Finite Group Databasebeta

← C3×#10599755 #14301616

Subgroups — C3×#10599755 #14301616

Order 1920.

1156 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4028C211yesmaximal, characteristic
2C3×#65820C211yesmaximal, characteristic, complemented
3C3×#65244C211yesmaximal, characteristic, complemented
4C3×#65244C211yesmaximal, characteristic, complemented
5C2×C3×#3938C211yesmaximal, characteristic, complemented
6C3×C5×#342C211yesmaximal, characteristic, complemented
7C3×#64747C211yesmaximal, characteristic, complemented
8#10599755C311yesmaximal, characteristic, direct product factor, Hall
9C24×C3×D5C2211yescharacteristic
10C22×C3×C22⋊D5C2211yescharacteristic
11C22×C3×D20—21no—
12C2×C3×C5×C22⋊C4C2211yescharacteristic
13C2×C3×C5×C22⋊C4C2211yescharacteristic
14C2×C3×C5×C22⋊C4C2211yescharacteristic
15C2×C3×C22⋊Dic5C2211yescharacteristic
16C2×C3×C22⋊Dic5C2211yescharacteristic
17C3×C4×C22⋊D5—22no—
18C2×C3×#685—21no—
19C2×C3×#685—22no—
20C2×C3×#685—22no—
21C3×#4028—22no—
22C3×#4028C414yescomplemented
23C3×#4027—22no—
24C3×D5×C22⋊C4—21no—
25C3×#4023—21no—
26C3×#3938—22no—
27C3×#1889—51nomaximal, Hall
28C2×#4028C2×C311yescharacteristic
29#65820C2×C311yescharacteristic, complemented
30#65244C2×C311yescharacteristic, complemented
31#65244C2×C311yescharacteristic, complemented
32C2×#3938C2×C311yescharacteristic, complemented
33C5×#342C2×C311yescharacteristic, complemented
34#64747C2×C311yescharacteristic, complemented
35C24×C3×C5C2311yescharacteristic
36C23×C3×D5—21no—
37C23×C3×D5—21no—
38C23×C3×D5—21no—
39C23×C3×D5—21no—
40C23×C3×D5—21no—
41C23×C3×D5—21no—
42C23×C3×D5—44no—
43C23×C3×D5C2×C412yescomplemented
44C23×C3×D5D411yescharacteristic
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5D412yescomplemented
48C2×C3×C22⋊D5—21no—
49C2×C3×C22⋊D5—21no—
50C2×C3×C22⋊D5—21no—
51C2×C3×C22⋊D5—21no—
52C2×C3×C22⋊D5C2×C412yes—
53C2×C3×C22⋊D5D412yes—
54C22×C3×Dic5—21no—
55C22×C3×Dic5D411yescharacteristic
56C22×C3×Dic5D412yescomplemented
57C2×C3×D20—24no—
58C2×C3×D20—44no—
59C2×C3×C4×D5—22no—
60C3×C5×C22⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C5×C22⋊C4—21no—
63C3×C5×C22⋊C4—21no—
64C3×C5×C22⋊C4—21no—
65C3×C5×C22⋊C4C2×C412yescomplemented
66C3×C22⋊Dic5—21no—
67C3×C22⋊Dic5—21no—
68C3×C22⋊Dic5—21no—
69C3×C22⋊Dic5—21no—
70C3×#685—22no—
71C3×#685—24no—
72C3×#685—44no—
73C3×#685—44no—
74C3×#683—22no—
75C3×C4×Dic5—22no—
76C2×C3×#120—51no—
77C3×#518—51no—
78C2×C3×#99—51no—
79C3×#379—51no—
80C3×#379—51no—
81C3×#342—51no—
82C3×#342D511yescharacteristic, complemented
83C24×D5C22×C311yescharacteristic
84C22×C22⋊D5C22×C311yescharacteristic
85C22×D20—21no—
86C2×C5×C22⋊C4C22×C311yescharacteristic
87C2×C5×C22⋊C4C22×C311yescharacteristic
88C2×C5×C22⋊C4C22×C311yescharacteristic
89C2×C22⋊Dic5C22×C311yescharacteristic
90C2×C22⋊Dic5C22×C311yescharacteristic
91C4×C22⋊D5—22no—
92C2×#685—21no—
93C2×#685—22no—
94C2×#685—22no—
95#4028—22no—
96#4028C3×C414yescomplemented
97#4027—22no—
98D5×C22⋊C4—21no—
99#4023—21no—
100#3938—22no—
101#1889—51noSylow
102C23×C3×C5—42no—
103C23×C3×C5C22×C411yescharacteristic
104C23×C3×C5C2×D411yescharacteristic
105C23×C3×C5C2×D411yescharacteristic
106C23×C3×C5C2×D411yescharacteristic
107C23×C3×C5C2×D411yescharacteristic
108C23×C3×C5#4111yescharacteristic
109C23×C3×C5#4111yescharacteristic
110C22×C3×D5—24no—
111C22×C3×D5—24no—
112C22×C3×D5—24no—
113C22×C3×D5—22no—
114C22×C3×D5—22no—
115C22×C3×D5—22no—
116C22×C3×D5—21no—
117C22×C3×D5—21no—
118C22×C3×D5—44no—
119C22×C3×D5—44no—
120C22×C3×D5—44no—
121C22×C3×D5—44no—
122C22×C3×D5—42no—
123C22×C3×D5—42no—
124C22×C3×D5—42no—
125C22×C3×D5—41no—
126C22×C3×D5—41no—
127C22×C3×D5—41no—
128C22×C3×D5—41no—
129C22×C3×D5—42no—
130C22×C3×D5C22⋊C412yescomplemented
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—42no—
134C2×C3×C4×C5—42no—
135C2×C3×C4×C5—42no—
136C2×C3×C4×C5—41no—
137C3×C22⋊D5—22no—
138C3×C22⋊D5—22no—
139C3×C22⋊D5—22no—
140C3×C22⋊D5—22no—
141C2×C3×Dic5—21no—
142C2×C3×Dic5—21no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—42no—
145C2×C3×Dic5—42no—
146C2×C3×Dic5—41no—
147C2×C3×Dic5C22⋊C412yescomplemented
148C3×D20—48no—
149C3×C4×D5—82no—
150C25×C3—51no—
151C22×C3×D4—51no—
152C22×C3×D4—101no—
153C3×#120—54no—
154C3×#120—102no—
155C3×C4×D4—102no—
156C3×C4×D4—102no—
157C3×#117—101no—
158C2×C3×C22⋊C4—51no—
159C2×C3×C22⋊C4—51no—
160C2×C3×C22⋊C4—101no—
161C2×C3×C22⋊C4—101no—
162C2×C3×C22⋊C4—102no—
163C2×C3×C22⋊C4—102no—
164C2×C3×C22⋊C4C2×D511yescharacteristic
165C2×C3×C22⋊C4C2×D511yescharacteristic
166C2×C3×C22⋊C4C2×D511yescharacteristic
167C3×#99—102no—
168C24×C5C23×C311yescharacteristic
169C23×D5—21no—
170C23×D5—21no—
171C23×D5—21no—
172C23×D5—21no—
173C23×D5—21no—
174C23×D5—21no—
175C23×D5—44no—
176C23×D5C2×C3×C412yescomplemented
177C23×D5C3×D411yescharacteristic
178C22×C4×C5—21no—
179C22×C4×C5—21no—
180C22×C4×C5C3×D412yescomplemented
181C2×C22⋊D5—21no—
182C2×C22⋊D5—21no—
183C2×C22⋊D5—21no—
184C2×C22⋊D5—21no—
185C2×C22⋊D5C2×C3×C412yes—
186C2×C22⋊D5C3×D412yes—
187C22×Dic5—21no—
188C22×Dic5C3×D411yescharacteristic
189C22×Dic5C3×D412yescomplemented
190C2×D20—24no—
191C2×D20—44no—
192C2×C4×D5—22no—
193C5×C22⋊C4—21no—
194C5×C22⋊C4—21no—
195C5×C22⋊C4—21no—
196C5×C22⋊C4—21no—
197C5×C22⋊C4—21no—
198C5×C22⋊C4C2×C3×C412yescomplemented
199C22⋊Dic5—21no—
200C22⋊Dic5—21no—
201C22⋊Dic5—21no—
202C22⋊Dic5—21no—
203#685—22no—
204#685—24no—
205#685—44no—
206#685—44no—
207#683—22no—
208C4×Dic5—22no—
209C2×#120—51no—
210#518—51no—
211C2×#99—51no—
212#379—51no—
213#379—51no—
214#342—51no—
215#342C3×D511yescharacteristic, complemented
216C22×C3×C5—22no—
217C22×C3×C5—22no—
218C22×C3×C5—22no—
219C22×C3×C5—41no—
220C22×C3×C5—41no—
221C22×C3×C5—41no—
222C22×C3×C5—41no—
223C22×C3×C5C2×C22⋊C411yescharacteristic
224C22×C3×C5C4×D411yescharacteristic
225C22×C3×C5C4×D411yescharacteristic
226C22×C3×C5#12011yescharacteristic
227C22×C3×C5#12111yescharacteristic
228C22×C3×C5#12111yescharacteristic
229C22×C3×C5#12311yescharacteristic
230C2×C3×D5—24no—
231C2×C3×D5—24no—
232C2×C3×D5—24no—
233C2×C3×D5—22no—
234C2×C3×D5—44no—
235C2×C3×D5—42no—
236C2×C3×D5—44no—
237C2×C3×D5—42no—
238C2×C3×D5—42no—
239C2×C3×D5—44no—
240C2×C3×D5—42no—
241C2×C3×D5—42no—
242C2×C3×D5—44no—
243C2×C3×D5—42no—
244C2×C3×D5—81no—
245C3×C4×C5—42no—
246C3×C4×C5—42no—
247C3×C4×C5—81no—
248C3×Dic5—22no—
249C3×Dic5—42no—
250C3×Dic5—81no—
251C24×C3—51no—
252C24×C3—52no—
253C24×C3—101no—
254C24×C3—101no—
255C24×C3—101no—
256C24×C3—101no—
257C24×C3—101no—
258C24×C3—101no—
259C24×C3—204no—
260C24×C3C22×D511yescharacteristic
261C2×C3×D4—52no—
262C2×C3×D4—52no—
263C2×C3×D4—104no—
264C2×C3×D4—101no—
265C2×C3×D4—101no—
266C2×C3×D4—101no—
267C2×C3×D4—101no—
268C2×C3×D4—204no—
269C22×C3×C4—21no—
270C22×C3×C4—21no—
271C22×C3×C4—51no—
272C22×C3×C4—52no—
273C22×C3×C4—102no—
274C22×C3×C4—101no—
275C22×C3×C4D2012yescomplemented
276C3×C4⋊C4—102no—
277C3×C22⋊C4—21no—
278C3×C22⋊C4—21no—
279C3×C22⋊C4—21no—
280C3×C22⋊C4—21no—
281C3×C22⋊C4—21no—
282C3×C22⋊C4—104no—
283C3×C22⋊C4—102no—
284C3×C22⋊C4—101no—
285C3×C22⋊C4—101no—
286C3×C22⋊C4—101no—
287C3×C22⋊C4—101no—
288C3×C22⋊C4—204no—
289C3×C22⋊C4—204no—
290C3×C22⋊C4C4×D512yescomplemented
291C3×C42—102no—
292C23×C5—42no—
293C23×C5C22×C3×C411yescharacteristic
294C23×C5C2×C3×D411yescharacteristic
295C23×C5C2×C3×D411yescharacteristic
296C23×C5C2×C3×D411yescharacteristic
297C23×C5C2×C3×D411yescharacteristic
298C23×C5C3×#4111yescharacteristic
299C23×C5C3×#4111yescharacteristic
300C22×D5—24no—
301C22×D5—24no—
302C22×D5—24no—
303C22×D5—22no—
304C22×D5—22no—
305C22×D5—22no—
306C22×D5—21no—
307C22×D5—21no—
308C22×D5—44no—
309C22×D5—44no—
310C22×D5—44no—
311C22×D5—44no—
312C22×D5—42no—
313C22×D5—42no—
314C22×D5—42no—
315C22×D5—41no—
316C22×D5—41no—
317C22×D5—41no—
318C22×D5—41no—
319C22×D5—42no—
320C22×D5C3×C22⋊C412yescomplemented
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—42no—
324C2×C4×C5—42no—
325C2×C4×C5—42no—
326C2×C4×C5—41no—
327C22⋊D5—22no—
328C22⋊D5—22no—
329C22⋊D5—22no—
330C22⋊D5—22no—
331C2×Dic5—21no—
332C2×Dic5—21no—
333C2×Dic5—22no—
334C2×Dic5—42no—
335C2×Dic5—42no—
336C2×Dic5—41no—
337C2×Dic5C3×C22⋊C412yescomplemented
338D20—48no—
339C4×D5—82no—
340C25—51no—
341C22×D4—51no—
342C22×D4—101no—
343#120—54no—
344#120—102no—
345C4×D4—102no—
346C4×D4—102no—
347#117—101no—
348C2×C22⋊C4—51no—
349C2×C22⋊C4—51no—
350C2×C22⋊C4—101no—
351C2×C22⋊C4—101no—
352C2×C22⋊C4—102no—
353C2×C22⋊C4—102no—
354C2×C22⋊C4C2×C3×D511yescharacteristic
355C2×C22⋊C4C2×C3×D511yescharacteristic
356C2×C22⋊C4C2×C3×D511yescharacteristic
357#99—102no—
358C2×C3×C5—21no—
359C2×C3×C5—21no—
360C2×C3×C5—21no—
361C2×C3×C5—21no—
362C2×C3×C5—21no—
363C2×C3×C5—21no—
364C2×C3×C5#38611yescharacteristic
365C2×C3×C5B(4,𝔽2)12yes—
366C3×D5—44no—
367C3×D5—42no—
368C23×C3—42no—
369C23×C3—52no—
370C23×C3—104no—
371C23×C3—104no—
372C23×C3—104no—
373C23×C3—102no—
374C23×C3—102no—
375C23×C3—101no—
376C23×C3—102no—
377C23×C3—101no—
378C23×C3—202no—
379C23×C3—202no—
380C23×C3—202no—
381C23×C3—202no—
382C23×C3—204no—
383C23×C3—204no—
384C23×C3—204no—
385C23×C3—204no—
386C23×C3—201no—
387C23×C3—201no—
388C23×C3—201no—
389C23×C3—201no—
390C23×C3C2×C4×D511yescharacteristic
391C23×C3C2×D2011yescharacteristic
392C23×C3#70911yescharacteristic
393C23×C3D4×D511yescharacteristic
394C23×C3D4×D511yescharacteristic
395C23×C3D4×D511yescharacteristic
396C23×C3#71311yescharacteristic
397C3×D4—102no—
398C3×D4—102no—
399C3×D4—102no—
400C3×D4—102no—
401C3×D4—208no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—41no—
408C2×C3×C4—52no—
409C2×C3×C4—101no—
410C2×C3×C4—101no—
411C2×C3×C4—102no—
412C2×C3×C4—202no—
413C2×C3×C4—202no—
414C2×C3×C4—201no—
415C2×C3×C4—402no—
416C22×C5—22no—
417C22×C5—22no—
418C22×C5—22no—
419C22×C5—41no—
420C22×C5—41no—
421C22×C5—41no—
422C22×C5—41no—
423C22×C5C2×C3×C22⋊C411yescharacteristic
424C22×C5C3×C4×D411yescharacteristic
425C22×C5C3×C4×D411yescharacteristic
426C22×C5C3×#12011yescharacteristic
427C22×C5C3×#12111yescharacteristic
428C22×C5C3×#12111yescharacteristic
429C22×C5C3×#12311yescharacteristic
430C2×D5—24no—
431C2×D5—24no—
432C2×D5—24no—
433C2×D5—22no—
434C2×D5—44no—
435C2×D5—42no—
436C2×D5—44no—
437C2×D5—42no—
438C2×D5—42no—
439C2×D5—44no—
440C2×D5—42no—
441C2×D5—42no—
442C2×D5—44no—
443C2×D5—42no—
444C2×D5—81no—
445C4×C5—42no—
446C4×C5—42no—
447C4×C5—81no—
448Dic5—22no—
449Dic5—42no—
450Dic5—81no—
451C24—51no—
452C24—52no—
453C24—101no—
454C24—101no—
455C24—101no—
456C24—101no—
457C24—101no—
458C24—101no—
459C24—204no—
460C24C22×C3×D511yescharacteristic
461C2×D4—52no—
462C2×D4—52no—
463C2×D4—104no—
464C2×D4—101no—
465C2×D4—101no—
466C2×D4—101no—
467C2×D4—101no—
468C2×D4—204no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—51no—
472C22×C4—52no—
473C22×C4—102no—
474C22×C4—101no—
475C22×C4C3×D2012yescomplemented
476C4⋊C4—102no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—104no—
483C22⋊C4—102no—
484C22⋊C4—101no—
485C22⋊C4—101no—
486C22⋊C4—101no—
487C22⋊C4—101no—
488C22⋊C4—204no—
489C22⋊C4—204no—
490C22⋊C4C3×C4×D512yescomplemented
491C42—102no—
492C3×C5#188911yescharacteristic, complemented, Hall
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—104no—
501C22×C3—104no—
502C22×C3—104no—
503C22×C3—102no—
504C22×C3—202no—
505C22×C3—202no—
506C22×C3—204no—
507C22×C3—204no—
508C22×C3—202no—
509C22×C3—202no—
510C22×C3—204no—
511C22×C3—202no—
512C22×C3—204no—
513C22×C3—202no—
514C22×C3—401no—
515C22×C3C4×D2011yescharacteristic
516C22×C3D5×C22⋊C411yescharacteristic
517C22×C3#402811yescharacteristic
518C22×C3#403011yescharacteristic
519C22×C3#403911yescharacteristic
520C22×C3#404011yescharacteristic
521C22×C3#404111yescharacteristic
522C3×C4—42no—
523C3×C4—42no—
524C3×C4—81no—
525C3×C4—102no—
526C3×C4—202no—
527C3×C4—401no—
528C2×C5—21no—
529C2×C5—21no—
530C2×C5—21no—
531C2×C5—21no—
532C2×C5—21no—
533C2×C5—21no—
534C2×C5C3×#38611yescharacteristic
535C2×C5C3×B(4,𝔽2)12yes—
536D5—44no—
537D5—42no—
538C23—42no—
539C23—52no—
540C23—104no—
541C23—104no—
542C23—104no—
543C23—102no—
544C23—102no—
545C23—101no—
546C23—102no—
547C23—101no—
548C23—202no—
549C23—202no—
550C23—202no—
551C23—202no—
552C23—204no—
553C23—204no—
554C23—204no—
555C23—204no—
556C23—201no—
557C23—201no—
558C23—201no—
559C23—201no—
560C23C2×C3×C4×D511yescharacteristic
561C23C2×C3×D2011yescharacteristic
562C23C3×#70911yescharacteristic
563C23C3×D4×D511yescharacteristic
564C23C3×D4×D511yescharacteristic
565C23C3×D4×D511yescharacteristic
566C23C3×#71311yescharacteristic
567D4—102no—
568D4—102no—
569D4—102no—
570D4—102no—
571D4—208no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—52no—
579C2×C4—101no—
580C2×C4—101no—
581C2×C4—102no—
582C2×C4—202no—
583C2×C4—202no—
584C2×C4—201no—
585C2×C4—402no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—204no—
593C2×C3—202no—
594C2×C3#6495411yescharacteristic
595C2×C3#6503012yes—
596C5C3×#188911yescharacteristic, complemented, Sylow
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—104no—
605C22—104no—
606C22—104no—
607C22—102no—
608C22—202no—
609C22—202no—
610C22—204no—
611C22—204no—
612C22—202no—
613C22—202no—
614C22—204no—
615C22—202no—
616C22—204no—
617C22—202no—
618C22—401no—
619C22C3×C4×D2011yescharacteristic
620C22C3×D5×C22⋊C411yescharacteristic
621C22C3×#402811yescharacteristic
622C22C3×#403011yescharacteristic
623C22C3×#403911yescharacteristic
624C22C3×#404011yescharacteristic
625C22C3×#404111yescharacteristic
626C4—42no—
627C4—42no—
628C4—81no—
629C4—102no—
630C4—202no—
631C4—401no—
632C3#1059975511yescharacteristic, direct product factor, Sylow
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—204no—
640C2—202no—
641C2C3×#6495411yescharacteristic
642C2C3×#6503012yes—
subgroupG/H
centerC22×C3#4028
commutatorC23×C5C22×C3×C4
FrattiniC24C22×C3×D5
FittingC3×C5×#342C2
socleC22×C3×C5#120
prime radicalC2×C3×#4028C2
cosocleC24×C5C23×C3
hypercenterC3×#342D5
hypocenterC5C3×#1889
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C3×#342

Complements

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

NH ≅ G/Ntype
2. C3×#65820C2semidirect
3. C3×#65244C2semidirect
4. C3×#65244C2semidirect
5. C2×C3×#3938C2semidirect
6. C3×C5×#342C2semidirect
7. C3×#64747C2semidirect
8. #10599755C3direct (N × H)
22. C3×#4028C4semidirect
29. #65820C2×C3semidirect
30. #65244C2×C3semidirect
31. #65244C2×C3semidirect
32. C2×#3938C2×C3semidirect
33. C5×#342C2×C3semidirect
34. #64747C2×C3semidirect
43. C23×C3×D5C2×C4semidirect
47. C22×C3×C4×C5D4semidirect
56. C22×C3×Dic5D4semidirect
65. C3×C5×C22⋊C4C2×C4semidirect
82. C3×#342D5semidirect
96. #4028C3×C4semidirect
130. C22×C3×D5C22⋊C4semidirect
147. C2×C3×Dic5C22⋊C4semidirect
176. C23×D5C2×C3×C4semidirect
180. C22×C4×C5C3×D4semidirect
189. C22×Dic5C3×D4semidirect
198. C5×C22⋊C4C2×C3×C4semidirect
215. #342C3×D5semidirect
275. C22×C3×C4D20semidirect
290. C3×C22⋊C4C4×D5semidirect
320. C22×D5C3×C22⋊C4semidirect
337. C2×Dic5C3×C22⋊C4semidirect
475. C22×C4C3×D20semidirect
490. C22⋊C4C3×C4×D5semidirect
492. C3×C5#1889semidirect
596. C5C3×#1889semidirect
632. C3#10599755direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40282
C3×#658202
C3×#652442
C2×C3×#39382
C3×C5×#3422
C3×#647472
#105997553
C3×#18895