Finite Group Databasebeta

← C3×#10610410 #14424547

Subgroups — C3×#10610410 #14424547

Order 1920.

1400 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4150C211yesmaximal, characteristic
2C3×#66159C211yesmaximal, characteristic, complemented
3C2×C3×#4055C211yesmaximal, characteristic, complemented
4C22×C3×Dic20C211yesmaximal, characteristic, complemented
5C3×C5×#408C211yesmaximal, characteristic, complemented
6C3×#65409C211yesmaximal, characteristic, complemented
7C3×#65402C211yesmaximal, characteristic, complemented
8C3×#65028C214yesmaximal, complemented
9C3×#65022C214yesmaximal, complemented
10#10610410C311yesmaximal, characteristic, direct product factor, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C3×#4150—24no—
13C3×#4150—24no—
14C2×C3×Q8×D5—21no—
15C2×C3×#711—22no—
16C2×C3×#711—21no—
17C2×C3×#709C2212yes—
18C2×C3×#709C2211yescharacteristic
19C22×C3×Dic10C2212yes—
20C22×C3×Dic10C2211yescharacteristic
21C22×C3×Dic10C2211yescharacteristic
22C2×C3×C5×M16C2211yescharacteristic
23C22×C3×C5×C8C2211yescharacteristic
24C3×#4090—21no—
25C3×#4082—21no—
26C3×#4081—22no—
27C3×#4072C2211yescharacteristic
28C3×#4070C2214yes—
29C3×#4055—24no—
30C2×C3×Dic20—24no—
31C2×C3×Dic20C2212yes—
32C2×C3×Dic20C2214yes—
33C2×C3×#677C2212yes—
34C3×C5×C22⋊C8C2214yes—
35C3×#3953C2212yes—
36C3×#3948C2212yes—
37C3×#3948C2214yes—
38C3×#3011—51nomaximal, Hall
39C2×#4150C2×C311yescharacteristic
40#66159C2×C311yescharacteristic, complemented
41C2×#4055C2×C311yescharacteristic, complemented
42C22×Dic20C2×C311yescharacteristic, complemented
43C5×#408C2×C311yescharacteristic, complemented
44#65409C2×C311yescharacteristic, complemented
45#65402C2×C311yescharacteristic, complemented
46#65028C2×C314yescomplemented
47#65022C2×C314yescomplemented
48C3×C5×#41—22no—
49C3×C5×#41—22no—
50C2×C3×C5×Q8D411yescharacteristic
51C2×C3×C5×D4D412yes—
52C2×C3×C5×D4D411yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C2×C3×C22⋊D5—22no—
56C2×C3×C22⋊D5—21no—
57C22×C3×Dic5—22no—
58C22×C3×Dic5—21no—
59C22×C3×Dic5—21no—
60C3×Q8×D5—28no—
61C3×#711—216no—
62C3×#711—28no—
63C3×#709—22no—
64C3×#709—24no—
65C3×#709—24no—
66C3×#709D414yes—
67C2×C3×D20C2311yescharacteristic
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5—21no—
70C2×C3×Dic10—22no—
71C2×C3×Dic10—21no—
72C2×C3×Dic10—22no—
73C2×C3×Dic10—24no—
74C2×C3×Dic10—24no—
75C2×C3×Dic10—22no—
76C2×C3×Dic10—22no—
77C2×C3×Dic10—21no—
78C2×C3×Dic10D414yes—
79C2×C3×Dic10C2312yes—
80C2×C3×Dic10C2311yescharacteristic
81C2×C3×Dic10C2312yes—
82C3×C5×M16—22no—
83C2×C3×C5×C8—22no—
84C2×C3×C5×C8C2312yes—
85C2×C3×C5×C8C2312yes—
86C3×C22⋊Dic5—21no—
87C3×C22⋊Dic5—24no—
88C3×C4⋊Dic5C2312yes—
89C3×#683—24no—
90C3×C4×Dic5—21no—
91C3×Dic20—28no—
92C3×Dic20—24no—
93C3×#677—24no—
94C2×C3×2−1+4—51no—
95C2×C3×#137—51no—
96C22×C3×Q16—51no—
97C3×#536—51no—
98C3×#452—54no—
99C3×#451—54no—
100C3×#419—51no—
101C3×#416—51no—
102C3×#408D511yescharacteristic, complemented
103C2×C5×#41C22×C311yescharacteristic
104#4150—24no—
105#4150—24no—
106C2×Q8×D5—21no—
107C2×#711—22no—
108C2×#711—21no—
109C2×#709C22×C312yes—
110C2×#709C22×C311yescharacteristic
111C22×Dic10C22×C312yes—
112C22×Dic10C22×C311yescharacteristic
113C22×Dic10C22×C311yescharacteristic
114C2×C5×M16C22×C311yescharacteristic
115C22×C5×C8C22×C311yescharacteristic
116#4090—21no—
117#4082—21no—
118#4081—22no—
119#4072C22×C311yescharacteristic
120#4070C22×C314yes—
121#4055—24no—
122C2×Dic20—24no—
123C2×Dic20C22×C312yes—
124C2×Dic20C22×C314yes—
125C2×#677C22×C312yes—
126C5×C22⋊C8C22×C314yes—
127#3953C22×C312yes—
128#3948C22×C312yes—
129#3948C22×C314yes—
130#3011—51noSylow
131C23×C3×C5C2×D412yes—
132C23×C3×C5C2×D411yescharacteristic
133C22×C3×D5—21no—
134C3×C5×Q8—22no—
135C3×C5×D4—24no—
136C3×C5×D4—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2411yescharacteristic
144C3×C22⋊D5—28no—
145C3×C22⋊D5—24no—
146C2×C3×Dic5—28no—
147C2×C3×Dic5—24no—
148C2×C3×Dic5—21no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—22no—
152C2×C3×Dic5—42no—
153C3×D20—21no—
154C3×D20C2×D412yes—
155C3×C4×D5—28no—
156C3×C4×D5—24no—
157C3×Dic10—28no—
158C3×Dic10—24no—
159C3×Dic10—22no—
160C3×Dic10—21no—
161C3×Dic10—22no—
162C3×Dic10—22no—
163C3×Dic10—42no—
164C3×Dic10C2×D412yes—
165C3×Dic10C2×D414yes—
166C3×C5×C8—22no—
167C3×C5×C8—22no—
168C3×2−1+4—104no—
169C3×2−1+4—104no—
170C2×C3×#41—51no—
171C2×C3×#41—52no—
172C2×C3×#41—102no—
173C2×C3×#41—101no—
174C2×C3×#41C2×D511yescharacteristic
175C22×C3×Q8—51no—
176C22×C3×Q8—51no—
177C22×C3×Q8—52no—
178C22×C3×Q8—101no—
179C3×#137—104no—
180C2×C3×Q16—54no—
181C2×C3×Q16—52no—
182C2×C3×Q16—104no—
183C2×C3×SD16—52no—
184C2×C3×M16C2×D511yescharacteristic
185C22×C3×C8C2×D511yescharacteristic
186C3×C4⋊Q8—101no—
187C3×#124—101no—
188C3×#123—102no—
189C3×C22⋊Q8—54no—
190C3×#117—51no—
191C3×#103—52no—
192C3×#103—54no—
193C3×#102—52no—
194C3×C22⋊C8C2×D514yes—
195C5×#41—22no—
196C5×#41—22no—
197C2×C5×Q8C3×D411yescharacteristic
198C2×C5×D4C3×D412yes—
199C2×C5×D4C3×D411yescharacteristic
200C22×C4×C5C23×C312yes—
201C22×C4×C5C23×C311yescharacteristic
202C2×C22⋊D5—22no—
203C2×C22⋊D5—21no—
204C22×Dic5—22no—
205C22×Dic5—21no—
206C22×Dic5—21no—
207Q8×D5—28no—
208#711—216no—
209#711—28no—
210#709—22no—
211#709—24no—
212#709—24no—
213#709C3×D414yes—
214C2×D20C23×C311yescharacteristic
215C2×C4×D5—22no—
216C2×C4×D5—21no—
217C2×Dic10—22no—
218C2×Dic10—21no—
219C2×Dic10—22no—
220C2×Dic10—24no—
221C2×Dic10—24no—
222C2×Dic10—22no—
223C2×Dic10—22no—
224C2×Dic10—21no—
225C2×Dic10C3×D414yes—
226C2×Dic10C23×C312yes—
227C2×Dic10C23×C311yescharacteristic
228C2×Dic10C23×C312yes—
229C5×M16—22no—
230C2×C5×C8—22no—
231C2×C5×C8C23×C312yes—
232C2×C5×C8C23×C312yes—
233C22⋊Dic5—24no—
234C22⋊Dic5—21no—
235C4⋊Dic5C23×C312yes—
236#683—24no—
237C4×Dic5—21no—
238Dic20—28no—
239Dic20—24no—
240#677—24no—
241C2×2−1+4—51no—
242C2×#137—51no—
243C22×Q16—51no—
244#536—51no—
245#452—54no—
246#451—54no—
247#419—51no—
248#416—51no—
249#408C3×D511yescharacteristic, complemented
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—24no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×C3×D5—22no—
256C2×C3×D5—42no—
257C3×C4×C5—22no—
258C3×C4×C5#12012yes—
259C3×C4×C5C22×D411yescharacteristic
260C3×C4×C5C22×D411yescharacteristic
261C3×Dic5—22no—
262C3×Dic5—24no—
263C3×Dic5—42no—
264C3×Dic5—42no—
265C3×#41—22no—
266C3×#41—22no—
267C3×#41—54no—
268C3×#41—104no—
269C3×#41—104no—
270C3×#41—1016no—
271C3×#41—108no—
272C3×#41—102no—
273C2×C3×Q8—54no—
274C2×C3×Q8—52no—
275C2×C3×Q8—51no—
276C2×C3×Q8—52no—
277C2×C3×Q8—104no—
278C2×C3×Q8—104no—
279C2×C3×Q8—108no—
280C2×C3×Q8—102no—
281C2×C3×Q8—101no—
282C2×C3×Q8—101no—
283C2×C3×Q8—102no—
284C2×C3×Q8—102no—
285C2×C3×Q8—102no—
286C2×C3×Q8D2011yescharacteristic
287C2×C3×D4—51no—
288C2×C3×D4—102no—
289C2×C3×D4—101no—
290C2×C3×D4D2012yes—
291C2×C3×D4D2011yescharacteristic
292C22×C3×C4—102no—
293C22×C3×C4—101no—
294C22×C3×C4—102no—
295C22×C3×C4—101no—
296C22×C3×C4—101no—
297C22×C3×C4C22×D512yes—
298C22×C3×C4C22×D511yescharacteristic
299C3×Q16—108no—
300C3×Q16—104no—
301C3×SD16—104no—
302C3×M16—22no—
303C2×C3×C8—22no—
304C2×C3×C8C22×D512yes—
305C2×C3×C8C22×D512yes—
306C3×C4⋊C4—52no—
307C3×C4⋊C4—104no—
308C3×C22⋊C4—101no—
309C3×C22⋊C4—104no—
310C3×C42—101no—
311C23×C5C2×C3×D412yes—
312C23×C5C2×C3×D411yescharacteristic
313C22×D5—21no—
314C5×Q8—22no—
315C5×D4—24no—
316C5×D4—22no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5C2×C3×D412yes—
320C2×C4×C5C2×C3×D411yescharacteristic
321C2×C4×C5C2×C3×D412yes—
322C2×C4×C5C2×C3×D412yes—
323C2×C4×C5C24×C311yescharacteristic
324C22⋊D5—28no—
325C22⋊D5—24no—
326C2×Dic5—28no—
327C2×Dic5—24no—
328C2×Dic5—21no—
329C2×Dic5—22no—
330C2×Dic5—22no—
331C2×Dic5—22no—
332C2×Dic5—42no—
333D20—21no—
334D20C2×C3×D412yes—
335C4×D5—28no—
336C4×D5—24no—
337Dic10—28no—
338Dic10—24no—
339Dic10—22no—
340Dic10—21no—
341Dic10—22no—
342Dic10—22no—
343Dic10—42no—
344Dic10C2×C3×D412yes—
345Dic10C2×C3×D414yes—
346C5×C8—22no—
347C5×C8—22no—
3482−1+4—104no—
3492−1+4—104no—
350C2×#41—51no—
351C2×#41—52no—
352C2×#41—102no—
353C2×#41—101no—
354C2×#41C2×C3×D511yescharacteristic
355C22×Q8—51no—
356C22×Q8—51no—
357C22×Q8—52no—
358C22×Q8—101no—
359#137—104no—
360C2×Q16—54no—
361C2×Q16—52no—
362C2×Q16—104no—
363C2×SD16—52no—
364C2×M16C2×C3×D511yescharacteristic
365C22×C8C2×C3×D511yescharacteristic
366C4⋊Q8—101no—
367#124—101no—
368#123—102no—
369C22⋊Q8—54no—
370#117—51no—
371#103—52no—
372#103—54no—
373#102—52no—
374C22⋊C8C2×C3×D514yes—
375C2×C3×C5—22no—
376C2×C3×C5—42no—
377C2×C3×C5C2×#12011yescharacteristic
378C2×C3×C5#57812yes—
379C3×D5—42no—
380C23×C3—101no—
381C23×C3C2×D2012yes—
382C23×C3C2×D2011yescharacteristic
383C3×Q8—22no—
384C3×Q8—54no—
385C3×Q8—52no—
386C3×Q8—108no—
387C3×Q8—102no—
388C3×Q8—101no—
389C3×Q8—104no—
390C3×Q8—102no—
391C3×Q8—102no—
392C3×Q8—202no—
393C3×D4—24no—
394C3×D4—22no—
395C3×D4—52no—
396C3×D4—108no—
397C3×D4—104no—
398C3×D4—101no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—108no—
402C2×C3×C4—108no—
403C2×C3×C4—102no—
404C2×C3×C4—101no—
405C2×C3×C4—104no—
406C2×C3×C4—104no—
407C2×C3×C4—102no—
408C2×C3×C4—102no—
409C2×C3×C4—202no—
410C2×C3×C4C2×D2012yes—
411C2×C3×C4C2×D2011yescharacteristic
412C2×C3×C4D4×D512yes—
413C2×C3×C4D4×D512yes—
414C2×C3×C4C23×D511yescharacteristic
415C3×C8—22no—
416C3×C8—22no—
417C22×C5—22no—
418C22×C5—24no—
419C22×C5—22no—
420C22×C5C3×#12012yes—
421C22×C5C22×C3×D411yescharacteristic
422C2×D5—22no—
423C2×D5—42no—
424C4×C5—22no—
425C4×C5C3×#12012yes—
426C4×C5C22×C3×D411yescharacteristic
427C4×C5C22×C3×D411yescharacteristic
428Dic5—22no—
429Dic5—24no—
430Dic5—42no—
431Dic5—42no—
432#41—22no—
433#41—22no—
434#41—54no—
435#41—104no—
436#41—104no—
437#41—1016no—
438#41—108no—
439#41—102no—
440C2×Q8—54no—
441C2×Q8—52no—
442C2×Q8—51no—
443C2×Q8—52no—
444C2×Q8—104no—
445C2×Q8—104no—
446C2×Q8—108no—
447C2×Q8—102no—
448C2×Q8—101no—
449C2×Q8—101no—
450C2×Q8—102no—
451C2×Q8—102no—
452C2×Q8—102no—
453C2×Q8C3×D2011yescharacteristic
454C2×D4—51no—
455C2×D4—102no—
456C2×D4—101no—
457C2×D4C3×D2012yes—
458C2×D4C3×D2011yescharacteristic
459C22×C4—102no—
460C22×C4—101no—
461C22×C4—102no—
462C22×C4—101no—
463C22×C4—101no—
464C22×C4C22×C3×D512yes—
465C22×C4C22×C3×D511yescharacteristic
466Q16—108no—
467Q16—104no—
468SD16—104no—
469M16—22no—
470C2×C8—22no—
471C2×C8C22×C3×D512yes—
472C2×C8C22×C3×D512yes—
473C4⋊C4—52no—
474C4⋊C4—104no—
475C22⋊C4—101no—
476C22⋊C4—104no—
477C42—101no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—102no—
483C22×C3—202no—
484C22×C3#402812yes—
485C22×C3C22×D2011yescharacteristic
486C3×C4—22no—
487C3×C4—104no—
488C3×C4—102no—
489C3×C4—202no—
490C3×C4—202no—
491C3×C4#402812yes—
492C3×C4C2×D4×D511yescharacteristic
493C3×C4C2×D4×D511yescharacteristic
494C2×C5—22no—
495C2×C5—42no—
496C2×C5C2×C3×#12011yescharacteristic
497C2×C5C3×#57812yes—
498D5—42no—
499C23—101no—
500C23C2×C3×D2012yes—
501C23C2×C3×D2011yescharacteristic
502Q8—22no—
503Q8—54no—
504Q8—52no—
505Q8—108no—
506Q8—102no—
507Q8—101no—
508Q8—104no—
509Q8—102no—
510Q8—102no—
511Q8—202no—
512D4—24no—
513D4—22no—
514D4—52no—
515D4—108no—
516D4—104no—
517D4—101no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—108no—
521C2×C4—108no—
522C2×C4—102no—
523C2×C4—101no—
524C2×C4—104no—
525C2×C4—104no—
526C2×C4—102no—
527C2×C4—102no—
528C2×C4—202no—
529C2×C4C2×C3×D2012yes—
530C2×C4C2×C3×D2011yescharacteristic
531C2×C4C3×D4×D512yes—
532C2×C4C3×D4×D512yes—
533C2×C4C23×C3×D511yescharacteristic
534C8—22no—
535C8—22no—
536C2×C3—22no—
537C2×C3—42no—
538C2×C3—202no—
539C2×C3C2×#402811yescharacteristic
540C2×C3#6608712yes—
541C5C3×#301111yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—24no—
545C22—102no—
546C22—202no—
547C22C3×#402812yes—
548C22C22×C3×D2011yescharacteristic
549C4—22no—
550C4—104no—
551C4—102no—
552C4—202no—
553C4—202no—
554C4C3×#402812yes—
555C4C2×C3×D4×D511yescharacteristic
556C4C2×C3×D4×D511yescharacteristic
557C3#1061041011yescharacteristic, direct product factor, Sylow
558C2—22no—
559C2—42no—
560C2—202no—
561C2C2×C3×#402811yescharacteristic
562C2C3×#6608712yes—
subgroupG/H
centerC22×C3C22×D20
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#408C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4150C2
hypercenterC3×#408D5
hypocenterC5C3×#3011
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C3×#408

Complements

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

NH ≅ G/Ntype
2. C3×#66159C2semidirect
3. C2×C3×#4055C2semidirect
4. C22×C3×Dic20C2semidirect
5. C3×C5×#408C2semidirect
6. C3×#65409C2semidirect
7. C3×#65402C2semidirect
8. C3×#65028C2semidirect
9. C3×#65022C2semidirect
10. #10610410C3direct (N × H)
40. #66159C2×C3semidirect
41. C2×#4055C2×C3semidirect
42. C22×Dic20C2×C3semidirect
43. C5×#408C2×C3semidirect
44. #65409C2×C3semidirect
45. #65402C2×C3semidirect
46. #65028C2×C3semidirect
47. #65022C2×C3semidirect
102. C3×#408D5semidirect
249. #408C3×D5semidirect
478. C3×C5#3011semidirect
541. C5C3×#3011semidirect
557. C3#10610410direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41502
C3×#661592
C2×C3×#40552
C22×C3×Dic202
C3×C5×#4082
C3×#654092
C3×#654022
C3×#650282
C3×#650222
#106104103
C3×#30115