Finite Group Databasebeta

← C3×#10604567 #14305890

Subgroups — C3×#10604567 #14305890

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5×#124C211yesmaximal, characteristic
2C2×C3×#4009C212yesmaximal, complemented
3C2×C3×#4007C212yesmaximal
4C3×#65688C211yesmaximal, characteristic, complemented
5C3×#65676C211yesmaximal, characteristic, complemented
6#10604567C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C2×C3×D4×D5C2211yescharacteristic
9C2×C3×C4⋊AGL(1,𝔽5)—22no—
10C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
11C2×C3×D5⋊C8C2212yes—
12C3×C5×#124C411yescharacteristic, complemented
13C3×#4093C411yescharacteristic, complemented
14C3×#4082C411yescharacteristic, complemented
15C3×#4031—22no—
16C3×D5×C22⋊C4—22no—
17C3×#4021C411yescharacteristic, complemented
18C3×C42×D5C2211yescharacteristic
19C3×#4009—24no—
20C3×#4007—24no—
21C3×#4002—22no—
22C3×#1959—51nomaximal, Hall
23D5×#124C2×C311yescharacteristic
24C2×#4009C2×C312yescomplemented
25C2×#4007C2×C312yes—
26#65688C2×C311yescharacteristic, complemented
27#65676C2×C311yescharacteristic, complemented
28C23×C3×D5—21no—
29C22×C3×AGL(1,𝔽5)—22no—
30C2×C3×C5×Q8C2×C411yescharacteristic
31C2×C3×C5×D4C2×C411yescharacteristic
32C2×C3×C22⋊D5—21no—
33C3×Q8×D5—22no—
34C3×Q8×D5—22no—
35C3×D4×D5—22no—
36C3×D4×D5—22no—
37C2×C3×D20C2×C411yescharacteristic
38C2×C3×C4×D5—21no—
39C2×C3×C4×D5D412yes—
40C2×C3×C4×D5C2311yescharacteristic
41C2×C3×Dic10—21no—
42C2×C3×Dic10C2×C411yescharacteristic
43C2×C3×#170—22no—
44C3×C4⋊AGL(1,𝔽5)—22no—
45C3×C4⋊AGL(1,𝔽5)—22no—
46C3×C4⋊AGL(1,𝔽5)—44no—
47C3×D5⋊C8—22no—
48C3×D5⋊C8D414yes—
49C3×C5×C22⋊C4—22no—
50C3×C42×C5C2×C411yescharacteristic
51C3×C22⋊Dic5—22no—
52C3×#685—22no—
53C3×#685—22no—
54C3×C4×Dic5C2×C411yescharacteristic
55C3×C4×Dic5D412yes—
56C2×C3×#124—51no—
57C2×C3×#103—52no—
58C2×C3×#102—52no—
59C2×C3×#97—51no—
60C3×#384—51no—
61C2×Q8×D5C22×C311yescharacteristic
62C2×D4×D5C22×C311yescharacteristic
63C2×C4⋊AGL(1,𝔽5)—22no—
64C2×C4⋊AGL(1,𝔽5)C22×C312yes—
65C2×D5⋊C8C22×C312yes—
66C5×#124C3×C411yescharacteristic, complemented
67#4093C3×C411yescharacteristic, complemented
68#4082C3×C411yescharacteristic, complemented
69#4031—22no—
70D5×C22⋊C4—22no—
71#4021C3×C411yescharacteristic, complemented
72C42×D5C22×C311yescharacteristic
73#4009—24no—
74#4007—24no—
75#4002—22no—
76#1959—51noSylow
77C23×C3×C5—21no—
78C22×C3×D5—21no—
79C22×C3×D5—41no—
80C22×C3×D5—41no—
81C22×C3×D5—42no—
82C22×C3×D5—42no—
83C22×C3×D5C2×D411yescharacteristic
84C2×C3×AGL(1,𝔽5)—42no—
85C2×C3×AGL(1,𝔽5)—44no—
86C3×C5×Q8—22no—
87C3×C5×D4—22no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5C22⋊C412yes—
90C2×C3×C4×C5C22×C411yescharacteristic
91C3×C22⋊D5—42no—
92C2×C3×Dic5—21no—
93C2×C3×Dic5C22⋊C412yes—
94C2×C3×Dic5C2×D411yescharacteristic
95C3×D20—22no—
96C3×C4×D5—22no—
97C3×C4×D5—22no—
98C3×C4×D5—41no—
99C3×C4×D5—41no—
100C3×C4×D5C2×D412yes—
101C3×C4×D5#4112yes—
102C3×Dic10—22no—
103C3×Dic10—42no—
104C3×#170—24no—
105C22×C3×Q8—51no—
106C22×C3×D4—51no—
107C22×C3×C8—52no—
108C3×#124—51no—
109C3×#124—51no—
110C3×#124—51no—
111C3×#124—102no—
112C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
113C2×C3×C4⋊C4—52no—
114C2×C3×C4⋊C4—102no—
115C2×C3×C22⋊C4—102no—
116C2×C3×C42—51no—
117C3×#103—104no—
118C3×#102—104no—
119C3×#97—102no—
120C23×D5—21no—
121C22×AGL(1,𝔽5)—22no—
122C2×C5×Q8C2×C3×C411yescharacteristic
123C2×C5×D4C2×C3×C411yescharacteristic
124C2×C22⋊D5—21no—
125Q8×D5—22no—
126Q8×D5—22no—
127D4×D5—22no—
128D4×D5—22no—
129C2×D20C2×C3×C411yescharacteristic
130C2×C4×D5—21no—
131C2×C4×D5C3×D412yes—
132C2×C4×D5C23×C311yescharacteristic
133C2×Dic10—21no—
134C2×Dic10C2×C3×C411yescharacteristic
135C2×#170—22no—
136C4⋊AGL(1,𝔽5)—22no—
137C4⋊AGL(1,𝔽5)—22no—
138C4⋊AGL(1,𝔽5)—44no—
139D5⋊C8—22no—
140D5⋊C8C3×D414yes—
141C5×C22⋊C4—22no—
142C42×C5C2×C3×C411yescharacteristic
143C22⋊Dic5—22no—
144#685—22no—
145#685—22no—
146C4×Dic5C2×C3×C411yescharacteristic
147C4×Dic5C3×D412yes—
148C2×#124—51no—
149C2×#103—52no—
150C2×#102—52no—
151C2×#97—51no—
152#384—51no—
153C22×C3×C5—41no—
154C22×C3×C5—42no—
155C22×C3×C5C2×C22⋊C411yescharacteristic
156C2×C3×D5—41no—
157C2×C3×D5—42no—
158C2×C3×D5—82no—
159C2×C3×D5—81no—
160C2×C3×D5—81no—
161C2×C3×D5#12411yescharacteristic
162C2×C3×D5#12711yescharacteristic
163C2×C3×D5AGL(1,ℤ/8)12yes—
164C2×C3×D5#13712yes—
165C3×AGL(1,𝔽5)—82no—
166C3×C4×C5—22no—
167C3×C4×C5—41no—
168C3×C4×C5C4×D412yes—
169C3×Dic5—22no—
170C3×Dic5—41no—
171C3×Dic5#12112yes—
172C24×C3—101no—
173C2×C3×Q8—51no—
174C2×C3×Q8—101no—
175C2×C3×Q8—102no—
176C2×C3×Q8—102no—
177C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
178C2×C3×D4—51no—
179C2×C3×D4—102no—
180C2×C3×D4—102no—
181C2×C3×D4—101no—
182C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
183C22×C3×C4—51no—
184C22×C3×C4—52no—
185C22×C3×C4—101no—
186C22×C3×C4—102no—
187C2×C3×C8—54no—
188C2×C3×C8—102no—
189C2×C3×C8—102no—
190C3×C4⋊C4—102no—
191C3×C4⋊C4—102no—
192C3×C4⋊C4—204no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—102no—
195C3×C22⋊C4—102no—
196C3×C22⋊C4—102no—
197C3×C42—52no—
198C3×C42—51no—
199C3×C42C2×AGL(1,𝔽5)11yescharacteristic
200C23×C5—21no—
201C22×D5—21no—
202C22×D5—41no—
203C22×D5—41no—
204C22×D5—42no—
205C22×D5—42no—
206C22×D5C2×C3×D411yescharacteristic
207C2×AGL(1,𝔽5)—42no—
208C2×AGL(1,𝔽5)—44no—
209C5×Q8—22no—
210C5×D4—22no—
211C2×C4×C5—21no—
212C2×C4×C5C3×C22⋊C412yes—
213C2×C4×C5C22×C3×C411yescharacteristic
214C22⋊D5—42no—
215C2×Dic5—21no—
216C2×Dic5C3×C22⋊C412yes—
217C2×Dic5C2×C3×D411yescharacteristic
218D20—22no—
219C4×D5—22no—
220C4×D5—22no—
221C4×D5—41no—
222C4×D5—41no—
223C4×D5C2×C3×D412yes—
224C4×D5C3×#4112yes—
225Dic10—22no—
226Dic10—42no—
227#170—24no—
228C22×Q8—51no—
229C22×D4—51no—
230C22×C8—52no—
231#124—51no—
232#124—51no—
233#124—51no—
234#124—102no—
235#124C3×AGL(1,𝔽5)11yescharacteristic, complemented
236C2×C4⋊C4—52no—
237C2×C4⋊C4—102no—
238C2×C22⋊C4—102no—
239C2×C42—51no—
240#103—104no—
241#102—104no—
242#97—102no—
243C2×C3×C5—81no—
244C2×C3×C5#39011yescharacteristic
245C2×C3×C5#41712yes—
246C3×D5—81no—
247C3×D5#48912yes—
248C3×D5#49611yescharacteristic
249C3×D5#49711yescharacteristic
250C23×C3—21no—
251C23×C3—51no—
252C23×C3—101no—
253C23×C3—201no—
254C23×C3—201no—
255C23×C3—202no—
256C23×C3—202no—
257C3×Q8—22no—
258C3×Q8—102no—
259C3×Q8—202no—
260C3×D4—22no—
261C3×D4—102no—
262C3×D4—202no—
263C2×C3×C4—21no—
264C2×C3×C4—51no—
265C2×C3×C4—52no—
266C2×C3×C4—52no—
267C2×C3×C4—52no—
268C2×C3×C4—102no—
269C2×C3×C4—102no—
270C2×C3×C4—101no—
271C2×C3×C4—201no—
272C2×C3×C4—201no—
273C2×C3×C4—202no—
274C2×C3×C4—204no—
275C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
276C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
277C3×C8—104no—
278C22×C5—41no—
279C22×C5—42no—
280C22×C5C2×C3×C22⋊C411yescharacteristic
281C2×D5—41no—
282C2×D5—42no—
283C2×D5—82no—
284C2×D5—81no—
285C2×D5—81no—
286C2×D5C3×#12411yescharacteristic
287C2×D5C3×#12711yescharacteristic
288C2×D5C3×AGL(1,ℤ/8)12yes—
289C2×D5C3×#13712yes—
290AGL(1,𝔽5)—82no—
291C4×C5—22no—
292C4×C5—41no—
293C4×C5C3×C4×D412yes—
294Dic5—22no—
295Dic5—41no—
296Dic5C3×#12112yes—
297C24—101no—
298C2×Q8—51no—
299C2×Q8—101no—
300C2×Q8—102no—
301C2×Q8—102no—
302C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
303C2×D4—51no—
304C2×D4—102no—
305C2×D4—102no—
306C2×D4—101no—
307C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
308C22×C4—51no—
309C22×C4—52no—
310C22×C4—101no—
311C22×C4—102no—
312C2×C8—54no—
313C2×C8—102no—
314C2×C8—102no—
315C4⋊C4—102no—
316C4⋊C4—102no—
317C4⋊C4—204no—
318C22⋊C4—22no—
319C22⋊C4—102no—
320C22⋊C4—102no—
321C22⋊C4—102no—
322C42—52no—
323C42—51no—
324C42C2×C3×AGL(1,𝔽5)11yescharacteristic
325C3×C5#195911yescharacteristic, complemented, Hall
326C22×C3—42no—
327C22×C3—41no—
328C22×C3—52no—
329C22×C3—52no—
330C22×C3—51no—
331C22×C3—51no—
332C22×C3—202no—
333C22×C3—201no—
334C22×C3—401no—
335C22×C3—402no—
336C22×C3—401no—
337C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
338C3×C4—22no—
339C3×C4—41no—
340C3×C4—52no—
341C3×C4—102no—
342C3×C4—201no—
343C3×C4—402no—
344C3×C4D4×AGL(1,𝔽5)12yes—
345C2×C5—81no—
346C2×C5C3×#39011yescharacteristic
347C2×C5C3×#41712yes—
348D5—81no—
349D5C3×#48912yes—
350D5C3×#49611yescharacteristic
351D5C3×#49711yescharacteristic
352C23—21no—
353C23—51no—
354C23—101no—
355C23—201no—
356C23—201no—
357C23—202no—
358C23—202no—
359Q8—22no—
360Q8—102no—
361Q8—202no—
362D4—22no—
363D4—102no—
364D4—202no—
365C2×C4—21no—
366C2×C4—51no—
367C2×C4—52no—
368C2×C4—52no—
369C2×C4—52no—
370C2×C4—102no—
371C2×C4—102no—
372C2×C4—101no—
373C2×C4—201no—
374C2×C4—201no—
375C2×C4—202no—
376C2×C4—204no—
377C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
378C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
379C8—104no—
380C2×C3—51no—
381C2×C3—52no—
382C2×C3—51no—
383C2×C3—81no—
384C2×C3—401no—
385C2×C3#6578011yescharacteristic
386C2×C3#6579312yes—
387C5C3×#195911yescharacteristic, complemented, Sylow
388C22—41no—
389C22—42no—
390C22—51no—
391C22—51no—
392C22—52no—
393C22—52no—
394C22—201no—
395C22—202no—
396C22—401no—
397C22—402no—
398C22—401no—
399C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
400C4—22no—
401C4—41no—
402C4—52no—
403C4—102no—
404C4—201no—
405C4—402no—
406C4C3×D4×AGL(1,𝔽5)12yes—
407C3#1060456711yescharacteristic, direct product factor, Sylow
408C2—51no—
409C2—52no—
410C2—51no—
411C2—81no—
412C2—401no—
413C2C3×#6578011yescharacteristic
414C2C3×#6579312yes—
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#124C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×D4×D5C22
cosocleC2×C4×D5C23×C3
hypercenterC3×#124AGL(1,𝔽5)
hypocenterC5C3×#1959
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4009C2semidirect
4. C3×#65688C2semidirect
5. C3×#65676C2semidirect
6. #10604567C3direct (N × H)
12. C3×C5×#124C4semidirect
13. C3×#4093C4semidirect
14. C3×#4082C4semidirect
17. C3×#4021C4semidirect
24. C2×#4009C2×C3semidirect
26. #65688C2×C3semidirect
27. #65676C2×C3semidirect
66. C5×#124C3×C4semidirect
67. #4093C3×C4semidirect
68. #4082C3×C4semidirect
71. #4021C3×C4semidirect
112. C3×#124AGL(1,𝔽5)semidirect
235. #124C3×AGL(1,𝔽5)semidirect
325. C3×C5#1959semidirect
387. C5C3×#1959semidirect
407. C3#10604567direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×#1242
C2×C3×#40092
C2×C3×#40072
C3×#656882
C3×#656762
#106045673
C3×#19595