Finite Group Databasebeta

← C2×C3×#65015 #14411625

Subgroups — C2×C3×#65015 #14411625

Order 1920.

2016 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×D5C211yesmaximal, characteristic
2C22×C3×#676C212yesmaximal, complemented
3C2×C3×C5×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×C3×C22⋊#168C211yesmaximal, characteristic, complemented
5C2×C3×#3952C212yesmaximal, complemented
6C3×#65015C218yesmaximal, direct product factor
7C2×#65015C311yesmaximal, characteristic, direct product factor, Hall
8C24×C3×D5C411yescharacteristic
9C23×C3×C4×C5C2211yescharacteristic
10C23×C3×Dic5C411yescharacteristic
11C22×C3×C4×D5—28no—
12C22×C3×C4×D5C411yescharacteristic
13C22×C3×C4×D5C411yescharacteristic
14C22×C3×C4×D5C414yes—
15C22×C3×C4×D5C2211yescharacteristic
16C22×C3×C4×D5C2214yes—
17C22×C3×C4×D5C2211yescharacteristic
18C22×C3×C5×C8C2212yescomplemented
19C22×C3×#168C2212yescomplemented
20C2×C3×#676—28no—
21C2×C3×#676C2218yescomplemented
22C3×C5×C22⋊C8C2214yescomplemented
23C3×C22⋊#168C2214yescomplemented
24C3×#3952C2218yescomplemented
25C2×C3×#407—51nomaximal, Hall
26C23×C4×D5C2×C311yescharacteristic
27C22×#676C2×C312yescomplemented
28C2×C5×C22⋊C8C2×C311yescharacteristic, complemented
29C2×C22⋊#168C2×C311yescharacteristic, complemented
30C2×#3952C2×C312yescomplemented
31#65015C2×C318yesdirect product factor
32C24×C3×C5C2×C411yescharacteristic
33C23×C3×D5—24no—
34C23×C3×D5—24no—
35C23×C3×D5—24no—
36C23×C3×D5C2×C414yes—
37C23×C3×D5C2×C411yescharacteristic
38C23×C3×D5C2×C411yescharacteristic
39C22×C3×C4×C5—24no—
40C22×C3×C4×C5C2×C411yescharacteristic
41C22×C3×C4×C5C2×C412yes—
42C22×C3×C4×C5C2312yes—
43C22×C3×C4×C5C2311yescharacteristic
44C22×C3×Dic5—24no—
45C22×C3×Dic5C2×C411yescharacteristic
46C22×C3×Dic5C2×C414yes—
47C22×C3×Dic5C2×C411yescharacteristic
48C2×C3×C4×D5—24no—
49C2×C3×C4×D5—24no—
50C2×C3×C4×D5—24no—
51C2×C3×C4×D5—216no—
52C2×C3×C4×D5—216no—
53C2×C3×C4×D5C2×C414yes—
54C2×C3×C4×D5C2×C412yes—
55C2×C3×C4×D5C2×C414yes—
56C2×C3×C4×D5C2×C412yes—
57C2×C3×C4×D5D418yes—
58C2×C3×C4×D5C2312yes—
59C2×C3×C4×D5C2312yes—
60C2×C3×C5×C8—24no—
61C2×C3×C5×C8C2314yescomplemented
62C2×C3×#168—24no—
63C2×C3×#168C2314yescomplemented
64C3×#676—216no—
65C24×C3×C4—51no—
66C22×C3×M16—52no—
67C3×#407—58no—
68C2×C3×C22⋊C8—51no—
69C2×C3×C22⋊C8—52no—
70C2×C3×C22⋊C8D511yescharacteristic, complemented
71C24×D5C3×C411yescharacteristic
72C23×C4×C5C22×C311yescharacteristic
73C23×Dic5C3×C411yescharacteristic
74C22×C4×D5—28no—
75C22×C4×D5C3×C411yescharacteristic
76C22×C4×D5C3×C411yescharacteristic
77C22×C4×D5C3×C414yes—
78C22×C4×D5C22×C311yescharacteristic
79C22×C4×D5C22×C314yes—
80C22×C4×D5C22×C311yescharacteristic
81C22×C5×C8C22×C312yescomplemented
82C22×#168C22×C312yescomplemented
83C2×#676—28no—
84C2×#676C22×C318yescomplemented
85C5×C22⋊C8C22×C314yescomplemented
86C22⋊#168C22×C314yescomplemented
87#3952C22×C318yescomplemented
88C2×#407—51noSylow
89C23×C3×C5—22no—
90C23×C3×C5—22no—
91C23×C3×C5M1614yescomplemented
92C23×C3×C5C22×C412yes—
93C23×C3×C5C22×C411yescharacteristic
94C22×C3×D5—232no—
95C22×C3×D5—22no—
96C22×C3×D5—22no—
97C22×C3×D5—22no—
98C22×C3×D5—22no—
99C22×C3×D5—28no—
100C22×C3×D5—28no—
101C22×C3×D5—28no—
102C22×C3×D5C22⋊C414yes—
103C22×C3×D5M1614yescomplemented
104C22×C3×D5C22×C412yes—
105C22×C3×D5C22×C412yes—
106C2×C3×C4×C5—22no—
107C2×C3×C4×C5—24no—
108C2×C3×C4×C5—24no—
109C2×C3×C4×C5C22×C412yes—
110C2×C3×C4×C5C22×C411yescharacteristic
111C2×C3×C4×C5C2×D414yes—
112C2×C3×C4×C5C2411yescharacteristic
113C2×C3×Dic5—22no—
114C2×C3×Dic5—28no—
115C2×C3×Dic5C22⋊C414yes—
116C2×C3×Dic5C22×C412yes—
117C2×C3×Dic5C22×C412yes—
118C3×C4×D5—28no—
119C3×C4×D5—24no—
120C3×C4×D5—24no—
121C3×C4×D5—28no—
122C3×C4×D5C22⋊C418yes—
123C3×C4×D5C2×D418yes—
124C3×C5×C8—24no—
125C3×#168—24no—
126C25×C3—51no—
127C23×C3×C4—51no—
128C23×C3×C4—51no—
129C23×C3×C4—51no—
130C23×C3×C4—54no—
131C23×C3×C4—51no—
132C23×C3×C4—54no—
133C23×C3×C4—51no—
134C23×C3×C4—108no—
135C23×C3×C4C2×D511yescharacteristic
136C2×C3×M16—58no—
137C2×C3×M16—108no—
138C22×C3×C8—52no—
139C22×C3×C8C2×D512yescomplemented
140C3×C22⋊C8—54no—
141C3×C22⋊C8—58no—
142C3×C22⋊C8C2×D514yescomplemented
143C24×C5C2×C3×C411yescharacteristic
144C23×D5—24no—
145C23×D5—24no—
146C23×D5—24no—
147C23×D5C2×C3×C414yes—
148C23×D5C2×C3×C411yescharacteristic
149C23×D5C2×C3×C411yescharacteristic
150C22×C4×C5—24no—
151C22×C4×C5C2×C3×C411yescharacteristic
152C22×C4×C5C2×C3×C412yes—
153C22×C4×C5C23×C312yes—
154C22×C4×C5C23×C311yescharacteristic
155C22×Dic5—24no—
156C22×Dic5C2×C3×C411yescharacteristic
157C22×Dic5C2×C3×C414yes—
158C22×Dic5C2×C3×C411yescharacteristic
159C2×C4×D5—24no—
160C2×C4×D5—24no—
161C2×C4×D5—24no—
162C2×C4×D5—216no—
163C2×C4×D5—216no—
164C2×C4×D5C2×C3×C414yes—
165C2×C4×D5C2×C3×C412yes—
166C2×C4×D5C2×C3×C414yes—
167C2×C4×D5C2×C3×C412yes—
168C2×C4×D5C3×D418yes—
169C2×C4×D5C23×C312yes—
170C2×C4×D5C23×C312yes—
171C2×C5×C8—24no—
172C2×C5×C8C23×C314yescomplemented
173C2×#168—24no—
174C2×#168C23×C314yescomplemented
175#676—216no—
176C24×C4—51no—
177C22×M16—52no—
178#407—58no—
179C2×C22⋊C8—51no—
180C2×C22⋊C8—52no—
181C2×C22⋊C8C3×D511yescharacteristic, complemented
182C22×C3×C5—22no—
183C22×C3×C5—22no—
184C22×C3×C5—28no—
185C22×C3×C5C2×C22⋊C412yes—
186C22×C3×C5C2×M1612yes—
187C22×C3×C5C2×M1612yes—
188C22×C3×C5C2×M1614yescomplemented
189C22×C3×C5C23×C411yescharacteristic
190C2×C3×D5—232no—
191C2×C3×D5—22no—
192C2×C3×D5—22no—
193C2×C3×D5—22no—
194C2×C3×D5—22no—
195C2×C3×D5—28no—
196C2×C3×D5—28no—
197C2×C3×D5C2×C22⋊C414yes—
198C2×C3×D5C2×M1614yescomplemented
199C3×C4×C5—22no—
200C3×C4×C5C2×C22⋊C412yes—
201C3×C4×C5C22×D412yes—
202C3×Dic5—22no—
203C3×Dic5C2×C22⋊C414yes—
204C24×C3—51no—
205C24×C3—51no—
206C24×C3—54no—
207C24×C3—104no—
208C24×C3—104no—
209C24×C3—104no—
210C24×C3C4×D511yescharacteristic
211C22×C3×C4—24no—
212C22×C3×C4—51no—
213C22×C3×C4—52no—
214C22×C3×C4—58no—
215C22×C3×C4—52no—
216C22×C3×C4—54no—
217C22×C3×C4—51no—
218C22×C3×C4—54no—
219C22×C3×C4—54no—
220C22×C3×C4—52no—
221C22×C3×C4—52no—
222C22×C3×C4—104no—
223C22×C3×C4—1016no—
224C22×C3×C4—1016no—
225C22×C3×C4—104no—
226C22×C3×C4—104no—
227C22×C3×C4—104no—
228C22×C3×C4C4×D511yescharacteristic
229C22×C3×C4C4×D512yes—
230C22×C3×C4C22×D512yes—
231C22×C3×C4C22×D511yescharacteristic
232C3×M16—1016no—
233C2×C3×C8—24no—
234C2×C3×C8—54no—
235C2×C3×C8—104no—
236C2×C3×C8C22×D514yescomplemented
237C23×C5—22no—
238C23×C5—22no—
239C23×C5C3×M1614yescomplemented
240C23×C5C22×C3×C412yes—
241C23×C5C22×C3×C411yescharacteristic
242C22×D5—232no—
243C22×D5—22no—
244C22×D5—22no—
245C22×D5—22no—
246C22×D5—22no—
247C22×D5—28no—
248C22×D5—28no—
249C22×D5—28no—
250C22×D5C3×C22⋊C414yes—
251C22×D5C3×M1614yescomplemented
252C22×D5C22×C3×C412yes—
253C22×D5C22×C3×C412yes—
254C2×C4×C5—22no—
255C2×C4×C5—24no—
256C2×C4×C5—24no—
257C2×C4×C5C22×C3×C412yes—
258C2×C4×C5C22×C3×C411yescharacteristic
259C2×C4×C5C2×C3×D414yes—
260C2×C4×C5C24×C311yescharacteristic
261C2×Dic5—22no—
262C2×Dic5—28no—
263C2×Dic5C3×C22⋊C414yes—
264C2×Dic5C22×C3×C412yes—
265C2×Dic5C22×C3×C412yes—
266C4×D5—28no—
267C4×D5—24no—
268C4×D5—24no—
269C4×D5—28no—
270C4×D5C3×C22⋊C418yes—
271C4×D5C2×C3×D418yes—
272C5×C8—24no—
273#168—24no—
274C25—51no—
275C23×C4—51no—
276C23×C4—51no—
277C23×C4—51no—
278C23×C4—54no—
279C23×C4—51no—
280C23×C4—51no—
281C23×C4—54no—
282C23×C4—108no—
283C23×C4C2×C3×D511yescharacteristic
284C2×M16—58no—
285C2×M16—108no—
286C22×C8—52no—
287C22×C8C2×C3×D512yescomplemented
288C22⋊C8—54no—
289C22⋊C8—58no—
290C22⋊C8C2×C3×D514yescomplemented
291C2×C3×C5—24no—
292C2×C3×C5#40714yescomplemented
293C2×C3×C5C22×C22⋊C411yescharacteristic
294C2×C3×C5C22×M1611yescharacteristic
295C2×C3×C5C22×M1611yescharacteristic
296C3×D5—24no—
297C3×D5—24no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—52no—
301C23×C3—52no—
302C23×C3—54no—
303C23×C3—54no—
304C23×C3—108no—
305C23×C3—108no—
306C23×C3—1032no—
307C23×C3—102no—
308C23×C3—102no—
309C23×C3—102no—
310C23×C3—108no—
311C23×C3—102no—
312C23×C3#67614yescomplemented
313C23×C3C2×C4×D512yes—
314C23×C3C2×C4×D511yescharacteristic
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—52no—
319C2×C3×C4—54no—
320C2×C3×C4—58no—
321C2×C3×C4—58no—
322C2×C3×C4—52no—
323C2×C3×C4—108no—
324C2×C3×C4—108no—
325C2×C3×C4—108no—
326C2×C3×C4—102no—
327C2×C3×C4—104no—
328C2×C3×C4—104no—
329C2×C3×C4C2×C4×D512yes—
330C2×C3×C4C2×C4×D511yescharacteristic
331C2×C3×C4D4×D514yes—
332C2×C3×C4C23×D511yescharacteristic
333C3×C8—24no—
334C3×C8—104no—
335C22×C5—22no—
336C22×C5—28no—
337C22×C5—22no—
338C22×C5C2×C3×C22⋊C412yes—
339C22×C5C2×C3×M1612yes—
340C22×C5C2×C3×M1612yes—
341C22×C5C2×C3×M1614yescomplemented
342C22×C5C23×C3×C411yescharacteristic
343C2×D5—232no—
344C2×D5—22no—
345C2×D5—22no—
346C2×D5—22no—
347C2×D5—28no—
348C2×D5—28no—
349C2×D5—22no—
350C2×D5C2×C3×C22⋊C414yes—
351C2×D5C2×C3×M1614yescomplemented
352C4×C5—22no—
353C4×C5C2×C3×C22⋊C412yes—
354C4×C5C22×C3×D412yes—
355Dic5—22no—
356Dic5C2×C3×C22⋊C414yes—
357C24—51no—
358C24—51no—
359C24—54no—
360C24—104no—
361C24—104no—
362C24—104no—
363C24C3×C4×D511yescharacteristic
364C22×C4—24no—
365C22×C4—51no—
366C22×C4—52no—
367C22×C4—58no—
368C22×C4—52no—
369C22×C4—54no—
370C22×C4—51no—
371C22×C4—54no—
372C22×C4—54no—
373C22×C4—52no—
374C22×C4—52no—
375C22×C4—104no—
376C22×C4—1016no—
377C22×C4—1016no—
378C22×C4—104no—
379C22×C4—104no—
380C22×C4—104no—
381C22×C4C3×C4×D511yescharacteristic
382C22×C4C3×C4×D512yes—
383C22×C4C22×C3×D512yes—
384C22×C4C22×C3×D511yescharacteristic
385M16—1016no—
386C2×C8—24no—
387C2×C8—54no—
388C2×C8—104no—
389C2×C8C22×C3×D514yescomplemented
390C3×C5C2×#40711yescharacteristic, complemented, Hall
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—28no—
394C22×C3—54no—
395C22×C3—54no—
396C22×C3—1032no—
397C22×C3—102no—
398C22×C3—102no—
399C22×C3—102no—
400C22×C3—102no—
401C22×C3—108no—
402C22×C3—108no—
403C22×C3D5×C22⋊C412yes—
404C22×C3C2×#67612yes—
405C22×C3C2×#67614yescomplemented
406C22×C3D5×M1612yes—
407C22×C3C22×C4×D511yescharacteristic
408C3×C4—22no—
409C3×C4—54no—
410C3×C4—102no—
411C3×C4D5×C22⋊C412yes—
412C3×C4C2×D4×D512yes—
413C2×C5—24no—
414C2×C5C3×#40714yescomplemented
415C2×C5C22×C3×C22⋊C411yescharacteristic
416C2×C5C22×C3×M1611yescharacteristic
417C2×C5C22×C3×M1611yescharacteristic
418D5—24no—
419D5—24no—
420C23—22no—
421C23—22no—
422C23—52no—
423C23—52no—
424C23—54no—
425C23—54no—
426C23—108no—
427C23—108no—
428C23—1032no—
429C23—102no—
430C23—102no—
431C23—102no—
432C23—108no—
433C23—102no—
434C23C3×#67614yescomplemented
435C23C2×C3×C4×D512yes—
436C23C2×C3×C4×D511yescharacteristic
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—52no—
441C2×C4—54no—
442C2×C4—58no—
443C2×C4—58no—
444C2×C4—52no—
445C2×C4—108no—
446C2×C4—108no—
447C2×C4—108no—
448C2×C4—102no—
449C2×C4—104no—
450C2×C4—104no—
451C2×C4C2×C3×C4×D512yes—
452C2×C4C2×C3×C4×D511yescharacteristic
453C2×C4C3×D4×D514yes—
454C2×C4C23×C3×D511yescharacteristic
455C8—24no—
456C8—104no—
457C2×C3—24no—
458C2×C3—104no—
459C2×C3—104no—
460C2×C3#6501514yesdirect product factor
461C2×C3C2×D5×C22⋊C411yescharacteristic
462C2×C3C22×#67611yescharacteristic
463C2×C3C2×D5×M1611yescharacteristic
464C5C2×C3×#40711yescharacteristic, complemented, Sylow
465C22—22no—
466C22—22no—
467C22—28no—
468C22—54no—
469C22—54no—
470C22—1032no—
471C22—102no—
472C22—102no—
473C22—102no—
474C22—102no—
475C22—108no—
476C22—108no—
477C22C3×D5×C22⋊C412yes—
478C22C2×C3×#67612yes—
479C22C2×C3×#67614yescomplemented
480C22C3×D5×M1612yes—
481C22C22×C3×C4×D511yescharacteristic
482C4—22no—
483C4—54no—
484C4—102no—
485C4C3×D5×C22⋊C412yes—
486C4C2×C3×D4×D512yes—
487C3C2×#6501511yescharacteristic, direct product factor, Sylow
488C2—24no—
489C2—104no—
490C2—104no—
491C2C3×#6501514yesdirect product factor
492C2C2×C3×D5×C22⋊C411yescharacteristic
493C2C22×C3×#67611yescharacteristic
494C2C2×C3×D5×M1611yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D5
commutatorC22×C5C23×C3×C4
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×C22⋊C8C2
socleC23×C3×C5C22×C4
prime radicalC24×C3×D5C4
cosocleC2×C4×C5C24×C3
hypercenterC2×C3×C22⋊C8D5
hypocenterC5C2×C3×#407
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×C3×#676C2semidirect
3. C2×C3×C5×C22⋊C8C2semidirect
4. C2×C3×C22⋊#168C2semidirect
5. C2×C3×#3952C2semidirect
6. C3×#65015C2direct (N × H)
7. C2×#65015C3direct (N × H)
18. C22×C3×C5×C8C22semidirect
19. C22×C3×#168C22semidirect
21. C2×C3×#676C22semidirect
22. C3×C5×C22⋊C8C22semidirect
23. C3×C22⋊#168C22semidirect
24. C3×#3952C22semidirect
27. C22×#676C2×C3semidirect
28. C2×C5×C22⋊C8C2×C3semidirect
29. C2×C22⋊#168C2×C3semidirect
30. C2×#3952C2×C3semidirect
31. #65015C2×C3direct (N × H)
61. C2×C3×C5×C8C23semidirect
63. C2×C3×#168C23semidirect
70. C2×C3×C22⋊C8D5semidirect
81. C22×C5×C8C22×C3semidirect
82. C22×#168C22×C3semidirect
84. C2×#676C22×C3semidirect
85. C5×C22⋊C8C22×C3semidirect
86. C22⋊#168C22×C3semidirect
87. #3952C22×C3semidirect
91. C23×C3×C5M16semidirect
103. C22×C3×D5M16semidirect
139. C22×C3×C8C2×D5semidirect
142. C3×C22⋊C8C2×D5semidirect
172. C2×C5×C8C23×C3semidirect
174. C2×#168C23×C3semidirect
181. C2×C22⋊C8C3×D5semidirect
188. C22×C3×C5C2×M16semidirect
198. C2×C3×D5C2×M16semidirect
236. C2×C3×C8C22×D5semidirect
239. C23×C5C3×M16semidirect
251. C22×D5C3×M16semidirect
287. C22×C8C2×C3×D5semidirect
290. C22⋊C8C2×C3×D5semidirect
292. C2×C3×C5#407semidirect
312. C23×C3#676semidirect
341. C22×C5C2×C3×M16semidirect
351. C2×D5C2×C3×M16semidirect
389. C2×C8C22×C3×D5semidirect
390. C3×C5C2×#407semidirect
405. C22×C3C2×#676semidirect
414. C2×C5C3×#407semidirect
434. C23C3×#676semidirect
460. C2×C3#65015direct (N × H)
464. C5C2×C3×#407semidirect
479. C22C2×C3×#676semidirect
487. C3C2×#65015direct (N × H)
491. C2C3×#65015direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×D52
C22×C3×#6762
C2×C3×C5×C22⋊C82
C2×C3×C22⋊#1682
C2×C3×#39522
C3×#650152
C2×#650153
C2×C3×#4075