Finite Group Databasebeta

← C3×#10604380 #14298546

Subgroups — C3×#10604380 #14298546

Order 1920.

612 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5×#117C211yesmaximal, characteristic
2C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C3×#65756C211yesmaximal, characteristic
4C3×#64875C212yesmaximal, complemented
5C3×#64875C212yesmaximal, complemented
6#10604380C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C2×C3×C22⋊AGL(1,𝔽5)—21no—
9C2×C3×C4×AGL(1,𝔽5)C2212yes—
10C2×C3×#700C2212yes—
11C3×C5×#117C411yescharacteristic, complemented
12C3×#4072C411yescharacteristic, complemented
13C3×#4038C2211yescharacteristic
14C3×#4038C2211yescharacteristic
15C3×D5×C4⋊C4C411yescharacteristic, complemented
16C3×D5×C4⋊C4C411yescharacteristic, complemented
17C3×D5×C22⋊C4—21no—
18C3×#4023—21no—
19C3×#4018—21no—
20C3×C42×D5—21no—
21C3×C22⋊#170—21no—
22C3×(C2×AGL(1,𝔽5))⋊C4—21no—
23C3×#4003—21no—
24C3×#1826—51nomaximal, Hall
25D5×#117C2×C311yescharacteristic
26C4×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic
27#65756C2×C311yescharacteristic
28#64875C2×C312yescomplemented
29#64875C2×C312yescomplemented
30C23×C3×D5D411yescharacteristic
31C22×C3×AGL(1,𝔽5)—22no—
32C22×C3×C4×C5C2×C411yescharacteristic
33C22×C3×Dic5D411yescharacteristic
34C2×C3×C4×D5—24no—
35C2×C3×C4×D5—21no—
36C2×C3×C4×D5—21no—
37C2×C3×C4×D5C2×C411yescharacteristic
38C2×C3×C4×D5D411yescharacteristic
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5C2311yescharacteristic
41C3×C22⋊AGL(1,𝔽5)—24no—
42C2×C3×#170—22no—
43C3×C4×AGL(1,𝔽5)—24no—
44C3×#700—24no—
45C3×C5×C4⋊C4C2×C411yescharacteristic
46C3×C5×C4⋊C4C2×C411yescharacteristic
47C3×C5×C22⋊C4—21no—
48C3×C42×C5—21no—
49C3×C22⋊Dic5—21no—
50C3×#685—21no—
51C3×#685—21no—
52C3×C4⋊Dic5C2×C411yescharacteristic
53C3×C4⋊Dic5C2×C411yescharacteristic
54C3×#683—21no—
55C3×#683—21no—
56C3×C4×Dic5—21no—
57C3×C4×Dic5—21no—
58C3×C4×Dic5—21no—
59C2×C3×#117—51no—
60C3×#407—51no—
61C3×C4×C22⋊C4—51no—
62C3×#339—52no—
63C3×#339—52no—
64C22×C4×D5C22×C311yescharacteristic
65C2×C22⋊AGL(1,𝔽5)—21no—
66C2×C4×AGL(1,𝔽5)C22×C312yes—
67C2×#700C22×C312yes—
68C5×#117C3×C411yescharacteristic, complemented
69#4072C3×C411yescharacteristic, complemented
70#4038C22×C311yescharacteristic
71#4038C22×C311yescharacteristic
72D5×C4⋊C4C3×C411yescharacteristic, complemented
73D5×C4⋊C4C3×C411yescharacteristic, complemented
74D5×C22⋊C4—21no—
75#4023—21no—
76#4018—21no—
77C42×D5—21no—
78C22⋊#170—21no—
79(C2×AGL(1,𝔽5))⋊C4—21no—
80#4003—21no—
81#1826—51noSylow
82C23×C3×C5C22⋊C411yescharacteristic
83C22×C3×D5—22no—
84C22×C3×D5—22no—
85C22×C3×D5—41no—
86C22×C3×D5C22⋊C411yescharacteristic
87C22×C3×D5C2×D411yescharacteristic
88C2×C3×AGL(1,𝔽5)—24no—
89C2×C3×AGL(1,𝔽5)—42no—
90C2×C3×AGL(1,𝔽5)—42no—
91C2×C3×C4×C5—22no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C2×C3×C4×C5C22×C411yescharacteristic
96C2×C3×Dic5—22no—
97C2×C3×Dic5—21no—
98C2×C3×Dic5—21no—
99C2×C3×Dic5C22⋊C411yescharacteristic
100C2×C3×Dic5C2×D411yescharacteristic
101C3×C4×D5—22no—
102C3×C4×D5—22no—
103C3×C4×D5—22no—
104C3×C4×D5—42no—
105C3×C4×D5—42no—
106C3×C4×D5#4114yes—
107C3×#170—42no—
108C23×C3×C4—51no—
109C2×C3×M16—52no—
110C3×#117—51no—
111C3×#117—51no—
112C3×#117—51no—
113C3×#117—101no—
114C3×#117—101no—
115C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
116C2×C3×C4⋊C4—51no—
117C2×C3×C4⋊C4—51no—
118C2×C3×C22⋊C4—101no—
119C2×C3×C22⋊C4—101no—
120C2×C3×C42—52no—
121C2×C3×C42—101no—
122C3×C22⋊C8—101no—
123C3×C22⋊C8—101no—
124C3×#95—101no—
125C23×D5C3×D411yescharacteristic
126C22×AGL(1,𝔽5)—22no—
127C22×C4×C5C2×C3×C411yescharacteristic
128C22×Dic5C3×D411yescharacteristic
129C2×C4×D5—24no—
130C2×C4×D5—21no—
131C2×C4×D5—21no—
132C2×C4×D5C2×C3×C411yescharacteristic
133C2×C4×D5C3×D411yescharacteristic
134C2×C4×D5C3×D411yescharacteristic
135C2×C4×D5C23×C311yescharacteristic
136C22⋊AGL(1,𝔽5)—24no—
137C2×#170—22no—
138C4×AGL(1,𝔽5)—24no—
139#700—24no—
140C5×C4⋊C4C2×C3×C411yescharacteristic
141C5×C4⋊C4C2×C3×C411yescharacteristic
142C5×C22⋊C4—21no—
143C42×C5—21no—
144C22⋊Dic5—21no—
145#685—21no—
146#685—21no—
147C4⋊Dic5C2×C3×C411yescharacteristic
148C4⋊Dic5C2×C3×C411yescharacteristic
149#683—21no—
150#683—21no—
151C4×Dic5—21no—
152C4×Dic5—21no—
153C4×Dic5—21no—
154C2×#117—51no—
155#407—51no—
156C4×C22⋊C4—51no—
157#339—52no—
158#339—52no—
159C22×C3×C5—21no—
160C22×C3×C5—21no—
161C22×C3×C5—21no—
162C22×C3×C5C2×C22⋊C411yescharacteristic
163C2×C3×D5—21no—
164C2×C3×D5—21no—
165C2×C3×D5—21no—
166C2×C3×D5—21no—
167C2×C3×D5—44no—
168C2×C3×D5#10412yes—
169C2×C3×D5#12312yes—
170C3×AGL(1,𝔽5)—44no—
171C3×C4×C5—21no—
172C3×C4×C5—41no—
173C3×C4×C5—41no—
174C3×C4×C5#11712yes—
175C3×Dic5—21no—
176C3×Dic5—41no—
177C3×Dic5—41no—
178C3×Dic5#12312yes—
179C24×C3—51no—
180C22×C3×C4—51no—
181C22×C3×C4—51no—
182C22×C3×C4—51no—
183C22×C3×C4—51no—
184C22×C3×C4—51no—
185C22×C3×C4—104no—
186C22×C3×C4—101no—
187C22×C3×C4—101no—
188C22×C3×C4—102no—
189C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
190C3×M16—104no—
191C2×C3×C8—102no—
192C3×C4⋊C4—51no—
193C3×C4⋊C4—51no—
194C3×C4⋊C4—101no—
195C3×C4⋊C4—101no—
196C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
197C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
198C3×C22⋊C4—21no—
199C3×C22⋊C4—101no—
200C3×C22⋊C4—101no—
201C3×C22⋊C4—104no—
202C3×C22⋊C4—101no—
203C3×C42—21no—
204C3×C42—101no—
205C3×C42—101no—
206C3×C42—104no—
207C3×C42—101no—
208C23×C5C3×C22⋊C411yescharacteristic
209C22×D5—22no—
210C22×D5—22no—
211C22×D5—41no—
212C22×D5C3×C22⋊C411yescharacteristic
213C22×D5C2×C3×D411yescharacteristic
214C2×AGL(1,𝔽5)—24no—
215C2×AGL(1,𝔽5)—42no—
216C2×AGL(1,𝔽5)—42no—
217C2×C4×C5—22no—
218C2×C4×C5—21no—
219C2×C4×C5—21no—
220C2×C4×C5C3×C22⋊C411yescharacteristic
221C2×C4×C5C22×C3×C411yescharacteristic
222C2×Dic5—22no—
223C2×Dic5—21no—
224C2×Dic5—21no—
225C2×Dic5C3×C22⋊C411yescharacteristic
226C2×Dic5C2×C3×D411yescharacteristic
227C4×D5—22no—
228C4×D5—22no—
229C4×D5—22no—
230C4×D5—42no—
231C4×D5—42no—
232C4×D5C3×#4114yes—
233#170—42no—
234C23×C4—51no—
235C2×M16—52no—
236#117—51no—
237#117—51no—
238#117—51no—
239#117—101no—
240#117—101no—
241#117C3×AGL(1,𝔽5)11yescharacteristic, complemented
242C2×C4⋊C4—51no—
243C2×C4⋊C4—51no—
244C2×C22⋊C4—101no—
245C2×C22⋊C4—101no—
246C2×C42—52no—
247C2×C42—101no—
248C22⋊C8—101no—
249C22⋊C8—101no—
250#95—101no—
251C2×C3×C5—41no—
252C2×C3×C5#38111yescharacteristic
253C2×C3×C5C2×#10411yescharacteristic
254C2×C3×C5#42111yescharacteristic
255C3×D5—22no—
256C3×D5—22no—
257C23×C3—51no—
258C23×C3—51no—
259C23×C3—102no—
260C23×C3—102no—
261C23×C3—201no—
262C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—51no—
267C2×C3×C4—54no—
268C2×C3×C4—51no—
269C2×C3×C4—102no—
270C2×C3×C4—102no—
271C2×C3×C4—102no—
272C2×C3×C4—102no—
273C2×C3×C4—104no—
274C2×C3×C4—101no—
275C2×C3×C4—101no—
276C2×C3×C4—202no—
277C2×C3×C4—202no—
278C2×C3×C4—202no—
279C2×C3×C4—202no—
280C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
281C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
282C3×C8—202no—
283C22×C5—21no—
284C22×C5—21no—
285C22×C5—21no—
286C22×C5C2×C3×C22⋊C411yescharacteristic
287C2×D5—21no—
288C2×D5—21no—
289C2×D5—21no—
290C2×D5—21no—
291C2×D5—44no—
292C2×D5C3×#10412yes—
293C2×D5C3×#12312yes—
294AGL(1,𝔽5)—44no—
295C4×C5—21no—
296C4×C5—41no—
297C4×C5—41no—
298C4×C5C3×#11712yes—
299Dic5—21no—
300Dic5—41no—
301Dic5—41no—
302Dic5C3×#12312yes—
303C24—51no—
304C22×C4—51no—
305C22×C4—51no—
306C22×C4—51no—
307C22×C4—51no—
308C22×C4—51no—
309C22×C4—104no—
310C22×C4—101no—
311C22×C4—101no—
312C22×C4—102no—
313C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
314M16—104no—
315C2×C8—102no—
316C4⋊C4—51no—
317C4⋊C4—51no—
318C4⋊C4—101no—
319C4⋊C4—101no—
320C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
321C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—101no—
324C22⋊C4—101no—
325C22⋊C4—104no—
326C22⋊C4—101no—
327C42—21no—
328C42—101no—
329C42—101no—
330C42—104no—
331C42—101no—
332C3×C5#182611yescharacteristic, complemented, Hall
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—52no—
337C22×C3—52no—
338C22×C3—101no—
339C22×C3—101no—
340C22×C3—101no—
341C22×C3—101no—
342C22×C3—204no—
343C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
344C3×C4—21no—
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—52no—
348C3×C4—101no—
349C3×C4—204no—
350C3×C4—201no—
351C3×C4—201no—
352C3×C4#413012yes—
353C2×C5—41no—
354C2×C5C3×#38111yescharacteristic
355C2×C5C2×C3×#10411yescharacteristic
356C2×C5C3×#42111yescharacteristic
357D5—22no—
358D5—22no—
359C23—51no—
360C23—51no—
361C23—102no—
362C23—102no—
363C23—201no—
364C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—51no—
369C2×C4—54no—
370C2×C4—51no—
371C2×C4—102no—
372C2×C4—102no—
373C2×C4—102no—
374C2×C4—102no—
375C2×C4—104no—
376C2×C4—101no—
377C2×C4—101no—
378C2×C4—202no—
379C2×C4—202no—
380C2×C4—202no—
381C2×C4—202no—
382C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
383C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
384C8—202no—
385C2×C3—41no—
386C2×C3—102no—
387C2×C3—102no—
388C2×C3#6576411yescharacteristic
389C2×C3D5⋊#10411yescharacteristic
390C2×C3#6579511yescharacteristic
391C5C3×#182611yescharacteristic, complemented, Sylow
392C22—21no—
393C22—21no—
394C22—21no—
395C22—52no—
396C22—52no—
397C22—101no—
398C22—101no—
399C22—101no—
400C22—101no—
401C22—204no—
402C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
403C4—21no—
404C4—41no—
405C4—41no—
406C4—52no—
407C4—101no—
408C4—204no—
409C4—201no—
410C4—201no—
411C4C3×#413012yes—
412C3#1060438011yescharacteristic, direct product factor, Sylow
413C2—41no—
414C2—102no—
415C2—102no—
416C2C3×#6576411yescharacteristic
417C2C3×D5⋊#10411yescharacteristic
418C2C3×#6579511yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#117C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C4×D5C23×C3
hypercenterC3×#117AGL(1,𝔽5)
hypocenterC5C3×#1826
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
4. C3×#64875C2semidirect
5. C3×#64875C2semidirect
6. #10604380C3direct (N × H)
11. C3×C5×#117C4semidirect
12. C3×#4072C4semidirect
15. C3×D5×C4⋊C4C4semidirect
16. C3×D5×C4⋊C4C4semidirect
28. #64875C2×C3semidirect
29. #64875C2×C3semidirect
68. C5×#117C3×C4semidirect
69. #4072C3×C4semidirect
72. D5×C4⋊C4C3×C4semidirect
73. D5×C4⋊C4C3×C4semidirect
115. C3×#117AGL(1,𝔽5)semidirect
241. #117C3×AGL(1,𝔽5)semidirect
332. C3×C5#1826semidirect
391. C5C3×#1826semidirect
412. C3#10604380direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×#1172
C3×C4×C22⋊AGL(1,𝔽5)2
C3×#657562
C3×#648752
#106043803
C3×#18265