Finite Group Databasebeta

← C3×#10618187 #14495601

Subgroups — C3×#10618187 #14495601

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×#533C211yesmaximal, characteristic, complemented
2C3×#66069C214yesmaximal, complemented
3C3×#66034C211yesmaximal, characteristic, complemented
4C3×#66033C212yesmaximal, complemented
5C3×C5⋊#564C211yesmaximal, characteristic, complemented
6C3×#65981C214yesmaximal, complemented
7C3×#65980C214yesmaximal, complemented
8C3×#65914C214yesmaximal, complemented
9C3×#65912C214yesmaximal, complemented
10C3×#65862C214yesmaximal, complemented
11C3×#65813C212yesmaximal, complemented
12#10618187C311yesmaximal, characteristic, direct product factor, Hall
13C3×C5×C4⋊Q8C2212yes—
14C3×C5×#125C2211yescharacteristic
15C3×C5×#125C2211yescharacteristic
16C3×C5×C22⋊Q8C2214yes—
17C3×C4×C5×Q8C2214yes—
18C3×C5×#117C2212yes—
19C2×C3×C42×C5C2211yescharacteristic
20C3×#4093C2214yes—
21C3×#4092C2214yes—
22C3×#4092C2214yes—
23C3×#4090C2214yes—
24C3×#4076C2212yes—
25C3×#4075C2214yes—
26C3×C4×C22⋊D5C2214yes—
27C3×#4070C2212yes—
28C3×#4044C2214yes—
29C3×#4044C2214yes—
30C3×#4044C2214yes—
31C3×#4043C2214yes—
32C3×#4042C2214yes—
33C3×#4042C2214yes—
34C3×#4042C2214yes—
35C3×#4041C2214yes—
36C3×#4040C2214yes—
37C3×#4040C2214yes—
38C3×#4040C2214yes—
39C3×#4036C2214yes—
40C3×#4035C2214yes—
41C3×#4035C2214yes—
42C3×#4035C2214yes—
43C3×#4034C2214yes—
44C3×#4031C2214yes—
45C3×#4030C2214yes—
46C3×#4029C2214yes—
47C3×#4025C2214yes—
48C3×#4022C2214yes—
49C3×#4021C2212yes—
50C3×C4×D20C2214yes—
51C3×C4×D20C2212yes—
52C3×#4018C2214yes—
53C3×#4018C2214yes—
54C3×C42×D5C2214yes—
55C3×#4016C2212yes—
56C3×C4×Dic10C2214yes—
57C3×C4×Dic10C2212yes—
58C3×#3531—51nomaximal, Hall
59C5×#533C2×C311yescharacteristic, complemented
60#66069C2×C314yescomplemented
61#66034C2×C311yescharacteristic, complemented
62#66033C2×C312yescomplemented
63C5⋊#564C2×C311yescharacteristic, complemented
64#65981C2×C314yescomplemented
65#65980C2×C314yescomplemented
66#65914C2×C314yescomplemented
67#65912C2×C314yescomplemented
68#65862C2×C314yescomplemented
69#65813C2×C312yescomplemented
70C2×C3×C5×Q8C2314yes—
71C22×C3×C4×C5C2311yescharacteristic
72C22×C3×C4×C5C2312yes—
73C2×C3×C22⋊D5C2314yes—
74C2×C3×D20C2314yes—
75C2×C3×D20C2312yes—
76C2×C3×C4×D5C2314yes—
77C2×C3×C4×D5C2314yes—
78C2×C3×C4×D5C2314yes—
79C2×C3×Dic10C2314yes—
80C2×C3×Dic10C2312yes—
81C3×C5×C4⋊C4C2312yes—
82C3×C5×C4⋊C4C2312yes—
83C3×C5×C4⋊C4C2314yes—
84C3×C5×C4⋊C4C2314yes—
85C3×C5×C4⋊C4C2314yes—
86C3×C5×C22⋊C4C2314yes—
87C3×C42×C5C2312yes—
88C3×C42×C5C2314yes—
89C3×C42×C5C2311yescharacteristic
90C3×C42×C5C2311yescharacteristic
91C3×C22⋊Dic5C2314yes—
92C3×#685C2314yes—
93C3×#685C2314yes—
94C3×#685C2314yes—
95C3×#685C2314yes—
96C3×#685C2314yes—
97C3×#685C2314yes—
98C3×#685C2314yes—
99C3×#685C2314yes—
100C3×C4⋊Dic5C2314yes—
101C3×C4⋊Dic5C2314yes—
102C3×C4⋊Dic5C2312yes—
103C3×C4⋊Dic5C2312yes—
104C3×#683C2314yes—
105C3×#683C2314yes—
106C3×#683C2314yes—
107C3×#683C2314yes—
108C3×#683C2314yes—
109C3×#683C2314yes—
110C3×#683C2314yes—
111C3×#683C2314yes—
112C3×C4×Dic5C2314yes—
113C3×C4×Dic5C2314yes—
114C3×C4×Dic5C2314yes—
115C3×#564—51no—
116C3×#563—54no—
117C3×#563—52no—
118C3×#563—54no—
119C3×#562—51no—
120C3×#562—54no—
121C3×#533—54no—
122C3×#533D511yescharacteristic, complemented
123C3×#529—54no—
124C3×#529—54no—
125C3×#529—52no—
126C5×C4⋊Q8C22×C312yes—
127C5×#125C22×C311yescharacteristic
128C5×#125C22×C311yescharacteristic
129C5×C22⋊Q8C22×C314yes—
130C4×C5×Q8C22×C314yes—
131C5×#117C22×C312yes—
132C2×C42×C5C22×C311yescharacteristic
133#4093C22×C314yes—
134#4092C22×C314yes—
135#4092C22×C314yes—
136#4090C22×C314yes—
137#4076C22×C312yes—
138#4075C22×C314yes—
139C4×C22⋊D5C22×C314yes—
140#4070C22×C312yes—
141#4044C22×C314yes—
142#4044C22×C314yes—
143#4044C22×C314yes—
144#4043C22×C314yes—
145#4042C22×C314yes—
146#4042C22×C314yes—
147#4042C22×C314yes—
148#4041C22×C314yes—
149#4040C22×C314yes—
150#4040C22×C314yes—
151#4040C22×C314yes—
152#4036C22×C314yes—
153#4035C22×C314yes—
154#4035C22×C314yes—
155#4035C22×C314yes—
156#4034C22×C314yes—
157#4031C22×C314yes—
158#4030C22×C314yes—
159#4029C22×C314yes—
160#4025C22×C314yes—
161#4022C22×C314yes—
162#4021C22×C312yes—
163C4×D20C22×C312yes—
164C4×D20C22×C314yes—
165#4018C22×C314yes—
166#4018C22×C314yes—
167C42×D5C22×C314yes—
168#4016C22×C312yes—
169C4×Dic10C22×C312yes—
170C4×Dic10C22×C314yes—
171#3531—51noSylow
172C23×C3×C5C2411yescharacteristic
173C22×C3×D5C2414yes—
174C3×C5×Q8—44no—
175C2×C3×C4×C5—22no—
176C2×C3×C4×C5—42no—
177C2×C3×C4×C5C2414yes—
178C2×C3×C4×C5C2414yes—
179C2×C3×C4×C5C2411yescharacteristic
180C2×C3×C4×C5C2412yes—
181C2×C3×C4×C5C2412yes—
182C2×C3×C4×C5C2411yescharacteristic
183C3×C22⋊D5—44no—
184C2×C3×Dic5C2414yes—
185C2×C3×Dic5C2414yes—
186C2×C3×Dic5C2414yes—
187C3×D20—42no—
188C3×D20—44no—
189C3×C4×D5—28no—
190C3×C4×D5—44no—
191C3×C4×D5—44no—
192C3×Dic10—42no—
193C3×Dic10—44no—
194C3×C4⋊Q8—54no—
195C3×C4⋊Q8—54no—
196C3×C4⋊Q8C2×D512yes—
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—52no—
207C3×#125C2×D511yescharacteristic
208C3×#125C2×D511yescharacteristic
209C3×#124—54no—
210C3×#124—54no—
211C3×#124—52no—
212C3×#123—54no—
213C3×#123—54no—
214C3×#123—54no—
215C3×#123—54no—
216C3×#123—54no—
217C3×C22⋊Q8—54no—
218C3×C22⋊Q8—54no—
219C3×C22⋊Q8—52no—
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×#121—54no—
228C3×C4×Q8—54no—
229C3×C4×Q8—52no—
230C3×C4×Q8C2×D514yes—
231C3×C4×D4—54no—
232C3×C4×D4—54no—
233C3×C4×D4—52no—
234C3×#117—54no—
235C3×#117—54no—
236C3×#117C2×D512yes—
237C2×C3×C42—54no—
238C2×C3×C42C2×D511yescharacteristic
239C2×C5×Q8C23×C314yes—
240C22×C4×C5C23×C311yescharacteristic
241C22×C4×C5C23×C312yes—
242C2×C22⋊D5C23×C314yes—
243C2×D20C23×C312yes—
244C2×D20C23×C314yes—
245C2×C4×D5C23×C314yes—
246C2×C4×D5C23×C314yes—
247C2×C4×D5C23×C314yes—
248C2×Dic10C23×C312yes—
249C2×Dic10C23×C314yes—
250C5×C4⋊C4C23×C312yes—
251C5×C4⋊C4C23×C312yes—
252C5×C4⋊C4C23×C314yes—
253C5×C4⋊C4C23×C314yes—
254C5×C4⋊C4C23×C314yes—
255C5×C22⋊C4C23×C314yes—
256C42×C5C23×C312yes—
257C42×C5C23×C311yescharacteristic
258C42×C5C23×C311yescharacteristic
259C42×C5C23×C314yes—
260C22⋊Dic5C23×C314yes—
261#685C23×C314yes—
262#685C23×C314yes—
263#685C23×C314yes—
264#685C23×C314yes—
265#685C23×C314yes—
266#685C23×C314yes—
267#685C23×C314yes—
268#685C23×C314yes—
269C4⋊Dic5C23×C312yes—
270C4⋊Dic5C23×C314yes—
271C4⋊Dic5C23×C312yes—
272C4⋊Dic5C23×C314yes—
273#683C23×C314yes—
274#683C23×C314yes—
275#683C23×C314yes—
276#683C23×C314yes—
277#683C23×C314yes—
278#683C23×C314yes—
279#683C23×C314yes—
280#683C23×C314yes—
281C4×Dic5C23×C314yes—
282C4×Dic5C23×C314yes—
283C4×Dic5C23×C314yes—
284#564—51no—
285#563—54no—
286#563—52no—
287#563—54no—
288#562—51no—
289#562—54no—
290#533—54no—
291#533C3×D511yescharacteristic, complemented
292#529—54no—
293#529—54no—
294#529—52no—
295C22×C3×C5—21no—
296C22×C3×C5—22no—
297C22×C3×C5C2511yescharacteristic
298C2×C3×D5—24no—
299C2×C3×D5—24no—
300C2×C3×D5—24no—
301C3×C4×C5—21no—
302C3×C4×C5—24no—
303C3×C4×C5—22no—
304C3×C4×C5—24no—
305C3×C4×C5—22no—
306C3×C4×C52−1+412yes—
307C3×Dic5—24no—
308C3×Dic5—24no—
309C3×Dic5—24no—
310C2×C3×Q8—52no—
311C2×C3×Q8—54no—
312C2×C3×Q8C22×D514yes—
313C2×C3×D4—54no—
314C2×C3×D4—54no—
315C2×C3×D4—52no—
316C22×C3×C4—54no—
317C22×C3×C4—54no—
318C22×C3×C4—54no—
319C22×C3×C4C22×D511yescharacteristic
320C22×C3×C4C22×D512yes—
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—52no—
332C3×C4⋊C4—54no—
333C3×C4⋊C4C22×D512yes—
334C3×C4⋊C4C22×D512yes—
335C3×C4⋊C4C22×D514yes—
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⋊C4—54no—
346C3×C22⋊C4—54no—
347C3×C22⋊C4C22×D514yes—
348C3×C42—54no—
349C3×C42—54no—
350C3×C42—54no—
351C3×C42C22×D511yescharacteristic
352C3×C42C22×D511yescharacteristic
353C3×C42C22×D514yes—
354C3×C42C22×D512yes—
355C23×C5C24×C311yescharacteristic
356C22×D5C24×C314yes—
357C5×Q8—44no—
358C2×C4×C5—22no—
359C2×C4×C5—42no—
360C2×C4×C5C24×C314yes—
361C2×C4×C5C24×C314yes—
362C2×C4×C5C24×C311yescharacteristic
363C2×C4×C5C24×C312yes—
364C2×C4×C5C24×C312yes—
365C2×C4×C5C24×C311yescharacteristic
366C22⋊D5—44no—
367C2×Dic5C24×C314yes—
368C2×Dic5C24×C314yes—
369C2×Dic5C24×C314yes—
370D20—42no—
371D20—44no—
372C4×D5—28no—
373C4×D5—44no—
374C4×D5—44no—
375Dic10—42no—
376Dic10—44no—
377C4⋊Q8—54no—
378C4⋊Q8—54no—
379C4⋊Q8C2×C3×D512yes—
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#125C2×C3×D511yescharacteristic
391#125C2×C3×D511yescharacteristic
392#124—54no—
393#124—54no—
394#124—52no—
395#123—54no—
396#123—54no—
397#123—54no—
398#123—54no—
399#123—54no—
400C22⋊Q8—54no—
401C22⋊Q8—54no—
402C22⋊Q8—52no—
403C22⋊Q8—54no—
404C22⋊Q8—54no—
405C22⋊Q8—54no—
406C22⋊Q8—54no—
407C22⋊Q8C2×C3×D514yes—
408#121—52no—
409#121—54no—
410#121—54no—
411C4×Q8—54no—
412C4×Q8—52no—
413C4×Q8C2×C3×D514yes—
414C4×D4—54no—
415C4×D4—54no—
416C4×D4—52no—
417#117—54no—
418#117—54no—
419#117C2×C3×D512yes—
420C2×C42—54no—
421C2×C42C2×C3×D511yescharacteristic
422C2×C3×C5—41no—
423C2×C3×C5C2×2−1+411yescharacteristic
424C2×C3×C5#58512yes—
425C3×D5—44no—
426C23×C3—54no—
427C23×C3C23×D511yescharacteristic
428C3×Q8—44no—
429C3×Q8—202no—
430C3×Q8—204no—
431C3×D4—202no—
432C3×D4—204no—
433C3×D4—204no—
434C2×C3×C4—22no—
435C2×C3×C4—42no—
436C2×C3×C4—54no—
437C2×C3×C4—54no—
438C2×C3×C4—54no—
439C2×C3×C4—108no—
440C2×C3×C4—204no—
441C2×C3×C4—204no—
442C2×C3×C4C23×D514yes—
443C2×C3×C4C23×D514yes—
444C2×C3×C4C23×D511yescharacteristic
445C2×C3×C4C23×D512yes—
446C2×C3×C4C23×D512yes—
447C2×C3×C4C23×D511yescharacteristic
448C22×C5—21no—
449C22×C5—22no—
450C22×C5C25×C311yescharacteristic
451C2×D5—24no—
452C2×D5—24no—
453C2×D5—24no—
454C4×C5—21no—
455C4×C5—24no—
456C4×C5—22no—
457C4×C5—24no—
458C4×C5—22no—
459C4×C5C3×2−1+412yes—
460Dic5—24no—
461Dic5—24no—
462Dic5—24no—
463C2×Q8—52no—
464C2×Q8—54no—
465C2×Q8C22×C3×D514yes—
466C2×D4—54no—
467C2×D4—54no—
468C2×D4—52no—
469C22×C4—54no—
470C22×C4—54no—
471C22×C4—54no—
472C22×C4C22×C3×D511yescharacteristic
473C22×C4C22×C3×D512yes—
474C4⋊C4—54no—
475C4⋊C4—54no—
476C4⋊C4—54no—
477C4⋊C4—54no—
478C4⋊C4—54no—
479C4⋊C4—54no—
480C4⋊C4—52no—
481C4⋊C4—54no—
482C4⋊C4—52no—
483C4⋊C4—54no—
484C4⋊C4—54no—
485C4⋊C4—54no—
486C4⋊C4C22×C3×D512yes—
487C4⋊C4C22×C3×D512yes—
488C4⋊C4C22×C3×D514yes—
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⋊C4—54no—
499C22⋊C4—54no—
500C22⋊C4C22×C3×D514yes—
501C42—54no—
502C42—54no—
503C42—54no—
504C42C22×C3×D514yes—
505C42C22×C3×D512yes—
506C42C22×C3×D511yescharacteristic
507C42C22×C3×D511yescharacteristic
508C3×C5#353111yescharacteristic, complemented, Hall
509C22×C3—21no—
510C22×C3—22no—
511C22×C3—104no—
512C22×C3—104no—
513C22×C3—104no—
514C22×C3C24×D511yescharacteristic
515C3×C4—21no—
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—24no—
519C3×C4—24no—
520C3×C4—104no—
521C3×C4—104no—
522C3×C4—104no—
523C3×C4#414712yes—
524C2×C5—41no—
525C2×C5C2×C3×2−1+411yescharacteristic
526C2×C5C3×#58512yes—
527D5—44no—
528C23—54no—
529C23C23×C3×D511yescharacteristic
530Q8—44no—
531Q8—202no—
532Q8—204no—
533D4—202no—
534D4—204no—
535D4—204no—
536C2×C4—22no—
537C2×C4—42no—
538C2×C4—54no—
539C2×C4—54no—
540C2×C4—54no—
541C2×C4—108no—
542C2×C4—204no—
543C2×C4—204no—
544C2×C4C23×C3×D514yes—
545C2×C4C23×C3×D514yes—
546C2×C4C23×C3×D511yescharacteristic
547C2×C4C23×C3×D512yes—
548C2×C4C23×C3×D512yes—
549C2×C4C23×C3×D511yescharacteristic
550C2×C3—41no—
551C2×C3—204no—
552C2×C3C2×#414711yescharacteristic
553C2×C3#6628312yes—
554C5C3×#353111yescharacteristic, complemented, Sylow
555C22—21no—
556C22—22no—
557C22—104no—
558C22—104no—
559C22—104no—
560C22C24×C3×D511yescharacteristic
561C4—21no—
562C4—22no—
563C4—22no—
564C4—24no—
565C4—24no—
566C4—104no—
567C4—104no—
568C4—104no—
569C4C3×#414712yes—
570C3#1061818711yescharacteristic, direct product factor, Sylow
571C2—41no—
572C2—204no—
573C2C2×C3×#414711yescharacteristic
574C2C3×#6628312yes—
subgroupG/H
centerC2×C3×C4C23×D5
commutator, cosocleC22×C5C25×C3
FrattiniC22C24×C3×D5
FittingC3×C5×#533C2
socleC22×C3×C5C25
hypercenterC3×#533D5
hypocenterC5C3×#3531
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C3×#533

Complements

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

NH ≅ G/Ntype
1. C3×C5×#533C2semidirect
2. C3×#66069C2semidirect
3. C3×#66034C2semidirect
4. C3×#66033C2semidirect
5. C3×C5⋊#564C2semidirect
6. C3×#65981C2semidirect
7. C3×#65980C2semidirect
8. C3×#65914C2semidirect
9. C3×#65912C2semidirect
10. C3×#65862C2semidirect
11. C3×#65813C2semidirect
12. #10618187C3direct (N × H)
59. C5×#533C2×C3semidirect
60. #66069C2×C3semidirect
61. #66034C2×C3semidirect
62. #66033C2×C3semidirect
63. C5⋊#564C2×C3semidirect
64. #65981C2×C3semidirect
65. #65980C2×C3semidirect
66. #65914C2×C3semidirect
67. #65912C2×C3semidirect
68. #65862C2×C3semidirect
69. #65813C2×C3semidirect
122. C3×#533D5semidirect
291. #533C3×D5semidirect
508. C3×C5#3531semidirect
554. C5C3×#3531semidirect
570. C3#10618187direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5332
C3×#660692
C3×#660342
C3×#660332
C3×C5⋊#5642
C3×#659812
C3×#659802
C3×#659142
C3×#659122
C3×#658622
C3×#658132
#106181873
C3×#35315