Finite Group Databasebeta

← C2×C3×#64958 #14320439

Subgroups — C2×C3×#64958 #14320439

Order 1920.

1856 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#685C211yesmaximal, characteristic
2C22×C3×#685C211yesmaximal, characteristic
3C22×C3×#685C211yesmaximal, characteristic
4C22×C3×#683C211yesmaximal, characteristic, complemented
5C22×C3×C4×Dic5C211yesmaximal, characteristic, complemented
6C2×C3×C5×#95C211yesmaximal, characteristic, complemented
7C2×C3×#3963C211yesmaximal, characteristic, complemented
8C3×#64958C218yesmaximal, direct product factor
9C2×#64958C311yesmaximal, characteristic, direct product factor, Hall
10C24×C3×D5C2211yescharacteristic
11C23×C3×C4×C5C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C23×C3×Dic5C2211yescharacteristic
15C23×C3×Dic5C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C2×C3×#685—24no—
18C2×C3×#685—24no—
19C2×C3×#685C418yescomplemented
20C2×C3×#685C2214yes—
21C2×C3×#685C2214yes—
22C2×C3×#685C2214yes—
23C2×C3×#683—24no—
24C2×C3×#683C2214yescomplemented
25C2×C3×C4×Dic5—24no—
26C2×C3×C4×Dic5C2214yescomplemented
27C3×C5×#95C2214yescomplemented
28C3×#3963C2214yescomplemented
29C2×C3×#388—51nomaximal, Hall
30C22×#685C2×C311yescharacteristic
31C22×#685C2×C311yescharacteristic
32C22×#685C2×C311yescharacteristic
33C22×#683C2×C311yescharacteristic, complemented
34C22×C4×Dic5C2×C311yescharacteristic, complemented
35C2×C5×#95C2×C311yescharacteristic, complemented
36C2×#3963C2×C311yescharacteristic, complemented
37#64958C2×C318yesdirect product factor
38C24×C3×C5C2311yescharacteristic
39C23×C3×D5—22no—
40C23×C3×D5—22no—
41C23×C3×D5—22no—
42C23×C3×D5—22no—
43C23×C3×D5—22no—
44C23×C3×D5—22no—
45C23×C3×D5C2×C414yes—
46C23×C3×D5C2312yes—
47C22×C3×C4×C5—22no—
48C22×C3×C4×C5—22no—
49C22×C3×C4×C5—22no—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—22no—
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—22no—
55C22×C3×C4×C5C2×C414yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2312yes—
59C22×C3×Dic5—22no—
60C22×C3×Dic5—22no—
61C22×C3×Dic5—22no—
62C22×C3×Dic5—22no—
63C22×C3×Dic5—22no—
64C22×C3×Dic5—22no—
65C22×C3×Dic5C2×C414yes—
66C22×C3×Dic5D414yes—
67C22×C3×Dic5D414yes—
68C22×C3×Dic5C2312yes—
69C22×C3×Dic5C2312yes—
70C22×C3×Dic5C2312yes—
71C3×#685—28no—
72C3×#685—28no—
73C3×#685C2×C4116yescomplemented
74C3×#683—28no—
75C3×C4×Dic5—28no—
76C22×C3×C4⋊C4—51no—
77C22×C3×C22⋊C4—51no—
78C22×C3×C22⋊C4—51no—
79C22×C3×C22⋊C4—51no—
80C22×C3×C42—51no—
81C3×#388—58no—
82C2×C3×#95—51no—
83C2×C3×#95D511yescharacteristic, complemented
84C24×D5C22×C311yescharacteristic
85C23×C4×C5C22×C311yescharacteristic
86C23×C4×C5C22×C311yescharacteristic
87C23×C4×C5C22×C311yescharacteristic
88C23×Dic5C22×C311yescharacteristic
89C23×Dic5C22×C311yescharacteristic
90C23×Dic5C22×C311yescharacteristic
91C2×#685—24no—
92C2×#685—24no—
93C2×#685C3×C418yescomplemented
94C2×#685C22×C314yes—
95C2×#685C22×C314yes—
96C2×#685C22×C314yes—
97C2×#683—24no—
98C2×#683C22×C314yescomplemented
99C2×C4×Dic5—24no—
100C2×C4×Dic5C22×C314yescomplemented
101C5×#95C22×C314yescomplemented
102#3963C22×C314yescomplemented
103C2×#388—51noSylow
104C23×C3×C5C22×C412yes—
105C23×C3×C5C2×D412yes—
106C23×C3×C5C2×D412yes—
107C23×C3×C5#4112yes—
108C23×C3×C5#4112yes—
109C23×C3×C5#4112yes—
110C23×C3×C5#4112yes—
111C23×C3×C5C2411yescharacteristic
112C22×C3×D5—28no—
113C22×C3×D5—28no—
114C22×C3×D5—28no—
115C22×C3×D5—22no—
116C22×C3×D5—22no—
117C22×C3×D5—22no—
118C22×C3×D5—22no—
119C22×C3×D5—22no—
120C22×C3×D5—22no—
121C22×C3×D5—44no—
122C22×C3×D5—44no—
123C22×C3×D5—44no—
124C22×C3×D5—44no—
125C22×C3×D5C22×C414yes—
126C2×C3×C4×C5—28no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—44no—
136C2×C3×C4×C5—44no—
137C2×C3×C4×C5C22×C414yes—
138C2×C3×Dic5—28no—
139C2×C3×Dic5—28no—
140C2×C3×Dic5—28no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—22no—
145C2×C3×Dic5—22no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5C22×C414yes—
148C2×C3×Dic5C2×D414yes—
149C2×C3×Dic5C2×D414yes—
150C25×C3—51no—
151C23×C3×C4—51no—
152C23×C3×C4—51no—
153C23×C3×C4—51no—
154C23×C3×C4C2×D511yescharacteristic
155C23×C3×C4C2×D511yescharacteristic
156C23×C3×C4C2×D511yescharacteristic
157C2×C3×C4⋊C4—54no—
158C2×C3×C4⋊C4—104no—
159C2×C3×C22⋊C4—58no—
160C2×C3×C22⋊C4—54no—
161C2×C3×C22⋊C4—54no—
162C2×C3×C22⋊C4—54no—
163C2×C3×C22⋊C4—104no—
164C2×C3×C22⋊C4—104no—
165C2×C3×C42—54no—
166C2×C3×C42—104no—
167C3×#95—54no—
168C3×#95C2×D514yescomplemented
169C24×C5C23×C311yescharacteristic
170C23×D5—22no—
171C23×D5—22no—
172C23×D5—22no—
173C23×D5—22no—
174C23×D5—22no—
175C23×D5—22no—
176C23×D5C2×C3×C414yes—
177C23×D5C23×C312yes—
178C22×C4×C5—22no—
179C22×C4×C5—22no—
180C22×C4×C5—22no—
181C22×C4×C5—22no—
182C22×C4×C5—22no—
183C22×C4×C5—22no—
184C22×C4×C5—22no—
185C22×C4×C5—22no—
186C22×C4×C5C2×C3×C414yes—
187C22×C4×C5C23×C312yes—
188C22×C4×C5C23×C312yes—
189C22×C4×C5C23×C312yes—
190C22×Dic5—22no—
191C22×Dic5—22no—
192C22×Dic5—22no—
193C22×Dic5—22no—
194C22×Dic5—22no—
195C22×Dic5—22no—
196C22×Dic5C2×C3×C414yes—
197C22×Dic5C3×D414yes—
198C22×Dic5C3×D414yes—
199C22×Dic5C23×C312yes—
200C22×Dic5C23×C312yes—
201C22×Dic5C23×C312yes—
202#685—28no—
203#685—28no—
204#685C2×C3×C4116yescomplemented
205#683—28no—
206C4×Dic5—28no—
207C22×C4⋊C4—51no—
208C22×C22⋊C4—51no—
209C22×C22⋊C4—51no—
210C22×C22⋊C4—51no—
211C22×C42—51no—
212#388—58no—
213C2×#95—51no—
214C2×#95C3×D511yescharacteristic, complemented
215C22×C3×C5#11714yes—
216C22×C3×C5C4×D414yes—
217C22×C3×C5C4×D414yes—
218C22×C3×C5#12114yes—
219C22×C3×C5#12314yes—
220C22×C3×C5#12414yes—
221C22×C3×C5#12614yes—
222C22×C3×C5C23×C411yescharacteristic
223C22×C3×C5C22×D411yescharacteristic
224C22×C3×C5C22×D411yescharacteristic
225C22×C3×C5C2×#4111yescharacteristic
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C2×C3×D5—24no—
230C2×C3×D5—24no—
231C2×C3×D5—24no—
232C2×C3×D5—42no—
233C2×C3×D5—42no—
234C2×C3×D5—42no—
235C2×C3×D5—42no—
236C2×C3×D5—416no—
237C3×C4×C5—24no—
238C3×C4×C5—42no—
239C3×C4×C5—42no—
240C3×Dic5—24no—
241C3×Dic5—24no—
242C3×Dic5—24no—
243C24×C3—54no—
244C24×C3—52no—
245C24×C3—102no—
246C24×C3—102no—
247C24×C3—102no—
248C24×C3—102no—
249C24×C3—102no—
250C24×C3—102no—
251C24×C3C22×D511yescharacteristic
252C22×C3×C4—22no—
253C22×C3×C4—22no—
254C22×C3×C4—22no—
255C22×C3×C4—22no—
256C22×C3×C4—22no—
257C22×C3×C4—22no—
258C22×C3×C4—22no—
259C22×C3×C4—22no—
260C22×C3×C4—54no—
261C22×C3×C4—54no—
262C22×C3×C4—54no—
263C22×C3×C4—52no—
264C22×C3×C4—52no—
265C22×C3×C4—52no—
266C22×C3×C4—102no—
267C22×C3×C4—102no—
268C22×C3×C4—102no—
269C22×C3×C4—102no—
270C22×C3×C4—102no—
271C22×C3×C4—102no—
272C22×C3×C4C4×D514yes—
273C22×C3×C4C22×D512yes—
274C22×C3×C4C22×D512yes—
275C22×C3×C4C22×D512yes—
276C3×C4⋊C4—108no—
277C3×C22⋊C4—516no—
278C3×C22⋊C4—108no—
279C3×C22⋊C4—108no—
280C3×C42—108no—
281C23×C5C22×C3×C412yes—
282C23×C5C2×C3×D412yes—
283C23×C5C2×C3×D412yes—
284C23×C5C3×#4112yes—
285C23×C5C3×#4112yes—
286C23×C5C3×#4112yes—
287C23×C5C3×#4112yes—
288C23×C5C24×C311yescharacteristic
289C22×D5—28no—
290C22×D5—28no—
291C22×D5—28no—
292C22×D5—22no—
293C22×D5—22no—
294C22×D5—22no—
295C22×D5—22no—
296C22×D5—22no—
297C22×D5—22no—
298C22×D5—44no—
299C22×D5—44no—
300C22×D5—44no—
301C22×D5—44no—
302C22×D5C22×C3×C414yes—
303C2×C4×C5—28no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—22no—
312C2×C4×C5—44no—
313C2×C4×C5—44no—
314C2×C4×C5C22×C3×C414yes—
315C2×Dic5—28no—
316C2×Dic5—28no—
317C2×Dic5—28no—
318C2×Dic5—22no—
319C2×Dic5—22no—
320C2×Dic5—22no—
321C2×Dic5—22no—
322C2×Dic5—22no—
323C2×Dic5—22no—
324C2×Dic5C22×C3×C414yes—
325C2×Dic5C2×C3×D414yes—
326C2×Dic5C2×C3×D414yes—
327C25—51no—
328C23×C4—51no—
329C23×C4—51no—
330C23×C4—51no—
331C23×C4C2×C3×D511yescharacteristic
332C23×C4C2×C3×D511yescharacteristic
333C23×C4C2×C3×D511yescharacteristic
334C2×C4⋊C4—54no—
335C2×C4⋊C4—104no—
336C2×C22⋊C4—58no—
337C2×C22⋊C4—54no—
338C2×C22⋊C4—54no—
339C2×C22⋊C4—54no—
340C2×C22⋊C4—104no—
341C2×C22⋊C4—104no—
342C2×C42—54no—
343C2×C42—104no—
344#95—54no—
345#95C2×C3×D514yescomplemented
346C2×C3×C5#38818yescomplemented
347C2×C3×C5C2×#11711yescharacteristic
348C2×C3×C5C2×C4×D411yescharacteristic
349C2×C3×C5C2×C4×D411yescharacteristic
350C2×C3×C5C2×#12111yescharacteristic
351C2×C3×C5C2×#12311yescharacteristic
352C2×C3×C5C2×#12411yescharacteristic
353C2×C3×C5C2×#12611yescharacteristic
354C3×D5—44no—
355C23×C3—54no—
356C23×C3—102no—
357C23×C3—102no—
358C23×C3—102no—
359C23×C3—102no—
360C23×C3—102no—
361C23×C3—102no—
362C23×C3—108no—
363C23×C3—108no—
364C23×C3—108no—
365C23×C3—204no—
366C23×C3—204no—
367C23×C3—204no—
368C23×C3—204no—
369C23×C3C2×C4×D512yes—
370C23×C3#70912yes—
371C23×C3#70912yes—
372C23×C3D4×D512yes—
373C23×C3D4×D512yes—
374C23×C3#71112yes—
375C23×C3#71312yes—
376C23×C3C23×D511yescharacteristic
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—28no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—44no—
387C2×C3×C4—44no—
388C2×C3×C4—54no—
389C2×C3×C4—54no—
390C2×C3×C4—54no—
391C2×C3×C4—102no—
392C2×C3×C4—102no—
393C2×C3×C4—102no—
394C2×C3×C4—102no—
395C2×C3×C4—102no—
396C2×C3×C4—102no—
397C2×C3×C4—108no—
398C2×C3×C4—108no—
399C2×C3×C4—108no—
400C2×C3×C4C2×C4×D514yes—
401C22×C5C3×#11714yes—
402C22×C5C3×C4×D414yes—
403C22×C5C3×C4×D414yes—
404C22×C5C3×#12114yes—
405C22×C5C3×#12314yes—
406C22×C5C3×#12414yes—
407C22×C5C3×#12614yes—
408C22×C5C23×C3×C411yescharacteristic
409C22×C5C22×C3×D411yescharacteristic
410C22×C5C22×C3×D411yescharacteristic
411C22×C5C2×C3×#4111yescharacteristic
412C22×C5C2×C3×#4111yescharacteristic
413C22×C5C2×C3×#4111yescharacteristic
414C22×C5C2×C3×#4111yescharacteristic
415C2×D5—24no—
416C2×D5—24no—
417C2×D5—24no—
418C2×D5—416no—
419C2×D5—42no—
420C2×D5—42no—
421C2×D5—42no—
422C2×D5—42no—
423C4×C5—24no—
424C4×C5—42no—
425C4×C5—42no—
426Dic5—24no—
427Dic5—24no—
428Dic5—24no—
429C24—54no—
430C24—52no—
431C24—102no—
432C24—102no—
433C24—102no—
434C24—102no—
435C24—102no—
436C24—102no—
437C24C22×C3×D511yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—54no—
447C22×C4—54no—
448C22×C4—54no—
449C22×C4—52no—
450C22×C4—52no—
451C22×C4—52no—
452C22×C4—102no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4C3×C4×D514yes—
459C22×C4C22×C3×D512yes—
460C22×C4C22×C3×D512yes—
461C22×C4C22×C3×D512yes—
462C4⋊C4—108no—
463C22⋊C4—516no—
464C22⋊C4—108no—
465C22⋊C4—108no—
466C42—108no—
467C3×C5C2×#38811yescharacteristic, complemented, Hall
468C22×C3—104no—
469C22×C3—104no—
470C22×C3—104no—
471C22×C3—202no—
472C22×C3—202no—
473C22×C3—202no—
474C22×C3—202no—
475C22×C3—2016no—
476C22×C3#401814yes—
477C22×C3#402714yes—
478C22×C3#403014yes—
479C22×C3#403114yes—
480C22×C3#403914yes—
481C22×C3#404014yes—
482C22×C3#404414yes—
483C22×C3C22×C4×D511yescharacteristic
484C22×C3C2×#70911yescharacteristic
485C22×C3C2×#70911yescharacteristic
486C22×C3C2×D4×D511yescharacteristic
487C22×C3C2×D4×D511yescharacteristic
488C22×C3C2×#71111yescharacteristic
489C22×C3C2×#71311yescharacteristic
490C3×C4—24no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—104no—
494C3×C4—104no—
495C3×C4—104no—
496C2×C5C3×#38818yescomplemented
497C2×C5C2×C3×#11711yescharacteristic
498C2×C5C2×C3×C4×D411yescharacteristic
499C2×C5C2×C3×C4×D411yescharacteristic
500C2×C5C2×C3×#12111yescharacteristic
501C2×C5C2×C3×#12311yescharacteristic
502C2×C5C2×C3×#12411yescharacteristic
503C2×C5C2×C3×#12611yescharacteristic
504D5—44no—
505C23—54no—
506C23—102no—
507C23—102no—
508C23—102no—
509C23—102no—
510C23—102no—
511C23—102no—
512C23—108no—
513C23—108no—
514C23—108no—
515C23—204no—
516C23—204no—
517C23—204no—
518C23—204no—
519C23C2×C3×C4×D512yes—
520C23C3×#70912yes—
521C23C3×#70912yes—
522C23C3×D4×D512yes—
523C23C3×D4×D512yes—
524C23C3×#71112yes—
525C23C3×#71312yes—
526C23C23×C3×D511yescharacteristic
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—54no—
539C2×C4—54no—
540C2×C4—54no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—102no—
547C2×C4—108no—
548C2×C4—108no—
549C2×C4—108no—
550C2×C4C2×C3×C4×D514yes—
551C2×C3—204no—
552C2×C3#6495818yesdirect product factor
553C2×C3C2×#401811yescharacteristic
554C2×C3C2×#402711yescharacteristic
555C2×C3C2×#403011yescharacteristic
556C2×C3C2×#403111yescharacteristic
557C2×C3C2×#403911yescharacteristic
558C2×C3C2×#404011yescharacteristic
559C2×C3C2×#404411yescharacteristic
560C5C2×C3×#38811yescharacteristic, complemented, Sylow
561C22—104no—
562C22—104no—
563C22—104no—
564C22—202no—
565C22—2016no—
566C22—202no—
567C22—202no—
568C22—202no—
569C22C3×#401814yes—
570C22C3×#402714yes—
571C22C3×#403014yes—
572C22C3×#403114yes—
573C22C3×#403914yes—
574C22C3×#404014yes—
575C22C3×#404414yes—
576C22C22×C3×C4×D511yescharacteristic
577C22C2×C3×#70911yescharacteristic
578C22C2×C3×#70911yescharacteristic
579C22C2×C3×D4×D511yescharacteristic
580C22C2×C3×D4×D511yescharacteristic
581C22C2×C3×#71111yescharacteristic
582C22C2×C3×#71311yescharacteristic
583C4—24no—
584C4—42no—
585C4—42no—
586C4—104no—
587C4—104no—
588C4—104no—
589C3C2×#6495811yescharacteristic, direct product factor, Sylow
590C2—204no—
591C2C3×#6495818yesdirect product factor
592C2C2×C3×#401811yescharacteristic
593C2C2×C3×#402711yescharacteristic
594C2C2×C3×#403011yescharacteristic
595C2C2×C3×#403111yescharacteristic
596C2C2×C3×#403911yescharacteristic
597C2C2×C3×#404011yescharacteristic
598C2C2×C3×#404411yescharacteristic
subgroupG/H
centerC24×C3C22×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC2×C3×C5×#95C2
socleC24×C3×C5C23
prime radicalC24×C3×D5C22
cosocleC23×C5C24×C3
hypercenterC2×C3×#95D5
hypocenterC5C2×C3×#388
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C24×C3 ◁ C2×C3×#95

Complements

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

NH ≅ G/Ntype
4. C22×C3×#683C2semidirect
5. C22×C3×C4×Dic5C2semidirect
6. C2×C3×C5×#95C2semidirect
7. C2×C3×#3963C2semidirect
8. C3×#64958C2direct (N × H)
9. C2×#64958C3direct (N × H)
19. C2×C3×#685C4semidirect
24. C2×C3×#683C22semidirect
26. C2×C3×C4×Dic5C22semidirect
27. C3×C5×#95C22semidirect
28. C3×#3963C22semidirect
33. C22×#683C2×C3semidirect
34. C22×C4×Dic5C2×C3semidirect
35. C2×C5×#95C2×C3semidirect
36. C2×#3963C2×C3semidirect
37. #64958C2×C3direct (N × H)
73. C3×#685C2×C4semidirect
83. C2×C3×#95D5semidirect
93. C2×#685C3×C4semidirect
98. C2×#683C22×C3semidirect
100. C2×C4×Dic5C22×C3semidirect
101. C5×#95C22×C3semidirect
102. #3963C22×C3semidirect
168. C3×#95C2×D5semidirect
204. #685C2×C3×C4semidirect
214. C2×#95C3×D5semidirect
345. #95C2×C3×D5semidirect
346. C2×C3×C5#388semidirect
467. C3×C5C2×#388semidirect
496. C2×C5C3×#388semidirect
552. C2×C3#64958direct (N × H)
560. C5C2×C3×#388semidirect
589. C3C2×#64958direct (N × H)
591. C2C3×#64958direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6852
C22×C3×#6832
C22×C3×C4×Dic52
C2×C3×C5×#952
C2×C3×#39632
C3×#649582
C2×#649583
C2×C3×#3885