Finite Group Databasebeta

← C3×#10617703 #14484784

Subgroups — C3×#10617703 #14484784

Order 1920.

2224 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C3×#66151C212yesmaximal, complemented
3C3×#65919C213yesmaximal, complemented
4C3×#65918C213yesmaximal, complemented
5C3×#65914C216yesmaximal, complemented
6C3×#65911C211yesmaximal, characteristic, complemented
7C3×C4×#713C212yesmaximal, complemented
8C3×#65904C211yesmaximal, characteristic, complemented
9C3×#65848C216yesmaximal, complemented
10C3×#65811C213yesmaximal, complemented
11C3×C4×#709C213yesmaximal, complemented
12#10617703C311yesmaximal, characteristic, direct product factor, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C2×C3×#713C2212yes—
15C2×C3×#709C2213yes—
16C3×C4×C5×Q8C2211yescharacteristic
17C3×C4×C5×Q8C2211yescharacteristic
18C3×C4×C5×Q8C2213yes—
19C3×C4×C5×Q8C2213yes—
20C2×C3×C5×C4⋊C4C2213yes—
21C2×C3×C42×C5C2213yes—
22C3×#4093C2216yes—
23C3×#4092C2216yes—
24C3×Q8×Dic5C2212yes—
25C3×#4076C2216yes—
26C3×#4076C2216yes—
27C3×C4×C22⋊D5C2216yes—
28C3×C4×C22⋊D5C2216yes—
29C3×#4072C2211yescharacteristic
30C3×#4044C22112yes—
31C3×#4041C2216yes—
32C3×#4040C22112yes—
33C3×#4039C2216yes—
34C3×#4038C2216yes—
35C3×#4036C2216yes—
36C3×#4021C2216yes—
37C3×#4020C2213yes—
38C3×C4×D20C2216yes—
39C3×C4×D20C2213yes—
40C3×C4×D20C2216yes—
41C3×C4×D20C2213yes—
42C3×#4018C2216yes—
43C3×C42×D5C2216yes—
44C3×#4015C2213yes—
45C3×C4×Dic10C2213yes—
46C3×C4×Dic10C2213yes—
47C3×#3480—51nomaximal, Hall
48C2×C4×C5×Q8C2×C311yescharacteristic, complemented
49#66151C2×C312yescomplemented
50#65919C2×C313yescomplemented
51#65918C2×C313yescomplemented
52#65914C2×C316yescomplemented
53#65911C2×C311yescharacteristic, complemented
54C4×#713C2×C312yescomplemented
55#65904C2×C311yescharacteristic, complemented
56#65848C2×C316yescomplemented
57#65811C2×C313yescomplemented
58C4×#709C2×C313yescomplemented
59C2×C3×C5×Q8—24no—
60C2×C3×C5×Q8C2313yes—
61C2×C3×C5×Q8C2311yescharacteristic
62C22×C3×C4×C5C2313yes—
63C22×C3×C4×C5C2313yes—
64C22×C3×C4×C5C2311yescharacteristic
65C2×C3×C22⋊D5C2316yes—
66C3×#713—28no—
67C3×#709—212no—
68C2×C3×D20C2316yes—
69C2×C3×D20C2313yes—
70C2×C3×D20C2316yes—
71C2×C3×C4×D5C2316yes—
72C2×C3×C4×D5C2316yes—
73C2×C3×C4×D5C2316yes—
74C2×C3×Dic10C2313yes—
75C3×C5×C4⋊C4C2313yes—
76C3×C5×C4⋊C4C2313yes—
77C3×C5×C4⋊C4C2316yes—
78C3×C42×C5C2313yes—
79C3×C42×C5C2313yes—
80C3×C42×C5C2313yes—
81C3×C42×C5C2313yes—
82C3×C22⋊Dic5C2312yes—
83C3×#685C23112yes—
84C3×#685C2316yes—
85C3×#685C2316yes—
86C3×#685C23112yes—
87C3×C4⋊Dic5C2316yes—
88C3×C4⋊Dic5C2311yescharacteristic
89C3×C4⋊Dic5C2311yescharacteristic
90C3×C4⋊Dic5C2316yes—
91C3×#683C2316yes—
92C3×#683C2316yes—
93C3×C4×Dic5C2316yes—
94C3×C4×Dic5C2316yes—
95C3×C4×Dic5C2312yes—
96C3×#559—53no—
97C3×#557—51no—
98C3×#556—53no—
99C3×#553—56no—
100C3×#550—51no—
101C3×#548—52no—
102C3×#532—53no—
103C3×#529—56no—
104C3×C4×#41—52no—
105C3×C4×#41—53no—
106C2×C3×C4×Q8D511yescharacteristic, complemented
107C22×C5×Q8C22×C311yescharacteristic
108C2×#713C22×C312yes—
109C2×#709C22×C313yes—
110C4×C5×Q8C22×C311yescharacteristic
111C4×C5×Q8C22×C311yescharacteristic
112C4×C5×Q8C22×C313yes—
113C4×C5×Q8C22×C313yes—
114C2×C5×C4⋊C4C22×C313yes—
115C2×C42×C5C22×C313yes—
116#4093C22×C316yes—
117#4092C22×C316yes—
118Q8×Dic5C22×C312yes—
119#4076C22×C316yes—
120#4076C22×C316yes—
121C4×C22⋊D5C22×C316yes—
122C4×C22⋊D5C22×C316yes—
123#4072C22×C311yescharacteristic
124#4044C22×C3112yes—
125#4041C22×C316yes—
126#4040C22×C3112yes—
127#4039C22×C316yes—
128#4038C22×C316yes—
129#4036C22×C316yes—
130#4021C22×C316yes—
131#4020C22×C313yes—
132C4×D20C22×C313yes—
133C4×D20C22×C313yes—
134C4×D20C22×C316yes—
135C4×D20C22×C316yes—
136#4018C22×C316yes—
137C42×D5C22×C316yes—
138#4015C22×C313yes—
139C4×Dic10C22×C313yes—
140C4×Dic10C22×C313yes—
141#3480—51noSylow
142C23×C3×C5C2411yescharacteristic
143C22×C3×D5C2416yes—
144C3×C5×Q8—26no—
145C3×C5×Q8#4114yescomplemented
146C2×C3×C4×C5—26no—
147C2×C3×C4×C5—43no—
148C2×C3×C4×C5#4114yescomplemented
149C2×C3×C4×C5C2413yes—
150C2×C3×C4×C5C2413yes—
151C2×C3×C4×C5C2413yes—
152C2×C3×C4×C5C2413yes—
153C2×C3×C4×C5C2411yescharacteristic
154C2×C3×C4×C5C2411yescharacteristic
155C3×C22⋊D5—212no—
156C2×C3×Dic5C2412yes—
157C2×C3×Dic5C2412yes—
158C2×C3×Dic5C2416yes—
159C3×D20—26no—
160C3×D20—212no—
161C3×D20—46no—
162C3×C4×D5—212no—
163C3×C4×D5—212no—
164C3×C4×D5—46no—
165C3×Dic10—26no—
166C2×C3×#41—52no—
167C2×C3×#41—53no—
168C22×C3×Q8C2×D511yescharacteristic
169C3×C4⋊Q8—53no—
170C3×#127—53no—
171C3×#126—512no—
172C3×#125—56no—
173C3×#124—56no—
174C3×#124—56no—
175C3×#123—512no—
176C3×C22⋊Q8—56no—
177C3×#121—56no—
178C3×#121—56no—
179C3×#121—56no—
180C3×C4×Q8—52no—
181C3×C4×Q8—53no—
182C3×C4×Q8—53no—
183C3×C4×Q8C2×D513yes—
184C3×C4×Q8C2×D511yescharacteristic
185C3×C4×Q8C2×D513yes—
186C3×C4×Q8C2×D511yescharacteristic
187C3×C4×D4—56no—
188C3×C4×D4—56no—
189C3×C4×D4—56no—
190C3×C4×D4—53no—
191C3×C4×D4—56no—
192C3×C4×D4—53no—
193C3×C4×D4—56no—
194C3×#117—51no—
195C3×#117—56no—
196C3×#117—56no—
197C2×C3×C4⋊C4C2×D513yes—
198C2×C3×C42—56no—
199C2×C3×C42C2×D513yes—
200C2×C5×Q8—24no—
201C2×C5×Q8C23×C313yes—
202C2×C5×Q8C23×C311yescharacteristic
203C22×C4×C5C23×C313yes—
204C22×C4×C5C23×C313yes—
205C22×C4×C5C23×C311yescharacteristic
206C2×C22⋊D5C23×C316yes—
207#713—28no—
208#709—212no—
209C2×D20C23×C313yes—
210C2×D20C23×C316yes—
211C2×D20C23×C316yes—
212C2×C4×D5C23×C316yes—
213C2×C4×D5C23×C316yes—
214C2×C4×D5C23×C316yes—
215C2×Dic10C23×C313yes—
216C5×C4⋊C4C23×C313yes—
217C5×C4⋊C4C23×C313yes—
218C5×C4⋊C4C23×C316yes—
219C42×C5C23×C313yes—
220C42×C5C23×C313yes—
221C42×C5C23×C313yes—
222C42×C5C23×C313yes—
223C22⋊Dic5C23×C312yes—
224#685C23×C316yes—
225#685C23×C316yes—
226#685C23×C3112yes—
227#685C23×C3112yes—
228C4⋊Dic5C23×C311yescharacteristic
229C4⋊Dic5C23×C311yescharacteristic
230C4⋊Dic5C23×C316yes—
231C4⋊Dic5C23×C316yes—
232#683C23×C316yes—
233#683C23×C316yes—
234C4×Dic5C23×C316yes—
235C4×Dic5C23×C316yes—
236C4×Dic5C23×C312yes—
237#559—53no—
238#557—51no—
239#556—53no—
240#553—56no—
241#550—51no—
242#548—52no—
243#532—53no—
244#529—56no—
245C4×#41—52no—
246C4×#41—53no—
247C2×C4×Q8C3×D511yescharacteristic, complemented
248C22×C3×C5—21no—
249C22×C3×C5—21no—
250C22×C3×C5C2×#4112yes—
251C22×C3×C5C2511yescharacteristic
252C2×C3×D5—26no—
253C2×C3×D5—26no—
254C2×C3×D5—26no—
255C3×C4×C5—23no—
256C3×C4×C5—23no—
257C3×C4×C5—23no—
258C3×C4×C5C2×#4116yes—
259C3×C4×C5C2×#4112yes—
260C3×C4×C5C2×#4112yes—
261C3×Dic5—26no—
262C3×Dic5—22no—
263C3×Dic5—22no—
264C3×#41—108no—
265C3×#41—1012no—
266C2×C3×Q8—24no—
267C2×C3×Q8—53no—
268C2×C3×Q8C22×D513yes—
269C2×C3×Q8C22×D511yescharacteristic
270C2×C3×D4—56no—
271C2×C3×D4—56no—
272C2×C3×D4—53no—
273C2×C3×D4—56no—
274C22×C3×C4—56no—
275C22×C3×C4—56no—
276C22×C3×C4—56no—
277C22×C3×C4C22×D513yes—
278C22×C3×C4C22×D513yes—
279C22×C3×C4C22×D511yescharacteristic
280C3×C4⋊C4—56no—
281C3×C4⋊C4—56no—
282C3×C4⋊C4—51no—
283C3×C4⋊C4—51no—
284C3×C4⋊C4—56no—
285C3×C4⋊C4—56no—
286C3×C4⋊C4C22×D513yes—
287C3×C4⋊C4C22×D513yes—
288C3×C4⋊C4C22×D516yes—
289C3×C22⋊C4—52no—
290C3×C22⋊C4—56no—
291C3×C22⋊C4—512no—
292C3×C22⋊C4—56no—
293C3×C22⋊C4—512no—
294C3×C42—56no—
295C3×C42—56no—
296C3×C42—52no—
297C3×C42C22×D513yes—
298C3×C42C22×D513yes—
299C3×C42C22×D513yes—
300C3×C42C22×D513yes—
301C23×C5C24×C311yescharacteristic
302C22×D5C24×C316yes—
303C5×Q8—26no—
304C5×Q8C3×#4114yescomplemented
305C2×C4×C5—26no—
306C2×C4×C5—43no—
307C2×C4×C5C3×#4114yescomplemented
308C2×C4×C5C24×C313yes—
309C2×C4×C5C24×C313yes—
310C2×C4×C5C24×C313yes—
311C2×C4×C5C24×C313yes—
312C2×C4×C5C24×C311yescharacteristic
313C2×C4×C5C24×C311yescharacteristic
314C22⋊D5—212no—
315C2×Dic5C24×C312yes—
316C2×Dic5C24×C312yes—
317C2×Dic5C24×C316yes—
318D20—26no—
319D20—212no—
320D20—46no—
321C4×D5—212no—
322C4×D5—212no—
323C4×D5—46no—
324Dic10—26no—
325C2×#41—52no—
326C2×#41—53no—
327C22×Q8C2×C3×D511yescharacteristic
328C4⋊Q8—53no—
329#127—53no—
330#126—512no—
331#125—56no—
332#124—56no—
333#124—56no—
334#123—512no—
335C22⋊Q8—56no—
336#121—56no—
337#121—56no—
338#121—56no—
339C4×Q8—52no—
340C4×Q8—53no—
341C4×Q8—53no—
342C4×Q8C2×C3×D511yescharacteristic
343C4×Q8C2×C3×D513yes—
344C4×Q8C2×C3×D511yescharacteristic
345C4×Q8C2×C3×D513yes—
346C4×D4—56no—
347C4×D4—56no—
348C4×D4—56no—
349C4×D4—56no—
350C4×D4—53no—
351C4×D4—53no—
352C4×D4—56no—
353#117—51no—
354#117—56no—
355#117—56no—
356C2×C4⋊C4C2×C3×D513yes—
357C2×C42—56no—
358C2×C42C2×C3×D513yes—
359C2×C3×C5—22no—
360C2×C3×C5C22×#4111yescharacteristic
361C2×C3×C5C22×#4111yescharacteristic
362C2×C3×C5#58511yescharacteristic
363C3×D5—46no—
364C23×C3—56no—
365C23×C3C23×D511yescharacteristic
366C3×Q8—26no—
367C3×Q8—106no—
368C3×Q8#70914yescomplemented
369C3×D4—1012no—
370C3×D4—1012no—
371C3×D4—106no—
372C3×D4—206no—
373C2×C3×C4—26no—
374C2×C3×C4—43no—
375C2×C3×C4—52no—
376C2×C3×C4—52no—
377C2×C3×C4—56no—
378C2×C3×C4—1012no—
379C2×C3×C4—1012no—
380C2×C3×C4—206no—
381C2×C3×C4#71314yescomplemented
382C2×C3×C4C23×D513yes—
383C2×C3×C4C23×D513yes—
384C2×C3×C4C23×D513yes—
385C2×C3×C4C23×D513yes—
386C2×C3×C4C23×D511yescharacteristic
387C2×C3×C4C23×D511yescharacteristic
388C22×C5—21no—
389C22×C5—21no—
390C22×C5C2×C3×#4112yes—
391C22×C5C25×C311yescharacteristic
392C2×D5—26no—
393C2×D5—26no—
394C2×D5—26no—
395C4×C5—23no—
396C4×C5—23no—
397C4×C5—23no—
398C4×C5C2×C3×#4116yes—
399C4×C5C2×C3×#4112yes—
400C4×C5C2×C3×#4112yes—
401Dic5—26no—
402Dic5—22no—
403Dic5—22no—
404#41—108no—
405#41—1012no—
406C2×Q8—24no—
407C2×Q8—53no—
408C2×Q8C22×C3×D513yes—
409C2×Q8C22×C3×D511yescharacteristic
410C2×D4—56no—
411C2×D4—56no—
412C2×D4—53no—
413C2×D4—56no—
414C22×C4—56no—
415C22×C4—56no—
416C22×C4—56no—
417C22×C4C22×C3×D513yes—
418C22×C4C22×C3×D513yes—
419C22×C4C22×C3×D511yescharacteristic
420C4⋊C4—51no—
421C4⋊C4—51no—
422C4⋊C4—56no—
423C4⋊C4—56no—
424C4⋊C4—56no—
425C4⋊C4—56no—
426C4⋊C4C22×C3×D513yes—
427C4⋊C4C22×C3×D513yes—
428C4⋊C4C22×C3×D516yes—
429C22⋊C4—52no—
430C22⋊C4—56no—
431C22⋊C4—512no—
432C22⋊C4—56no—
433C22⋊C4—512no—
434C42—52no—
435C42—56no—
436C42—56no—
437C42C22×C3×D513yes—
438C42C22×C3×D513yes—
439C42C22×C3×D513yes—
440C42C22×C3×D513yes—
441C3×C5#348011yescharacteristic, complemented, Hall
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—106no—
445C22×C3—106no—
446C22×C3—106no—
447C22×C3C2×#71312yes—
448C22×C3C24×D511yescharacteristic
449C3×C4—23no—
450C3×C4—23no—
451C3×C4—23no—
452C3×C4—102no—
453C3×C4—106no—
454C3×C4—102no—
455C3×C4C2×#70916yes—
456C3×C4C2×#71312yes—
457C3×C4C2×#71312yes—
458C2×C5—22no—
459C2×C5C22×C3×#4111yescharacteristic
460C2×C5C22×C3×#4111yescharacteristic
461C2×C5C3×#58511yescharacteristic
462D5—46no—
463C23—56no—
464C23C23×C3×D511yescharacteristic
465Q8—26no—
466Q8—106no—
467Q8C3×#70914yescomplemented
468D4—1012no—
469D4—1012no—
470D4—106no—
471D4—206no—
472C2×C4—26no—
473C2×C4—43no—
474C2×C4—52no—
475C2×C4—52no—
476C2×C4—56no—
477C2×C4—1012no—
478C2×C4—1012no—
479C2×C4—206no—
480C2×C4C3×#71314yescomplemented
481C2×C4C23×C3×D513yes—
482C2×C4C23×C3×D513yes—
483C2×C4C23×C3×D513yes—
484C2×C4C23×C3×D513yes—
485C2×C4C23×C3×D511yescharacteristic
486C2×C4C23×C3×D511yescharacteristic
487C2×C3—22no—
488C2×C3—206no—
489C2×C3C22×#70911yescharacteristic
490C2×C3C22×#71311yescharacteristic
491C2×C3#6628311yescharacteristic
492C5C3×#348011yescharacteristic, complemented, Sylow
493C22—21no—
494C22—21no—
495C22—106no—
496C22—106no—
497C22—106no—
498C22C2×C3×#71312yes—
499C22C24×C3×D511yescharacteristic
500C4—23no—
501C4—23no—
502C4—23no—
503C4—102no—
504C4—106no—
505C4—102no—
506C4C2×C3×#70916yes—
507C4C2×C3×#71312yes—
508C4C2×C3×#71312yes—
509C3#1061770311yescharacteristic, direct product factor, Sylow
510C2—22no—
511C2—206no—
512C2C22×C3×#70911yescharacteristic
513C2C22×C3×#71311yescharacteristic
514C2C3×#6628311yescharacteristic
subgroupG/H
centerC2×C3×C4C23×D5
commutator, cosocleC22×C5C25×C3
FrattiniC22C24×C3×D5
FittingC2×C3×C4×C5×Q8C2
socleC22×C3×C5C25
hypercenterC2×C3×C4×Q8D5
hypocenterC5C3×#3480
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×Q8C2semidirect
2. C3×#66151C2semidirect
3. C3×#65919C2semidirect
4. C3×#65918C2semidirect
5. C3×#65914C2semidirect
6. C3×#65911C2semidirect
7. C3×C4×#713C2semidirect
8. C3×#65904C2semidirect
9. C3×#65848C2semidirect
10. C3×#65811C2semidirect
11. C3×C4×#709C2semidirect
12. #10617703C3direct (N × H)
48. C2×C4×C5×Q8C2×C3semidirect
49. #66151C2×C3semidirect
50. #65919C2×C3semidirect
51. #65918C2×C3semidirect
52. #65914C2×C3semidirect
53. #65911C2×C3semidirect
54. C4×#713C2×C3semidirect
55. #65904C2×C3semidirect
56. #65848C2×C3semidirect
57. #65811C2×C3semidirect
58. C4×#709C2×C3semidirect
106. C2×C3×C4×Q8D5semidirect
145. C3×C5×Q8#41semidirect
148. C2×C3×C4×C5#41semidirect
247. C2×C4×Q8C3×D5semidirect
304. C5×Q8C3×#41semidirect
307. C2×C4×C5C3×#41semidirect
368. C3×Q8#709semidirect
381. C2×C3×C4#713semidirect
441. C3×C5#3480semidirect
467. Q8C3×#709semidirect
480. C2×C4C3×#713semidirect
492. C5C3×#3480semidirect
509. C3#10617703direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×Q82
C3×#661512
C3×#659192
C3×#659182
C3×#659142
C3×#659112
C3×C4×#7132
C3×#659042
C3×#658482
C3×#658112
C3×C4×#7092
#106177033
C3×#34805