D4—22no—
217C2×D4—102no—
218C2×D4C2×C3×AGL(1,𝔽5)12yes—
219C22×C4—51no—
220C22×C4—52no—
221C22×C4—102no—
222C22×C4—102no—
223M16—104no—
224C2×C8—102no—
225C4⋊C4—102no—
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—44no—
235C22×C3—42no—
236C22×C3—52no—
237C22×C3—102no—
238C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
239C3×C4—51no—
240C3×C4—51no—
241C3×C4—108no—
242C3×C4—102no—
243C3×C4—202no—
244C3×C4#400814yescomplemented
245C3×C4D4×AGL(1,𝔽5)12yes—
246C2×C5—82no—
247C2×C5C3×#39011yescharacteristic
248C2×C5C2×C3×#10412yes—
249D5—22no—
250C23—22no—
251C23—51no—
252Q8—104no—
253D4—24no—
254D4—44no—
255D4—204no—
256C2×C4—51no—
257C2×C4—52no—
258C2×C4—52no—
259C2×C4—52no—
260C2×C4—104no—
261C2×C4—108no—
262C2×C4—104no—
263C2×C4—102no—
264C2×C4—202no—
265C2×C4—202no—
266C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
267C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
268C8—202no—
269C2×C3—82no—
270C2×C3—102no—
271C2×C3C2×#400812yes—
272C2×C3#6578011yescharacteristic
273C5C3×#196211yescharacteristic, complemented, Sylow
274C22—44no—
275C22—42no—
276C22—52no—
277C22—102no—
278C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
279C4—51no—
280C4—51no—
281C4—108no—
282C4—102no—
283C4—202no—
284C4C3×#400814yescomplemented
285C4C3×D4×AGL(1,𝔽5)12yes—
286C3#1060461111yescharacteristic, direct product factor, Sylow
287C2—82no—
288C2—102no—
289C2C2×C3×#400812yes—
290C2C3×#6578011yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#127C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66050C2
cosocleC2×C4×D5C23×C3
hypercenterC3×#127AGL(1,𝔽5)
hypocenterC5C3×#1962
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4008C2semidirect
3. C3×C42×AGL(1,𝔽5)C2semidirect
4. C3×#65681C2semidirect
5. #10604611C3direct (N × H)
10. C3×C5×#127C4semidirect
12. C3×#4082C4semidirect
15. C3×#4015C4semidirect
20. C2×#4008C2×C3semidirect
21. C42×AGL(1,𝔽5)C2×C3semidirect
22. #65681C2×C3semidirect
36. C3×C4×AGL(1,𝔽5)D4semidirect
51. C5×#127C3×C4semidirect
53. #4082C3×C4semidirect
56. #4015C3×C4semidirect
82. C3×#127AGL(1,𝔽5)semidirect
104. C4×AGL(1,𝔽5)C3×D4semidirect
121. C3×C4×C5#104semidirect
169. #127C3×AGL(1,𝔽5)semidirect
207. C4×C5C3×#104semidirect
233. C3×C5#1962semidirect
244. C3×C4#4008semidirect
273. C5C3×#1962semidirect
284. C4C3×#4008semidirect
286. C3#10604611direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660502
C2×C3×#40082
C3×C42×AGL(1,𝔽5)2
C3×#656812
#106046113
C3×#19625