Finite Group Databasebeta

← C3×#10604670 #14301489

Subgroups — C3×#10604670 #14301489

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66262C211yesmaximal, characteristic
2C3×D5×C4⋊Q8C211yesmaximal, characteristic
3C3×D5⋊#101C211yesmaximal, characteristic
4C3×#65782C212yesmaximal
5C3×#64862C212yesmaximal
6#10604670C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C2×C3×Q8×D5C2211yescharacteristic
9C3×Q8×AGL(1,𝔽5)—24no—
10C3×#4130—22no—
11C3×#4130C2212yes—
12C3×D5⋊M16C2212yes—
13C3×C5×C4⋊Q8C411yescharacteristic, complemented
14C3×#4090C411yescharacteristic, complemented
15C3×#4090C411yescharacteristic, complemented
16C3×D5×C4⋊C4—22no—
17C3×#4034—22no—
18C3×C42×D5C2211yescharacteristic
19C3×#4015C411yescharacteristic, complemented
20C3×#4009—24no—
21C3×#4005—22no—
22C3×#1886—51nomaximal, Hall
23#66262C2×C311yescharacteristic
24D5×C4⋊Q8C2×C311yescharacteristic
25D5⋊#101C2×C311yescharacteristic
26#65782C2×C312yes—
27#64862C2×C312yes—
28C2×C3×C5×Q8C2×C411yescharacteristic
29C2×C3×C5×Q8C2×C411yescharacteristic
30C3×Q8×D5—22no—
31C3×Q8×D5—22no—
32C3×Q8×D5—22no—
33C3×Q8×D5—22no—
34C2×C3×C4×D5—21no—
35C2×C3×C4×D5—21no—
36C2×C3×C4×D5D412yes—
37C2×C3×C4×D5C2311yescharacteristic
38C2×C3×Dic10—21no—
39C2×C3×Dic10C2×C411yescharacteristic
40C2×C3×Dic10C2×C411yescharacteristic
41C2×C3×Dic10D412yes—
42C3×C22⋊AGL(1,𝔽5)—22no—
43C3×#704—22no—
44C3×C4⋊AGL(1,𝔽5)—24no—
45C3×C4⋊AGL(1,𝔽5)D414yes—
46C3×C4×AGL(1,𝔽5)—24no—
47C3×C4×AGL(1,𝔽5)—22no—
48C3×#700—22no—
49C3×D5⋊C8—22no—
50C3×C5×C4⋊C4—22no—
51C3×C42×C5C2×C411yescharacteristic
52C3×C4⋊Dic5—22no—
53C3×#683—22no—
54C3×#683—22no—
55C3×C4×Dic5—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C2×C3×C4⋊Q8—51no—
58C3×#519—51no—
59C3×#419—52no—
60C2×C3×#101—51no—
61C3×#339—52no—
62C2×Q8×D5C22×C311yescharacteristic
63C2×Q8×D5C22×C311yescharacteristic
64Q8×AGL(1,𝔽5)—24no—
65#4130—22no—
66#4130C22×C312yes—
67D5⋊M16C22×C312yes—
68C5×C4⋊Q8C3×C411yescharacteristic, complemented
69#4090C3×C411yescharacteristic, complemented
70#4090C3×C411yescharacteristic, complemented
71D5×C4⋊C4—22no—
72#4034—22no—
73C42×D5C22×C311yescharacteristic
74#4015C3×C411yescharacteristic, complemented
75#4009—24no—
76#4005—22no—
77#1886—51noSylow
78C22×C3×D5C2×D411yescharacteristic
79C2×C3×AGL(1,𝔽5)—24no—
80C3×C5×Q8—22no—
81C3×C5×Q8—22no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5C22⋊C412yes—
85C2×C3×C4×C5C22×C411yescharacteristic
86C2×C3×Dic5—21no—
87C2×C3×Dic5—21no—
88C2×C3×Dic5C22⋊C412yes—
89C2×C3×Dic5C2×D411yescharacteristic
90C3×C4×D5—24no—
91C3×C4×D5—22no—
92C3×C4×D5—22no—
93C3×C4×D5—41no—
94C3×C4×D5—41no—
95C3×C4×D5C2×D412yes—
96C3×C4×D5#4112yes—
97C3×Dic10—22no—
98C3×Dic10—24no—
99C3×Dic10—22no—
100C3×Dic10—42no—
101C3×#170—42no—
102C22×C3×Q8—51no—
103C22×C3×Q8—51no—
104C2×C3×M16—52no—
105C3×C4⋊Q8—51no—
106C3×C4⋊Q8—51no—
107C3×C4⋊Q8—51no—
108C3×C4⋊Q8—102no—
109C3×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
110C3×C4×Q8—104no—
111C3×#117—52no—
112C3×#117—102no—
113C2×C3×C4⋊C4—102no—
114C2×C3×C42—51no—
115C3×#103—104no—
116C3×#101—102no—
117C2×C5×Q8C2×C3×C411yescharacteristic
118C2×C5×Q8C2×C3×C411yescharacteristic
119Q8×D5—22no—
120Q8×D5—22no—
121Q8×D5—22no—
122Q8×D5—22no—
123C2×C4×D5—21no—
124C2×C4×D5—21no—
125C2×C4×D5C3×D412yes—
126C2×C4×D5C23×C311yescharacteristic
127C2×Dic10—21no—
128C2×Dic10C2×C3×C411yescharacteristic
129C2×Dic10C2×C3×C411yescharacteristic
130C2×Dic10C3×D412yes—
131C22⋊AGL(1,𝔽5)—22no—
132#704—22no—
133C4⋊AGL(1,𝔽5)—24no—
134C4⋊AGL(1,𝔽5)C3×D414yes—
135C4×AGL(1,𝔽5)—24no—
136C4×AGL(1,𝔽5)—22no—
137#700—22no—
138D5⋊C8—22no—
139C5×C4⋊C4—22no—
140C42×C5C2×C3×C411yescharacteristic
141C4⋊Dic5—22no—
142#683—22no—
143#683—22no—
144C4×Dic5—21no—
145C4×Dic5C2×C3×C411yescharacteristic
146C2×C4⋊Q8—51no—
147#519—51no—
148#419—52no—
149C2×#101—51no—
150#339—52no—
151C22×C3×C5C2×C22⋊C411yescharacteristic
152C2×C3×D5—22no—
153C2×C3×D5#12011yescharacteristic
154C2×C3×D5#12311yescharacteristic
155C3×AGL(1,𝔽5)—44no—
156C3×C4×C5—22no—
157C3×C4×C5—22no—
158C3×C4×C5—41no—
159C3×C4×C5C4×D412yes—
160C3×Dic5—22no—
161C3×Dic5—22no—
162C3×Dic5—41no—
163C3×Dic5#12112yes—
164C2×C3×Q8—52no—
165C2×C3×Q8—51no—
166C2×C3×Q8—51no—
167C2×C3×Q8—101no—
168C2×C3×Q8—102no—
169C2×C3×Q8—102no—
170C2×C3×Q8—102no—
171C2×C3×Q8—102no—
172C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
173C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
174C22×C3×C4—51no—
175C22×C3×C4—52no—
176C22×C3×C4—101no—
177C22×C3×C4—101no—
178C3×M16—102no—
179C3×M16—102no—
180C2×C3×C8—102no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—54no—
183C3×C4⋊C4—102no—
184C3×C4⋊C4—102no—
185C3×C4⋊C4—104no—
186C3×C4⋊C4—102no—
187C3×C22⋊C4—102no—
188C3×C42—51no—
189C3×C42—101no—
190C3×C42—102no—
191C3×C42—104no—
192C3×C42C2×AGL(1,𝔽5)11yescharacteristic
193C22×D5C2×C3×D411yescharacteristic
194C2×AGL(1,𝔽5)—24no—
195C5×Q8—22no—
196C5×Q8—22no—
197C2×C4×C5—21no—
198C2×C4×C5—21no—
199C2×C4×C5C3×C22⋊C412yes—
200C2×C4×C5C22×C3×C411yescharacteristic
201C2×Dic5—21no—
202C2×Dic5—21no—
203C2×Dic5C3×C22⋊C412yes—
204C2×Dic5C2×C3×D411yescharacteristic
205C4×D5—24no—
206C4×D5—22no—
207C4×D5—22no—
208C4×D5—41no—
209C4×D5—41no—
210C4×D5C2×C3×D412yes—
211C4×D5C3×#4112yes—
212Dic10—22no—
213Dic10—24no—
214Dic10—22no—
215Dic10—42no—
216#170—42no—
217C22×Q8—51no—
218C22×Q8—51no—
219C2×M16—52no—
220C4⋊Q8—51no—
221C4⋊Q8—51no—
222C4⋊Q8—51no—
223C4⋊Q8—102no—
224C4⋊Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
225C4×Q8—104no—
226#117—52no—
227#117—102no—
228C2×C4⋊C4—102no—
229C2×C42—51no—
230#103—104no—
231#101—102no—
232C2×C3×C5—21no—
233C2×C3×C5#38611yescharacteristic
234C3×D5—21no—
235C3×D5#45612yes—
236C23×C3—51no—
237C3×Q8—22no—
238C3×Q8—22no—
239C3×Q8—104no—
240C3×Q8—102no—
241C3×Q8—102no—
242C3×Q8—202no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—51no—
246C2×C3×C4—52no—
247C2×C3×C4—52no—
248C2×C3×C4—52no—
249C2×C3×C4—102no—
250C2×C3×C4—102no—
251C2×C3×C4—104no—
252C2×C3×C4—101no—
253C2×C3×C4—104no—
254C2×C3×C4—101no—
255C2×C3×C4—201no—
256C2×C3×C4—201no—
257C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
258C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
259C3×C8—202no—
260C22×C5C2×C3×C22⋊C411yescharacteristic
261C2×D5—22no—
262C2×D5C3×#12011yescharacteristic
263C2×D5C3×#12311yescharacteristic
264AGL(1,𝔽5)—44no—
265C4×C5—22no—
266C4×C5—22no—
267C4×C5—41no—
268C4×C5C3×C4×D412yes—
269Dic5—22no—
270Dic5—22no—
271Dic5—41no—
272Dic5C3×#12112yes—
273C2×Q8—52no—
274C2×Q8—51no—
275C2×Q8—51no—
276C2×Q8—101no—
277C2×Q8—102no—
278C2×Q8—102no—
279C2×Q8—102no—
280C2×Q8—102no—
281C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
282C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
283C22×C4—51no—
284C22×C4—52no—
285C22×C4—101no—
286C22×C4—101no—
287M16—102no—
288M16—102no—
289C2×C8—102no—
290C4⋊C4—22no—
291C4⋊C4—54no—
292C4⋊C4—102no—
293C4⋊C4—102no—
294C4⋊C4—104no—
295C4⋊C4—102no—
296C22⋊C4—102no—
297C42—51no—
298C42—101no—
299C42—102no—
300C42—104no—
301C42C2×C3×AGL(1,𝔽5)11yescharacteristic
302C3×C5#188611yescharacteristic, complemented, Hall
303C22×C3—51no—
304C22×C3—51no—
305C22×C3—102no—
306C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—41no—
310C3×C4—52no—
311C3×C4—102no—
312C3×C4—102no—
313C3×C4—204no—
314C3×C4—201no—
315C3×C4D4×AGL(1,𝔽5)12yes—
316C2×C5—21no—
317C2×C5C3×#38611yescharacteristic
318D5—21no—
319D5C3×#45612yes—
320C23—51no—
321Q8—22no—
322Q8—22no—
323Q8—104no—
324Q8—102no—
325Q8—102no—
326Q8—202no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—51no—
330C2×C4—52no—
331C2×C4—52no—
332C2×C4—52no—
333C2×C4—102no—
334C2×C4—102no—
335C2×C4—104no—
336C2×C4—101no—
337C2×C4—104no—
338C2×C4—101no—
339C2×C4—201no—
340C2×C4—201no—
341C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
342C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
343C8—202no—
344C2×C3—21no—
345C2×C3—52no—
346C2×C3—101no—
347C2×C3#6577911yescharacteristic
348C5C3×#188611yescharacteristic, complemented, Sylow
349C22—51no—
350C22—51no—
351C22—102no—
352C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
353C4—22no—
354C4—22no—
355C4—41no—
356C4—52no—
357C4—102no—
358C4—102no—
359C4—204no—
360C4—201no—
361C4C3×D4×AGL(1,𝔽5)12yes—
362C3#1060467011yescharacteristic, direct product factor, Sylow
363C2—21no—
364C2—52no—
365C2—101no—
366C2C3×#6577911yescharacteristic
subgroupG/H
centerC2×C3#65779
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C4⋊Q8C4
socleC2×C3×C5#386
prime radicalC22×C3×D5C2×D4
cosocleC2×C4×D5C23×C3
hypercenterC3×C4⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1886
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
6. #10604670C3direct (N × H)
13. C3×C5×C4⋊Q8C4semidirect
14. C3×#4090C4semidirect
15. C3×#4090C4semidirect
19. C3×#4015C4semidirect
68. C5×C4⋊Q8C3×C4semidirect
69. #4090C3×C4semidirect
70. #4090C3×C4semidirect
74. #4015C3×C4semidirect
109. C3×C4⋊Q8AGL(1,𝔽5)semidirect
224. C4⋊Q8C3×AGL(1,𝔽5)semidirect
302. C3×C5#1886semidirect
348. C5C3×#1886semidirect
362. C3#10604670direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662622
C3×D5×C4⋊Q82
C3×D5⋊#1012
C3×#657822
C3×#648622
#106046703
C3×#18865