Finite Group Databasebeta

← C2×C3×#65959 #14481200

Subgroups — C2×C3×#65959 #14481200

Order 1920.

2008 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×Q8×Dic5C211yesmaximal, characteristic, complemented
3C2×C3×#4090C211yesmaximal, characteristic, complemented
4C2×C3×#4072C211yesmaximal, characteristic
5C2×C3×#4070C211yesmaximal, characteristic
6C3×#65959C2116yesmaximal, direct product factor
7C2×C3×#4036C211yesmaximal, characteristic, complemented
8C2×C3×#4036C212yesmaximal, complemented
9C2×C3×#4035C212yesmaximal, complemented
10C2×C3×#4033C211yesmaximal, characteristic, complemented
11C2×C3×#4025C212yesmaximal
12C2×C3×#4025C212yesmaximal
13C2×#65959C311yesmaximal, characteristic, direct product factor, Hall
14C22×C3×C5×Q8C2211yescharacteristic
15C23×C3×C4×C5C2211yescharacteristic
16C22×C3×Dic10C2211yescharacteristic
17C3×C5×C22⋊Q8C2218yes—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C22⋊C4C2212yes—
21C2×C3×C22⋊Dic5C2212yes—
22C2×C3×C22⋊Dic5C2211yescharacteristic
23C2×C3×C22⋊Dic5C2211yescharacteristic
24C3×Q8×Dic5C2218yescomplemented
25C3×#4090C2218yescomplemented
26C3×#4072C2218yes—
27C2×C3×C4⋊Dic5C2211yescharacteristic
28C2×C3×C4⋊Dic5C2211yescharacteristic
29C2×C3×C4⋊Dic5C2211yescharacteristic
30C2×C3×C4⋊Dic5C2212yes—
31C2×C3×C4⋊Dic5C2212yes—
32C3×#4070C2218yes—
33C2×C3×#683C2212yes—
34C2×C3×#683C2212yes—
35C2×C3×#683C2212yes—
36C2×C3×#683C2212yes—
37C2×C3×#683C2212yes—
38C2×C3×C4×Dic5C2211yescharacteristic
39C2×C3×C4×Dic5C2211yescharacteristic
40C2×C3×C4×Dic5C2212yes—
41C2×C3×C4×Dic5C2212yes—
42C3×#4036C2218yescomplemented
43C3×#4036C22116yescomplemented
44C3×#4035C22116yescomplemented
45C3×#4033C2218yescomplemented
46C3×#4025C22116yes—
47C3×#4025C22116yes—
48C2×C3×#541—51nomaximal, Hall
49C2×C5×C22⋊Q8C2×C311yescharacteristic
50C2×Q8×Dic5C2×C311yescharacteristic, complemented
51C2×#4090C2×C311yescharacteristic, complemented
52C2×#4072C2×C311yescharacteristic
53C2×#4070C2×C311yescharacteristic
54#65959C2×C3116yesdirect product factor
55C2×#4036C2×C311yescharacteristic, complemented
56C2×#4036C2×C312yescomplemented
57C2×#4035C2×C312yescomplemented
58C2×#4033C2×C311yescharacteristic, complemented
59C2×#4025C2×C312yes—
60C2×#4025C2×C312yes—
61C24×C3×C5C2311yescharacteristic
62C2×C3×C5×Q8—24no—
63C2×C3×C5×Q8C2314yes—
64C22×C3×C4×C5—24no—
65C22×C3×C4×C5C2312yes—
66C22×C3×C4×C5C2312yes—
67C22×C3×C4×C5C2311yescharacteristic
68C22×C3×C4×C5C2314yes—
69C22×C3×C4×C5C2311yescharacteristic
70C22×C3×Dic5C2312yes—
71C22×C3×Dic5C2311yescharacteristic
72C22×C3×Dic5C2311yescharacteristic
73C22×C3×Dic5C2311yescharacteristic
74C22×C3×Dic5C2312yes—
75C22×C3×Dic5C2311yescharacteristic
76C2×C3×Dic10—24no—
77C2×C3×Dic10C2314yes—
78C3×C5×C4⋊C4C2314yes—
79C3×C5×C4⋊C4C2318yes—
80C3×C5×C22⋊C4C2318yes—
81C3×C22⋊Dic5C2318yes—
82C3×C22⋊Dic5C2314yes—
83C3×C22⋊Dic5C2314yes—
84C3×C4⋊Dic5C2318yes—
85C3×C4⋊Dic5C2314yes—
86C3×C4⋊Dic5C2318yes—
87C3×C4⋊Dic5C2314yes—
88C3×C4⋊Dic5C2314yes—
89C3×#683C2318yes—
90C3×#683C2318yes—
91C3×#683C2318yes—
92C3×#683C2318yes—
93C3×#683C2318yes—
94C3×C4×Dic5C2314yes—
95C3×C4×Dic5C2318yes—
96C3×C4×Dic5C2314yes—
97C3×C4×Dic5C2318yes—
98C3×#541—516no—
99C2×C3×C4⋊Q8—51no—
100C2×C3×#126—52no—
101C2×C3×#126—52no—
102C2×C3×#125—52no—
103C2×C3×#125—51no—
104C2×C3×#125—52no—
105C2×C3×C22⋊Q8—51no—
106C2×C3×C22⋊Q8D511yescharacteristic
107C2×C3×C4×Q8—51no—
108C2×C3×C4×Q8—51no—
109C2×C3×#117—51no—
110C22×C5×Q8C22×C311yescharacteristic
111C23×C4×C5C22×C311yescharacteristic
112C22×Dic10C22×C311yescharacteristic
113C5×C22⋊Q8C22×C318yes—
114C2×C5×C4⋊C4C22×C311yescharacteristic
115C2×C5×C4⋊C4C22×C312yes—
116C2×C5×C22⋊C4C22×C312yes—
117C2×C22⋊Dic5C22×C312yes—
118C2×C22⋊Dic5C22×C311yescharacteristic
119C2×C22⋊Dic5C22×C311yescharacteristic
120Q8×Dic5C22×C318yescomplemented
121#4090C22×C318yescomplemented
122#4072C22×C318yes—
123C2×C4⋊Dic5C22×C312yes—
124C2×C4⋊Dic5C22×C312yes—
125C2×C4⋊Dic5C22×C311yescharacteristic
126C2×C4⋊Dic5C22×C311yescharacteristic
127C2×C4⋊Dic5C22×C311yescharacteristic
128#4070C22×C318yes—
129C2×#683C22×C312yes—
130C2×#683C22×C312yes—
131C2×#683C22×C312yes—
132C2×#683C22×C312yes—
133C2×#683C22×C312yes—
134C2×C4×Dic5C22×C312yes—
135C2×C4×Dic5C22×C312yes—
136C2×C4×Dic5C22×C311yescharacteristic
137C2×C4×Dic5C22×C311yescharacteristic
138#4036C22×C3116yescomplemented
139#4036C22×C318yescomplemented
140#4035C22×C3116yescomplemented
141#4033C22×C318yescomplemented
142#4025C22×C3116yes—
143#4025C22×C3116yes—
144C2×#541—51noSylow
145C23×C3×C5—22no—
146C23×C3×C5—22no—
147C23×C3×C5—22no—
148C23×C3×C5C2412yes—
149C23×C3×C5C2411yescharacteristic
150C3×C5×Q8—28no—
151C2×C3×C4×C5—22no—
152C2×C3×C4×C5—24no—
153C2×C3×C4×C5—24no—
154C2×C3×C4×C5—28no—
155C2×C3×C4×C5#4114yes—
156C2×C3×C4×C5C2414yes—
157C2×C3×C4×C5C2414yes—
158C2×C3×C4×C5C2412yes—
159C2×C3×C4×C5C2412yes—
160C2×C3×Dic5—24no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5—24no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—22no—
166C2×C3×Dic5C2412yes—
167C2×C3×Dic5C2412yes—
168C2×C3×Dic5C2412yes—
169C2×C3×Dic5C2414yes—
170C2×C3×Dic5C2412yes—
171C2×C3×Dic5C2414yes—
172C3×Dic10—28no—
173C22×C3×Q8—51no—
174C22×C3×Q8C2×D511yescharacteristic
175C23×C3×C4C2×D511yescharacteristic
176C3×C4⋊Q8—58no—
177C3×#126—516no—
178C3×#126—516no—
179C3×#125—516no—
180C3×#125—58no—
181C3×#125—516no—
182C3×C22⋊Q8—58no—
183C3×C22⋊Q8C2×D518yes—
184C3×C4×Q8—58no—
185C3×C4×Q8—58no—
186C3×#117—58no—
187C2×C3×C4⋊C4—51no—
188C2×C3×C4⋊C4—51no—
189C2×C3×C4⋊C4—51no—
190C2×C3×C4⋊C4—52no—
191C2×C3×C4⋊C4—52no—
192C2×C3×C4⋊C4—52no—
193C2×C3×C4⋊C4—52no—
194C2×C3×C4⋊C4—52no—
195C2×C3×C4⋊C4—52no—
196C2×C3×C4⋊C4—52no—
197C2×C3×C4⋊C4C2×D511yescharacteristic
198C2×C3×C4⋊C4C2×D512yes—
199C2×C3×C22⋊C4—51no—
200C2×C3×C22⋊C4—51no—
201C2×C3×C22⋊C4—52no—
202C2×C3×C22⋊C4C2×D512yes—
203C2×C3×C42—51no—
204C2×C3×C42—51no—
205C2×C3×C42—52no—
206C2×C3×C42—52no—
207C24×C5C23×C311yescharacteristic
208C2×C5×Q8—24no—
209C2×C5×Q8C23×C314yes—
210C22×C4×C5—24no—
211C22×C4×C5C23×C312yes—
212C22×C4×C5C23×C312yes—
213C22×C4×C5C23×C311yescharacteristic
214C22×C4×C5C23×C314yes—
215C22×C4×C5C23×C311yescharacteristic
216C22×Dic5C23×C312yes—
217C22×Dic5C23×C311yescharacteristic
218C22×Dic5C23×C311yescharacteristic
219C22×Dic5C23×C311yescharacteristic
220C22×Dic5C23×C312yes—
221C22×Dic5C23×C311yescharacteristic
222C2×Dic10—24no—
223C2×Dic10C23×C314yes—
224C5×C4⋊C4C23×C314yes—
225C5×C4⋊C4C23×C318yes—
226C5×C22⋊C4C23×C318yes—
227C22⋊Dic5C23×C318yes—
228C22⋊Dic5C23×C314yes—
229C22⋊Dic5C23×C314yes—
230C4⋊Dic5C23×C318yes—
231C4⋊Dic5C23×C318yes—
232C4⋊Dic5C23×C314yes—
233C4⋊Dic5C23×C314yes—
234C4⋊Dic5C23×C314yes—
235#683C23×C318yes—
236#683C23×C318yes—
237#683C23×C318yes—
238#683C23×C318yes—
239#683C23×C318yes—
240C4×Dic5C23×C318yes—
241C4×Dic5C23×C318yes—
242C4×Dic5C23×C314yes—
243C4×Dic5C23×C314yes—
244#541—516no—
245C2×C4⋊Q8—51no—
246C2×#126—52no—
247C2×#126—52no—
248C2×#125—52no—
249C2×#125—51no—
250C2×#125—52no—
251C2×C22⋊Q8—51no—
252C2×C22⋊Q8C3×D511yescharacteristic
253C2×C4×Q8—51no—
254C2×C4×Q8—51no—
255C2×#117—51no—
256C22×C3×C5—22no—
257C22×C3×C5—22no—
258C22×C3×C5—22no—
259C22×C3×C5—44no—
260C22×C3×C5C2×#4112yes—
261C22×C3×C52−1+412yes—
262C22×C3×C52−1+412yes—
263C22×C3×C5C2511yescharacteristic
264C3×C4×C5—22no—
265C3×C4×C5—24no—
266C3×C4×C5—24no—
267C3×C4×C5C2×#4114yes—
268C3×Dic5—24no—
269C3×Dic5—22no—
270C3×Dic5—22no—
271C3×Dic5—24no—
272C3×Dic5—22no—
273C3×Dic5—22no—
274C24×C3C22×D511yescharacteristic
275C2×C3×Q8—24no—
276C2×C3×Q8—54no—
277C2×C3×Q8—104no—
278C2×C3×Q8C22×D514yes—
279C22×C3×C4—24no—
280C22×C3×C4—51no—
281C22×C3×C4—51no—
282C22×C3×C4—51no—
283C22×C3×C4—52no—
284C22×C3×C4—51no—
285C22×C3×C4—52no—
286C22×C3×C4C22×D514yes—
287C22×C3×C4C22×D512yes—
288C22×C3×C4C22×D512yes—
289C22×C3×C4C22×D511yescharacteristic
290C22×C3×C4C22×D511yescharacteristic
291C3×C4⋊C4—58no—
292C3×C4⋊C4—58no—
293C3×C4⋊C4—58no—
294C3×C4⋊C4—58no—
295C3×C4⋊C4—58no—
296C3×C4⋊C4—54no—
297C3×C4⋊C4—54no—
298C3×C4⋊C4—54no—
299C3×C4⋊C4—58no—
300C3×C4⋊C4—58no—
301C3×C4⋊C4C22×D514yes—
302C3×C4⋊C4C22×D518yes—
303C3×C22⋊C4—54no—
304C3×C22⋊C4—54no—
305C3×C22⋊C4—58no—
306C3×C22⋊C4C22×D518yes—
307C3×C42—54no—
308C3×C42—54no—
309C3×C42—58no—
310C3×C42—58no—
311C23×C5—22no—
312C23×C5—22no—
313C23×C5—22no—
314C23×C5C24×C312yes—
315C23×C5C24×C311yescharacteristic
316C5×Q8—28no—
317C2×C4×C5—22no—
318C2×C4×C5—24no—
319C2×C4×C5—24no—
320C2×C4×C5—28no—
321C2×C4×C5C3×#4114yes—
322C2×C4×C5C24×C314yes—
323C2×C4×C5C24×C314yes—
324C2×C4×C5C24×C312yes—
325C2×C4×C5C24×C312yes—
326C2×Dic5—24no—
327C2×Dic5—22no—
328C2×Dic5—24no—
329C2×Dic5—22no—
330C2×Dic5—22no—
331C2×Dic5—22no—
332C2×Dic5C24×C312yes—
333C2×Dic5C24×C312yes—
334C2×Dic5C24×C312yes—
335C2×Dic5C24×C314yes—
336C2×Dic5C24×C314yes—
337C2×Dic5C24×C312yes—
338Dic10—28no—
339C22×Q8—51no—
340C22×Q8C2×C3×D511yescharacteristic
341C23×C4C2×C3×D511yescharacteristic
342C4⋊Q8—58no—
343#126—516no—
344#126—516no—
345#125—516no—
346#125—58no—
347#125—516no—
348C22⋊Q8—58no—
349C22⋊Q8C2×C3×D518yes—
350C4×Q8—58no—
351C4×Q8—58no—
352#117—58no—
353C2×C4⋊C4—51no—
354C2×C4⋊C4—51no—
355C2×C4⋊C4—51no—
356C2×C4⋊C4—52no—
357C2×C4⋊C4—52no—
358C2×C4⋊C4—52no—
359C2×C4⋊C4—52no—
360C2×C4⋊C4—52no—
361C2×C4⋊C4—52no—
362C2×C4⋊C4—52no—
363C2×C4⋊C4C2×C3×D511yescharacteristic
364C2×C4⋊C4C2×C3×D512yes—
365C2×C22⋊C4—51no—
366C2×C22⋊C4—51no—
367C2×C22⋊C4—52no—
368C2×C22⋊C4C2×C3×D512yes—
369C2×C42—51no—
370C2×C42—51no—
371C2×C42—52no—
372C2×C42—52no—
373C2×C3×C5—42no—
374C2×C3×C5#54114yescomplemented
375C2×C3×C5C22×#4111yescharacteristic
376C2×C3×C5C2×2−1+411yescharacteristic
377C2×C3×C5C2×2−1+411yescharacteristic
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—22no—
381C23×C3C23×D512yes—
382C23×C3C23×D511yescharacteristic
383C3×Q8—28no—
384C3×Q8—108no—
385C2×C3×C4—22no—
386C2×C3×C4—24no—
387C2×C3×C4—24no—
388C2×C3×C4—28no—
389C2×C3×C4—52no—
390C2×C3×C4—52no—
391C2×C3×C4—52no—
392C2×C3×C4—54no—
393C2×C3×C4—52no—
394C2×C3×C4—54no—
395C2×C3×C4—104no—
396C2×C3×C4—102no—
397C2×C3×C4—102no—
398C2×C3×C4—104no—
399C2×C3×C4—102no—
400C2×C3×C4—102no—
401C2×C3×C4#71114yes—
402C2×C3×C4C23×D514yes—
403C2×C3×C4C23×D514yes—
404C2×C3×C4C23×D512yes—
405C2×C3×C4C23×D512yes—
406C22×C5—22no—
407C22×C5—22no—
408C22×C5—22no—
409C22×C5—44no—
410C22×C5C2×C3×#4112yes—
411C22×C5C3×2−1+412yes—
412C22×C5C3×2−1+412yes—
413C22×C5C25×C311yescharacteristic
414C4×C5—22no—
415C4×C5—24no—
416C4×C5—24no—
417C4×C5C2×C3×#4114yes—
418Dic5—24no—
419Dic5—22no—
420Dic5—24no—
421Dic5—22no—
422Dic5—22no—
423Dic5—22no—
424C24C22×C3×D511yescharacteristic
425C2×Q8—24no—
426C2×Q8—54no—
427C2×Q8—104no—
428C2×Q8C22×C3×D514yes—
429C22×C4—24no—
430C22×C4—51no—
431C22×C4—51no—
432C22×C4—51no—
433C22×C4—52no—
434C22×C4—51no—
435C22×C4—52no—
436C22×C4C22×C3×D512yes—
437C22×C4C22×C3×D512yes—
438C22×C4C22×C3×D511yescharacteristic
439C22×C4C22×C3×D514yes—
440C22×C4C22×C3×D511yescharacteristic
441C4⋊C4—54no—
442C4⋊C4—54no—
443C4⋊C4—54no—
444C4⋊C4—58no—
445C4⋊C4—58no—
446C4⋊C4—58no—
447C4⋊C4—58no—
448C4⋊C4—58no—
449C4⋊C4—58no—
450C4⋊C4—58no—
451C4⋊C4C22×C3×D514yes—
452C4⋊C4C22×C3×D518yes—
453C22⋊C4—54no—
454C22⋊C4—54no—
455C22⋊C4—58no—
456C22⋊C4C22×C3×D518yes—
457C42—54no—
458C42—54no—
459C42—58no—
460C42—58no—
461C3×C5C2×#54111yescharacteristic, complemented, Hall
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—44no—
466C22×C3C2×#71112yes—
467C22×C3#414712yes—
468C22×C3#415012yes—
469C22×C3C24×D511yescharacteristic
470C3×C4—22no—
471C3×C4—24no—
472C3×C4—24no—
473C3×C4—104no—
474C3×C4—102no—
475C3×C4—102no—
476C3×C4—104no—
477C3×C4—102no—
478C3×C4—102no—
479C3×C4C2×#71114yes—
480C2×C5—42no—
481C2×C5C3×#54114yescomplemented
482C2×C5C22×C3×#4111yescharacteristic
483C2×C5C2×C3×2−1+411yescharacteristic
484C2×C5C2×C3×2−1+411yescharacteristic
485C23—22no—
486C23—22no—
487C23—22no—
488C23C23×C3×D512yes—
489C23C23×C3×D511yescharacteristic
490Q8—28no—
491Q8—108no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—52no—
497C2×C4—52no—
498C2×C4—52no—
499C2×C4—54no—
500C2×C4—52no—
501C2×C4—54no—
502C2×C4—104no—
503C2×C4—102no—
504C2×C4—102no—
505C2×C4—104no—
506C2×C4—102no—
507C2×C4—102no—
508C2×C4C3×#71114yes—
509C2×C4C23×C3×D514yes—
510C2×C4C23×C3×D514yes—
511C2×C4C23×C3×D512yes—
512C2×C4C23×C3×D512yes—
513C2×C3—42no—
514C2×C3#6595914yesdirect product factor
515C2×C3C22×#71111yescharacteristic
516C2×C3C2×#414711yescharacteristic
517C2×C3C2×#415011yescharacteristic
518C5C2×C3×#54111yescharacteristic, complemented, Sylow
519C22—22no—
520C22—22no—
521C22—22no—
522C22—44no—
523C22C2×C3×#71112yes—
524C22C3×#414712yes—
525C22C3×#415012yes—
526C22C24×C3×D511yescharacteristic
527C4—22no—
528C4—24no—
529C4—24no—
530C4—104no—
531C4—102no—
532C4—102no—
533C4—104no—
534C4—102no—
535C4—102no—
536C4C2×C3×#71114yes—
537C3C2×#6595911yescharacteristic, direct product factor, Sylow
538C2—42no—
539C2C3×#6595914yesdirect product factor
540C2C22×C3×#71111yescharacteristic
541C2C2×C3×#414711yescharacteristic
542C2C2×C3×#415011yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutator, cosocleC22×C5C25×C3
FrattiniC22C24×C3×D5
FittingC2×C3×C5×C22⋊Q8C2
socleC23×C3×C5C24
prime radicalC24×C3×C5C23
hypercenterC2×C3×C22⋊Q8D5
hypocenterC5C2×C3×#541
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8×Dic5C2semidirect
3. C2×C3×#4090C2semidirect
6. C3×#65959C2direct (N × H)
7. C2×C3×#4036C2semidirect
8. C2×C3×#4036C2semidirect
9. C2×C3×#4035C2semidirect
10. C2×C3×#4033C2semidirect
13. C2×#65959C3direct (N × H)
24. C3×Q8×Dic5C22semidirect
25. C3×#4090C22semidirect
42. C3×#4036C22semidirect
43. C3×#4036C22semidirect
44. C3×#4035C22semidirect
45. C3×#4033C22semidirect
50. C2×Q8×Dic5C2×C3semidirect
51. C2×#4090C2×C3semidirect
54. #65959C2×C3direct (N × H)
55. C2×#4036C2×C3semidirect
56. C2×#4036C2×C3semidirect
57. C2×#4035C2×C3semidirect
58. C2×#4033C2×C3semidirect
120. Q8×Dic5C22×C3semidirect
121. #4090C22×C3semidirect
138. #4036C22×C3semidirect
139. #4036C22×C3semidirect
140. #4035C22×C3semidirect
141. #4033C22×C3semidirect
374. C2×C3×C5#541semidirect
461. C3×C5C2×#541semidirect
481. C2×C5C3×#541semidirect
514. C2×C3#65959direct (N × H)
518. C5C2×C3×#541semidirect
537. C3C2×#65959direct (N × H)
539. C2C3×#65959direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×C22⋊Q82
C2×C3×Q8×Dic52
C2×C3×#40902
C2×C3×#40722
C2×C3×#40702
C3×#659592
C2×C3×#40362
C2×C3×#40352
C2×C3×#40332
C2×C3×#40252
C2×#659593
C2×C3×#5415