Finite Group Databasebeta

← C3×#10601201 #14302876

Subgroups — C3×#10601201 #14302876

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66164C211yesmaximal, characteristic
2C3×#65855C211yesmaximal, characteristic, complemented
3C3×C5×#421C211yesmaximal, characteristic, complemented
4C3×#65525C211yesmaximal, characteristic, complemented
5C2×C3×#3955C211yesmaximal, characteristic, complemented
6C3×#64777C211yesmaximal, characteristic, complemented
7C3×#64751C211yesmaximal, characteristic, complemented
8#10601201C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C3×#4093C412yescomplemented
14C3×#4086C412yescomplemented
15C3×#4083—22no—
16C3×#4076—22no—
17C3×#4072C2211yescharacteristic
18C3×#4072C2211yescharacteristic
19C2×C3×#681C2211yescharacteristic
20C3×#4039—22no—
21C3×D5×C22⋊C4—22no—
22C3×C4×D20—22no—
23C3×C5×#104—22no—
24C3×#3969—22no—
25C3×#3955—22no—
26C3×#1911—51nomaximal, Hall
27#66164C2×C311yescharacteristic
28#65855C2×C311yescharacteristic, complemented
29C5×#421C2×C311yescharacteristic, complemented
30#65525C2×C311yescharacteristic, complemented
31C2×#3955C2×C311yescharacteristic, complemented
32#64777C2×C311yescharacteristic, complemented
33#64751C2×C311yescharacteristic, complemented
34C23×C3×D5D412yes—
35C3×C5×#41—21no—
36C3×C5×#41—21no—
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C2×C3×C5×Q8C2×C411yescharacteristic, complemented
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4C2×C411yescharacteristic, complemented
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2311yescharacteristic
44C2×C3×C22⋊D5—24no—
45C2×C3×D20—21no—
46C2×C3×D20—21no—
47C2×C3×D20—21no—
48C2×C3×D20—21no—
49C2×C3×D20C2×C412yes—
50C2×C3×D20D412yes—
51C2×C3×C4×D5—24no—
52C3×C5×M16—21no—
53C3×C5×M16—21no—
54C2×C3×C5×C8—21no—
55C3×C5×C4⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C42×C5D412yescomplemented
58C3×C22⋊Dic5—21no—
59C3×C22⋊Dic5—21no—
60C3×#685—24no—
61C3×#685—24no—
62C3×C4⋊Dic5—21no—
63C3×C4⋊Dic5—21no—
64C3×C4×Dic5C2×C412yescomplemented
65C3×C4×Dic5D412yescomplemented
66C3×#681—21no—
67C3×#681—21no—
68C2×C3×#168—21no—
69C3×#535—51no—
70C3×#518—51no—
71C3×#421—51no—
72C3×#421D511yescharacteristic, complemented
73C2×C3×#100—51no—
74C3×H(3,ℤ/4)—51no—
75C3×H(3,ℤ/4)—51no—
76C2×C5×#41C22×C311yescharacteristic
77C22×D20C22×C311yescharacteristic
78C2×C5×M16C22×C311yescharacteristic
79C5×#117C22×C311yescharacteristic
80#4093C3×C412yescomplemented
81#4086C3×C412yescomplemented
82#4083—22no—
83#4076—22no—
84#4072C22×C311yescharacteristic
85#4072C22×C311yescharacteristic
86C2×#681C22×C311yescharacteristic
87#4039—22no—
88D5×C22⋊C4—22no—
89C4×D20—22no—
90C5×#104—22no—
91#3969—22no—
92#3955—22no—
93#1911—51noSylow
94C23×C3×C5—21no—
95C23×C3×C5C2×D411yescharacteristic
96C22×C3×D5—22no—
97C22×C3×D5—22no—
98C22×C3×D5—82no—
99C22×C3×D5C22⋊C414yes—
100C3×C5×Q8—21no—
101C3×C5×Q8—21no—
102C3×C5×D4—21no—
103C3×C5×D4—21no—
104C3×C5×D4—41no—
105C3×C5×D4—41no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5C22×C411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×C4×C5C2×D411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5#4111yescharacteristic
115C2×C3×C4×C5#4111yescharacteristic
116C3×C22⋊D5—84no—
117C2×C3×Dic5—22no—
118C2×C3×Dic5—22no—
119C3×D20—22no—
120C3×D20—22no—
121C3×D20—22no—
122C3×D20—22no—
123C3×C4×D5—84no—
124C3×C5×C8—41no—
125C3×#168—41no—
126C2×C3×#41C2×D511yescharacteristic
127C22×C3×D4—51no—
128C2×C3×M16—51no—
129C2×C3×M16C2×D511yescharacteristic
130C3×#127—52no—
131C3×#124—52no—
132C3×#121—102no—
133C3×#120—102no—
134C3×C4×D4—102no—
135C3×C4×D4—102no—
136C3×#117—51no—
137C3×#117—51no—
138C3×#117C2×D511yescharacteristic
139C2×C3×C22⋊C4—102no—
140C3×#104—22no—
141C3×#104—102no—
142C3×#100—102no—
143C23×D5C3×D412yes—
144C5×#41—21no—
145C5×#41—21no—
146C5×#41—21no—
147C5×#41—21no—
148C2×C5×Q8C2×C3×C411yescharacteristic, complemented
149C2×C5×D4—21no—
150C2×C5×D4C2×C3×C411yescharacteristic, complemented
151C22×C4×C5—21no—
152C22×C4×C5C23×C311yescharacteristic
153C2×C22⋊D5—24no—
154C2×D20—21no—
155C2×D20—21no—
156C2×D20—21no—
157C2×D20—21no—
158C2×D20C2×C3×C412yes—
159C2×D20C3×D412yes—
160C2×C4×D5—24no—
161C5×M16—21no—
162C5×M16—21no—
163C2×C5×C8—21no—
164C5×C4⋊C4—21no—
165C5×C22⋊C4—21no—
166C42×C5C3×D412yescomplemented
167C22⋊Dic5—21no—
168C22⋊Dic5—21no—
169#685—24no—
170#685—24no—
171C4⋊Dic5—21no—
172C4⋊Dic5—21no—
173C4×Dic5C2×C3×C412yescomplemented
174C4×Dic5C3×D412yescomplemented
175#681—21no—
176#681—21no—
177C2×#168—21no—
178#535—51no—
179#518—51no—
180#421—51no—
181#421C3×D511yescharacteristic, complemented
182C2×#100—51no—
183H(3,ℤ/4)—51no—
184H(3,ℤ/4)—51no—
185C22×C3×C5—41no—
186C22×C3×C5—42no—
187C22×C3×C5—41no—
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C22×C3×C5#12011yescharacteristic
190C22×C3×C5#12311yescharacteristic
191C2×C3×D5—24no—
192C2×C3×D5—44no—
193C2×C3×D5—44no—
194C2×C3×D5—82no—
195C3×C4×C5—42no—
196C3×C4×C5—41no—
197C3×C4×C5C4×D411yescharacteristic
198C3×C4×C5C4×D411yescharacteristic
199C3×C4×C5#12111yescharacteristic
200C3×C4×C5#12111yescharacteristic
201C3×Dic5—42no—
202C3×Dic5—42no—
203C24×C3—52no—
204C3×#41—21no—
205C3×#41—21no—
206C3×#41—21no—
207C3×#41—21no—
208C2×C3×Q8C4×D511yescharacteristic, complemented
209C2×C3×D4—21no—
210C2×C3×D4—52no—
211C2×C3×D4—52no—
212C2×C3×D4—104no—
213C2×C3×D4—101no—
214C2×C3×D4—101no—
215C2×C3×D4—101no—
216C2×C3×D4—101no—
217C2×C3×D4C4×D511yescharacteristic, complemented
218C22×C3×C4—21no—
219C22×C3×C4—104no—
220C22×C3×C4C22×D511yescharacteristic
221C3×M16—21no—
222C3×M16—21no—
223C3×M16—101no—
224C3×M16—101no—
225C2×C3×C8—21no—
226C2×C3×C8—101no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—101no—
229C3×C4⋊C4—101no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—104no—
232C3×C22⋊C4—104no—
233C3×C22⋊C4—101no—
234C3×C22⋊C4—101no—
235C3×C42—52no—
236C3×C42—52no—
237C3×C42C22⋊D512yescomplemented
238C23×C5—21no—
239C23×C5C2×C3×D411yescharacteristic
240C22×D5—22no—
241C22×D5—22no—
242C22×D5—82no—
243C22×D5C3×C22⋊C414yes—
244C5×Q8—21no—
245C5×Q8—21no—
246C5×D4—21no—
247C5×D4—21no—
248C5×D4—41no—
249C5×D4—41no—
250C2×C4×C5—21no—
251C2×C4×C5—22no—
252C2×C4×C5—41no—
253C2×C4×C5—41no—
254C2×C4×C5C22×C3×C411yescharacteristic
255C2×C4×C5C2×C3×D411yescharacteristic
256C2×C4×C5C2×C3×D411yescharacteristic
257C2×C4×C5C2×C3×D411yescharacteristic
258C2×C4×C5C3×#4111yescharacteristic
259C2×C4×C5C3×#4111yescharacteristic
260C22⋊D5—84no—
261C2×Dic5—22no—
262C2×Dic5—22no—
263D20—22no—
264D20—22no—
265D20—22no—
266D20—22no—
267C4×D5—84no—
268C5×C8—41no—
269#168—41no—
270C2×#41C2×C3×D511yescharacteristic
271C22×D4—51no—
272C2×M16—51no—
273C2×M16C2×C3×D511yescharacteristic
274#127—52no—
275#124—52no—
276#121—102no—
277#120—102no—
278C4×D4—102no—
279C4×D4—102no—
280#117—51no—
281#117—51no—
282#117C2×C3×D511yescharacteristic
283C2×C22⋊C4—102no—
284#104—22no—
285#104—102no—
286#100—102no—
287C2×C3×C5—21no—
288C2×C3×C5—21no—
289C2×C3×C5—21no—
290C2×C3×C5—81no—
291C2×C3×C5#38611yescharacteristic
292C3×D5—44no—
293C23×C3—21no—
294C23×C3—54no—
295C23×C3—102no—
296C23×C3—102no—
297C23×C3—402no—
298C23×C3D4×D511yescharacteristic
299C3×Q8—21no—
300C3×Q8—21no—
301C3×D4—21no—
302C3×D4—21no—
303C3×D4—41no—
304C3×D4—41no—
305C3×D4—102no—
306C3×D4—102no—
307C3×D4—102no—
308C3×D4—102no—
309C3×D4—404no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—102no—
315C2×C3×C4—102no—
316C2×C3×C4—404no—
317C2×C3×C4C2×C4×D511yescharacteristic
318C2×C3×C4#70911yescharacteristic
319C2×C3×C4D4×D511yescharacteristic
320C2×C3×C4D4×D511yescharacteristic
321C2×C3×C4#71111yescharacteristic
322C2×C3×C4C2×C22⋊D511yescharacteristic
323C3×C8—41no—
324C3×C8—201no—
325C22×C5—41no—
326C22×C5—42no—
327C22×C5—41no—
328C22×C5C2×C3×C22⋊C411yescharacteristic
329C22×C5C3×#12011yescharacteristic
330C22×C5C3×#12311yescharacteristic
331C2×D5—24no—
332C2×D5—44no—
333C2×D5—44no—
334C2×D5—82no—
335C4×C5—42no—
336C4×C5—41no—
337C4×C5C3×C4×D411yescharacteristic
338C4×C5C3×C4×D411yescharacteristic
339C4×C5C3×#12111yescharacteristic
340C4×C5C3×#12111yescharacteristic
341Dic5—42no—
342Dic5—42no—
343C24—52no—
344#41—21no—
345#41—21no—
346#41—21no—
347#41—21no—
348C2×Q8C3×C4×D511yescharacteristic, complemented
349C2×D4—21no—
350C2×D4—52no—
351C2×D4—52no—
352C2×D4—104no—
353C2×D4—101no—
354C2×D4—101no—
355C2×D4—101no—
356C2×D4—101no—
357C2×D4C3×C4×D511yescharacteristic, complemented
358C22×C4—21no—
359C22×C4—104no—
360C22×C4C22×C3×D511yescharacteristic
361M16—21no—
362M16—21no—
363M16—101no—
364M16—101no—
365C2×C8—21no—
366C2×C8—101no—
367C4⋊C4—21no—
368C4⋊C4—101no—
369C4⋊C4—101no—
370C22⋊C4—21no—
371C22⋊C4—104no—
372C22⋊C4—104no—
373C22⋊C4—101no—
374C22⋊C4—101no—
375C42—52no—
376C42—52no—
377C42C3×C22⋊D512yescomplemented
378C3×C5#191111yescharacteristic, complemented, Hall
379C22×C3—41no—
380C22×C3—42no—
381C22×C3—41no—
382C22×C3—104no—
383C22×C3—204no—
384C22×C3—204no—
385C22×C3—402no—
386C22×C3D5×C22⋊C411yescharacteristic
387C22×C3#402911yescharacteristic
388C22×C3#408311yescharacteristic
389C3×C4—42no—
390C3×C4—41no—
391C3×C4—202no—
392C3×C4—202no—
393C3×C4#402711yescharacteristic
394C3×C4#403011yescharacteristic
395C3×C4C4×C22⋊D511yescharacteristic
396C3×C4#408411yescharacteristic
397C2×C5—21no—
398C2×C5—21no—
399C2×C5—21no—
400C2×C5—81no—
401C2×C5C3×#38611yescharacteristic
402D5—44no—
403C23—21no—
404C23—54no—
405C23—102no—
406C23—102no—
407C23—402no—
408C23C3×D4×D511yescharacteristic
409Q8—21no—
410Q8—21no—
411D4—21no—
412D4—21no—
413D4—41no—
414D4—41no—
415D4—102no—
416D4—102no—
417D4—102no—
418D4—102no—
419D4—404no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—102no—
425C2×C4—102no—
426C2×C4—404no—
427C2×C4C2×C3×C4×D511yescharacteristic
428C2×C4C3×#70911yescharacteristic
429C2×C4C3×D4×D511yescharacteristic
430C2×C4C3×D4×D511yescharacteristic
431C2×C4C3×#71111yescharacteristic
432C2×C4C2×C3×C22⋊D511yescharacteristic
433C8—41no—
434C8—201no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—81no—
439C2×C3—204no—
440C2×C3#6524511yescharacteristic
441C5C3×#191111yescharacteristic, complemented, Sylow
442C22—41no—
443C22—42no—
444C22—41no—
445C22—104no—
446C22—204no—
447C22—204no—
448C22—402no—
449C22C3×D5×C22⋊C411yescharacteristic
450C22C3×#402911yescharacteristic
451C22C3×#408311yescharacteristic
452C4—42no—
453C4—41no—
454C4—202no—
455C4—202no—
456C4C3×#402711yescharacteristic
457C4C3×#403011yescharacteristic
458C4C3×C4×C22⋊D511yescharacteristic
459C4C3×#408411yescharacteristic
460C3#1060120111yescharacteristic, direct product factor, Sylow
461C2—21no—
462C2—21no—
463C2—21no—
464C2—81no—
465C2—204no—
466C2C3×#6524511yescharacteristic
subgroupG/H
centerC2×C3#65245
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#421C2
socleC2×C3×C5#386
prime radicalC3×#66164C2
cosocleC22×C4×C5C23×C3
hypercenterC3×#421D5
hypocenterC5C3×#1911
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#421

Complements

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

NH ≅ G/Ntype
2. C3×#65855C2semidirect
3. C3×C5×#421C2semidirect
4. C3×#65525C2semidirect
5. C2×C3×#3955C2semidirect
6. C3×#64777C2semidirect
7. C3×#64751C2semidirect
8. #10601201C3direct (N × H)
13. C3×#4093C4semidirect
14. C3×#4086C4semidirect
28. #65855C2×C3semidirect
29. C5×#421C2×C3semidirect
30. #65525C2×C3semidirect
31. C2×#3955C2×C3semidirect
32. #64777C2×C3semidirect
33. #64751C2×C3semidirect
39. C2×C3×C5×Q8C2×C4semidirect
41. C2×C3×C5×D4C2×C4semidirect
57. C3×C42×C5D4semidirect
64. C3×C4×Dic5C2×C4semidirect
65. C3×C4×Dic5D4semidirect
72. C3×#421D5semidirect
80. #4093C3×C4semidirect
81. #4086C3×C4semidirect
148. C2×C5×Q8C2×C3×C4semidirect
150. C2×C5×D4C2×C3×C4semidirect
166. C42×C5C3×D4semidirect
173. C4×Dic5C2×C3×C4semidirect
174. C4×Dic5C3×D4semidirect
181. #421C3×D5semidirect
208. C2×C3×Q8C4×D5semidirect
217. C2×C3×D4C4×D5semidirect
237. C3×C42C22⋊D5semidirect
348. C2×Q8C3×C4×D5semidirect
357. C2×D4C3×C4×D5semidirect
377. C42C3×C22⋊D5semidirect
378. C3×C5#1911semidirect
441. C5C3×#1911semidirect
460. C3#10601201direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661642
C3×#658552
C3×C5×#4212
C3×#655252
C2×C3×#39552
C3×#647772
C3×#647512
#106012013
C3×#19115