Finite Group Databasebeta

← C3×#10605067 #14327499

Subgroups — C3×#10605067 #14327499

Order 1920.

1176 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C22⋊D5C211yesmaximal, characteristic
2C2×C3×C5×#95C211yesmaximal, characteristic, complemented
3C3×#65502C212yesmaximal
4C3×#65501C211yesmaximal, characteristic, complemented
5C3×C4×C22⋊Dic5C212yesmaximal, complemented
6C3×#64959C212yesmaximal, complemented
7C3×#64955C211yesmaximal, characteristic, complemented
8C3×#64953C211yesmaximal, characteristic, complemented
9C3×#64944C212yesmaximal, complemented
10C3×#64940C211yesmaximal, characteristic, complemented
11C3×#64937C211yesmaximal, characteristic, complemented
12#10605067C311yesmaximal, characteristic, direct product factor, Hall
13C23×C3×C4×C5C2212yes—
14C23×C3×C4×C5C2211yescharacteristic
15C22×C3×C22⋊D5C2211yescharacteristic
16C22×C3×C4×D5C2211yescharacteristic
17C2×C3×C22⋊Dic5C2212yes—
18C2×C3×C22⋊Dic5C2211yescharacteristic
19C3×C4×C22⋊D5C418yescomplemented
20C2×C3×#685C2212yes—
21C2×C3×#685C2212yes—
22C2×C3×#685C2211yescharacteristic
23C2×C3×C4⋊Dic5C2212yes—
24C2×C3×#683C2211yescharacteristic
25C2×C3×C4×Dic5C2212yes—
26C2×C3×C4×Dic5C2212yes—
27C2×C3×C4×Dic5C2211yescharacteristic
28C3×C5×#95C2211yescharacteristic
29C3×C5×#95C2211yescharacteristic
30C3×C5×#95C2212yes—
31C3×#3963C2212yes—
32C3×#3963C2211yescharacteristic
33C3×#3963C2212yes—
34C3×#3963C2211yescharacteristic
35C3×#3963C2212yes—
36C3×#3963C2212yes—
37C3×#2353—51nomaximal, Hall
38C2×C4×C22⋊D5C2×C311yescharacteristic
39C2×C5×#95C2×C311yescharacteristic, complemented
40#65502C2×C312yes—
41#65501C2×C311yescharacteristic, complemented
42C4×C22⋊Dic5C2×C312yescomplemented
43#64959C2×C312yescomplemented
44#64955C2×C311yescharacteristic, complemented
45#64953C2×C311yescharacteristic, complemented
46#64944C2×C312yescomplemented
47#64940C2×C311yescharacteristic, complemented
48#64937C2×C311yescharacteristic, complemented
49C24×C3×C5C2311yescharacteristic
50C23×C3×D5C2311yescharacteristic
51C22×C3×C4×C5—24no—
52C22×C3×C4×C5—24no—
53C22×C3×C4×C5—42no—
54C22×C3×C4×C5—42no—
55C22×C3×C4×C5C2×C414yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2312yes—
60C2×C3×C22⋊D5—22no—
61C2×C3×C22⋊D5—22no—
62C2×C3×C22⋊D5C2×C414yes—
63C22×C3×Dic5C2312yes—
64C22×C3×Dic5C2312yes—
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C2×C3×C4×D5—42no—
69C2×C3×C4×D5C2×C414yescomplemented
70C3×C22⋊Dic5—24no—
71C3×C22⋊Dic5C2×C414yes—
72C3×#685—24no—
73C3×#685—42no—
74C3×#685C2×C414yescomplemented
75C3×C4⋊Dic5—42no—
76C3×#683C2×C414yescomplemented
77C3×C4×Dic5—24no—
78C3×C4×Dic5—42no—
79C3×C4×Dic5C2×C414yescomplemented
80C2×C3×C4×D4—51no—
81C3×#388—52no—
82C3×#387—52no—
83C3×#386—52no—
84C3×#383—51no—
85C3×#381—51no—
86C3×#380—51no—
87C3×C4×C22⋊C4—52no—
88C3×C4×C22⋊C4—51no—
89C3×#376—51no—
90C2×C3×#95D511yescharacteristic, complemented
91C23×C4×C5C22×C312yes—
92C23×C4×C5C22×C311yescharacteristic
93C22×C22⋊D5C22×C311yescharacteristic
94C22×C4×D5C22×C311yescharacteristic
95C2×C22⋊Dic5C22×C312yes—
96C2×C22⋊Dic5C22×C311yescharacteristic
97C4×C22⋊D5C3×C418yescomplemented
98C2×#685C22×C312yes—
99C2×#685C22×C312yes—
100C2×#685C22×C311yescharacteristic
101C2×C4⋊Dic5C22×C312yes—
102C2×#683C22×C311yescharacteristic
103C2×C4×Dic5C22×C312yes—
104C2×C4×Dic5C22×C312yes—
105C2×C4×Dic5C22×C311yescharacteristic
106C5×#95C22×C311yescharacteristic
107C5×#95C22×C311yescharacteristic
108C5×#95C22×C312yes—
109#3963C22×C312yes—
110#3963C22×C311yescharacteristic
111#3963C22×C312yes—
112#3963C22×C311yescharacteristic
113#3963C22×C312yes—
114#3963C22×C312yes—
115#2353—51noSylow
116C23×C3×C5—22no—
117C23×C3×C5—22no—
118C23×C3×C5C22×C412yes—
119C23×C3×C5#4112yes—
120C23×C3×C5#4112yes—
121C23×C3×C5C2411yescharacteristic
122C22×C3×D5—22no—
123C22×C3×D5—21no—
124C22×C3×D5—21no—
125C22×C3×D5—22no—
126C22×C3×D5C22×C412yes—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—44no—
135C2×C3×C4×C5—48no—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5C22×C412yes—
138C2×C3×C4×C5#4114yes—
139C3×C22⋊D5—28no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—21no—
142C2×C3×Dic5—21no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—22no—
145C2×C3×Dic5—22no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5C22×C412yes—
151C2×C3×Dic5C22×C412yes—
152C2×C3×Dic5C22×C412yes—
153C3×C4×D5—44no—
154C22×C3×D4—51no—
155C23×C3×C4—51no—
156C23×C3×C4C2×D512yes—
157C23×C3×C4C2×D511yescharacteristic
158C3×C4×D4—58no—
159C2×C3×C4⋊C4—51no—
160C2×C3×C4⋊C4—52no—
161C2×C3×C22⋊C4—51no—
162C2×C3×C22⋊C4—51no—
163C2×C3×C22⋊C4—52no—
164C2×C3×C22⋊C4—52no—
165C2×C3×C22⋊C4—52no—
166C2×C3×C42—51no—
167C2×C3×C42—52no—
168C2×C3×C42—52no—
169C3×#95—51no—
170C3×#95—52no—
171C3×#95—51no—
172C3×#95—52no—
173C3×#95—52no—
174C3×#95—52no—
175C3×#95C2×D511yescharacteristic
176C3×#95C2×D511yescharacteristic
177C3×#95C2×D512yes—
178C24×C5C23×C311yescharacteristic
179C23×D5C23×C311yescharacteristic
180C22×C4×C5—24no—
181C22×C4×C5—24no—
182C22×C4×C5—42no—
183C22×C4×C5—42no—
184C22×C4×C5C2×C3×C414yes—
185C22×C4×C5C23×C312yes—
186C22×C4×C5C23×C311yescharacteristic
187C22×C4×C5C23×C311yescharacteristic
188C22×C4×C5C23×C312yes—
189C2×C22⋊D5—22no—
190C2×C22⋊D5—22no—
191C2×C22⋊D5C2×C3×C414yes—
192C22×Dic5C23×C312yes—
193C22×Dic5C23×C312yes—
194C22×Dic5C23×C311yescharacteristic
195C22×Dic5C23×C311yescharacteristic
196C22×Dic5C23×C311yescharacteristic
197C2×C4×D5—42no—
198C2×C4×D5C2×C3×C414yescomplemented
199C22⋊Dic5—24no—
200C22⋊Dic5C2×C3×C414yes—
201#685—24no—
202#685—42no—
203#685C2×C3×C414yescomplemented
204C4⋊Dic5—42no—
205#683C2×C3×C414yescomplemented
206C4×Dic5—24no—
207C4×Dic5—42no—
208C4×Dic5C2×C3×C414yescomplemented
209C2×C4×D4—51no—
210#388—52no—
211#387—52no—
212#386—52no—
213#383—51no—
214#381—51no—
215#380—51no—
216C4×C22⋊C4—52no—
217C4×C22⋊C4—51no—
218#376—51no—
219C2×#95C3×D511yescharacteristic, complemented
220C22×C3×C5—22no—
221C22×C3×C5—22no—
222C22×C3×C5—24no—
223C22×C3×C5—24no—
224C22×C3×C5#11714yes—
225C22×C3×C5C23×C411yescharacteristic
226C22×C3×C5C2×#4112yes—
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C22×C3×C5H(5,𝔽2)12yes—
230C2×C3×D5—22no—
231C2×C3×D5—22no—
232C2×C3×D5—22no—
233C2×C3×D5—42no—
234C2×C3×D5—42no—
235C3×C4×C5—22no—
236C3×C4×C5—22no—
237C3×C4×C5—24no—
238C3×C4×C5—42no—
239C3×Dic5—22no—
240C3×Dic5—22no—
241C3×Dic5—22no—
242C3×Dic5—42no—
243C3×Dic5—42no—
244C24×C3—51no—
245C24×C3C22×D511yescharacteristic
246C2×C3×D4—54no—
247C2×C3×D4—102no—
248C2×C3×D4—102no—
249C22×C3×C4—24no—
250C22×C3×C4—24no—
251C22×C3×C4—42no—
252C22×C3×C4—42no—
253C22×C3×C4—51no—
254C22×C3×C4—51no—
255C22×C3×C4—51no—
256C22×C3×C4—54no—
257C22×C3×C4—52no—
258C22×C3×C4—52no—
259C22×C3×C4—202no—
260C22×C3×C4C4×D514yes—
261C22×C3×C4C22×D512yes—
262C22×C3×C4C22×D511yescharacteristic
263C22×C3×C4C22×D511yescharacteristic
264C22×C3×C4C22×D512yes—
265C3×C4⋊C4—54no—
266C3×C4⋊C4—202no—
267C3×C22⋊C4—54no—
268C3×C22⋊C4—54no—
269C3×C22⋊C4—104no—
270C3×C22⋊C4—104no—
271C3×C22⋊C4—202no—
272C3×C42—54no—
273C3×C42—104no—
274C3×C42—202no—
275C23×C5—22no—
276C23×C5—22no—
277C23×C5C22×C3×C412yes—
278C23×C5C3×#4112yes—
279C23×C5C3×#4112yes—
280C23×C5C24×C311yescharacteristic
281C22×D5—22no—
282C22×D5—21no—
283C22×D5—21no—
284C22×D5—22no—
285C22×D5C22×C3×C412yes—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—44no—
294C2×C4×C5—48no—
295C2×C4×C5C22×C3×C412yes—
296C2×C4×C5C22×C3×C412yes—
297C2×C4×C5C3×#4114yes—
298C22⋊D5—28no—
299C2×Dic5—22no—
300C2×Dic5—21no—
301C2×Dic5—21no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—22no—
308C2×Dic5—22no—
309C2×Dic5C22×C3×C412yes—
310C2×Dic5C22×C3×C412yes—
311C2×Dic5C22×C3×C412yes—
312C4×D5—44no—
313C22×D4—51no—
314C23×C4—51no—
315C23×C4C2×C3×D512yes—
316C23×C4C2×C3×D511yescharacteristic
317C4×D4—58no—
318C2×C4⋊C4—51no—
319C2×C4⋊C4—52no—
320C2×C22⋊C4—51no—
321C2×C22⋊C4—51no—
322C2×C22⋊C4—52no—
323C2×C22⋊C4—52no—
324C2×C22⋊C4—52no—
325C2×C42—51no—
326C2×C42—52no—
327C2×C42—52no—
328#95—51no—
329#95—52no—
330#95—51no—
331#95—52no—
332#95—52no—
333#95—52no—
334#95C2×C3×D511yescharacteristic
335#95C2×C3×D511yescharacteristic
336#95C2×C3×D512yes—
337C2×C3×C5—24no—
338C2×C3×C5C2×#11711yescharacteristic
339C2×C3×C5C4×#4111yescharacteristic
340C2×C3×C5#51811yescharacteristic
341C2×C3×C5#55112yes—
342C2×C3×C5#55312yes—
343C3×D5—42no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—52no—
347C23×C3—102no—
348C23×C3—101no—
349C23×C3—102no—
350C23×C3—101no—
351C23×C3C2×C4×D512yes—
352C23×C3#71112yes—
353C23×C3#71312yes—
354C23×C3C23×D511yescharacteristic
355C3×D4—108no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—44no—
364C2×C3×C4—48no—
365C2×C3×C4—52no—
366C2×C3×C4—52no—
367C2×C3×C4—52no—
368C2×C3×C4—102no—
369C2×C3×C4—101no—
370C2×C3×C4—102no—
371C2×C3×C4—101no—
372C2×C3×C4—102no—
373C2×C3×C4—102no—
374C2×C3×C4—102no—
375C2×C3×C4—102no—
376C2×C3×C4—102no—
377C2×C3×C4—102no—
378C2×C3×C4—204no—
379C2×C3×C4C2×C4×D512yes—
380C2×C3×C4C2×C4×D512yes—
381C2×C3×C4#70914yes—
382C22×C5—22no—
383C22×C5—22no—
384C22×C5—24no—
385C22×C5—24no—
386C22×C5C3×#11714yes—
387C22×C5C23×C3×C411yescharacteristic
388C22×C5C2×C3×#4112yes—
389C22×C5C2×C3×#4111yescharacteristic
390C22×C5C2×C3×#4111yescharacteristic
391C22×C5C3×H(5,𝔽2)12yes—
392C2×D5—22no—
393C2×D5—22no—
394C2×D5—22no—
395C2×D5—42no—
396C2×D5—42no—
397C4×C5—22no—
398C4×C5—22no—
399C4×C5—24no—
400C4×C5—42no—
401Dic5—22no—
402Dic5—22no—
403Dic5—22no—
404Dic5—42no—
405Dic5—42no—
406C24—51no—
407C24C22×C3×D511yescharacteristic
408C2×D4—54no—
409C2×D4—102no—
410C2×D4—102no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—51no—
416C22×C4—51no—
417C22×C4—51no—
418C22×C4—54no—
419C22×C4—52no—
420C22×C4—52no—
421C22×C4—202no—
422C22×C4C3×C4×D514yes—
423C22×C4C22×C3×D512yes—
424C22×C4C22×C3×D511yescharacteristic
425C22×C4C22×C3×D511yescharacteristic
426C22×C4C22×C3×D512yes—
427C4⋊C4—54no—
428C4⋊C4—202no—
429C22⋊C4—54no—
430C22⋊C4—54no—
431C22⋊C4—104no—
432C22⋊C4—104no—
433C22⋊C4—202no—
434C42—54no—
435C42—104no—
436C42—202no—
437C3×C5#235311yescharacteristic, complemented, Hall
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—24no—
441C22×C3—24no—
442C22×C3—102no—
443C22×C3—102no—
444C22×C3—102no—
445C22×C3—202no—
446C22×C3—202no—
447C22×C3#403814yes—
448C22×C3C22×C4×D511yescharacteristic
449C22×C3C2×#70912yes—
450C22×C3C2×#71111yescharacteristic
451C22×C3#414412yes—
452C22×C3C2×#71311yescharacteristic
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—24no—
456C3×C4—42no—
457C3×C4—102no—
458C3×C4—102no—
459C3×C4—102no—
460C3×C4—202no—
461C3×C4—202no—
462C2×C5—24no—
463C2×C5C2×C3×#11711yescharacteristic
464C2×C5C3×C4×#4111yescharacteristic
465C2×C5C3×#51811yescharacteristic
466C2×C5C3×#55112yes—
467C2×C5C3×#55312yes—
468D5—42no—
469C23—22no—
470C23—22no—
471C23—52no—
472C23—102no—
473C23—101no—
474C23—102no—
475C23—101no—
476C23C2×C3×C4×D512yes—
477C23C3×#71112yes—
478C23C3×#71312yes—
479C23C23×C3×D511yescharacteristic
480D4—108no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—44no—
489C2×C4—48no—
490C2×C4—52no—
491C2×C4—52no—
492C2×C4—52no—
493C2×C4—102no—
494C2×C4—101no—
495C2×C4—102no—
496C2×C4—101no—
497C2×C4—102no—
498C2×C4—102no—
499C2×C4—102no—
500C2×C4—102no—
501C2×C4—102no—
502C2×C4—102no—
503C2×C4—204no—
504C2×C4C2×C3×C4×D512yes—
505C2×C4C2×C3×C4×D512yes—
506C2×C4C3×#70914yes—
507C2×C3—24no—
508C2×C3—202no—
509C2×C3C4×#70911yescharacteristic
510C2×C3#6582011yescharacteristic
511C2×C3#6582912yes—
512C2×C3C2×#403811yescharacteristic
513C2×C3#6584812yes—
514C5C3×#235311yescharacteristic, complemented, Sylow
515C22—24no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—102no—
520C22—102no—
521C22—102no—
522C22—202no—
523C22—202no—
524C22C3×#403814yes—
525C22C22×C3×C4×D511yescharacteristic
526C22C2×C3×#70912yes—
527C22C2×C3×#71111yescharacteristic
528C22C3×#414412yes—
529C22C2×C3×#71311yescharacteristic
530C4—22no—
531C4—22no—
532C4—24no—
533C4—42no—
534C4—102no—
535C4—102no—
536C4—102no—
537C4—202no—
538C4—202no—
539C3#1060506711yescharacteristic, direct product factor, Sylow
540C2—24no—
541C2—202no—
542C2C3×C4×#70911yescharacteristic
543C2C3×#6582011yescharacteristic
544C2C3×#6582912yes—
545C2C2×C3×#403811yescharacteristic
546C2C3×#6584812yes—
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC2×C3×C5×#95C2
socleC23×C3×C5C24
prime radicalC22×C3×C22⋊D5C22
cosocleC23×C5C24×C3
hypercenterC2×C3×#95D5
hypocenterC5C3×#2353
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×#95

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#95C2semidirect
4. C3×#65501C2semidirect
5. C3×C4×C22⋊Dic5C2semidirect
6. C3×#64959C2semidirect
7. C3×#64955C2semidirect
8. C3×#64953C2semidirect
9. C3×#64944C2semidirect
10. C3×#64940C2semidirect
11. C3×#64937C2semidirect
12. #10605067C3direct (N × H)
19. C3×C4×C22⋊D5C4semidirect
39. C2×C5×#95C2×C3semidirect
41. #65501C2×C3semidirect
42. C4×C22⋊Dic5C2×C3semidirect
43. #64959C2×C3semidirect
44. #64955C2×C3semidirect
45. #64953C2×C3semidirect
46. #64944C2×C3semidirect
47. #64940C2×C3semidirect
48. #64937C2×C3semidirect
69. C2×C3×C4×D5C2×C4semidirect
74. C3×#685C2×C4semidirect
76. C3×#683C2×C4semidirect
79. C3×C4×Dic5C2×C4semidirect
90. C2×C3×#95D5semidirect
97. C4×C22⋊D5C3×C4semidirect
198. C2×C4×D5C2×C3×C4semidirect
203. #685C2×C3×C4semidirect
205. #683C2×C3×C4semidirect
208. C4×Dic5C2×C3×C4semidirect
219. C2×#95C3×D5semidirect
437. C3×C5#2353semidirect
514. C5C3×#2353semidirect
539. C3#10605067direct (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×C22⋊D52
C2×C3×C5×#952
C3×#655022
C3×#655012
C3×C4×C22⋊Dic52
C3×#649592
C3×#649552
C3×#649532
C3×#649442
C3×#649402
C3×#649372
#106050673
C3×#23535