Finite Group Databasebeta

← C3×#10605377 #14327988

Subgroups — C3×#10605377 #14327988

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4039C211yesmaximal, characteristic
2C3×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#65277C211yesmaximal, characteristic
4C3×#65266C211yesmaximal, characteristic, complemented
5C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
6C3×#65229C212yesmaximal, complemented
7C3×#65228C211yesmaximal, characteristic
8C3×C4×#685C212yesmaximal, complemented
9C3×C4×#683C211yesmaximal, characteristic, complemented
10C3×#64958C212yesmaximal, complemented
11C3×#64953C212yesmaximal, complemented
12#10605377C311yesmaximal, characteristic, direct product factor, Hall
13C22×C3×D20C2211yescharacteristic
14C22×C3×C4×D5C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C42×C5C2211yescharacteristic
18C2×C3×C42×C5C2212yes—
19C2×C3×#685C2212yes—
20C2×C3×#685C2212yes—
21C2×C3×#685C2212yes—
22C2×C3×#685C2212yes—
23C2×C3×#685C2212yes—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×#683C2211yescharacteristic
26C2×C3×#683C2211yescharacteristic
27C2×C3×C4×Dic5C2211yescharacteristic
28C2×C3×C4×Dic5C2212yes—
29C2×C3×C4×Dic5C2211yescharacteristic
30C3×#4039C418yescomplemented
31C3×C5×#95C2212yes—
32C3×#3963C2211yescharacteristic
33C3×#3963C2212yes—
34C3×#3963C2212yes—
35C3×#3963C2212yes—
36C3×#3963C2211yescharacteristic
37C3×#2357—51nomaximal, Hall
38C2×#4039C2×C311yescharacteristic
39C4×C5×C4⋊C4C2×C311yescharacteristic, complemented
40#65277C2×C311yescharacteristic
41#65266C2×C311yescharacteristic, complemented
42C4⋊C4×Dic5C2×C311yescharacteristic, complemented
43#65229C2×C312yescomplemented
44#65228C2×C311yescharacteristic
45C4×#685C2×C312yescomplemented
46C4×#683C2×C311yescharacteristic, complemented
47#64958C2×C312yescomplemented
48#64953C2×C312yescomplemented
49C23×C3×D5C2312yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×Dic5C2311yescharacteristic
56C22×C3×Dic5C2312yes—
57C22×C3×Dic5C2311yescharacteristic
58C22×C3×Dic5C2311yescharacteristic
59C22×C3×Dic5C2311yescharacteristic
60C2×C3×D20—22no—
61C2×C3×D20—22no—
62C2×C3×D20C2×C414yes—
63C2×C3×C4×D5—42no—
64C2×C3×C4×D5—42no—
65C2×C3×C4×D5C2×C418yes—
66C3×C5×C4⋊C4—22no—
67C3×C5×C4⋊C4C2×C414yes—
68C3×C42×C5—22no—
69C3×C42×C5—42no—
70C3×#685—24no—
71C3×#685—24no—
72C3×#685—42no—
73C3×#685—42no—
74C3×#685C2×C418yes—
75C3×C4⋊Dic5—22no—
76C3×#683—22no—
77C3×#683—22no—
78C3×C4×Dic5—22no—
79C3×C4×Dic5—42no—
80C3×C4×Dic5C2×C414yes—
81C2×C3×C4×D4—51no—
82C3×#390—51no—
83C3×#390—51no—
84C3×#388—52no—
85C3×#388—52no—
86C3×#382—51no—
87C3×#381—52no—
88C3×C4×C4⋊C4—51no—
89C3×C4×C4⋊C4—51no—
90C3×C4×C4⋊C4D511yescharacteristic, complemented
91C3×C4×C22⋊C4—52no—
92C22×D20C22×C311yescharacteristic
93C22×C4×D5C22×C312yes—
94C2×C5×C4⋊C4C22×C311yescharacteristic
95C2×C5×C4⋊C4C22×C311yescharacteristic
96C2×C42×C5C22×C311yescharacteristic
97C2×C42×C5C22×C312yes—
98C2×#685C22×C312yes—
99C2×#685C22×C312yes—
100C2×#685C22×C312yes—
101C2×#685C22×C312yes—
102C2×#685C22×C312yes—
103C2×C4⋊Dic5C22×C311yescharacteristic
104C2×#683C22×C311yescharacteristic
105C2×#683C22×C311yescharacteristic
106C2×C4×Dic5C22×C311yescharacteristic
107C2×C4×Dic5C22×C312yes—
108C2×C4×Dic5C22×C311yescharacteristic
109#4039C3×C418yescomplemented
110C5×#95C22×C312yes—
111#3963C22×C311yescharacteristic
112#3963C22×C312yes—
113#3963C22×C312yes—
114#3963C22×C312yes—
115#3963C22×C311yescharacteristic
116#2357—51noSylow
117C23×C3×C5C2411yescharacteristic
118C22×C3×D5—22no—
119C22×C3×D5—22no—
120C22×C3×D5—22no—
121C22×C3×D5—22no—
122C22×C3×D5—22no—
123C22×C3×D5—22no—
124C22×C3×D5C22×C414yes—
125C2×C3×C4×C5—21no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5C22×C414yes—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5#4112yes—
138C2×C3×C4×C5#4112yes—
139C2×C3×C4×C5#4112yes—
140C2×C3×Dic5—21no—
141C2×C3×Dic5—21no—
142C2×C3×Dic5—21no—
143C2×C3×Dic5—21no—
144C2×C3×Dic5—21no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5C22×C412yes—
153C2×C3×Dic5C22×C412yes—
154C2×C3×Dic5#4112yes—
155C3×D20—28no—
156C3×C4×D5—48no—
157C22×C3×D4—51no—
158C23×C3×C4—52no—
159C3×C4×D4—58no—
160C2×C3×C4⋊C4—51no—
161C2×C3×C4⋊C4—51no—
162C2×C3×C4⋊C4—51no—
163C2×C3×C4⋊C4C2×D511yescharacteristic
164C2×C3×C4⋊C4C2×D511yescharacteristic
165C2×C3×C22⋊C4—52no—
166C2×C3×C22⋊C4—52no—
167C2×C3×C22⋊C4—52no—
168C2×C3×C22⋊C4—52no—
169C2×C3×C22⋊C4—52no—
170C2×C3×C42—51no—
171C2×C3×C42—51no—
172C2×C3×C42—52no—
173C2×C3×C42C2×D511yescharacteristic
174C2×C3×C42C2×D512yes—
175C3×#95—51no—
176C3×#95—52no—
177C3×#95—51no—
178C3×#95—52no—
179C3×#95—52no—
180C3×#95C2×D512yes—
181C23×D5C23×C312yes—
182C22×C4×C5C23×C312yes—
183C22×C4×C5C23×C311yescharacteristic
184C22×C4×C5C23×C312yes—
185C22×C4×C5C23×C311yescharacteristic
186C22×C4×C5C23×C311yescharacteristic
187C22×Dic5C23×C311yescharacteristic
188C22×Dic5C23×C312yes—
189C22×Dic5C23×C311yescharacteristic
190C22×Dic5C23×C311yescharacteristic
191C22×Dic5C23×C311yescharacteristic
192C2×D20—22no—
193C2×D20—22no—
194C2×D20C2×C3×C414yes—
195C2×C4×D5—42no—
196C2×C4×D5—42no—
197C2×C4×D5C2×C3×C418yes—
198C5×C4⋊C4—22no—
199C5×C4⋊C4C2×C3×C414yes—
200C42×C5—22no—
201C42×C5—42no—
202#685—24no—
203#685—24no—
204#685—42no—
205#685—42no—
206#685C2×C3×C418yes—
207C4⋊Dic5—22no—
208#683—22no—
209#683—22no—
210C4×Dic5—22no—
211C4×Dic5—42no—
212C4×Dic5C2×C3×C414yes—
213C2×C4×D4—51no—
214#390—51no—
215#390—51no—
216#388—52no—
217#388—52no—
218#382—51no—
219#381—52no—
220C4×C4⋊C4—51no—
221C4×C4⋊C4—51no—
222C4×C4⋊C4C3×D511yescharacteristic, complemented
223C4×C22⋊C4—52no—
224C22×C3×C5C23×C411yescharacteristic
225C22×C3×C5C2×#4111yescharacteristic
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C22×C3×C5H(5,𝔽2)11yescharacteristic
230C22×C3×C5H(5,𝔽2)11yescharacteristic
231C2×C3×D5—24no—
232C2×C3×D5—24no—
233C2×C3×D5—24no—
234C2×C3×D5—42no—
235C2×C3×D5—42no—
236C2×C3×D5—42no—
237C2×C3×D5—42no—
238C3×C4×C5—24no—
239C3×C4×C5—22no—
240C3×C4×C5—41no—
241C3×C4×C5—42no—
242C3×C4×C5#11714yes—
243C3×Dic5—22no—
244C3×Dic5—22no—
245C3×Dic5—22no—
246C3×Dic5—41no—
247C3×Dic5—42no—
248C24×C3—52no—
249C2×C3×D4—54no—
250C2×C3×D4—102no—
251C2×C3×D4—102no—
252C22×C3×C4—51no—
253C22×C3×C4—51no—
254C22×C3×C4—51no—
255C22×C3×C4—58no—
256C22×C3×C4—51no—
257C22×C3×C4—52no—
258C22×C3×C4—202no—
259C22×C3×C4—202no—
260C22×C3×C4C22×D512yes—
261C22×C3×C4C22×D511yescharacteristic
262C22×C3×C4C22×D512yes—
263C22×C3×C4C22×D511yescharacteristic
264C22×C3×C4C22×D511yescharacteristic
265C3×C4⋊C4—22no—
266C3×C4⋊C4—102no—
267C3×C4⋊C4—102no—
268C3×C4⋊C4—102no—
269C3×C4⋊C4C4×D514yes—
270C3×C22⋊C4—58no—
271C3×C22⋊C4—104no—
272C3×C22⋊C4—104no—
273C3×C22⋊C4—202no—
274C3×C22⋊C4—202no—
275C3×C42—22no—
276C3×C42—42no—
277C3×C42—54no—
278C3×C42—102no—
279C3×C42—202no—
280C23×C5C24×C311yescharacteristic
281C22×D5—22no—
282C22×D5—22no—
283C22×D5—22no—
284C22×D5—22no—
285C22×D5—22no—
286C22×D5—22no—
287C22×D5C22×C3×C414yes—
288C2×C4×C5—21no—
289C2×C4×C5—22no—
290C2×C4×C5—21no—
291C2×C4×C5—22no—
292C2×C4×C5—21no—
293C2×C4×C5—22no—
294C2×C4×C5—21no—
295C2×C4×C5—22no—
296C2×C4×C5—21no—
297C2×C4×C5—22no—
298C2×C4×C5C22×C3×C414yes—
299C2×C4×C5C22×C3×C412yes—
300C2×C4×C5C3×#4112yes—
301C2×C4×C5C3×#4112yes—
302C2×C4×C5C3×#4112yes—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—21no—
308C2×Dic5—21no—
309C2×Dic5—21no—
310C2×Dic5—21no—
311C2×Dic5—22no—
312C2×Dic5—22no—
313C2×Dic5—22no—
314C2×Dic5—21no—
315C2×Dic5C22×C3×C412yes—
316C2×Dic5C22×C3×C412yes—
317C2×Dic5C3×#4112yes—
318D20—28no—
319C4×D5—48no—
320C22×D4—51no—
321C23×C4—52no—
322C4×D4—58no—
323C2×C4⋊C4—51no—
324C2×C4⋊C4—51no—
325C2×C4⋊C4—51no—
326C2×C4⋊C4C2×C3×D511yescharacteristic
327C2×C4⋊C4C2×C3×D511yescharacteristic
328C2×C22⋊C4—52no—
329C2×C22⋊C4—52no—
330C2×C22⋊C4—52no—
331C2×C22⋊C4—52no—
332C2×C22⋊C4—52no—
333C2×C42—51no—
334C2×C42—51no—
335C2×C42—52no—
336C2×C42C2×C3×D511yescharacteristic
337C2×C42C2×C3×D512yes—
338#95—51no—
339#95—52no—
340#95—51no—
341#95—52no—
342#95—52no—
343#95C2×C3×D512yes—
344C2×C3×C5C2×#11711yescharacteristic
345C2×C3×C5C4×#4111yescharacteristic
346C2×C3×C5#51811yescharacteristic
347C2×C3×C5#55311yescharacteristic
348C2×C3×C5#55311yescharacteristic
349C2×C3×C5#55911yescharacteristic
350C2×C3×C5#55911yescharacteristic
351C3×D5—44no—
352C23×C3—54no—
353C23×C3—102no—
354C23×C3—102no—
355C23×C3—102no—
356C23×C3—102no—
357C23×C3—102no—
358C23×C3—102no—
359C23×C3C23×D511yescharacteristic
360C3×D4—108no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—52no—
372C2×C3×C4—52no—
373C2×C3×C4—52no—
374C2×C3×C4—101no—
375C2×C3×C4—101no—
376C2×C3×C4—101no—
377C2×C3×C4—101no—
378C2×C3×C4—101no—
379C2×C3×C4—101no—
380C2×C3×C4—101no—
381C2×C3×C4—102no—
382C2×C3×C4—101no—
383C2×C3×C4—102no—
384C2×C3×C4—101no—
385C2×C3×C4—102no—
386C2×C3×C4—208no—
387C2×C3×C4C2×C4×D514yes—
388C2×C3×C4C2×C4×D512yes—
389C2×C3×C4#70912yes—
390C2×C3×C4#70912yes—
391C2×C3×C4#71312yes—
392C22×C5C23×C3×C411yescharacteristic
393C22×C5C2×C3×#4111yescharacteristic
394C22×C5C2×C3×#4111yescharacteristic
395C22×C5C2×C3×#4111yescharacteristic
396C22×C5C2×C3×#4111yescharacteristic
397C22×C5C3×H(5,𝔽2)11yescharacteristic
398C22×C5C3×H(5,𝔽2)11yescharacteristic
399C2×D5—24no—
400C2×D5—24no—
401C2×D5—24no—
402C2×D5—42no—
403C2×D5—42no—
404C2×D5—42no—
405C2×D5—42no—
406C4×C5—24no—
407C4×C5—22no—
408C4×C5—42no—
409C4×C5—41no—
410C4×C5C3×#11714yes—
411Dic5—22no—
412Dic5—22no—
413Dic5—22no—
414Dic5—42no—
415Dic5—41no—
416C24—52no—
417C2×D4—54no—
418C2×D4—102no—
419C2×D4—102no—
420C22×C4—51no—
421C22×C4—51no—
422C22×C4—51no—
423C22×C4—58no—
424C22×C4—51no—
425C22×C4—52no—
426C22×C4—202no—
427C22×C4—202no—
428C22×C4C22×C3×D512yes—
429C22×C4C22×C3×D511yescharacteristic
430C22×C4C22×C3×D512yes—
431C22×C4C22×C3×D511yescharacteristic
432C22×C4C22×C3×D511yescharacteristic
433C4⋊C4—22no—
434C4⋊C4—102no—
435C4⋊C4—102no—
436C4⋊C4—102no—
437C4⋊C4C3×C4×D514yes—
438C22⋊C4—58no—
439C22⋊C4—104no—
440C22⋊C4—104no—
441C22⋊C4—202no—
442C22⋊C4—202no—
443C42—22no—
444C42—42no—
445C42—54no—
446C42—102no—
447C42—202no—
448C3×C5#235711yescharacteristic, complemented, Hall
449C22×C3—104no—
450C22×C3—104no—
451C22×C3—104no—
452C22×C3—202no—
453C22×C3—202no—
454C22×C3—202no—
455C22×C3—202no—
456C22×C3C22×C4×D511yescharacteristic
457C22×C3C2×#70911yescharacteristic
458C22×C3C2×#70911yescharacteristic
459C22×C3#414411yescharacteristic
460C22×C3C2×#71311yescharacteristic
461C22×C3D5×#4111yescharacteristic
462C22×C3#414911yescharacteristic
463C3×C4—22no—
464C3×C4—24no—
465C3×C4—41no—
466C3×C4—42no—
467C3×C4—102no—
468C3×C4—102no—
469C3×C4—102no—
470C3×C4—201no—
471C3×C4—202no—
472C3×C4#401814yes—
473C2×C5C2×C3×#11711yescharacteristic
474C2×C5C3×C4×#4111yescharacteristic
475C2×C5C3×#51811yescharacteristic
476C2×C5C3×#55311yescharacteristic
477C2×C5C3×#55311yescharacteristic
478C2×C5C3×#55911yescharacteristic
479C2×C5C3×#55911yescharacteristic
480D5—44no—
481C23—54no—
482C23—102no—
483C23—102no—
484C23—102no—
485C23—102no—
486C23—102no—
487C23—102no—
488C23C23×C3×D511yescharacteristic
489D4—108no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—52no—
501C2×C4—52no—
502C2×C4—52no—
503C2×C4—101no—
504C2×C4—101no—
505C2×C4—101no—
506C2×C4—101no—
507C2×C4—101no—
508C2×C4—101no—
509C2×C4—101no—
510C2×C4—102no—
511C2×C4—101no—
512C2×C4—102no—
513C2×C4—101no—
514C2×C4—102no—
515C2×C4—208no—
516C2×C4C2×C3×C4×D514yes—
517C2×C4C2×C3×C4×D512yes—
518C2×C4C3×#70912yes—
519C2×C4C3×#70912yes—
520C2×C4C3×#71312yes—
521C2×C3—204no—
522C2×C3C2×#401811yescharacteristic
523C2×C3#6584811yescharacteristic
524C2×C3#6586411yescharacteristic
525C2×C3#6587911yescharacteristic
526C2×C3#6589311yescharacteristic
527C2×C3C4×#71311yescharacteristic
528C2×C3#6591911yescharacteristic
529C5C3×#235711yescharacteristic, complemented, Sylow
530C22—104no—
531C22—104no—
532C22—104no—
533C22—202no—
534C22—202no—
535C22—202no—
536C22—202no—
537C22C22×C3×C4×D511yescharacteristic
538C22C2×C3×#70911yescharacteristic
539C22C2×C3×#70911yescharacteristic
540C22C3×#414411yescharacteristic
541C22C2×C3×#71311yescharacteristic
542C22C3×D5×#4111yescharacteristic
543C22C3×#414911yescharacteristic
544C4—22no—
545C4—24no—
546C4—41no—
547C4—42no—
548C4—102no—
549C4—102no—
550C4—102no—
551C4—201no—
552C4—202no—
553C4C3×#401814yes—
554C3#1060537711yescharacteristic, direct product factor, Sylow
555C2—204no—
556C2C2×C3×#401811yescharacteristic
557C2C3×#6584811yescharacteristic
558C2C3×#6586411yescharacteristic
559C2C3×#6587911yescharacteristic
560C2C3×#6589311yescharacteristic
561C2C3×C4×#71311yescharacteristic
562C2C3×#6591911yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C4×C5×C4⋊C4C2
socleC23×C3×C5C24
prime radicalC22×C3×D20C22
cosocleC23×C5C24×C3
hypercenterC3×C4×C4⋊C4D5
hypocenterC5C3×#2357
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C5×C4⋊C4C2semidirect
4. C3×#65266C2semidirect
5. C3×C4⋊C4×Dic5C2semidirect
6. C3×#65229C2semidirect
8. C3×C4×#685C2semidirect
9. C3×C4×#683C2semidirect
10. C3×#64958C2semidirect
11. C3×#64953C2semidirect
12. #10605377C3direct (N × H)
30. C3×#4039C4semidirect
39. C4×C5×C4⋊C4C2×C3semidirect
41. #65266C2×C3semidirect
42. C4⋊C4×Dic5C2×C3semidirect
43. #65229C2×C3semidirect
45. C4×#685C2×C3semidirect
46. C4×#683C2×C3semidirect
47. #64958C2×C3semidirect
48. #64953C2×C3semidirect
90. C3×C4×C4⋊C4D5semidirect
109. #4039C3×C4semidirect
222. C4×C4⋊C4C3×D5semidirect
448. C3×C5#2357semidirect
529. C5C3×#2357semidirect
554. C3#10605377direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40392
C3×C4×C5×C4⋊C42
C3×#652772
C3×#652662
C3×C4⋊C4×Dic52
C3×#652292
C3×#652282
C3×C4×#6852
C3×C4×#6832
C3×#649582
C3×#649532
#106053773
C3×#23575