Finite Group Databasebeta

← C2×C3×#65509 #14320288

Subgroups — C2×C3×#65509 #14320288

Order 1920.

3152 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C5×D4C211yesmaximal, characteristic
2C24×C3×Dic5C211yesmaximal, characteristic, complemented
3C22×C3×C22⋊Dic5C211yesmaximal, characteristic, complemented
4C22×C3×C22⋊Dic5C212yesmaximal, complemented
5C3×#65509C218yesmaximal, direct product factor
6C2×C3×#3963C212yesmaximal, complemented
7C2×#65509C311yesmaximal, characteristic, direct product factor, Hall
8C25×C3×C5C2211yescharacteristic
9C25×C3×C5C2211yescharacteristic
10C22×C3×C5×D4—24no—
11C22×C3×C5×D4—24no—
12C22×C3×C5×D4C418yescomplemented
13C22×C3×C5×D4C2214yes—
14C23×C3×C4×C5C2211yescharacteristic
15C23×C3×Dic5—24no—
16C23×C3×Dic5—28no—
17C23×C3×Dic5C2212yes—
18C23×C3×Dic5C2212yes—
19C23×C3×Dic5C2214yescomplemented
20C2×C3×C22⋊Dic5—24no—
21C2×C3×C22⋊Dic5—28no—
22C2×C3×C22⋊Dic5C2214yescomplemented
23C2×C3×C22⋊Dic5C2218yescomplemented
24C3×#3963C2218yescomplemented
25C2×C3×#386—51nomaximal, Hall
26C23×C5×D4C2×C311yescharacteristic
27C24×Dic5C2×C311yescharacteristic, complemented
28C22×C22⋊Dic5C2×C311yescharacteristic, complemented
29C22×C22⋊Dic5C2×C312yescomplemented
30#65509C2×C318yesdirect product factor
31C2×#3963C2×C312yescomplemented
32C24×C3×C5—24no—
33C24×C3×C5—24no—
34C24×C3×C5—24no—
35C24×C3×C5—22no—
36C24×C3×C5—22no—
37C24×C3×C5—24no—
38C24×C3×C5C2×C414yes—
39C24×C3×C5C2×C414yescomplemented
40C24×C3×C5D414yes—
41C24×C3×C5D414yes—
42C24×C3×C5C2312yes—
43C24×C3×C5C2312yes—
44C24×C3×C5C2311yescharacteristic
45C2×C3×C5×D4—232no—
46C2×C3×C5×D4—28no—
47C2×C3×C5×D4—28no—
48C2×C3×C5×D4C2×C4116yescomplemented
49C22×C3×C4×C5—22no—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5C2×C414yescomplemented
52C22×C3×C4×C5C2312yes—
53C22×C3×Dic5—232no—
54C22×C3×Dic5—24no—
55C22×C3×Dic5—24no—
56C22×C3×Dic5—28no—
57C22×C3×Dic5—216no—
58C22×C3×Dic5—24no—
59C22×C3×Dic5—24no—
60C22×C3×Dic5—24no—
61C22×C3×Dic5D418yescomplemented
62C22×C3×Dic5C2314yes—
63C22×C3×Dic5C2314yes—
64C3×C22⋊Dic5—28no—
65C3×C22⋊Dic5—216no—
66C23×C3×D4D511yescharacteristic
67C24×C3×C4—51no—
68C22×C3×C22⋊C4—52no—
69C22×C3×C22⋊C4—51no—
70C3×#386—58no—
71C2×C3×#95—52no—
72C25×C5C22×C311yescharacteristic
73C25×C5C22×C311yescharacteristic
74C22×C5×D4—24no—
75C22×C5×D4—24no—
76C22×C5×D4C3×C418yescomplemented
77C22×C5×D4C22×C314yes—
78C23×C4×C5C22×C311yescharacteristic
79C23×Dic5—24no—
80C23×Dic5—28no—
81C23×Dic5C22×C312yes—
82C23×Dic5C22×C312yes—
83C23×Dic5C22×C314yescomplemented
84C2×C22⋊Dic5—24no—
85C2×C22⋊Dic5—28no—
86C2×C22⋊Dic5C22×C314yescomplemented
87C2×C22⋊Dic5C22×C318yescomplemented
88#3963C22×C318yescomplemented
89C2×#386—51noSylow
90C23×C3×C5—216no—
91C23×C3×C5—28no—
92C23×C3×C5—216no—
93C23×C3×C5—28no—
94C23×C3×C5—24no—
95C23×C3×C5—24no—
96C23×C3×C5—28no—
97C23×C3×C5—216no—
98C23×C3×C5—24no—
99C23×C3×C5—22no—
100C23×C3×C5—22no—
101C23×C3×C5—24no—
102C23×C3×C5—44no—
103C23×C3×C5—48no—
104C23×C3×C5—44no—
105C23×C3×C5C22⋊C4116yescomplemented
106C23×C3×C5C22×C414yes—
107C23×C3×C5C22×C412yes—
108C23×C3×C5C22×C414yescomplemented
109C23×C3×C5C2×D414yes—
110C23×C3×C5C2×D414yes—
111C23×C3×C5C2×D412yes—
112C23×C3×C5C2×D412yes—
113C23×C3×C5C2×D414yes—
114C23×C3×C5#4114yes—
115C23×C3×C5C2411yescharacteristic
116C3×C5×D4—232no—
117C2×C3×C4×C5—28no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5C22×C414yescomplemented
121C2×C3×Dic5—232no—
122C2×C3×Dic5—216no—
123C2×C3×Dic5—24no—
124C2×C3×Dic5—24no—
125C2×C3×Dic5—24no—
126C2×C3×Dic5—24no—
127C2×C3×Dic5—24no—
128C2×C3×Dic5—48no—
129C2×C3×Dic5C2×D418yescomplemented
130C25×C3C2×D511yescharacteristic
131C25×C3C2×D511yescharacteristic
132C22×C3×D4—24no—
133C22×C3×D4—24no—
134C22×C3×D4Dic518yescomplemented
135C22×C3×D4C2×D514yes—
136C23×C3×C4—52no—
137C23×C3×C4—54no—
138C23×C3×C4—52no—
139C23×C3×C4—108no—
140C23×C3×C4—104no—
141C23×C3×C4C2×D511yescharacteristic
142C2×C3×C22⋊C4—58no—
143C2×C3×C22⋊C4—54no—
144C2×C3×C22⋊C4—108no—
145C2×C3×C22⋊C4—104no—
146C3×#95—58no—
147C24×C5—24no—
148C24×C5—24no—
149C24×C5—24no—
150C24×C5—22no—
151C24×C5—22no—
152C24×C5—24no—
153C24×C5C2×C3×C414yes—
154C24×C5C2×C3×C414yescomplemented
155C24×C5C3×D414yes—
156C24×C5C3×D414yes—
157C24×C5C23×C312yes—
158C24×C5C23×C312yes—
159C24×C5C23×C311yescharacteristic
160C2×C5×D4—232no—
161C2×C5×D4—28no—
162C2×C5×D4—28no—
163C2×C5×D4C2×C3×C4116yescomplemented
164C22×C4×C5—22no—
165C22×C4×C5—22no—
166C22×C4×C5C2×C3×C414yescomplemented
167C22×C4×C5C23×C312yes—
168C22×Dic5—232no—
169C22×Dic5—24no—
170C22×Dic5—24no—
171C22×Dic5—28no—
172C22×Dic5—216no—
173C22×Dic5—24no—
174C22×Dic5—24no—
175C22×Dic5—24no—
176C22×Dic5C3×D418yescomplemented
177C22×Dic5C23×C314yes—
178C22×Dic5C23×C314yes—
179C22⋊Dic5—28no—
180C22⋊Dic5—216no—
181C23×D4C3×D511yescharacteristic
182C24×C4—51no—
183C22×C22⋊C4—52no—
184C22×C22⋊C4—51no—
185#386—58no—
186C2×#95—52no—
187C22×C3×C5—28no—
188C22×C3×C5—24no—
189C22×C3×C5—28no—
190C22×C3×C5—24no—
191C22×C3×C5—232no—
192C22×C3×C5—24no—
193C22×C3×C5—28no—
194C22×C3×C5—42no—
195C22×C3×C5—44no—
196C22×C3×C5—42no—
197C22×C3×C5—416no—
198C22×C3×C5C2×C22⋊C414yes—
199C22×C3×C5C2×C22⋊C418yescomplemented
200C22×C3×C5C4×D418yes—
201C22×C3×C5#12014yes—
202C22×C3×C5#12118yes—
203C22×C3×C5#12314yes—
204C22×C3×C5C23×C411yescharacteristic
205C22×C3×C5C22×D412yes—
206C22×C3×C5C22×D411yescharacteristic
207C22×C3×C5C22×D411yescharacteristic
208C22×C3×C5C2×#4112yes—
209C3×C4×C5—24no—
210C3×Dic5—28no—
211C3×Dic5—44no—
212C24×C3—24no—
213C24×C3—24no—
214C24×C3—24no—
215C24×C3—24no—
216C24×C3—22no—
217C24×C3—22no—
218C24×C3C2×Dic514yes—
219C24×C3C2×Dic514yescomplemented
220C24×C3C22⋊D514yes—
221C24×C3C22⋊D514yes—
222C24×C3C22×D512yes—
223C24×C3C22×D512yes—
224C24×C3C22×D511yescharacteristic
225C2×C3×D4—232no—
226C2×C3×D4—28no—
227C2×C3×D4—28no—
228C2×C3×D4C2×Dic5116yescomplemented
229C22×C3×C4—22no—
230C22×C3×C4—22no—
231C22×C3×C4—58no—
232C22×C3×C4—54no—
233C22×C3×C4—54no—
234C22×C3×C4—1016no—
235C22×C3×C4—1032no—
236C22×C3×C4—104no—
237C22×C3×C4—108no—
238C22×C3×C4—104no—
239C22×C3×C4—104no—
240C22×C3×C4—104no—
241C22×C3×C4—104no—
242C22×C3×C4C2×Dic514yescomplemented
243C22×C3×C4C22×D512yes—
244C3×C22⋊C4—1016no—
245C3×C22⋊C4—108no—
246C23×C5—216no—
247C23×C5—28no—
248C23×C5—216no—
249C23×C5—28no—
250C23×C5—24no—
251C23×C5—24no—
252C23×C5—28no—
253C23×C5—216no—
254C23×C5—24no—
255C23×C5—22no—
256C23×C5—22no—
257C23×C5—24no—
258C23×C5—44no—
259C23×C5—48no—
260C23×C5—44no—
261C23×C5C3×C22⋊C4116yescomplemented
262C23×C5C22×C3×C414yes—
263C23×C5C22×C3×C412yes—
264C23×C5C22×C3×C414yescomplemented
265C23×C5C2×C3×D414yes—
266C23×C5C2×C3×D414yes—
267C23×C5C2×C3×D414yes—
268C23×C5C2×C3×D412yes—
269C23×C5C2×C3×D412yes—
270C23×C5C3×#4114yes—
271C23×C5C24×C311yescharacteristic
272C5×D4—232no—
273C2×C4×C5—28no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5C22×C3×C414yescomplemented
277C2×Dic5—232no—
278C2×Dic5—216no—
279C2×Dic5—24no—
280C2×Dic5—24no—
281C2×Dic5—24no—
282C2×Dic5—24no—
283C2×Dic5—24no—
284C2×Dic5—48no—
285C2×Dic5C2×C3×D418yescomplemented
286C25C2×C3×D511yescharacteristic
287C25C2×C3×D511yescharacteristic
288C22×D4—24no—
289C22×D4—24no—
290C22×D4C3×Dic518yescomplemented
291C22×D4C2×C3×D514yes—
292C23×C4—52no—
293C23×C4—52no—
294C23×C4—54no—
295C23×C4—108no—
296C23×C4—104no—
297C23×C4C2×C3×D511yescharacteristic
298C2×C22⋊C4—58no—
299C2×C22⋊C4—54no—
300C2×C22⋊C4—108no—
301C2×C22⋊C4—104no—
302#95—58no—
303C2×C3×C5—28no—
304C2×C3×C5—44no—
305C2×C3×C5#38618yescomplemented
306C2×C3×C5C22×C22⋊C411yescharacteristic
307C2×C3×C5C2×C4×D412yes—
308C2×C3×C5C2×#12011yescharacteristic
309C2×C3×C5C2×#12112yes—
310C2×C3×C5C2×#12311yescharacteristic
311C23×C3—24no—
312C23×C3—24no—
313C23×C3—216no—
314C23×C3—28no—
315C23×C3—28no—
316C23×C3—216no—
317C23×C3—28no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—216no—
323C23×C3—44no—
324C23×C3—44no—
325C23×C3—48no—
326C23×C3C22⋊Dic5116yescomplemented
327C23×C3D4×D514yes—
328C23×C3#71114yes—
329C23×C3C22×Dic514yes—
330C23×C3C22×Dic512yes—
331C23×C3C22×Dic514yescomplemented
332C23×C3C2×C22⋊D514yes—
333C23×C3C2×C22⋊D514yes—
334C23×C3C2×C22⋊D512yes—
335C23×C3C2×C22⋊D512yes—
336C23×C3C23×D511yescharacteristic
337C3×D4—232no—
338C2×C3×C4—28no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—58no—
342C2×C3×C4—1032no—
343C2×C3×C4—104no—
344C2×C3×C4—104no—
345C2×C3×C4—1016no—
346C2×C3×C4—104no—
347C2×C3×C4—104no—
348C2×C3×C4—104no—
349C2×C3×C4—208no—
350C2×C3×C4C22×Dic514yescomplemented
351C22×C5—232no—
352C22×C5—28no—
353C22×C5—24no—
354C22×C5—28no—
355C22×C5—24no—
356C22×C5—24no—
357C22×C5—28no—
358C22×C5—416no—
359C22×C5—42no—
360C22×C5—44no—
361C22×C5—42no—
362C22×C5C2×C3×C22⋊C414yes—
363C22×C5C2×C3×C22⋊C418yescomplemented
364C22×C5C3×C4×D418yes—
365C22×C5C3×#12014yes—
366C22×C5C3×#12118yes—
367C22×C5C3×#12314yes—
368C22×C5C23×C3×C411yescharacteristic
369C22×C5C22×C3×D412yes—
370C22×C5C22×C3×D411yescharacteristic
371C22×C5C22×C3×D411yescharacteristic
372C22×C5C2×C3×#4112yes—
373C4×C5—24no—
374Dic5—28no—
375Dic5—44no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—22no—
381C24—22no—
382C24C2×C3×Dic514yes—
383C24C2×C3×Dic514yescomplemented
384C24C3×C22⋊D514yes—
385C24C3×C22⋊D514yes—
386C24C22×C3×D512yes—
387C24C22×C3×D512yes—
388C24C22×C3×D511yescharacteristic
389C2×D4—232no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4C2×C3×Dic5116yescomplemented
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—58no—
396C22×C4—54no—
397C22×C4—54no—
398C22×C4—1016no—
399C22×C4—1032no—
400C22×C4—104no—
401C22×C4—108no—
402C22×C4—104no—
403C22×C4—104no—
404C22×C4—104no—
405C22×C4—104no—
406C22×C4C2×C3×Dic514yescomplemented
407C22×C4C22×C3×D512yes—
408C22⋊C4—1016no—
409C22⋊C4—108no—
410C3×C5C2×#38611yescharacteristic, complemented, Hall
411C22×C3—28no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—28no—
415C22×C3—232no—
416C22×C3—28no—
417C22×C3—24no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—44no—
421C22×C3—416no—
422C22×C3D4×Dic518yes—
423C22×C3#408114yes—
424C22×C3#408314yes—
425C22×C3#408518yes—
426C22×C3C2×C22⋊Dic514yes—
427C22×C3C2×C22⋊Dic518yescomplemented
428C22×C3C2×D4×D512yes—
429C22×C3C2×#71112yes—
430C22×C3C23×Dic511yescharacteristic
431C22×C3C22×C22⋊D511yescharacteristic
432C22×C3C22×C22⋊D511yescharacteristic
433C3×C4—24no—
434C3×C4—108no—
435C3×C4—204no—
436C2×C5—28no—
437C2×C5—44no—
438C2×C5C3×#38618yescomplemented
439C2×C5C22×C3×C22⋊C411yescharacteristic
440C2×C5C2×C3×C4×D412yes—
441C2×C5C2×C3×#12011yescharacteristic
442C2×C5C2×C3×#12112yes—
443C2×C5C2×C3×#12311yescharacteristic
444C23—216no—
445C23—28no—
446C23—28no—
447C23—216no—
448C23—24no—
449C23—24no—
450C23—216no—
451C23—28no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—44no—
457C23—44no—
458C23—48no—
459C23C3×C22⋊Dic5116yescomplemented
460C23C3×D4×D514yes—
461C23C3×#71114yes—
462C23C22×C3×Dic514yes—
463C23C22×C3×Dic512yes—
464C23C22×C3×Dic514yescomplemented
465C23C2×C3×C22⋊D514yes—
466C23C2×C3×C22⋊D514yes—
467C23C2×C3×C22⋊D512yes—
468C23C2×C3×C22⋊D512yes—
469C23C23×C3×D511yescharacteristic
470D4—232no—
471C2×C4—28no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—58no—
475C2×C4—1032no—
476C2×C4—1016no—
477C2×C4—104no—
478C2×C4—104no—
479C2×C4—104no—
480C2×C4—104no—
481C2×C4—104no—
482C2×C4—208no—
483C2×C4C22×C3×Dic514yescomplemented
484C2×C3—28no—
485C2×C3—44no—
486C2×C3#6550918yesdirect product factor
487C2×C3C2×D4×Dic512yes—
488C2×C3C2×#408111yescharacteristic
489C2×C3C2×#408311yescharacteristic
490C2×C3C2×#408512yes—
491C2×C3C22×C22⋊Dic511yescharacteristic
492C5C2×C3×#38611yescharacteristic, complemented, Sylow
493C22—28no—
494C22—24no—
495C22—24no—
496C22—28no—
497C22—232no—
498C22—28no—
499C22—24no—
500C22—42no—
501C22—42no—
502C22—44no—
503C22—416no—
504C22C3×D4×Dic518yes—
505C22C3×#408114yes—
506C22C3×#408314yes—
507C22C3×#408518yes—
508C22C2×C3×C22⋊Dic514yes—
509C22C2×C3×C22⋊Dic518yescomplemented
510C22C2×C3×D4×D512yes—
511C22C2×C3×#71112yes—
512C22C23×C3×Dic511yescharacteristic
513C22C22×C3×C22⋊D511yescharacteristic
514C22C22×C3×C22⋊D511yescharacteristic
515C4—24no—
516C4—108no—
517C4—204no—
518C3C2×#6550911yescharacteristic, direct product factor, Sylow
519C2—28no—
520C2—44no—
521C2C3×#6550918yesdirect product factor
522C2C2×C3×D4×Dic512yes—
523C2C2×C3×#408111yescharacteristic
524C2C2×C3×#408311yescharacteristic
525C2C2×C3×#408512yes—
526C2C22×C3×C22⋊Dic511yescharacteristic
subgroupG/H
centerC24×C3C22×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
Fitting, prime radicalC23×C3×C5×D4C2
socleC24×C3×C5C23
cosocleC23×C5C24×C3
hypercenterC23×C3×D4D5
hypocenterC5C2×C3×#386
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C24×C3 ◁ C23×C3×D4

Complements

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

NH ≅ G/Ntype
2. C24×C3×Dic5C2semidirect
3. C22×C3×C22⋊Dic5C2semidirect
4. C22×C3×C22⋊Dic5C2semidirect
5. C3×#65509C2direct (N × H)
6. C2×C3×#3963C2semidirect
7. C2×#65509C3direct (N × H)
12. C22×C3×C5×D4C4semidirect
19. C23×C3×Dic5C22semidirect
22. C2×C3×C22⋊Dic5C22semidirect
23. C2×C3×C22⋊Dic5C22semidirect
24. C3×#3963C22semidirect
27. C24×Dic5C2×C3semidirect
28. C22×C22⋊Dic5C2×C3semidirect
29. C22×C22⋊Dic5C2×C3semidirect
30. #65509C2×C3direct (N × H)
31. C2×#3963C2×C3semidirect
39. C24×C3×C5C2×C4semidirect
48. C2×C3×C5×D4C2×C4semidirect
51. C22×C3×C4×C5C2×C4semidirect
61. C22×C3×Dic5D4semidirect
76. C22×C5×D4C3×C4semidirect
83. C23×Dic5C22×C3semidirect
86. C2×C22⋊Dic5C22×C3semidirect
87. C2×C22⋊Dic5C22×C3semidirect
88. #3963C22×C3semidirect
105. C23×C3×C5C22⋊C4semidirect
108. C23×C3×C5C22×C4semidirect
120. C2×C3×C4×C5C22×C4semidirect
129. C2×C3×Dic5C2×D4semidirect
134. C22×C3×D4Dic5semidirect
154. C24×C5C2×C3×C4semidirect
163. C2×C5×D4C2×C3×C4semidirect
166. C22×C4×C5C2×C3×C4semidirect
176. C22×Dic5C3×D4semidirect
199. C22×C3×C5C2×C22⋊C4semidirect
219. C24×C3C2×Dic5semidirect
228. C2×C3×D4C2×Dic5semidirect
242. C22×C3×C4C2×Dic5semidirect
261. C23×C5C3×C22⋊C4semidirect
264. C23×C5C22×C3×C4semidirect
276. C2×C4×C5C22×C3×C4semidirect
285. C2×Dic5C2×C3×D4semidirect
290. C22×D4C3×Dic5semidirect
305. C2×C3×C5#386semidirect
326. C23×C3C22⋊Dic5semidirect
331. C23×C3C22×Dic5semidirect
350. C2×C3×C4C22×Dic5semidirect
363. C22×C5C2×C3×C22⋊C4semidirect
383. C24C2×C3×Dic5semidirect
392. C2×D4C2×C3×Dic5semidirect
406. C22×C4C2×C3×Dic5semidirect
410. C3×C5C2×#386semidirect
427. C22×C3C2×C22⋊Dic5semidirect
438. C2×C5C3×#386semidirect
459. C23C3×C22⋊Dic5semidirect
464. C23C22×C3×Dic5semidirect
483. C2×C4C22×C3×Dic5semidirect
486. C2×C3#65509direct (N × H)
492. C5C2×C3×#386semidirect
509. C22C2×C3×C22⋊Dic5semidirect
518. C3C2×#65509direct (N × H)
521. C2C3×#65509direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C5×D42
C24×C3×Dic52
C22×C3×C22⋊Dic52
C3×#655092
C2×C3×#39632
C2×#655093
C2×C3×#3865