Finite Group Databasebeta

← C2×C3×#65777 #14412111

Subgroups — C2×C3×#65777 #14412111

Order 1920.

1512 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#711C211yesmaximal, characteristic
2C22×C3×#704C212yesmaximal, complemented
3C2×C3×#4013C212yesmaximal, complemented
4C3×#65777C218yesmaximal, direct product factor
5C2×C3×#4005C212yesmaximal, complemented
6C2×#65777C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×D4C411yescharacteristic
8C22×C3×C22⋊D5C411yescharacteristic
9C22×C3×C22⋊D5C2211yescharacteristic
10C23×C3×Dic5C411yescharacteristic
11C23×C3×Dic5C2211yescharacteristic
12C2×C3×#711—22no—
13C2×C3×#711—22no—
14C2×C3×#711—24no—
15C2×C3×#711C414yes—
16C2×C3×#711C2214yes—
17C22×C3×C4×D5C411yescharacteristic
18C22×C3×Dic10C2211yescharacteristic
19C2×C3×#704—22no—
20C2×C3×#704—24no—
21C2×C3×#704C2214yes—
22C2×C3×#704C2218yescomplemented
23C22×C3×#170—22no—
24C3×#4013C2218yescomplemented
25C3×#4005C2218yescomplemented
26C2×C3×#413—51nomaximal, Hall
27C22×#711C2×C311yescharacteristic
28C22×#704C2×C312yescomplemented
29C2×#4013C2×C312yescomplemented
30#65777C2×C318yesdirect product factor
31C2×#4005C2×C312yescomplemented
32C24×C3×C5C2×C411yescharacteristic
33C24×C3×C5C2×C411yescharacteristic
34C23×C3×D5C2×C411yescharacteristic
35C2×C3×C5×D4—22no—
36C2×C3×C5×D4—22no—
37C2×C3×C5×D4C2×C414yes—
38C22×C3×C4×C5C2×C411yescharacteristic
39C2×C3×C22⋊D5—22no—
40C2×C3×C22⋊D5—22no—
41C2×C3×C22⋊D5—22no—
42C2×C3×C22⋊D5—22no—
43C2×C3×C22⋊D5C2×C412yes—
44C2×C3×C22⋊D5C2×C414yes—
45C2×C3×C22⋊D5C2312yes—
46C22×C3×Dic5—22no—
47C22×C3×Dic5—21no—
48C22×C3×Dic5—21no—
49C22×C3×Dic5C2×C411yescharacteristic
50C22×C3×Dic5C2×C414yes—
51C22×C3×Dic5C2×C411yescharacteristic
52C22×C3×Dic5C2×C412yes—
53C22×C3×Dic5D412yes—
54C22×C3×Dic5D412yes—
55C22×C3×Dic5D414yes—
56C22×C3×Dic5C2311yescharacteristic
57C22×C3×Dic5C2312yes—
58C3×#711—28no—
59C3×#711—28no—
60C3×#711—28no—
61C3×#711—28no—
62C2×C3×C4×D5—22no—
63C2×C3×C4×D5—21no—
64C2×C3×C4×D5—21no—
65C2×C3×C4×D5C2×C414yes—
66C2×C3×Dic10—22no—
67C2×C3×Dic10—22no—
68C2×C3×Dic10C2×C412yes—
69C2×C3×Dic10C2312yes—
70C3×#704—24no—
71C3×#704—28no—
72C3×#704C2318yes—
73C2×C3×#170—24no—
74C2×C3×#170—28no—
75C22×C3×#41—51no—
76C22×C3×M16—52no—
77C3×#413—58no—
78C2×C3×#101—52no—
79C2×C3×#100—52no—
80C22×C5×D4C3×C411yescharacteristic
81C22×C22⋊D5C3×C411yescharacteristic
82C22×C22⋊D5C22×C311yescharacteristic
83C23×Dic5C3×C411yescharacteristic
84C23×Dic5C22×C311yescharacteristic
85C2×#711—22no—
86C2×#711—22no—
87C2×#711—24no—
88C2×#711C3×C414yes—
89C2×#711C22×C314yes—
90C22×C4×D5C3×C411yescharacteristic
91C22×Dic10C22×C311yescharacteristic
92C2×#704—22no—
93C2×#704—24no—
94C2×#704C22×C314yes—
95C2×#704C22×C318yescomplemented
96C22×#170—22no—
97#4013C22×C318yescomplemented
98#4005C22×C318yescomplemented
99C2×#413—51noSylow
100C23×C3×C5—21no—
101C23×C3×C5—21no—
102C23×C3×C5—42no—
103C23×C3×C5—42no—
104C23×C3×C5C22⋊C412yes—
105C23×C3×C5C22⋊C412yes—
106C23×C3×C5C22×C411yescharacteristic
107C23×C3×C5C22×C412yes—
108C23×C3×C5C22×C412yes—
109C22×C3×D5—21no—
110C22×C3×D5—21no—
111C22×C3×D5—42no—
112C22×C3×D5C22×C412yes—
113C3×C5×D4—28no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5C22×C412yes—
117C3×C22⋊D5—28no—
118C3×C22⋊D5—24no—
119C3×C22⋊D5—24no—
120C2×C3×Dic5—21no—
121C2×C3×Dic5—21no—
122C2×C3×Dic5—24no—
123C2×C3×Dic5—24no—
124C2×C3×Dic5C22⋊C412yes—
125C2×C3×Dic5C22⋊C411yescharacteristic
126C2×C3×Dic5C22⋊C411yescharacteristic
127C2×C3×Dic5C22⋊C414yes—
128C2×C3×Dic5C22⋊C414yes—
129C2×C3×Dic5C22×C412yes—
130C2×C3×Dic5C22×C411yescharacteristic
131C2×C3×Dic5C22×C412yes—
132C2×C3×Dic5C2×D412yes—
133C2×C3×Dic5C2×D414yes—
134C2×C3×Dic5C2×D412yes—
135C2×C3×Dic5C2×D414yes—
136C2×C3×Dic5C2411yescharacteristic
137C3×C4×D5—24no—
138C3×C4×D5—24no—
139C3×Dic10—24no—
140C3×Dic10—24no—
141C3×#170—28no—
142C2×C3×#41—54no—
143C2×C3×#41—54no—
144C2×C3×#41—102no—
145C2×C3×#41—104no—
146C2×C3×#41—102no—
147C22×C3×Q8—51no—
148C22×C3×D4—51no—
149C22×C3×D4—51no—
150C22×C3×D4AGL(1,𝔽5)11yescharacteristic
151C23×C3×C4—51no—
152C23×C3×C4—51no—
153C23×C3×C4—51no—
154C2×C3×M16—54no—
155C2×C3×M16—58no—
156C2×C3×M16—104no—
157C2×C3×M16—102no—
158C22×C3×C8—102no—
159C3×#101—58no—
160C3×#100—58no—
161C24×C5C2×C3×C411yescharacteristic
162C24×C5C2×C3×C411yescharacteristic
163C23×D5C2×C3×C411yescharacteristic
164C2×C5×D4—22no—
165C2×C5×D4—22no—
166C2×C5×D4C2×C3×C414yes—
167C22×C4×C5C2×C3×C411yescharacteristic
168C2×C22⋊D5—22no—
169C2×C22⋊D5—22no—
170C2×C22⋊D5—22no—
171C2×C22⋊D5—22no—
172C2×C22⋊D5C2×C3×C412yes—
173C2×C22⋊D5C2×C3×C414yes—
174C2×C22⋊D5C23×C312yes—
175C22×Dic5—22no—
176C22×Dic5—21no—
177C22×Dic5—21no—
178C22×Dic5C2×C3×C411yescharacteristic
179C22×Dic5C2×C3×C414yes—
180C22×Dic5C2×C3×C411yescharacteristic
181C22×Dic5C2×C3×C412yes—
182C22×Dic5C3×D412yes—
183C22×Dic5C3×D412yes—
184C22×Dic5C3×D414yes—
185C22×Dic5C23×C311yescharacteristic
186C22×Dic5C23×C312yes—
187#711—28no—
188#711—28no—
189#711—28no—
190#711—28no—
191C2×C4×D5—22no—
192C2×C4×D5—21no—
193C2×C4×D5—21no—
194C2×C4×D5C2×C3×C414yes—
195C2×Dic10—22no—
196C2×Dic10—22no—
197C2×Dic10C2×C3×C412yes—
198C2×Dic10C23×C312yes—
199#704—24no—
200#704—28no—
201#704C23×C318yes—
202C2×#170—24no—
203C2×#170—28no—
204C22×#41—51no—
205C22×M16—52no—
206#413—58no—
207C2×#101—52no—
208C2×#100—52no—
209C22×C3×C5—22no—
210C22×C3×C5—24no—
211C22×C3×C5—22no—
212C22×C3×C5—42no—
213C22×C3×C5—42no—
214C22×C3×C5—42no—
215C22×C3×C5—42no—
216C22×C3×C5—42no—
217C22×C3×C5C2×C22⋊C411yescharacteristic
218C22×C3×C5C2×C22⋊C414yes—
219C22×C3×C5C2×C22⋊C411yescharacteristic
220C22×C3×C5C23×C411yescharacteristic
221C2×C3×D5—22no—
222C2×C3×D5—42no—
223C2×C3×D5—42no—
224C2×C3×D5—42no—
225C3×C4×C5—22no—
226C3×Dic5—22no—
227C3×Dic5C2×C22⋊C412yes—
228C3×Dic5C2×C22⋊C412yes—
229C3×Dic5C2×C22⋊C412yes—
230C3×Dic5C22×D412yes—
231C24×C3—51no—
232C24×C3C2×AGL(1,𝔽5)11yescharacteristic
233C24×C3C2×AGL(1,𝔽5)11yescharacteristic
234C3×#41—108no—
235C3×#41—108no—
236C3×#41—108no—
237C3×#41—108no—
238C2×C3×Q8—52no—
239C2×C3×Q8—52no—
240C2×C3×Q8—102no—
241C2×C3×Q8—102no—
242C2×C3×D4—22no—
243C2×C3×D4—22no—
244C2×C3×D4—52no—
245C2×C3×D4—52no—
246C2×C3×D4—54no—
247C2×C3×D4—102no—
248C2×C3×D4—102no—
249C2×C3×D4—102no—
250C2×C3×D4—102no—
251C2×C3×D4C2×AGL(1,𝔽5)14yes—
252C22×C3×C4—52no—
253C22×C3×C4—54no—
254C22×C3×C4—52no—
255C22×C3×C4—51no—
256C22×C3×C4—52no—
257C22×C3×C4—51no—
258C22×C3×C4—52no—
259C22×C3×C4—51no—
260C22×C3×C4—54no—
261C22×C3×C4—54no—
262C22×C3×C4—102no—
263C22×C3×C4—101no—
264C22×C3×C4—101no—
265C22×C3×C4—102no—
266C22×C3×C4—101no—
267C22×C3×C4—101no—
268C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
269C3×M16—58no—
270C3×M16—104no—
271C3×M16—108no—
272C2×C3×C8—108no—
273C2×C3×C8—104no—
274C23×C5—21no—
275C23×C5—21no—
276C23×C5—42no—
277C23×C5—42no—
278C23×C5C3×C22⋊C412yes—
279C23×C5C3×C22⋊C412yes—
280C23×C5C22×C3×C411yescharacteristic
281C23×C5C22×C3×C412yes—
282C23×C5C22×C3×C412yes—
283C22×D5—21no—
284C22×D5—21no—
285C22×D5—42no—
286C22×D5C22×C3×C412yes—
287C5×D4—28no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5C22×C3×C412yes—
291C22⋊D5—28no—
292C22⋊D5—24no—
293C22⋊D5—24no—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—24no—
297C2×Dic5—24no—
298C2×Dic5C3×C22⋊C412yes—
299C2×Dic5C3×C22⋊C411yescharacteristic
300C2×Dic5C3×C22⋊C411yescharacteristic
301C2×Dic5C3×C22⋊C414yes—
302C2×Dic5C3×C22⋊C414yes—
303C2×Dic5C22×C3×C412yes—
304C2×Dic5C22×C3×C411yescharacteristic
305C2×Dic5C22×C3×C412yes—
306C2×Dic5C2×C3×D412yes—
307C2×Dic5C2×C3×D414yes—
308C2×Dic5C2×C3×D412yes—
309C2×Dic5C2×C3×D414yes—
310C2×Dic5C24×C311yescharacteristic
311C4×D5—24no—
312C4×D5—24no—
313Dic10—24no—
314Dic10—24no—
315#170—28no—
316C2×#41—54no—
317C2×#41—54no—
318C2×#41—102no—
319C2×#41—102no—
320C2×#41—104no—
321C22×Q8—51no—
322C22×D4—51no—
323C22×D4—51no—
324C22×D4C3×AGL(1,𝔽5)11yescharacteristic
325C23×C4—51no—
326C23×C4—51no—
327C23×C4—51no—
328C2×M16—54no—
329C2×M16—58no—
330C2×M16—102no—
331C2×M16—104no—
332C22×C8—102no—
333#101—58no—
334#100—58no—
335C2×C3×C5—21no—
336C2×C3×C5—21no—
337C2×C3×C5—24no—
338C2×C3×C5—42no—
339C2×C3×C5#41312yescomplemented
340C2×C3×C5C22×C22⋊C411yescharacteristic
341C3×D5—42no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—42no—
345C23×C3—42no—
346C23×C3—52no—
347C23×C3—101no—
348C23×C3—101no—
349C23×C3—202no—
350C23×C3C22⋊AGL(1,𝔽5)12yes—
351C23×C3C22⋊AGL(1,𝔽5)12yes—
352C23×C3C22×AGL(1,𝔽5)11yescharacteristic
353C23×C3C22×AGL(1,𝔽5)12yes—
354C23×C3C22×AGL(1,𝔽5)12yes—
355C3×Q8—104no—
356C3×Q8—104no—
357C3×D4—28no—
358C3×D4—104no—
359C3×D4—104no—
360C3×D4—108no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—52no—
364C2×C3×C4—54no—
365C2×C3×C4—54no—
366C2×C3×C4—54no—
367C2×C3×C4—52no—
368C2×C3×C4—54no—
369C2×C3×C4—51no—
370C2×C3×C4—51no—
371C2×C3×C4—52no—
372C2×C3×C4—52no—
373C2×C3×C4—51no—
374C2×C3×C4—51no—
375C2×C3×C4—52no—
376C2×C3×C4—104no—
377C2×C3×C4—104no—
378C2×C3×C4—104no—
379C2×C3×C4—104no—
380C2×C3×C4—101no—
381C2×C3×C4—101no—
382C2×C3×C4C22×AGL(1,𝔽5)12yes—
383C3×C8—108no—
384C22×C5—22no—
385C22×C5—24no—
386C22×C5—22no—
387C22×C5—42no—
388C22×C5—42no—
389C22×C5—42no—
390C22×C5—42no—
391C22×C5—42no—
392C22×C5C2×C3×C22⋊C411yescharacteristic
393C22×C5C2×C3×C22⋊C414yes—
394C22×C5C2×C3×C22⋊C411yescharacteristic
395C22×C5C23×C3×C411yescharacteristic
396C2×D5—22no—
397C2×D5—42no—
398C2×D5—42no—
399C2×D5—42no—
400C4×C5—22no—
401Dic5—22no—
402Dic5C2×C3×C22⋊C412yes—
403Dic5C2×C3×C22⋊C412yes—
404Dic5C2×C3×C22⋊C412yes—
405Dic5C22×C3×D412yes—
406C24—51no—
407C24C2×C3×AGL(1,𝔽5)11yescharacteristic
408C24C2×C3×AGL(1,𝔽5)11yescharacteristic
409#41—108no—
410#41—108no—
411#41—108no—
412#41—108no—
413C2×Q8—52no—
414C2×Q8—52no—
415C2×Q8—102no—
416C2×Q8—102no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—52no—
420C2×D4—52no—
421C2×D4—54no—
422C2×D4—102no—
423C2×D4—102no—
424C2×D4—102no—
425C2×D4—102no—
426C2×D4C2×C3×AGL(1,𝔽5)14yes—
427C22×C4—52no—
428C22×C4—52no—
429C22×C4—54no—
430C22×C4—51no—
431C22×C4—52no—
432C22×C4—51no—
433C22×C4—52no—
434C22×C4—51no—
435C22×C4—54no—
436C22×C4—54no—
437C22×C4—102no—
438C22×C4—101no—
439C22×C4—101no—
440C22×C4—102no—
441C22×C4—101no—
442C22×C4—101no—
443C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
444M16—58no—
445M16—104no—
446M16—108no—
447C2×C8—104no—
448C2×C8—108no—
449C3×C5C2×#41311yescharacteristic, complemented, Hall
450C22×C3—22no—
451C22×C3—24no—
452C22×C3—22no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—102no—
459C22×C3—202no—
460C22×C3—202no—
461C22×C3—202no—
462C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
463C22×C3C2×C22⋊AGL(1,𝔽5)14yes—
464C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
465C22×C3C23×AGL(1,𝔽5)11yescharacteristic
466C3×C4—22no—
467C3×C4—52no—
468C3×C4—52no—
469C3×C4—52no—
470C3×C4—52no—
471C3×C4—102no—
472C2×C5—21no—
473C2×C5—21no—
474C2×C5—24no—
475C2×C5—42no—
476C2×C5C3×#41312yescomplemented
477C2×C5C22×C3×C22⋊C411yescharacteristic
478D5—42no—
479C23—21no—
480C23—21no—
481C23—42no—
482C23—42no—
483C23—52no—
484C23—101no—
485C23—101no—
486C23—202no—
487C23C3×C22⋊AGL(1,𝔽5)12yes—
488C23C3×C22⋊AGL(1,𝔽5)12yes—
489C23C22×C3×AGL(1,𝔽5)11yescharacteristic
490C23C22×C3×AGL(1,𝔽5)12yes—
491C23C22×C3×AGL(1,𝔽5)12yes—
492Q8—104no—
493Q8—104no—
494D4—28no—
495D4—104no—
496D4—104no—
497D4—108no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—52no—
501C2×C4—54no—
502C2×C4—52no—
503C2×C4—54no—
504C2×C4—52no—
505C2×C4—51no—
506C2×C4—51no—
507C2×C4—54no—
508C2×C4—54no—
509C2×C4—51no—
510C2×C4—52no—
511C2×C4—51no—
512C2×C4—52no—
513C2×C4—104no—
514C2×C4—104no—
515C2×C4—104no—
516C2×C4—104no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4C22×C3×AGL(1,𝔽5)12yes—
520C8—108no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—24no—
524C2×C3—42no—
525C2×C3—202no—
526C2×C3#6577712yesdirect product factor
527C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
528C5C2×C3×#41311yescharacteristic, complemented, Sylow
529C22—22no—
530C22—24no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—102no—
538C22—202no—
539C22—202no—
540C22—202no—
541C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
542C22C2×C3×C22⋊AGL(1,𝔽5)14yes—
543C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
544C22C23×C3×AGL(1,𝔽5)11yescharacteristic
545C4—22no—
546C4—52no—
547C4—52no—
548C4—52no—
549C4—52no—
550C4—102no—
551C3C2×#6577711yescharacteristic, direct product factor, Sylow
552C2—21no—
553C2—21no—
554C2—24no—
555C2—42no—
556C2—202no—
557C2C3×#6577712yesdirect product factor
558C2C22×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC22×C5C23×C3×C4
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC22×C3×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×#711C2
cosocleC2×Dic5C24×C3
hypercenterC22×C3×D4AGL(1,𝔽5)
hypocenterC5C2×C3×#413
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#704C2semidirect
3. C2×C3×#4013C2semidirect
4. C3×#65777C2direct (N × H)
5. C2×C3×#4005C2semidirect
6. C2×#65777C3direct (N × H)
22. C2×C3×#704C22semidirect
24. C3×#4013C22semidirect
25. C3×#4005C22semidirect
28. C22×#704C2×C3semidirect
29. C2×#4013C2×C3semidirect
30. #65777C2×C3direct (N × H)
31. C2×#4005C2×C3semidirect
95. C2×#704C22×C3semidirect
97. #4013C22×C3semidirect
98. #4005C22×C3semidirect
339. C2×C3×C5#413semidirect
449. C3×C5C2×#413semidirect
476. C2×C5C3×#413semidirect
526. C2×C3#65777direct (N × H)
528. C5C2×C3×#413semidirect
551. C3C2×#65777direct (N × H)
557. C2C3×#65777direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7112
C22×C3×#7042
C2×C3×#40132
C3×#657772
C2×C3×#40052
C2×#657773
C2×C3×#4135