Finite Group Databasebeta

← C3×#10604640 #14313420

Subgroups — C3×#10604640 #14313420

Order 1920.

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

normal subgroups only

This group has 516 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×#66059C211yesmaximal, characteristic
2C3×#65785C212yesmaximal, complemented
3C3×#64928C212yesmaximal, complemented
4C3×#64853C212yesmaximal, complemented
5#10604640C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×#713C411yescharacteristic, complemented
7C2×C3×#713C2211yescharacteristic
8C3×#4130—22no—
9C3×C5×C4⋊Q8C411yescharacteristic, complemented
10C3×#4093C411yescharacteristic, complemented
11C3×#4093C2211yescharacteristic
12C3×#4041—22no—
13C3×#4039—22no—
14C3×#4020C2211yescharacteristic
15C3×C42×D5C411yescharacteristic, complemented
16C3×#3999—22no—
17C3×#3999C2214yes—
18C3×#2126—51nomaximal, Hall
19#66059C2×C311yescharacteristic
20#65785C2×C312yescomplemented
21#64928C2×C312yescomplemented
22#64853C2×C312yescomplemented
23C2×C3×C5×Q8C2×C411yescharacteristic
24C2×C3×C5×Q8C2×C411yescharacteristic
25C3×#713—41no—
26C3×#713—41no—
27C3×#713—41no—
28C3×#713—41no—
29C2×C3×D20—21no—
30C2×C3×D20—22no—
31C2×C3×D20C2×C411yescharacteristic
32C2×C3×D20D412yes—
33C2×C3×D20C2311yescharacteristic
34C2×C3×C4×D5—21no—
35C2×C3×C4×D5—21no—
36C2×C3×C4×D5C2×C411yescharacteristic
37C2×C3×C4×D5D412yes—
38C3×C22⋊AGL(1,𝔽5)—24no—
39C3×C4⋊AGL(1,𝔽5)—42no—
40C3×C4×AGL(1,𝔽5)—42no—
41C3×C5×C4⋊C4—22no—
42C3×C42×C5C2×C411yescharacteristic
43C3×#685—22no—
44C3×#685—22no—
45C3×C4×Dic5—21no—
46C3×C4×Dic5C2×C411yescharacteristic
47C3×#532—51no—
48C3×#410—52no—
49C3×#354—52no—
50C3×#353—52no—
51C2×#713C3×C411yescharacteristic, complemented
52C2×#713C22×C311yescharacteristic
53#4130—22no—
54C5×C4⋊Q8C3×C411yescharacteristic, complemented
55#4093C3×C411yescharacteristic, complemented
56#4093C22×C311yescharacteristic
57#4041—22no—
58#4039—22no—
59#4020C22×C311yescharacteristic
60C42×D5C3×C411yescharacteristic, complemented
61#3999—22no—
62#3999C22×C314yes—
63#2126—51noSylow
64C22×C3×D5—21no—
65C22×C3×D5C22⋊C411yescharacteristic
66C22×C3×D5C2×D412yes—
67C2×C3×AGL(1,𝔽5)—44no—
68C3×C5×Q8—41no—
69C3×C5×Q8—41no—
70C2×C3×C4×C5—21no—
71C2×C3×C4×C5—21no—
72C2×C3×C4×C5C22⋊C412yes—
73C2×C3×C4×C5C22×C411yescharacteristic
74C2×C3×Dic5—21no—
75C2×C3×Dic5C22⋊C411yescharacteristic
76C3×D20—42no—
77C3×D20—41no—
78C3×D20—44no—
79C3×D20—41no—
80C3×D20—42no—
81C3×C4×D5—22no—
82C3×C4×D5—22no—
83C3×C4×D5—21no—
84C3×C4×D5—21no—
85C3×C4×D5—42no—
86C3×C4×D5—41no—
87C3×C4×D5—41no—
88C2×C3×#41—51no—
89C2×C3×#41—51no—
90C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
91C3×#127—51no—
92C3×#124—51no—
93C3×#124—51no—
94C3×#121—102no—
95C3×C4×D4—102no—
96C3×#117—102no—
97C2×C3×C42—51no—
98C3×#99—54no—
99C3×#99—102no—
100C2×C5×Q8C2×C3×C411yescharacteristic
101C2×C5×Q8C2×C3×C411yescharacteristic
102#713—41no—
103#713—41no—
104#713—41no—
105#713—41no—
106C2×D20—21no—
107C2×D20—22no—
108C2×D20C2×C3×C411yescharacteristic
109C2×D20C3×D412yes—
110C2×D20C23×C311yescharacteristic
111C2×C4×D5—21no—
112C2×C4×D5—21no—
113C2×C4×D5C2×C3×C411yescharacteristic
114C2×C4×D5C3×D412yes—
115C22⋊AGL(1,𝔽5)—24no—
116C4⋊AGL(1,𝔽5)—42no—
117C4×AGL(1,𝔽5)—42no—
118C5×C4⋊C4—22no—
119C42×C5C2×C3×C411yescharacteristic
120#685—22no—
121#685—22no—
122C4×Dic5—21no—
123C4×Dic5C2×C3×C411yescharacteristic
124#532—51no—
125#410—52no—
126#354—52no—
127#353—52no—
128C22×C3×C5C2×C22⋊C411yescharacteristic
129C2×C3×D5—21no—
130C2×C3×D5—22no—
131C2×C3×D5—41no—
132C2×C3×D5—42no—
133C2×C3×D5—42no—
134C2×C3×D5—41no—
135C3×AGL(1,𝔽5)—84no—
136C3×C4×C5—22no—
137C3×C4×C5—21no—
138C3×C4×C5—22no—
139C3×C4×C5—41no—
140C3×Dic5—41no—
141C3×Dic5#9911yescharacteristic
142C3×Dic5#9911yescharacteristic
143C3×#41—201no—
144C3×#41—201no—
145C3×#41—201no—
146C3×#41—201no—
147C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
148C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
149C2×C3×D4—52no—
150C2×C3×D4—51no—
151C2×C3×D4—51no—
152C2×C3×D4—102no—
153C2×C3×D4—101no—
154C22×C3×C4—52no—
155C22×C3×C4—51no—
156C22×C3×C4—101no—
157C22×C3×C4—101no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—202no—
160C3×C22⋊C4—102no—
161C3×C22⋊C4—102no—
162C3×C22⋊C4—104no—
163C3×C42—51no—
164C3×C42—101no—
165C3×C42—202no—
166C3×C42C2×AGL(1,𝔽5)11yescharacteristic
167C22×D5—21no—
168C22×D5C3×C22⋊C411yescharacteristic
169C22×D5C2×C3×D412yes—
170C2×AGL(1,𝔽5)—44no—
171C5×Q8—41no—
172C5×Q8—41no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5C3×C22⋊C412yes—
176C2×C4×C5C22×C3×C411yescharacteristic
177C2×Dic5—21no—
178C2×Dic5C3×C22⋊C411yescharacteristic
179D20—42no—
180D20—41no—
181D20—41no—
182D20—44no—
183D20—42no—
184C4×D5—22no—
185C4×D5—22no—
186C4×D5—21no—
187C4×D5—21no—
188C4×D5—42no—
189C4×D5—41no—
190C4×D5—41no—
191C2×#41—51no—
192C2×#41—51no—
193C4⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
194#127—51no—
195#124—51no—
196#124—51no—
197#121—102no—
198C4×D4—102no—
199#117—102no—
200C2×C42—51no—
201#99—54no—
202#99—102no—
203C2×C3×C5—21no—
204C2×C3×C5C2×#9911yescharacteristic
205C3×D5—42no—
206C3×D5—41no—
207C3×D5—81no—
208C23×C3—51no—
209C23×C3—52no—
210C23×C3—101no—
211C3×Q8—41no—
212C3×Q8—41no—
213C3×D4—202no—
214C3×D4—201no—
215C3×D4—201no—
216C3×D4—204no—
217C3×D4—202no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—51no—
221C2×C3×C4—102no—
222C2×C3×C4—102no—
223C2×C3×C4—101no—
224C2×C3×C4—101no—
225C2×C3×C4—101no—
226C2×C3×C4—202no—
227C2×C3×C4—201no—
228C2×C3×C4—201no—
229C2×C3×C4—204no—
230C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
231C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
232C22×C5C2×C3×C22⋊C411yescharacteristic
233C2×D5—21no—
234C2×D5—22no—
235C2×D5—41no—
236C2×D5—42no—
237C2×D5—42no—
238C2×D5—41no—
239AGL(1,𝔽5)—84no—
240C4×C5—22no—
241C4×C5—21no—
242C4×C5—22no—
243C4×C5—41no—
244Dic5—41no—
245Dic5C3×#9911yescharacteristic
246Dic5C3×#9911yescharacteristic
247#41—201no—
248#41—201no—
249#41—201no—
250#41—201no—
251C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
252C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
253C2×D4—52no—
254C2×D4—51no—
255C2×D4—51no—
256C2×D4—102no—
257C2×D4—101no—
258C22×C4—52no—
259C22×C4—51no—
260C22×C4—101no—
261C22×C4—101no—
262C4⋊C4—22no—
263C4⋊C4—202no—
264C22⋊C4—102no—
265C22⋊C4—102no—
266C22⋊C4—104no—
267C42—51no—
268C42—101no—
269C42—202no—
270C42C2×C3×AGL(1,𝔽5)11yescharacteristic
271C3×C5#212611yescharacteristic, complemented, Hall
272C22×C3—102no—
273C22×C3—101no—
274C22×C3—201no—
275C22×C3—202no—
276C22×C3—202no—
277C22×C3—201no—
278C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—51no—
284C3×C4—51no—
285C3×C4—201no—
286C3×C4—404no—
287C2×C5—21no—
288C2×C5C2×C3×#9911yescharacteristic
289D5—42no—
290D5—41no—
291D5—81no—
292C23—52no—
293C23—51no—
294C23—101no—
295Q8—41no—
296Q8—41no—
297D4—202no—
298D4—201no—
299D4—201no—
300D4—204no—
301D4—202no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—51no—
305C2×C4—102no—
306C2×C4—102no—
307C2×C4—101no—
308C2×C4—101no—
309C2×C4—101no—
310C2×C4—202no—
311C2×C4—201no—
312C2×C4—201no—
313C2×C4—204no—
314C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
315C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
316C2×C3—21no—
317C2×C3—202no—
318C2×C3—201no—
319C2×C3—401no—
320C2×C3D5⋊#9911yescharacteristic
321C5C3×#212611yescharacteristic, complemented, Sylow
322C22—102no—
323C22—101no—
324C22—201no—
325C22—202no—
326C22—202no—
327C22—201no—
328C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
329C4—22no—
330C4—22no—
331C4—21no—
332C4—41no—
333C4—51no—
334C4—51no—
335C4—201no—
336C4—404no—
337C3#1060464011yescharacteristic, direct product factor, Sylow
338C2—21no—
339C2—202no—
340C2—201no—
341C2—401no—
342C2C3×D5⋊#9911yescharacteristic
subgroupG/H
centerC2×C3D5⋊#99
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C4⋊Q8C4
socleC2×C3×C5C2×#99
prime radicalC3×#66059C2
cosocleC2×D20C23×C3
hypercenterC3×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#2126
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#65785C2semidirect
3. C3×#64928C2semidirect
4. C3×#64853C2semidirect
5. #10604640C3direct (N × H)
6. C2×C3×#713C4semidirect
9. C3×C5×C4⋊Q8C4semidirect
10. C3×#4093C4semidirect
15. C3×C42×D5C4semidirect
20. #65785C2×C3semidirect
21. #64928C2×C3semidirect
22. #64853C2×C3semidirect
51. C2×#713C3×C4semidirect
54. C5×C4⋊Q8C3×C4semidirect
55. #4093C3×C4semidirect
60. C42×D5C3×C4semidirect
90. C3×C4⋊Q8AGL(1,𝔽5)semidirect
193. C4⋊Q8C3×AGL(1,𝔽5)semidirect
271. C3×C5#2126semidirect
321. C5C3×#2126semidirect
337. C3#10604640direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660592
C3×#657852
C3×#649282
C3×#648532
#106046403
C3×#21265