Finite Group Databasebeta

← C3×#10604563 #14301579

Subgroups — C3×#10604563 #14301579

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66262C211yesmaximal, characteristic, complemented
2C3×D5×#124C211yesmaximal, characteristic
3C3×D5⋊#101C211yesmaximal, characteristic, complemented
4C3×#65767C212yesmaximal
5C3×#64862C212yesmaximal, complemented
6#10604563C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C2×C3×D4×D5C2211yescharacteristic
9C3×Q8×AGL(1,𝔽5)—24no—
10C3×#4130—22no—
11C3×#4130C2212yes—
12C3×D5⋊M16C2212yes—
13C3×C5×#124C411yescharacteristic, complemented
14C3×#4093C411yescharacteristic, complemented
15C3×#4082C411yescharacteristic, complemented
16C3×#4031—22no—
17C3×D5×C22⋊C4—22no—
18C3×#4021C411yescharacteristic, complemented
19C3×C42×D5C2211yescharacteristic
20C3×#4007—24no—
21C3×#4005—22no—
22C3×#1888—51nomaximal, Hall
23#66262C2×C311yescharacteristic, complemented
24D5×#124C2×C311yescharacteristic
25D5⋊#101C2×C311yescharacteristic, complemented
26#65767C2×C312yes—
27#64862C2×C312yescomplemented
28C23×C3×D5—21no—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C2×C3×C5×D4C2×C411yescharacteristic
31C2×C3×C22⋊D5—21no—
32C3×Q8×D5—22no—
33C3×Q8×D5—22no—
34C3×D4×D5—22no—
35C3×D4×D5—22no—
36C2×C3×D20C2×C411yescharacteristic
37C2×C3×C4×D5—21no—
38C2×C3×C4×D5D412yes—
39C2×C3×C4×D5C2311yescharacteristic
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×C22⋊C4—22no—
51C3×C42×C5C2×C411yescharacteristic
52C3×C22⋊Dic5—22no—
53C3×#685—22no—
54C3×#685—22no—
55C3×C4×Dic5—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C2×C3×#124—51no—
58C3×#519—51no—
59C3×#418—52no—
60C2×C3×#101—51no—
61C3×#339—52no—
62C2×Q8×D5C22×C311yescharacteristic
63C2×D4×D5C22×C311yescharacteristic
64Q8×AGL(1,𝔽5)—24no—
65#4130—22no—
66#4130C22×C312yes—
67D5⋊M16C22×C312yes—
68C5×#124C3×C411yescharacteristic, complemented
69#4093C3×C411yescharacteristic, complemented
70#4082C3×C411yescharacteristic, complemented
71#4031—22no—
72D5×C22⋊C4—22no—
73#4021C3×C411yescharacteristic, complemented
74C42×D5C22×C311yescharacteristic
75#4007—24no—
76#4005—22no—
77#1888—51noSylow
78C23×C3×C5—21no—
79C22×C3×D5—21no—
80C22×C3×D5—44no—
81C22×C3×D5—41no—
82C22×C3×D5—41no—
83C22×C3×D5C2×D411yescharacteristic
84C2×C3×AGL(1,𝔽5)—24no—
85C3×C5×Q8—22no—
86C3×C5×D4—22no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5C22⋊C412yes—
89C2×C3×C4×C5C22×C411yescharacteristic
90C3×C22⋊D5—42no—
91C2×C3×Dic5—21no—
92C2×C3×Dic5C22⋊C412yes—
93C2×C3×Dic5C2×D411yescharacteristic
94C3×D20—22no—
95C3×C4×D5—22no—
96C3×C4×D5—22no—
97C3×C4×D5—42no—
98C3×C4×D5C2×D412yes—
99C3×C4×D5#4112yes—
100C3×Dic10—22no—
101C3×Dic10—24no—
102C3×#170—42no—
103C22×C3×Q8—51no—
104C22×C3×D4—51no—
105C2×C3×M16—52no—
106C3×#124—51no—
107C3×#124—51no—
108C3×#124—51no—
109C3×#124—102no—
110C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
111C3×C4×Q8—104no—
112C3×#117—52no—
113C3×#117—102no—
114C2×C3×C22⋊C4—102no—
115C2×C3×C42—51no—
116C3×#102—104no—
117C3×#101—102no—
118C23×D5—21no—
119C2×C5×Q8C2×C3×C411yescharacteristic
120C2×C5×D4C2×C3×C411yescharacteristic
121C2×C22⋊D5—21no—
122Q8×D5—22no—
123Q8×D5—22no—
124D4×D5—22no—
125D4×D5—22no—
126C2×D20C2×C3×C411yescharacteristic
127C2×C4×D5—21no—
128C2×C4×D5C3×D412yes—
129C2×C4×D5C23×C311yescharacteristic
130C2×Dic10C2×C3×C411yescharacteristic
131C2×Dic10C3×D412yes—
132C22⋊AGL(1,𝔽5)—22no—
133#704—22no—
134C4⋊AGL(1,𝔽5)—24no—
135C4⋊AGL(1,𝔽5)C3×D414yes—
136C4×AGL(1,𝔽5)—24no—
137C4×AGL(1,𝔽5)—22no—
138#700—22no—
139D5⋊C8—22no—
140C5×C22⋊C4—22no—
141C42×C5C2×C3×C411yescharacteristic
142C22⋊Dic5—22no—
143#685—22no—
144#685—22no—
145C4×Dic5—21no—
146C4×Dic5C2×C3×C411yescharacteristic
147C2×#124—51no—
148#519—51no—
149#418—52no—
150C2×#101—51no—
151#339—52no—
152C22×C3×C5—42no—
153C22×C3×C5—41no—
154C22×C3×C5C2×C22⋊C411yescharacteristic
155C2×C3×D5—22no—
156C2×C3×D5—42no—
157C2×C3×D5—41no—
158C2×C3×D5—82no—
159C2×C3×D5—82no—
160C2×C3×D5#12011yescharacteristic
161C2×C3×D5#12311yescharacteristic
162C3×AGL(1,𝔽5)—44no—
163C3×C4×C5—22no—
164C3×C4×C5—41no—
165C3×C4×C5C4×D412yes—
166C3×Dic5—22no—
167C3×Dic5—41no—
168C3×Dic5#12112yes—
169C24×C3—101no—
170C2×C3×Q8—52no—
171C2×C3×Q8—51no—
172C2×C3×Q8—102no—
173C2×C3×Q8—102no—
174C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
175C2×C3×D4—51no—
176C2×C3×D4—102no—
177C2×C3×D4—102no—
178C2×C3×D4—101no—
179C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
180C22×C3×C4—51no—
181C22×C3×C4—52no—
182C22×C3×C4—101no—
183C3×M16—102no—
184C3×M16—102no—
185C2×C3×C8—102no—
186C3×C4⋊C4—54no—
187C3×C4⋊C4—104no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4—102no—
190C3×C22⋊C4—102no—
191C3×C22⋊C4—102no—
192C3×C22⋊C4—102no—
193C3×C42—51no—
194C3×C42—101no—
195C3×C42—102no—
196C3×C42—104no—
197C3×C42C2×AGL(1,𝔽5)11yescharacteristic
198C23×C5—21no—
199C22×D5—21no—
200C22×D5—44no—
201C22×D5—41no—
202C22×D5—41no—
203C22×D5C2×C3×D411yescharacteristic
204C2×AGL(1,𝔽5)—24no—
205C5×Q8—22no—
206C5×D4—22no—
207C2×C4×C5—21no—
208C2×C4×C5C3×C22⋊C412yes—
209C2×C4×C5C22×C3×C411yescharacteristic
210C22⋊D5—42no—
211C2×Dic5—21no—
212C2×Dic5C3×C22⋊C412yes—
213C2×Dic5C2×C3×D411yescharacteristic
214D20—22no—
215C4×D5—22no—
216C4×D5—22no—
217C4×D5—42no—
218C4×D5C2×C3×D412yes—
219C4×D5C3×#4112yes—
220Dic10—22no—
221Dic10—24no—
222#170—42no—
223C22×Q8—51no—
224C22×D4—51no—
225C2×M16—52no—
226#124—51no—
227#124—51no—
228#124—51no—
229#124—102no—
230#124C3×AGL(1,𝔽5)11yescharacteristic, complemented
231C4×Q8—104no—
232#117—52no—
233#117—102no—
234C2×C22⋊C4—102no—
235C2×C42—51no—
236#102—104no—
237#101—102no—
238C2×C3×C5—21no—
239C2×C3×C5—81no—
240C2×C3×C5#38611yescharacteristic
241C3×D5—21no—
242C3×D5—81no—
243C3×D5#45412yes—
244C23×C3—21no—
245C23×C3—51no—
246C23×C3—101no—
247C23×C3—201no—
248C23×C3—201no—
249C23×C3—204no—
250C3×Q8—22no—
251C3×Q8—104no—
252C3×Q8—102no—
253C3×D4—22no—
254C3×D4—102no—
255C3×D4—202no—
256C2×C3×C4—21no—
257C2×C3×C4—51no—
258C2×C3×C4—52no—
259C2×C3×C4—52no—
260C2×C3×C4—52no—
261C2×C3×C4—102no—
262C2×C3×C4—102no—
263C2×C3×C4—104no—
264C2×C3×C4—101no—
265C2×C3×C4—202no—
266C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
267C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
268C3×C8—202no—
269C22×C5—42no—
270C22×C5—41no—
271C22×C5C2×C3×C22⋊C411yescharacteristic
272C2×D5—22no—
273C2×D5—42no—
274C2×D5—41no—
275C2×D5—82no—
276C2×D5—82no—
277C2×D5C3×#12011yescharacteristic
278C2×D5C3×#12311yescharacteristic
279AGL(1,𝔽5)—44no—
280C4×C5—22no—
281C4×C5—41no—
282C4×C5C3×C4×D412yes—
283Dic5—22no—
284Dic5—41no—
285Dic5C3×#12112yes—
286C24—101no—
287C2×Q8—52no—
288C2×Q8—51no—
289C2×Q8—102no—
290C2×Q8—102no—
291C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
292C2×D4—51no—
293C2×D4—102no—
294C2×D4—102no—
295C2×D4—101no—
296C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
297C22×C4—51no—
298C22×C4—52no—
299C22×C4—101no—
300M16—102no—
301M16—102no—
302C2×C8—102no—
303C4⋊C4—54no—
304C4⋊C4—104no—
305C22⋊C4—22no—
306C22⋊C4—102no—
307C22⋊C4—102no—
308C22⋊C4—102no—
309C22⋊C4—102no—
310C42—51no—
311C42—101no—
312C42—102no—
313C42—104no—
314C42C2×C3×AGL(1,𝔽5)11yescharacteristic
315C3×C5#188811yescharacteristic, complemented, Hall
316C22×C3—42no—
317C22×C3—41no—
318C22×C3—51no—
319C22×C3—51no—
320C22×C3—102no—
321C22×C3—202no—
322C22×C3—201no—
323C22×C3—402no—
324C22×C3—402no—
325C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
326C3×C4—22no—
327C3×C4—41no—
328C3×C4—52no—
329C3×C4—102no—
330C3×C4—201no—
331C3×C4—204no—
332C3×C4D4×AGL(1,𝔽5)12yes—
333C2×C5—21no—
334C2×C5—81no—
335C2×C5C3×#38611yescharacteristic
336D5—21no—
337D5—81no—
338D5C3×#45412yes—
339C23—21no—
340C23—51no—
341C23—101no—
342C23—201no—
343C23—201no—
344C23—204no—
345Q8—22no—
346Q8—104no—
347Q8—102no—
348D4—22no—
349D4—102no—
350D4—202no—
351C2×C4—21no—
352C2×C4—51no—
353C2×C4—52no—
354C2×C4—52no—
355C2×C4—52no—
356C2×C4—102no—
357C2×C4—102no—
358C2×C4—104no—
359C2×C4—101no—
360C2×C4—202no—
361C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
362C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
363C8—202no—
364C2×C3—21no—
365C2×C3—52no—
366C2×C3—81no—
367C2×C3—101no—
368C2×C3—401no—
369C2×C3#6577911yescharacteristic
370C5C3×#188811yescharacteristic, complemented, Sylow
371C22—42no—
372C22—41no—
373C22—51no—
374C22—51no—
375C22—102no—
376C22—202no—
377C22—201no—
378C22—402no—
379C22—402no—
380C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
381C4—22no—
382C4—41no—
383C4—52no—
384C4—102no—
385C4—201no—
386C4—204no—
387C4C3×D4×AGL(1,𝔽5)12yes—
388C3#1060456311yescharacteristic, direct product factor, Sylow
389C2—21no—
390C2—52no—
391C2—81no—
392C2—101no—
393C2—401no—
394C2C3×#6577911yescharacteristic
subgroupG/H
centerC2×C3#65779
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#124C4
socleC2×C3×C5#386
prime radicalC2×C3×D4×D5C22
cosocleC2×C4×D5C23×C3
hypercenterC3×#124AGL(1,𝔽5)
hypocenterC5C3×#1888
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C3×#66262C2semidirect
3. C3×D5⋊#101C2semidirect
5. C3×#64862C2semidirect
6. #10604563C3direct (N × H)
13. C3×C5×#124C4semidirect
14. C3×#4093C4semidirect
15. C3×#4082C4semidirect
18. C3×#4021C4semidirect
23. #66262C2×C3semidirect
25. D5⋊#101C2×C3semidirect
27. #64862C2×C3semidirect
68. C5×#124C3×C4semidirect
69. #4093C3×C4semidirect
70. #4082C3×C4semidirect
73. #4021C3×C4semidirect
110. C3×#124AGL(1,𝔽5)semidirect
230. #124C3×AGL(1,𝔽5)semidirect
315. C3×C5#1888semidirect
370. C5C3×#1888semidirect
388. C3#10604563direct (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×#1242
C3×D5⋊#1012
C3×#657672
C3×#648622
#106045633
C3×#18885