Finite Group Databasebeta

← C3×#10604547 #14313509

Subgroups — C3×#10604547 #14313509

Order 1920.

540 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 104 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×#66008C211yesmaximal, characteristic
2C3×#65785C212yesmaximal, complemented
3C3×#64922C212yesmaximal, complemented
4C3×#64853C212yesmaximal, complemented
5#10604547C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×#713C2211yescharacteristic
7C2×C3×D4×D5C411yescharacteristic, complemented
8C3×#4130—22no—
9C3×C5×#124C411yescharacteristic, complemented
10C3×#4093C411yescharacteristic, complemented
11C3×#4086C2211yescharacteristic
12C3×#4030—22no—
13C3×#4028—22no—
14C3×#4020C2211yescharacteristic
15C3×#4018C411yescharacteristic, complemented
16C3×#3999—22no—
17C3×#3999C2214yescomplemented
18C3×#2128—51nomaximal, Hall
19#66008C2×C311yescharacteristic
20#65785C2×C312yescomplemented
21#64922C2×C312yescomplemented
22#64853C2×C312yescomplemented
23C23×C3×D5—21no—
24C2×C3×C5×Q8C2×C411yescharacteristic
25C2×C3×C5×D4C2×C411yescharacteristic
26C2×C3×C22⋊D5—22no—
27C2×C3×C22⋊D5—21no—
28C3×#713—42no—
29C3×D4×D5—42no—
30C2×C3×D20—22no—
31C2×C3×D20C2×C411yescharacteristic
32C2×C3×D20D412yes—
33C2×C3×D20C2311yescharacteristic
34C2×C3×C4×D5C2×C411yescharacteristic
35C2×C3×C4×D5D412yes—
36C3×C22⋊AGL(1,𝔽5)—24no—
37C3×C4⋊AGL(1,𝔽5)—42no—
38C3×C4×AGL(1,𝔽5)—42no—
39C3×C5×C22⋊C4—22no—
40C3×C42×C5C2×C411yescharacteristic
41C3×#685—21no—
42C3×#685—22no—
43C3×#683—21no—
44C3×C4×Dic5C2×C411yescharacteristic
45C3×#535—51no—
46C3×#410—52no—
47C3×#353—52no—
48C3×#353—52no—
49C2×#713C22×C311yescharacteristic
50C2×D4×D5C3×C411yescharacteristic, complemented
51#4130—22no—
52C5×#124C3×C411yescharacteristic, complemented
53#4093C3×C411yescharacteristic, complemented
54#4086C22×C311yescharacteristic
55#4030—22no—
56#4028—22no—
57#4020C22×C311yescharacteristic
58#4018C3×C411yescharacteristic, complemented
59#3999—22no—
60#3999C22×C314yescomplemented
61#2128—51noSylow
62C23×C3×C5—21no—
63C22×C3×D5—21no—
64C22×C3×D5—42no—
65C22×C3×D5—82no—
66C22×C3×D5C22⋊C411yescharacteristic
67C22×C3×D5C2×D412yes—
68C2×C3×AGL(1,𝔽5)—44no—
69C3×C5×Q8—41no—
70C3×C5×D4—41no—
71C2×C3×C4×C5—21no—
72C2×C3×C4×C5C22⋊C412yes—
73C2×C3×C4×C5C22×C411yescharacteristic
74C3×C22⋊D5—42no—
75C3×C22⋊D5—82no—
76C2×C3×Dic5—21no—
77C2×C3×Dic5C22⋊C411yescharacteristic
78C3×D20—42no—
79C3×D20—41no—
80C3×D20—41no—
81C3×D20—82no—
82C3×C4×D5—22no—
83C3×C4×D5—22no—
84C3×C4×D5—22no—
85C2×C3×#41—51no—
86C22×C3×D4—51no—
87C3×#127—51no—
88C3×#127—51no—
89C3×#124—51no—
90C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
91C3×#121—102no—
92C3×#120—102no—
93C3×#117—51no—
94C3×#117—102no—
95C3×#99—54no—
96C3×#99—102no—
97C23×D5—21no—
98C2×C5×Q8C2×C3×C411yescharacteristic
99C2×C5×D4C2×C3×C411yescharacteristic
100C2×C22⋊D5—22no—
101C2×C22⋊D5—21no—
102#713—42no—
103D4×D5—42no—
104C2×D20—22no—
105C2×D20C2×C3×C411yescharacteristic
106C2×D20C3×D412yes—
107C2×D20C23×C311yescharacteristic
108C2×C4×D5C2×C3×C411yescharacteristic
109C2×C4×D5C3×D412yes—
110C22⋊AGL(1,𝔽5)—24no—
111C4⋊AGL(1,𝔽5)—42no—
112C4×AGL(1,𝔽5)—42no—
113C5×C22⋊C4—22no—
114C42×C5C2×C3×C411yescharacteristic
115#685—21no—
116#685—22no—
117#683—21no—
118C4×Dic5C2×C3×C411yescharacteristic
119#535—51no—
120#410—52no—
121#353—52no—
122#353—52no—
123C22×C3×C5—42no—
124C22×C3×C5—41no—
125C22×C3×C5C2×C22⋊C411yescharacteristic
126C2×C3×D5—22no—
127C2×C3×D5—21no—
128C2×C3×D5—42no—
129C2×C3×D5—41no—
130C2×C3×D5—42no—
131C2×C3×D5—41no—
132C2×C3×D5—84no—
133C3×AGL(1,𝔽5)—84no—
134C3×C4×C5—21no—
135C3×C4×C5—22no—
136C3×C4×C5—41no—
137C3×Dic5—41no—
138C3×Dic5#9912yes—
139C24×C3—101no—
140C3×#41—202no—
141C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
142C2×C3×D4—52no—
143C2×C3×D4—51no—
144C2×C3×D4—51no—
145C2×C3×D4—102no—
146C2×C3×D4—101no—
147C2×C3×D4—102no—
148C2×C3×D4—202no—
149C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
150C22×C3×C4—52no—
151C22×C3×C4—51no—
152C3×C4⋊C4—101no—
153C3×C4⋊C4—202no—
154C3×C22⋊C4—22no—
155C3×C22⋊C4—101no—
156C3×C22⋊C4—102no—
157C3×C22⋊C4—104no—
158C3×C42—51no—
159C3×C42—202no—
160C3×C42C2×AGL(1,𝔽5)11yescharacteristic
161C23×C5—21no—
162C22×D5—21no—
163C22×D5—42no—
164C22×D5—82no—
165C22×D5C3×C22⋊C411yescharacteristic
166C22×D5C2×C3×D412yes—
167C2×AGL(1,𝔽5)—44no—
168C5×Q8—41no—
169C5×D4—41no—
170C2×C4×C5—21no—
171C2×C4×C5C3×C22⋊C412yes—
172C2×C4×C5C22×C3×C411yescharacteristic
173C22⋊D5—42no—
174C22⋊D5—82no—
175C2×Dic5—21no—
176C2×Dic5C3×C22⋊C411yescharacteristic
177D20—42no—
178D20—41no—
179D20—41no—
180D20—82no—
181C4×D5—22no—
182C4×D5—22no—
183C4×D5—22no—
184C2×#41—51no—
185C22×D4—51no—
186#127—51no—
187#127—51no—
188#124—51no—
189#124C3×AGL(1,𝔽5)11yescharacteristic, complemented
190#121—102no—
191#120—102no—
192#117—51no—
193#117—102no—
194#99—54no—
195#99—102no—
196C2×C3×C5—21no—
197C2×C3×C5—81no—
198C2×C3×C5C2×#9911yescharacteristic
199C3×D5—42no—
200C3×D5—41no—
201C3×D5—81no—
202C23×C3—21no—
203C23×C3—51no—
204C23×C3—52no—
205C23×C3—101no—
206C23×C3—202no—
207C23×C3—402no—
208C3×Q8—41no—
209C3×D4—41no—
210C3×D4—202no—
211C3×D4—201no—
212C3×D4—201no—
213C3×D4—202no—
214C3×D4—402no—
215C3×D4—402no—
216C2×C3×C4—21no—
217C2×C3×C4—51no—
218C2×C3×C4—102no—
219C2×C3×C4—102no—
220C2×C3×C4—102no—
221C2×C3×C4—101no—
222C2×C3×C4—204no—
223C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
224C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
225C22×C5—42no—
226C22×C5—41no—
227C22×C5C2×C3×C22⋊C411yescharacteristic
228C2×D5—22no—
229C2×D5—21no—
230C2×D5—42no—
231C2×D5—41no—
232C2×D5—42no—
233C2×D5—41no—
234C2×D5—84no—
235AGL(1,𝔽5)—84no—
236C4×C5—21no—
237C4×C5—22no—
238C4×C5—41no—
239Dic5—41no—
240Dic5C3×#9912yes—
241C24—101no—
242#41—202no—
243C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
244C2×D4—52no—
245C2×D4—51no—
246C2×D4—51no—
247C2×D4—102no—
248C2×D4—101no—
249C2×D4—102no—
250C2×D4—202no—
251C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
252C22×C4—52no—
253C22×C4—51no—
254C4⋊C4—101no—
255C4⋊C4—202no—
256C22⋊C4—22no—
257C22⋊C4—101no—
258C22⋊C4—102no—
259C22⋊C4—104no—
260C42—51no—
261C42—202no—
262C42C2×C3×AGL(1,𝔽5)11yescharacteristic
263C3×C5#212811yescharacteristic, complemented, Hall
264C22×C3—42no—
265C22×C3—41no—
266C22×C3—102no—
267C22×C3—101no—
268C22×C3—202no—
269C22×C3—201no—
270C22×C3—202no—
271C22×C3—201no—
272C22×C3—404no—
273C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
274C3×C4—22no—
275C3×C4—21no—
276C3×C4—41no—
277C3×C4—52no—
278C3×C4—201no—
279C3×C4—404no—
280C2×C5—21no—
281C2×C5—81no—
282C2×C5C2×C3×#9911yescharacteristic
283D5—42no—
284D5—41no—
285D5—81no—
286C23—21no—
287C23—52no—
288C23—51no—
289C23—101no—
290C23—202no—
291C23—402no—
292Q8—41no—
293D4—41no—
294D4—202no—
295D4—201no—
296D4—201no—
297D4—202no—
298D4—402no—
299D4—402no—
300C2×C4—21no—
301C2×C4—51no—
302C2×C4—102no—
303C2×C4—102no—
304C2×C4—102no—
305C2×C4—101no—
306C2×C4—204no—
307C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
308C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
309C2×C3—21no—
310C2×C3—81no—
311C2×C3—202no—
312C2×C3—201no—
313C2×C3—401no—
314C2×C3D5⋊#9911yescharacteristic
315C5C3×#212811yescharacteristic, complemented, Sylow
316C22—42no—
317C22—41no—
318C22—102no—
319C22—101no—
320C22—202no—
321C22—201no—
322C22—202no—
323C22—201no—
324C22—404no—
325C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
326C4—22no—
327C4—21no—
328C4—41no—
329C4—52no—
330C4—201no—
331C4—404no—
332C3#1060454711yescharacteristic, direct product factor, Sylow
333C2—21no—
334C2—81no—
335C2—202no—
336C2—201no—
337C2—401no—
338C2C3×D5⋊#9911yescharacteristic
subgroupG/H
centerC2×C3D5⋊#99
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#124C4
socleC2×C3×C5C2×#99
prime radicalC3×#66008C2
cosocleC2×D20C23×C3
hypercenterC3×#124AGL(1,𝔽5)
hypocenterC5C3×#2128
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. C3×#65785C2semidirect
3. C3×#64922C2semidirect
4. C3×#64853C2semidirect
5. #10604547C3direct (N × H)
7. C2×C3×D4×D5C4semidirect
9. C3×C5×#124C4semidirect
10. C3×#4093C4semidirect
15. C3×#4018C4semidirect
17. C3×#3999C22semidirect
20. #65785C2×C3semidirect
21. #64922C2×C3semidirect
22. #64853C2×C3semidirect
50. C2×D4×D5C3×C4semidirect
52. C5×#124C3×C4semidirect
53. #4093C3×C4semidirect
58. #4018C3×C4semidirect
60. #3999C22×C3semidirect
90. C3×#124AGL(1,𝔽5)semidirect
189. #124C3×AGL(1,𝔽5)semidirect
263. C3×C5#2128semidirect
315. C5C3×#2128semidirect
332. C3#10604547direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660082
C3×#657852
C3×#649222
C3×#648532
#106045473
C3×#21285