Finite Group Databasebeta

← C3×#10610598 #14416842

Subgroups — C3×#10610598 #14416842

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×Dic5C211yesmaximal, characteristic
2C2×C3×#4097C211yesmaximal, characteristic
3C2×C3×#4033C211yesmaximal, characteristic, complemented
4C3×C5×#417C211yesmaximal, characteristic
5C3×#65409C211yesmaximal, characteristic, complemented
6C3×M16×Dic5C211yesmaximal, characteristic, complemented
7C3×#65253C211yesmaximal, characteristic, complemented
8C3×#65079C212yesmaximal, complemented
9C3×#65078C212yesmaximal, complemented
10C3×#65047C212yesmaximal, complemented
11C3×#65046C212yesmaximal, complemented
12#10610598C311yesmaximal, characteristic, direct product factor, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C2×C3×C5×M16C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C3×#4097C418yescomplemented
18C3×Q8×Dic5C2211yescharacteristic
19C3×Q8×Dic5C2211yescharacteristic
20C2×C3×#689C2212yes—
21C3×D4×Dic5—21no—
22C3×D4×Dic5—21no—
23C3×D4×Dic5C2211yescharacteristic
24C3×D4×Dic5C2211yescharacteristic
25C2×C3×#687C2212yes—
26C3×#4072—21no—
27C3×#4072C2211yescharacteristic
28C2×C3×#683—21no—
29C2×C3×C4×Dic5—21no—
30C2×C3×C4×Dic5—21no—
31C2×C3×C4×Dic5C2211yescharacteristic
32C2×C3×#681C2211yescharacteristic
33C3×#4033—21no—
34C3×#4033—21no—
35C3×#4033C2212yes—
36C3×#4033C2212yes—
37C3×C5×#103C2212yes—
38C3×C5×#102C2212yes—
39C3×#3948C2212yes—
40C3×#3948C2212yes—
41C3×#3947C2212yes—
42C3×C8×Dic5C2212yes—
43C3×#3940C2212yes—
44C3×#3939C2212yes—
45C3×C4×#137—51nomaximal, Hall
46#41×Dic5C2×C311yescharacteristic
47C2×#4097C2×C311yescharacteristic
48C2×#4033C2×C311yescharacteristic, complemented
49C5×#417C2×C311yescharacteristic
50#65409C2×C311yescharacteristic, complemented
51M16×Dic5C2×C311yescharacteristic, complemented
52#65253C2×C311yescharacteristic, complemented
53#65079C2×C312yescomplemented
54#65078C2×C312yescomplemented
55#65047C2×C312yescomplemented
56#65046C2×C312yescomplemented
57C3×C5×#41—22no—
58C3×C5×#41C2×C414yes—
59C2×C3×C5×Q8C2311yescharacteristic
60C2×C3×C5×D4—21no—
61C2×C3×C5×D4C2311yescharacteristic
62C22×C3×C4×C5—21no—
63C22×C3×C4×C5—21no—
64C22×C3×C4×C5C2311yescharacteristic
65C22×C3×Dic5—21no—
66C22×C3×Dic5—21no—
67C22×C3×Dic5—21no—
68C22×C3×Dic5D411yescharacteristic
69C22×C3×Dic5D411yescharacteristic
70C2×C3×Dic10—21no—
71C2×C3×Dic10—22no—
72C2×C3×Dic10C2×C414yescomplemented
73C2×C3×Dic10C2312yes—
74C3×C5×M16—22no—
75C2×C3×C5×C8C2312yes—
76C3×C5×C4⋊C4—21no—
77C3×C5×C4⋊C4C2312yes—
78C3×C22⋊Dic5—21no—
79C3×C22⋊Dic5—21no—
80C3×C22⋊Dic5—21no—
81C3×#689C2×C418yescomplemented
82C3×#687C2×C418yescomplemented
83C3×C4⋊Dic5—21no—
84C3×C4⋊Dic5—21no—
85C3×C4⋊Dic5C2311yescharacteristic
86C3×C4⋊Dic5C2311yescharacteristic
87C3×#683—22no—
88C3×#683—22no—
89C3×C4×Dic5—21no—
90C3×C4×Dic5—21no—
91C3×C4×Dic5—21no—
92C3×C4×Dic5—22no—
93C3×C4×Dic5—22no—
94C3×C4×Dic5D411yescharacteristic
95C3×C4×Dic5D411yescharacteristic
96C3×C4×Dic5C2311yescharacteristic
97C3×C4×Dic5C2311yescharacteristic
98C3×#681C2×C414yescomplemented
99C2×C3×#168C2312yes—
100C2×C3×#137—51no—
101C3×C4×#41—51no—
102C2×C3×C4×Q8—51no—
103C3×#441—52no—
104C3×#440—52no—
105C3×C4×Q16—52no—
106C3×C4×SD16—52no—
107C3×#428—51no—
108C3×#419—51no—
109C3×#417D511yescharacteristic
110C3×C4×M16—51no—
111C2×C5×#41C22×C311yescharacteristic
112C22×Dic10C22×C311yescharacteristic
113C2×C5×M16C22×C311yescharacteristic
114C2×C5×C4⋊C4C22×C311yescharacteristic
115#4097C3×C418yescomplemented
116Q8×Dic5C22×C311yescharacteristic
117Q8×Dic5C22×C311yescharacteristic
118C2×#689C22×C312yes—
119D4×Dic5—21no—
120D4×Dic5—21no—
121D4×Dic5C22×C311yescharacteristic
122D4×Dic5C22×C311yescharacteristic
123C2×#687C22×C312yes—
124#4072—21no—
125#4072C22×C311yescharacteristic
126C2×#683—21no—
127C2×C4×Dic5—21no—
128C2×C4×Dic5—21no—
129C2×C4×Dic5C22×C311yescharacteristic
130C2×#681C22×C311yescharacteristic
131#4033—21no—
132#4033—21no—
133#4033C22×C312yes—
134#4033C22×C312yes—
135C5×#103C22×C312yes—
136C5×#102C22×C312yes—
137#3948C22×C312yes—
138#3948C22×C312yes—
139#3947C22×C312yes—
140C8×Dic5C22×C312yes—
141#3940C22×C312yes—
142#3939C22×C312yes—
143C4×#137—51noSylow
144C23×C3×C5—21no—
145C23×C3×C5C2×D411yescharacteristic
146C3×C5×Q8—21no—
147C3×C5×Q8C22×C412yes—
148C3×C5×D4—24no—
149C3×C5×D4—21no—
150C3×C5×D4C22×C412yes—
151C2×C3×C4×C5—21no—
152C2×C3×C4×C5—24no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—42no—
155C2×C3×C4×C5C22×C412yes—
156C2×C3×C4×C5C2×D411yescharacteristic
157C2×C3×C4×C5#4112yes—
158C2×C3×C4×C5C2411yescharacteristic
159C2×C3×Dic5—24no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5—21no—
163C2×C3×Dic5—21no—
164C2×C3×Dic5—41no—
165C2×C3×Dic5—42no—
166C2×C3×Dic5—42no—
167C2×C3×Dic5C2×D411yescharacteristic
168C2×C3×Dic5C2×D411yescharacteristic
169C2×C3×Dic5C2×D411yescharacteristic
170C2×C3×Dic5C2×D411yescharacteristic
171C3×Dic10—24no—
172C3×Dic10—22no—
173C3×Dic10C22×C414yes—
174C3×C5×C8—22no—
175C3×#168C22×C414yes—
176C2×C3×#41C2×D511yescharacteristic
177C22×C3×Q8—51no—
178C3×#137—58no—
179C2×C3×Q16—52no—
180C2×C3×SD16—52no—
181C2×C3×M16—51no—
182C2×C3×M16C2×D511yescharacteristic
183C3×C4×Q8—51no—
184C3×C4×Q8—51no—
185C3×C4×Q8—52no—
186C3×C4×Q8—52no—
187C3×C4×Q8—101no—
188C3×C4×Q8—101no—
189C3×C4×D4—51no—
190C3×C4×D4—51no—
191C3×C4×D4—101no—
192C3×C4×D4—101no—
193C3×#117—51no—
194C3×#117—101no—
195C2×C3×C4⋊C4—101no—
196C2×C3×C4⋊C4C2×D511yescharacteristic
197C2×C3×C42—51no—
198C2×C3×C42—101no—
199C2×C3×C42—101no—
200C3×#107—52no—
201C3×#106—52no—
202C3×#103—52no—
203C3×#103—52no—
204C3×#103C2×D512yes—
205C3×#102C2×D512yes—
206C3×#97—52no—
207C3×C4×C8—52no—
208C5×#41—22no—
209C5×#41C2×C3×C414yes—
210C2×C5×Q8C23×C311yescharacteristic
211C2×C5×D4—21no—
212C2×C5×D4C23×C311yescharacteristic
213C22×C4×C5—21no—
214C22×C4×C5—21no—
215C22×C4×C5C23×C311yescharacteristic
216C22×Dic5—21no—
217C22×Dic5—21no—
218C22×Dic5—21no—
219C22×Dic5C3×D411yescharacteristic
220C22×Dic5C3×D411yescharacteristic
221C2×Dic10—21no—
222C2×Dic10—22no—
223C2×Dic10C2×C3×C414yescomplemented
224C2×Dic10C23×C312yes—
225C5×M16—22no—
226C2×C5×C8C23×C312yes—
227C5×C4⋊C4—21no—
228C5×C4⋊C4C23×C312yes—
229C22⋊Dic5—21no—
230C22⋊Dic5—21no—
231C22⋊Dic5—21no—
232#689C2×C3×C418yescomplemented
233#687C2×C3×C418yescomplemented
234C4⋊Dic5—21no—
235C4⋊Dic5—21no—
236C4⋊Dic5C23×C311yescharacteristic
237C4⋊Dic5C23×C311yescharacteristic
238#683—22no—
239#683—22no—
240C4×Dic5—21no—
241C4×Dic5—22no—
242C4×Dic5—22no—
243C4×Dic5—21no—
244C4×Dic5—21no—
245C4×Dic5C3×D411yescharacteristic
246C4×Dic5C3×D411yescharacteristic
247C4×Dic5C23×C311yescharacteristic
248C4×Dic5C23×C311yescharacteristic
249#681C2×C3×C414yescomplemented
250C2×#168C23×C312yes—
251C2×#137—51no—
252C4×#41—51no—
253C2×C4×Q8—51no—
254#441—52no—
255#440—52no—
256C4×Q16—52no—
257C4×SD16—52no—
258#428—51no—
259#419—51no—
260#417C3×D511yescharacteristic
261C4×M16—51no—
262C22×C3×C5—21no—
263C22×C3×C5—21no—
264C22×C3×C5—22no—
265C22×C3×C5—41no—
266C22×C3×C5—41no—
267C22×C3×C5C4×D412yes—
268C22×C3×C5C22×D411yescharacteristic
269C3×C4×C5—22no—
270C3×C4×C5—42no—
271C3×C4×C5C4×D412yes—
272C3×C4×C5C23×C411yescharacteristic
273C3×C4×C5C2×#4111yescharacteristic
274C3×Dic5—24no—
275C3×Dic5—21no—
276C3×Dic5—21no—
277C3×Dic5—21no—
278C3×Dic5—41no—
279C3×Dic5—41no—
280C3×Dic5#13712yescomplemented
281C3×#41—22no—
282C3×#41C4×D514yes—
283C2×C3×Q8—54no—
284C2×C3×Q8—52no—
285C2×C3×Q8—101no—
286C2×C3×Q8—102no—
287C2×C3×Q8C22×D511yescharacteristic
288C2×C3×D4—21no—
289C2×C3×D4C22×D511yescharacteristic
290C22×C3×C4—21no—
291C22×C3×C4—21no—
292C22×C3×C4—51no—
293C22×C3×C4—51no—
294C22×C3×C4—101no—
295C22×C3×C4—101no—
296C22×C3×C4—101no—
297C22×C3×C4C22×D511yescharacteristic
298C3×Q16—58no—
299C3×SD16—58no—
300C3×M16—22no—
301C3×M16—54no—
302C2×C3×C8—52no—
303C2×C3×C8C22×D512yes—
304C3×C4⋊C4—21no—
305C3×C4⋊C4—51no—
306C3×C4⋊C4—51no—
307C3×C4⋊C4—102no—
308C3×C4⋊C4—102no—
309C3×C4⋊C4—101no—
310C3×C4⋊C4—101no—
311C3×C4⋊C4C22×D512yes—
312C3×C22⋊C4—101no—
313C3×C22⋊C4—101no—
314C3×C22⋊C4—101no—
315C3×C42—51no—
316C3×C42—51no—
317C3×C42—51no—
318C3×C42—51no—
319C3×C42—101no—
320C3×C42—101no—
321C3×C42—102no—
322C3×C42—102no—
323C3×C42—101no—
324C23×C5—21no—
325C23×C5C2×C3×D411yescharacteristic
326C5×Q8—21no—
327C5×Q8C22×C3×C412yes—
328C5×D4—24no—
329C5×D4—21no—
330C5×D4C22×C3×C412yes—
331C2×C4×C5—21no—
332C2×C4×C5—24no—
333C2×C4×C5—22no—
334C2×C4×C5—42no—
335C2×C4×C5C22×C3×C412yes—
336C2×C4×C5C2×C3×D411yescharacteristic
337C2×C4×C5C3×#4112yes—
338C2×C4×C5C24×C311yescharacteristic
339C2×Dic5—24no—
340C2×Dic5—22no—
341C2×Dic5—22no—
342C2×Dic5—21no—
343C2×Dic5—21no—
344C2×Dic5—41no—
345C2×Dic5—42no—
346C2×Dic5—42no—
347C2×Dic5C2×C3×D411yescharacteristic
348C2×Dic5C2×C3×D411yescharacteristic
349C2×Dic5C2×C3×D411yescharacteristic
350C2×Dic5C2×C3×D411yescharacteristic
351Dic10—24no—
352Dic10—22no—
353Dic10C22×C3×C414yes—
354C5×C8—22no—
355#168C22×C3×C414yes—
356C2×#41C2×C3×D511yescharacteristic
357C22×Q8—51no—
358#137—58no—
359C2×Q16—52no—
360C2×SD16—52no—
361C2×M16—51no—
362C2×M16C2×C3×D511yescharacteristic
363C4×Q8—51no—
364C4×Q8—51no—
365C4×Q8—52no—
366C4×Q8—52no—
367C4×Q8—101no—
368C4×Q8—101no—
369C4×D4—51no—
370C4×D4—51no—
371C4×D4—101no—
372C4×D4—101no—
373#117—51no—
374#117—101no—
375C2×C4⋊C4—101no—
376C2×C4⋊C4C2×C3×D511yescharacteristic
377C2×C42—51no—
378C2×C42—101no—
379C2×C42—101no—
380#107—52no—
381#106—52no—
382#103—52no—
383#103—52no—
384#103C2×C3×D512yes—
385#102C2×C3×D512yes—
386#97—52no—
387C4×C8—52no—
388C2×C3×C5—22no—
389C2×C3×C5—42no—
390C2×C3×C5C2×C4×D411yescharacteristic
391C2×C3×C5C2×#13711yescharacteristic
392C2×C3×C5#57511yescharacteristic
393C23×C3—21no—
394C23×C3D4×D511yescharacteristic
395C3×Q8—21no—
396C3×Q8—54no—
397C3×Q8—102no—
398C3×Q8—104no—
399C3×Q8C2×C4×D512yes—
400C3×D4—24no—
401C3×D4—21no—
402C3×D4C2×C4×D512yes—
403C2×C3×C4—21no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—42no—
407C2×C3×C4—51no—
408C2×C3×C4—51no—
409C2×C3×C4—51no—
410C2×C3×C4—51no—
411C2×C3×C4—102no—
412C2×C3×C4—102no—
413C2×C3×C4—104no—
414C2×C3×C4—101no—
415C2×C3×C4—101no—
416C2×C3×C4—202no—
417C2×C3×C4—201no—
418C2×C3×C4—202no—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4D4×D511yescharacteristic
421C2×C3×C4#71112yes—
422C2×C3×C4C23×D511yescharacteristic
423C3×C8—22no—
424C3×C8—54no—
425C22×C5—21no—
426C22×C5—21no—
427C22×C5—22no—
428C22×C5—41no—
429C22×C5—41no—
430C22×C5C3×C4×D412yes—
431C22×C5C22×C3×D411yescharacteristic
432C4×C5—22no—
433C4×C5—42no—
434C4×C5C3×C4×D412yes—
435C4×C5C23×C3×C411yescharacteristic
436C4×C5C2×C3×#4111yescharacteristic
437Dic5—21no—
438Dic5—21no—
439Dic5—21no—
440Dic5—24no—
441Dic5—41no—
442Dic5—41no—
443Dic5C3×#13712yescomplemented
444#41—22no—
445#41C3×C4×D514yes—
446C2×Q8—54no—
447C2×Q8—52no—
448C2×Q8—101no—
449C2×Q8—102no—
450C2×Q8C22×C3×D511yescharacteristic
451C2×D4—21no—
452C2×D4C22×C3×D511yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—51no—
456C22×C4—51no—
457C22×C4—101no—
458C22×C4—101no—
459C22×C4—101no—
460C22×C4C22×C3×D511yescharacteristic
461Q16—58no—
462SD16—58no—
463M16—22no—
464M16—54no—
465C2×C8—52no—
466C2×C8C22×C3×D512yes—
467C4⋊C4—21no—
468C4⋊C4—51no—
469C4⋊C4—51no—
470C4⋊C4—102no—
471C4⋊C4—102no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4C22×C3×D512yes—
475C22⋊C4—101no—
476C22⋊C4—101no—
477C22⋊C4—101no—
478C42—51no—
479C42—51no—
480C42—51no—
481C42—51no—
482C42—101no—
483C42—101no—
484C42—102no—
485C42—102no—
486C42—101no—
487C3×C5C4×#13711yescharacteristic, complemented, Hall
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—22no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3#402712yes—
494C22×C3C2×D4×D511yescharacteristic
495C3×C4—22no—
496C3×C4—42no—
497C3×C4—52no—
498C3×C4—104no—
499C3×C4—101no—
500C3×C4—101no—
501C3×C4—101no—
502C3×C4—201no—
503C3×C4—201no—
504C3×C4#402712yes—
505C3×C4C22×C4×D511yescharacteristic
506C3×C4C2×#71111yescharacteristic
507C2×C5—22no—
508C2×C5—42no—
509C2×C5C2×C3×C4×D411yescharacteristic
510C2×C5C2×C3×#13711yescharacteristic
511C2×C5C3×#57511yescharacteristic
512C23—21no—
513C23C3×D4×D511yescharacteristic
514Q8—21no—
515Q8—54no—
516Q8—102no—
517Q8—104no—
518Q8C2×C3×C4×D512yes—
519D4—24no—
520D4—21no—
521D4C2×C3×C4×D512yes—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—42no—
526C2×C4—51no—
527C2×C4—51no—
528C2×C4—51no—
529C2×C4—51no—
530C2×C4—102no—
531C2×C4—102no—
532C2×C4—104no—
533C2×C4—101no—
534C2×C4—101no—
535C2×C4—202no—
536C2×C4—201no—
537C2×C4—202no—
538C2×C4C2×C3×C4×D512yes—
539C2×C4C3×D4×D511yescharacteristic
540C2×C4C3×#71112yes—
541C2×C4C23×C3×D511yescharacteristic
542C8—22no—
543C8—54no—
544C2×C3—22no—
545C2×C3—42no—
546C2×C3C2×#402711yescharacteristic
547C2×C3#6610711yescharacteristic
548C2×C3D5×#13711yescharacteristic
549C5C3×C4×#13711yescharacteristic, complemented, Sylow
550C22—21no—
551C22—21no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22C3×#402712yes—
556C22C2×C3×D4×D511yescharacteristic
557C4—22no—
558C4—42no—
559C4—52no—
560C4—101no—
561C4—101no—
562C4—101no—
563C4—104no—
564C4—201no—
565C4—201no—
566C4C3×#402712yes—
567C4C22×C3×C4×D511yescharacteristic
568C4C2×C3×#71111yescharacteristic
569C3#1061059811yescharacteristic, direct product factor, Sylow
570C2—22no—
571C2—42no—
572C2C2×C3×#402711yescharacteristic
573C2C3×#6610711yescharacteristic
574C2C3×D5×#13711yescharacteristic
subgroupG/H
centerC22×C3C2×D4×D5
commutatorC4×C5C23×C3×C4
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#417C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×C5×#41C22
cosocleC2×C4×C5C24×C3
hypercenterC3×#417D5
hypocenterC5C3×C4×#137
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#417

Complements

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

NH ≅ G/Ntype
3. C2×C3×#4033C2semidirect
5. C3×#65409C2semidirect
6. C3×M16×Dic5C2semidirect
7. C3×#65253C2semidirect
8. C3×#65079C2semidirect
9. C3×#65078C2semidirect
10. C3×#65047C2semidirect
11. C3×#65046C2semidirect
12. #10610598C3direct (N × H)
17. C3×#4097C4semidirect
48. C2×#4033C2×C3semidirect
50. #65409C2×C3semidirect
51. M16×Dic5C2×C3semidirect
52. #65253C2×C3semidirect
53. #65079C2×C3semidirect
54. #65078C2×C3semidirect
55. #65047C2×C3semidirect
56. #65046C2×C3semidirect
72. C2×C3×Dic10C2×C4semidirect
81. C3×#689C2×C4semidirect
82. C3×#687C2×C4semidirect
98. C3×#681C2×C4semidirect
115. #4097C3×C4semidirect
223. C2×Dic10C2×C3×C4semidirect
232. #689C2×C3×C4semidirect
233. #687C2×C3×C4semidirect
249. #681C2×C3×C4semidirect
280. C3×Dic5#137semidirect
443. Dic5C3×#137semidirect
487. C3×C5C4×#137semidirect
549. C5C3×C4×#137semidirect
569. C3#10610598direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#41×Dic52
C2×C3×#40972
C2×C3×#40332
C3×C5×#4172
C3×#654092
C3×M16×Dic52
C3×#652532
C3×#650792
C3×#650782
C3×#650472
C3×#650462
#106105983
C3×C4×#1375