Finite Group Databasebeta

← C2×C3×#66172 #14510128

Subgroups — C2×C3×#66172 #14510128

Order 1920.

2816 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×2−1+4C211yesmaximal, characteristic, complemented
2C2×C3×#4150C211yesmaximal, characteristic, complemented
3C3×#66172C2116yesmaximal, direct product factor
4C2×C3×#4097C213yesmaximal, complemented
5C2×C3×#4096C213yesmaximal, complemented
6C2×C3×#4094C211yesmaximal, characteristic, complemented
7C22×C3×#689C213yesmaximal, complemented
8C2×C3×#4088C213yesmaximal, complemented
9C2×#66172C311yesmaximal, characteristic, direct product factor, Hall
10C3×C5×2−1+4—24no—
11C3×C5×2−1+4C2218yescomplemented
12C2×C3×C5×#41—23no—
13C2×C3×C5×#41C2213yes—
14C2×C3×C5×#41C2211yescharacteristic
15C22×C3×C5×Q8—21no—
16C22×C3×C5×Q8C2213yes—
17C3×#4150—24no—
18C3×#4150C2218yescomplemented
19C2×C3×Q8×D5—21no—
20C2×C3×#711—23no—
21C2×C3×#709C2213yes—
22C22×C3×Dic10C2213yes—
23C3×#4097C22124yescomplemented
24C3×#4096C22124yescomplemented
25C3×#4094C2218yescomplemented
26C2×C3×#689C2216yes—
27C2×C3×#689C2213yes—
28C2×C3×#689C22124yescomplemented
29C3×#4088C22124yescomplemented
30C2×C3×#688C2213yes—
31C2×C3×#687C2213yes—
32C2×C3×#686C2211yescharacteristic
33C2×C3×#681C2213yes—
34C22×C3×#168C2213yes—
35C2×C3×#578—51nomaximal, Hall
36C2×C5×2−1+4C2×C311yescharacteristic, complemented
37C2×#4150C2×C311yescharacteristic, complemented
38#66172C2×C3116yesdirect product factor
39C2×#4097C2×C313yescomplemented
40C2×#4096C2×C313yescomplemented
41C2×#4094C2×C311yescharacteristic, complemented
42C22×#689C2×C313yescomplemented
43C2×#4088C2×C313yescomplemented
44C3×C5×#41—224no—
45C3×C5×#41—26no—
46C3×C5×#41D414yes—
47C3×C5×#41C23112yes—
48C3×C5×#41C2314yes—
49C2×C3×C5×Q8—28no—
50C2×C3×C5×Q8—23no—
51C2×C3×C5×Q8—26no—
52C2×C3×C5×Q8D411yescharacteristic
53C2×C3×C5×Q8C2313yes—
54C2×C3×C5×Q8C23112yes—
55C2×C3×C5×D4—23no—
56C2×C3×C5×D4D413yes—
57C2×C3×C5×D4C2311yescharacteristic
58C22×C3×C4×C5—23no—
59C22×C3×C4×C5—23no—
60C22×C3×C4×C5C2313yes—
61C2×C3×C22⋊D5—23no—
62C22×C3×Dic5—23no—
63C3×Q8×D5—28no—
64C3×#711—224no—
65C3×#709—26no—
66C3×#709C23112yes—
67C2×C3×D20C2311yescharacteristic
68C2×C3×C4×D5—23no—
69C2×C3×Dic10—23no—
70C2×C3×Dic10—26no—
71C2×C3×Dic10C2313yes—
72C2×C3×Dic10C23112yes—
73C3×#689C23124yes—
74C3×#689C23112yes—
75C3×#688C23112yes—
76C3×#687C23112yes—
77C3×#686C2314yes—
78C3×#681C23112yes—
79C2×C3×#168C2311yescharacteristic
80C2×C3×#168C2313yes—
81C2×C3×#168C23112yes—
82C2×C3×2−1+4—51no—
83C2×C3×2−1+4D511yescharacteristic, complemented
84C3×#578—516no—
85C2×C3×#137—53no—
86C2×C3×#137—53no—
87C2×C3×#135—53no—
88C22×C3×Q16—53no—
89C2×C3×#131—51no—
90C5×2−1+4—24no—
91C5×2−1+4C22×C318yescomplemented
92C2×C5×#41—23no—
93C2×C5×#41C22×C313yes—
94C2×C5×#41C22×C311yescharacteristic
95C22×C5×Q8—21no—
96C22×C5×Q8C22×C313yes—
97#4150—24no—
98#4150C22×C318yescomplemented
99C2×Q8×D5—21no—
100C2×#711—23no—
101C2×#709C22×C313yes—
102C22×Dic10C22×C313yes—
103#4097C22×C3124yescomplemented
104#4096C22×C3124yescomplemented
105#4094C22×C318yescomplemented
106C2×#689C22×C316yes—
107C2×#689C22×C313yes—
108C2×#689C22×C3124yescomplemented
109#4088C22×C3124yescomplemented
110C2×#688C22×C313yes—
111C2×#687C22×C313yes—
112C2×#686C22×C311yescharacteristic
113C2×#681C22×C313yes—
114C22×#168C22×C313yes—
115C2×#578—51noSylow
116C23×C3×C5—21no—
117C23×C3×C5C2×D413yes—
118C22×C3×D5—21no—
119C3×C5×Q8—23no—
120C3×C5×Q8—212no—
121C3×C5×Q8C2×D414yes—
122C3×C5×Q8C2416yes—
123C3×C5×D4—212no—
124C3×C5×D4—21no—
125C3×C5×D4C2×D4112yes—
126C3×C5×D4C2412yes—
127C2×C3×C4×C5—23no—
128C2×C3×C4×C5—212no—
129C2×C3×C4×C5—212no—
130C2×C3×C4×C5C2×D413yes—
131C2×C3×C4×C5C2×D416yes—
132C2×C3×C4×C5C2411yescharacteristic
133C2×C3×C4×C5C2416yes—
134C3×C22⋊D5—212no—
135C2×C3×Dic5—23no—
136C2×C3×Dic5—212no—
137C3×D20—21no—
138C3×D20C2412yes—
139C3×C4×D5—212no—
140C3×Dic10—23no—
141C3×Dic10—212no—
142C3×Dic10C2416yes—
143C3×#168C2412yes—
144C3×#168C2416yes—
145C3×2−1+4—24no—
146C3×2−1+4—58no—
147C3×2−1+4—104no—
148C3×2−1+4C2×D518yescomplemented
149C2×C3×#41—23no—
150C2×C3×#41—53no—
151C2×C3×#41—103no—
152C2×C3×#41C2×D513yes—
153C2×C3×#41C2×D511yescharacteristic
154C22×C3×Q8—21no—
155C22×C3×Q8—53no—
156C22×C3×Q8—101no—
157C22×C3×Q8C2×D513yes—
158C3×#137—524no—
159C3×#137—524no—
160C3×#135—524no—
161C2×C3×Q16—53no—
162C2×C3×Q16—56no—
163C2×C3×Q16—524no—
164C2×C3×SD16—53no—
165C2×C3×SD16—53no—
166C2×C3×D8—51no—
167C3×#131—58no—
168C2×C3×M16—53no—
169C22×C3×C8—53no—
170C5×#41—224no—
171C5×#41—26no—
172C5×#41C3×D414yes—
173C5×#41C23×C3112yes—
174C5×#41C23×C314yes—
175C2×C5×Q8—28no—
176C2×C5×Q8—23no—
177C2×C5×Q8—26no—
178C2×C5×Q8C3×D411yescharacteristic
179C2×C5×Q8C23×C313yes—
180C2×C5×Q8C23×C3112yes—
181C2×C5×D4—23no—
182C2×C5×D4C3×D413yes—
183C2×C5×D4C23×C311yescharacteristic
184C22×C4×C5—23no—
185C22×C4×C5—23no—
186C22×C4×C5C23×C313yes—
187C2×C22⋊D5—23no—
188C22×Dic5—23no—
189Q8×D5—28no—
190#711—224no—
191#709—26no—
192#709C23×C3112yes—
193C2×D20C23×C311yescharacteristic
194C2×C4×D5—23no—
195C2×Dic10—23no—
196C2×Dic10—26no—
197C2×Dic10C23×C3112yes—
198C2×Dic10C23×C313yes—
199#689C23×C3112yes—
200#689C23×C3124yes—
201#688C23×C3112yes—
202#687C23×C3112yes—
203#686C23×C314yes—
204#681C23×C3112yes—
205C2×#168C23×C3112yes—
206C2×#168C23×C311yescharacteristic
207C2×#168C23×C313yes—
208C2×2−1+4—51no—
209C2×2−1+4C3×D511yescharacteristic, complemented
210#578—516no—
211C2×#137—53no—
212C2×#137—53no—
213C2×#135—53no—
214C22×Q16—53no—
215C2×#131—51no—
216C22×C3×C5—26no—
217C22×C3×C5—22no—
218C22×C3×C5—42no—
219C22×C3×C5C22×D416yes—
220C22×C3×C5C22×D411yescharacteristic
221C2×C3×D5—22no—
222C2×C3×D5—42no—
223C3×C4×C5—26no—
224C3×C4×C5C22×D411yescharacteristic
225C3×C4×C5C22×D416yes—
226C3×C4×C5C2511yescharacteristic
227C3×Dic5—26no—
228C3×#41—224no—
229C3×#41—26no—
230C3×#41—512no—
231C3×#41—1024no—
232C3×#41—106no—
233C3×#41C22⋊D514yes—
234C3×#41C22×D5112yes—
235C3×#41C22×D514yes—
236C2×C3×Q8—28no—
237C2×C3×Q8—23no—
238C2×C3×Q8—26no—
239C2×C3×Q8—53no—
240C2×C3×Q8—512no—
241C2×C3×Q8—108no—
242C2×C3×Q8—103no—
243C2×C3×Q8—106no—
244C2×C3×Q8C22⋊D511yescharacteristic
245C2×C3×Q8C22×D513yes—
246C2×C3×Q8C22×D5112yes—
247C2×C3×D4—23no—
248C2×C3×D4—51no—
249C2×C3×D4—103no—
250C2×C3×D4C22⋊D513yes—
251C2×C3×D4C22×D511yescharacteristic
252C22×C3×C4—23no—
253C22×C3×C4—23no—
254C22×C3×C4—103no—
255C22×C3×C4—103no—
256C22×C3×C4C22×D513yes—
257C3×Q16—512no—
258C3×Q16—524no—
259C3×SD16—512no—
260C3×SD16—512no—
261C3×D8—54no—
262C3×M16—512no—
263C2×C3×C8—51no—
264C2×C3×C8—53no—
265C2×C3×C8—512no—
266C23×C5—21no—
267C23×C5C2×C3×D413yes—
268C22×D5—21no—
269C5×Q8—23no—
270C5×Q8—212no—
271C5×Q8C2×C3×D414yes—
272C5×Q8C24×C316yes—
273C5×D4—212no—
274C5×D4—21no—
275C5×D4C2×C3×D4112yes—
276C5×D4C24×C312yes—
277C2×C4×C5—23no—
278C2×C4×C5—212no—
279C2×C4×C5—212no—
280C2×C4×C5C2×C3×D413yes—
281C2×C4×C5C2×C3×D416yes—
282C2×C4×C5C24×C311yescharacteristic
283C2×C4×C5C24×C316yes—
284C22⋊D5—212no—
285C2×Dic5—23no—
286C2×Dic5—212no—
287D20—21no—
288D20C24×C312yes—
289C4×D5—212no—
290Dic10—23no—
291Dic10—212no—
292Dic10C24×C316yes—
293#168C24×C312yes—
294#168C24×C316yes—
2952−1+4—24no—
2962−1+4—58no—
2972−1+4—104no—
2982−1+4C2×C3×D518yescomplemented
299C2×#41—23no—
300C2×#41—53no—
301C2×#41—103no—
302C2×#41C2×C3×D513yes—
303C2×#41C2×C3×D511yescharacteristic
304C22×Q8—21no—
305C22×Q8—53no—
306C22×Q8—101no—
307C22×Q8C2×C3×D513yes—
308#137—524no—
309#137—524no—
310#135—524no—
311C2×Q16—53no—
312C2×Q16—56no—
313C2×Q16—524no—
314C2×SD16—53no—
315C2×SD16—53no—
316C2×D8—51no—
317#131—58no—
318C2×M16—53no—
319C22×C8—53no—
320C2×C3×C5—26no—
321C2×C3×C5—42no—
322C2×C3×C5#57812yescomplemented
323C2×C3×C5C23×D411yescharacteristic
324C3×D5—42no—
325C23×C3—21no—
326C23×C3—101no—
327C23×C3C2×C22⋊D513yes—
328C3×Q8—212no—
329C3×Q8—23no—
330C3×Q8—56no—
331C3×Q8—1012no—
332C3×Q8—103no—
333C3×Q8C2×C22⋊D514yes—
334C3×Q8C23×D516yes—
335C3×D4—212no—
336C3×D4—21no—
337C3×D4—52no—
338C3×D4—1012no—
339C3×D4—101no—
340C3×D4C2×C22⋊D5112yes—
341C3×D4C23×D512yes—
342C2×C3×C4—212no—
343C2×C3×C4—23no—
344C2×C3×C4—212no—
345C2×C3×C4—1012no—
346C2×C3×C4—103no—
347C2×C3×C4—1012no—
348C2×C3×C4C2×C22⋊D516yes—
349C2×C3×C4C2×C22⋊D513yes—
350C2×C3×C4C23×D516yes—
351C2×C3×C4C23×D511yescharacteristic
352C3×C8—52no—
353C3×C8—56no—
354C22×C5—26no—
355C22×C5—22no—
356C22×C5—42no—
357C22×C5C22×C3×D416yes—
358C22×C5C22×C3×D411yescharacteristic
359C2×D5—22no—
360C2×D5—42no—
361C4×C5—26no—
362C4×C5C22×C3×D411yescharacteristic
363C4×C5C22×C3×D416yes—
364C4×C5C25×C311yescharacteristic
365Dic5—26no—
366#41—224no—
367#41—26no—
368#41—512no—
369#41—1024no—
370#41—106no—
371#41C3×C22⋊D514yes—
372#41C22×C3×D5112yes—
373#41C22×C3×D514yes—
374C2×Q8—28no—
375C2×Q8—23no—
376C2×Q8—26no—
377C2×Q8—53no—
378C2×Q8—512no—
379C2×Q8—108no—
380C2×Q8—103no—
381C2×Q8—106no—
382C2×Q8C3×C22⋊D511yescharacteristic
383C2×Q8C22×C3×D513yes—
384C2×Q8C22×C3×D5112yes—
385C2×D4—23no—
386C2×D4—51no—
387C2×D4—103no—
388C2×D4C3×C22⋊D513yes—
389C2×D4C22×C3×D511yescharacteristic
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—103no—
393C22×C4—103no—
394C22×C4C22×C3×D513yes—
395Q16—512no—
396Q16—524no—
397SD16—512no—
398SD16—512no—
399D8—54no—
400M16—512no—
401C2×C8—51no—
402C2×C8—53no—
403C2×C8—512no—
404C3×C5C2×#57811yescharacteristic, complemented, Hall
405C22×C3—26no—
406C22×C3—22no—
407C22×C3—42no—
408C22×C3—102no—
409C22×C3—202no—
410C22×C3C22×C22⋊D516yes—
411C22×C3C22×C22⋊D511yescharacteristic
412C3×C4—26no—
413C3×C4—106no—
414C3×C4C22×C22⋊D516yes—
415C3×C4C22×C22⋊D511yescharacteristic
416C3×C4C24×D511yescharacteristic
417C2×C5—26no—
418C2×C5—42no—
419C2×C5C3×#57812yescomplemented
420C2×C5C23×C3×D411yescharacteristic
421D5—42no—
422C23—21no—
423C23—101no—
424C23C2×C3×C22⋊D513yes—
425Q8—212no—
426Q8—23no—
427Q8—56no—
428Q8—1012no—
429Q8—103no—
430Q8C2×C3×C22⋊D514yes—
431Q8C23×C3×D516yes—
432D4—212no—
433D4—21no—
434D4—52no—
435D4—1012no—
436D4—101no—
437D4C2×C3×C22⋊D5112yes—
438D4C23×C3×D512yes—
439C2×C4—23no—
440C2×C4—212no—
441C2×C4—212no—
442C2×C4—103no—
443C2×C4—1012no—
444C2×C4—1012no—
445C2×C4C2×C3×C22⋊D513yes—
446C2×C4C2×C3×C22⋊D516yes—
447C2×C4C23×C3×D511yescharacteristic
448C2×C4C23×C3×D516yes—
449C8—52no—
450C8—56no—
451C2×C3—26no—
452C2×C3—42no—
453C2×C3—202no—
454C2×C3#6617212yesdirect product factor
455C2×C3C23×C22⋊D511yescharacteristic
456C5C2×C3×#57811yescharacteristic, complemented, Sylow
457C22—26no—
458C22—22no—
459C22—42no—
460C22—102no—
461C22—202no—
462C22C22×C3×C22⋊D516yes—
463C22C22×C3×C22⋊D511yescharacteristic
464C4—26no—
465C4—106no—
466C4C22×C3×C22⋊D516yes—
467C4C22×C3×C22⋊D511yescharacteristic
468C4C24×C3×D511yescharacteristic
469C3C2×#6617211yescharacteristic, direct product factor, Sylow
470C2—26no—
471C2—42no—
472C2—202no—
473C2C3×#6617212yesdirect product factor
474C2C23×C3×C22⋊D511yescharacteristic
subgroupG/H
centerC22×C3C22×C22⋊D5
commutator, cosocleC4×C5C25×C3
FrattiniC4C24×C3×D5
FittingC2×C3×C5×2−1+4C2
socleC22×C3×C5C22×D4
hypercenterC2×C3×2−1+4D5
hypocenterC5C2×C3×#578
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×2−1+4C2semidirect
2. C2×C3×#4150C2semidirect
3. C3×#66172C2direct (N × H)
4. C2×C3×#4097C2semidirect
5. C2×C3×#4096C2semidirect
6. C2×C3×#4094C2semidirect
7. C22×C3×#689C2semidirect
8. C2×C3×#4088C2semidirect
9. C2×#66172C3direct (N × H)
11. C3×C5×2−1+4C22semidirect
18. C3×#4150C22semidirect
23. C3×#4097C22semidirect
24. C3×#4096C22semidirect
25. C3×#4094C22semidirect
28. C2×C3×#689C22semidirect
29. C3×#4088C22semidirect
36. C2×C5×2−1+4C2×C3semidirect
37. C2×#4150C2×C3semidirect
38. #66172C2×C3direct (N × H)
39. C2×#4097C2×C3semidirect
40. C2×#4096C2×C3semidirect
41. C2×#4094C2×C3semidirect
42. C22×#689C2×C3semidirect
43. C2×#4088C2×C3semidirect
83. C2×C3×2−1+4D5semidirect
91. C5×2−1+4C22×C3semidirect
98. #4150C22×C3semidirect
103. #4097C22×C3semidirect
104. #4096C22×C3semidirect
105. #4094C22×C3semidirect
108. C2×#689C22×C3semidirect
109. #4088C22×C3semidirect
148. C3×2−1+4C2×D5semidirect
209. C2×2−1+4C3×D5semidirect
298. 2−1+4C2×C3×D5semidirect
322. C2×C3×C5#578semidirect
404. C3×C5C2×#578semidirect
419. C2×C5C3×#578semidirect
454. C2×C3#66172direct (N × H)
456. C5C2×C3×#578semidirect
469. C3C2×#66172direct (N × H)
473. C2C3×#66172direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×2−1+42
C2×C3×#41502
C3×#661722
C2×C3×#40972
C2×C3×#40962
C2×C3×#40942
C22×C3×#6892
C2×C3×#40882
C2×#661723
C2×C3×#5785