Finite Group Databasebeta

← C3×#10598705 #14277355

Subgroups — C3×#10598705 #14277355

Order 1920.

792 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 96 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×#65800C212yesmaximal
2C3×#65244C211yesmaximal, characteristic
3#10598705C311yesmaximal, characteristic, direct product factor, Hall
4C24×C3×D5C2211yescharacteristic
5C2×C3×C22⋊AGL(1,𝔽5)—22no—
6C2×C3×C22⋊AGL(1,𝔽5)—22no—
7C2×C3×C22⋊AGL(1,𝔽5)C414yescomplemented
8C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
9C2×C3×C22⋊Dic5C411yescharacteristic, complemented
10C2×C3×#685—22no—
11C3×#1304—51nomaximal, Hall
12#65800C2×C312yes—
13#65244C2×C311yescharacteristic
14C24×C3×C5C2×C411yescharacteristic
15C23×C3×D5—44no—
16C23×C3×D5—41no—
17C23×C3×D5—41no—
18C23×C3×D5C2×C412yes—
19C23×C3×D5D411yescharacteristic
20C23×C3×D5D412yes—
21C23×C3×D5Q811yescharacteristic
22C22×C3×AGL(1,𝔽5)—24no—
23C22×C3×C4×C5—21no—
24C22×C3×Dic5—21no—
25C3×C22⋊AGL(1,𝔽5)—42no—
26C3×C22⋊AGL(1,𝔽5)—42no—
27C3×C22⋊AGL(1,𝔽5)—44no—
28C3×C22⋊AGL(1,𝔽5)—44no—
29C3×C5×C22⋊C4—41no—
30C3×C22⋊Dic5—41no—
31C3×#685—44no—
32C3×#379—52no—
33C3×#379—51no—
34C24×D5C22×C311yescharacteristic
35C2×C22⋊AGL(1,𝔽5)—22no—
36C2×C22⋊AGL(1,𝔽5)—22no—
37C2×C22⋊AGL(1,𝔽5)C3×C414yescomplemented
38C2×C5×C22⋊C4C3×C411yescharacteristic, complemented
39C2×C22⋊Dic5C3×C411yescharacteristic, complemented
40C2×#685—22no—
41#1304—51noSylow
42C23×C3×C5—21no—
43C23×C3×C5—21no—
44C23×C3×C5—42no—
45C23×C3×C5C4211yescharacteristic
46C23×C3×C5C22⋊C411yescharacteristic
47C23×C3×C5C4⋊C411yescharacteristic
48C22×C3×D5—22no—
49C22×C3×D5—22no—
50C22×C3×D5—24no—
51C22×C3×D5—24no—
52C22×C3×D5—24no—
53C22×C3×D5—42no—
54C22×C3×D5—42no—
55C22×C3×D5—41no—
56C22×C3×D5—42no—
57C22×C3×D5—42no—
58C22×C3×D5—41no—
59C22×C3×D5—48no—
60C22×C3×D5—48no—
61C22×C3×D5C22⋊C412yes—
62C22×C3×D5C4⋊C412yes—
63C2×C3×AGL(1,𝔽5)—44no—
64C2×C3×AGL(1,𝔽5)—44no—
65C2×C3×AGL(1,𝔽5)—44no—
66C2×C3×C4×C5—42no—
67C2×C3×C4×C5—41no—
68C2×C3×Dic5—42no—
69C2×C3×Dic5—41no—
70C25×C3—51no—
71C2×C3×C22⋊C4—54no—
72C2×C3×C22⋊C4—51no—
73C2×C3×C22⋊C4—102no—
74C2×C3×C22⋊C4—102no—
75C2×C3×C22⋊C4—102no—
76C2×C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
77C24×C5C2×C3×C411yescharacteristic
78C23×D5—44no—
79C23×D5—41no—
80C23×D5—41no—
81C23×D5C2×C3×C412yes—
82C23×D5C3×D411yescharacteristic
83C23×D5C3×D412yes—
84C23×D5C3×Q811yescharacteristic
85C22×AGL(1,𝔽5)—24no—
86C22×C4×C5—21no—
87C22×Dic5—21no—
88C22⋊AGL(1,𝔽5)—42no—
89C22⋊AGL(1,𝔽5)—42no—
90C22⋊AGL(1,𝔽5)—44no—
91C22⋊AGL(1,𝔽5)—44no—
92C5×C22⋊C4—41no—
93C22⋊Dic5—41no—
94#685—44no—
95#379—52no—
96#379—51no—
97C22×C3×C5—22no—
98C22×C3×C5—22no—
99C22×C3×C5—22no—
100C22×C3×C5—21no—
101C22×C3×C5—21no—
102C22×C3×C5—42no—
103C22×C3×C5—42no—
104C22×C3×C5#9511yescharacteristic
105C22×C3×C5#9911yescharacteristic
106C22×C3×C5#9911yescharacteristic
107C2×C3×D5—22no—
108C2×C3×D5—22no—
109C2×C3×D5—22no—
110C2×C3×D5—22no—
111C2×C3×D5—22no—
112C2×C3×D5—44no—
113C2×C3×D5—44no—
114C2×C3×D5—44no—
115C2×C3×D5—48no—
116C2×C3×D5—44no—
117C2×C3×D5#9914yes—
118C3×AGL(1,𝔽5)—84no—
119C3×C4×C5—81no—
120C3×Dic5—81no—
121C24×C3—51no—
122C24×C3—52no—
123C24×C3—52no—
124C24×C3—51no—
125C24×C3—201no—
126C24×C3—201no—
127C24×C3—204no—
128C24×C3C2×AGL(1,𝔽5)11yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4—104no—
131C22×C3×C4—101no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—204no—
134C3×C22⋊C4—204no—
135C3×C22⋊C4—202no—
136C3×C22⋊C4—202no—
137C3×C22⋊C4—201no—
138C3×C22⋊C4—204no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—42no—
142C23×C5C3×C4211yescharacteristic
143C23×C5C3×C22⋊C411yescharacteristic
144C23×C5C3×C4⋊C411yescharacteristic
145C22×D5—22no—
146C22×D5—22no—
147C22×D5—24no—
148C22×D5—24no—
149C22×D5—24no—
150C22×D5—42no—
151C22×D5—42no—
152C22×D5—41no—
153C22×D5—42no—
154C22×D5—42no—
155C22×D5—41no—
156C22×D5—48no—
157C22×D5—48no—
158C22×D5C3×C22⋊C412yes—
159C22×D5C3×C4⋊C412yes—
160C2×AGL(1,𝔽5)—44no—
161C2×AGL(1,𝔽5)—44no—
162C2×AGL(1,𝔽5)—44no—
163C2×C4×C5—42no—
164C2×C4×C5—41no—
165C2×Dic5—42no—
166C2×Dic5—41no—
167C25—51no—
168C2×C22⋊C4—54no—
169C2×C22⋊C4—51no—
170C2×C22⋊C4—102no—
171C2×C22⋊C4—102no—
172C2×C22⋊C4—102no—
173C2×C22⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
174C2×C3×C5—21no—
175C2×C3×C5—21no—
176C2×C3×C5—41no—
177C2×C3×C5—41no—
178C2×C3×C5#34212yes—
179C2×C3×C5#34211yescharacteristic
180C3×D5—24no—
181C3×D5—42no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—42no—
185C23×C3—52no—
186C23×C3—52no—
187C23×C3—104no—
188C23×C3—104no—
189C23×C3—104no—
190C23×C3—102no—
191C23×C3—102no—
192C23×C3—202no—
193C23×C3—202no—
194C23×C3—202no—
195C23×C3—202no—
196C23×C3—201no—
197C23×C3—201no—
198C23×C3—208no—
199C23×C3—208no—
200C23×C3C4×AGL(1,𝔽5)11yescharacteristic
201C23×C3C4⋊AGL(1,𝔽5)11yescharacteristic
202C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—202no—
206C2×C3×C4—204no—
207C2×C3×C4—204no—
208C2×C3×C4—204no—
209C2×C3×C4—201no—
210C22×C5—22no—
211C22×C5—22no—
212C22×C5—22no—
213C22×C5—21no—
214C22×C5—21no—
215C22×C5—42no—
216C22×C5—42no—
217C22×C5C3×#9511yescharacteristic
218C22×C5C3×#9911yescharacteristic
219C22×C5C3×#9911yescharacteristic
220C2×D5—22no—
221C2×D5—22no—
222C2×D5—22no—
223C2×D5—22no—
224C2×D5—22no—
225C2×D5—44no—
226C2×D5—44no—
227C2×D5—44no—
228C2×D5—48no—
229C2×D5—44no—
230C2×D5C3×#9914yes—
231AGL(1,𝔽5)—84no—
232C4×C5—81no—
233Dic5—81no—
234C24—51no—
235C24—52no—
236C24—52no—
237C24—51no—
238C24—201no—
239C24—201no—
240C24—204no—
241C24C2×C3×AGL(1,𝔽5)11yescharacteristic
242C22×C4—21no—
243C22×C4—104no—
244C22×C4—101no—
245C22⋊C4—41no—
246C22⋊C4—204no—
247C22⋊C4—204no—
248C22⋊C4—202no—
249C22⋊C4—202no—
250C22⋊C4—201no—
251C22⋊C4—204no—
252C3×C5#130411yescharacteristic, complemented, Hall
253C22×C3—22no—
254C22×C3—22no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—42no—
259C22×C3—42no—
260C22×C3—54no—
261C22×C3—102no—
262C22×C3—102no—
263C22×C3—102no—
264C22×C3—102no—
265C22×C3—102no—
266C22×C3—204no—
267C22×C3—204no—
268C22×C3—204no—
269C22×C3—208no—
270C22×C3—204no—
271C22×C3#399911yescharacteristic
272C22×C3(C2×AGL(1,𝔽5))⋊C411yescharacteristic
273C22×C3#401111yescharacteristic
274C3×C4—81no—
275C3×C4—401no—
276C3×C4—404no—
277C2×C5—21no—
278C2×C5—21no—
279C2×C5—41no—
280C2×C5—41no—
281C2×C5C3×#34212yes—
282C2×C5C3×#34211yescharacteristic
283D5—24no—
284D5—42no—
285C23—21no—
286C23—21no—
287C23—42no—
288C23—52no—
289C23—52no—
290C23—104no—
291C23—104no—
292C23—104no—
293C23—102no—
294C23—102no—
295C23—202no—
296C23—202no—
297C23—202no—
298C23—202no—
299C23—201no—
300C23—201no—
301C23—208no—
302C23—208no—
303C23C3×C4×AGL(1,𝔽5)11yescharacteristic
304C23C3×C4⋊AGL(1,𝔽5)11yescharacteristic
305C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—202no—
309C2×C4—204no—
310C2×C4—204no—
311C2×C4—204no—
312C2×C4—201no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—104no—
318C2×C3—202no—
319C2×C3#6486612yes—
320C2×C3#6491711yescharacteristic
321C5C3×#130411yescharacteristic, complemented, Sylow
322C22—22no—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—21no—
327C22—42no—
328C22—42no—
329C22—54no—
330C22—102no—
331C22—102no—
332C22—102no—
333C22—102no—
334C22—102no—
335C22—204no—
336C22—204no—
337C22—204no—
338C22—208no—
339C22—204no—
340C22C3×#399911yescharacteristic
341C22C3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
342C22C3×#401111yescharacteristic
343C4—81no—
344C4—401no—
345C4—404no—
346C3#1059870511yescharacteristic, direct product factor, Sylow
347C2—21no—
348C2—21no—
349C2—41no—
350C2—41no—
351C2—104no—
352C2—202no—
353C2C3×#6486612yes—
354C2C3×#6491711yescharacteristic
subgroupG/H
centerC22×C3(C2×AGL(1,𝔽5))⋊C4
commutatorC23×C5C3×C42
FrattiniC24C2×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C22⋊C4C4
socleC22×C3×C5#95
prime radicalC24×C3×D5C22
cosocleC24×D5C22×C3
hypercenterC2×C3×C22⋊C4AGL(1,𝔽5)
hypocenterC5C3×#1304
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #10598705C3direct (N × H)
7. C2×C3×C22⋊AGL(1,𝔽5)C4semidirect
8. C2×C3×C5×C22⋊C4C4semidirect
9. C2×C3×C22⋊Dic5C4semidirect
37. C2×C22⋊AGL(1,𝔽5)C3×C4semidirect
38. C2×C5×C22⋊C4C3×C4semidirect
39. C2×C22⋊Dic5C3×C4semidirect
76. C2×C3×C22⋊C4AGL(1,𝔽5)semidirect
173. C2×C22⋊C4C3×AGL(1,𝔽5)semidirect
252. C3×C5#1304semidirect
321. C5C3×#1304semidirect
346. C3#10598705direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658002
C3×#652442
#105987053
C3×#13045