Finite Group Databasebeta

← C2×C3×#65016 #14411678

Subgroups — C2×C3×#65016 #14411678

Order 1920.

1584 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#709C211yesmaximal, characteristic
2C22×C3×#676C211yesmaximal, characteristic, complemented
3C22×C3×C8×D5C211yesmaximal, characteristic, complemented
4C2×C3×C5×C22⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×C22⋊#168C211yesmaximal, characteristic, complemented
6C2×C3×#3952C211yesmaximal, characteristic, complemented
7C2×C3×#3952C211yesmaximal, characteristic, complemented
8C3×#65016C218yesmaximal, direct product factor
9C2×#65016C311yesmaximal, characteristic, direct product factor, Hall
10C23×C3×C4×C5C2211yescharacteristic
11C22×C3×C22⋊D5C411yescharacteristic
12C22×C3×C22⋊D5C411yescharacteristic
13C2×C3×#709—24no—
14C2×C3×#709—24no—
15C2×C3×#709C414yes—
16C2×C3×#709C2214yes—
17C22×C3×D20C411yescharacteristic
18C22×C3×C4×D5C2211yescharacteristic
19C22×C3×C4×D5C2211yescharacteristic
20C22×C3×Dic10C411yescharacteristic
21C22×C3×C5×C8C2211yescharacteristic
22C22×C3×C5×C8C2211yescharacteristic
23C22×C3×#168C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C2×C3×#676—24no—
26C2×C3×#676C2214yescomplemented
27C2×C3×C8×D5—24no—
28C2×C3×C8×D5C2214yescomplemented
29C3×C5×C22⋊C8C2214yescomplemented
30C3×C22⋊#168C2214yescomplemented
31C3×#3952C2214yescomplemented
32C3×#3952C2214yescomplemented
33C2×C3×#408—51nomaximal, Hall
34C22×#709C2×C311yescharacteristic
35C22×#676C2×C311yescharacteristic, complemented
36C22×C8×D5C2×C311yescharacteristic, complemented
37C2×C5×C22⋊C8C2×C311yescharacteristic, complemented
38C2×C22⋊#168C2×C311yescharacteristic, complemented
39C2×#3952C2×C311yescharacteristic, complemented
40C2×#3952C2×C311yescharacteristic, complemented
41#65016C2×C318yesdirect product factor
42C24×C3×C5C2×C411yescharacteristic
43C23×C3×D5C2×C411yescharacteristic
44C23×C3×D5C2×C411yescharacteristic
45C22×C3×C4×C5—22no—
46C22×C3×C4×C5—22no—
47C22×C3×C4×C5C2×C411yescharacteristic
48C22×C3×C4×C5C2×C412yes—
49C22×C3×C4×C5C2312yes—
50C22×C3×C4×C5C2311yescharacteristic
51C2×C3×C22⋊D5—24no—
52C2×C3×C22⋊D5—24no—
53C2×C3×C22⋊D5C2×C414yes—
54C2×C3×C22⋊D5C2×C414yes—
55C22×C3×Dic5C2×C411yescharacteristic
56C22×C3×Dic5C2×C411yescharacteristic
57C3×#709—216no—
58C3×#709—28no—
59C3×#709—28no—
60C2×C3×D20—24no—
61C2×C3×D20C2×C414yes—
62C2×C3×C4×D5—22no—
63C2×C3×C4×D5—22no—
64C2×C3×C4×D5C2×C412yes—
65C2×C3×C4×D5C2×C412yes—
66C2×C3×C4×D5D414yes—
67C2×C3×C4×D5D414yes—
68C2×C3×C4×D5C2312yes—
69C2×C3×C4×D5C2312yes—
70C2×C3×Dic10—24no—
71C2×C3×Dic10C2×C414yes—
72C2×C3×C5×C8—22no—
73C2×C3×C5×C8—22no—
74C2×C3×C5×C8C2312yes—
75C2×C3×C5×C8C2312yes—
76C2×C3×#168—22no—
77C2×C3×#168—22no—
78C2×C3×#168C2312yes—
79C2×C3×#168C2312yes—
80C3×#676—28no—
81C3×C8×D5—28no—
82C22×C3×#41—51no—
83C22×C3×M16—51no—
84C23×C3×C8—51no—
85C3×#408—58no—
86C2×C3×C22⋊C8—51no—
87C2×C3×C22⋊C8—51no—
88C2×C3×C22⋊C8—51no—
89C2×C3×C22⋊C8D511yescharacteristic, complemented
90C23×C4×C5C22×C311yescharacteristic
91C22×C22⋊D5C3×C411yescharacteristic
92C22×C22⋊D5C3×C411yescharacteristic
93C2×#709—24no—
94C2×#709—24no—
95C2×#709C3×C414yes—
96C2×#709C22×C314yes—
97C22×D20C3×C411yescharacteristic
98C22×C4×D5C22×C311yescharacteristic
99C22×C4×D5C22×C311yescharacteristic
100C22×Dic10C3×C411yescharacteristic
101C22×C5×C8C22×C311yescharacteristic
102C22×C5×C8C22×C311yescharacteristic
103C22×#168C22×C311yescharacteristic
104C22×#168C22×C311yescharacteristic
105C2×#676—24no—
106C2×#676C22×C314yescomplemented
107C2×C8×D5—24no—
108C2×C8×D5C22×C314yescomplemented
109C5×C22⋊C8C22×C314yescomplemented
110C22⋊#168C22×C314yescomplemented
111#3952C22×C314yescomplemented
112#3952C22×C314yescomplemented
113C2×#408—51noSylow
114C23×C3×C5—22no—
115C23×C3×C5—22no—
116C23×C3×C5—22no—
117C23×C3×C5C22×C412yes—
118C23×C3×C5C22×C411yescharacteristic
119C22×C3×D5—22no—
120C22×C3×D5—22no—
121C22×C3×D5—22no—
122C22×C3×D5—22no—
123C22×C3×D5—22no—
124C22×C3×D5C22⋊C414yes—
125C22×C3×D5C22×C412yes—
126C22×C3×D5C22×C412yes—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—24no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5C22×C412yes—
132C2×C3×C4×C5C22×C411yescharacteristic
133C2×C3×C4×C5C2×D412yes—
134C2×C3×C4×C5C2×D412yes—
135C2×C3×C4×C5C2411yescharacteristic
136C3×C22⋊D5—28no—
137C3×C22⋊D5—28no—
138C2×C3×Dic5—22no—
139C2×C3×Dic5C22⋊C414yes—
140C2×C3×Dic5C22×C412yes—
141C2×C3×Dic5C22×C412yes—
142C3×D20—28no—
143C3×C4×D5—24no—
144C3×C4×D5—22no—
145C3×C4×D5—22no—
146C3×C4×D5C22⋊C418yes—
147C3×C4×D5C2×D414yes—
148C3×C4×D5C2×D414yes—
149C3×Dic10—28no—
150C3×C5×C8—22no—
151C3×C5×C8—22no—
152C3×#168—22no—
153C3×#168—22no—
154C2×C3×#41—54no—
155C2×C3×#41—54no—
156C2×C3×#41—104no—
157C2×C3×#41—104no—
158C22×C3×Q8—51no—
159C22×C3×D4—51no—
160C22×C3×D4—51no—
161C22×C3×D4—51no—
162C23×C3×C4—51no—
163C23×C3×C4—51no—
164C23×C3×C4C2×D511yescharacteristic
165C2×C3×M16—54no—
166C2×C3×M16—104no—
167C22×C3×C8—51no—
168C22×C3×C8—51no—
169C22×C3×C8—54no—
170C22×C3×C8—104no—
171C22×C3×C8C2×D511yescharacteristic
172C22×C3×C8C2×D511yescharacteristic
173C3×C22⋊C8—54no—
174C3×C22⋊C8—54no—
175C3×C22⋊C8—54no—
176C3×C22⋊C8C2×D514yescomplemented
177C24×C5C2×C3×C411yescharacteristic
178C23×D5C2×C3×C411yescharacteristic
179C23×D5C2×C3×C411yescharacteristic
180C22×C4×C5—22no—
181C22×C4×C5—22no—
182C22×C4×C5C2×C3×C411yescharacteristic
183C22×C4×C5C2×C3×C412yes—
184C22×C4×C5C23×C312yes—
185C22×C4×C5C23×C311yescharacteristic
186C2×C22⋊D5—24no—
187C2×C22⋊D5—24no—
188C2×C22⋊D5C2×C3×C414yes—
189C2×C22⋊D5C2×C3×C414yes—
190C22×Dic5C2×C3×C411yescharacteristic
191C22×Dic5C2×C3×C411yescharacteristic
192#709—216no—
193#709—28no—
194#709—28no—
195C2×D20—24no—
196C2×D20C2×C3×C414yes—
197C2×C4×D5—22no—
198C2×C4×D5—22no—
199C2×C4×D5C2×C3×C412yes—
200C2×C4×D5C2×C3×C412yes—
201C2×C4×D5C3×D414yes—
202C2×C4×D5C3×D414yes—
203C2×C4×D5C23×C312yes—
204C2×C4×D5C23×C312yes—
205C2×Dic10—24no—
206C2×Dic10C2×C3×C414yes—
207C2×C5×C8—22no—
208C2×C5×C8—22no—
209C2×C5×C8C23×C312yes—
210C2×C5×C8C23×C312yes—
211C2×#168—22no—
212C2×#168—22no—
213C2×#168C23×C312yes—
214C2×#168C23×C312yes—
215#676—28no—
216C8×D5—28no—
217C22×#41—51no—
218C22×M16—51no—
219C23×C8—51no—
220#408—58no—
221C2×C22⋊C8—51no—
222C2×C22⋊C8—51no—
223C2×C22⋊C8—51no—
224C2×C22⋊C8C3×D511yescharacteristic, complemented
225C22×C3×C5—22no—
226C22×C3×C5—22no—
227C22×C3×C5—22no—
228C22×C3×C5—44no—
229C22×C3×C5C2×C22⋊C412yes—
230C22×C3×C5#13112yes—
231C22×C3×C5#13112yes—
232C22×C3×C5C23×C411yescharacteristic
233C2×C3×D5—22no—
234C2×C3×D5—22no—
235C2×C3×D5—28no—
236C2×C3×D5—22no—
237C2×C3×D5—22no—
238C2×C3×D5—22no—
239C2×C3×D5—44no—
240C2×C3×D5C2×C22⋊C414yes—
241C3×C4×C5—22no—
242C3×C4×C5C2×C22⋊C412yes—
243C3×C4×C5C22×D411yescharacteristic
244C3×C4×C5C22×D411yescharacteristic
245C3×Dic5—22no—
246C3×Dic5C2×C22⋊C414yes—
247C24×C3—51no—
248C24×C3—51no—
249C24×C3C4×D511yescharacteristic
250C3×#41—108no—
251C3×#41—108no—
252C3×#41—1016no—
253C2×C3×Q8—54no—
254C2×C3×Q8—104no—
255C2×C3×D4—54no—
256C2×C3×D4—54no—
257C2×C3×D4—54no—
258C2×C3×D4—104no—
259C2×C3×D4—104no—
260C2×C3×D4—104no—
261C22×C3×C4—22no—
262C22×C3×C4—22no—
263C22×C3×C4—51no—
264C22×C3×C4—52no—
265C22×C3×C4—54no—
266C22×C3×C4—54no—
267C22×C3×C4—52no—
268C22×C3×C4—51no—
269C22×C3×C4—52no—
270C22×C3×C4—52no—
271C22×C3×C4—102no—
272C22×C3×C4—102no—
273C22×C3×C4C4×D511yescharacteristic
274C22×C3×C4C4×D512yes—
275C22×C3×C4C22×D512yes—
276C22×C3×C4C22×D511yescharacteristic
277C3×M16—108no—
278C2×C3×C8—22no—
279C2×C3×C8—22no—
280C2×C3×C8—52no—
281C2×C3×C8—52no—
282C2×C3×C8—108no—
283C2×C3×C8—102no—
284C2×C3×C8—102no—
285C2×C3×C8C22×D512yes—
286C2×C3×C8C22×D512yes—
287C23×C5—22no—
288C23×C5—22no—
289C23×C5—22no—
290C23×C5C22×C3×C412yes—
291C23×C5C22×C3×C411yescharacteristic
292C22×D5—22no—
293C22×D5—22no—
294C22×D5—22no—
295C22×D5—22no—
296C22×D5—22no—
297C22×D5C3×C22⋊C414yes—
298C22×D5C22×C3×C412yes—
299C22×D5C22×C3×C412yes—
300C2×C4×C5—22no—
301C2×C4×C5—24no—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5C22×C3×C412yes—
305C2×C4×C5C22×C3×C411yescharacteristic
306C2×C4×C5C2×C3×D412yes—
307C2×C4×C5C2×C3×D412yes—
308C2×C4×C5C24×C311yescharacteristic
309C22⋊D5—28no—
310C22⋊D5—28no—
311C2×Dic5—22no—
312C2×Dic5C3×C22⋊C414yes—
313C2×Dic5C22×C3×C412yes—
314C2×Dic5C22×C3×C412yes—
315D20—28no—
316C4×D5—24no—
317C4×D5—22no—
318C4×D5—22no—
319C4×D5C3×C22⋊C418yes—
320C4×D5C2×C3×D414yes—
321C4×D5C2×C3×D414yes—
322Dic10—28no—
323C5×C8—22no—
324C5×C8—22no—
325#168—22no—
326#168—22no—
327C2×#41—54no—
328C2×#41—54no—
329C2×#41—104no—
330C2×#41—104no—
331C22×Q8—51no—
332C22×D4—51no—
333C22×D4—51no—
334C22×D4—51no—
335C23×C4—51no—
336C23×C4—51no—
337C23×C4C2×C3×D511yescharacteristic
338C2×M16—54no—
339C2×M16—104no—
340C22×C8—51no—
341C22×C8—51no—
342C22×C8—54no—
343C22×C8—104no—
344C22×C8C2×C3×D511yescharacteristic
345C22×C8C2×C3×D511yescharacteristic
346C22⋊C8—54no—
347C22⋊C8—54no—
348C22⋊C8—54no—
349C22⋊C8C2×C3×D514yescomplemented
350C2×C3×C5—42no—
351C2×C3×C5#40814yescomplemented
352C2×C3×C5C22×C22⋊C411yescharacteristic
353C2×C3×C5C2×#13111yescharacteristic
354C2×C3×C5C2×#13111yescharacteristic
355C3×D5—24no—
356C3×D5—42no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—52no—
361C23×C3—54no—
362C23×C3—52no—
363C23×C3—102no—
364C23×C3—102no—
365C23×C3—102no—
366C23×C3—102no—
367C23×C3—102no—
368C23×C3C2×C4×D512yes—
369C23×C3C2×C4×D511yescharacteristic
370C3×Q8—108no—
371C3×D4—108no—
372C3×D4—108no—
373C3×D4—108no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—24no—
378C2×C3×C4—52no—
379C2×C3×C4—54no—
380C2×C3×C4—54no—
381C2×C3×C4—54no—
382C2×C3×C4—58no—
383C2×C3×C4—52no—
384C2×C3×C4—102no—
385C2×C3×C4—102no—
386C2×C3×C4—102no—
387C2×C3×C4—104no—
388C2×C3×C4C2×C4×D512yes—
389C2×C3×C4C2×C4×D511yescharacteristic
390C2×C3×C4D4×D512yes—
391C2×C3×C4D4×D512yes—
392C2×C3×C4C23×D511yescharacteristic
393C3×C8—22no—
394C3×C8—22no—
395C3×C8—102no—
396C3×C8—102no—
397C22×C5—22no—
398C22×C5—22no—
399C22×C5—22no—
400C22×C5—44no—
401C22×C5C2×C3×C22⋊C412yes—
402C22×C5C3×#13112yes—
403C22×C5C3×#13112yes—
404C22×C5C23×C3×C411yescharacteristic
405C2×D5—22no—
406C2×D5—22no—
407C2×D5—28no—
408C2×D5—22no—
409C2×D5—22no—
410C2×D5—22no—
411C2×D5—44no—
412C2×D5C2×C3×C22⋊C414yes—
413C4×C5—22no—
414C4×C5C2×C3×C22⋊C412yes—
415C4×C5C22×C3×D411yescharacteristic
416C4×C5C22×C3×D411yescharacteristic
417Dic5—22no—
418Dic5C2×C3×C22⋊C414yes—
419C24—51no—
420C24—51no—
421C24C3×C4×D511yescharacteristic
422#41—108no—
423#41—108no—
424#41—1016no—
425C2×Q8—54no—
426C2×Q8—104no—
427C2×D4—54no—
428C2×D4—54no—
429C2×D4—54no—
430C2×D4—104no—
431C2×D4—104no—
432C2×D4—104no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—51no—
436C22×C4—52no—
437C22×C4—54no—
438C22×C4—54no—
439C22×C4—52no—
440C22×C4—51no—
441C22×C4—52no—
442C22×C4—52no—
443C22×C4—102no—
444C22×C4—102no—
445C22×C4C3×C4×D511yescharacteristic
446C22×C4C3×C4×D512yes—
447C22×C4C22×C3×D512yes—
448C22×C4C22×C3×D511yescharacteristic
449M16—108no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—52no—
453C2×C8—52no—
454C2×C8—108no—
455C2×C8—102no—
456C2×C8—102no—
457C2×C8C22×C3×D512yes—
458C2×C8C22×C3×D512yes—
459C3×C5C2×#40811yescharacteristic, complemented, Hall
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—44no—
464C22×C3—54no—
465C22×C3—102no—
466C22×C3—102no—
467C22×C3—108no—
468C22×C3—102no—
469C22×C3—102no—
470C22×C3—102no—
471C22×C3—204no—
472C22×C3D5×C22⋊C412yes—
473C22×C3#404712yes—
474C22×C3#405312yes—
475C22×C3C22×C4×D511yescharacteristic
476C3×C4—22no—
477C3×C4—54no—
478C3×C4—102no—
479C3×C4D5×C22⋊C412yes—
480C3×C4C2×D4×D511yescharacteristic
481C3×C4C2×D4×D511yescharacteristic
482C2×C5—42no—
483C2×C5C3×#40814yescomplemented
484C2×C5C22×C3×C22⋊C411yescharacteristic
485C2×C5C2×C3×#13111yescharacteristic
486C2×C5C2×C3×#13111yescharacteristic
487D5—24no—
488D5—42no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—52no—
493C23—54no—
494C23—52no—
495C23—102no—
496C23—102no—
497C23—102no—
498C23—102no—
499C23—102no—
500C23C2×C3×C4×D512yes—
501C23C2×C3×C4×D511yescharacteristic
502Q8—108no—
503D4—108no—
504D4—108no—
505D4—108no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—52no—
511C2×C4—54no—
512C2×C4—54no—
513C2×C4—54no—
514C2×C4—58no—
515C2×C4—52no—
516C2×C4—102no—
517C2×C4—102no—
518C2×C4—102no—
519C2×C4—104no—
520C2×C4C2×C3×C4×D512yes—
521C2×C4C2×C3×C4×D511yescharacteristic
522C2×C4C3×D4×D512yes—
523C2×C4C3×D4×D512yes—
524C2×C4C23×C3×D511yescharacteristic
525C8—22no—
526C8—22no—
527C8—102no—
528C8—102no—
529C2×C3—42no—
530C2×C3—104no—
531C2×C3—202no—
532C2×C3#6501614yesdirect product factor
533C2×C3C2×D5×C22⋊C411yescharacteristic
534C2×C3C2×#404711yescharacteristic
535C2×C3C2×#405311yescharacteristic
536C5C2×C3×#40811yescharacteristic, complemented, Sylow
537C22—22no—
538C22—22no—
539C22—22no—
540C22—44no—
541C22—54no—
542C22—102no—
543C22—102no—
544C22—108no—
545C22—102no—
546C22—102no—
547C22—102no—
548C22—204no—
549C22C3×D5×C22⋊C412yes—
550C22C3×#404712yes—
551C22C3×#405312yes—
552C22C22×C3×C4×D511yescharacteristic
553C4—22no—
554C4—54no—
555C4—102no—
556C4C3×D5×C22⋊C412yes—
557C4C2×C3×D4×D511yescharacteristic
558C4C2×C3×D4×D511yescharacteristic
559C3C2×#6501611yescharacteristic, direct product factor, Sylow
560C2—42no—
561C2—104no—
562C2—202no—
563C2C3×#6501614yesdirect product factor
564C2C2×C3×D5×C22⋊C411yescharacteristic
565C2C2×C3×#404711yescharacteristic
566C2C2×C3×#405311yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D5
commutatorC22×C5C23×C3×C4
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×C22⋊C8C2
socleC23×C3×C5C22×C4
prime radicalC22×C3×#709C2
cosocleC2×C4×C5C24×C3
hypercenterC2×C3×C22⋊C8D5
hypocenterC5C2×C3×#408
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×C3×#676C2semidirect
3. C22×C3×C8×D5C2semidirect
4. C2×C3×C5×C22⋊C8C2semidirect
5. C2×C3×C22⋊#168C2semidirect
6. C2×C3×#3952C2semidirect
7. C2×C3×#3952C2semidirect
8. C3×#65016C2direct (N × H)
9. C2×#65016C3direct (N × H)
26. C2×C3×#676C22semidirect
28. C2×C3×C8×D5C22semidirect
29. C3×C5×C22⋊C8C22semidirect
30. C3×C22⋊#168C22semidirect
31. C3×#3952C22semidirect
32. C3×#3952C22semidirect
35. C22×#676C2×C3semidirect
36. C22×C8×D5C2×C3semidirect
37. C2×C5×C22⋊C8C2×C3semidirect
38. C2×C22⋊#168C2×C3semidirect
39. C2×#3952C2×C3semidirect
40. C2×#3952C2×C3semidirect
41. #65016C2×C3direct (N × H)
89. C2×C3×C22⋊C8D5semidirect
106. C2×#676C22×C3semidirect
108. C2×C8×D5C22×C3semidirect
109. C5×C22⋊C8C22×C3semidirect
110. C22⋊#168C22×C3semidirect
111. #3952C22×C3semidirect
112. #3952C22×C3semidirect
176. C3×C22⋊C8C2×D5semidirect
224. C2×C22⋊C8C3×D5semidirect
349. C22⋊C8C2×C3×D5semidirect
351. C2×C3×C5#408semidirect
459. C3×C5C2×#408semidirect
483. C2×C5C3×#408semidirect
532. C2×C3#65016direct (N × H)
536. C5C2×C3×#408semidirect
559. C3C2×#65016direct (N × H)
563. C2C3×#65016direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7092
C22×C3×#6762
C22×C3×C8×D52
C2×C3×C5×C22⋊C82
C2×C3×C22⋊#1682
C2×C3×#39522
C3×#650162
C2×#650163
C2×C3×#4085