Finite Group Databasebeta

← C3×#10604942 #14296867

Subgroups — C3×#10604942 #14296867

Order 1920.

1224 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 172 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×#711C211yesmaximal, characteristic
2C2×C3×#4011C214yesmaximal, complemented
3C3×#65764C212yesmaximal, complemented
4#10604942C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C5×D4C411yescharacteristic, complemented
6C22×C3×C22⋊D5C2212yes—
7C23×C3×Dic5C412yescomplemented
8C2×C3×#711—28no—
9C2×C3×#711—24no—
10C22×C3×C4×D5C2211yescharacteristic
11C22×C3×Dic10C411yescharacteristic, complemented
12C2×C3×C22⋊AGL(1,𝔽5)C2214yes—
13C3×#4011—28no—
14C3×(C2×AGL(1,𝔽5))⋊C4—24no—
15C3×#1782—51nomaximal, Hall
16C22×#711C2×C311yescharacteristic
17C2×#4011C2×C314yescomplemented
18#65764C2×C312yescomplemented
19C24×C3×C5C2×C412yes—
20C23×C3×D5C2311yescharacteristic
21C22×C3×AGL(1,𝔽5)—24no—
22C2×C3×C5×D4—24no—
23C2×C3×C5×D4—22no—
24C22×C3×C4×C5C2×C411yescharacteristic
25C2×C3×C22⋊D5—24no—
26C2×C3×C22⋊D5—24no—
27C2×C3×C22⋊D5D418yes—
28C22×C3×Dic5—24no—
29C22×C3×Dic5—24no—
30C22×C3×Dic5—24no—
31C22×C3×Dic5C2×C412yes—
32C22×C3×Dic5C2×C411yescharacteristic
33C3×#711—232no—
34C2×C3×C4×D5—24no—
35C2×C3×C4×D5D414yes—
36C2×C3×Dic10—22no—
37C2×C3×Dic10—24no—
38C3×C22⋊AGL(1,𝔽5)—28no—
39C22×C3×#41—51no—
40C2×C3×#99—54no—
41C3×#381—52no—
42C22×C5×D4C3×C411yescharacteristic, complemented
43C22×C22⋊D5C22×C312yes—
44C23×Dic5C3×C412yescomplemented
45C2×#711—28no—
46C2×#711—24no—
47C22×C4×D5C22×C311yescharacteristic
48C22×Dic10C3×C411yescharacteristic, complemented
49C2×C22⋊AGL(1,𝔽5)C22×C314yes—
50#4011—28no—
51(C2×AGL(1,𝔽5))⋊C4—24no—
52#1782—51noSylow
53C23×C3×C5—22no—
54C23×C3×C5—22no—
55C23×C3×C5—44no—
56C23×C3×C5C22⋊C414yes—
57C23×C3×C5C22×C411yescharacteristic
58C22×C3×D5—42no—
59C22×C3×D5C2×D412yes—
60C22×C3×D5C2×D414yes—
61C2×C3×AGL(1,𝔽5)—48no—
62C2×C3×AGL(1,𝔽5)—44no—
63C3×C5×D4—28no—
64C2×C3×C4×C5—22no—
65C2×C3×C4×C5C22⋊C412yes—
66C3×C22⋊D5—216no—
67C2×C3×Dic5—216no—
68C2×C3×Dic5—22no—
69C2×C3×Dic5—22no—
70C2×C3×Dic5—22no—
71C2×C3×Dic5C22⋊C414yes—
72C2×C3×Dic5C22⋊C412yes—
73C3×C4×D5—28no—
74C3×Dic10—28no—
75C2×C3×#41—104no—
76C2×C3×#41—108no—
77C22×C3×Q8—51no—
78C22×C3×D4—52no—
79C22×C3×D4AGL(1,𝔽5)11yescharacteristic, complemented
80C23×C3×C4—51no—
81C23×C3×C4—52no—
82C2×C3×C22⋊C4—54no—
83C3×#99—108no—
84C3×#95—104no—
85C24×C5C2×C3×C412yes—
86C23×D5C23×C311yescharacteristic
87C22×AGL(1,𝔽5)—24no—
88C2×C5×D4—24no—
89C2×C5×D4—22no—
90C22×C4×C5C2×C3×C411yescharacteristic
91C2×C22⋊D5—24no—
92C2×C22⋊D5—24no—
93C2×C22⋊D5C3×D418yes—
94C22×Dic5—24no—
95C22×Dic5—24no—
96C22×Dic5—24no—
97C22×Dic5C2×C3×C412yes—
98C22×Dic5C2×C3×C411yescharacteristic
99#711—232no—
100C2×C4×D5—24no—
101C2×C4×D5C3×D414yes—
102C2×Dic10—22no—
103C2×Dic10—24no—
104C22⋊AGL(1,𝔽5)—28no—
105C22×#41—51no—
106C2×#99—54no—
107#381—52no—
108C22×C3×C5—22no—
109C22×C3×C5—24no—
110C22×C3×C5—44no—
111C22×C3×C5—44no—
112C22×C3×C5—44no—
113C22×C3×C5C2×C22⋊C412yes—
114C22×C3×C5C2×C22⋊C411yescharacteristic
115C2×C3×D5—24no—
116C2×C3×D5—44no—
117C2×C3×D5#12014yes—
118C3×AGL(1,𝔽5)—84no—
119C3×C4×C5—22no—
120C3×Dic5—24no—
121C3×Dic5—22no—
122C24×C3—51no—
123C24×C3C2×AGL(1,𝔽5)12yes—
124C3×#41—1032no—
125C2×C3×Q8—104no—
126C2×C3×Q8—102no—
127C2×C3×D4—24no—
128C2×C3×D4—22no—
129C2×C3×D4—58no—
130C2×C3×D4—104no—
131C2×C3×D4—104no—
132C22×C3×C4—51no—
133C22×C3×C4—52no—
134C22×C3×C4—54no—
135C22×C3×C4—104no—
136C22×C3×C4—104no—
137C22×C3×C4—104no—
138C22×C3×C4—104no—
139C22×C3×C4—104no—
140C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
141C3×C22⋊C4—108no—
142C23×C5—22no—
143C23×C5—22no—
144C23×C5—44no—
145C23×C5C3×C22⋊C414yes—
146C23×C5C22×C3×C411yescharacteristic
147C22×D5—42no—
148C22×D5C2×C3×D412yes—
149C22×D5C2×C3×D414yes—
150C2×AGL(1,𝔽5)—48no—
151C2×AGL(1,𝔽5)—44no—
152C5×D4—28no—
153C2×C4×C5—22no—
154C2×C4×C5C3×C22⋊C412yes—
155C22⋊D5—216no—
156C2×Dic5—216no—
157C2×Dic5—22no—
158C2×Dic5—22no—
159C2×Dic5—22no—
160C2×Dic5C3×C22⋊C414yes—
161C2×Dic5C3×C22⋊C412yes—
162C4×D5—28no—
163Dic10—28no—
164C2×#41—104no—
165C2×#41—108no—
166C22×Q8—51no—
167C22×D4—52no—
168C22×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
169C23×C4—51no—
170C23×C4—52no—
171C2×C22⋊C4—54no—
172#99—108no—
173#95—104no—
174C2×C3×C5—22no—
175C2×C3×C5—44no—
176C2×C3×C5#37911yescharacteristic
177C2×C3×C5#41012yes—
178C3×D5—24no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—44no—
182C23×C3—52no—
183C23×C3—54no—
184C23×C3—202no—
185C23×C3C22⋊AGL(1,𝔽5)14yes—
186C23×C3C22×AGL(1,𝔽5)11yescharacteristic
187C3×Q8—108no—
188C3×D4—28no—
189C3×D4—1016no—
190C2×C3×C4—22no—
191C2×C3×C4—54no—
192C2×C3×C4—52no—
193C2×C3×C4—108no—
194C2×C3×C4—1016no—
195C2×C3×C4—102no—
196C2×C3×C4—102no—
197C2×C3×C4—102no—
198C2×C3×C4—204no—
199C2×C3×C4—208no—
200C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
201C22×C5—22no—
202C22×C5—24no—
203C22×C5—44no—
204C22×C5—44no—
205C22×C5—44no—
206C22×C5C2×C3×C22⋊C412yes—
207C22×C5C2×C3×C22⋊C411yescharacteristic
208C2×D5—24no—
209C2×D5—44no—
210C2×D5C3×#12014yes—
211AGL(1,𝔽5)—84no—
212C4×C5—22no—
213Dic5—24no—
214Dic5—22no—
215C24—51no—
216C24C2×C3×AGL(1,𝔽5)12yes—
217#41—1032no—
218C2×Q8—104no—
219C2×Q8—102no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—58no—
223C2×D4—104no—
224C2×D4—104no—
225C22×C4—51no—
226C22×C4—52no—
227C22×C4—54no—
228C22×C4—104no—
229C22×C4—104no—
230C22×C4—104no—
231C22×C4—104no—
232C22×C4—104no—
233C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
234C22⋊C4—108no—
235C3×C5#178211yescharacteristic, complemented, Hall
236C22×C3—22no—
237C22×C3—24no—
238C22×C3—44no—
239C22×C3—44no—
240C22×C3—44no—
241C22×C3—54no—
242C22×C3—104no—
243C22×C3—204no—
244C22×C3C2×C22⋊AGL(1,𝔽5)12yes—
245C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
246C3×C4—22no—
247C3×C4—102no—
248C3×C4—104no—
249C3×C4—404no—
250C2×C5—22no—
251C2×C5—44no—
252C2×C5C3×#37911yescharacteristic
253C2×C5C3×#41012yes—
254D5—24no—
255C23—22no—
256C23—22no—
257C23—44no—
258C23—52no—
259C23—54no—
260C23—202no—
261C23C3×C22⋊AGL(1,𝔽5)14yes—
262C23C22×C3×AGL(1,𝔽5)11yescharacteristic
263Q8—108no—
264D4—28no—
265D4—1016no—
266C2×C4—22no—
267C2×C4—52no—
268C2×C4—54no—
269C2×C4—108no—
270C2×C4—1016no—
271C2×C4—102no—
272C2×C4—102no—
273C2×C4—102no—
274C2×C4—204no—
275C2×C4—208no—
276C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
277C2×C3—22no—
278C2×C3—44no—
279C2×C3—104no—
280C2×C3#6577112yes—
281C2×C3#6580011yescharacteristic
282C5C3×#178211yescharacteristic, complemented, Sylow
283C22—22no—
284C22—24no—
285C22—44no—
286C22—44no—
287C22—44no—
288C22—54no—
289C22—104no—
290C22—204no—
291C22C2×C3×C22⋊AGL(1,𝔽5)12yes—
292C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
293C4—22no—
294C4—102no—
295C4—104no—
296C4—404no—
297C3#1060494211yescharacteristic, direct product factor, Sylow
298C2—22no—
299C2—44no—
300C2—104no—
301C2C3×#6577112yes—
302C2C3×#6580011yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC22×C3×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×#711C2
cosocleC23×D5C23×C3
hypercenterC22×C3×D4AGL(1,𝔽5)
hypocenterC5C3×#1782
LCSG ▷ C23×C5 ▷ C2×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. C2×C3×#4011C2semidirect
3. C3×#65764C2semidirect
4. #10604942C3direct (N × H)
5. C22×C3×C5×D4C4semidirect
7. C23×C3×Dic5C4semidirect
11. C22×C3×Dic10C4semidirect
17. C2×#4011C2×C3semidirect
18. #65764C2×C3semidirect
42. C22×C5×D4C3×C4semidirect
44. C23×Dic5C3×C4semidirect
48. C22×Dic10C3×C4semidirect
79. C22×C3×D4AGL(1,𝔽5)semidirect
168. C22×D4C3×AGL(1,𝔽5)semidirect
235. C3×C5#1782semidirect
282. C5C3×#1782semidirect
297. C3#10604942direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7112
C2×C3×#40112
C3×#657642
#106049423
C3×#17825