Finite Group Databasebeta

← C2×C3×#65342 #14425628

Subgroups — C2×C3×#65342 #14425628

Order 1920.

1588 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4144C211yesmaximal, characteristic
2C2×C3×C5×#123C211yesmaximal, characteristic, complemented
3C2×C3×#4075C212yesmaximal, complemented
4C2×C3×#3966C211yesmaximal, characteristic, complemented
5C3×#65342C218yesmaximal, direct product factor
6C2×C3×#3938C212yesmaximal, complemented
7C2×#65342C311yesmaximal, characteristic, direct product factor, Hall
8C22×C3×C5×D4C2211yescharacteristic
9C23×C3×C4×C5C2211yescharacteristic, complemented
10C22×C3×C22⋊D5—21no—
11C22×C3×C22⋊D5C2212yes—
12C3×#4144—22no—
13C3×#4144—24no—
14C3×#4144C2214yes—
15C2×C3×#711—22no—
16C2×C3×D4×D5—22no—
17C2×C3×#709—21no—
18C3×C5×#123—22no—
19C3×C5×#123C2214yescomplemented
20C2×C3×C5×C4⋊C4—21no—
21C2×C3×C5×C22⋊C4—21no—
22C2×C3×C5×C22⋊C4C2211yescharacteristic, complemented
23C2×C3×C22⋊Dic5C2212yescomplemented
24C3×#4075—24no—
25C3×#4075C2218yescomplemented
26C2×C3×#685—22no—
27C2×C3×#683—22no—
28C3×#3966C2214yescomplemented
29C3×#3938C2218yescomplemented
30C2×C3×#458—51nomaximal, Hall
31C2×#4144C2×C311yescharacteristic
32C2×C5×#123C2×C311yescharacteristic, complemented
33C2×#4075C2×C312yescomplemented
34C2×#3966C2×C311yescharacteristic, complemented
35#65342C2×C318yesdirect product factor
36C2×#3938C2×C312yescomplemented
37C24×C3×C5D411yescharacteristic
38C24×C3×C5C2311yescharacteristic
39C23×C3×D5—21no—
40C23×C3×D5D412yes—
41C2×C3×C5×D4—22no—
42C2×C3×C5×D4—22no—
43C2×C3×C5×D4D412yes—
44C2×C3×C5×D4C2312yes—
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5—22no—
49C22×C3×C4×C5—22no—
50C22×C3×C4×C5D411yescharacteristic
51C22×C3×C4×C5C2312yescomplemented
52C2×C3×C22⋊D5—22no—
53C2×C3×C22⋊D5—24no—
54C2×C3×C22⋊D5—22no—
55C2×C3×C22⋊D5—22no—
56C2×C3×C22⋊D5—22no—
57C2×C3×C22⋊D5—22no—
58C2×C3×C22⋊D5—24no—
59C2×C3×C22⋊D5D414yes—
60C2×C3×C22⋊D5C2314yes—
61C22×C3×Dic5—21no—
62C22×C3×Dic5—22no—
63C22×C3×Dic5D412yes—
64C3×#711—28no—
65C3×#711—28no—
66C3×D4×D5—28no—
67C3×D4×D5—28no—
68C3×#709—24no—
69C3×#709—24no—
70C2×C3×D20—21no—
71C2×C3×C4×D5—22no—
72C2×C3×Dic10—21no—
73C3×C5×C4⋊C4—24no—
74C3×C5×C22⋊C4—24no—
75C3×C5×C22⋊C4—21no—
76C3×C5×C22⋊C4C2312yescomplemented
77C3×C22⋊Dic5—22no—
78C3×C22⋊Dic5C2314yescomplemented
79C3×#685—28no—
80C3×#683—28no—
81C2×C3×H(5,𝔽2)—51no—
82C2×C3×#123—52no—
83C2×C3×#123D511yescharacteristic, complemented
84C3×#458—58no—
85C2×C3×#99—51no—
86C2×C3×#99—52no—
87C22×C5×D4C22×C311yescharacteristic
88C23×C4×C5C22×C311yescharacteristic, complemented
89C22×C22⋊D5—21no—
90C22×C22⋊D5C22×C312yes—
91#4144—22no—
92#4144—24no—
93#4144C22×C314yes—
94C2×#711—22no—
95C2×D4×D5—22no—
96C2×#709—21no—
97C5×#123—22no—
98C5×#123C22×C314yescomplemented
99C2×C5×C4⋊C4—21no—
100C2×C5×C22⋊C4—21no—
101C2×C5×C22⋊C4C22×C311yescharacteristic, complemented
102C2×C22⋊Dic5C22×C312yescomplemented
103#4075—24no—
104#4075C22×C318yescomplemented
105C2×#685—22no—
106C2×#683—22no—
107#3966C22×C314yescomplemented
108#3938C22×C318yescomplemented
109C2×#458—51noSylow
110C23×C3×C5—21no—
111C23×C3×C5—21no—
112C23×C3×C5—42no—
113C23×C3×C5—42no—
114C23×C3×C5C2×D412yes—
115C23×C3×C5C2×D411yescharacteristic
116C23×C3×C5C2×D412yes—
117C23×C3×C5C2×D412yes—
118C23×C3×C5C2×D411yescharacteristic
119C23×C3×C5C2411yescharacteristic
120C22×C3×D5—22no—
121C22×C3×D5—24no—
122C22×C3×D5—22no—
123C22×C3×D5—44no—
124C22×C3×D5—82no—
125C22×C3×D5C2×D414yes—
126C3×C5×D4—24no—
127C3×C5×D4—24no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—24no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—42no—
134C2×C3×C4×C5—42no—
135C2×C3×C4×C5—41no—
136C2×C3×C4×C5—41no—
137C2×C3×C4×C5C2×D412yes—
138C3×C22⋊D5—28no—
139C3×C22⋊D5—24no—
140C3×C22⋊D5—24no—
141C3×C22⋊D5—24no—
142C3×C22⋊D5—24no—
143C2×C3×Dic5—24no—
144C2×C3×Dic5—22no—
145C2×C3×Dic5—22no—
146C2×C3×Dic5—24no—
147C2×C3×Dic5—44no—
148C2×C3×Dic5C2×D414yes—
149C3×D20—24no—
150C3×C4×D5—28no—
151C3×Dic10—24no—
152C3×H(5,𝔽2)—54no—
153C3×H(5,𝔽2)—104no—
154C3×H(5,𝔽2)—102no—
155C2×C3×#41—102no—
156C2×C3×#41—101no—
157C22×C3×D4—52no—
158C22×C3×D4—102no—
159C22×C3×D4—101no—
160C22×C3×D4C2×D511yescharacteristic
161C23×C3×C4C2×D511yescharacteristic, complemented
162C3×#123—22no—
163C3×#123—58no—
164C3×#123—104no—
165C3×#123C2×D514yescomplemented
166C2×C3×C4⋊C4—21no—
167C2×C3×C4⋊C4—102no—
168C2×C3×C22⋊C4—21no—
169C2×C3×C22⋊C4—52no—
170C2×C3×C22⋊C4—102no—
171C2×C3×C22⋊C4C2×D511yescharacteristic, complemented
172C3×#99—54no—
173C3×#99—58no—
174C24×C5C3×D411yescharacteristic
175C24×C5C23×C311yescharacteristic
176C23×D5—21no—
177C23×D5C3×D412yes—
178C2×C5×D4—22no—
179C2×C5×D4—22no—
180C2×C5×D4C3×D412yes—
181C2×C5×D4C23×C312yes—
182C22×C4×C5—21no—
183C22×C4×C5—21no—
184C22×C4×C5—21no—
185C22×C4×C5—22no—
186C22×C4×C5—22no—
187C22×C4×C5C3×D411yescharacteristic
188C22×C4×C5C23×C312yescomplemented
189C2×C22⋊D5—22no—
190C2×C22⋊D5—24no—
191C2×C22⋊D5—22no—
192C2×C22⋊D5—22no—
193C2×C22⋊D5—22no—
194C2×C22⋊D5—22no—
195C2×C22⋊D5—24no—
196C2×C22⋊D5C3×D414yes—
197C2×C22⋊D5C23×C314yes—
198C22×Dic5—21no—
199C22×Dic5—22no—
200C22×Dic5C3×D412yes—
201#711—28no—
202#711—28no—
203D4×D5—28no—
204D4×D5—28no—
205#709—24no—
206#709—24no—
207C2×D20—21no—
208C2×C4×D5—22no—
209C2×Dic10—21no—
210C5×C4⋊C4—24no—
211C5×C22⋊C4—24no—
212C5×C22⋊C4—21no—
213C5×C22⋊C4C23×C312yescomplemented
214C22⋊Dic5—22no—
215C22⋊Dic5C23×C314yescomplemented
216#685—28no—
217#683—28no—
218C2×H(5,𝔽2)—51no—
219C2×#123—52no—
220C2×#123C3×D511yescharacteristic, complemented
221#458—58no—
222C2×#99—51no—
223C2×#99—52no—
224C22×C3×C5—24no—
225C22×C3×C5—22no—
226C22×C3×C5—22no—
227C22×C3×C5—42no—
228C22×C3×C5—41no—
229C22×C3×C5—41no—
230C22×C3×C5—42no—
231C22×C3×C5—42no—
232C22×C3×C5—81no—
233C22×C3×C5#12012yes—
234C22×C3×C5#12011yescharacteristic
235C22×C3×C5#12011yescharacteristic
236C22×C3×C5C22×D412yes—
237C22×C3×C5C22×D411yescharacteristic
238C2×C3×D5—24no—
239C2×C3×D5—44no—
240C2×C3×D5—44no—
241C2×C3×D5—82no—
242C2×C3×D5—84no—
243C3×C4×C5—22no—
244C3×C4×C5—42no—
245C3×C4×C5—42no—
246C3×Dic5—24no—
247C3×Dic5—44no—
248C24×C3—52no—
249C24×C3—101no—
250C24×C3C22⋊D511yescharacteristic
251C24×C3C22×D511yescharacteristic
252C3×#41—104no—
253C3×#41—108no—
254C3×#41—108no—
255C3×#41—104no—
256C2×C3×Q8—101no—
257C2×C3×D4—22no—
258C2×C3×D4—22no—
259C2×C3×D4—54no—
260C2×C3×D4—54no—
261C2×C3×D4—101no—
262C2×C3×D4—108no—
263C2×C3×D4—108no—
264C2×C3×D4—102no—
265C2×C3×D4—104no—
266C2×C3×D4—102no—
267C2×C3×D4—102no—
268C2×C3×D4—102no—
269C2×C3×D4—102no—
270C2×C3×D4—104no—
271C2×C3×D4C22⋊D512yes—
272C2×C3×D4C22×D512yes—
273C22×C3×C4—22no—
274C22×C3×C4—22no—
275C22×C3×C4—21no—
276C22×C3×C4—21no—
277C22×C3×C4—21no—
278C22×C3×C4—52no—
279C22×C3×C4—102no—
280C22×C3×C4—101no—
281C22×C3×C4—102no—
282C22×C3×C4C22⋊D511yescharacteristic
283C22×C3×C4C22×D512yescomplemented
284C3×C4⋊C4—24no—
285C3×C4⋊C4—108no—
286C3×C22⋊C4—24no—
287C3×C22⋊C4—21no—
288C3×C22⋊C4—54no—
289C3×C22⋊C4—108no—
290C3×C22⋊C4—102no—
291C3×C22⋊C4C22×D512yescomplemented
292C23×C5—21no—
293C23×C5—21no—
294C23×C5—42no—
295C23×C5—42no—
296C23×C5C2×C3×D412yes—
297C23×C5C2×C3×D412yes—
298C23×C5C2×C3×D411yescharacteristic
299C23×C5C2×C3×D412yes—
300C23×C5C2×C3×D411yescharacteristic
301C23×C5C24×C311yescharacteristic
302C22×D5—22no—
303C22×D5—24no—
304C22×D5—22no—
305C22×D5—44no—
306C22×D5—82no—
307C22×D5C2×C3×D414yes—
308C5×D4—24no—
309C5×D4—24no—
310C2×C4×C5—21no—
311C2×C4×C5—21no—
312C2×C4×C5—24no—
313C2×C4×C5—22no—
314C2×C4×C5—22no—
315C2×C4×C5—42no—
316C2×C4×C5—42no—
317C2×C4×C5—41no—
318C2×C4×C5—41no—
319C2×C4×C5C2×C3×D412yes—
320C22⋊D5—28no—
321C22⋊D5—24no—
322C22⋊D5—24no—
323C22⋊D5—24no—
324C22⋊D5—24no—
325C2×Dic5—24no—
326C2×Dic5—22no—
327C2×Dic5—22no—
328C2×Dic5—24no—
329C2×Dic5—44no—
330C2×Dic5C2×C3×D414yes—
331D20—24no—
332C4×D5—28no—
333Dic10—24no—
334H(5,𝔽2)—54no—
335H(5,𝔽2)—104no—
336H(5,𝔽2)—102no—
337C2×#41—102no—
338C2×#41—101no—
339C22×D4—52no—
340C22×D4—102no—
341C22×D4—101no—
342C22×D4C2×C3×D511yescharacteristic
343C23×C4C2×C3×D511yescharacteristic, complemented
344#123—22no—
345#123—58no—
346#123—104no—
347#123C2×C3×D514yescomplemented
348C2×C4⋊C4—21no—
349C2×C4⋊C4—102no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—52no—
352C2×C22⋊C4—102no—
353C2×C22⋊C4C2×C3×D511yescharacteristic, complemented
354#99—54no—
355#99—58no—
356C2×C3×C5—22no—
357C2×C3×C5—21no—
358C2×C3×C5—22no—
359C2×C3×C5—21no—
360C2×C3×C5—42no—
361C2×C3×C5#45812yescomplemented
362C2×C3×C5C2×#12011yescharacteristic
363C3×D5—44no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—42no—
367C23×C3—42no—
368C23×C3—54no—
369C23×C3—104no—
370C23×C3—102no—
371C23×C3—102no—
372C23×C3—204no—
373C23×C3—402no—
374C23×C3D4×D512yes—
375C23×C3D4×D512yes—
376C23×C3C2×C22⋊D511yescharacteristic
377C23×C3C2×C22⋊D512yes—
378C23×C3C2×C22⋊D511yescharacteristic
379C23×C3C23×D511yescharacteristic
380C3×Q8—104no—
381C3×D4—24no—
382C3×D4—24no—
383C3×D4—104no—
384C3×D4—108no—
385C3×D4—104no—
386C3×D4—104no—
387C3×D4—104no—
388C3×D4—104no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—24no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—42no—
395C2×C3×C4—41no—
396C2×C3×C4—42no—
397C2×C3×C4—41no—
398C2×C3×C4—54no—
399C2×C3×C4—108no—
400C2×C3×C4—104no—
401C2×C3×C4—102no—
402C2×C3×C4—102no—
403C2×C3×C4—104no—
404C2×C3×C4—204no—
405C2×C3×C4C2×C22⋊D512yes—
406C22×C5—24no—
407C22×C5—22no—
408C22×C5—22no—
409C22×C5—42no—
410C22×C5—41no—
411C22×C5—41no—
412C22×C5—42no—
413C22×C5—42no—
414C22×C5—81no—
415C22×C5C3×#12012yes—
416C22×C5C3×#12011yescharacteristic
417C22×C5C3×#12011yescharacteristic
418C22×C5C22×C3×D412yes—
419C22×C5C22×C3×D411yescharacteristic
420C2×D5—24no—
421C2×D5—44no—
422C2×D5—44no—
423C2×D5—82no—
424C2×D5—84no—
425C4×C5—22no—
426C4×C5—42no—
427C4×C5—42no—
428Dic5—24no—
429Dic5—44no—
430C24—52no—
431C24—101no—
432C24C3×C22⋊D511yescharacteristic
433C24C22×C3×D511yescharacteristic
434#41—104no—
435#41—108no—
436#41—108no—
437#41—104no—
438C2×Q8—101no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—54no—
442C2×D4—54no—
443C2×D4—101no—
444C2×D4—108no—
445C2×D4—108no—
446C2×D4—102no—
447C2×D4—104no—
448C2×D4—102no—
449C2×D4—102no—
450C2×D4—102no—
451C2×D4—102no—
452C2×D4—104no—
453C2×D4C3×C22⋊D512yes—
454C2×D4C22×C3×D512yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—52no—
461C22×C4—102no—
462C22×C4—101no—
463C22×C4—102no—
464C22×C4C3×C22⋊D511yescharacteristic
465C22×C4C22×C3×D512yescomplemented
466C4⋊C4—24no—
467C4⋊C4—108no—
468C22⋊C4—24no—
469C22⋊C4—21no—
470C22⋊C4—54no—
471C22⋊C4—108no—
472C22⋊C4—102no—
473C22⋊C4C22×C3×D512yescomplemented
474C3×C5C2×#45811yescharacteristic, complemented, Hall
475C22×C3—22no—
476C22×C3—24no—
477C22×C3—22no—
478C22×C3—42no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—81no—
484C22×C3—104no—
485C22×C3—204no—
486C22×C3—204no—
487C22×C3—402no—
488C22×C3—404no—
489C22×C3#408311yescharacteristic
490C22×C3#408312yes—
491C22×C3#408311yescharacteristic
492C22×C3C2×D4×D512yes—
493C22×C3C22×C22⋊D511yescharacteristic
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—104no—
498C3×C4—204no—
499C2×C5—22no—
500C2×C5—21no—
501C2×C5—22no—
502C2×C5—21no—
503C2×C5—42no—
504C2×C5C3×#45812yescomplemented
505C2×C5C2×C3×#12011yescharacteristic
506D5—44no—
507C23—21no—
508C23—21no—
509C23—42no—
510C23—42no—
511C23—54no—
512C23—104no—
513C23—102no—
514C23—102no—
515C23—204no—
516C23—402no—
517C23C3×D4×D512yes—
518C23C3×D4×D512yes—
519C23C2×C3×C22⋊D511yescharacteristic
520C23C2×C3×C22⋊D512yes—
521C23C2×C3×C22⋊D511yescharacteristic
522C23C23×C3×D511yescharacteristic
523Q8—104no—
524D4—24no—
525D4—24no—
526D4—104no—
527D4—108no—
528D4—104no—
529D4—104no—
530D4—104no—
531D4—104no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—54no—
542C2×C4—108no—
543C2×C4—104no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—104no—
547C2×C4—204no—
548C2×C4C2×C3×C22⋊D512yes—
549C2×C3—21no—
550C2×C3—22no—
551C2×C3—21no—
552C2×C3—22no—
553C2×C3—42no—
554C2×C3—204no—
555C2×C3#6534212yesdirect product factor
556C2×C3C2×#408311yescharacteristic
557C5C2×C3×#45811yescharacteristic, complemented, Sylow
558C22—22no—
559C22—24no—
560C22—22no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—104no—
568C22—204no—
569C22—204no—
570C22—402no—
571C22—404no—
572C22C3×#408311yescharacteristic
573C22C3×#408312yes—
574C22C3×#408311yescharacteristic
575C22C2×C3×D4×D512yes—
576C22C22×C3×C22⋊D511yescharacteristic
577C4—22no—
578C4—42no—
579C4—42no—
580C4—104no—
581C4—204no—
582C3C2×#6534211yescharacteristic, direct product factor, Sylow
583C2—21no—
584C2—22no—
585C2—21no—
586C2—22no—
587C2—42no—
588C2—204no—
589C2C3×#6534212yesdirect product factor
590C2C2×C3×#408311yescharacteristic
subgroupG/H
centerC22×C3#4083
commutator, cosocleC23×C5C24×C3
FrattiniC23C23×C3×D5
FittingC2×C3×C5×#123C2
socleC22×C3×C5#120
prime radicalC2×C3×#4144C2
hypercenterC2×C3×#123D5
hypocenterC5C2×C3×#458
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C2×C3×#123

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#123C2semidirect
3. C2×C3×#4075C2semidirect
4. C2×C3×#3966C2semidirect
5. C3×#65342C2direct (N × H)
6. C2×C3×#3938C2semidirect
7. C2×#65342C3direct (N × H)
9. C23×C3×C4×C5C22semidirect
19. C3×C5×#123C22semidirect
22. C2×C3×C5×C22⋊C4C22semidirect
23. C2×C3×C22⋊Dic5C22semidirect
25. C3×#4075C22semidirect
28. C3×#3966C22semidirect
29. C3×#3938C22semidirect
32. C2×C5×#123C2×C3semidirect
33. C2×#4075C2×C3semidirect
34. C2×#3966C2×C3semidirect
35. #65342C2×C3direct (N × H)
36. C2×#3938C2×C3semidirect
51. C22×C3×C4×C5C23semidirect
76. C3×C5×C22⋊C4C23semidirect
78. C3×C22⋊Dic5C23semidirect
83. C2×C3×#123D5semidirect
88. C23×C4×C5C22×C3semidirect
98. C5×#123C22×C3semidirect
101. C2×C5×C22⋊C4C22×C3semidirect
102. C2×C22⋊Dic5C22×C3semidirect
104. #4075C22×C3semidirect
107. #3966C22×C3semidirect
108. #3938C22×C3semidirect
161. C23×C3×C4C2×D5semidirect
165. C3×#123C2×D5semidirect
171. C2×C3×C22⋊C4C2×D5semidirect
188. C22×C4×C5C23×C3semidirect
213. C5×C22⋊C4C23×C3semidirect
215. C22⋊Dic5C23×C3semidirect
220. C2×#123C3×D5semidirect
283. C22×C3×C4C22×D5semidirect
291. C3×C22⋊C4C22×D5semidirect
343. C23×C4C2×C3×D5semidirect
347. #123C2×C3×D5semidirect
353. C2×C22⋊C4C2×C3×D5semidirect
361. C2×C3×C5#458semidirect
465. C22×C4C22×C3×D5semidirect
473. C22⋊C4C22×C3×D5semidirect
474. C3×C5C2×#458semidirect
504. C2×C5C3×#458semidirect
555. C2×C3#65342direct (N × H)
557. C5C2×C3×#458semidirect
582. C3C2×#65342direct (N × H)
589. C2C3×#65342direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41442
C2×C3×C5×#1232
C2×C3×#40752
C2×C3×#39662
C3×#653422
C2×C3×#39382
C2×#653423
C2×C3×#4585