Finite Group Databasebeta

← C3×#10610421 #14425668

Subgroups — C3×#10610421 #14425668

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×#66162C211yesmaximal, characteristic, complemented
3C3×#65858C212yesmaximal, complemented
4C3×C5×#410C211yesmaximal, characteristic, complemented
5C3×#65295C212yesmaximal, complemented
6C3×#65031C214yesmaximal, complemented
7C3×#65030C214yesmaximal, complemented
8#10610421C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C3×#4150—22no—
11C3×#4149—22no—
12C3×D5×#41—24no—
13C3×#4147—21no—
14C3×#4144—21no—
15C3×#4144C2214yes—
16C2×C3×#709—22no—
17C2×C3×#709C2212yes—
18C22×C3×C4×D5C2211yescharacteristic, complemented
19C3×C5×#117C2212yescomplemented
20C3×#4092—21no—
21C3×#4084—21no—
22C3×#4083C2212yescomplemented
23C3×#4081C2212yescomplemented
24C3×C4×C22⋊D5—22no—
25C3×#4072C2211yescharacteristic, complemented
26C3×#4043—22no—
27C3×#4041—22no—
28C3×#4032C2214yescomplemented
29C3×#4028C2214yescomplemented
30C3×C4×D20—24no—
31C3×C5×#99C2214yescomplemented
32C3×#3938C2218yescomplemented
33C3×#3025—51nomaximal, Hall
34#66283C2×C311yescharacteristic, complemented
35#66162C2×C311yescharacteristic, complemented
36#65858C2×C312yescomplemented
37C5×#410C2×C311yescharacteristic, complemented
38#65295C2×C312yescomplemented
39#65031C2×C314yescomplemented
40#65030C2×C314yescomplemented
41C23×C3×D5C2311yescharacteristic
42C3×C5×#41—22no—
43C3×C5×#41—22no—
44C2×C3×C5×Q8D411yescharacteristic, complemented
45C2×C3×C5×D4D411yescharacteristic, complemented
46C2×C3×C5×D4C2312yes—
47C22×C3×C4×C5D412yescomplemented
48C22×C3×C4×C5C2311yescharacteristic
49C2×C3×C22⋊D5—24no—
50C2×C3×C22⋊D5—22no—
51C2×C3×C22⋊D5C2314yes—
52C22×C3×Dic5C2311yescharacteristic
53C3×#713—24no—
54C3×Q8×D5—24no—
55C3×#711—24no—
56C3×#711—28no—
57C3×D4×D5—24no—
58C3×D4×D5—28no—
59C3×#709—24no—
60C3×#709—24no—
61C3×#709—22no—
62C3×#709—24no—
63C3×#709—22no—
64C3×#709—22no—
65C3×#709—22no—
66C2×C3×D20—22no—
67C2×C3×D20—24no—
68C2×C3×D20D412yescomplemented
69C2×C3×C4×D5—24no—
70C2×C3×C4×D5—21no—
71C2×C3×C4×D5—22no—
72C2×C3×C4×D5—22no—
73C2×C3×C4×D5—21no—
74C2×C3×C4×D5D414yescomplemented
75C2×C3×Dic10—22no—
76C2×C3×Dic10D412yescomplemented
77C3×C5×C4⋊C4—22no—
78C3×C5×C22⋊C4C2314yes—
79C3×C42×C5—22no—
80C3×C22⋊Dic5—22no—
81C3×C22⋊Dic5C2312yes—
82C3×#685—24no—
83C3×#685—22no—
84C3×#685—24no—
85C3×C4⋊Dic5—21no—
86C3×C4⋊Dic5—24no—
87C3×#683—22no—
88C3×C4×Dic5—21no—
89C3×#585—51no—
90C3×#525—51no—
91C3×#525—52no—
92C3×#458—54no—
93C3×B(4,𝔽2)—54no—
94C3×#410—52no—
95C3×#410D511yescharacteristic, complemented
96C2×C5×#41C22×C311yescharacteristic
97#4150—22no—
98#4149—22no—
99D5×#41—24no—
100#4147—21no—
101#4144—21no—
102#4144C22×C314yes—
103C2×#709—22no—
104C2×#709C22×C312yes—
105C22×C4×D5C22×C311yescharacteristic, complemented
106C5×#117C22×C312yescomplemented
107#4092—21no—
108#4084—21no—
109#4083C22×C312yescomplemented
110#4081C22×C312yescomplemented
111C4×C22⋊D5—22no—
112#4072C22×C311yescharacteristic, complemented
113#4043—22no—
114#4041—22no—
115#4032C22×C314yescomplemented
116#4028C22×C314yescomplemented
117C4×D20—24no—
118C5×#99C22×C314yescomplemented
119#3938C22×C318yescomplemented
120#3025—51noSylow
121C23×C3×C5C2×D412yes—
122C23×C3×C5C2411yescharacteristic
123C22×C3×D5—22no—
124C22×C3×D5—21no—
125C22×C3×D5—24no—
126C22×C3×D5—42no—
127C22×C3×D5C2×D414yes—
128C3×C5×Q8—22no—
129C3×C5×D4—24no—
130C3×C5×D4—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—24no—
134C2×C3×C4×C5C2×D412yes—
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2×D412yes—
139C3×C22⋊D5—28no—
140C3×C22⋊D5—24no—
141C3×C22⋊D5—24no—
142C3×C22⋊D5—44no—
143C2×C3×Dic5—24no—
144C2×C3×Dic5—22no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5C2×D414yes—
148C3×D20—24no—
149C3×D20—22no—
150C3×D20—22no—
151C3×D20—48no—
152C3×C4×D5—28no—
153C3×C4×D5—24no—
154C3×C4×D5—24no—
155C3×C4×D5—41no—
156C3×C4×D5—42no—
157C3×C4×D5—41no—
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×C4—51no—
171C3×#123—52no—
172C3×#123—54no—
173C3×C22⋊Q8—102no—
174C3×C22⋊Q8—101no—
175C3×#121—102no—
176C3×#121—101no—
177C3×#120—52no—
178C3×#120—54no—
179C3×C4×D4—104no—
180C3×C4×D4—102no—
181C3×#117—51no—
182C3×#117C2×D512yescomplemented
183C3×#99—58no—
184C3×#99C2×D514yescomplemented
185C23×D5C23×C311yescharacteristic
186C5×#41—22no—
187C5×#41—22no—
188C2×C5×Q8C3×D411yescharacteristic, complemented
189C2×C5×D4C3×D411yescharacteristic, complemented
190C2×C5×D4C23×C312yes—
191C22×C4×C5C3×D412yescomplemented
192C22×C4×C5C23×C311yescharacteristic
193C2×C22⋊D5—24no—
194C2×C22⋊D5—22no—
195C2×C22⋊D5C23×C314yes—
196C22×Dic5C23×C311yescharacteristic
197#713—24no—
198Q8×D5—24no—
199#711—24no—
200#711—28no—
201D4×D5—24no—
202D4×D5—28no—
203#709—24no—
204#709—24no—
205#709—22no—
206#709—24no—
207#709—22no—
208#709—22no—
209#709—22no—
210C2×D20—22no—
211C2×D20—24no—
212C2×D20C3×D412yescomplemented
213C2×C4×D5—24no—
214C2×C4×D5—22no—
215C2×C4×D5—22no—
216C2×C4×D5—21no—
217C2×C4×D5—21no—
218C2×C4×D5C3×D414yescomplemented
219C2×Dic10—22no—
220C2×Dic10C3×D412yescomplemented
221C5×C4⋊C4—22no—
222C5×C22⋊C4C23×C314yes—
223C42×C5—22no—
224C22⋊Dic5—22no—
225C22⋊Dic5C23×C312yes—
226#685—24no—
227#685—22no—
228#685—24no—
229C4⋊Dic5—21no—
230C4⋊Dic5—24no—
231#683—22no—
232C4×Dic5—21no—
233#585—51no—
234#525—51no—
235#525—52no—
236#458—54no—
237B(4,𝔽2)—54no—
238#410—52no—
239#410C3×D511yescharacteristic, complemented
240C22×C3×C5—22no—
241C22×C3×C5—41no—
242C22×C3×C5—42no—
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C22×D412yes—
245C2×C3×D5—24no—
246C2×C3×D5—22no—
247C2×C3×D5—24no—
248C2×C3×D5—41no—
249C2×C3×D5—42no—
250C2×C3×D5—41no—
251C2×C3×D5—44no—
252C2×C3×D5—84no—
253C3×C4×C5—22no—
254C3×C4×C5—44no—
255C3×C4×C5#12011yescharacteristic
256C3×C4×C5#12012yes—
257C3×C4×C5#12011yescharacteristic
258C3×Dic5—24no—
259C3×Dic5—42no—
260C3×Dic5—41no—
261C24×C3—51no—
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—104no—
272C3×#41—102no—
273C3×#41—102no—
274C2×C3×Q8—52no—
275C2×C3×Q8—102no—
276C2×C3×Q8—104no—
277C2×C3×Q8D2011yescharacteristic, complemented
278C2×C3×D4—54no—
279C2×C3×D4—52no—
280C2×C3×D4—102no—
281C2×C3×D4—104no—
282C2×C3×D4—108no—
283C2×C3×D4—104no—
284C2×C3×D4—104no—
285C2×C3×D4—102no—
286C2×C3×D4D2011yescharacteristic, complemented
287C2×C3×D4C22×D512yes—
288C22×C3×C4—54no—
289C22×C3×C4—51no—
290C22×C3×C4—104no—
291C22×C3×C4—101no—
292C22×C3×C4—101no—
293C22×C3×C4—102no—
294C22×C3×C4—102no—
295C22×C3×C4D2012yescomplemented
296C22×C3×C4C22×D511yescharacteristic
297C3×C4⋊C4—22no—
298C3×C4⋊C4—104no—
299C3×C4⋊C4—102no—
300C3×C4⋊C4—101no—
301C3×C22⋊C4—52no—
302C3×C22⋊C4—104no—
303C3×C22⋊C4—104no—
304C3×C22⋊C4—102no—
305C3×C22⋊C4—102no—
306C3×C22⋊C4C22×D514yes—
307C3×C42—22no—
308C3×C42—101no—
309C23×C5C2×C3×D412yes—
310C23×C5C24×C311yescharacteristic
311C22×D5—22no—
312C22×D5—21no—
313C22×D5—24no—
314C22×D5—42no—
315C22×D5C2×C3×D414yes—
316C5×Q8—22no—
317C5×D4—24no—
318C5×D4—22no—
319C2×C4×C5—22no—
320C2×C4×C5—22no—
321C2×C4×C5—24no—
322C2×C4×C5C2×C3×D412yes—
323C2×C4×C5C2×C3×D411yescharacteristic
324C2×C4×C5C2×C3×D411yescharacteristic
325C2×C4×C5C2×C3×D412yes—
326C2×C4×C5C2×C3×D412yes—
327C22⋊D5—28no—
328C22⋊D5—24no—
329C22⋊D5—24no—
330C22⋊D5—44no—
331C2×Dic5—24no—
332C2×Dic5—22no—
333C2×Dic5—21no—
334C2×Dic5—22no—
335C2×Dic5C2×C3×D414yes—
336D20—24no—
337D20—22no—
338D20—22no—
339D20—48no—
340C4×D5—28no—
341C4×D5—24no—
342C4×D5—24no—
343C4×D5—41no—
344C4×D5—42no—
345C4×D5—41no—
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×C4—51no—
359#123—52no—
360#123—54no—
361C22⋊Q8—102no—
362C22⋊Q8—101no—
363#121—102no—
364#121—101no—
365#120—52no—
366#120—54no—
367C4×D4—104no—
368C4×D4—102no—
369#117—51no—
370#117C2×C3×D512yescomplemented
371#99—58no—
372#99C2×C3×D514yescomplemented
373C2×C3×C5—21no—
374C2×C3×C5—22no—
375C2×C3×C5—42no—
376C2×C3×C5C2×#12011yescharacteristic
377C3×D5—42no—
378C3×D5—44no—
379C23×C3—54no—
380C23×C3—104no—
381C23×C3—101no—
382C23×C3—102no—
383C23×C3—202no—
384C23×C3C2×D2012yes—
385C23×C3C23×D511yescharacteristic
386C3×Q8—22no—
387C3×Q8—104no—
388C3×Q8—102no—
389C3×Q8—102no—
390C3×D4—24no—
391C3×D4—22no—
392C3×D4—104no—
393C3×D4—108no—
394C3×D4—104no—
395C3×D4—104no—
396C3×D4—102no—
397C3×D4—102no—
398C3×D4—208no—
399C3×D4—204no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—24no—
403C2×C3×C4—54no—
404C2×C3×C4—108no—
405C2×C3×C4—104no—
406C2×C3×C4—104no—
407C2×C3×C4—104no—
408C2×C3×C4—102no—
409C2×C3×C4—101no—
410C2×C3×C4—102no—
411C2×C3×C4—201no—
412C2×C3×C4—202no—
413C2×C3×C4—201no—
414C2×C3×C4C2×D2012yes—
415C2×C3×C4C2×D2011yescharacteristic
416C2×C3×C4C2×D2011yescharacteristic
417C2×C3×C4D4×D512yes—
418C2×C3×C4D4×D512yes—
419C22×C5—22no—
420C22×C5—41no—
421C22×C5—42no—
422C22×C5C22×C3×D411yescharacteristic
423C22×C5C22×C3×D412yes—
424C2×D5—24no—
425C2×D5—22no—
426C2×D5—24no—
427C2×D5—41no—
428C2×D5—42no—
429C2×D5—41no—
430C2×D5—44no—
431C2×D5—84no—
432C4×C5—22no—
433C4×C5—44no—
434C4×C5C3×#12011yescharacteristic
435C4×C5C3×#12012yes—
436C4×C5C3×#12011yescharacteristic
437Dic5—24no—
438Dic5—42no—
439Dic5—41no—
440C24—51no—
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—104no—
451#41—102no—
452#41—102no—
453C2×Q8—52no—
454C2×Q8—102no—
455C2×Q8—104no—
456C2×Q8C3×D2011yescharacteristic, complemented
457C2×D4—54no—
458C2×D4—52no—
459C2×D4—102no—
460C2×D4—104no—
461C2×D4—108no—
462C2×D4—104no—
463C2×D4—104no—
464C2×D4—102no—
465C2×D4C3×D2011yescharacteristic, complemented
466C2×D4C22×C3×D512yes—
467C22×C4—54no—
468C22×C4—51no—
469C22×C4—104no—
470C22×C4—101no—
471C22×C4—101no—
472C22×C4—102no—
473C22×C4—102no—
474C22×C4C3×D2012yescomplemented
475C22×C4C22×C3×D511yescharacteristic
476C4⋊C4—22no—
477C4⋊C4—104no—
478C4⋊C4—101no—
479C4⋊C4—102no—
480C22⋊C4—52no—
481C22⋊C4—104no—
482C22⋊C4—104no—
483C22⋊C4—102no—
484C22⋊C4—102no—
485C22⋊C4C22×C3×D514yes—
486C42—22no—
487C42—101no—
488C3×C5#302511yescharacteristic, complemented, Hall
489C22×C3—22no—
490C22×C3—41no—
491C22×C3—42no—
492C22×C3—104no—
493C22×C3—102no—
494C22×C3—104no—
495C22×C3—204no—
496C22×C3—202no—
497C22×C3—201no—
498C22×C3—201no—
499C22×C3—404no—
500C22×C3C22×D2011yescharacteristic
501C22×C3C2×D4×D512yes—
502C3×C4—22no—
503C3×C4—44no—
504C3×C4—104no—
505C3×C4—202no—
506C3×C4—201no—
507C3×C4#402811yescharacteristic
508C3×C4#402812yes—
509C3×C4#402811yescharacteristic
510C2×C5—21no—
511C2×C5—22no—
512C2×C5—42no—
513C2×C5C2×C3×#12011yescharacteristic
514D5—42no—
515D5—44no—
516C23—54no—
517C23—104no—
518C23—101no—
519C23—102no—
520C23—202no—
521C23C2×C3×D2012yes—
522C23C23×C3×D511yescharacteristic
523Q8—22no—
524Q8—104no—
525Q8—102no—
526Q8—102no—
527D4—24no—
528D4—22no—
529D4—104no—
530D4—108no—
531D4—104no—
532D4—104no—
533D4—102no—
534D4—102no—
535D4—208no—
536D4—204no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—54no—
541C2×C4—108no—
542C2×C4—104no—
543C2×C4—104no—
544C2×C4—104no—
545C2×C4—102no—
546C2×C4—101no—
547C2×C4—102no—
548C2×C4—201no—
549C2×C4—201no—
550C2×C4—202no—
551C2×C4C2×C3×D2012yes—
552C2×C4C2×C3×D2011yescharacteristic
553C2×C4C2×C3×D2011yescharacteristic
554C2×C4C3×D4×D512yes—
555C2×C4C3×D4×D512yes—
556C2×C3—22no—
557C2×C3—21no—
558C2×C3—42no—
559C2×C3—202no—
560C2×C3—204no—
561C2×C3C2×#402811yescharacteristic
562C5C3×#302511yescharacteristic, complemented, Sylow
563C22—22no—
564C22—41no—
565C22—42no—
566C22—104no—
567C22—102no—
568C22—104no—
569C22—204no—
570C22—202no—
571C22—201no—
572C22—201no—
573C22—404no—
574C22C22×C3×D2011yescharacteristic
575C22C2×C3×D4×D512yes—
576C4—22no—
577C4—44no—
578C4—104no—
579C4—202no—
580C4—201no—
581C4C3×#402811yescharacteristic
582C4C3×#402811yescharacteristic
583C4C3×#402812yes—
584C3#1061042111yescharacteristic, direct product factor, Sylow
585C2—22no—
586C2—21no—
587C2—42no—
588C2—202no—
589C2—204no—
590C2C2×C3×#402811yescharacteristic
subgroupG/H
centerC3×C4#4028
commutator, cosocleC23×C5C24×C3
FrattiniC23C23×C3×D5
FittingC3×C5×#410C2
socleC2×C3×C5C2×#120
hypercenterC3×#410D5
hypocenterC5C3×#3025
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
1. C3×#66283C2semidirect
2. C3×#66162C2semidirect
3. C3×#65858C2semidirect
4. C3×C5×#410C2semidirect
5. C3×#65295C2semidirect
6. C3×#65031C2semidirect
7. C3×#65030C2semidirect
8. #10610421C3direct (N × H)
18. C22×C3×C4×D5C22semidirect
19. C3×C5×#117C22semidirect
22. C3×#4083C22semidirect
23. C3×#4081C22semidirect
25. C3×#4072C22semidirect
28. C3×#4032C22semidirect
29. C3×#4028C22semidirect
31. C3×C5×#99C22semidirect
32. C3×#3938C22semidirect
34. #66283C2×C3semidirect
35. #66162C2×C3semidirect
36. #65858C2×C3semidirect
37. C5×#410C2×C3semidirect
38. #65295C2×C3semidirect
39. #65031C2×C3semidirect
40. #65030C2×C3semidirect
44. C2×C3×C5×Q8D4semidirect
45. C2×C3×C5×D4D4semidirect
47. C22×C3×C4×C5D4semidirect
68. C2×C3×D20D4semidirect
74. C2×C3×C4×D5D4semidirect
76. C2×C3×Dic10D4semidirect
95. C3×#410D5semidirect
105. C22×C4×D5C22×C3semidirect
106. C5×#117C22×C3semidirect
109. #4083C22×C3semidirect
110. #4081C22×C3semidirect
112. #4072C22×C3semidirect
115. #4032C22×C3semidirect
116. #4028C22×C3semidirect
118. C5×#99C22×C3semidirect
119. #3938C22×C3semidirect
182. C3×#117C2×D5semidirect
184. C3×#99C2×D5semidirect
188. C2×C5×Q8C3×D4semidirect
189. C2×C5×D4C3×D4semidirect
191. C22×C4×C5C3×D4semidirect
212. C2×D20C3×D4semidirect
218. C2×C4×D5C3×D4semidirect
220. C2×Dic10C3×D4semidirect
239. #410C3×D5semidirect
277. C2×C3×Q8D20semidirect
286. C2×C3×D4D20semidirect
295. C22×C3×C4D20semidirect
370. #117C2×C3×D5semidirect
372. #99C2×C3×D5semidirect
456. C2×Q8C3×D20semidirect
465. C2×D4C3×D20semidirect
474. C22×C4C3×D20semidirect
488. C3×C5#3025semidirect
562. C5C3×#3025semidirect
584. C3#10610421direct (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×#661622
C3×#658582
C3×C5×#4102
C3×#652952
C3×#650312
C3×#650302
#106104213
C3×#30255