Finite Group Databasebeta

← C3×#10605599 #14322708

Subgroups — C3×#10605599 #14322708

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4072C211yesmaximal, characteristic
2C3×C5×#380C211yesmaximal, characteristic
3C3×#65501C211yesmaximal, characteristic
4C3×#65266C211yesmaximal, characteristic, complemented
5C3×#65265C211yesmaximal, characteristic, complemented
6C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
7C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
8C3×#65234C212yesmaximal
9C3×#65233C212yesmaximal
10C3×#64943C212yesmaximal, complemented
11C3×#64942C212yesmaximal, complemented
12#10605599C311yesmaximal, characteristic, direct product factor, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×C3×C22⋊Dic5C2212yes—
18C2×C3×C22⋊Dic5C2211yescharacteristic
19C2×C3×C22⋊Dic5C2211yescharacteristic
20C3×#4072C418yescomplemented
21C2×C3×C4⋊Dic5C2211yescharacteristic
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×#683C2212yes—
26C2×C3×#683C2212yes—
27C2×C3×C4×Dic5C2212yes—
28C2×C3×C4×Dic5C2212yes—
29C2×C3×C4×Dic5C2211yescharacteristic
30C2×C3×C4×Dic5C2211yescharacteristic
31C3×C5×#95C2212yes—
32C3×#3963C2212yes—
33C3×#3963C2212yes—
34C3×#3963C2212yes—
35C3×#3963C2212yes—
36C3×#3963C2212yes—
37C3×#2314—51nomaximal, Hall
38C2×#4072C2×C311yescharacteristic
39C5×#380C2×C311yescharacteristic
40#65501C2×C311yescharacteristic
41#65266C2×C311yescharacteristic, complemented
42#65265C2×C311yescharacteristic, complemented
43C4⋊C4×Dic5C2×C311yescharacteristic, complemented
44C4⋊C4×Dic5C2×C311yescharacteristic, complemented
45#65234C2×C312yes—
46#65233C2×C312yes—
47#64943C2×C312yescomplemented
48#64942C2×C312yescomplemented
49C24×C3×C5C2311yescharacteristic
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—22no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×Dic5C2312yes—
58C22×C3×Dic5C2312yes—
59C22×C3×Dic5C2311yescharacteristic
60C22×C3×Dic5C2311yescharacteristic
61C22×C3×Dic5C2311yescharacteristic
62C22×C3×Dic5C2311yescharacteristic
63C3×C5×C4⋊C4—22no—
64C3×C5×C4⋊C4—22no—
65C3×C5×C22⋊C4—42no—
66C3×C22⋊Dic5—42no—
67C3×C22⋊Dic5C2×C414yes—
68C3×C22⋊Dic5C2×C414yes—
69C3×C4⋊Dic5—22no—
70C3×C4⋊Dic5—22no—
71C3×C4⋊Dic5C2×C414yes—
72C3×C4⋊Dic5C2×C414yes—
73C3×#683—42no—
74C3×#683—42no—
75C3×C4×Dic5—42no—
76C3×C4×Dic5—42no—
77C3×C4×Dic5C2×C414yes—
78C3×C4×Dic5C2×C414yes—
79C2×C3×#117—51no—
80C3×#388—52no—
81C3×#388—52no—
82C3×#387—52no—
83C3×#387—52no—
84C3×#384—51no—
85C3×#382—51no—
86C3×#380—51no—
87C3×#380D511yescharacteristic
88C3×C4×C4⋊C4—51no—
89C3×C4×C4⋊C4—51no—
90C23×C4×C5C22×C311yescharacteristic
91C2×C5×C4⋊C4C22×C311yescharacteristic
92C2×C5×C4⋊C4C22×C311yescharacteristic
93C2×C5×C22⋊C4C22×C312yes—
94C2×C22⋊Dic5C22×C312yes—
95C2×C22⋊Dic5C22×C311yescharacteristic
96C2×C22⋊Dic5C22×C311yescharacteristic
97#4072C3×C418yescomplemented
98C2×C4⋊Dic5C22×C311yescharacteristic
99C2×C4⋊Dic5C22×C311yescharacteristic
100C2×C4⋊Dic5C22×C311yescharacteristic
101C2×C4⋊Dic5C22×C311yescharacteristic
102C2×#683C22×C312yes—
103C2×#683C22×C312yes—
104C2×C4×Dic5C22×C312yes—
105C2×C4×Dic5C22×C312yes—
106C2×C4×Dic5C22×C311yescharacteristic
107C2×C4×Dic5C22×C311yescharacteristic
108C5×#95C22×C312yes—
109#3963C22×C312yes—
110#3963C22×C312yes—
111#3963C22×C312yes—
112#3963C22×C312yes—
113#3963C22×C312yes—
114#2314—51noSylow
115C23×C3×C5—21no—
116C23×C3×C5—21no—
117C23×C3×C5—21no—
118C23×C3×C5—21no—
119C23×C3×C5—21no—
120C23×C3×C5—21no—
121C23×C3×C5C22×C412yes—
122C23×C3×C5C2411yescharacteristic
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—28no—
126C2×C3×C4×C5—22no—
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×C5C22×C412yes—
133C2×C3×C4×C5C22×C412yes—
134C2×C3×C4×C5#4112yes—
135C2×C3×C4×C5#4112yes—
136C2×C3×Dic5—21no—
137C2×C3×Dic5—21no—
138C2×C3×Dic5—21no—
139C2×C3×Dic5—21no—
140C2×C3×Dic5—21no—
141C2×C3×Dic5—21no—
142C2×C3×Dic5—21no—
143C2×C3×Dic5—21no—
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—
153C2×C3×Dic5C22×C412yes—
154C23×C3×C4C2×D511yescharacteristic
155C3×#117—58no—
156C2×C3×C4⋊C4—51no—
157C2×C3×C4⋊C4—51no—
158C2×C3×C4⋊C4—51no—
159C2×C3×C4⋊C4—51no—
160C2×C3×C4⋊C4—52no—
161C2×C3×C4⋊C4—52no—
162C2×C3×C4⋊C4C2×D511yescharacteristic
163C2×C3×C4⋊C4C2×D511yescharacteristic
164C2×C3×C22⋊C4—51no—
165C2×C3×C22⋊C4—51no—
166C2×C3×C22⋊C4—52no—
167C2×C3×C22⋊C4C2×D512yes—
168C2×C3×C42—51no—
169C2×C3×C42—51no—
170C2×C3×C42—52no—
171C2×C3×C42—52no—
172C3×#95—52no—
173C3×#95—52no—
174C3×#95—52no—
175C3×#95—52no—
176C3×#95—52no—
177C3×#95C2×D512yes—
178C24×C5C23×C311yescharacteristic
179C22×C4×C5—22no—
180C22×C4×C5—22no—
181C22×C4×C5C2×C3×C414yes—
182C22×C4×C5C23×C312yes—
183C22×C4×C5C23×C312yes—
184C22×C4×C5C23×C311yescharacteristic
185C22×C4×C5C23×C311yescharacteristic
186C22×Dic5C23×C312yes—
187C22×Dic5C23×C312yes—
188C22×Dic5C23×C311yescharacteristic
189C22×Dic5C23×C311yescharacteristic
190C22×Dic5C23×C311yescharacteristic
191C22×Dic5C23×C311yescharacteristic
192C5×C4⋊C4—22no—
193C5×C4⋊C4—22no—
194C5×C22⋊C4—42no—
195C22⋊Dic5—42no—
196C22⋊Dic5C2×C3×C414yes—
197C22⋊Dic5C2×C3×C414yes—
198C4⋊Dic5—22no—
199C4⋊Dic5—22no—
200C4⋊Dic5C2×C3×C414yes—
201C4⋊Dic5C2×C3×C414yes—
202#683—42no—
203#683—42no—
204C4×Dic5—42no—
205C4×Dic5—42no—
206C4×Dic5C2×C3×C414yes—
207C4×Dic5C2×C3×C414yes—
208C2×#117—51no—
209#388—52no—
210#388—52no—
211#387—52no—
212#387—52no—
213#384—51no—
214#382—51no—
215#380—51no—
216#380C3×D511yescharacteristic
217C4×C4⋊C4—51no—
218C4×C4⋊C4—51no—
219C22×C3×C5—22no—
220C22×C3×C5—22no—
221C22×C3×C5—22no—
222C22×C3×C5—41no—
223C22×C3×C5—41no—
224C22×C3×C5—41no—
225C22×C3×C5—41no—
226C22×C3×C5C23×C411yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C22×C3×C5H(5,𝔽2)11yescharacteristic
230C22×C3×C5H(5,𝔽2)11yescharacteristic
231C22×C3×C52−1+411yescharacteristic
232C22×C3×C52−1+411yescharacteristic
233C3×C4×C5—22no—
234C3×C4×C5—42no—
235C3×C4×C5—42no—
236C3×C4×C5#11714yes—
237C3×Dic5—22no—
238C3×Dic5—22no—
239C3×Dic5—22no—
240C3×Dic5—22no—
241C3×Dic5—42no—
242C3×Dic5—42no—
243C24×C3C22×D511yescharacteristic
244C22×C3×C4—22no—
245C22×C3×C4—22no—
246C22×C3×C4—51no—
247C22×C3×C4—51no—
248C22×C3×C4—51no—
249C22×C3×C4—51no—
250C22×C3×C4—52no—
251C22×C3×C4—52no—
252C22×C3×C4C4×D514yes—
253C22×C3×C4C22×D512yes—
254C22×C3×C4C22×D512yes—
255C22×C3×C4C22×D511yescharacteristic
256C22×C3×C4C22×D511yescharacteristic
257C3×C4⋊C4—22no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—54no—
260C3×C4⋊C4—54no—
261C3×C4⋊C4—102no—
262C3×C4⋊C4—102no—
263C3×C4⋊C4—202no—
264C3×C4⋊C4—202no—
265C3×C22⋊C4—42no—
266C3×C22⋊C4—54no—
267C3×C22⋊C4—54no—
268C3×C22⋊C4—202no—
269C3×C42—54no—
270C3×C42—54no—
271C3×C42—202no—
272C3×C42—202no—
273C23×C5—21no—
274C23×C5—21no—
275C23×C5—21no—
276C23×C5—21no—
277C23×C5—21no—
278C23×C5—21no—
279C23×C5C22×C3×C412yes—
280C23×C5C24×C311yescharacteristic
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—28no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5C22×C3×C412yes—
291C2×C4×C5C22×C3×C412yes—
292C2×C4×C5C3×#4112yes—
293C2×C4×C5C3×#4112yes—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—21no—
299C2×Dic5—21no—
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×Dic5C22×C3×C412yes—
309C2×Dic5C22×C3×C412yes—
310C2×Dic5C22×C3×C412yes—
311C2×Dic5C22×C3×C412yes—
312C23×C4C2×C3×D511yescharacteristic
313#117—58no—
314C2×C4⋊C4—51no—
315C2×C4⋊C4—51no—
316C2×C4⋊C4—51no—
317C2×C4⋊C4—51no—
318C2×C4⋊C4—52no—
319C2×C4⋊C4—52no—
320C2×C4⋊C4C2×C3×D511yescharacteristic
321C2×C4⋊C4C2×C3×D511yescharacteristic
322C2×C22⋊C4—51no—
323C2×C22⋊C4—51no—
324C2×C22⋊C4—52no—
325C2×C22⋊C4C2×C3×D512yes—
326C2×C42—51no—
327C2×C42—51no—
328C2×C42—52no—
329C2×C42—52no—
330#95—52no—
331#95—52no—
332#95—52no—
333#95—52no—
334#95—52no—
335#95C2×C3×D512yes—
336C2×C3×C5—42no—
337C2×C3×C5C2×#11711yescharacteristic
338C2×C3×C5#52011yescharacteristic
339C2×C3×C5#52011yescharacteristic
340C2×C3×C5#54011yescharacteristic
341C2×C3×C5#54111yescharacteristic
342C2×C3×C5#54211yescharacteristic
343C2×C3×C5#54211yescharacteristic
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3C2×C4×D512yes—
351C23×C3C23×D511yescharacteristic
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—28no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—52no—
362C2×C3×C4—52no—
363C2×C3×C4—52no—
364C2×C3×C4—52no—
365C2×C3×C4—101no—
366C2×C3×C4—101no—
367C2×C3×C4—101no—
368C2×C3×C4—101no—
369C2×C3×C4—101no—
370C2×C3×C4—101no—
371C2×C3×C4—101no—
372C2×C3×C4—101no—
373C2×C3×C4—102no—
374C2×C3×C4—102no—
375C2×C3×C4—102no—
376C2×C3×C4—102no—
377C2×C3×C4—102no—
378C2×C3×C4—102no—
379C2×C3×C4C2×C4×D512yes—
380C2×C3×C4C2×C4×D512yes—
381C2×C3×C4#71112yes—
382C2×C3×C4#71112yes—
383C22×C5—22no—
384C22×C5—22no—
385C22×C5—22no—
386C22×C5—41no—
387C22×C5—41no—
388C22×C5—41no—
389C22×C5—41no—
390C22×C5C23×C3×C411yescharacteristic
391C22×C5C2×C3×#4111yescharacteristic
392C22×C5C2×C3×#4111yescharacteristic
393C22×C5C3×H(5,𝔽2)11yescharacteristic
394C22×C5C3×H(5,𝔽2)11yescharacteristic
395C22×C5C3×2−1+411yescharacteristic
396C22×C5C3×2−1+411yescharacteristic
397C4×C5—22no—
398C4×C5—42no—
399C4×C5—42no—
400C4×C5C3×#11714yes—
401Dic5—22no—
402Dic5—22no—
403Dic5—22no—
404Dic5—22no—
405Dic5—42no—
406Dic5—42no—
407C24C22×C3×D511yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—51no—
411C22×C4—51no—
412C22×C4—51no—
413C22×C4—51no—
414C22×C4—52no—
415C22×C4—52no—
416C22×C4C3×C4×D514yes—
417C22×C4C22×C3×D512yes—
418C22×C4C22×C3×D512yes—
419C22×C4C22×C3×D511yescharacteristic
420C22×C4C22×C3×D511yescharacteristic
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—54no—
424C4⋊C4—54no—
425C4⋊C4—102no—
426C4⋊C4—102no—
427C4⋊C4—202no—
428C4⋊C4—202no—
429C22⋊C4—42no—
430C22⋊C4—54no—
431C22⋊C4—54no—
432C22⋊C4—202no—
433C42—54no—
434C42—54no—
435C42—202no—
436C42—202no—
437C3×C5#231411yescharacteristic, complemented, Hall
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3C22×C4×D511yescharacteristic
446C22×C3C2×#71111yescharacteristic
447C22×C3C2×#71111yescharacteristic
448C22×C3#414411yescharacteristic
449C22×C3#414711yescharacteristic
450C22×C3#414911yescharacteristic
451C22×C3#415011yescharacteristic
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—102no—
456C3×C4—102no—
457C3×C4—102no—
458C3×C4—102no—
459C3×C4—202no—
460C3×C4—202no—
461C3×C4#402314yes—
462C2×C5—42no—
463C2×C5C2×C3×#11711yescharacteristic
464C2×C5C3×#52011yescharacteristic
465C2×C5C3×#52011yescharacteristic
466C2×C5C3×#54011yescharacteristic
467C2×C5C3×#54111yescharacteristic
468C2×C5C3×#54211yescharacteristic
469C2×C5C3×#54211yescharacteristic
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23C2×C3×C4×D512yes—
477C23C23×C3×D511yescharacteristic
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—28no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—52no—
488C2×C4—52no—
489C2×C4—52no—
490C2×C4—52no—
491C2×C4—101no—
492C2×C4—101no—
493C2×C4—101no—
494C2×C4—101no—
495C2×C4—101no—
496C2×C4—101no—
497C2×C4—101no—
498C2×C4—101no—
499C2×C4—102no—
500C2×C4—102no—
501C2×C4—102no—
502C2×C4—102no—
503C2×C4—102no—
504C2×C4—102no—
505C2×C4C2×C3×C4×D512yes—
506C2×C4C2×C3×C4×D512yes—
507C2×C4C3×#71112yes—
508C2×C4C3×#71112yes—
509C2×C3—42no—
510C2×C3C2×#402311yescharacteristic
511C2×C3#6583811yescharacteristic
512C2×C3#6585711yescharacteristic
513C2×C3#6593711yescharacteristic
514C2×C3#6595311yescharacteristic
515C2×C3#6595911yescharacteristic
516C2×C3#6597711yescharacteristic
517C5C3×#231411yescharacteristic, complemented, Sylow
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22C22×C3×C4×D511yescharacteristic
526C22C2×C3×#71111yescharacteristic
527C22C2×C3×#71111yescharacteristic
528C22C3×#414411yescharacteristic
529C22C3×#414711yescharacteristic
530C22C3×#414911yescharacteristic
531C22C3×#415011yescharacteristic
532C4—22no—
533C4—42no—
534C4—42no—
535C4—102no—
536C4—102no—
537C4—102no—
538C4—102no—
539C4—202no—
540C4—202no—
541C4C3×#402314yes—
542C3#1060559911yescharacteristic, direct product factor, Sylow
543C2—42no—
544C2C2×C3×#402311yescharacteristic
545C2C3×#6583811yescharacteristic
546C2C3×#6585711yescharacteristic
547C2C3×#6593711yescharacteristic
548C2C3×#6595311yescharacteristic
549C2C3×#6595911yescharacteristic
550C2C3×#6597711yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C5×#380C2
socleC23×C3×C5C24
prime radicalC24×C3×C5C23
cosocleC23×C5C24×C3
hypercenterC3×#380D5
hypocenterC5C3×#2314
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×#380

Complements

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

NH ≅ G/Ntype
4. C3×#65266C2semidirect
5. C3×#65265C2semidirect
6. C3×C4⋊C4×Dic5C2semidirect
7. C3×C4⋊C4×Dic5C2semidirect
10. C3×#64943C2semidirect
11. C3×#64942C2semidirect
12. #10605599C3direct (N × H)
20. C3×#4072C4semidirect
41. #65266C2×C3semidirect
42. #65265C2×C3semidirect
43. C4⋊C4×Dic5C2×C3semidirect
44. C4⋊C4×Dic5C2×C3semidirect
47. #64943C2×C3semidirect
48. #64942C2×C3semidirect
97. #4072C3×C4semidirect
437. C3×C5#2314semidirect
517. C5C3×#2314semidirect
542. C3#10605599direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40722
C3×C5×#3802
C3×#655012
C3×#652662
C3×#652652
C3×C4⋊C4×Dic52
C3×#652342
C3×#652332
C3×#649432
C3×#649422
#106055993
C3×#23145