Finite Group Databasebeta

← C3×C4×#4086 #14322246

Subgroups — C3×C4×#4086 #14322246

Order 1920.

1848 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C5×D4C211yesmaximal, characteristic, complemented
2C2×C3×#4086C211yesmaximal, characteristic
3C2×C3×C4×C22⋊D5C214yesmaximal, complemented
4C2×C3×C4×D20C211yesmaximal, characteristic, complemented
5C3×#65510C211yesmaximal, characteristic, complemented
6C3×#65277C211yesmaximal, characteristic, complemented
7C3×#65265C211yesmaximal, characteristic, complemented
8C3×#65246C214yesmaximal, complemented
9C3×C42×Dic5C211yesmaximal, characteristic, complemented
10C4×#4086C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2212yes—
13C22×C3×C22⋊D5C2214yes—
14C22×C3×D20C2211yescharacteristic
15C22×C3×C4×D5C2212yes—
16C3×C4×C5×D4—24no—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2212yes—
19C2×C3×C42×C5C2211yescharacteristic
20C2×C3×C22⋊Dic5C2214yes—
21C3×#4086C418yesdirect product factor
22C3×C4×C22⋊D5—216no—
23C2×C3×#685C2214yes—
24C2×C3×#685C2212yes—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×#683C2214yes—
27C2×C3×C4×Dic5C2212yes—
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×C3×C4×Dic5C2211yescharacteristic
30C2×C3×C4×Dic5C2212yes—
31C3×C4×D20—24no—
32C3×C4×#127—51nomaximal, Hall
33C2×C4×C5×D4C2×C311yescharacteristic, complemented
34C2×#4086C2×C311yescharacteristic
35C2×C4×C22⋊D5C2×C314yescomplemented
36C2×C4×D20C2×C311yescharacteristic, complemented
37#65510C2×C311yescharacteristic, complemented
38#65277C2×C311yescharacteristic, complemented
39#65265C2×C311yescharacteristic, complemented
40#65246C2×C314yescomplemented
41C42×Dic5C2×C311yescharacteristic, complemented
42C24×C3×C5C2312yes—
43C23×C3×D5C2312yes—
44C2×C3×C5×D4—22no—
45C2×C3×C5×D4—22no—
46C2×C3×C5×D4C2×C414yescomplemented
47C22×C3×C4×C5—24no—
48C22×C3×C4×C5—28no—
49C22×C3×C4×C5C2312yes—
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C2×C3×C22⋊D5—28no—
54C2×C3×C22⋊D5—28no—
55C2×C3×C22⋊D5C2×C4116yescomplemented
56C22×C3×Dic5C2312yes—
57C22×C3×Dic5C2312yes—
58C22×C3×Dic5C2312yes—
59C2×C3×D20—22no—
60C2×C3×D20—22no—
61C2×C3×D20C2×C414yescomplemented
62C2×C3×C4×D5—24no—
63C2×C3×C4×D5—28no—
64C3×C5×C4⋊C4—22no—
65C3×C5×C22⋊C4—24no—
66C3×C42×C5D414yescomplemented
67C3×C22⋊Dic5—28no—
68C3×#685—24no—
69C3×#685—28no—
70C3×C4⋊Dic5—22no—
71C3×#683—28no—
72C3×C4×Dic5—22no—
73C3×C4×Dic5—24no—
74C3×C4×Dic5C2×C414yescomplemented
75C3×C4×Dic5D418yescomplemented
76C2×C3×#127—51no—
77C2×C3×C4×D4—54no—
78C2×C3×C4×D4—51no—
79C2×C3×C4×D4D511yescharacteristic, complemented
80C3×#390—51no—
81C3×#390—51no—
82C3×#390—54no—
83C3×#384—51no—
84C3×C43—51no—
85C22×C5×D4C22×C311yescharacteristic
86C23×C4×C5C22×C312yes—
87C22×C22⋊D5C22×C314yes—
88C22×D20C22×C311yescharacteristic
89C22×C4×D5C22×C312yes—
90C4×C5×D4—24no—
91C2×C5×C4⋊C4C22×C311yescharacteristic
92C2×C5×C22⋊C4C22×C312yes—
93C2×C42×C5C22×C311yescharacteristic
94C2×C22⋊Dic5C22×C314yes—
95#4086C3×C418yesdirect product factor
96C4×C22⋊D5—216no—
97C2×#685C22×C312yes—
98C2×#685C22×C314yes—
99C2×C4⋊Dic5C22×C311yescharacteristic
100C2×#683C22×C314yes—
101C2×C4×Dic5C22×C312yes—
102C2×C4×Dic5C22×C311yescharacteristic
103C2×C4×Dic5C22×C311yescharacteristic
104C2×C4×Dic5C22×C312yes—
105C4×D20—24no—
106C4×#127—51noSylow
107C23×C3×C5—24no—
108C23×C3×C5—22no—
109C23×C3×C5—24no—
110C23×C3×C5—22no—
111C23×C3×C5C22×C414yes—
112C23×C3×C5C2411yescharacteristic
113C22×C3×D5—22no—
114C22×C3×D5—24no—
115C22×C3×D5—22no—
116C22×C3×D5—24no—
117C22×C3×D5C22×C414yes—
118C3×C5×D4—28no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—24no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—48no—
123C2×C3×C4×C5C22×C412yes—
124C2×C3×C4×C5C2×D412yes—
125C2×C3×C4×C5C2×D412yes—
126C2×C3×C4×C5C2×D414yes—
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5#4112yes—
129C3×C22⋊D5—232no—
130C2×C3×Dic5—22no—
131C2×C3×Dic5—22no—
132C2×C3×Dic5—24no—
133C2×C3×Dic5—22no—
134C2×C3×Dic5C22×C414yes—
135C2×C3×Dic5C2×D414yes—
136C2×C3×Dic5C2×D414yes—
137C2×C3×Dic5#4114yes—
138C3×D20—28no—
139C3×C4×D5—48no—
140C22×C3×D4—54no—
141C22×C3×D4—51no—
142C22×C3×D4C2×D511yescharacteristic
143C23×C3×C4—52no—
144C23×C3×C4C2×D512yes—
145C3×#127—58no—
146C3×C4×D4—24no—
147C3×C4×D4—1016no—
148C3×C4×D4—104no—
149C2×C3×C4⋊C4—54no—
150C2×C3×C4⋊C4—51no—
151C2×C3×C4⋊C4C2×D511yescharacteristic
152C2×C3×C22⋊C4—54no—
153C2×C3×C22⋊C4—54no—
154C2×C3×C22⋊C4—52no—
155C2×C3×C22⋊C4C2×D512yes—
156C2×C3×C42—51no—
157C2×C3×C42—52no—
158C2×C3×C42—52no—
159C2×C3×C42—51no—
160C2×C3×C42C2×D511yescharacteristic
161C24×C5C23×C312yes—
162C23×D5C23×C312yes—
163C2×C5×D4—22no—
164C2×C5×D4—22no—
165C2×C5×D4C2×C3×C414yescomplemented
166C22×C4×C5—24no—
167C22×C4×C5—28no—
168C22×C4×C5C23×C312yes—
169C22×C4×C5C23×C311yescharacteristic
170C22×C4×C5C23×C311yescharacteristic
171C22×C4×C5C23×C311yescharacteristic
172C2×C22⋊D5—28no—
173C2×C22⋊D5—28no—
174C2×C22⋊D5C2×C3×C4116yescomplemented
175C22×Dic5C23×C312yes—
176C22×Dic5C23×C312yes—
177C22×Dic5C23×C312yes—
178C2×D20—22no—
179C2×D20—22no—
180C2×D20C2×C3×C414yescomplemented
181C2×C4×D5—24no—
182C2×C4×D5—28no—
183C5×C4⋊C4—22no—
184C5×C22⋊C4—24no—
185C42×C5C3×D414yescomplemented
186C22⋊Dic5—28no—
187#685—24no—
188#685—28no—
189C4⋊Dic5—22no—
190#683—28no—
191C4×Dic5—22no—
192C4×Dic5—24no—
193C4×Dic5C2×C3×C414yescomplemented
194C4×Dic5C3×D418yescomplemented
195C2×#127—51no—
196C2×C4×D4—54no—
197C2×C4×D4—51no—
198C2×C4×D4C3×D511yescharacteristic, complemented
199#390—51no—
200#390—51no—
201#390—54no—
202#384—51no—
203C43—51no—
204C22×C3×C5—28no—
205C22×C3×C5—24no—
206C22×C3×C5—42no—
207C22×C3×C5—44no—
208C22×C3×C5—42no—
209C22×C3×C5C23×C411yescharacteristic
210C22×C3×C5C22×D412yes—
211C22×C3×C5C22×D411yescharacteristic
212C22×C3×C5C2×#4111yescharacteristic
213C22×C3×C5C2×#4112yes—
214C2×C3×D5—24no—
215C2×C3×D5—28no—
216C2×C3×D5—42no—
217C2×C3×D5—44no—
218C2×C3×D5—42no—
219C3×C4×C5—22no—
220C3×C4×C5—42no—
221C3×C4×C5C4×D414yescomplemented
222C3×C4×C5#12714yescomplemented
223C3×Dic5—24no—
224C3×Dic5—42no—
225C3×Dic5C4×D418yescomplemented
226C24×C3—52no—
227C24×C3C22×D512yes—
228C2×C3×D4—22no—
229C2×C3×D4—22no—
230C2×C3×D4—516no—
231C2×C3×D4—54no—
232C2×C3×D4—108no—
233C2×C3×D4—108no—
234C2×C3×D4—102no—
235C2×C3×D4—102no—
236C2×C3×D4C4×D514yescomplemented
237C22×C3×C4—28no—
238C22×C3×C4—24no—
239C22×C3×C4—52no—
240C22×C3×C4—52no—
241C22×C3×C4—52no—
242C22×C3×C4—108no—
243C22×C3×C4—104no—
244C22×C3×C4C22×D512yes—
245C22×C3×C4C22×D511yescharacteristic
246C22×C3×C4C22×D511yescharacteristic
247C22×C3×C4C22×D511yescharacteristic
248C3×C4⋊C4—22no—
249C3×C4⋊C4—108no—
250C3×C4⋊C4—102no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—108no—
253C3×C22⋊C4—108no—
254C3×C22⋊C4—104no—
255C3×C42—54no—
256C3×C42—58no—
257C3×C42—104no—
258C3×C42—102no—
259C3×C42C22⋊D514yescomplemented
260C23×C5—24no—
261C23×C5—22no—
262C23×C5—24no—
263C23×C5—22no—
264C23×C5C22×C3×C414yes—
265C23×C5C24×C311yescharacteristic
266C22×D5—22no—
267C22×D5—24no—
268C22×D5—22no—
269C22×D5—24no—
270C22×D5C22×C3×C414yes—
271C5×D4—28no—
272C2×C4×C5—22no—
273C2×C4×C5—24no—
274C2×C4×C5—22no—
275C2×C4×C5—48no—
276C2×C4×C5C22×C3×C412yes—
277C2×C4×C5C2×C3×D414yes—
278C2×C4×C5C2×C3×D412yes—
279C2×C4×C5C2×C3×D412yes—
280C2×C4×C5C2×C3×D412yes—
281C2×C4×C5C3×#4112yes—
282C22⋊D5—232no—
283C2×Dic5—22no—
284C2×Dic5—22no—
285C2×Dic5—24no—
286C2×Dic5—22no—
287C2×Dic5C22×C3×C414yes—
288C2×Dic5C2×C3×D414yes—
289C2×Dic5C2×C3×D414yes—
290C2×Dic5C3×#4114yes—
291D20—28no—
292C4×D5—48no—
293C22×D4—54no—
294C22×D4—51no—
295C22×D4C2×C3×D511yescharacteristic
296C23×C4—52no—
297C23×C4C2×C3×D512yes—
298#127—58no—
299C4×D4—24no—
300C4×D4—1016no—
301C4×D4—104no—
302C2×C4⋊C4—54no—
303C2×C4⋊C4—51no—
304C2×C4⋊C4C2×C3×D511yescharacteristic
305C2×C22⋊C4—54no—
306C2×C22⋊C4—54no—
307C2×C22⋊C4—52no—
308C2×C22⋊C4C2×C3×D512yes—
309C2×C42—52no—
310C2×C42—51no—
311C2×C42—52no—
312C2×C42—51no—
313C2×C42C2×C3×D511yescharacteristic
314C2×C3×C5—44no—
315C2×C3×C5C2×C4×D411yescharacteristic
316C2×C3×C5C2×C4×D412yes—
317C2×C3×C5C2×#12711yescharacteristic
318C2×C3×C5#53212yes—
319C2×C3×C5#53211yescharacteristic
320C3×D5—44no—
321C23×C3—22no—
322C23×C3—24no—
323C23×C3—24no—
324C23×C3—22no—
325C23×C3—54no—
326C23×C3—102no—
327C23×C3—104no—
328C23×C3—104no—
329C23×C3—102no—
330C23×C3C2×C4×D514yes—
331C23×C3C23×D511yescharacteristic
332C3×D4—28no—
333C3×D4—1032no—
334C3×D4—108no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—48no—
339C2×C3×C4—54no—
340C2×C3×C4—54no—
341C2×C3×C4—54no—
342C2×C3×C4—54no—
343C2×C3×C4—102no—
344C2×C3×C4—102no—
345C2×C3×C4—104no—
346C2×C3×C4—102no—
347C2×C3×C4—208no—
348C2×C3×C4C2×C4×D512yes—
349C2×C3×C4#70912yes—
350C2×C3×C4D4×D514yes—
351C2×C3×C4C2×C22⋊D512yes—
352C2×C3×C4C2×C22⋊D512yes—
353C2×C3×C4C2×C22⋊D512yes—
354C22×C5—28no—
355C22×C5—24no—
356C22×C5—42no—
357C22×C5—44no—
358C22×C5—42no—
359C22×C5C23×C3×C411yescharacteristic
360C22×C5C22×C3×D412yes—
361C22×C5C22×C3×D411yescharacteristic
362C22×C5C2×C3×#4111yescharacteristic
363C22×C5C2×C3×#4112yes—
364C2×D5—24no—
365C2×D5—28no—
366C2×D5—42no—
367C2×D5—44no—
368C2×D5—42no—
369C4×C5—22no—
370C4×C5—42no—
371C4×C5C3×C4×D414yescomplemented
372C4×C5C3×#12714yescomplemented
373Dic5—24no—
374Dic5—42no—
375Dic5C3×C4×D418yescomplemented
376C24—52no—
377C24C22×C3×D512yes—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—516no—
381C2×D4—54no—
382C2×D4—108no—
383C2×D4—108no—
384C2×D4—102no—
385C2×D4—102no—
386C2×D4C3×C4×D514yescomplemented
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—52no—
390C22×C4—52no—
391C22×C4—52no—
392C22×C4—108no—
393C22×C4—104no—
394C22×C4C22×C3×D512yes—
395C22×C4C22×C3×D511yescharacteristic
396C22×C4C22×C3×D511yescharacteristic
397C22×C4C22×C3×D511yescharacteristic
398C4⋊C4—22no—
399C4⋊C4—108no—
400C4⋊C4—102no—
401C22⋊C4—24no—
402C22⋊C4—108no—
403C22⋊C4—108no—
404C22⋊C4—104no—
405C42—54no—
406C42—58no—
407C42—104no—
408C42—102no—
409C42C3×C22⋊D514yescomplemented
410C3×C5C4×#12711yescharacteristic, complemented, Hall
411C22×C3—24no—
412C22×C3—28no—
413C22×C3—42no—
414C22×C3—44no—
415C22×C3—42no—
416C22×C3—104no—
417C22×C3—108no—
418C22×C3—202no—
419C22×C3—204no—
420C22×C3—202no—
421C22×C3C22×C4×D511yescharacteristic
422C22×C3C2×#70911yescharacteristic
423C22×C3C2×D4×D512yes—
424C22×C3D5×#4112yes—
425C22×C3C22×C22⋊D511yescharacteristic
426C3×C4—22no—
427C3×C4—42no—
428C3×C4—58no—
429C3×C4—104no—
430C3×C4—202no—
431C3×C4C4×C22⋊D514yescomplemented
432C3×C4#408614yesdirect product factor
433C2×C5—44no—
434C2×C5C2×C3×C4×D411yescharacteristic
435C2×C5C2×C3×C4×D412yes—
436C2×C5C2×C3×#12711yescharacteristic
437C2×C5C3×#53212yes—
438C2×C5C3×#53211yescharacteristic
439D5—44no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—54no—
445C23—102no—
446C23—104no—
447C23—104no—
448C23—102no—
449C23C2×C3×C4×D514yes—
450C23C23×C3×D511yescharacteristic
451D4—28no—
452D4—1032no—
453D4—108no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—54no—
459C2×C4—54no—
460C2×C4—54no—
461C2×C4—54no—
462C2×C4—102no—
463C2×C4—102no—
464C2×C4—104no—
465C2×C4—102no—
466C2×C4—208no—
467C2×C4C2×C3×C4×D512yes—
468C2×C4C3×#70912yes—
469C2×C4C3×D4×D514yes—
470C2×C4C2×C3×C22⋊D512yes—
471C2×C4C2×C3×C22⋊D512yes—
472C2×C4C2×C3×C22⋊D512yes—
473C2×C3—44no—
474C2×C3—204no—
475C2×C3C4×D4×D512yes—
476C2×C3#6588212yes—
477C2×C3C2×C4×C22⋊D511yescharacteristic
478C2×C3C2×#408611yescharacteristic
479C2×C3#6616611yescharacteristic
480C5C3×C4×#12711yescharacteristic, complemented, Sylow
481C22—24no—
482C22—28no—
483C22—42no—
484C22—44no—
485C22—42no—
486C22—104no—
487C22—108no—
488C22—202no—
489C22—204no—
490C22—202no—
491C22C22×C3×C4×D511yescharacteristic
492C22C2×C3×#70911yescharacteristic
493C22C2×C3×D4×D512yes—
494C22C3×D5×#4112yes—
495C22C22×C3×C22⋊D511yescharacteristic
496C4—22no—
497C4—42no—
498C4—58no—
499C4—104no—
500C4—202no—
501C4C3×C4×C22⋊D514yescomplemented
502C4C3×#408614yesdirect product factor
503C3C4×#408611yescharacteristic, direct product factor, Sylow
504C2—44no—
505C2—204no—
506C2C3×C4×D4×D512yes—
507C2C3×#6588212yes—
508C2C2×C3×C4×C22⋊D511yescharacteristic
509C2C2×C3×#408611yescharacteristic
510C2C3×#6616611yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC2×C3×C4×C5×D4C2
socleC23×C3×C5C24
prime radicalC2×C3×#4086C2
cosocleC23×C5C24×C3
hypercenterC2×C3×C4×D4D5
hypocenterC5C3×C4×#127
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×D4C2semidirect
3. C2×C3×C4×C22⋊D5C2semidirect
4. C2×C3×C4×D20C2semidirect
5. C3×#65510C2semidirect
6. C3×#65277C2semidirect
7. C3×#65265C2semidirect
8. C3×#65246C2semidirect
9. C3×C42×Dic5C2semidirect
10. C4×#4086C3direct (N × H)
21. C3×#4086C4direct (N × H)
33. C2×C4×C5×D4C2×C3semidirect
35. C2×C4×C22⋊D5C2×C3semidirect
36. C2×C4×D20C2×C3semidirect
37. #65510C2×C3semidirect
38. #65277C2×C3semidirect
39. #65265C2×C3semidirect
40. #65246C2×C3semidirect
41. C42×Dic5C2×C3semidirect
46. C2×C3×C5×D4C2×C4semidirect
55. C2×C3×C22⋊D5C2×C4semidirect
61. C2×C3×D20C2×C4semidirect
66. C3×C42×C5D4semidirect
74. C3×C4×Dic5C2×C4semidirect
75. C3×C4×Dic5D4semidirect
79. C2×C3×C4×D4D5semidirect
95. #4086C3×C4direct (N × H)
165. C2×C5×D4C2×C3×C4semidirect
174. C2×C22⋊D5C2×C3×C4semidirect
180. C2×D20C2×C3×C4semidirect
185. C42×C5C3×D4semidirect
193. C4×Dic5C2×C3×C4semidirect
194. C4×Dic5C3×D4semidirect
198. C2×C4×D4C3×D5semidirect
221. C3×C4×C5C4×D4semidirect
222. C3×C4×C5#127semidirect
225. C3×Dic5C4×D4semidirect
236. C2×C3×D4C4×D5semidirect
259. C3×C42C22⋊D5semidirect
371. C4×C5C3×C4×D4semidirect
372. C4×C5C3×#127semidirect
375. Dic5C3×C4×D4semidirect
386. C2×D4C3×C4×D5semidirect
409. C42C3×C22⋊D5semidirect
410. C3×C5C4×#127semidirect
431. C3×C4C4×C22⋊D5semidirect
432. C3×C4#4086direct (N × H)
480. C5C3×C4×#127semidirect
501. C4C3×C4×C22⋊D5semidirect
502. C4C3×#4086direct (N × H)
503. C3C4×#4086direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×D42
C2×C3×#40862
C2×C3×C4×C22⋊D52
C2×C3×C4×D202
C3×#655102
C3×#652772
C3×#652652
C3×#652462
C3×C42×Dic52
C4×#40863
C3×C4×#1275