Finite Group Databasebeta

← C3×#10604666 #14307451

Subgroups — C3×#10604666 #14307451

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66062C211yesmaximal, characteristic
2C3×#65686C211yesmaximal, characteristic
3C3×#65682C211yesmaximal, characteristic
4C3×#64875C214yesmaximal
5#10604666C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×C4×AGL(1,𝔽5)C2212yes—
7C2×C3×#700C2212yes—
8C3×C5×C4⋊Q8C411yescharacteristic, complemented
9C3×#4092—21no—
10C3×Q8×Dic5—21no—
11C3×#4043—21no—
12C3×#4038C2212yes—
13C3×#4036C412yescomplemented
14C3×#4033—21no—
15C3×C42×D5C2211yescharacteristic
16C3×#4015C411yescharacteristic, complemented
17C3×(C2×AGL(1,𝔽5))⋊C4—21no—
18C3×(C2×AGL(1,𝔽5))⋊C4—21no—
19C3×#4004—21no—
20C3×#4004—21no—
21C3×#1995—51nomaximal, Hall
22#66062C2×C311yescharacteristic
23#65686C2×C311yescharacteristic
24#65682C2×C311yescharacteristic
25#64875C2×C314yes—
26C22×C3×AGL(1,𝔽5)—22no—
27C2×C3×C5×Q8—21no—
28C2×C3×C4×D5D411yescharacteristic
29C2×C3×C4×D5D411yescharacteristic
30C2×C3×C4×D5C2311yescharacteristic
31C2×C3×Dic10—21no—
32C2×C3×#170—22no—
33C3×C4×AGL(1,𝔽5)—22no—
34C3×C4×AGL(1,𝔽5)Q814yescomplemented
35C3×#700—24no—
36C3×C5×C4⋊C4—21no—
37C3×C5×C4⋊C4C2×C412yes—
38C3×C42×C5C2×C411yescharacteristic
39C3×#685—22no—
40C3×C4⋊Dic5—21no—
41C3×C4⋊Dic5—22no—
42C3×C4⋊Dic5C2×C412yes—
43C3×#683—22no—
44C3×C4×Dic5—22no—
45C3×C4×Dic5C2×C411yescharacteristic
46C3×C4×Dic5D411yescharacteristic
47C3×C4×Dic5D411yescharacteristic
48C3×#533—51no—
49C3×#423—51no—
50C3×#376—51no—
51C3×#339—54no—
52C2×C4×AGL(1,𝔽5)C22×C312yes—
53C2×#700C22×C312yes—
54C5×C4⋊Q8C3×C411yescharacteristic, complemented
55#4092—21no—
56Q8×Dic5—21no—
57#4043—21no—
58#4038C22×C312yes—
59#4036C3×C412yescomplemented
60#4033—21no—
61C42×D5C22×C311yescharacteristic
62#4015C3×C411yescharacteristic, complemented
63(C2×AGL(1,𝔽5))⋊C4—21no—
64(C2×AGL(1,𝔽5))⋊C4—21no—
65#4004—21no—
66#4004—21no—
67#1995—51noSylow
68C22×C3×D5C2×D411yescharacteristic
69C2×C3×AGL(1,𝔽5)—24no—
70C2×C3×AGL(1,𝔽5)—42no—
71C2×C3×AGL(1,𝔽5)—42no—
72C3×C5×Q8—42no—
73C2×C3×C4×C5—22no—
74C2×C3×C4×C5C22⋊C411yescharacteristic
75C2×C3×C4×C5C22⋊C411yescharacteristic
76C2×C3×C4×C5C22×C411yescharacteristic
77C2×C3×Dic5—22no—
78C2×C3×Dic5C22⋊C411yescharacteristic
79C2×C3×Dic5C22⋊C411yescharacteristic
80C2×C3×Dic5C2×D411yescharacteristic
81C3×C4×D5—41no—
82C3×C4×D5—41no—
83C3×C4×D5C2×Q812yes—
84C3×C4×D5#4112yes—
85C3×Dic10—42no—
86C3×#170—42no—
87C2×C3×M16—52no—
88C3×C4⋊Q8—51no—
89C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
90C3×#125—52no—
91C3×C22⋊Q8—101no—
92C3×C22⋊Q8—101no—
93C3×C4×Q8—101no—
94C3×C4×Q8—101no—
95C3×#117—52no—
96C2×C3×C42—51no—
97C2×C3×C42—52no—
98C3×C4⋊C8—101no—
99C3×C4⋊C8—101no—
100C3×#95—101no—
101C3×#95—101no—
102C22×AGL(1,𝔽5)—22no—
103C2×C5×Q8—21no—
104C2×C4×D5C3×D411yescharacteristic
105C2×C4×D5C3×D411yescharacteristic
106C2×C4×D5C23×C311yescharacteristic
107C2×Dic10—21no—
108C2×#170—22no—
109C4×AGL(1,𝔽5)—22no—
110C4×AGL(1,𝔽5)C3×Q814yescomplemented
111#700—24no—
112C5×C4⋊C4—21no—
113C5×C4⋊C4C2×C3×C412yes—
114C42×C5C2×C3×C411yescharacteristic
115#685—22no—
116C4⋊Dic5—21no—
117C4⋊Dic5—22no—
118C4⋊Dic5C2×C3×C412yes—
119#683—22no—
120C4×Dic5—22no—
121C4×Dic5C2×C3×C411yescharacteristic
122C4×Dic5C3×D411yescharacteristic
123C4×Dic5C3×D411yescharacteristic
124#533—51no—
125#423—51no—
126#376—51no—
127#339—54no—
128C22×C3×C5C2×C22⋊C411yescharacteristic
129C2×C3×D5—21no—
130C2×C3×D5—21no—
131C2×C3×D5C22⋊Q812yes—
132C3×AGL(1,𝔽5)—44no—
133C3×C4×C5—21no—
134C3×C4×C5—21no—
135C3×C4×C5—42no—
136C3×C4×C5C4×Q812yes—
137C3×Dic5—21no—
138C3×Dic5—21no—
139C3×Dic5—42no—
140C3×Dic5#12411yescharacteristic
141C3×Dic5C4⋊Q811yescharacteristic
142C2×C3×Q8—21no—
143C2×C3×Q8—101no—
144C22×C3×C4—51no—
145C22×C3×C4—51no—
146C22×C3×C4—51no—
147C22×C3×C4—102no—
148C3×M16—104no—
149C2×C3×C8—102no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—52no—
152C3×C4⋊C4—102no—
153C3×C4⋊C4—102no—
154C3×C4⋊C4—101no—
155C3×C4⋊C4C2×AGL(1,𝔽5)12yes—
156C3×C22⋊C4—102no—
157C3×C42—51no—
158C3×C42—51no—
159C3×C42—51no—
160C3×C42—54no—
161C3×C42—102no—
162C3×C42—102no—
163C3×C42C2×AGL(1,𝔽5)11yescharacteristic
164C22×D5C2×C3×D411yescharacteristic
165C2×AGL(1,𝔽5)—24no—
166C2×AGL(1,𝔽5)—42no—
167C2×AGL(1,𝔽5)—42no—
168C5×Q8—42no—
169C2×C4×C5—22no—
170C2×C4×C5C3×C22⋊C411yescharacteristic
171C2×C4×C5C3×C22⋊C411yescharacteristic
172C2×C4×C5C22×C3×C411yescharacteristic
173C2×Dic5—22no—
174C2×Dic5C3×C22⋊C411yescharacteristic
175C2×Dic5C3×C22⋊C411yescharacteristic
176C2×Dic5C2×C3×D411yescharacteristic
177C4×D5—41no—
178C4×D5—41no—
179C4×D5C2×C3×Q812yes—
180C4×D5C3×#4112yes—
181Dic10—42no—
182#170—42no—
183C2×M16—52no—
184C4⋊Q8—51no—
185C4⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
186#125—52no—
187C22⋊Q8—101no—
188C22⋊Q8—101no—
189C4×Q8—101no—
190C4×Q8—101no—
191#117—52no—
192C2×C42—51no—
193C2×C42—52no—
194C4⋊C8—101no—
195C4⋊C8—101no—
196#95—101no—
197#95—101no—
198C2×C3×C5#39111yescharacteristic
199C2×C3×C5#42111yescharacteristic
200C2×C3×C5#42111yescharacteristic
201C3×D5—22no—
202C23×C3—51no—
203C3×Q8—42no—
204C3×Q8—202no—
205C2×C3×C4—22no—
206C2×C3×C4—51no—
207C2×C3×C4—52no—
208C2×C3×C4—52no—
209C2×C3×C4—51no—
210C2×C3×C4—51no—
211C2×C3×C4—104no—
212C2×C3×C4—102no—
213C2×C3×C4—201no—
214C2×C3×C4—201no—
215C2×C3×C4—202no—
216C2×C3×C4—202no—
217C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
218C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
219C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
220C3×C8—202no—
221C22×C5C2×C3×C22⋊C411yescharacteristic
222C2×D5—21no—
223C2×D5—21no—
224C2×D5C3×C22⋊Q812yes—
225AGL(1,𝔽5)—44no—
226C4×C5—21no—
227C4×C5—21no—
228C4×C5—42no—
229C4×C5C3×C4×Q812yes—
230Dic5—21no—
231Dic5—21no—
232Dic5—42no—
233Dic5C3×#12411yescharacteristic
234Dic5C3×C4⋊Q811yescharacteristic
235C2×Q8—21no—
236C2×Q8—101no—
237C22×C4—51no—
238C22×C4—51no—
239C22×C4—51no—
240C22×C4—102no—
241M16—104no—
242C2×C8—102no—
243C4⋊C4—21no—
244C4⋊C4—52no—
245C4⋊C4—102no—
246C4⋊C4—102no—
247C4⋊C4—101no—
248C4⋊C4C2×C3×AGL(1,𝔽5)12yes—
249C22⋊C4—102no—
250C42—51no—
251C42—51no—
252C42—51no—
253C42—54no—
254C42—102no—
255C42—102no—
256C42C2×C3×AGL(1,𝔽5)11yescharacteristic
257C3×C5#199511yescharacteristic, complemented, Hall
258C22×C3—52no—
259C22×C3—101no—
260C22×C3—101no—
261C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—42no—
265C3×C4—51no—
266C3×C4—51no—
267C3×C4—101no—
268C3×C4—101no—
269C3×C4—204no—
270C3×C4—202no—
271C3×C4Q8×AGL(1,𝔽5)12yes—
272C2×C5C3×#39111yescharacteristic
273C2×C5C3×#42111yescharacteristic
274C2×C5C3×#42111yescharacteristic
275D5—22no—
276C23—51no—
277Q8—42no—
278Q8—202no—
279C2×C4—22no—
280C2×C4—51no—
281C2×C4—52no—
282C2×C4—52no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—104no—
286C2×C4—102no—
287C2×C4—201no—
288C2×C4—201no—
289C2×C4—202no—
290C2×C4—202no—
291C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
292C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
293C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
294C8—202no—
295C2×C3—102no—
296C2×C3#6576911yescharacteristic
297C2×C3#6578411yescharacteristic
298C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
299C5C3×#199511yescharacteristic, complemented, Sylow
300C22—52no—
301C22—101no—
302C22—101no—
303C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
304C4—21no—
305C4—21no—
306C4—42no—
307C4—51no—
308C4—51no—
309C4—101no—
310C4—101no—
311C4—204no—
312C4—202no—
313C4C3×Q8×AGL(1,𝔽5)12yes—
314C3#1060466611yescharacteristic, direct product factor, Sylow
315C2—102no—
316C2C3×#6576911yescharacteristic
317C2C3×#6578411yescharacteristic
318C2C3×(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C4⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C4×D5C23×C3
hypercenterC3×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1995
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
5. #10604666C3direct (N × H)
8. C3×C5×C4⋊Q8C4semidirect
13. C3×#4036C4semidirect
16. C3×#4015C4semidirect
34. C3×C4×AGL(1,𝔽5)Q8semidirect
54. C5×C4⋊Q8C3×C4semidirect
59. #4036C3×C4semidirect
62. #4015C3×C4semidirect
89. C3×C4⋊Q8AGL(1,𝔽5)semidirect
110. C4×AGL(1,𝔽5)C3×Q8semidirect
185. C4⋊Q8C3×AGL(1,𝔽5)semidirect
257. C3×C5#1995semidirect
299. C5C3×#1995semidirect
314. C3#10604666direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660622
C3×#656862
C3×#656822
C3×#648752
#106046663
C3×#19955