Finite Group Databasebeta

← C3×#10602501 #14282968

Subgroups — C3×#10602501 #14282968

Order 1920.

668 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#525C211yesmaximal, characteristic
2C2×C3×#3963C211yesmaximal, characteristic, complemented
3C3×#65496C211yesmaximal, characteristic, complemented
4C3×#64758C212yesmaximal, complemented
5C3×#64757C212yesmaximal, complemented
6#10602501C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41C411yescharacteristic, complemented
8C23×C3×C4×C5C2211yescharacteristic
9C23×C3×Dic5—21no—
10C3×C5×#123—21no—
11C3×C5×C22⋊Q8C411yescharacteristic, complemented
12C3×C5×C22⋊Q8C2211yescharacteristic
13C3×C5×#121C411yescharacteristic, complemented
14C3×C5×#121C2211yescharacteristic
15C3×C5×#120—21no—
16C3×C4×C5×D4—21no—
17C3×C4×C5×D4—21no—
18C3×C5×#117C411yescharacteristic, complemented
19C2×C3×#681—21no—
20C3×#3963—21no—
21C3×#3963C2212yes—
22C3×C22⋊#168—21no—
23C3×C22⋊#168C2212yes—
24C3×#1504—51nomaximal, Hall
25C5×#525C2×C311yescharacteristic
26C2×#3963C2×C311yescharacteristic, complemented
27#65496C2×C311yescharacteristic, complemented
28#64758C2×C312yescomplemented
29#64757C2×C312yescomplemented
30C24×C3×C5D411yescharacteristic
31C3×C5×#41—42no—
32C2×C3×C5×Q8C2×C411yescharacteristic, complemented
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4C2×C411yescharacteristic, complemented
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5—42no—
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×C4×C5C2×C411yescharacteristic
41C22×C3×C4×C5D411yescharacteristic
42C22×C3×C4×C5D411yescharacteristic
43C22×C3×C4×C5D411yescharacteristic
44C22×C3×C4×C5C2311yescharacteristic
45C22×C3×Dic5—24no—
46C22×C3×Dic5—22no—
47C22×C3×Dic5—42no—
48C22×C3×Dic5—42no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4C2×C411yescharacteristic, complemented
52C3×C5×C4⋊C4C2×C411yescharacteristic, complemented
53C3×C5×C22⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C42×C5—21no—
59C3×#681—42no—
60C2×C3×#168—22no—
61C3×#525D511yescharacteristic
62C3×#407—51no—
63C2×C3×#95—51no—
64C3×#328—52no—
65C3×#327—52no—
66C2×C5×#41C3×C411yescharacteristic, complemented
67C23×C4×C5C22×C311yescharacteristic
68C23×Dic5—21no—
69C5×#123—21no—
70C5×C22⋊Q8C3×C411yescharacteristic, complemented
71C5×C22⋊Q8C22×C311yescharacteristic
72C5×#121C3×C411yescharacteristic, complemented
73C5×#121C22×C311yescharacteristic
74C5×#120—21no—
75C4×C5×D4—21no—
76C4×C5×D4—21no—
77C5×#117C3×C411yescharacteristic, complemented
78C2×#681—21no—
79#3963—21no—
80#3963C22×C312yes—
81C22⋊#168—21no—
82C22⋊#168C22×C312yes—
83#1504—51noSylow
84C23×C3×C5—22no—
85C23×C3×C5—22no—
86C23×C3×C5—21no—
87C23×C3×C5—21no—
88C23×C3×C5—21no—
89C23×C3×C5C22⋊C411yescharacteristic
90C23×C3×C5C22⋊C411yescharacteristic
91C23×C3×C5C2×D411yescharacteristic
92C3×C5×Q8—41no—
93C3×C5×D4—41no—
94C3×C5×D4—42no—
95C3×C5×D4—42no—
96C3×C5×D4—41no—
97C3×C5×D4—41no—
98C2×C3×C4×C5—22no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—42no—
108C2×C3×C4×C5C22⋊C411yescharacteristic
109C2×C3×C4×C5C22⋊C411yescharacteristic
110C2×C3×C4×C5C22×C411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×Dic5—24no—
113C2×C3×Dic5—48no—
114C2×C3×Dic5—42no—
115C2×C3×Dic5—42no—
116C3×#168—42no—
117C2×C3×#41Dic511yescharacteristic, complemented
118C23×C3×C4—101no—
119C23×C3×C4C2×D511yescharacteristic
120C2×C3×M16—101no—
121C3×#123—21no—
122C3×C22⋊Q8Dic511yescharacteristic, complemented
123C3×C22⋊Q8C2×D511yescharacteristic
124C3×#121Dic511yescharacteristic, complemented
125C3×#121C2×D511yescharacteristic
126C3×#120—21no—
127C3×C4×D4—21no—
128C3×C4×D4—21no—
129C3×#117Dic511yescharacteristic, complemented
130C3×C22⋊C8—52no—
131C3×C22⋊C8—101no—
132C3×#95—52no—
133C3×#95—101no—
134C24×C5C3×D411yescharacteristic
135C5×#41—42no—
136C2×C5×Q8C2×C3×C411yescharacteristic, complemented
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4C2×C3×C411yescharacteristic, complemented
141C22×C4×C5—21no—
142C22×C4×C5—42no—
143C22×C4×C5C2×C3×C411yescharacteristic
144C22×C4×C5C2×C3×C411yescharacteristic
145C22×C4×C5C3×D411yescharacteristic
146C22×C4×C5C3×D411yescharacteristic
147C22×C4×C5C3×D411yescharacteristic
148C22×C4×C5C23×C311yescharacteristic
149C22×Dic5—24no—
150C22×Dic5—22no—
151C22×Dic5—42no—
152C22×Dic5—42no—
153C5×C4⋊C4—21no—
154C5×C4⋊C4—21no—
155C5×C4⋊C4C2×C3×C411yescharacteristic, complemented
156C5×C4⋊C4C2×C3×C411yescharacteristic, complemented
157C5×C22⋊C4—21no—
158C5×C22⋊C4—21no—
159C5×C22⋊C4—21no—
160C5×C22⋊C4—21no—
161C5×C22⋊C4—21no—
162C42×C5—21no—
163#681—42no—
164C2×#168—22no—
165#525C3×D511yescharacteristic
166#407—51no—
167C2×#95—51no—
168#328—52no—
169#327—52no—
170C22×C3×C5—24no—
171C22×C3×C5—24no—
172C22×C3×C5—21no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5—21no—
176C22×C3×C5—21no—
177C22×C3×C5—21no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5—41no—
181C22×C3×C5#9912yes—
182C22×C3×C5#10412yes—
183C22×C3×C5C2×C22⋊C411yescharacteristic
184C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
185C22×C3×C5#13711yescharacteristic
186C3×C4×C5—21no—
187C3×C4×C5—21no—
188C3×C4×C5—21no—
189C3×C4×C5—21no—
190C3×C4×C5—41no—
191C3×C4×C5—41no—
192C3×C4×C5—41no—
193C3×Dic5—44no—
194C24×C3C22⋊D511yescharacteristic
195C3×#41—42no—
196C2×C3×Q8C2×Dic511yescharacteristic, complemented
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4C2×Dic511yescharacteristic, complemented
201C22×C3×C4—21no—
202C22×C3×C4—42no—
203C22×C3×C4—104no—
204C22×C3×C4—102no—
205C22×C3×C4—202no—
206C22×C3×C4—202no—
207C22×C3×C4C2×Dic511yescharacteristic
208C22×C3×C4C2×Dic511yescharacteristic
209C22×C3×C4C22⋊D511yescharacteristic
210C22×C3×C4C22⋊D511yescharacteristic
211C22×C3×C4C22⋊D511yescharacteristic
212C22×C3×C4C22×D511yescharacteristic
213C3×M16—202no—
214C2×C3×C8—102no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4C2×Dic511yescharacteristic, complemented
218C3×C4⋊C4C2×Dic511yescharacteristic, complemented
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—21no—
224C3×C42—21no—
225C23×C5—22no—
226C23×C5—22no—
227C23×C5—21no—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5C3×C22⋊C411yescharacteristic
231C23×C5C3×C22⋊C411yescharacteristic
232C23×C5C2×C3×D411yescharacteristic
233C5×Q8—41no—
234C5×D4—41no—
235C5×D4—42no—
236C5×D4—42no—
237C5×D4—41no—
238C5×D4—41no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—42no—
249C2×C4×C5C3×C22⋊C411yescharacteristic
250C2×C4×C5C3×C22⋊C411yescharacteristic
251C2×C4×C5C22×C3×C411yescharacteristic
252C2×C4×C5C2×C3×D411yescharacteristic
253C2×Dic5—24no—
254C2×Dic5—48no—
255C2×Dic5—42no—
256C2×Dic5—42no—
257#168—42no—
258C2×#41C3×Dic511yescharacteristic, complemented
259C23×C4—101no—
260C23×C4C2×C3×D511yescharacteristic
261C2×M16—101no—
262#123—21no—
263C22⋊Q8C3×Dic511yescharacteristic, complemented
264C22⋊Q8C2×C3×D511yescharacteristic
265#121C3×Dic511yescharacteristic, complemented
266#121C2×C3×D511yescharacteristic
267#120—21no—
268C4×D4—21no—
269C4×D4—21no—
270#117C3×Dic511yescharacteristic, complemented
271C22⋊C8—52no—
272C22⋊C8—101no—
273#95—52no—
274#95—101no—
275C2×C3×C5—21no—
276C2×C3×C5—22no—
277C2×C3×C5—22no—
278C2×C3×C5—21no—
279C2×C3×C5—81no—
280C2×C3×C5C2×#9911yescharacteristic
281C2×C3×C5#41711yescharacteristic
282C2×C3×C5C2×#10411yescharacteristic
283C23×C3—21no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3C22⋊Dic511yescharacteristic
289C23×C3C22⋊Dic511yescharacteristic
290C23×C3C2×C22⋊D511yescharacteristic
291C3×Q8—41no—
292C3×D4—41no—
293C3×D4—41no—
294C3×D4—42no—
295C3×D4—42no—
296C3×D4—41no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—104no—
308C2×C3×C4—208no—
309C2×C3×C4—202no—
310C2×C3×C4—202no—
311C2×C3×C4C22⋊Dic511yescharacteristic
312C2×C3×C4C22⋊Dic511yescharacteristic
313C2×C3×C4C22×Dic511yescharacteristic
314C2×C3×C4C2×C22⋊D511yescharacteristic
315C3×C8—202no—
316C22×C5—24no—
317C22×C5—24no—
318C22×C5—21no—
319C22×C5—21no—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5—41no—
325C22×C5—41no—
326C22×C5—41no—
327C22×C5C3×#9912yes—
328C22×C5C3×#10412yes—
329C22×C5C2×C3×C22⋊C411yescharacteristic
330C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
331C22×C5C3×#13711yescharacteristic
332C4×C5—21no—
333C4×C5—21no—
334C4×C5—21no—
335C4×C5—21no—
336C4×C5—41no—
337C4×C5—41no—
338C4×C5—41no—
339Dic5—44no—
340C24C3×C22⋊D511yescharacteristic
341#41—42no—
342C2×Q8C2×C3×Dic511yescharacteristic, complemented
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C2×C3×Dic511yescharacteristic, complemented
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—104no—
350C22×C4—102no—
351C22×C4—202no—
352C22×C4—202no—
353C22×C4C2×C3×Dic511yescharacteristic
354C22×C4C2×C3×Dic511yescharacteristic
355C22×C4C3×C22⋊D511yescharacteristic
356C22×C4C3×C22⋊D511yescharacteristic
357C22×C4C3×C22⋊D511yescharacteristic
358C22×C4C22×C3×D511yescharacteristic
359M16—202no—
360C2×C8—102no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4C2×C3×Dic511yescharacteristic, complemented
364C4⋊C4C2×C3×Dic511yescharacteristic, complemented
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C42—21no—
371C3×C5#150411yescharacteristic, complemented, Hall
372C22×C3—24no—
373C22×C3—24no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3#396612yes—
384C22×C3#396912yes—
385C22×C3#409511yescharacteristic
386C22×C3#409711yescharacteristic
387C22×C3C2×C22⋊Dic511yescharacteristic
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—41no—
395C3×C4—204no—
396C2×C5—21no—
397C2×C5—22no—
398C2×C5—22no—
399C2×C5—21no—
400C2×C5—81no—
401C2×C5C2×C3×#9911yescharacteristic
402C2×C5C3×#41711yescharacteristic
403C2×C5C2×C3×#10411yescharacteristic
404C23—21no—
405C23—21no—
406C23—22no—
407C23—22no—
408C23—21no—
409C23C3×C22⋊Dic511yescharacteristic
410C23C3×C22⋊Dic511yescharacteristic
411C23C2×C3×C22⋊D511yescharacteristic
412Q8—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416D4—42no—
417D4—42no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—104no—
429C2×C4—208no—
430C2×C4—202no—
431C2×C4—202no—
432C2×C4C3×C22⋊Dic511yescharacteristic
433C2×C4C3×C22⋊Dic511yescharacteristic
434C2×C4C22×C3×Dic511yescharacteristic
435C2×C4C2×C3×C22⋊D511yescharacteristic
436C8—202no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—22no—
440C2×C3—22no—
441C2×C3—81no—
442C2×C3C2×#396611yescharacteristic
443C2×C3#6552111yescharacteristic
444C2×C3C2×#396911yescharacteristic
445C5C3×#150411yescharacteristic, complemented, Sylow
446C22—24no—
447C22—24no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22C3×#396612yes—
458C22C3×#396912yes—
459C22C3×#409511yescharacteristic
460C22C3×#409711yescharacteristic
461C22C2×C3×C22⋊Dic511yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—204no—
470C3#1060250111yescharacteristic, direct product factor, Sylow
471C2—21no—
472C2—21no—
473C2—22no—
474C2—22no—
475C2—81no—
476C2C2×C3×#396611yescharacteristic
477C2C3×#6552111yescharacteristic
478C2C2×C3×#396911yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊Dic5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
Fitting, prime radicalC3×C5×#525C2
socleC22×C3×C5C2×C22⋊C4
cosocleC22×C4×C5C23×C3
hypercenterC3×#525D5
hypocenterC5C3×#1504
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C3×#525

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3963C2semidirect
3. C3×#65496C2semidirect
4. C3×#64758C2semidirect
5. C3×#64757C2semidirect
6. #10602501C3direct (N × H)
7. C2×C3×C5×#41C4semidirect
11. C3×C5×C22⋊Q8C4semidirect
13. C3×C5×#121C4semidirect
18. C3×C5×#117C4semidirect
26. C2×#3963C2×C3semidirect
27. #65496C2×C3semidirect
28. #64758C2×C3semidirect
29. #64757C2×C3semidirect
32. C2×C3×C5×Q8C2×C4semidirect
36. C2×C3×C5×D4C2×C4semidirect
51. C3×C5×C4⋊C4C2×C4semidirect
52. C3×C5×C4⋊C4C2×C4semidirect
66. C2×C5×#41C3×C4semidirect
70. C5×C22⋊Q8C3×C4semidirect
72. C5×#121C3×C4semidirect
77. C5×#117C3×C4semidirect
117. C2×C3×#41Dic5semidirect
122. C3×C22⋊Q8Dic5semidirect
124. C3×#121Dic5semidirect
129. C3×#117Dic5semidirect
136. C2×C5×Q8C2×C3×C4semidirect
140. C2×C5×D4C2×C3×C4semidirect
155. C5×C4⋊C4C2×C3×C4semidirect
156. C5×C4⋊C4C2×C3×C4semidirect
196. C2×C3×Q8C2×Dic5semidirect
200. C2×C3×D4C2×Dic5semidirect
217. C3×C4⋊C4C2×Dic5semidirect
218. C3×C4⋊C4C2×Dic5semidirect
258. C2×#41C3×Dic5semidirect
263. C22⋊Q8C3×Dic5semidirect
265. #121C3×Dic5semidirect
270. #117C3×Dic5semidirect
342. C2×Q8C2×C3×Dic5semidirect
346. C2×D4C2×C3×Dic5semidirect
363. C4⋊C4C2×C3×Dic5semidirect
364. C4⋊C4C2×C3×Dic5semidirect
371. C3×C5#1504semidirect
445. C5C3×#1504semidirect
470. C3#10602501direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5252
C2×C3×#39632
C3×#654962
C3×#647582
C3×#647572
#106025013
C3×#15045