Finite Group Databasebeta

← C3×D5×#391 #14320920

Subgroups — C3×D5×#391 #14320920

Order 1920.

1856 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
1C22×C3×Q8×D5C211yesmaximal, characteristic
2C2×C3×D5×C4⋊C4C211yesmaximal, characteristic
3C2×C3×C42×D5C211yesmaximal, characteristic
4C3×C5×#391C211yesmaximal, characteristic, complemented
5C3×#65519C211yesmaximal, characteristic, complemented
6C3×#65263C211yesmaximal, characteristic, complemented
7C3×#65218C211yesmaximal, characteristic, complemented
8C3×D5×#95C214yesmaximal
9C3×#64935C214yesmaximal, complemented
10D5×#391C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C2×C3×Q8×D5—24no—
13C2×C3×Q8×D5—24no—
14C2×C3×Q8×D5C418yescomplemented
15C22×C3×C4×D5C2212yes—
16C22×C3×C4×D5C2212yes—
17C22×C3×C4×D5C2212yes—
18C22×C3×C4×D5C2211yescharacteristic
19C22×C3×Dic10C2212yes—
20C22×C3×Dic10C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×C3×C42×C5C2211yescharacteristic
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C2×C3×#683C2212yes—
25C2×C3×C4×Dic5C2212yes—
26C2×C3×C4×Dic5C2211yescharacteristic
27C3×D5×C4⋊C4—24no—
28C3×C42×D5—24no—
29C3×C5×#95C2214yes—
30C3×#3963C2214yes—
31C3×#3963C2214yes—
32C3×#3963C2214yes—
33C2×C3×#391—51nomaximal, Hall
34C22×Q8×D5C2×C311yescharacteristic
35C2×D5×C4⋊C4C2×C311yescharacteristic
36C2×C42×D5C2×C311yescharacteristic
37C5×#391C2×C311yescharacteristic, complemented
38#65519C2×C311yescharacteristic, complemented
39#65263C2×C311yescharacteristic, complemented
40#65218C2×C311yescharacteristic, complemented
41D5×#95C2×C314yes—
42#64935C2×C314yescomplemented
43C23×C3×D5C2311yescharacteristic
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8C2×C414yescomplemented
47C22×C3×C4×C5C2312yes—
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2312yes—
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×Dic5C2312yes—
52C22×C3×Dic5C2312yes—
53C22×C3×Dic5C2312yes—
54C22×C3×Dic5C2311yescharacteristic
55C3×Q8×D5—232no—
56C2×C3×C4×D5—24no—
57C2×C3×C4×D5—24no—
58C2×C3×C4×D5—24no—
59C2×C3×C4×D5—24no—
60C2×C3×C4×D5—24no—
61C2×C3×C4×D5—28no—
62C2×C3×C4×D5C2×C418yes—
63C2×C3×C4×D5C2×C414yes—
64C2×C3×C4×D5D414yes—
65C2×C3×C4×D5D414yes—
66C2×C3×C4×D5Q818yescomplemented
67C2×C3×Dic10—22no—
68C2×C3×Dic10—22no—
69C2×C3×Dic10—24no—
70C2×C3×Dic10—24no—
71C2×C3×Dic10C2×C414yescomplemented
72C2×C3×Dic10C2×C418yescomplemented
73C3×C5×C4⋊C4—22no—
74C3×C42×C5—22no—
75C3×C4⋊Dic5—22no—
76C3×#683—24no—
77C3×C4×Dic5—22no—
78C3×C4×Dic5—24no—
79C23×C3×Q8—51no—
80C22×C3×C4⋊C4—51no—
81C22×C3×C42—51no—
82C3×#391—51no—
83C3×#391—51no—
84C3×#391—51no—
85C3×#391—54no—
86C3×#391D511yescharacteristic, direct product factor
87C2×C3×#95—54no—
88C22×C5×Q8C22×C311yescharacteristic
89C2×Q8×D5—24no—
90C2×Q8×D5—24no—
91C2×Q8×D5C3×C418yescomplemented
92C22×C4×D5C22×C312yes—
93C22×C4×D5C22×C312yes—
94C22×C4×D5C22×C312yes—
95C22×C4×D5C22×C311yescharacteristic
96C22×Dic10C22×C312yes—
97C22×Dic10C22×C311yescharacteristic
98C2×C5×C4⋊C4C22×C311yescharacteristic
99C2×C42×C5C22×C311yescharacteristic
100C2×C4⋊Dic5C22×C311yescharacteristic
101C2×#683C22×C312yes—
102C2×C4×Dic5C22×C312yes—
103C2×C4×Dic5C22×C311yescharacteristic
104D5×C4⋊C4—24no—
105C42×D5—24no—
106C5×#95C22×C314yes—
107#3963C22×C314yes—
108#3963C22×C314yes—
109#3963C22×C314yes—
110C2×#391—51noSylow
111C23×C3×C5C2411yescharacteristic
112C22×C3×D5C22×C412yes—
113C22×C3×D5C2×D412yes—
114C22×C3×D5C2×D412yes—
115C22×C3×D5C2×Q814yes—
116C22×C3×D5#4114yes—
117C3×C5×Q8—28no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—24no—
124C2×C3×C4×C5C22×C414yes—
125C2×C3×C4×C5C22×C412yes—
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5C2×Q814yescomplemented
129C2×C3×Dic5—22no—
130C2×C3×Dic5—22no—
131C2×C3×Dic5—22no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5—22no—
134C2×C3×Dic5—24no—
135C2×C3×Dic5C22×C414yes—
136C2×C3×Dic5C22×C412yes—
137C2×C3×Dic5C2×D412yes—
138C2×C3×Dic5C2×D412yes—
139C2×C3×Dic5C2×Q814yescomplemented
140C3×C4×D5—216no—
141C3×C4×D5—216no—
142C3×C4×D5—48no—
143C3×C4×D5C22⋊C4116yes—
144C3×Dic10—28no—
145C3×Dic10—216no—
146C22×C3×Q8—58no—
147C22×C3×Q8—52no—
148C22×C3×Q8—51no—
149C22×C3×Q8—104no—
150C22×C3×Q8—104no—
151C22×C3×Q8C2×D511yescharacteristic
152C23×C3×C4—51no—
153C23×C3×C4—52no—
154C23×C3×C4—52no—
155C23×C3×C4—52no—
156C2×C3×C4⋊C4—52no—
157C2×C3×C4⋊C4—51no—
158C2×C3×C4⋊C4—104no—
159C2×C3×C4⋊C4C2×D511yescharacteristic
160C2×C3×C42—52no—
161C2×C3×C42—51no—
162C2×C3×C42—104no—
163C2×C3×C42C2×D511yescharacteristic
164C3×#95—54no—
165C3×#95—54no—
166C3×#95—54no—
167C3×#95C2×D514yes—
168C23×D5C23×C311yescharacteristic
169C2×C5×Q8—22no—
170C2×C5×Q8—22no—
171C2×C5×Q8C2×C3×C414yescomplemented
172C22×C4×C5C23×C312yes—
173C22×C4×C5C23×C312yes—
174C22×C4×C5C23×C312yes—
175C22×C4×C5C23×C311yescharacteristic
176C22×Dic5C23×C312yes—
177C22×Dic5C23×C312yes—
178C22×Dic5C23×C312yes—
179C22×Dic5C23×C311yescharacteristic
180Q8×D5—232no—
181C2×C4×D5—24no—
182C2×C4×D5—24no—
183C2×C4×D5—24no—
184C2×C4×D5—24no—
185C2×C4×D5—24no—
186C2×C4×D5—28no—
187C2×C4×D5C2×C3×C418yes—
188C2×C4×D5C2×C3×C414yes—
189C2×C4×D5C3×D414yes—
190C2×C4×D5C3×D414yes—
191C2×C4×D5C3×Q818yescomplemented
192C2×Dic10—22no—
193C2×Dic10—22no—
194C2×Dic10—24no—
195C2×Dic10—24no—
196C2×Dic10C2×C3×C414yescomplemented
197C2×Dic10C2×C3×C418yescomplemented
198C5×C4⋊C4—22no—
199C42×C5—22no—
200C4⋊Dic5—22no—
201#683—24no—
202C4×Dic5—22no—
203C4×Dic5—24no—
204C23×Q8—51no—
205C22×C4⋊C4—51no—
206C22×C42—51no—
207#391—51no—
208#391—51no—
209#391—51no—
210#391—54no—
211#391C3×D511yescharacteristic, direct product factor
212C2×#95—54no—
213C22×C3×C5C23×C411yescharacteristic
214C22×C3×C5C22×D411yescharacteristic
215C22×C3×C5C22×D411yescharacteristic
216C22×C3×C5C22×Q812yes—
217C22×C3×C5C2×#4112yes—
218C2×C3×D5C2×C22⋊C414yes—
219C2×C3×D5C4×Q818yes—
220C2×C3×D5C22⋊Q818yes—
221C2×C3×D5#12414yes—
222C2×C3×D5C4⋊Q814yes—
223C3×C4×C5—24no—
224C3×C4×C5—24no—
225C3×C4×C5—42no—
226C3×C4×C5C2×C22⋊C414yes—
227C3×Dic5—24no—
228C3×Dic5—24no—
229C3×Dic5—42no—
230C3×Dic5C2×C22⋊C414yes—
231C24×C3—51no—
232C2×C3×Q8—22no—
233C2×C3×Q8—22no—
234C2×C3×Q8—58no—
235C2×C3×Q8—54no—
236C2×C3×Q8—104no—
237C2×C3×Q8—104no—
238C2×C3×Q8—1032no—
239C2×C3×Q8—102no—
240C2×C3×Q8—102no—
241C2×C3×Q8C4×D514yescomplemented
242C22×C3×C4—51no—
243C22×C3×C4—54no—
244C22×C3×C4—54no—
245C22×C3×C4—54no—
246C22×C3×C4—52no—
247C22×C3×C4—52no—
248C22×C3×C4—58no—
249C22×C3×C4—58no—
250C22×C3×C4—52no—
251C22×C3×C4—104no—
252C22×C3×C4—104no—
253C22×C3×C4—104no—
254C22×C3×C4—104no—
255C22×C3×C4—104no—
256C22×C3×C4—108no—
257C22×C3×C4C22×D512yes—
258C22×C3×C4C22×D512yes—
259C22×C3×C4C22×D512yes—
260C22×C3×C4C22×D511yescharacteristic
261C3×C4⋊C4—22no—
262C3×C4⋊C4—104no—
263C3×C4⋊C4—102no—
264C3×C42—22no—
265C3×C42—104no—
266C3×C42—102no—
267C23×C5C24×C311yescharacteristic
268C22×D5C22×C3×C412yes—
269C22×D5C2×C3×D412yes—
270C22×D5C2×C3×D412yes—
271C22×D5C2×C3×Q814yes—
272C22×D5C3×#4114yes—
273C5×Q8—28no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—24no—
280C2×C4×C5C22×C3×C414yes—
281C2×C4×C5C22×C3×C412yes—
282C2×C4×C5C2×C3×D412yes—
283C2×C4×C5C2×C3×D412yes—
284C2×C4×C5C2×C3×Q814yescomplemented
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—22no—
288C2×Dic5—22no—
289C2×Dic5—22no—
290C2×Dic5—24no—
291C2×Dic5C22×C3×C414yes—
292C2×Dic5C22×C3×C412yes—
293C2×Dic5C2×C3×D412yes—
294C2×Dic5C2×C3×D412yes—
295C2×Dic5C2×C3×Q814yescomplemented
296C4×D5—216no—
297C4×D5—216no—
298C4×D5—48no—
299C4×D5C3×C22⋊C4116yes—
300Dic10—28no—
301Dic10—216no—
302C22×Q8—58no—
303C22×Q8—52no—
304C22×Q8—51no—
305C22×Q8—104no—
306C22×Q8—104no—
307C22×Q8C2×C3×D511yescharacteristic
308C23×C4—51no—
309C23×C4—52no—
310C23×C4—52no—
311C23×C4—52no—
312C2×C4⋊C4—52no—
313C2×C4⋊C4—51no—
314C2×C4⋊C4—104no—
315C2×C4⋊C4C2×C3×D511yescharacteristic
316C2×C42—52no—
317C2×C42—51no—
318C2×C42—104no—
319C2×C42C2×C3×D511yescharacteristic
320#95—54no—
321#95—54no—
322#95—54no—
323#95C2×C3×D514yes—
324C2×C3×C5C22×C22⋊C411yescharacteristic
325C2×C3×C5C2×C4×Q812yes—
326C2×C3×C5C2×C22⋊Q812yes—
327C2×C3×C5C2×#12411yescharacteristic
328C2×C3×C5C2×C4⋊Q811yescharacteristic
329C3×D5#39118yesdirect product factor
330C23×C3—52no—
331C23×C3—52no—
332C23×C3—52no—
333C23×C3—54no—
334C23×C3—54no—
335C23×C3C23×D511yescharacteristic
336C3×Q8—28no—
337C3×Q8—1016no—
338C3×Q8—108no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—52no—
346C2×C3×C4—52no—
347C2×C3×C4—52no—
348C2×C3×C4—516no—
349C2×C3×C4—54no—
350C2×C3×C4—54no—
351C2×C3×C4—1016no—
352C2×C3×C4—1016no—
353C2×C3×C4—102no—
354C2×C3×C4—102no—
355C2×C3×C4—102no—
356C2×C3×C4—102no—
357C2×C3×C4—102no—
358C2×C3×C4—104no—
359C2×C3×C4—208no—
360C2×C3×C4C2×C4×D514yes—
361C2×C3×C4C2×C4×D512yes—
362C2×C3×C4D4×D512yes—
363C2×C3×C4D4×D512yes—
364C2×C3×C4Q8×D514yescomplemented
365C22×C5C23×C3×C411yescharacteristic
366C22×C5C22×C3×D411yescharacteristic
367C22×C5C22×C3×D411yescharacteristic
368C22×C5C22×C3×Q812yes—
369C22×C5C2×C3×#4112yes—
370C2×D5C2×C3×C22⋊C414yes—
371C2×D5C3×C4×Q818yes—
372C2×D5C3×C22⋊Q818yes—
373C2×D5C3×#12414yes—
374C2×D5C3×C4⋊Q814yes—
375C4×C5—24no—
376C4×C5—24no—
377C4×C5—42no—
378C4×C5C2×C3×C22⋊C414yes—
379Dic5—24no—
380Dic5—24no—
381Dic5—42no—
382Dic5C2×C3×C22⋊C414yes—
383C24—51no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—58no—
387C2×Q8—54no—
388C2×Q8—104no—
389C2×Q8—104no—
390C2×Q8—1032no—
391C2×Q8—102no—
392C2×Q8—102no—
393C2×Q8C3×C4×D514yescomplemented
394C22×C4—51no—
395C22×C4—54no—
396C22×C4—54no—
397C22×C4—54no—
398C22×C4—52no—
399C22×C4—52no—
400C22×C4—58no—
401C22×C4—58no—
402C22×C4—52no—
403C22×C4—104no—
404C22×C4—104no—
405C22×C4—104no—
406C22×C4—104no—
407C22×C4—104no—
408C22×C4—108no—
409C22×C4C22×C3×D512yes—
410C22×C4C22×C3×D512yes—
411C22×C4C22×C3×D512yes—
412C22×C4C22×C3×D511yescharacteristic
413C4⋊C4—22no—
414C4⋊C4—104no—
415C4⋊C4—102no—
416C42—22no—
417C42—104no—
418C42—102no—
419C3×C5C2×#39111yescharacteristic, complemented, Hall
420C22×C3—54no—
421C22×C3—54no—
422C22×C3—58no—
423C22×C3—58no—
424C22×C3—54no—
425C22×C3C22×C4×D511yescharacteristic
426C22×C3C2×D4×D511yescharacteristic
427C22×C3C2×D4×D511yescharacteristic
428C22×C3C2×Q8×D512yes—
429C22×C3D5×#4112yes—
430C3×C4—24no—
431C3×C4—24no—
432C3×C4—42no—
433C3×C4—54no—
434C3×C4—104no—
435C3×C4—104no—
436C3×C4—202no—
437C3×C4D5×C22⋊C414yes—
438C2×C5C22×C3×C22⋊C411yescharacteristic
439C2×C5C2×C3×C4×Q812yes—
440C2×C5C2×C3×C22⋊Q812yes—
441C2×C5C2×C3×#12411yescharacteristic
442C2×C5C2×C3×C4⋊Q811yescharacteristic
443D5C3×#39118yesdirect product factor
444C23—52no—
445C23—52no—
446C23—52no—
447C23—54no—
448C23—54no—
449C23C23×C3×D511yescharacteristic
450Q8—28no—
451Q8—1016no—
452Q8—108no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—52no—
460C2×C4—52no—
461C2×C4—52no—
462C2×C4—516no—
463C2×C4—54no—
464C2×C4—54no—
465C2×C4—1016no—
466C2×C4—1016no—
467C2×C4—102no—
468C2×C4—102no—
469C2×C4—102no—
470C2×C4—102no—
471C2×C4—102no—
472C2×C4—104no—
473C2×C4—208no—
474C2×C4C2×C3×C4×D514yes—
475C2×C4C2×C3×C4×D512yes—
476C2×C4C3×D4×D512yes—
477C2×C4C3×D4×D512yes—
478C2×C4C3×Q8×D514yescomplemented
479C2×C3—58no—
480C2×C3C2×D5×C22⋊C411yescharacteristic
481C2×C3C4×Q8×D512yes—
482C2×C3D5×C22⋊Q812yes—
483C2×C3D5×#12411yescharacteristic
484C2×C3D5×C4⋊Q811yescharacteristic
485C5C2×C3×#39111yescharacteristic, complemented, Sylow
486C22—58no—
487C22—58no—
488C22—54no—
489C22—54no—
490C22—54no—
491C22C22×C3×C4×D511yescharacteristic
492C22C2×C3×D4×D511yescharacteristic
493C22C2×C3×D4×D511yescharacteristic
494C22C2×C3×Q8×D512yes—
495C22C3×D5×#4112yes—
496C4—24no—
497C4—24no—
498C4—42no—
499C4—54no—
500C4—104no—
501C4—104no—
502C4—202no—
503C4C3×D5×C22⋊C414yes—
504C3D5×#39111yescharacteristic, direct product factor, Sylow
505C2—58no—
506C2C2×C3×D5×C22⋊C411yescharacteristic
507C2C3×C4×Q8×D512yes—
508C2C3×D5×C22⋊Q812yes—
509C2C3×D5×#12411yescharacteristic
510C2C3×D5×C4⋊Q811yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C5×#391C2
socleC23×C3×C5C24
prime radicalC23×C3×D5C23
cosocleC23×C5C24×C3
hypercenterC3×#391D5
hypocenterC5C2×C3×#391
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×#391

Complements

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

NH ≅ G/Ntype
4. C3×C5×#391C2semidirect
5. C3×#65519C2semidirect
6. C3×#65263C2semidirect
7. C3×#65218C2semidirect
9. C3×#64935C2semidirect
10. D5×#391C3direct (N × H)
14. C2×C3×Q8×D5C4semidirect
37. C5×#391C2×C3semidirect
38. #65519C2×C3semidirect
39. #65263C2×C3semidirect
40. #65218C2×C3semidirect
42. #64935C2×C3semidirect
46. C2×C3×C5×Q8C2×C4semidirect
66. C2×C3×C4×D5Q8semidirect
71. C2×C3×Dic10C2×C4semidirect
72. C2×C3×Dic10C2×C4semidirect
86. C3×#391D5direct (N × H)
91. C2×Q8×D5C3×C4semidirect
128. C2×C3×C4×C5C2×Q8semidirect
139. C2×C3×Dic5C2×Q8semidirect
171. C2×C5×Q8C2×C3×C4semidirect
191. C2×C4×D5C3×Q8semidirect
196. C2×Dic10C2×C3×C4semidirect
197. C2×Dic10C2×C3×C4semidirect
211. #391C3×D5direct (N × H)
241. C2×C3×Q8C4×D5semidirect
284. C2×C4×C5C2×C3×Q8semidirect
295. C2×Dic5C2×C3×Q8semidirect
329. C3×D5#391direct (N × H)
364. C2×C3×C4Q8×D5semidirect
393. C2×Q8C3×C4×D5semidirect
419. C3×C5C2×#391semidirect
443. D5C3×#391direct (N × H)
478. C2×C4C3×Q8×D5semidirect
485. C5C2×C3×#391semidirect
504. C3D5×#391direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8×D52
C2×C3×D5×C4⋊C42
C2×C3×C42×D52
C3×C5×#3912
C3×#655192
C3×#652632
C3×#652182
C3×D5×#952
C3×#649352
D5×#3913
C2×C3×#3915