Finite Group Databasebeta

← C3×#10615848 #14424541

Subgroups — C3×#10615848 #14424541

Order 1920.

1400 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×C5×2−1+4C211yesmaximal, characteristic
2C3×#66159C211yesmaximal, characteristic, complemented
3C2×C3×#4097C211yesmaximal, characteristic, complemented
4C22×C3×#689C211yesmaximal, characteristic, complemented
5C3×#65528C214yesmaximal, complemented
6C3×#65523C211yesmaximal, characteristic, complemented
7C3×#65522C211yesmaximal, characteristic, complemented
8C3×#65517C214yesmaximal, complemented
9C3×#65514C211yesmaximal, characteristic, complemented
10#10615848C311yesmaximal, characteristic, direct product factor, Hall
11C3×C5×2−1+4—24no—
12C3×C5×2−1+4—24no—
13C2×C3×C5×#41—22no—
14C2×C3×C5×#41—21no—
15C2×C3×C5×#41C2212yes—
16C2×C3×C5×#41C2211yescharacteristic
17C2×C3×C5×#41C2211yescharacteristic
18C22×C3×C5×Q8—21no—
19C22×C3×C5×Q8C2212yes—
20C22×C3×C5×Q8C2211yescharacteristic
21C22×C3×Dic10C2211yescharacteristic
22C3×#4097—24no—
23C3×#4090—21no—
24C2×C3×#689—24no—
25C2×C3×#689C2214yes—
26C2×C3×#689C2212yes—
27C3×#4082—21no—
28C3×#4081—22no—
29C2×C3×#687C2212yes—
30C3×#4072C2211yescharacteristic
31C3×#4070C2214yes—
32C2×C3×#681C2211yescharacteristic
33C22×C3×#168C2211yescharacteristic
34C3×#3967C2214yes—
35C3×#3967C2212yes—
36C3×#3964C2212yes—
37C3×C22⋊#168C2214yes—
38C3×#3011—51nomaximal, Hall
39C2×C5×2−1+4C2×C311yescharacteristic
40#66159C2×C311yescharacteristic, complemented
41C2×#4097C2×C311yescharacteristic, complemented
42C22×#689C2×C311yescharacteristic, complemented
43#65528C2×C314yescomplemented
44#65523C2×C311yescharacteristic, complemented
45#65522C2×C311yescharacteristic, complemented
46#65517C2×C314yescomplemented
47#65514C2×C311yescharacteristic, complemented
48C3×C5×#41—216no—
49C3×C5×#41—28no—
50C3×C5×#41—24no—
51C3×C5×#41—24no—
52C3×C5×#41—22no—
53C3×C5×#41—22no—
54C3×C5×#41—22no—
55C3×C5×#41D414yes—
56C2×C3×C5×Q8—28no—
57C2×C3×C5×Q8—24no—
58C2×C3×C5×Q8—24no—
59C2×C3×C5×Q8—22no—
60C2×C3×C5×Q8—21no—
61C2×C3×C5×Q8—22no—
62C2×C3×C5×Q8D414yes—
63C2×C3×C5×Q8D411yescharacteristic
64C2×C3×C5×Q8C2311yescharacteristic
65C2×C3×C5×Q8C2312yes—
66C2×C3×C5×D4—22no—
67C2×C3×C5×D4—21no—
68C2×C3×C5×D4D412yes—
69C2×C3×C5×D4D411yescharacteristic
70C2×C3×C5×D4C2311yescharacteristic
71C22×C3×C4×C5—22no—
72C22×C3×C4×C5—21no—
73C22×C3×C4×C5—22no—
74C22×C3×C4×C5—21no—
75C22×C3×C4×C5C2312yes—
76C22×C3×C4×C5C2311yescharacteristic
77C22×C3×Dic5—21no—
78C2×C3×Dic10—22no—
79C2×C3×Dic10—22no—
80C2×C3×Dic10—21no—
81C2×C3×Dic10C2312yes—
82C3×C22⋊Dic5—21no—
83C3×C22⋊Dic5—24no—
84C3×#689—28no—
85C3×#689—24no—
86C3×#687—24no—
87C3×C4⋊Dic5C2312yes—
88C3×#683—24no—
89C3×C4×Dic5—21no—
90C3×#681—22no—
91C2×C3×#168—22no—
92C2×C3×#168C2312yes—
93C2×C3×#168C2312yes—
94C2×C3×2−1+4D511yescharacteristic
95C2×C3×#137—51no—
96C22×C3×Q16—51no—
97C3×#536—51no—
98C3×#452—54no—
99C3×#451—54no—
100C3×#419—51no—
101C3×#416—51no—
102C3×#408—51no—
103C5×2−1+4—24no—
104C5×2−1+4—24no—
105C2×C5×#41—22no—
106C2×C5×#41—21no—
107C2×C5×#41C22×C312yes—
108C2×C5×#41C22×C311yescharacteristic
109C2×C5×#41C22×C311yescharacteristic
110C22×C5×Q8—21no—
111C22×C5×Q8C22×C312yes—
112C22×C5×Q8C22×C311yescharacteristic
113C22×Dic10C22×C311yescharacteristic
114#4097—24no—
115#4090—21no—
116C2×#689—24no—
117C2×#689C22×C314yes—
118C2×#689C22×C312yes—
119#4082—21no—
120#4081—22no—
121C2×#687C22×C312yes—
122#4072C22×C311yescharacteristic
123#4070C22×C314yes—
124C2×#681C22×C311yescharacteristic
125C22×#168C22×C311yescharacteristic
126#3967C22×C314yes—
127#3967C22×C312yes—
128#3964C22×C312yes—
129C22⋊#168C22×C314yes—
130#3011—51noSylow
131C23×C3×C5—21no—
132C23×C3×C5C2×D412yes—
133C23×C3×C5C2×D411yescharacteristic
134C3×C5×Q8—28no—
135C3×C5×Q8—24no—
136C3×C5×Q8—22no—
137C3×C5×Q8—21no—
138C3×C5×Q8—22no—
139C3×C5×Q8C2×D412yes—
140C3×C5×Q8C2×D414yes—
141C3×C5×D4—28no—
142C3×C5×D4—24no—
143C3×C5×D4—24no—
144C3×C5×D4—22no—
145C3×C5×D4—21no—
146C3×C5×D4C2×D412yes—
147C2×C3×C4×C5—28no—
148C2×C3×C4×C5—28no—
149C2×C3×C4×C5—24no—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5—21no—
152C2×C3×C4×C5—24no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5C2×D412yes—
156C2×C3×C4×C5C2×D411yescharacteristic
157C2×C3×C4×C5C2×D412yes—
158C2×C3×C4×C5C2×D412yes—
159C2×C3×C4×C5C2411yescharacteristic
160C2×C3×Dic5—22no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5—42no—
163C3×Dic10—22no—
164C3×Dic10—22no—
165C3×Dic10—42no—
166C3×#168—22no—
167C3×#168—22no—
168C3×2−1+4—24no—
169C3×2−1+4—24no—
170C2×C3×#41—22no—
171C2×C3×#41—21no—
172C2×C3×#41C2×D512yes—
173C2×C3×#41C2×D511yescharacteristic
174C2×C3×#41C2×D511yescharacteristic
175C22×C3×Q8—21no—
176C22×C3×Q8—51no—
177C22×C3×Q8C2×D512yes—
178C22×C3×Q8C2×D511yescharacteristic
179C3×#137—104no—
180C2×C3×Q16—54no—
181C2×C3×Q16—52no—
182C2×C3×Q16—104no—
183C2×C3×SD16—52no—
184C2×C3×M16—51no—
185C22×C3×C8—51no—
186C3×C4⋊Q8—101no—
187C3×#124—101no—
188C3×#123—102no—
189C3×C22⋊Q8—54no—
190C3×#117—51no—
191C3×#103—52no—
192C3×#103—54no—
193C3×#102—52no—
194C3×C22⋊C8—54no—
195C5×#41—216no—
196C5×#41—28no—
197C5×#41—24no—
198C5×#41—24no—
199C5×#41—22no—
200C5×#41—22no—
201C5×#41—22no—
202C5×#41C3×D414yes—
203C2×C5×Q8—28no—
204C2×C5×Q8—24no—
205C2×C5×Q8—24no—
206C2×C5×Q8—22no—
207C2×C5×Q8—21no—
208C2×C5×Q8—22no—
209C2×C5×Q8C3×D414yes—
210C2×C5×Q8C3×D411yescharacteristic
211C2×C5×Q8C23×C311yescharacteristic
212C2×C5×Q8C23×C312yes—
213C2×C5×D4—22no—
214C2×C5×D4—21no—
215C2×C5×D4C3×D412yes—
216C2×C5×D4C3×D411yescharacteristic
217C2×C5×D4C23×C311yescharacteristic
218C22×C4×C5—22no—
219C22×C4×C5—21no—
220C22×C4×C5—22no—
221C22×C4×C5—21no—
222C22×C4×C5C23×C312yes—
223C22×C4×C5C23×C311yescharacteristic
224C22×Dic5—21no—
225C2×Dic10—22no—
226C2×Dic10—22no—
227C2×Dic10—21no—
228C2×Dic10C23×C312yes—
229C22⋊Dic5—24no—
230C22⋊Dic5—21no—
231#689—24no—
232#689—28no—
233#687—24no—
234C4⋊Dic5C23×C312yes—
235#683—24no—
236C4×Dic5—21no—
237#681—22no—
238C2×#168—22no—
239C2×#168C23×C312yes—
240C2×#168C23×C312yes—
241C2×2−1+4C3×D511yescharacteristic
242C2×#137—51no—
243C22×Q16—51no—
244#536—51no—
245#452—54no—
246#451—54no—
247#419—51no—
248#416—51no—
249#408—51no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—22no—
253C22×C3×C5—24no—
254C22×C3×C5—42no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C3×C4×C5—24no—
258C3×C4×C5—22no—
259C3×C4×C5—22no—
260C3×C4×C5#12012yes—
261C3×C4×C5C22×D411yescharacteristic
262C3×C4×C5C22×D411yescharacteristic
263C3×Dic5—42no—
264C3×Dic5—42no—
265C3×#41—216no—
266C3×#41—28no—
267C3×#41—24no—
268C3×#41—24no—
269C3×#41—22no—
270C3×#41—22no—
271C3×#41—22no—
272C3×#41C22⋊D514yes—
273C2×C3×Q8—28no—
274C2×C3×Q8—24no—
275C2×C3×Q8—24no—
276C2×C3×Q8—22no—
277C2×C3×Q8—21no—
278C2×C3×Q8—22no—
279C2×C3×Q8—52no—
280C2×C3×Q8—101no—
281C2×C3×Q8—102no—
282C2×C3×Q8—102no—
283C2×C3×Q8C22⋊D514yes—
284C2×C3×Q8C22⋊D511yescharacteristic
285C2×C3×Q8C22×D511yescharacteristic
286C2×C3×Q8C22×D512yes—
287C2×C3×D4—22no—
288C2×C3×D4—21no—
289C2×C3×D4C22⋊D512yes—
290C2×C3×D4C22⋊D511yescharacteristic
291C2×C3×D4C22×D511yescharacteristic
292C22×C3×C4—22no—
293C22×C3×C4—21no—
294C22×C3×C4—22no—
295C22×C3×C4—21no—
296C22×C3×C4—101no—
297C22×C3×C4C22×D512yes—
298C22×C3×C4C22×D511yescharacteristic
299C3×Q16—108no—
300C3×Q16—104no—
301C3×SD16—104no—
302C3×M16—102no—
303C2×C3×C8—52no—
304C2×C3×C8—52no—
305C2×C3×C8—102no—
306C3×C4⋊C4—52no—
307C3×C4⋊C4—104no—
308C3×C22⋊C4—101no—
309C3×C22⋊C4—104no—
310C3×C42—101no—
311C23×C5—21no—
312C23×C5C2×C3×D412yes—
313C23×C5C2×C3×D411yescharacteristic
314C5×Q8—28no—
315C5×Q8—24no—
316C5×Q8—22no—
317C5×Q8—21no—
318C5×Q8—22no—
319C5×Q8C2×C3×D412yes—
320C5×Q8C2×C3×D414yes—
321C5×D4—28no—
322C5×D4—24no—
323C5×D4—24no—
324C5×D4—22no—
325C5×D4—21no—
326C5×D4C2×C3×D412yes—
327C2×C4×C5—28no—
328C2×C4×C5—28no—
329C2×C4×C5—24no—
330C2×C4×C5—22no—
331C2×C4×C5—21no—
332C2×C4×C5—24no—
333C2×C4×C5—22no—
334C2×C4×C5—22no—
335C2×C4×C5C2×C3×D412yes—
336C2×C4×C5C2×C3×D411yescharacteristic
337C2×C4×C5C2×C3×D412yes—
338C2×C4×C5C2×C3×D412yes—
339C2×C4×C5C24×C311yescharacteristic
340C2×Dic5—22no—
341C2×Dic5—22no—
342C2×Dic5—42no—
343Dic10—22no—
344Dic10—22no—
345Dic10—42no—
346#168—22no—
347#168—22no—
3482−1+4—24no—
3492−1+4—24no—
350C2×#41—22no—
351C2×#41—21no—
352C2×#41C2×C3×D512yes—
353C2×#41C2×C3×D511yescharacteristic
354C2×#41C2×C3×D511yescharacteristic
355C22×Q8—21no—
356C22×Q8—51no—
357C22×Q8C2×C3×D512yes—
358C22×Q8C2×C3×D511yescharacteristic
359#137—104no—
360C2×Q16—54no—
361C2×Q16—52no—
362C2×Q16—104no—
363C2×SD16—52no—
364C2×M16—51no—
365C22×C8—51no—
366C4⋊Q8—101no—
367#124—101no—
368#123—102no—
369C22⋊Q8—54no—
370#117—51no—
371#103—52no—
372#103—54no—
373#102—52no—
374C22⋊C8—54no—
375C2×C3×C5—22no—
376C2×C3×C5—42no—
377C2×C3×C5—42no—
378C2×C3×C5C2×#12011yescharacteristic
379C2×C3×C5#57812yes—
380C23×C3—21no—
381C23×C3C2×C22⋊D512yes—
382C23×C3C2×C22⋊D511yescharacteristic
383C3×Q8—28no—
384C3×Q8—24no—
385C3×Q8—22no—
386C3×Q8—21no—
387C3×Q8—22no—
388C3×Q8—102no—
389C3×Q8—102no—
390C3×Q8—202no—
391C3×Q8C2×C22⋊D512yes—
392C3×Q8C2×C22⋊D514yes—
393C3×D4—28no—
394C3×D4—24no—
395C3×D4—24no—
396C3×D4—22no—
397C3×D4—21no—
398C3×D4C2×C22⋊D512yes—
399C2×C3×C4—28no—
400C2×C3×C4—28no—
401C2×C3×C4—24no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—102no—
408C2×C3×C4—102no—
409C2×C3×C4—202no—
410C2×C3×C4C2×C22⋊D512yes—
411C2×C3×C4C2×C22⋊D511yescharacteristic
412C2×C3×C4C2×C22⋊D512yes—
413C2×C3×C4C2×C22⋊D512yes—
414C2×C3×C4C23×D511yescharacteristic
415C3×C8—102no—
416C3×C8—102no—
417C22×C5—22no—
418C22×C5—24no—
419C22×C5—22no—
420C22×C5—22no—
421C22×C5—42no—
422C22×C5C3×#12012yes—
423C22×C5C22×C3×D411yescharacteristic
424C4×C5—24no—
425C4×C5—22no—
426C4×C5—22no—
427C4×C5C3×#12012yes—
428C4×C5C22×C3×D411yescharacteristic
429C4×C5C22×C3×D411yescharacteristic
430Dic5—42no—
431Dic5—42no—
432#41—216no—
433#41—28no—
434#41—24no—
435#41—24no—
436#41—22no—
437#41—22no—
438#41—22no—
439#41C3×C22⋊D514yes—
440C2×Q8—28no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—52no—
447C2×Q8—101no—
448C2×Q8—102no—
449C2×Q8—102no—
450C2×Q8C3×C22⋊D514yes—
451C2×Q8C3×C22⋊D511yescharacteristic
452C2×Q8C22×C3×D511yescharacteristic
453C2×Q8C22×C3×D512yes—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4C3×C22⋊D512yes—
457C2×D4C3×C22⋊D511yescharacteristic
458C2×D4C22×C3×D511yescharacteristic
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—101no—
464C22×C4C22×C3×D512yes—
465C22×C4C22×C3×D511yescharacteristic
466Q16—108no—
467Q16—104no—
468SD16—104no—
469M16—102no—
470C2×C8—52no—
471C2×C8—52no—
472C2×C8—102no—
473C4⋊C4—52no—
474C4⋊C4—104no—
475C22⋊C4—101no—
476C22⋊C4—104no—
477C42—101no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3—24no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—42no—
484C22×C3#409912yes—
485C22×C3C22×C22⋊D511yescharacteristic
486C3×C4—24no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—202no—
490C3×C4—202no—
491C3×C4#409912yes—
492C3×C4C22×C22⋊D511yescharacteristic
493C3×C4C22×C22⋊D511yescharacteristic
494C2×C5—22no—
495C2×C5—42no—
496C2×C5—42no—
497C2×C5C2×C3×#12011yescharacteristic
498C2×C5C3×#57812yes—
499C23—21no—
500C23C2×C3×C22⋊D512yes—
501C23C2×C3×C22⋊D511yescharacteristic
502Q8—28no—
503Q8—24no—
504Q8—22no—
505Q8—21no—
506Q8—22no—
507Q8—102no—
508Q8—102no—
509Q8—202no—
510Q8C2×C3×C22⋊D512yes—
511Q8C2×C3×C22⋊D514yes—
512D4—28no—
513D4—24no—
514D4—24no—
515D4—22no—
516D4—21no—
517D4C2×C3×C22⋊D512yes—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—102no—
527C2×C4—102no—
528C2×C4—202no—
529C2×C4C2×C3×C22⋊D512yes—
530C2×C4C2×C3×C22⋊D511yescharacteristic
531C2×C4C2×C3×C22⋊D512yes—
532C2×C4C2×C3×C22⋊D512yes—
533C2×C4C23×C3×D511yescharacteristic
534C8—102no—
535C8—102no—
536C2×C3—22no—
537C2×C3—42no—
538C2×C3—42no—
539C2×C3#6617212yes—
540C2×C3C2×#409911yescharacteristic
541C5C3×#301111yescharacteristic, complemented, Sylow
542C22—22no—
543C22—24no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22C3×#409912yes—
548C22C22×C3×C22⋊D511yescharacteristic
549C4—24no—
550C4—22no—
551C4—22no—
552C4—202no—
553C4—202no—
554C4C3×#409912yes—
555C4C22×C3×C22⋊D511yescharacteristic
556C4C22×C3×C22⋊D511yescharacteristic
557C3#1061584811yescharacteristic, direct product factor, Sylow
558C2—22no—
559C2—42no—
560C2—42no—
561C2C3×#6617212yes—
562C2C2×C3×#409911yescharacteristic
subgroupG/H
centerC22×C3C22×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
Fitting, prime radicalC2×C3×C5×2−1+4C2
socleC22×C3×C5C22×D4
hypercenterC2×C3×2−1+4D5
hypocenterC5C3×#3011
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×2−1+4

Complements

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

NH ≅ G/Ntype
2. C3×#66159C2semidirect
3. C2×C3×#4097C2semidirect
4. C22×C3×#689C2semidirect
5. C3×#65528C2semidirect
6. C3×#65523C2semidirect
7. C3×#65522C2semidirect
8. C3×#65517C2semidirect
9. C3×#65514C2semidirect
10. #10615848C3direct (N × H)
40. #66159C2×C3semidirect
41. C2×#4097C2×C3semidirect
42. C22×#689C2×C3semidirect
43. #65528C2×C3semidirect
44. #65523C2×C3semidirect
45. #65522C2×C3semidirect
46. #65517C2×C3semidirect
47. #65514C2×C3semidirect
478. C3×C5#3011semidirect
541. C5C3×#3011semidirect
557. C3#10615848direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×2−1+42
C3×#661592
C2×C3×#40972
C22×C3×#6892
C3×#655282
C3×#655232
C3×#655222
C3×#655172
C3×#655142
#106158483
C3×#30115