Finite Group Databasebeta

← C3×#10605604 #14324039

Subgroups — C3×#10605604 #14324039

Order 1920.

1392 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4076C211yesmaximal, characteristic
2C2×C3×#4027C212yesmaximal, complemented
3C3×C5×#380C211yesmaximal, characteristic, complemented
4C2×C3×#3963C211yesmaximal, characteristic, complemented
5C3×#65279C212yesmaximal, complemented
6C3×#65278C212yesmaximal, complemented
7C3×#65247C212yesmaximal, complemented
8C3×#64954C212yesmaximal, complemented
9C3×#64943C212yesmaximal, complemented
10#10605604C311yesmaximal, characteristic, direct product factor, Hall
11C23×C3×C4×C5C2211yescharacteristic
12C22×C3×C22⋊D5C2212yes—
13C23×C3×Dic5C2211yescharacteristic
14C23×C3×Dic5C2211yescharacteristic
15C22×C3×D20C2211yescharacteristic
16C22×C3×C4×D5C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C22⋊C4C2212yes—
20C3×#4076C418yescomplemented
21C2×C3×#685C2212yes—
22C2×C3×#685C2212yes—
23C2×C3×#685C2212yes—
24C2×C3×#685C2212yes—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×#683C2212yes—
27C2×C3×C4×Dic5C2212yes—
28C3×#4027—28no—
29C3×C5×#95C2212yes—
30C3×#3963C2212yes—
31C3×#3963C2212yes—
32C3×#3963C2212yes—
33C3×#3963C2212yes—
34C3×#2325—51nomaximal, Hall
35C2×#4076C2×C311yescharacteristic
36C2×#4027C2×C312yescomplemented
37C5×#380C2×C311yescharacteristic, complemented
38C2×#3963C2×C311yescharacteristic, complemented
39#65279C2×C312yescomplemented
40#65278C2×C312yescomplemented
41#65247C2×C312yescomplemented
42#64954C2×C312yescomplemented
43#64943C2×C312yescomplemented
44C24×C3×C5C2311yescharacteristic
45C23×C3×D5C2312yes—
46C22×C3×C4×C5—41no—
47C22×C3×C4×C5—41no—
48C22×C3×C4×C5C2×C414yescomplemented
49C22×C3×C4×C5C2312yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C2×C3×C22⋊D5—24no—
54C2×C3×C22⋊D5—24no—
55C2×C3×C22⋊D5C2×C418yescomplemented
56C22×C3×Dic5—22no—
57C22×C3×Dic5—41no—
58C22×C3×Dic5—41no—
59C22×C3×Dic5—41no—
60C22×C3×Dic5—41no—
61C22×C3×Dic5D414yescomplemented
62C22×C3×Dic5C2312yes—
63C22×C3×Dic5C2312yes—
64C22×C3×Dic5C2312yes—
65C2×C3×D20—41no—
66C2×C3×D20—41no—
67C2×C3×D20C2×C414yescomplemented
68C2×C3×C4×D5—24no—
69C2×C3×C4×D5—42no—
70C2×C3×C4×D5—42no—
71C3×C5×C4⋊C4—41no—
72C3×C5×C4⋊C4—41no—
73C3×C5×C22⋊C4—24no—
74C3×#685—24no—
75C3×#685—42no—
76C3×#685—42no—
77C3×#685C2×C418yescomplemented
78C3×C4⋊Dic5C2×C414yescomplemented
79C3×#683—24no—
80C3×C4×Dic5—24no—
81C2×C3×#121—51no—
82C2×C3×C4×D4—52no—
83C3×#388—52no—
84C3×#387—52no—
85C3×#386—52no—
86C3×#386—52no—
87C3×#385—52no—
88C3×#380D511yescharacteristic, complemented
89C2×C3×#95—51no—
90C23×C4×C5C22×C311yescharacteristic
91C22×C22⋊D5C22×C312yes—
92C23×Dic5C22×C311yescharacteristic
93C23×Dic5C22×C311yescharacteristic
94C22×D20C22×C311yescharacteristic
95C22×C4×D5C22×C312yes—
96C2×C5×C4⋊C4C22×C311yescharacteristic
97C2×C5×C4⋊C4C22×C311yescharacteristic
98C2×C5×C22⋊C4C22×C312yes—
99#4076C3×C418yescomplemented
100C2×#685C22×C312yes—
101C2×#685C22×C312yes—
102C2×#685C22×C312yes—
103C2×#685C22×C312yes—
104C2×C4⋊Dic5C22×C311yescharacteristic
105C2×#683C22×C312yes—
106C2×C4×Dic5C22×C312yes—
107#4027—28no—
108C5×#95C22×C312yes—
109#3963C22×C312yes—
110#3963C22×C312yes—
111#3963C22×C312yes—
112#3963C22×C312yes—
113#2325—51noSylow
114C23×C3×C5—21no—
115C23×C3×C5—21no—
116C23×C3×C5—21no—
117C23×C3×C5—21no—
118C23×C3×C5C22×C412yes—
119C23×C3×C5C2×D412yes—
120C23×C3×C5#4112yes—
121C23×C3×C5C2411yescharacteristic
122C22×C3×D5—22no—
123C22×C3×D5—22no—
124C22×C3×D5—22no—
125C22×C3×D5—22no—
126C22×C3×D5—22no—
127C22×C3×D5—22no—
128C22×C3×D5C22×C414yes—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—44no—
140C2×C3×C4×C5C22×C412yes—
141C2×C3×C4×C5C22×C412yes—
142C3×C22⋊D5—216no—
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×Dic5—44no—
151C2×C3×Dic5—44no—
152C2×C3×Dic5C22×C414yes—
153C2×C3×Dic5C2×D414yes—
154C3×D20—44no—
155C3×C4×D5—48no—
156C22×C3×D4—52no—
157C22×C3×D4—51no—
158C23×C3×C4—51no—
159C23×C3×C4—51no—
160C23×C3×C4—52no—
161C23×C3×C4C2×D511yescharacteristic
162C3×#121—58no—
163C3×C4×D4—108no—
164C2×C3×C4⋊C4—51no—
165C2×C3×C4⋊C4—52no—
166C2×C3×C4⋊C4C2×D511yescharacteristic
167C2×C3×C4⋊C4C2×D511yescharacteristic
168C2×C3×C22⋊C4—52no—
169C2×C3×C22⋊C4—52no—
170C2×C3×C22⋊C4—52no—
171C2×C3×C22⋊C4—52no—
172C2×C3×C22⋊C4C2×D512yes—
173C2×C3×C42—52no—
174C3×#95—52no—
175C3×#95—52no—
176C3×#95—52no—
177C3×#95—52no—
178C3×#95C2×D512yes—
179C24×C5C23×C311yescharacteristic
180C23×D5C23×C312yes—
181C22×C4×C5—41no—
182C22×C4×C5—41no—
183C22×C4×C5C2×C3×C414yescomplemented
184C22×C4×C5C23×C312yes—
185C22×C4×C5C23×C312yes—
186C22×C4×C5C23×C311yescharacteristic
187C22×C4×C5C23×C311yescharacteristic
188C2×C22⋊D5—24no—
189C2×C22⋊D5—24no—
190C2×C22⋊D5C2×C3×C418yescomplemented
191C22×Dic5—22no—
192C22×Dic5—41no—
193C22×Dic5—41no—
194C22×Dic5—41no—
195C22×Dic5—41no—
196C22×Dic5C3×D414yescomplemented
197C22×Dic5C23×C312yes—
198C22×Dic5C23×C312yes—
199C22×Dic5C23×C312yes—
200C2×D20—41no—
201C2×D20—41no—
202C2×D20C2×C3×C414yescomplemented
203C2×C4×D5—24no—
204C2×C4×D5—42no—
205C2×C4×D5—42no—
206C5×C4⋊C4—41no—
207C5×C4⋊C4—41no—
208C5×C22⋊C4—24no—
209#685—24no—
210#685—42no—
211#685—42no—
212#685C2×C3×C418yescomplemented
213C4⋊Dic5C2×C3×C414yescomplemented
214#683—24no—
215C4×Dic5—24no—
216C2×#121—51no—
217C2×C4×D4—52no—
218#388—52no—
219#387—52no—
220#386—52no—
221#386—52no—
222#385—52no—
223#380C3×D511yescharacteristic, complemented
224C2×#95—51no—
225C22×C3×C5—22no—
226C22×C3×C5—22no—
227C22×C3×C5—22no—
228C22×C3×C5—22no—
229C22×C3×C5—22no—
230C22×C3×C5—22no—
231C22×C3×C5C4×D414yes—
232C22×C3×C5C23×C411yescharacteristic
233C22×C3×C5C22×D411yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5H(5,𝔽2)11yescharacteristic
236C22×C3×C5H(5,𝔽2)11yescharacteristic
237C22×C3×C5H(5,𝔽2)11yescharacteristic
238C22×C3×C52−1+411yescharacteristic
239C2×C3×D5—24no—
240C2×C3×D5—24no—
241C2×C3×D5—24no—
242C2×C3×D5—42no—
243C2×C3×D5—42no—
244C2×C3×D5—42no—
245C2×C3×D5—42no—
246C3×C4×C5—22no—
247C3×C4×C5—22no—
248C3×C4×C5—42no—
249C3×C4×C5—42no—
250C3×Dic5—24no—
251C3×Dic5—24no—
252C3×Dic5—42no—
253C24×C3—52no—
254C24×C3C22×D511yescharacteristic
255C2×C3×D4—58no—
256C2×C3×D4—54no—
257C2×C3×D4—104no—
258C2×C3×D4—104no—
259C2×C3×D4—201no—
260C2×C3×D4—201no—
261C22×C3×C4—41no—
262C22×C3×C4—41no—
263C22×C3×C4—52no—
264C22×C3×C4—52no—
265C22×C3×C4—54no—
266C22×C3×C4—52no—
267C22×C3×C4—104no—
268C22×C3×C4—102no—
269C22×C3×C4—202no—
270C22×C3×C4—202no—
271C22×C3×C4—201no—
272C22×C3×C4—201no—
273C22×C3×C4—201no—
274C22×C3×C4—201no—
275C22×C3×C4C4×D514yescomplemented
276C22×C3×C4C22×D512yes—
277C22×C3×C4C22×D512yes—
278C22×C3×C4C22×D511yescharacteristic
279C22×C3×C4C22×D511yescharacteristic
280C3×C4⋊C4—41no—
281C3×C4⋊C4—41no—
282C3×C4⋊C4—54no—
283C3×C4⋊C4—104no—
284C3×C22⋊C4—24no—
285C3×C22⋊C4—58no—
286C3×C22⋊C4—104no—
287C3×C22⋊C4—202no—
288C3×C22⋊C4—202no—
289C3×C42—104no—
290C23×C5—21no—
291C23×C5—21no—
292C23×C5—21no—
293C23×C5—21no—
294C23×C5C22×C3×C412yes—
295C23×C5C2×C3×D412yes—
296C23×C5C3×#4112yes—
297C23×C5C24×C311yescharacteristic
298C22×D5—22no—
299C22×D5—22no—
300C22×D5—22no—
301C22×D5—22no—
302C22×D5—22no—
303C22×D5—22no—
304C22×D5C22×C3×C414yes—
305C2×C4×C5—21no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—22no—
312C2×C4×C5—22no—
313C2×C4×C5—22no—
314C2×C4×C5—22no—
315C2×C4×C5—44no—
316C2×C4×C5C22×C3×C412yes—
317C2×C4×C5C22×C3×C412yes—
318C22⋊D5—216no—
319C2×Dic5—22no—
320C2×Dic5—22no—
321C2×Dic5—22no—
322C2×Dic5—22no—
323C2×Dic5—22no—
324C2×Dic5—22no—
325C2×Dic5—22no—
326C2×Dic5—44no—
327C2×Dic5—44no—
328C2×Dic5C22×C3×C414yes—
329C2×Dic5C2×C3×D414yes—
330D20—44no—
331C4×D5—48no—
332C22×D4—52no—
333C22×D4—51no—
334C23×C4—51no—
335C23×C4—51no—
336C23×C4—52no—
337C23×C4C2×C3×D511yescharacteristic
338#121—58no—
339C4×D4—108no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4—52no—
342C2×C4⋊C4C2×C3×D511yescharacteristic
343C2×C4⋊C4C2×C3×D511yescharacteristic
344C2×C22⋊C4—52no—
345C2×C22⋊C4—52no—
346C2×C22⋊C4—52no—
347C2×C22⋊C4—52no—
348C2×C22⋊C4C2×C3×D512yes—
349C2×C42—52no—
350#95—52no—
351#95—52no—
352#95—52no—
353#95—52no—
354#95C2×C3×D512yes—
355C2×C3×C5—24no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#51811yescharacteristic
358C2×C3×C5#52011yescharacteristic
359C2×C3×C5#53411yescharacteristic
360C2×C3×C5#53711yescharacteristic
361C2×C3×C5#53811yescharacteristic
362C2×C3×C5#53911yescharacteristic
363C3×D5—44no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—54no—
369C23×C3—102no—
370C23×C3—102no—
371C23×C3—102no—
372C23×C3—102no—
373C23×C3—102no—
374C23×C3—102no—
375C23×C3C2×C4×D512yes—
376C23×C3D4×D512yes—
377C23×C3#71312yes—
378C23×C3C23×D511yescharacteristic
379C3×D4—1016no—
380C3×D4—204no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—44no—
392C2×C3×C4—54no—
393C2×C3×C4—54no—
394C2×C3×C4—102no—
395C2×C3×C4—102no—
396C2×C3×C4—102no—
397C2×C3×C4—102no—
398C2×C3×C4—102no—
399C2×C3×C4—102no—
400C2×C3×C4—102no—
401C2×C3×C4—208no—
402C2×C3×C4—204no—
403C2×C3×C4—204no—
404C2×C3×C4C2×C4×D512yes—
405C2×C3×C4C2×C4×D512yes—
406C22×C5—22no—
407C22×C5—22no—
408C22×C5—22no—
409C22×C5—22no—
410C22×C5—22no—
411C22×C5—22no—
412C22×C5C3×C4×D414yes—
413C22×C5C23×C3×C411yescharacteristic
414C22×C5C22×C3×D411yescharacteristic
415C22×C5C2×C3×#4111yescharacteristic
416C22×C5C3×H(5,𝔽2)11yescharacteristic
417C22×C5C3×H(5,𝔽2)11yescharacteristic
418C22×C5C3×H(5,𝔽2)11yescharacteristic
419C22×C5C3×2−1+411yescharacteristic
420C2×D5—24no—
421C2×D5—24no—
422C2×D5—24no—
423C2×D5—42no—
424C2×D5—42no—
425C2×D5—42no—
426C2×D5—42no—
427C4×C5—22no—
428C4×C5—22no—
429C4×C5—42no—
430C4×C5—42no—
431Dic5—24no—
432Dic5—24no—
433Dic5—42no—
434C24—52no—
435C24C22×C3×D511yescharacteristic
436C2×D4—58no—
437C2×D4—54no—
438C2×D4—104no—
439C2×D4—104no—
440C2×D4—201no—
441C2×D4—201no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—52no—
445C22×C4—52no—
446C22×C4—54no—
447C22×C4—52no—
448C22×C4—104no—
449C22×C4—102no—
450C22×C4—202no—
451C22×C4—202no—
452C22×C4—201no—
453C22×C4—201no—
454C22×C4—201no—
455C22×C4—201no—
456C22×C4C3×C4×D514yescomplemented
457C22×C4C22×C3×D512yes—
458C22×C4C22×C3×D512yes—
459C22×C4C22×C3×D511yescharacteristic
460C22×C4C22×C3×D511yescharacteristic
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—54no—
464C4⋊C4—104no—
465C22⋊C4—24no—
466C22⋊C4—58no—
467C22⋊C4—104no—
468C22⋊C4—202no—
469C22⋊C4—202no—
470C42—104no—
471C3×C5#232511yescharacteristic, complemented, Hall
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—104no—
479C22×C3—104no—
480C22×C3—104no—
481C22×C3—202no—
482C22×C3—202no—
483C22×C3—202no—
484C22×C3—202no—
485C22×C3#403914yes—
486C22×C3C22×C4×D511yescharacteristic
487C22×C3C2×D4×D511yescharacteristic
488C22×C3#414411yescharacteristic
489C22×C3#414411yescharacteristic
490C22×C3C2×#71311yescharacteristic
491C22×C3#414911yescharacteristic
492C22×C3#415011yescharacteristic
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—104no—
498C3×C4—104no—
499C3×C4—202no—
500C2×C5—24no—
501C2×C5C2×C3×C4×D411yescharacteristic
502C2×C5C3×#51811yescharacteristic
503C2×C5C3×#52011yescharacteristic
504C2×C5C3×#53411yescharacteristic
505C2×C5C3×#53711yescharacteristic
506C2×C5C3×#53811yescharacteristic
507C2×C5C3×#53911yescharacteristic
508D5—44no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—54no—
514C23—102no—
515C23—102no—
516C23—102no—
517C23—102no—
518C23—102no—
519C23—102no—
520C23C2×C3×C4×D512yes—
521C23C3×D4×D512yes—
522C23C3×#71312yes—
523C23C23×C3×D511yescharacteristic
524D4—1016no—
525D4—204no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—44no—
537C2×C4—54no—
538C2×C4—54no—
539C2×C4—102no—
540C2×C4—102no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—208no—
547C2×C4—204no—
548C2×C4—204no—
549C2×C4C2×C3×C4×D512yes—
550C2×C4C2×C3×C4×D512yes—
551C2×C3—24no—
552C2×C3—204no—
553C2×C3#6582011yescharacteristic
554C2×C3C2×#403911yescharacteristic
555C2×C3#6585711yescharacteristic
556C2×C3#6593111yescharacteristic
557C2×C3#6593911yescharacteristic
558C2×C3#6597811yescharacteristic
559C2×C3#6597911yescharacteristic
560C5C3×#232511yescharacteristic, complemented, Sylow
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—104no—
568C22—104no—
569C22—104no—
570C22—202no—
571C22—202no—
572C22—202no—
573C22—202no—
574C22C3×#403914yes—
575C22C22×C3×C4×D511yescharacteristic
576C22C2×C3×D4×D511yescharacteristic
577C22C3×#414411yescharacteristic
578C22C3×#414411yescharacteristic
579C22C2×C3×#71311yescharacteristic
580C22C3×#414911yescharacteristic
581C22C3×#415011yescharacteristic
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—104no—
587C4—104no—
588C4—202no—
589C3#1060560411yescharacteristic, direct product factor, Sylow
590C2—24no—
591C2—204no—
592C2C3×#6582011yescharacteristic
593C2C2×C3×#403911yescharacteristic
594C2C3×#6585711yescharacteristic
595C2C3×#6593111yescharacteristic
596C2C3×#6593911yescharacteristic
597C2C3×#6597811yescharacteristic
598C2C3×#6597911yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C5×#380C2
socleC23×C3×C5C24
prime radicalC2×C3×#4076C2
cosocleC23×C5C24×C3
hypercenterC3×#380D5
hypocenterC5C3×#2325
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×#380

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4027C2semidirect
3. C3×C5×#380C2semidirect
4. C2×C3×#3963C2semidirect
5. C3×#65279C2semidirect
6. C3×#65278C2semidirect
7. C3×#65247C2semidirect
8. C3×#64954C2semidirect
9. C3×#64943C2semidirect
10. #10605604C3direct (N × H)
20. C3×#4076C4semidirect
36. C2×#4027C2×C3semidirect
37. C5×#380C2×C3semidirect
38. C2×#3963C2×C3semidirect
39. #65279C2×C3semidirect
40. #65278C2×C3semidirect
41. #65247C2×C3semidirect
42. #64954C2×C3semidirect
43. #64943C2×C3semidirect
48. C22×C3×C4×C5C2×C4semidirect
55. C2×C3×C22⋊D5C2×C4semidirect
61. C22×C3×Dic5D4semidirect
67. C2×C3×D20C2×C4semidirect
77. C3×#685C2×C4semidirect
78. C3×C4⋊Dic5C2×C4semidirect
88. C3×#380D5semidirect
99. #4076C3×C4semidirect
183. C22×C4×C5C2×C3×C4semidirect
190. C2×C22⋊D5C2×C3×C4semidirect
196. C22×Dic5C3×D4semidirect
202. C2×D20C2×C3×C4semidirect
212. #685C2×C3×C4semidirect
213. C4⋊Dic5C2×C3×C4semidirect
223. #380C3×D5semidirect
275. C22×C3×C4C4×D5semidirect
456. C22×C4C3×C4×D5semidirect
471. C3×C5#2325semidirect
560. C5C3×#2325semidirect
589. C3#10605604direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40762
C2×C3×#40272
C3×C5×#3802
C2×C3×#39632
C3×#652792
C3×#652782
C3×#652472
C3×#649542
C3×#649432
#106056043
C3×#23255