Finite Group Databasebeta

← C2×C3×#66058 #14480477

Subgroups — C2×C3×#66058 #14480477

Order 1920.

3024 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#713C211yesmaximal, characteristic
2C22×C3×Q8×D5C211yesmaximal, characteristic, complemented
3C2×C3×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
4C2×C3×#4093C211yesmaximal, characteristic, complemented
5C2×C3×#4090C211yesmaximal, characteristic, complemented
6C3×#66058C2116yesmaximal, direct product factor
7C2×C3×#4042C214yesmaximal, complemented
8C2×C3×#4040C214yesmaximal, complemented
9C2×C3×#4021C211yesmaximal, characteristic, complemented
10C2×C3×#4018C211yesmaximal, characteristic, complemented
11C2×#66058C311yesmaximal, characteristic, direct product factor, Hall
12C22×C3×C5×Q8C2211yescharacteristic
13C22×C3×C5×Q8C2211yescharacteristic
14C2×C3×#713—28no—
15C2×C3×#713C2218yes—
16C2×C3×Q8×D5—28no—
17C2×C3×Q8×D5C2218yescomplemented
18C22×C3×D20C2211yescharacteristic
19C22×C3×D20C2212yes—
20C22×C3×C4×D5C2212yes—
21C22×C3×C4×D5C2212yes—
22C22×C3×C4×D5C2211yescharacteristic
23C22×C3×Dic10C2211yescharacteristic
24C22×C3×Dic10C2212yes—
25C3×C5×C4⋊Q8C2218yescomplemented
26C2×C3×C5×C4⋊C4C2214yes—
27C2×C3×C42×C5C2211yescharacteristic
28C3×#4093C2218yescomplemented
29C3×#4090C2218yescomplemented
30C2×C3×#685C2214yes—
31C2×C3×#685C2212yes—
32C2×C3×#685C2214yes—
33C2×C3×#683C2214yes—
34C2×C3×#683C2212yes—
35C2×C3×C4×Dic5C2211yescharacteristic
36C3×#4042C22132yescomplemented
37C3×#4040C22132yescomplemented
38C3×#4021C2218yescomplemented
39C3×#4018C2218yescomplemented
40C2×C3×#536—51nomaximal, Hall
41C22×#713C2×C311yescharacteristic
42C22×Q8×D5C2×C311yescharacteristic, complemented
43C2×C5×C4⋊Q8C2×C311yescharacteristic, complemented
44C2×#4093C2×C311yescharacteristic, complemented
45C2×#4090C2×C311yescharacteristic, complemented
46#66058C2×C3116yesdirect product factor
47C2×#4042C2×C314yescomplemented
48C2×#4040C2×C314yescomplemented
49C2×#4021C2×C311yescharacteristic, complemented
50C2×#4018C2×C311yescharacteristic, complemented
51C23×C3×D5C2312yes—
52C23×C3×D5C2311yescharacteristic
53C2×C3×C5×Q8—24no—
54C2×C3×C5×Q8—24no—
55C2×C3×C5×Q8C2314yes—
56C2×C3×C5×Q8C2314yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2311yescharacteristic
61C22×C3×Dic5C2312yes—
62C22×C3×Dic5C2312yes—
63C22×C3×Dic5C2311yescharacteristic
64C3×#713—232no—
65C3×Q8×D5—232no—
66C2×C3×D20—28no—
67C2×C3×D20—24no—
68C2×C3×D20C2314yes—
69C2×C3×D20C2318yes—
70C2×C3×C4×D5—28no—
71C2×C3×C4×D5—28no—
72C2×C3×C4×D5D418yes—
73C2×C3×C4×D5C2318yes—
74C2×C3×C4×D5C2318yes—
75C2×C3×C4×D5C2314yes—
76C2×C3×Dic10—24no—
77C2×C3×Dic10—28no—
78C2×C3×Dic10C2314yes—
79C2×C3×Dic10C2318yes—
80C3×C5×C4⋊C4C23116yes—
81C3×C42×C5C2314yes—
82C3×#685C23116yes—
83C3×#685C2318yes—
84C3×#685C23116yes—
85C3×#683C23116yes—
86C3×#683C2318yes—
87C3×C4×Dic5C2314yes—
88C22×C3×#41—51no—
89C23×C3×Q8—51no—
90C3×#536—516no—
91C2×C3×C4⋊Q8—51no—
92C2×C3×C4⋊Q8D511yescharacteristic, complemented
93C2×C3×#124—51no—
94C2×C3×#124—51no—
95C2×C3×#123—54no—
96C2×C3×C22⋊Q8—54no—
97C2×C3×#117—51no—
98C22×C5×Q8C22×C311yescharacteristic
99C22×C5×Q8C22×C311yescharacteristic
100C2×#713—28no—
101C2×#713C22×C318yes—
102C2×Q8×D5—28no—
103C2×Q8×D5C22×C318yescomplemented
104C22×D20C22×C311yescharacteristic
105C22×D20C22×C312yes—
106C22×C4×D5C22×C312yes—
107C22×C4×D5C22×C312yes—
108C22×C4×D5C22×C311yescharacteristic
109C22×Dic10C22×C311yescharacteristic
110C22×Dic10C22×C312yes—
111C5×C4⋊Q8C22×C318yescomplemented
112C2×C5×C4⋊C4C22×C314yes—
113C2×C42×C5C22×C311yescharacteristic
114#4093C22×C318yescomplemented
115#4090C22×C318yescomplemented
116C2×#685C22×C312yes—
117C2×#685C22×C314yes—
118C2×#685C22×C314yes—
119C2×#683C22×C312yes—
120C2×#683C22×C314yes—
121C2×C4×Dic5C22×C311yescharacteristic
122#4042C22×C3132yescomplemented
123#4040C22×C3132yescomplemented
124#4021C22×C318yescomplemented
125#4018C22×C318yescomplemented
126C2×#536—51noSylow
127C23×C3×C5C2411yescharacteristic
128C22×C3×D5—24no—
129C22×C3×D5—24no—
130C22×C3×D5—28no—
131C22×C3×D5C2×D414yes—
132C22×C3×D5C2414yes—
133C22×C3×D5C2412yes—
134C3×C5×Q8—28no—
135C3×C5×Q8—28no—
136C2×C3×C4×C5—24no—
137C2×C3×C4×C5—24no—
138C2×C3×C4×C5—24no—
139C2×C3×C4×C5C2×D414yes—
140C2×C3×C4×C5C2414yes—
141C2×C3×C4×C5C2414yes—
142C2×C3×C4×C5C2414yes—
143C2×C3×C4×C5C2412yes—
144C2×C3×Dic5—24no—
145C2×C3×Dic5—24no—
146C2×C3×Dic5C2×D414yes—
147C2×C3×Dic5C2414yes—
148C2×C3×Dic5C2414yes—
149C2×C3×Dic5C2412yes—
150C3×D20—216no—
151C3×D20—28no—
152C3×C4×D5—216no—
153C3×C4×D5—216no—
154C3×C4×D5C2×D4116yes—
155C3×Dic10—28no—
156C3×Dic10—216no—
157C2×C3×#41—58no—
158C2×C3×#41—108no—
159C22×C3×Q8—58no—
160C22×C3×Q8—52no—
161C22×C3×Q8—51no—
162C22×C3×Q8—108no—
163C22×C3×Q8C2×D511yescharacteristic
164C22×C3×Q8C2×D511yescharacteristic
165C22×C3×D4—52no—
166C22×C3×D4—51no—
167C23×C3×C4—51no—
168C23×C3×C4—52no—
169C23×C3×C4—52no—
170C3×C4⋊Q8—58no—
171C3×C4⋊Q8C2×D518yescomplemented
172C3×#124—58no—
173C3×#124—58no—
174C3×#123—532no—
175C3×C22⋊Q8—532no—
176C3×#117—58no—
177C2×C3×C4⋊C4—52no—
178C2×C3×C4⋊C4—54no—
179C2×C3×C4⋊C4C2×D514yes—
180C2×C3×C22⋊C4—52no—
181C2×C3×C22⋊C4—54no—
182C2×C3×C22⋊C4—54no—
183C2×C3×C42—51no—
184C2×C3×C42C2×D511yescharacteristic
185C23×D5C23×C312yes—
186C23×D5C23×C311yescharacteristic
187C2×C5×Q8—24no—
188C2×C5×Q8—24no—
189C2×C5×Q8C23×C314yes—
190C2×C5×Q8C23×C314yes—
191C22×C4×C5C23×C312yes—
192C22×C4×C5C23×C312yes—
193C22×C4×C5C23×C312yes—
194C22×C4×C5C23×C311yescharacteristic
195C22×Dic5C23×C312yes—
196C22×Dic5C23×C312yes—
197C22×Dic5C23×C311yescharacteristic
198#713—232no—
199Q8×D5—232no—
200C2×D20—28no—
201C2×D20—24no—
202C2×D20C23×C314yes—
203C2×D20C23×C318yes—
204C2×C4×D5—28no—
205C2×C4×D5—28no—
206C2×C4×D5C3×D418yes—
207C2×C4×D5C23×C318yes—
208C2×C4×D5C23×C318yes—
209C2×C4×D5C23×C314yes—
210C2×Dic10—24no—
211C2×Dic10—28no—
212C2×Dic10C23×C314yes—
213C2×Dic10C23×C318yes—
214C5×C4⋊C4C23×C3116yes—
215C42×C5C23×C314yes—
216#685C23×C318yes—
217#685C23×C3116yes—
218#685C23×C3116yes—
219#683C23×C318yes—
220#683C23×C3116yes—
221C4×Dic5C23×C314yes—
222C22×#41—51no—
223C23×Q8—51no—
224#536—516no—
225C2×C4⋊Q8—51no—
226C2×C4⋊Q8C3×D511yescharacteristic, complemented
227C2×#124—51no—
228C2×#124—51no—
229C2×#123—54no—
230C2×C22⋊Q8—54no—
231C2×#117—51no—
232C22×C3×C5C22×D412yes—
233C22×C3×C52−1+414yes—
234C22×C3×C5C2511yescharacteristic
235C2×C3×D5—28no—
236C2×C3×D5—24no—
237C2×C3×D5—24no—
238C2×C3×D5—28no—
239C2×C3×D5—48no—
240C2×C3×D5C22×D414yes—
241C3×C4×C5—24no—
242C3×C4×C5—24no—
243C3×C4×C5—24no—
244C3×C4×C5C22×D414yes—
245C3×Dic5—24no—
246C3×Dic5—24no—
247C3×Dic5C22×D414yes—
248C24×C3—51no—
249C24×C3—52no—
250C3×#41—1032no—
251C2×C3×Q8—24no—
252C2×C3×Q8—24no—
253C2×C3×Q8—58no—
254C2×C3×Q8—54no—
255C2×C3×Q8—108no—
256C2×C3×Q8—1032no—
257C2×C3×Q8—104no—
258C2×C3×Q8C22×D514yes—
259C2×C3×Q8C22×D514yes—
260C2×C3×D4—58no—
261C2×C3×D4—54no—
262C2×C3×D4—108no—
263C2×C3×D4—104no—
264C22×C3×C4—51no—
265C22×C3×C4—58no—
266C22×C3×C4—54no—
267C22×C3×C4—52no—
268C22×C3×C4—52no—
269C22×C3×C4—58no—
270C22×C3×C4—58no—
271C22×C3×C4—108no—
272C22×C3×C4—108no—
273C22×C3×C4C22×D512yes—
274C22×C3×C4C22×D512yes—
275C22×C3×C4C22×D512yes—
276C22×C3×C4C22×D511yescharacteristic
277C3×C4⋊C4—58no—
278C3×C4⋊C4—516no—
279C3×C4⋊C4C22×D5116yes—
280C3×C22⋊C4—58no—
281C3×C22⋊C4—516no—
282C3×C22⋊C4—516no—
283C3×C42—54no—
284C3×C42C22×D514yes—
285C23×C5C24×C311yescharacteristic
286C22×D5—24no—
287C22×D5—24no—
288C22×D5—28no—
289C22×D5C2×C3×D414yes—
290C22×D5C24×C314yes—
291C22×D5C24×C312yes—
292C5×Q8—28no—
293C5×Q8—28no—
294C2×C4×C5—24no—
295C2×C4×C5—24no—
296C2×C4×C5—24no—
297C2×C4×C5C2×C3×D414yes—
298C2×C4×C5C24×C314yes—
299C2×C4×C5C24×C314yes—
300C2×C4×C5C24×C314yes—
301C2×C4×C5C24×C312yes—
302C2×Dic5—24no—
303C2×Dic5—24no—
304C2×Dic5C2×C3×D414yes—
305C2×Dic5C24×C314yes—
306C2×Dic5C24×C314yes—
307C2×Dic5C24×C312yes—
308D20—216no—
309D20—28no—
310C4×D5—216no—
311C4×D5—216no—
312C4×D5C2×C3×D4116yes—
313Dic10—28no—
314Dic10—216no—
315C2×#41—58no—
316C2×#41—108no—
317C22×Q8—58no—
318C22×Q8—52no—
319C22×Q8—51no—
320C22×Q8—108no—
321C22×Q8C2×C3×D511yescharacteristic
322C22×Q8C2×C3×D511yescharacteristic
323C22×D4—52no—
324C22×D4—51no—
325C23×C4—51no—
326C23×C4—52no—
327C23×C4—52no—
328C4⋊Q8—58no—
329C4⋊Q8C2×C3×D518yescomplemented
330#124—58no—
331#124—58no—
332#123—532no—
333C22⋊Q8—532no—
334#117—58no—
335C2×C4⋊C4—52no—
336C2×C4⋊C4—54no—
337C2×C4⋊C4C2×C3×D514yes—
338C2×C22⋊C4—52no—
339C2×C22⋊C4—54no—
340C2×C22⋊C4—54no—
341C2×C42—51no—
342C2×C42C2×C3×D511yescharacteristic
343C2×C3×C5#53614yescomplemented
344C2×C3×C5C23×D411yescharacteristic
345C2×C3×C5C2×2−1+412yes—
346C3×D5—24no—
347C3×D5—44no—
348C23×C3—54no—
349C23×C3—52no—
350C23×C3—54no—
351C23×C3—104no—
352C23×C3—104no—
353C23×C3—108no—
354C23×C3C23×D511yescharacteristic
355C3×Q8—28no—
356C3×Q8—28no—
357C3×Q8—1016no—
358C3×Q8—108no—
359C3×D4—1016no—
360C3×D4—108no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—54no—
365C2×C3×C4—52no—
366C2×C3×C4—516no—
367C2×C3×C4—54no—
368C2×C3×C4—54no—
369C2×C3×C4—1016no—
370C2×C3×C4—1016no—
371C2×C3×C4—104no—
372C2×C3×C4—104no—
373C2×C3×C4D4×D514yes—
374C2×C3×C4C23×D514yes—
375C2×C3×C4C23×D514yes—
376C2×C3×C4C23×D514yes—
377C2×C3×C4C23×D512yes—
378C22×C5C22×C3×D412yes—
379C22×C5C3×2−1+414yes—
380C22×C5C25×C311yescharacteristic
381C2×D5—24no—
382C2×D5—28no—
383C2×D5—24no—
384C2×D5—28no—
385C2×D5—48no—
386C2×D5C22×C3×D414yes—
387C4×C5—24no—
388C4×C5—24no—
389C4×C5—24no—
390C4×C5C22×C3×D414yes—
391Dic5—24no—
392Dic5—24no—
393Dic5C22×C3×D414yes—
394C24—51no—
395C24—52no—
396#41—1032no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—58no—
400C2×Q8—54no—
401C2×Q8—108no—
402C2×Q8—1032no—
403C2×Q8—104no—
404C2×Q8C22×C3×D514yes—
405C2×Q8C22×C3×D514yes—
406C2×D4—58no—
407C2×D4—54no—
408C2×D4—108no—
409C2×D4—104no—
410C22×C4—51no—
411C22×C4—58no—
412C22×C4—54no—
413C22×C4—52no—
414C22×C4—52no—
415C22×C4—58no—
416C22×C4—58no—
417C22×C4—108no—
418C22×C4—108no—
419C22×C4C22×C3×D512yes—
420C22×C4C22×C3×D512yes—
421C22×C4C22×C3×D512yes—
422C22×C4C22×C3×D511yescharacteristic
423C4⋊C4—58no—
424C4⋊C4—516no—
425C4⋊C4C22×C3×D5116yes—
426C22⋊C4—58no—
427C22⋊C4—516no—
428C22⋊C4—516no—
429C42—54no—
430C42C22×C3×D514yes—
431C3×C5C2×#53611yescharacteristic, complemented, Hall
432C22×C3—54no—
433C22×C3—108no—
434C22×C3—104no—
435C22×C3—104no—
436C22×C3—108no—
437C22×C3—208no—
438C22×C3C2×D4×D512yes—
439C22×C3#414714yes—
440C22×C3C24×D511yescharacteristic
441C3×C4—24no—
442C3×C4—24no—
443C3×C4—24no—
444C3×C4—54no—
445C3×C4—104no—
446C3×C4—104no—
447C3×C4C2×D4×D514yes—
448C2×C5C3×#53614yescomplemented
449C2×C5C23×C3×D411yescharacteristic
450C2×C5C2×C3×2−1+412yes—
451D5—24no—
452D5—44no—
453C23—54no—
454C23—52no—
455C23—54no—
456C23—104no—
457C23—104no—
458C23—108no—
459C23C23×C3×D511yescharacteristic
460Q8—28no—
461Q8—28no—
462Q8—1016no—
463Q8—108no—
464D4—1016no—
465D4—108no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—54no—
470C2×C4—52no—
471C2×C4—516no—
472C2×C4—54no—
473C2×C4—54no—
474C2×C4—1016no—
475C2×C4—1016no—
476C2×C4—104no—
477C2×C4—104no—
478C2×C4C3×D4×D514yes—
479C2×C4C23×C3×D514yes—
480C2×C4C23×C3×D514yes—
481C2×C4C23×C3×D514yes—
482C2×C4C23×C3×D512yes—
483C2×C3—104no—
484C2×C3—204no—
485C2×C3#6605814yesdirect product factor
486C2×C3C22×D4×D511yescharacteristic
487C2×C3C2×#414712yes—
488C5C2×C3×#53611yescharacteristic, complemented, Sylow
489C22—54no—
490C22—104no—
491C22—108no—
492C22—104no—
493C22—108no—
494C22—208no—
495C22C2×C3×D4×D512yes—
496C22C3×#414714yes—
497C22C24×C3×D511yescharacteristic
498C4—24no—
499C4—24no—
500C4—24no—
501C4—54no—
502C4—104no—
503C4—104no—
504C4C2×C3×D4×D514yes—
505C3C2×#6605811yescharacteristic, direct product factor, Sylow
506C2—104no—
507C2—204no—
508C2C3×#6605814yesdirect product factor
509C2C22×C3×D4×D511yescharacteristic
510C2C2×C3×#414712yes—
subgroupG/H
centerC23×C3C23×D5
commutator, cosocleC22×C5C25×C3
FrattiniC22C24×C3×D5
FittingC2×C3×C5×C4⋊Q8C2
socleC23×C3×C5C24
prime radicalC22×C3×#713C2
hypercenterC2×C3×C4⋊Q8D5
hypocenterC5C2×C3×#536
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×C3×Q8×D5C2semidirect
3. C2×C3×C5×C4⋊Q8C2semidirect
4. C2×C3×#4093C2semidirect
5. C2×C3×#4090C2semidirect
6. C3×#66058C2direct (N × H)
7. C2×C3×#4042C2semidirect
8. C2×C3×#4040C2semidirect
9. C2×C3×#4021C2semidirect
10. C2×C3×#4018C2semidirect
11. C2×#66058C3direct (N × H)
17. C2×C3×Q8×D5C22semidirect
25. C3×C5×C4⋊Q8C22semidirect
28. C3×#4093C22semidirect
29. C3×#4090C22semidirect
36. C3×#4042C22semidirect
37. C3×#4040C22semidirect
38. C3×#4021C22semidirect
39. C3×#4018C22semidirect
42. C22×Q8×D5C2×C3semidirect
43. C2×C5×C4⋊Q8C2×C3semidirect
44. C2×#4093C2×C3semidirect
45. C2×#4090C2×C3semidirect
46. #66058C2×C3direct (N × H)
47. C2×#4042C2×C3semidirect
48. C2×#4040C2×C3semidirect
49. C2×#4021C2×C3semidirect
50. C2×#4018C2×C3semidirect
92. C2×C3×C4⋊Q8D5semidirect
103. C2×Q8×D5C22×C3semidirect
111. C5×C4⋊Q8C22×C3semidirect
114. #4093C22×C3semidirect
115. #4090C22×C3semidirect
122. #4042C22×C3semidirect
123. #4040C22×C3semidirect
124. #4021C22×C3semidirect
125. #4018C22×C3semidirect
171. C3×C4⋊Q8C2×D5semidirect
226. C2×C4⋊Q8C3×D5semidirect
329. C4⋊Q8C2×C3×D5semidirect
343. C2×C3×C5#536semidirect
431. C3×C5C2×#536semidirect
448. C2×C5C3×#536semidirect
485. C2×C3#66058direct (N × H)
488. C5C2×C3×#536semidirect
505. C3C2×#66058direct (N × H)
508. C2C3×#66058direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7132
C22×C3×Q8×D52
C2×C3×C5×C4⋊Q82
C2×C3×#40932
C2×C3×#40902
C3×#660582
C2×C3×#40422
C2×C3×#40402
C2×C3×#40212
C2×C3×#40182
C2×#660583
C2×C3×#5365