Finite Group Databasebeta

← C3×#10614667 #14425659

Subgroups — C3×#10614667 #14425659

Order 1920.

1532 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66283C211yesmaximal, characteristic, complemented
2C3×C5×#525C211yesmaximal, characteristic, complemented
3C3×#66125C212yesmaximal, complemented
4C3×#65529C211yesmaximal, characteristic, complemented
5C3×#65342C214yesmaximal, complemented
6C3×#65321C214yesmaximal, complemented
7C3×#65295C212yesmaximal, complemented
8#10614667C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C23×C3×C4×C5C2211yescharacteristic, complemented
11C3×#4150—22no—
12C3×#4149—22no—
13C3×D5×#41—24no—
14C3×#4147—21no—
15C3×#4144—21no—
16C3×#4144C2214yes—
17C2×C3×#709—22no—
18C2×C3×#709C2212yes—
19C3×C5×#123C2212yescomplemented
20C3×C5×C22⋊Q8—21no—
21C3×C5×#121—21no—
22C3×C5×#120C2212yescomplemented
23C3×C4×C5×D4—22no—
24C3×C5×#117C2211yescharacteristic, complemented
25C3×#4099C2214yescomplemented
26C3×#4076—22no—
27C3×#4075C2214yescomplemented
28C3×C4×C22⋊D5—24no—
29C3×#4072C2212yescomplemented
30C3×#4070—22no—
31C3×#3966C2214yescomplemented
32C3×#3938C2218yescomplemented
33C3×#3025—51nomaximal, Hall
34#66283C2×C311yescharacteristic, complemented
35C5×#525C2×C311yescharacteristic, complemented
36#66125C2×C312yescomplemented
37#65529C2×C311yescharacteristic, complemented
38#65342C2×C314yescomplemented
39#65321C2×C314yescomplemented
40#65295C2×C312yescomplemented
41C24×C3×C5C2311yescharacteristic
42C3×C5×#41—22no—
43C3×C5×#41—22no—
44C2×C3×C5×Q8D411yescharacteristic, complemented
45C2×C3×C5×D4—22no—
46C2×C3×C5×D4D411yescharacteristic, complemented
47C2×C3×C5×D4C2312yes—
48C22×C3×C4×C5—21no—
49C22×C3×C4×C5—21no—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—22no—
52C22×C3×C4×C5D412yescomplemented
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C2×C3×C22⋊D5—24no—
56C2×C3×C22⋊D5—24no—
57C2×C3×C22⋊D5C2314yes—
58C3×#713—24no—
59C3×Q8×D5—24no—
60C3×#711—24no—
61C3×#711—28no—
62C3×D4×D5—24no—
63C3×D4×D5—28no—
64C3×#709—24no—
65C3×#709—24no—
66C3×#709—22no—
67C3×#709—24no—
68C3×#709—22no—
69C3×#709—22no—
70C3×#709—22no—
71C2×C3×D20—22no—
72C2×C3×D20D412yescomplemented
73C2×C3×C4×D5—24no—
74C2×C3×C4×D5D414yescomplemented
75C2×C3×Dic10—22no—
76C2×C3×Dic10D412yescomplemented
77C3×C5×C4⋊C4—21no—
78C3×C5×C4⋊C4—22no—
79C3×C5×C22⋊C4—22no—
80C3×C5×C22⋊C4—22no—
81C3×C5×C22⋊C4C2312yes—
82C3×C42×C5—21no—
83C3×C22⋊Dic5—24no—
84C3×C22⋊Dic5C2314yes—
85C3×#685—24no—
86C3×C4⋊Dic5—22no—
87C3×#683—24no—
88C3×C4×Dic5—22no—
89C3×#585—51no—
90C3×#525—52no—
91C3×#525D511yescharacteristic, complemented
92C3×#458—54no—
93C3×B(4,𝔽2)—54no—
94C3×#410—51no—
95C3×#410—52no—
96C2×C5×#41C22×C311yescharacteristic
97C23×C4×C5C22×C311yescharacteristic, complemented
98#4150—22no—
99#4149—22no—
100D5×#41—24no—
101#4147—21no—
102#4144—21no—
103#4144C22×C314yes—
104C2×#709—22no—
105C2×#709C22×C312yes—
106C5×#123C22×C312yescomplemented
107C5×C22⋊Q8—21no—
108C5×#121—21no—
109C5×#120C22×C312yescomplemented
110C4×C5×D4—22no—
111C5×#117C22×C311yescharacteristic, complemented
112#4099C22×C314yescomplemented
113#4076—22no—
114#4075C22×C314yescomplemented
115C4×C22⋊D5—24no—
116#4072C22×C312yescomplemented
117#4070—22no—
118#3966C22×C314yescomplemented
119#3938C22×C318yescomplemented
120#3025—51noSylow
121C23×C3×C5—22no—
122C23×C3×C5—21no—
123C23×C3×C5—42no—
124C23×C3×C5C2×D412yes—
125C23×C3×C5C2411yescharacteristic
126C22×C3×D5—24no—
127C22×C3×D5C2×D414yes—
128C3×C5×Q8—22no—
129C3×C5×D4—24no—
130C3×C5×D4—22no—
131C3×C5×D4—44no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—41no—
138C2×C3×C4×C5—42no—
139C2×C3×C4×C5—41no—
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D411yescharacteristic
145C3×C22⋊D5—28no—
146C3×C22⋊D5—24no—
147C3×C22⋊D5—24no—
148C3×C22⋊D5—48no—
149C2×C3×Dic5—24no—
150C2×C3×Dic5—24no—
151C2×C3×Dic5C2×D414yes—
152C3×D20—24no—
153C3×D20—22no—
154C3×D20—22no—
155C3×C4×D5—28no—
156C3×C4×D5—24no—
157C3×C4×D5—24no—
158C3×Dic10—24no—
159C3×Dic10—22no—
160C3×Dic10—22no—
161C3×2−1+4—102no—
162C3×2−1+4—101no—
163C3×H(5,𝔽2)—54no—
164C3×H(5,𝔽2)—102no—
165C3×H(5,𝔽2)—101no—
166C2×C3×#41—52no—
167C2×C3×#41—104no—
168C2×C3×#41—102no—
169C2×C3×#41C2×D511yescharacteristic
170C23×C3×C4C2×D511yescharacteristic, complemented
171C3×#123—54no—
172C3×#123C2×D512yescomplemented
173C3×C22⋊Q8—21no—
174C3×C22⋊Q8—102no—
175C3×#121—21no—
176C3×#121—102no—
177C3×#120—54no—
178C3×#120C2×D512yescomplemented
179C3×C4×D4—22no—
180C3×C4×D4—104no—
181C3×#117—52no—
182C3×#117C2×D511yescharacteristic, complemented
183C3×#99—54no—
184C3×#99—58no—
185C24×C5C23×C311yescharacteristic
186C5×#41—22no—
187C5×#41—22no—
188C2×C5×Q8C3×D411yescharacteristic, complemented
189C2×C5×D4—22no—
190C2×C5×D4C3×D411yescharacteristic, complemented
191C2×C5×D4C23×C312yes—
192C22×C4×C5—21no—
193C22×C4×C5—21no—
194C22×C4×C5—22no—
195C22×C4×C5—22no—
196C22×C4×C5C3×D412yescomplemented
197C22×C4×C5C23×C311yescharacteristic
198C22×C4×C5C23×C311yescharacteristic
199C2×C22⋊D5—24no—
200C2×C22⋊D5—24no—
201C2×C22⋊D5C23×C314yes—
202#713—24no—
203Q8×D5—24no—
204#711—24no—
205#711—28no—
206D4×D5—24no—
207D4×D5—28no—
208#709—24no—
209#709—24no—
210#709—22no—
211#709—24no—
212#709—22no—
213#709—22no—
214#709—22no—
215C2×D20—22no—
216C2×D20C3×D412yescomplemented
217C2×C4×D5—24no—
218C2×C4×D5C3×D414yescomplemented
219C2×Dic10—22no—
220C2×Dic10C3×D412yescomplemented
221C5×C4⋊C4—21no—
222C5×C4⋊C4—22no—
223C5×C22⋊C4—22no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4C23×C312yes—
226C42×C5—21no—
227C22⋊Dic5—24no—
228C22⋊Dic5C23×C314yes—
229#685—24no—
230C4⋊Dic5—22no—
231#683—24no—
232C4×Dic5—22no—
233#585—51no—
234#525—52no—
235#525C3×D511yescharacteristic, complemented
236#458—54no—
237B(4,𝔽2)—54no—
238#410—51no—
239#410—52no—
240C22×C3×C5—24no—
241C22×C3×C5—22no—
242C22×C3×C5—22no—
243C22×C3×C5—41no—
244C22×C3×C5—42no—
245C22×C3×C5—42no—
246C22×C3×C5—41no—
247C22×C3×C5—41no—
248C22×C3×C5C22×D412yes—
249C22×C3×C5C22×D411yescharacteristic
250C2×C3×D5—24no—
251C2×C3×D5—44no—
252C2×C3×D5—84no—
253C3×C4×C5—22no—
254C3×C4×C5—42no—
255C3×C4×C5—41no—
256C3×C4×C5#12011yescharacteristic
257C3×C4×C5#12012yes—
258C3×C4×C5#12011yescharacteristic
259C3×Dic5—24no—
260C3×Dic5—44no—
261C24×C3C22×D511yescharacteristic
262C3×#41—22no—
263C3×#41—22no—
264C3×#41—104no—
265C3×#41—104no—
266C3×#41—108no—
267C3×#41—104no—
268C3×#41—104no—
269C3×#41—102no—
270C3×#41—102no—
271C3×#41—102no—
272C3×#41—102no—
273C3×#41—104no—
274C2×C3×Q8—52no—
275C2×C3×Q8—102no—
276C2×C3×Q8—104no—
277C2×C3×Q8C22⋊D511yescharacteristic, complemented
278C2×C3×D4—22no—
279C2×C3×D4—54no—
280C2×C3×D4—52no—
281C2×C3×D4—102no—
282C2×C3×D4—108no—
283C2×C3×D4—104no—
284C2×C3×D4—104no—
285C2×C3×D4—104no—
286C2×C3×D4C22⋊D511yescharacteristic, complemented
287C2×C3×D4C22×D512yes—
288C22×C3×C4—22no—
289C22×C3×C4—22no—
290C22×C3×C4—21no—
291C22×C3×C4—21no—
292C22×C3×C4—54no—
293C22×C3×C4—104no—
294C22×C3×C4C22⋊D512yescomplemented
295C22×C3×C4C22×D511yescharacteristic
296C22×C3×C4C22×D511yescharacteristic
297C3×C4⋊C4—21no—
298C3×C4⋊C4—22no—
299C3×C4⋊C4—104no—
300C3×C4⋊C4—102no—
301C3×C22⋊C4—22no—
302C3×C22⋊C4—22no—
303C3×C22⋊C4—54no—
304C3×C22⋊C4—104no—
305C3×C22⋊C4—104no—
306C3×C22⋊C4C22×D512yes—
307C3×C42—21no—
308C3×C42—102no—
309C23×C5—22no—
310C23×C5—21no—
311C23×C5—42no—
312C23×C5C2×C3×D412yes—
313C23×C5C24×C311yescharacteristic
314C22×D5—24no—
315C22×D5C2×C3×D414yes—
316C5×Q8—22no—
317C5×D4—24no—
318C5×D4—22no—
319C5×D4—44no—
320C2×C4×C5—22no—
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—22no—
324C2×C4×C5—21no—
325C2×C4×C5—41no—
326C2×C4×C5—42no—
327C2×C4×C5—41no—
328C2×C4×C5C2×C3×D412yes—
329C2×C4×C5C2×C3×D412yes—
330C2×C4×C5C2×C3×D412yes—
331C2×C4×C5C2×C3×D411yescharacteristic
332C2×C4×C5C2×C3×D411yescharacteristic
333C22⋊D5—28no—
334C22⋊D5—24no—
335C22⋊D5—24no—
336C22⋊D5—48no—
337C2×Dic5—24no—
338C2×Dic5—24no—
339C2×Dic5C2×C3×D414yes—
340D20—24no—
341D20—22no—
342D20—22no—
343C4×D5—28no—
344C4×D5—24no—
345C4×D5—24no—
346Dic10—24no—
347Dic10—22no—
348Dic10—22no—
3492−1+4—102no—
3502−1+4—101no—
351H(5,𝔽2)—54no—
352H(5,𝔽2)—102no—
353H(5,𝔽2)—101no—
354C2×#41—52no—
355C2×#41—104no—
356C2×#41—102no—
357C2×#41C2×C3×D511yescharacteristic
358C23×C4C2×C3×D511yescharacteristic, complemented
359#123—54no—
360#123C2×C3×D512yescomplemented
361C22⋊Q8—21no—
362C22⋊Q8—102no—
363#121—21no—
364#121—102no—
365#120—54no—
366#120C2×C3×D512yescomplemented
367C4×D4—22no—
368C4×D4—104no—
369#117—52no—
370#117C2×C3×D511yescharacteristic, complemented
371#99—54no—
372#99—58no—
373C2×C3×C5—22no—
374C2×C3×C5—21no—
375C2×C3×C5—42no—
376C2×C3×C5—42no—
377C2×C3×C5C2×#12011yescharacteristic
378C3×D5—44no—
379C23×C3—22no—
380C23×C3—21no—
381C23×C3—42no—
382C23×C3—54no—
383C23×C3—104no—
384C23×C3C2×C22⋊D512yes—
385C23×C3C23×D511yescharacteristic
386C3×Q8—22no—
387C3×Q8—104no—
388C3×Q8—102no—
389C3×Q8—102no—
390C3×D4—24no—
391C3×D4—22no—
392C3×D4—44no—
393C3×D4—104no—
394C3×D4—108no—
395C3×D4—104no—
396C3×D4—104no—
397C3×D4—102no—
398C3×D4—102no—
399C3×D4—208no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—21no—
405C2×C3×C4—41no—
406C2×C3×C4—42no—
407C2×C3×C4—41no—
408C2×C3×C4—54no—
409C2×C3×C4—108no—
410C2×C3×C4—104no—
411C2×C3×C4—104no—
412C2×C3×C4—104no—
413C2×C3×C4—104no—
414C2×C3×C4D4×D512yes—
415C2×C3×C4D4×D512yes—
416C2×C3×C4C2×C22⋊D512yes—
417C2×C3×C4C2×C22⋊D511yescharacteristic
418C2×C3×C4C2×C22⋊D511yescharacteristic
419C22×C5—24no—
420C22×C5—22no—
421C22×C5—22no—
422C22×C5—41no—
423C22×C5—42no—
424C22×C5—42no—
425C22×C5—41no—
426C22×C5—41no—
427C22×C5C22×C3×D412yes—
428C22×C5C22×C3×D411yescharacteristic
429C2×D5—24no—
430C2×D5—44no—
431C2×D5—84no—
432C4×C5—22no—
433C4×C5—42no—
434C4×C5—41no—
435C4×C5C3×#12011yescharacteristic
436C4×C5C3×#12012yes—
437C4×C5C3×#12011yescharacteristic
438Dic5—24no—
439Dic5—44no—
440C24C22×C3×D511yescharacteristic
441#41—22no—
442#41—22no—
443#41—104no—
444#41—104no—
445#41—108no—
446#41—104no—
447#41—104no—
448#41—102no—
449#41—102no—
450#41—102no—
451#41—102no—
452#41—104no—
453C2×Q8—52no—
454C2×Q8—102no—
455C2×Q8—104no—
456C2×Q8C3×C22⋊D511yescharacteristic, complemented
457C2×D4—22no—
458C2×D4—54no—
459C2×D4—52no—
460C2×D4—102no—
461C2×D4—108no—
462C2×D4—104no—
463C2×D4—104no—
464C2×D4—104no—
465C2×D4C3×C22⋊D511yescharacteristic, complemented
466C2×D4C22×C3×D512yes—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—54no—
472C22×C4—104no—
473C22×C4C3×C22⋊D512yescomplemented
474C22×C4C22×C3×D511yescharacteristic
475C22×C4C22×C3×D511yescharacteristic
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—104no—
479C4⋊C4—102no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—54no—
483C22⋊C4—104no—
484C22⋊C4—104no—
485C22⋊C4C22×C3×D512yes—
486C42—21no—
487C42—102no—
488C3×C5#302511yescharacteristic, complemented, Hall
489C22×C3—22no—
490C22×C3—24no—
491C22×C3—22no—
492C22×C3—41no—
493C22×C3—42no—
494C22×C3—41no—
495C22×C3—42no—
496C22×C3—41no—
497C22×C3—104no—
498C22×C3—204no—
499C22×C3—404no—
500C22×C3C2×D4×D512yes—
501C22×C3C22×C22⋊D511yescharacteristic
502C3×C4—22no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—104no—
506C3×C4—204no—
507C3×C4#408311yescharacteristic
508C3×C4#408312yes—
509C3×C4#408311yescharacteristic
510C2×C5—22no—
511C2×C5—21no—
512C2×C5—42no—
513C2×C5—42no—
514C2×C5C2×C3×#12011yescharacteristic
515D5—44no—
516C23—22no—
517C23—21no—
518C23—42no—
519C23—54no—
520C23—104no—
521C23C2×C3×C22⋊D512yes—
522C23C23×C3×D511yescharacteristic
523Q8—22no—
524Q8—104no—
525Q8—102no—
526Q8—102no—
527D4—24no—
528D4—22no—
529D4—44no—
530D4—104no—
531D4—108no—
532D4—104no—
533D4—104no—
534D4—102no—
535D4—102no—
536D4—208no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—54no—
546C2×C4—108no—
547C2×C4—104no—
548C2×C4—104no—
549C2×C4—104no—
550C2×C4—104no—
551C2×C4C3×D4×D512yes—
552C2×C4C3×D4×D512yes—
553C2×C4C2×C3×C22⋊D512yes—
554C2×C4C2×C3×C22⋊D511yescharacteristic
555C2×C4C2×C3×C22⋊D511yescharacteristic
556C2×C3—21no—
557C2×C3—22no—
558C2×C3—42no—
559C2×C3—42no—
560C2×C3—204no—
561C2×C3C2×#408311yescharacteristic
562C5C3×#302511yescharacteristic, complemented, Sylow
563C22—22no—
564C22—24no—
565C22—22no—
566C22—41no—
567C22—42no—
568C22—41no—
569C22—42no—
570C22—41no—
571C22—104no—
572C22—204no—
573C22—404no—
574C22C2×C3×D4×D512yes—
575C22C22×C3×C22⋊D511yescharacteristic
576C4—22no—
577C4—42no—
578C4—41no—
579C4—104no—
580C4—204no—
581C4C3×#408311yescharacteristic
582C4C3×#408311yescharacteristic
583C4C3×#408312yes—
584C3#1061466711yescharacteristic, direct product factor, Sylow
585C2—21no—
586C2—22no—
587C2—42no—
588C2—42no—
589C2—204no—
590C2C2×C3×#408311yescharacteristic
subgroupG/H
centerC3×C4#4083
commutator, cosocleC23×C5C24×C3
FrattiniC23C23×C3×D5
FittingC3×C5×#525C2
socleC2×C3×C5C2×#120
hypercenterC3×#525D5
hypocenterC5C3×#3025
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#525

Complements

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

NH ≅ G/Ntype
1. C3×#66283C2semidirect
2. C3×C5×#525C2semidirect
3. C3×#66125C2semidirect
4. C3×#65529C2semidirect
5. C3×#65342C2semidirect
6. C3×#65321C2semidirect
7. C3×#65295C2semidirect
8. #10614667C3direct (N × H)
10. C23×C3×C4×C5C22semidirect
19. C3×C5×#123C22semidirect
22. C3×C5×#120C22semidirect
24. C3×C5×#117C22semidirect
25. C3×#4099C22semidirect
27. C3×#4075C22semidirect
29. C3×#4072C22semidirect
31. C3×#3966C22semidirect
32. C3×#3938C22semidirect
34. #66283C2×C3semidirect
35. C5×#525C2×C3semidirect
36. #66125C2×C3semidirect
37. #65529C2×C3semidirect
38. #65342C2×C3semidirect
39. #65321C2×C3semidirect
40. #65295C2×C3semidirect
44. C2×C3×C5×Q8D4semidirect
46. C2×C3×C5×D4D4semidirect
52. C22×C3×C4×C5D4semidirect
72. C2×C3×D20D4semidirect
74. C2×C3×C4×D5D4semidirect
76. C2×C3×Dic10D4semidirect
91. C3×#525D5semidirect
97. C23×C4×C5C22×C3semidirect
106. C5×#123C22×C3semidirect
109. C5×#120C22×C3semidirect
111. C5×#117C22×C3semidirect
112. #4099C22×C3semidirect
114. #4075C22×C3semidirect
116. #4072C22×C3semidirect
118. #3966C22×C3semidirect
119. #3938C22×C3semidirect
170. C23×C3×C4C2×D5semidirect
172. C3×#123C2×D5semidirect
178. C3×#120C2×D5semidirect
182. C3×#117C2×D5semidirect
188. C2×C5×Q8C3×D4semidirect
190. C2×C5×D4C3×D4semidirect
196. C22×C4×C5C3×D4semidirect
216. C2×D20C3×D4semidirect
218. C2×C4×D5C3×D4semidirect
220. C2×Dic10C3×D4semidirect
235. #525C3×D5semidirect
277. C2×C3×Q8C22⋊D5semidirect
286. C2×C3×D4C22⋊D5semidirect
294. C22×C3×C4C22⋊D5semidirect
358. C23×C4C2×C3×D5semidirect
360. #123C2×C3×D5semidirect
366. #120C2×C3×D5semidirect
370. #117C2×C3×D5semidirect
456. C2×Q8C3×C22⋊D5semidirect
465. C2×D4C3×C22⋊D5semidirect
473. C22×C4C3×C22⋊D5semidirect
488. C3×C5#3025semidirect
562. C5C3×#3025semidirect
584. C3#10614667direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662832
C3×C5×#5252
C3×#661252
C3×#655292
C3×#653422
C3×#653212
C3×#652952
#106146673
C3×#30255