Finite Group Databasebeta

← C2×C3×#65849 #14480933

Subgroups — C2×C3×#65849 #14480933

Order 1920.

2288 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×#4076C211yesmaximal, characteristic
3C2×C3×#4075C212yesmaximal, complemented
4C2×C3×C4×C22⋊D5C212yesmaximal, complemented
5C2×C3×#4070C211yesmaximal, characteristic, complemented
6C3×#65849C2116yesmaximal, direct product factor
7C2×C3×#4044C212yesmaximal, complemented
8C2×C3×#4042C212yesmaximal, complemented
9C2×C3×#4040C212yesmaximal, complemented
10C2×C3×#4035C212yesmaximal, complemented
11C2×#65849C311yesmaximal, characteristic, direct product factor, Hall
12C23×C3×C4×C5C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C22×C3×C22⋊D5C2212yes—
16C22×C3×D20C2211yescharacteristic
17C22×C3×C4×D5C2212yes—
18C22×C3×Dic10C2211yescharacteristic
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C4⋊C4C2212yes—
21C2×C3×C5×C4⋊C4C2218yescomplemented
22C2×C3×C22⋊Dic5C2212yes—
23C3×#4076C2218yes—
24C3×#4075C22116yescomplemented
25C3×C4×C22⋊D5C22116yescomplemented
26C2×C3×#685C2212yes—
27C2×C3×#685C2212yes—
28C2×C3×#685C2212yes—
29C2×C3×#685C2212yes—
30C2×C3×C4⋊Dic5C2211yescharacteristic
31C2×C3×C4⋊Dic5C2211yescharacteristic
32C3×#4070C2218yescomplemented
33C2×C3×#683C2212yes—
34C2×C3×#683C2212yes—
35C2×C3×#683C2212yes—
36C2×C3×#683C2212yes—
37C2×C3×C4×Dic5C2212yes—
38C3×#4044C22116yescomplemented
39C3×#4042C22116yescomplemented
40C3×#4040C22116yescomplemented
41C3×#4035C22116yescomplemented
42C2×C3×#539—51nomaximal, Hall
43C22×C5×C4⋊C4C2×C311yescharacteristic, complemented
44C2×#4076C2×C311yescharacteristic
45C2×#4075C2×C312yescomplemented
46C2×C4×C22⋊D5C2×C312yescomplemented
47C2×#4070C2×C311yescharacteristic, complemented
48#65849C2×C3116yesdirect product factor
49C2×#4044C2×C312yescomplemented
50C2×#4042C2×C312yescomplemented
51C2×#4040C2×C312yescomplemented
52C2×#4035C2×C312yescomplemented
53C24×C3×C5C2311yescharacteristic
54C23×C3×D5C2312yes—
55C22×C3×C4×C5—24no—
56C22×C3×C4×C5—42no—
57C22×C3×C4×C5—42no—
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2311yescharacteristic
61C22×C3×C4×C5C2312yes—
62C22×C3×C4×C5C2314yes—
63C22×C3×C4×C5C2314yes—
64C22×C3×C4×C5C2314yes—
65C2×C3×C22⋊D5—28no—
66C2×C3×C22⋊D5C2318yes—
67C22×C3×Dic5C2312yes—
68C22×C3×Dic5C2312yes—
69C22×C3×Dic5C2312yes—
70C2×C3×D20—42no—
71C2×C3×D20C2314yes—
72C2×C3×C4×D5—44no—
73C2×C3×C4×D5C2318yes—
74C2×C3×Dic10—42no—
75C2×C3×Dic10C2314yes—
76C3×C5×C4⋊C4C2318yes—
77C3×C5×C4⋊C4C2318yes—
78C3×C22⋊Dic5C2318yes—
79C3×#685C2318yes—
80C3×#685C2318yes—
81C3×#685C2318yes—
82C3×#685C2318yes—
83C3×C4⋊Dic5C2314yes—
84C3×C4⋊Dic5C2314yes—
85C3×#683C2318yes—
86C3×#683C2318yes—
87C3×#683C2318yes—
88C3×#683C2318yes—
89C3×C4×Dic5C2318yes—
90C3×#539—516no—
91C2×C3×#126—52no—
92C2×C3×#125—52no—
93C2×C3×#123—52no—
94C2×C3×#123—52no—
95C2×C3×C22⋊Q8—51no—
96C2×C3×C22⋊Q8—52no—
97C2×C3×#121—51no—
98C2×C3×C4×D4—52no—
99C22×C3×C4⋊C4D511yescharacteristic, complemented
100C23×C4×C5C22×C311yescharacteristic
101C23×C4×C5C22×C311yescharacteristic
102C23×C4×C5C22×C311yescharacteristic
103C22×C22⋊D5C22×C312yes—
104C22×D20C22×C311yescharacteristic
105C22×C4×D5C22×C312yes—
106C22×Dic10C22×C311yescharacteristic
107C2×C5×C4⋊C4C22×C312yes—
108C2×C5×C4⋊C4C22×C312yes—
109C2×C5×C4⋊C4C22×C318yescomplemented
110C2×C22⋊Dic5C22×C312yes—
111#4076C22×C318yes—
112#4075C22×C3116yescomplemented
113C4×C22⋊D5C22×C3116yescomplemented
114C2×#685C22×C312yes—
115C2×#685C22×C312yes—
116C2×#685C22×C312yes—
117C2×#685C22×C312yes—
118C2×C4⋊Dic5C22×C311yescharacteristic
119C2×C4⋊Dic5C22×C311yescharacteristic
120#4070C22×C318yescomplemented
121C2×#683C22×C312yes—
122C2×#683C22×C312yes—
123C2×#683C22×C312yes—
124C2×#683C22×C312yes—
125C2×C4×Dic5C22×C312yes—
126#4044C22×C3116yescomplemented
127#4042C22×C3116yescomplemented
128#4040C22×C3116yescomplemented
129#4035C22×C3116yescomplemented
130C2×#539—51noSylow
131C23×C3×C5—22no—
132C23×C3×C5—22no—
133C23×C3×C5#4114yes—
134C23×C3×C5C2412yes—
135C23×C3×C5C2411yescharacteristic
136C22×C3×D5—24no—
137C22×C3×D5—24no—
138C22×C3×D5—24no—
139C22×C3×D5C2414yes—
140C2×C3×C4×C5—28no—
141C2×C3×C4×C5—24no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—24no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—44no—
146C2×C3×C4×C5—44no—
147C2×C3×C4×C5C2414yes—
148C2×C3×C4×C5C2412yes—
149C2×C3×C4×C5C2414yes—
150C2×C3×C4×C5C2412yes—
151C3×C22⋊D5—216no—
152C2×C3×Dic5—24no—
153C2×C3×Dic5—24no—
154C2×C3×Dic5—24no—
155C2×C3×Dic5C2414yes—
156C2×C3×Dic5C2414yes—
157C2×C3×Dic5C2414yes—
158C3×D20—44no—
159C3×C4×D5—48no—
160C3×Dic10—44no—
161C22×C3×Q8—51no—
162C22×C3×D4—52no—
163C22×C3×D4—51no—
164C23×C3×C4—52no—
165C23×C3×C4C2×D511yescharacteristic
166C23×C3×C4C2×D511yescharacteristic
167C23×C3×C4C2×D511yescharacteristic
168C3×#126—516no—
169C3×#125—516no—
170C3×#123—516no—
171C3×#123—516no—
172C3×C22⋊Q8—58no—
173C3×C22⋊Q8—516no—
174C3×#121—58no—
175C3×C4×D4—516no—
176C2×C3×C4⋊C4—52no—
177C2×C3×C4⋊C4—51no—
178C2×C3×C4⋊C4—52no—
179C2×C3×C4⋊C4—52no—
180C2×C3×C4⋊C4—51no—
181C2×C3×C4⋊C4—52no—
182C2×C3×C4⋊C4C2×D512yes—
183C2×C3×C4⋊C4C2×D512yes—
184C2×C3×C4⋊C4C2×D518yescomplemented
185C2×C3×C22⋊C4—52no—
186C2×C3×C22⋊C4—52no—
187C2×C3×C22⋊C4—52no—
188C2×C3×C22⋊C4—52no—
189C2×C3×C22⋊C4—52no—
190C2×C3×C42—52no—
191C24×C5C23×C311yescharacteristic
192C23×D5C23×C312yes—
193C22×C4×C5—24no—
194C22×C4×C5—42no—
195C22×C4×C5—42no—
196C22×C4×C5C23×C311yescharacteristic
197C22×C4×C5C23×C312yes—
198C22×C4×C5C23×C311yescharacteristic
199C22×C4×C5C23×C312yes—
200C22×C4×C5C23×C314yes—
201C22×C4×C5C23×C314yes—
202C22×C4×C5C23×C314yes—
203C2×C22⋊D5—28no—
204C2×C22⋊D5C23×C318yes—
205C22×Dic5C23×C312yes—
206C22×Dic5C23×C312yes—
207C22×Dic5C23×C312yes—
208C2×D20—42no—
209C2×D20C23×C314yes—
210C2×C4×D5—44no—
211C2×C4×D5C23×C318yes—
212C2×Dic10—42no—
213C2×Dic10C23×C314yes—
214C5×C4⋊C4C23×C318yes—
215C5×C4⋊C4C23×C318yes—
216C22⋊Dic5C23×C318yes—
217#685C23×C318yes—
218#685C23×C318yes—
219#685C23×C318yes—
220#685C23×C318yes—
221C4⋊Dic5C23×C314yes—
222C4⋊Dic5C23×C314yes—
223#683C23×C318yes—
224#683C23×C318yes—
225#683C23×C318yes—
226#683C23×C318yes—
227C4×Dic5C23×C318yes—
228#539—516no—
229C2×#126—52no—
230C2×#125—52no—
231C2×#123—52no—
232C2×#123—52no—
233C2×C22⋊Q8—51no—
234C2×C22⋊Q8—52no—
235C2×#121—51no—
236C2×C4×D4—52no—
237C22×C4⋊C4C3×D511yescharacteristic, complemented
238C22×C3×C5—28no—
239C22×C3×C5—22no—
240C22×C3×C5—22no—
241C22×C3×C5C2×#4114yes—
242C22×C3×C5C2×#4112yes—
243C22×C3×C5H(5,𝔽2)12yes—
244C22×C3×C52−1+412yes—
245C22×C3×C5C2511yescharacteristic
246C2×C3×D5—24no—
247C2×C3×D5—24no—
248C2×C3×D5—24no—
249C2×C3×D5—48no—
250C3×C4×C5—24no—
251C3×C4×C5—22no—
252C3×C4×C5—24no—
253C3×C4×C5—22no—
254C3×Dic5—24no—
255C3×Dic5—24no—
256C3×Dic5—24no—
257C24×C3—52no—
258C24×C3C22×D511yescharacteristic
259C2×C3×Q8—54no—
260C2×C3×Q8—202no—
261C2×C3×D4—58no—
262C2×C3×D4—54no—
263C2×C3×D4—108no—
264C2×C3×D4—202no—
265C22×C3×C4—24no—
266C22×C3×C4—42no—
267C22×C3×C4—42no—
268C22×C3×C4—52no—
269C22×C3×C4—52no—
270C22×C3×C4—52no—
271C22×C3×C4—58no—
272C22×C3×C4—204no—
273C22×C3×C4C22×D514yes—
274C22×C3×C4C22×D512yes—
275C22×C3×C4C22×D512yes—
276C22×C3×C4C22×D511yescharacteristic
277C22×C3×C4C22×D511yescharacteristic
278C22×C3×C4C22×D514yes—
279C22×C3×C4C22×D514yes—
280C3×C4⋊C4—58no—
281C3×C4⋊C4—58no—
282C3×C4⋊C4—58no—
283C3×C4⋊C4—58no—
284C3×C4⋊C4—54no—
285C3×C4⋊C4—54no—
286C3×C4⋊C4C22×D518yes—
287C3×C4⋊C4C22×D518yes—
288C3×C22⋊C4—58no—
289C3×C22⋊C4—58no—
290C3×C22⋊C4—58no—
291C3×C22⋊C4—58no—
292C3×C22⋊C4—58no—
293C3×C42—58no—
294C23×C5—22no—
295C23×C5—22no—
296C23×C5C3×#4114yes—
297C23×C5C24×C312yes—
298C23×C5C24×C311yescharacteristic
299C22×D5—24no—
300C22×D5—24no—
301C22×D5—24no—
302C22×D5C24×C314yes—
303C2×C4×C5—28no—
304C2×C4×C5—24no—
305C2×C4×C5—22no—
306C2×C4×C5—24no—
307C2×C4×C5—22no—
308C2×C4×C5—44no—
309C2×C4×C5—44no—
310C2×C4×C5C24×C314yes—
311C2×C4×C5C24×C312yes—
312C2×C4×C5C24×C314yes—
313C2×C4×C5C24×C312yes—
314C22⋊D5—216no—
315C2×Dic5—24no—
316C2×Dic5—24no—
317C2×Dic5—24no—
318C2×Dic5C24×C314yes—
319C2×Dic5C24×C314yes—
320C2×Dic5C24×C314yes—
321D20—44no—
322C4×D5—48no—
323Dic10—44no—
324C22×Q8—51no—
325C22×D4—52no—
326C22×D4—51no—
327C23×C4—52no—
328C23×C4C2×C3×D511yescharacteristic
329C23×C4C2×C3×D511yescharacteristic
330C23×C4C2×C3×D511yescharacteristic
331#126—516no—
332#125—516no—
333#123—516no—
334#123—516no—
335C22⋊Q8—58no—
336C22⋊Q8—516no—
337#121—58no—
338C4×D4—516no—
339C2×C4⋊C4—52no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4—52no—
342C2×C4⋊C4—52no—
343C2×C4⋊C4—51no—
344C2×C4⋊C4—52no—
345C2×C4⋊C4C2×C3×D512yes—
346C2×C4⋊C4C2×C3×D512yes—
347C2×C4⋊C4C2×C3×D518yescomplemented
348C2×C22⋊C4—52no—
349C2×C22⋊C4—52no—
350C2×C22⋊C4—52no—
351C2×C22⋊C4—52no—
352C2×C22⋊C4—52no—
353C2×C42—52no—
354C2×C3×C5—24no—
355C2×C3×C5#53914yescomplemented
356C2×C3×C5C22×#4111yescharacteristic
357C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
358C2×C3×C5C2×2−1+411yescharacteristic
359C3×D5—44no—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3—54no—
363C23×C3—104no—
364C23×C3—104no—
365C23×C3—104no—
366C23×C3#70914yes—
367C23×C3C23×D512yes—
368C23×C3C23×D511yescharacteristic
369C3×Q8—204no—
370C3×D4—1016no—
371C3×D4—204no—
372C2×C3×C4—28no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—44no—
378C2×C3×C4—44no—
379C2×C3×C4—54no—
380C2×C3×C4—54no—
381C2×C3×C4—54no—
382C2×C3×C4—104no—
383C2×C3×C4—104no—
384C2×C3×C4—104no—
385C2×C3×C4—208no—
386C2×C3×C4C23×D514yes—
387C2×C3×C4C23×D514yes—
388C2×C3×C4C23×D512yes—
389C2×C3×C4C23×D512yes—
390C22×C5—22no—
391C22×C5—22no—
392C22×C5—28no—
393C22×C5C2×C3×#4114yes—
394C22×C5C2×C3×#4112yes—
395C22×C5C3×H(5,𝔽2)12yes—
396C22×C5C3×2−1+412yes—
397C22×C5C25×C311yescharacteristic
398C2×D5—24no—
399C2×D5—24no—
400C2×D5—24no—
401C2×D5—48no—
402C4×C5—24no—
403C4×C5—22no—
404C4×C5—24no—
405C4×C5—22no—
406Dic5—24no—
407Dic5—24no—
408Dic5—24no—
409C24—52no—
410C24C22×C3×D511yescharacteristic
411C2×Q8—54no—
412C2×Q8—202no—
413C2×D4—58no—
414C2×D4—54no—
415C2×D4—108no—
416C2×D4—202no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—52no—
421C22×C4—52no—
422C22×C4—52no—
423C22×C4—58no—
424C22×C4—204no—
425C22×C4C22×C3×D512yes—
426C22×C4C22×C3×D512yes—
427C22×C4C22×C3×D511yescharacteristic
428C22×C4C22×C3×D511yescharacteristic
429C22×C4C22×C3×D514yes—
430C22×C4C22×C3×D514yes—
431C22×C4C22×C3×D514yes—
432C4⋊C4—58no—
433C4⋊C4—54no—
434C4⋊C4—58no—
435C4⋊C4—58no—
436C4⋊C4—54no—
437C4⋊C4—58no—
438C4⋊C4C22×C3×D518yes—
439C4⋊C4C22×C3×D518yes—
440C22⋊C4—58no—
441C22⋊C4—58no—
442C22⋊C4—58no—
443C22⋊C4—58no—
444C22⋊C4—58no—
445C42—58no—
446C3×C5C2×#53911yescharacteristic, complemented, Hall
447C22×C3—28no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—104no—
451C22×C3—104no—
452C22×C3—104no—
453C22×C3—208no—
454C22×C3C2×#70914yes—
455C22×C3C2×#70912yes—
456C22×C3#414412yes—
457C22×C3#414712yes—
458C22×C3C24×D511yescharacteristic
459C3×C4—24no—
460C3×C4—24no—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—104no—
464C3×C4—104no—
465C3×C4—104no—
466C2×C5—24no—
467C2×C5C3×#53914yescomplemented
468C2×C5C22×C3×#4111yescharacteristic
469C2×C5C2×C3×H(5,𝔽2)11yescharacteristic
470C2×C5C2×C3×2−1+411yescharacteristic
471D5—44no—
472C23—22no—
473C23—22no—
474C23—54no—
475C23—104no—
476C23—104no—
477C23—104no—
478C23C3×#70914yes—
479C23C23×C3×D512yes—
480C23C23×C3×D511yescharacteristic
481Q8—204no—
482D4—1016no—
483D4—204no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—54no—
492C2×C4—54no—
493C2×C4—54no—
494C2×C4—104no—
495C2×C4—104no—
496C2×C4—104no—
497C2×C4—208no—
498C2×C4C23×C3×D514yes—
499C2×C4C23×C3×D514yes—
500C2×C4C23×C3×D512yes—
501C2×C4C23×C3×D512yes—
502C2×C3—24no—
503C2×C3—204no—
504C2×C3#6584914yesdirect product factor
505C2×C3C22×#70911yescharacteristic
506C2×C3C2×#414411yescharacteristic
507C2×C3C2×#414711yescharacteristic
508C5C2×C3×#53911yescharacteristic, complemented, Sylow
509C22—28no—
510C22—22no—
511C22—22no—
512C22—104no—
513C22—104no—
514C22—104no—
515C22—208no—
516C22C2×C3×#70914yes—
517C22C2×C3×#70912yes—
518C22C3×#414412yes—
519C22C3×#414712yes—
520C22C24×C3×D511yescharacteristic
521C4—24no—
522C4—24no—
523C4—22no—
524C4—22no—
525C4—104no—
526C4—104no—
527C4—104no—
528C3C2×#6584911yescharacteristic, direct product factor, Sylow
529C2—24no—
530C2—204no—
531C2C3×#6584914yesdirect product factor
532C2C22×C3×#70911yescharacteristic
533C2C2×C3×#414411yescharacteristic
534C2C2×C3×#414711yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutator, cosocleC22×C5C25×C3
FrattiniC22C24×C3×D5
FittingC22×C3×C5×C4⋊C4C2
socleC23×C3×C5C24
prime radicalC2×C3×#4076C2
hypercenterC22×C3×C4⋊C4D5
hypocenterC5C2×C3×#539
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×C4⋊C4C2semidirect
3. C2×C3×#4075C2semidirect
4. C2×C3×C4×C22⋊D5C2semidirect
5. C2×C3×#4070C2semidirect
6. C3×#65849C2direct (N × H)
7. C2×C3×#4044C2semidirect
8. C2×C3×#4042C2semidirect
9. C2×C3×#4040C2semidirect
10. C2×C3×#4035C2semidirect
11. C2×#65849C3direct (N × H)
21. C2×C3×C5×C4⋊C4C22semidirect
24. C3×#4075C22semidirect
25. C3×C4×C22⋊D5C22semidirect
32. C3×#4070C22semidirect
38. C3×#4044C22semidirect
39. C3×#4042C22semidirect
40. C3×#4040C22semidirect
41. C3×#4035C22semidirect
43. C22×C5×C4⋊C4C2×C3semidirect
45. C2×#4075C2×C3semidirect
46. C2×C4×C22⋊D5C2×C3semidirect
47. C2×#4070C2×C3semidirect
48. #65849C2×C3direct (N × H)
49. C2×#4044C2×C3semidirect
50. C2×#4042C2×C3semidirect
51. C2×#4040C2×C3semidirect
52. C2×#4035C2×C3semidirect
99. C22×C3×C4⋊C4D5semidirect
109. C2×C5×C4⋊C4C22×C3semidirect
112. #4075C22×C3semidirect
113. C4×C22⋊D5C22×C3semidirect
120. #4070C22×C3semidirect
126. #4044C22×C3semidirect
127. #4042C22×C3semidirect
128. #4040C22×C3semidirect
129. #4035C22×C3semidirect
184. C2×C3×C4⋊C4C2×D5semidirect
237. C22×C4⋊C4C3×D5semidirect
347. C2×C4⋊C4C2×C3×D5semidirect
355. C2×C3×C5#539semidirect
446. C3×C5C2×#539semidirect
467. C2×C5C3×#539semidirect
504. C2×C3#65849direct (N × H)
508. C5C2×C3×#539semidirect
528. C3C2×#65849direct (N × H)
531. C2C3×#65849direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×C4⋊C42
C2×C3×#40762
C2×C3×#40752
C2×C3×C4×C22⋊D52
C2×C3×#40702
C3×#658492
C2×C3×#40442
C2×C3×#40422
C2×C3×#40402
C2×C3×#40352
C2×#658493
C2×C3×#5395