Finite Group Databasebeta

← C3×#10604761 #14297946

Subgroups — C3×#10604761 #14297946

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D5×M16C211yesmaximal, characteristic
2C3×#65765C212yesmaximal
3C2×C3×#3994C211yesmaximal, characteristic, complemented
4C2×C3×#3993C211yesmaximal, characteristic, complemented
5C3×#64893C212yesmaximal, complemented
6#10604761C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C2×C3×C22⋊AGL(1,𝔽5)—22no—
9C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
10C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
11C2×C3×C5×M16C411yescharacteristic, complemented
12C2×C3×#681C411yescharacteristic, complemented
13C3×D5×M16—22no—
14C3×D5×M16—22no—
15C2×C3×#676C411yescharacteristic, complemented
16C2×C3×#676C411yescharacteristic, complemented
17C2×C3×C8×D5C2211yescharacteristic
18C2×C3×C8×D5C2211yescharacteristic
19C3×(C2×AGL(1,𝔽5))⋊C4—22no—
20C3×#3994—22no—
21C3×#3993—22no—
22C3×#1809—51nomaximal, Hall
23C2×D5×M16C2×C311yescharacteristic
24#65765C2×C312yes—
25C2×#3994C2×C311yescharacteristic, complemented
26C2×#3993C2×C311yescharacteristic, complemented
27#64893C2×C312yescomplemented
28C23×C3×D5D411yescharacteristic
29C22×C3×AGL(1,𝔽5)—22no—
30C22×C3×AGL(1,𝔽5)—22no—
31C22×C3×C4×C5C2×C411yescharacteristic
32C22×C3×Dic5Q811yescharacteristic
33C2×C3×C4×D5—21no—
34C2×C3×C4×D5—21no—
35C2×C3×C4×D5—21no—
36C2×C3×C4×D5—21no—
37C2×C3×C4×D5C2×C411yescharacteristic
38C2×C3×C4×D5D411yescharacteristic
39C2×C3×C4×D5Q811yescharacteristic
40C2×C3×C4×D5C2311yescharacteristic
41C3×C22⋊AGL(1,𝔽5)—44no—
42C3×C4⋊AGL(1,𝔽5)—22no—
43C3×C4⋊AGL(1,𝔽5)—22no—
44C3×C4⋊AGL(1,𝔽5)—22no—
45C3×C4⋊AGL(1,𝔽5)—22no—
46C3×C5×M16—22no—
47C2×C3×C5×C8C2×C411yescharacteristic
48C2×C3×C5×C8C2×C411yescharacteristic
49C3×#681—22no—
50C2×C3×#168C2×C411yescharacteristic
51C2×C3×#168C2×C411yescharacteristic
52C3×#676—22no—
53C3×#676—22no—
54C3×C8×D5—21no—
55C3×C8×D5—21no—
56C3×C8×D5D412yes—
57C3×C8×D5D412yes—
58C22×C3×M16—51no—
59C2×C3×#107—51no—
60C2×C3×#106—51no—
61C3×#380—52no—
62C3×#340—52no—
63C22×C4×D5C22×C311yescharacteristic
64C2×C22⋊AGL(1,𝔽5)—22no—
65C2×C4⋊AGL(1,𝔽5)C22×C312yes—
66C2×C4⋊AGL(1,𝔽5)C22×C312yes—
67C2×C5×M16C3×C411yescharacteristic, complemented
68C2×#681C3×C411yescharacteristic, complemented
69D5×M16—22no—
70D5×M16—22no—
71C2×#676C3×C411yescharacteristic, complemented
72C2×#676C3×C411yescharacteristic, complemented
73C2×C8×D5C22×C311yescharacteristic
74C2×C8×D5C22×C311yescharacteristic
75(C2×AGL(1,𝔽5))⋊C4—22no—
76#3994—22no—
77#3993—22no—
78#1809—51noSylow
79C23×C3×C5C4⋊C411yescharacteristic
80C22×C3×D5—21no—
81C22×C3×D5—21no—
82C22×C3×D5—22no—
83C22×C3×D5—22no—
84C22×C3×D5C4⋊C411yescharacteristic
85C22×C3×D5C2×D411yescharacteristic
86C2×C3×AGL(1,𝔽5)—42no—
87C2×C3×AGL(1,𝔽5)—42no—
88C2×C3×AGL(1,𝔽5)—42no—
89C2×C3×AGL(1,𝔽5)—42no—
90C2×C3×AGL(1,𝔽5)—42no—
91C2×C3×AGL(1,𝔽5)—42no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5C4⋊C411yescharacteristic
95C2×C3×C4×C5C22×C411yescharacteristic
96C2×C3×Dic5—21no—
97C2×C3×Dic5—21no—
98C2×C3×Dic5C4⋊C411yescharacteristic
99C2×C3×Dic5C2×Q811yescharacteristic
100C3×C4×D5—21no—
101C3×C4×D5—21no—
102C3×C4×D5—21no—
103C3×C4×D5—21no—
104C3×C4×D5—21no—
105C3×C4×D5—21no—
106C3×C4×D5C2×D411yescharacteristic
107C3×C4×D5C2×D411yescharacteristic
108C3×C4×D5#4111yescharacteristic
109C3×C4×D5#4111yescharacteristic
110C3×C5×C8—21no—
111C3×C5×C8C22⋊C412yescomplemented
112C3×#168—21no—
113C3×#168C22⋊C412yescomplemented
114C23×C3×C4—51no—
115C2×C3×M16—51no—
116C2×C3×M16—51no—
117C2×C3×M16—51no—
118C2×C3×M16—102no—
119C2×C3×M16—102no—
120C2×C3×M16AGL(1,𝔽5)11yescharacteristic, complemented
121C22×C3×C8—51no—
122C22×C3×C8—51no—
123C2×C3×C4⋊C4—52no—
124C2×C3×C4⋊C4—52no—
125C2×C3×C22⋊C4—102no—
126C3×#107—102no—
127C3×#106—102no—
128C3×#95—102no—
129C23×D5C3×D411yescharacteristic
130C22×AGL(1,𝔽5)—22no—
131C22×AGL(1,𝔽5)—22no—
132C22×C4×C5C2×C3×C411yescharacteristic
133C22×Dic5C3×Q811yescharacteristic
134C2×C4×D5—21no—
135C2×C4×D5—21no—
136C2×C4×D5—21no—
137C2×C4×D5—21no—
138C2×C4×D5C2×C3×C411yescharacteristic
139C2×C4×D5C3×D411yescharacteristic
140C2×C4×D5C3×Q811yescharacteristic
141C2×C4×D5C23×C311yescharacteristic
142C22⋊AGL(1,𝔽5)—44no—
143C4⋊AGL(1,𝔽5)—22no—
144C4⋊AGL(1,𝔽5)—22no—
145C4⋊AGL(1,𝔽5)—22no—
146C4⋊AGL(1,𝔽5)—22no—
147C5×M16—22no—
148C2×C5×C8C2×C3×C411yescharacteristic
149C2×C5×C8C2×C3×C411yescharacteristic
150#681—22no—
151C2×#168C2×C3×C411yescharacteristic
152C2×#168C2×C3×C411yescharacteristic
153#676—22no—
154#676—22no—
155C8×D5—21no—
156C8×D5—21no—
157C8×D5C3×D412yes—
158C8×D5C3×D412yes—
159C22×M16—51no—
160C2×#107—51no—
161C2×#106—51no—
162#380—52no—
163#340—52no—
164C22×C3×C5—21no—
165C22×C3×C5—22no—
166C22×C3×C5C2×C4⋊C411yescharacteristic
167C2×C3×D5—21no—
168C2×C3×D5—22no—
169C2×C3×D5—42no—
170C2×C3×D5—41no—
171C2×C3×D5—41no—
172C2×C3×D5#12111yescharacteristic
173C2×C3×D5#12111yescharacteristic
174C2×C3×D5AGL(1,ℤ/8)12yes—
175C2×C3×D5#13712yes—
176C3×AGL(1,𝔽5)—82no—
177C3×AGL(1,𝔽5)—82no—
178C3×C4×C5—21no—
179C3×C4×C5C2×C22⋊C411yescharacteristic
180C3×C4×C5#11711yescharacteristic
181C3×Dic5—21no—
182C3×Dic5C22⋊Q811yescharacteristic
183C3×Dic5C22⋊Q811yescharacteristic
184C24×C3—51no—
185C22×C3×C4—51no—
186C22×C3×C4—51no—
187C22×C3×C4—51no—
188C22×C3×C4—51no—
189C22×C3×C4—51no—
190C22×C3×C4—101no—
191C22×C3×C4—101no—
192C22×C3×C4—101no—
193C22×C3×C4—101no—
194C22×C3×C4—102no—
195C22×C3×C4—102no—
196C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
197C3×M16—22no—
198C3×M16—102no—
199C3×M16—102no—
200C3×M16—102no—
201C2×C3×C8—51no—
202C2×C3×C8—52no—
203C2×C3×C8—52no—
204C2×C3×C8—51no—
205C2×C3×C8—101no—
206C2×C3×C8—101no—
207C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
208C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
209C3×C4⋊C4—102no—
210C3×C4⋊C4—102no—
211C3×C4⋊C4—102no—
212C3×C4⋊C4—102no—
213C3×C22⋊C4—204no—
214C23×C5C3×C4⋊C411yescharacteristic
215C22×D5—21no—
216C22×D5—21no—
217C22×D5—22no—
218C22×D5—22no—
219C22×D5C3×C4⋊C411yescharacteristic
220C22×D5C2×C3×D411yescharacteristic
221C2×AGL(1,𝔽5)—42no—
222C2×AGL(1,𝔽5)—42no—
223C2×AGL(1,𝔽5)—42no—
224C2×AGL(1,𝔽5)—42no—
225C2×AGL(1,𝔽5)—42no—
226C2×AGL(1,𝔽5)—42no—
227C2×C4×C5—21no—
228C2×C4×C5—21no—
229C2×C4×C5C3×C4⋊C411yescharacteristic
230C2×C4×C5C22×C3×C411yescharacteristic
231C2×Dic5—21no—
232C2×Dic5—21no—
233C2×Dic5C3×C4⋊C411yescharacteristic
234C2×Dic5C2×C3×Q811yescharacteristic
235C4×D5—21no—
236C4×D5—21no—
237C4×D5—21no—
238C4×D5—21no—
239C4×D5—21no—
240C4×D5—21no—
241C4×D5C2×C3×D411yescharacteristic
242C4×D5C2×C3×D411yescharacteristic
243C4×D5C3×#4111yescharacteristic
244C4×D5C3×#4111yescharacteristic
245C5×C8—21no—
246C5×C8C3×C22⋊C412yescomplemented
247#168—21no—
248#168C3×C22⋊C412yescomplemented
249C23×C4—51no—
250C2×M16—51no—
251C2×M16—51no—
252C2×M16—51no—
253C2×M16—102no—
254C2×M16—102no—
255C2×M16C3×AGL(1,𝔽5)11yescharacteristic, complemented
256C22×C8—51no—
257C22×C8—51no—
258C2×C4⋊C4—52no—
259C2×C4⋊C4—52no—
260C2×C22⋊C4—102no—
261#107—102no—
262#106—102no—
263#95—102no—
264C2×C3×C5—41no—
265C2×C3×C5#38011yescharacteristic
266C2×C3×C5#42812yes—
267C3×D5—41no—
268C3×D5#46812yes—
269C3×D5#46911yescharacteristic
270C3×D5#47011yescharacteristic
271C23×C3—51no—
272C23×C3—51no—
273C23×C3—102no—
274C23×C3—101no—
275C23×C3—101no—
276C23×C3—102no—
277C23×C3C4⋊AGL(1,𝔽5)11yescharacteristic
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—51no—
281C2×C3×C4—51no—
282C2×C3×C4—51no—
283C2×C3×C4—51no—
284C2×C3×C4—51no—
285C2×C3×C4—51no—
286C2×C3×C4—101no—
287C2×C3×C4—101no—
288C2×C3×C4—101no—
289C2×C3×C4—101no—
290C2×C3×C4—101no—
291C2×C3×C4—101no—
292C2×C3×C4—101no—
293C2×C3×C4—101no—
294C2×C3×C4—202no—
295C2×C3×C4—202no—
296C2×C3×C4—202no—
297C2×C3×C4—202no—
298C2×C3×C4—202no—
299C2×C3×C4—202no—
300C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic
301C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
302C3×C8—21no—
303C3×C8—52no—
304C3×C8—101no—
305C3×C8C22⋊AGL(1,𝔽5)12yescomplemented
306C22×C5—21no—
307C22×C5—22no—
308C22×C5C2×C3×C4⋊C411yescharacteristic
309C2×D5—21no—
310C2×D5—22no—
311C2×D5—42no—
312C2×D5—41no—
313C2×D5—41no—
314C2×D5C3×#12111yescharacteristic
315C2×D5C3×#12111yescharacteristic
316C2×D5C3×AGL(1,ℤ/8)12yes—
317C2×D5C3×#13712yes—
318AGL(1,𝔽5)—82no—
319AGL(1,𝔽5)—82no—
320C4×C5—21no—
321C4×C5C2×C3×C22⋊C411yescharacteristic
322C4×C5C3×#11711yescharacteristic
323Dic5—21no—
324Dic5C3×C22⋊Q811yescharacteristic
325Dic5C3×C22⋊Q811yescharacteristic
326C24—51no—
327C22×C4—51no—
328C22×C4—51no—
329C22×C4—51no—
330C22×C4—51no—
331C22×C4—51no—
332C22×C4—101no—
333C22×C4—101no—
334C22×C4—101no—
335C22×C4—101no—
336C22×C4—102no—
337C22×C4—102no—
338C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
339M16—22no—
340M16—102no—
341M16—102no—
342M16—102no—
343C2×C8—51no—
344C2×C8—52no—
345C2×C8—52no—
346C2×C8—51no—
347C2×C8—101no—
348C2×C8—101no—
349C2×C8C2×C3×AGL(1,𝔽5)11yescharacteristic
350C2×C8C2×C3×AGL(1,𝔽5)11yescharacteristic
351C4⋊C4—102no—
352C4⋊C4—102no—
353C4⋊C4—102no—
354C4⋊C4—102no—
355C22⋊C4—204no—
356C3×C5#180911yescharacteristic, complemented, Hall
357C22×C3—22no—
358C22×C3—21no—
359C22×C3—52no—
360C22×C3—52no—
361C22×C3—51no—
362C22×C3—51no—
363C22×C3—102no—
364C22×C3—101no—
365C22×C3—201no—
366C22×C3—202no—
367C22×C3—201no—
368C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
369C3×C4—21no—
370C3×C4—51no—
371C3×C4—51no—
372C3×C4—101no—
373C3×C4—402no—
374C3×C4—402no—
375C3×C4#413011yescharacteristic
376C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
377C2×C5—41no—
378C2×C5C3×#38011yescharacteristic
379C2×C5C3×#42812yes—
380D5—41no—
381D5C3×#46812yes—
382D5C3×#46911yescharacteristic
383D5C3×#47011yescharacteristic
384C23—51no—
385C23—51no—
386C23—102no—
387C23—101no—
388C23—101no—
389C23—102no—
390C23C3×C4⋊AGL(1,𝔽5)11yescharacteristic
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—51no—
394C2×C4—51no—
395C2×C4—51no—
396C2×C4—51no—
397C2×C4—51no—
398C2×C4—51no—
399C2×C4—101no—
400C2×C4—101no—
401C2×C4—101no—
402C2×C4—101no—
403C2×C4—101no—
404C2×C4—101no—
405C2×C4—101no—
406C2×C4—101no—
407C2×C4—202no—
408C2×C4—202no—
409C2×C4—202no—
410C2×C4—202no—
411C2×C4—202no—
412C2×C4—202no—
413C2×C4C3×C4⋊AGL(1,𝔽5)11yescharacteristic
414C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
415C8—21no—
416C8—52no—
417C8—101no—
418C8C3×C22⋊AGL(1,𝔽5)12yescomplemented
419C2×C3—41no—
420C2×C3—51no—
421C2×C3—52no—
422C2×C3—51no—
423C2×C3—201no—
424C2×C3#6572712yes—
425C2×C3#6576511yescharacteristic
426C5C3×#180911yescharacteristic, complemented, Sylow
427C22—22no—
428C22—21no—
429C22—52no—
430C22—52no—
431C22—51no—
432C22—51no—
433C22—102no—
434C22—101no—
435C22—201no—
436C22—202no—
437C22—201no—
438C22C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
439C4—21no—
440C4—51no—
441C4—51no—
442C4—101no—
443C4—402no—
444C4—402no—
445C4C3×#413011yescharacteristic
446C4C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
447C3#1060476111yescharacteristic, direct product factor, Sylow
448C2—41no—
449C2—51no—
450C2—52no—
451C2—51no—
452C2—201no—
453C2C3×#6572712yes—
454C2C3×#6576511yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×M16C4
socleC22×C3×C5C2×C4⋊C4
prime radicalC23×C3×D5D4
cosocleC2×C4×D5C23×C3
hypercenterC2×C3×M16AGL(1,𝔽5)
hypocenterC5C3×#1809
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×M16

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3994C2semidirect
4. C2×C3×#3993C2semidirect
5. C3×#64893C2semidirect
6. #10604761C3direct (N × H)
11. C2×C3×C5×M16C4semidirect
12. C2×C3×#681C4semidirect
15. C2×C3×#676C4semidirect
16. C2×C3×#676C4semidirect
25. C2×#3994C2×C3semidirect
26. C2×#3993C2×C3semidirect
27. #64893C2×C3semidirect
67. C2×C5×M16C3×C4semidirect
68. C2×#681C3×C4semidirect
71. C2×#676C3×C4semidirect
72. C2×#676C3×C4semidirect
111. C3×C5×C8C22⋊C4semidirect
113. C3×#168C22⋊C4semidirect
120. C2×C3×M16AGL(1,𝔽5)semidirect
246. C5×C8C3×C22⋊C4semidirect
248. #168C3×C22⋊C4semidirect
255. C2×M16C3×AGL(1,𝔽5)semidirect
305. C3×C8C22⋊AGL(1,𝔽5)semidirect
356. C3×C5#1809semidirect
418. C8C3×C22⋊AGL(1,𝔽5)semidirect
426. C5C3×#1809semidirect
447. C3#10604761direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D5×M162
C3×#657652
C2×C3×#39942
C2×C3×#39932
C3×#648932
#106047613
C3×#18095