Finite Group Databasebeta

← C3×#10618995 #14495592

Subgroups — C3×#10618995 #14495592

Order 1920.

1860 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#562C211yesmaximal, characteristic, complemented
2C3×#66022C211yesmaximal, characteristic, complemented
3C3×C5⋊#564C211yesmaximal, characteristic, complemented
4C3×#66014C212yesmaximal, complemented
5C3×#66006C212yesmaximal, complemented
6C3×#65985C214yesmaximal, complemented
7C3×#65983C214yesmaximal, complemented
8C3×#65980C214yesmaximal, complemented
9C3×#65956C214yesmaximal, complemented
10C3×#65955C214yesmaximal, complemented
11C3×#65934C214yesmaximal, complemented
12#10618995C311yesmaximal, characteristic, direct product factor, Hall
13C3×C5×#125C2211yescharacteristic
14C3×C5×#124C2212yes—
15C3×C5×#123C2214yes—
16C3×C5×C22⋊Q8C2214yes—
17C3×C5×#121C2214yes—
18C3×#4092C2212yes—
19C3×Q8×Dic5C2212yes—
20C3×#4090C2214yes—
21C3×#4085C2214yes—
22C3×#4084C2212yes—
23C3×#4082C2214yes—
24C3×#4081C2214yes—
25C3×#4081C2214yes—
26C3×D4×Dic5C2214yes—
27C3×D4×Dic5C2212yes—
28C3×#4075C2214yes—
29C3×#4070C2214yes—
30C3×#4070C2214yes—
31C2×C3×C4×Dic5C2214yes—
32C3×#4044C2214yes—
33C3×#4042C2214yes—
34C3×#4038C2212yes—
35C3×#4036C2214yes—
36C3×#4035C2214yes—
37C3×#4035C2214yes—
38C3×#4035C2214yes—
39C3×#4034C2214yes—
40C3×#4034C2212yes—
41C3×#4033C2214yes—
42C3×#4033C2214yes—
43C3×#4031C2214yes—
44C3×#4029C2214yes—
45C3×#4027C2214yes—
46C3×#4025C2214yes—
47C3×#4025C2214yes—
48C3×#4025C2214yes—
49C3×#4025C2214yes—
50C3×#4024C2214yes—
51C3×#4024C2214yes—
52C3×#4024C2214yes—
53C3×#4023C2214yes—
54C3×#4023C2214yes—
55C3×C42×D5C2211yescharacteristic
56C3×#4016C2211yescharacteristic
57C3×#4016C2212yes—
58C3×#3531—51nomaximal, Hall
59C5×#562C2×C311yescharacteristic, complemented
60#66022C2×C311yescharacteristic, complemented
61C5⋊#564C2×C311yescharacteristic, complemented
62#66014C2×C312yescomplemented
63#66006C2×C312yescomplemented
64#65985C2×C314yescomplemented
65#65983C2×C314yescomplemented
66#65980C2×C314yescomplemented
67#65956C2×C314yescomplemented
68#65955C2×C314yescomplemented
69#65934C2×C314yescomplemented
70C2×C3×C5×Q8C2312yes—
71C2×C3×C5×D4C2312yes—
72C2×C3×C5×D4C2314yes—
73C22×C3×C4×C5C2314yes—
74C2×C3×C22⋊D5C2314yes—
75C22×C3×Dic5C2314yes—
76C22×C3×Dic5C2314yes—
77C2×C3×C4×D5C2311yescharacteristic
78C2×C3×C4×D5C2312yes—
79C2×C3×Dic10C2314yes—
80C2×C3×Dic10C2314yes—
81C3×C5×C4⋊C4C2314yes—
82C3×C5×C4⋊C4C2314yes—
83C3×C5×C4⋊C4C2312yes—
84C3×C5×C22⋊C4C2314yes—
85C3×C5×C22⋊C4C2314yes—
86C3×C5×C22⋊C4C2314yes—
87C3×C42×C5C2311yescharacteristic
88C3×C22⋊Dic5C2314yes—
89C3×C22⋊Dic5C2314yes—
90C3×C22⋊Dic5C2314yes—
91C3×C22⋊Dic5C2314yes—
92C3×C22⋊Dic5C2314yes—
93C3×#685C2314yes—
94C3×#685C2314yes—
95C3×C4⋊Dic5C2312yes—
96C3×C4⋊Dic5C2314yes—
97C3×C4⋊Dic5C2312yes—
98C3×C4⋊Dic5C2314yes—
99C3×C4⋊Dic5C2312yes—
100C3×#683C2314yes—
101C3×#683C2314yes—
102C3×#683C2314yes—
103C3×#683C2314yes—
104C3×#683C2314yes—
105C3×#683C2314yes—
106C3×#683C2314yes—
107C3×#683C2314yes—
108C3×#683C2314yes—
109C3×C4×Dic5C2312yes—
110C3×C4×Dic5C2314yes—
111C3×C4×Dic5C2314yes—
112C3×C4×Dic5C2314yes—
113C3×C4×Dic5C2314yes—
114C3×C4×Dic5C2311yescharacteristic
115C3×#564—51no—
116C3×#563—52no—
117C3×#563—54no—
118C3×#563—54no—
119C3×#562—54no—
120C3×#562D511yescharacteristic, complemented
121C3×#533—51no—
122C3×#533—54no—
123C3×#529—52no—
124C3×#529—54no—
125C3×#529—54no—
126C5×#125C22×C311yescharacteristic
127C5×#124C22×C312yes—
128C5×#123C22×C314yes—
129C5×C22⋊Q8C22×C314yes—
130C5×#121C22×C314yes—
131#4092C22×C312yes—
132Q8×Dic5C22×C312yes—
133#4090C22×C314yes—
134#4085C22×C314yes—
135#4084C22×C312yes—
136#4082C22×C314yes—
137#4081C22×C314yes—
138#4081C22×C314yes—
139D4×Dic5C22×C314yes—
140D4×Dic5C22×C312yes—
141#4075C22×C314yes—
142#4070C22×C314yes—
143#4070C22×C314yes—
144C2×C4×Dic5C22×C314yes—
145#4044C22×C314yes—
146#4042C22×C314yes—
147#4038C22×C312yes—
148#4036C22×C314yes—
149#4035C22×C314yes—
150#4035C22×C314yes—
151#4035C22×C314yes—
152#4034C22×C314yes—
153#4034C22×C312yes—
154#4033C22×C314yes—
155#4033C22×C314yes—
156#4031C22×C314yes—
157#4029C22×C314yes—
158#4027C22×C314yes—
159#4025C22×C314yes—
160#4025C22×C314yes—
161#4025C22×C314yes—
162#4025C22×C314yes—
163#4024C22×C314yes—
164#4024C22×C314yes—
165#4024C22×C314yes—
166#4023C22×C314yes—
167#4023C22×C314yes—
168C42×D5C22×C311yescharacteristic
169#4016C22×C311yescharacteristic
170#4016C22×C312yes—
171#3531—51noSylow
172C23×C3×C5C2414yes—
173C22×C3×D5C2411yescharacteristic
174C3×C5×Q8—42no—
175C3×C5×D4—44no—
176C3×C5×D4—42no—
177C2×C3×C4×C5—44no—
178C2×C3×C4×C5C2414yes—
179C2×C3×C4×C5C2414yes—
180C2×C3×C4×C5C2411yescharacteristic
181C2×C3×C4×C5C2412yes—
182C3×C22⋊D5—44no—
183C2×C3×Dic5—28no—
184C2×C3×Dic5—44no—
185C2×C3×Dic5C2414yes—
186C2×C3×Dic5C2414yes—
187C2×C3×Dic5C2414yes—
188C2×C3×Dic5C2412yes—
189C2×C3×Dic5C2411yescharacteristic
190C3×C4×D5—22no—
191C3×C4×D5—42no—
192C3×Dic10—44no—
193C3×Dic10—44no—
194C3×C4⋊Q8—54no—
195C3×C4⋊Q8—54no—
196C3×C4⋊Q8—52no—
197C3×#126—54no—
198C3×#126—54no—
199C3×#126—54no—
200C3×#126—54no—
201C3×#126—54no—
202C3×#125—54no—
203C3×#125—54no—
204C3×#125—54no—
205C3×#125—54no—
206C3×#125—51no—
207C3×#125—52no—
208C3×#125C2×D511yescharacteristic
209C3×#124—54no—
210C3×#124—54no—
211C3×#124C2×D512yes—
212C3×#123—54no—
213C3×#123—54no—
214C3×#123—54no—
215C3×#123—54no—
216C3×#123C2×D514yes—
217C3×C22⋊Q8—52no—
218C3×C22⋊Q8—54no—
219C3×C22⋊Q8—54no—
220C3×C22⋊Q8—54no—
221C3×C22⋊Q8—54no—
222C3×C22⋊Q8—54no—
223C3×C22⋊Q8—54no—
224C3×C22⋊Q8C2×D514yes—
225C3×#121—52no—
226C3×#121—54no—
227C3×#121C2×D514yes—
228C3×C4×Q8—52no—
229C3×C4×Q8—54no—
230C3×C4×Q8—54no—
231C3×C4×D4—52no—
232C3×C4×D4—54no—
233C3×C4×D4—54no—
234C3×#117—52no—
235C3×#117—54no—
236C3×#117—54no—
237C2×C3×C42—54no—
238C2×C3×C42—51no—
239C2×C5×Q8C23×C312yes—
240C2×C5×D4C23×C314yes—
241C2×C5×D4C23×C312yes—
242C22×C4×C5C23×C314yes—
243C2×C22⋊D5C23×C314yes—
244C22×Dic5C23×C314yes—
245C22×Dic5C23×C314yes—
246C2×C4×D5C23×C311yescharacteristic
247C2×C4×D5C23×C312yes—
248C2×Dic10C23×C314yes—
249C2×Dic10C23×C314yes—
250C5×C4⋊C4C23×C314yes—
251C5×C4⋊C4C23×C314yes—
252C5×C4⋊C4C23×C312yes—
253C5×C22⋊C4C23×C314yes—
254C5×C22⋊C4C23×C314yes—
255C5×C22⋊C4C23×C314yes—
256C42×C5C23×C311yescharacteristic
257C22⋊Dic5C23×C314yes—
258C22⋊Dic5C23×C314yes—
259C22⋊Dic5C23×C314yes—
260C22⋊Dic5C23×C314yes—
261C22⋊Dic5C23×C314yes—
262#685C23×C314yes—
263#685C23×C314yes—
264C4⋊Dic5C23×C312yes—
265C4⋊Dic5C23×C312yes—
266C4⋊Dic5C23×C312yes—
267C4⋊Dic5C23×C314yes—
268C4⋊Dic5C23×C314yes—
269#683C23×C314yes—
270#683C23×C314yes—
271#683C23×C314yes—
272#683C23×C314yes—
273#683C23×C314yes—
274#683C23×C314yes—
275#683C23×C314yes—
276#683C23×C314yes—
277#683C23×C314yes—
278C4×Dic5C23×C312yes—
279C4×Dic5C23×C311yescharacteristic
280C4×Dic5C23×C314yes—
281C4×Dic5C23×C314yes—
282C4×Dic5C23×C314yes—
283C4×Dic5C23×C314yes—
284#564—51no—
285#563—52no—
286#563—54no—
287#563—54no—
288#562—54no—
289#562C3×D511yescharacteristic, complemented
290#533—51no—
291#533—54no—
292#529—52no—
293#529—54no—
294#529—54no—
295C22×C3×C5—24no—
296C22×C3×C5—24no—
297C22×C3×C5—24no—
298C22×C3×C5C2511yescharacteristic
299C2×C3×D5—21no—
300C2×C3×D5—22no—
301C3×C4×C5—21no—
302C3×C4×C5—24no—
303C3×C4×C5—24no—
304C3×C4×C5—22no—
305C3×Dic5—24no—
306C3×Dic5—22no—
307C3×Dic5—24no—
308C3×Dic5—24no—
309C3×Dic52−1+412yes—
310C2×C3×Q8—54no—
311C2×C3×Q8—54no—
312C2×C3×Q8C22×D512yes—
313C2×C3×D4—54no—
314C2×C3×D4C22×D512yes—
315C2×C3×D4C22×D514yes—
316C22×C3×C4—54no—
317C22×C3×C4—54no—
318C22×C3×C4—51no—
319C22×C3×C4—52no—
320C22×C3×C4C22×D514yes—
321C3×C4⋊C4—54no—
322C3×C4⋊C4—54no—
323C3×C4⋊C4—54no—
324C3×C4⋊C4—54no—
325C3×C4⋊C4—54no—
326C3×C4⋊C4—54no—
327C3×C4⋊C4—54no—
328C3×C4⋊C4—54no—
329C3×C4⋊C4—54no—
330C3×C4⋊C4—52no—
331C3×C4⋊C4—54no—
332C3×C4⋊C4—52no—
333C3×C4⋊C4—52no—
334C3×C4⋊C4—54no—
335C3×C4⋊C4C22×D512yes—
336C3×C4⋊C4C22×D514yes—
337C3×C4⋊C4C22×D514yes—
338C3×C22⋊C4—54no—
339C3×C22⋊C4—54no—
340C3×C22⋊C4—54no—
341C3×C22⋊C4—54no—
342C3×C22⋊C4—54no—
343C3×C22⋊C4—54no—
344C3×C22⋊C4—54no—
345C3×C22⋊C4C22×D514yes—
346C3×C22⋊C4C22×D514yes—
347C3×C22⋊C4C22×D514yes—
348C3×C42—54no—
349C3×C42—52no—
350C3×C42—54no—
351C3×C42—54no—
352C3×C42—54no—
353C3×C42—51no—
354C3×C42C22×D511yescharacteristic
355C23×C5C24×C314yes—
356C22×D5C24×C311yescharacteristic
357C5×Q8—42no—
358C5×D4—44no—
359C5×D4—42no—
360C2×C4×C5—44no—
361C2×C4×C5C24×C314yes—
362C2×C4×C5C24×C314yes—
363C2×C4×C5C24×C311yescharacteristic
364C2×C4×C5C24×C312yes—
365C22⋊D5—44no—
366C2×Dic5—28no—
367C2×Dic5—44no—
368C2×Dic5C24×C314yes—
369C2×Dic5C24×C314yes—
370C2×Dic5C24×C314yes—
371C2×Dic5C24×C312yes—
372C2×Dic5C24×C311yescharacteristic
373C4×D5—22no—
374C4×D5—42no—
375Dic10—44no—
376Dic10—44no—
377C4⋊Q8—54no—
378C4⋊Q8—54no—
379C4⋊Q8—52no—
380#126—54no—
381#126—54no—
382#126—54no—
383#126—54no—
384#126—54no—
385#125—54no—
386#125—54no—
387#125—54no—
388#125—54no—
389#125—52no—
390#125—51no—
391#125C2×C3×D511yescharacteristic
392#124—54no—
393#124—54no—
394#124C2×C3×D512yes—
395#123—54no—
396#123—54no—
397#123—54no—
398#123—54no—
399#123C2×C3×D514yes—
400C22⋊Q8—52no—
401C22⋊Q8—54no—
402C22⋊Q8—54no—
403C22⋊Q8—54no—
404C22⋊Q8—54no—
405C22⋊Q8—54no—
406C22⋊Q8—54no—
407C22⋊Q8C2×C3×D514yes—
408#121—52no—
409#121—54no—
410#121C2×C3×D514yes—
411C4×Q8—52no—
412C4×Q8—54no—
413C4×Q8—54no—
414C4×D4—52no—
415C4×D4—54no—
416C4×D4—54no—
417#117—52no—
418#117—54no—
419#117—54no—
420C2×C42—54no—
421C2×C42—51no—
422C2×C3×C5—44no—
423C2×C3×C5C2×2−1+411yescharacteristic
424C2×C3×C5#58512yes—
425C3×D5—41no—
426C23×C3—51no—
427C23×C3C23×D514yes—
428C3×Q8—42no—
429C3×Q8—204no—
430C3×Q8—204no—
431C3×D4—44no—
432C3×D4—42no—
433C3×D4—204no—
434C2×C3×C4—44no—
435C2×C3×C4—51no—
436C2×C3×C4—54no—
437C2×C3×C4—54no—
438C2×C3×C4—52no—
439C2×C3×C4—54no—
440C2×C3×C4—102no—
441C2×C3×C4—108no—
442C2×C3×C4—202no—
443C2×C3×C4—204no—
444C2×C3×C4C23×D514yes—
445C2×C3×C4C23×D514yes—
446C2×C3×C4C23×D511yescharacteristic
447C2×C3×C4C23×D512yes—
448C22×C5—24no—
449C22×C5—24no—
450C22×C5—24no—
451C22×C5C25×C311yescharacteristic
452C2×D5—21no—
453C2×D5—22no—
454C4×C5—21no—
455C4×C5—24no—
456C4×C5—24no—
457C4×C5—22no—
458Dic5—24no—
459Dic5—22no—
460Dic5—24no—
461Dic5—24no—
462Dic5C3×2−1+412yes—
463C2×Q8—54no—
464C2×Q8—54no—
465C2×Q8C22×C3×D512yes—
466C2×D4—54no—
467C2×D4C22×C3×D512yes—
468C2×D4C22×C3×D514yes—
469C22×C4—54no—
470C22×C4—54no—
471C22×C4—51no—
472C22×C4—52no—
473C22×C4C22×C3×D514yes—
474C4⋊C4—54no—
475C4⋊C4—54no—
476C4⋊C4—52no—
477C4⋊C4—54no—
478C4⋊C4—54no—
479C4⋊C4—54no—
480C4⋊C4—54no—
481C4⋊C4—52no—
482C4⋊C4—54no—
483C4⋊C4—52no—
484C4⋊C4—54no—
485C4⋊C4—54no—
486C4⋊C4—54no—
487C4⋊C4—54no—
488C4⋊C4C22×C3×D512yes—
489C4⋊C4C22×C3×D514yes—
490C4⋊C4C22×C3×D514yes—
491C22⋊C4—54no—
492C22⋊C4—54no—
493C22⋊C4—54no—
494C22⋊C4—54no—
495C22⋊C4—54no—
496C22⋊C4—54no—
497C22⋊C4—54no—
498C22⋊C4C22×C3×D514yes—
499C22⋊C4C22×C3×D514yes—
500C22⋊C4C22×C3×D514yes—
501C42—54no—
502C42—52no—
503C42—54no—
504C42—54no—
505C42—54no—
506C42—51no—
507C42C22×C3×D511yescharacteristic
508C3×C5#353111yescharacteristic, complemented, Hall
509C22×C3—24no—
510C22×C3—24no—
511C22×C3—24no—
512C22×C3—102no—
513C22×C3—101no—
514C22×C3C24×D511yescharacteristic
515C3×C4—24no—
516C3×C4—24no—
517C3×C4—22no—
518C3×C4—21no—
519C3×C4—52no—
520C3×C4—104no—
521C3×C4—102no—
522C3×C4—104no—
523C3×C4—104no—
524C2×C5—44no—
525C2×C5C2×C3×2−1+411yescharacteristic
526C2×C5C3×#58512yes—
527D5—41no—
528C23—51no—
529C23C23×C3×D514yes—
530Q8—42no—
531Q8—204no—
532Q8—204no—
533D4—44no—
534D4—42no—
535D4—204no—
536C2×C4—44no—
537C2×C4—51no—
538C2×C4—54no—
539C2×C4—52no—
540C2×C4—54no—
541C2×C4—54no—
542C2×C4—102no—
543C2×C4—108no—
544C2×C4—202no—
545C2×C4—204no—
546C2×C4C23×C3×D514yes—
547C2×C4C23×C3×D514yes—
548C2×C4C23×C3×D511yescharacteristic
549C2×C4C23×C3×D512yes—
550C2×C3—44no—
551C2×C3—201no—
552C2×C3#6628512yes—
553C2×C3D5×2−1+411yescharacteristic
554C5C3×#353111yescharacteristic, complemented, Sylow
555C22—24no—
556C22—24no—
557C22—24no—
558C22—102no—
559C22—101no—
560C22C24×C3×D511yescharacteristic
561C4—24no—
562C4—24no—
563C4—22no—
564C4—21no—
565C4—52no—
566C4—104no—
567C4—102no—
568C4—104no—
569C4—104no—
570C3#1061899511yescharacteristic, direct product factor, Sylow
571C2—44no—
572C2—201no—
573C2C3×#6628512yes—
574C2C3×D5×2−1+411yescharacteristic
subgroupG/H
centerC22×C3C24×D5
commutator, cosocleC22×C5C25×C3
FrattiniC22C24×C3×D5
FittingC3×C5×#562C2
socleC22×C3×C5C25
hypercenterC3×#562D5
hypocenterC5C3×#3531
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×#562

Complements

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

NH ≅ G/Ntype
1. C3×C5×#562C2semidirect
2. C3×#66022C2semidirect
3. C3×C5⋊#564C2semidirect
4. C3×#66014C2semidirect
5. C3×#66006C2semidirect
6. C3×#65985C2semidirect
7. C3×#65983C2semidirect
8. C3×#65980C2semidirect
9. C3×#65956C2semidirect
10. C3×#65955C2semidirect
11. C3×#65934C2semidirect
12. #10618995C3direct (N × H)
59. C5×#562C2×C3semidirect
60. #66022C2×C3semidirect
61. C5⋊#564C2×C3semidirect
62. #66014C2×C3semidirect
63. #66006C2×C3semidirect
64. #65985C2×C3semidirect
65. #65983C2×C3semidirect
66. #65980C2×C3semidirect
67. #65956C2×C3semidirect
68. #65955C2×C3semidirect
69. #65934C2×C3semidirect
120. C3×#562D5semidirect
289. #562C3×D5semidirect
508. C3×C5#3531semidirect
554. C5C3×#3531semidirect
570. C3#10618995direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5622
C3×#660222
C3×C5⋊#5642
C3×#660142
C3×#660062
C3×#659852
C3×#659832
C3×#659802
C3×#659562
C3×#659552
C3×#659342
#106189953
C3×#35315