Finite Group Databasebeta

← C3×#10610931 #14416824

Subgroups — C3×#10610931 #14416824

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
1C2×C3×#4055C211yesmaximal, characteristic
2C3×#65856C211yesmaximal, characteristic
3C2×C3×#4033C211yesmaximal, characteristic, complemented
4C3×C5×#428C211yesmaximal, characteristic, complemented
5C3×M16×Dic5C211yesmaximal, characteristic, complemented
6C3×#65293C211yesmaximal, characteristic, complemented
7C3×#65255C211yesmaximal, characteristic
8C3×#65178C212yesmaximal, complemented
9C3×#65162C212yesmaximal, complemented
10C3×#65142C212yesmaximal, complemented
11C3×#65141C212yesmaximal, complemented
12#10610931C311yesmaximal, characteristic, direct product factor, Hall
13C2×C3×#709C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C2×C3×C5×M16C2211yescharacteristic
16C3×C5×#117C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×#683—21no—
19C2×C3×C4×Dic5—21no—
20C2×C3×C4×Dic5C2211yescharacteristic
21C2×C3×#681C2211yescharacteristic
22C3×#4055C418yescomplemented
23C2×C3×Dic20C2212yes—
24C2×C3×#677C2212yes—
25C3×#4039C2211yescharacteristic
26C3×#4039C2211yescharacteristic
27C3×#4033—21no—
28C3×#4033—21no—
29C3×#4033C2211yescharacteristic
30C3×#4033C2211yescharacteristic
31C3×#4033C2212yes—
32C3×#4033C2212yes—
33C3×#4027—21no—
34C3×#4027—21no—
35C3×#4018—21no—
36C3×C42×D5—21no—
37C3×C5×#107C2212yes—
38C3×C5×#106C2212yes—
39C3×#3947C2212yes—
40C3×C8×Dic5C2212yes—
41C3×#3942C2212yes—
42C3×#3942C2212yes—
43C3×#3942C2212yes—
44C3×#3941C2212yes—
45C3×C4×#137—51nomaximal, Hall
46C2×#4055C2×C311yescharacteristic
47#65856C2×C311yescharacteristic
48C2×#4033C2×C311yescharacteristic, complemented
49C5×#428C2×C311yescharacteristic, complemented
50M16×Dic5C2×C311yescharacteristic, complemented
51#65293C2×C311yescharacteristic, complemented
52#65255C2×C311yescharacteristic
53#65178C2×C312yescomplemented
54#65162C2×C312yescomplemented
55#65142C2×C312yescomplemented
56#65141C2×C312yescomplemented
57C22×C3×C4×C5—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C2×C3×C22⋊D5—21no—
60C22×C3×Dic5—21no—
61C22×C3×Dic5D411yescharacteristic
62C22×C3×Dic5D411yescharacteristic
63C3×#709—22no—
64C3×#709C2×C414yes—
65C2×C3×D20C2311yescharacteristic
66C2×C3×C4×D5—21no—
67C2×C3×C4×D5—21no—
68C2×C3×C4×D5—21no—
69C2×C3×Dic10—21no—
70C2×C3×Dic10—22no—
71C2×C3×Dic10C2×C414yescomplemented
72C2×C3×Dic10C2312yes—
73C2×C3×Dic10C2311yescharacteristic
74C3×C5×M16C2×C414yescomplemented
75C2×C3×C5×C8C2312yes—
76C3×C5×C4⋊C4—21no—
77C3×C5×C4⋊C4C2312yes—
78C3×C5×C4⋊C4C2311yescharacteristic
79C3×C5×C4⋊C4C2311yescharacteristic
80C3×C5×C22⋊C4—21no—
81C3×C42×C5—21no—
82C3×#685—21no—
83C3×#685—21no—
84C3×#683—21no—
85C3×#683—21no—
86C3×#683—22no—
87C3×#683—22no—
88C3×C4×Dic5—21no—
89C3×C4×Dic5—21no—
90C3×C4×Dic5—22no—
91C3×C4×Dic5—22no—
92C3×C4×Dic5D411yescharacteristic
93C3×C4×Dic5D411yescharacteristic
94C3×C4×Dic5C2311yescharacteristic
95C3×C4×Dic5C2311yescharacteristic
96C3×#681—22no—
97C2×C3×#168C2312yes—
98C3×Dic20C2×C418yescomplemented
99C3×#677C2×C418yescomplemented
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×#428D511yescharacteristic, complemented
108C3×#419—51no—
109C3×#417—51no—
110C3×C4×M16—51no—
111C2×#709C22×C311yescharacteristic
112C22×Dic10C22×C311yescharacteristic
113C2×C5×M16C22×C311yescharacteristic
114C5×#117C22×C311yescharacteristic
115C2×C5×C4⋊C4C22×C311yescharacteristic
116C2×#683—21no—
117C2×C4×Dic5—21no—
118C2×C4×Dic5C22×C311yescharacteristic
119C2×#681C22×C311yescharacteristic
120#4055C3×C418yescomplemented
121C2×Dic20C22×C312yes—
122C2×#677C22×C312yes—
123#4039C22×C311yescharacteristic
124#4039C22×C311yescharacteristic
125#4033—21no—
126#4033—21no—
127#4033C22×C312yes—
128#4033C22×C312yes—
129#4033C22×C311yescharacteristic
130#4033C22×C311yescharacteristic
131#4027—21no—
132#4027—21no—
133#4018—21no—
134C42×D5—21no—
135C5×#107C22×C312yes—
136C5×#106C22×C312yes—
137#3947C22×C312yes—
138C8×Dic5C22×C312yes—
139#3942C22×C312yes—
140#3942C22×C312yes—
141#3942C22×C312yes—
142#3941C22×C312yes—
143C4×#137—51noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×C3×D5—21no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—21no—
148C2×C3×C4×C5—21no—
149C2×C3×C4×C5—42no—
150C2×C3×C4×C5C22×C412yes—
151C2×C3×C4×C5C2×D411yescharacteristic
152C2×C3×C4×C5#4112yes—
153C2×C3×C4×C5C2411yescharacteristic
154C3×C22⋊D5—24no—
155C2×C3×Dic5—24no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—41no—
160C2×C3×Dic5C2×D411yescharacteristic
161C2×C3×Dic5C2×D411yescharacteristic
162C2×C3×Dic5C2×D411yescharacteristic
163C2×C3×Dic5C2×D411yescharacteristic
164C3×D20—21no—
165C3×D20C22×C412yes—
166C3×C4×D5—24no—
167C3×C4×D5—42no—
168C3×C4×D5—42no—
169C3×Dic10—24no—
170C3×Dic10—22no—
171C3×Dic10—21no—
172C3×Dic10C22×C414yes—
173C3×Dic10C22×C412yes—
174C3×C5×C8C22×C414yes—
175C3×#168—22no—
176C2×C3×#41—51no—
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—52no—
184C3×C4×Q8—51no—
185C3×C4×Q8—52no—
186C3×C4×Q8—51no—
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—101no—
194C3×#117C2×D511yescharacteristic
195C2×C3×C4⋊C4—101no—
196C2×C3×C4⋊C4C2×D511yescharacteristic
197C2×C3×C42—51no—
198C2×C3×C42—101no—
199C2×C3×C42—101no—
200C3×#107C2×D512yes—
201C3×#106C2×D512yes—
202C3×#103—52no—
203C3×#103—52no—
204C3×#103—52no—
205C3×#102—52no—
206C3×#97—52no—
207C3×C4×C8—52no—
208C22×C4×C5—21no—
209C22×C4×C5C23×C311yescharacteristic
210C2×C22⋊D5—21no—
211C22×Dic5—21no—
212C22×Dic5C3×D411yescharacteristic
213C22×Dic5C3×D411yescharacteristic
214#709—22no—
215#709C2×C3×C414yes—
216C2×D20C23×C311yescharacteristic
217C2×C4×D5—21no—
218C2×C4×D5—21no—
219C2×C4×D5—21no—
220C2×Dic10—21no—
221C2×Dic10—22no—
222C2×Dic10C2×C3×C414yescomplemented
223C2×Dic10C23×C312yes—
224C2×Dic10C23×C311yescharacteristic
225C5×M16C2×C3×C414yescomplemented
226C2×C5×C8C23×C312yes—
227C5×C4⋊C4—21no—
228C5×C4⋊C4C23×C312yes—
229C5×C4⋊C4C23×C311yescharacteristic
230C5×C4⋊C4C23×C311yescharacteristic
231C5×C22⋊C4—21no—
232C42×C5—21no—
233#685—21no—
234#685—21no—
235#683—21no—
236#683—21no—
237#683—22no—
238#683—22no—
239C4×Dic5—21no—
240C4×Dic5—21no—
241C4×Dic5—22no—
242C4×Dic5—22no—
243C4×Dic5C3×D411yescharacteristic
244C4×Dic5C3×D411yescharacteristic
245C4×Dic5C23×C311yescharacteristic
246C4×Dic5C23×C311yescharacteristic
247#681—22no—
248C2×#168C23×C312yes—
249Dic20C2×C3×C418yescomplemented
250#677C2×C3×C418yescomplemented
251C2×#137—51no—
252C4×#41—51no—
253C2×C4×Q8—51no—
254#441—52no—
255#440—52no—
256C4×Q16—52no—
257C4×SD16—52no—
258#428C3×D511yescharacteristic, complemented
259#419—51no—
260#417—51no—
261C4×M16—51no—
262C22×C3×C5—21no—
263C22×C3×C5—21no—
264C22×C3×C5C4×D412yes—
265C22×C3×C5C22×D411yescharacteristic
266C2×C3×D5—22no—
267C2×C3×D5—41no—
268C2×C3×D5—41no—
269C3×C4×C5—41no—
270C3×C4×C5—41no—
271C3×C4×C5—42no—
272C3×C4×C5C4×D412yes—
273C3×C4×C5C23×C411yescharacteristic
274C3×C4×C5C2×#4111yescharacteristic
275C3×Dic5—24no—
276C3×Dic5—22no—
277C3×Dic5—21no—
278C3×Dic5—21no—
279C3×Dic5—21no—
280C3×Dic5#13712yescomplemented
281C3×#41—54no—
282C3×#41—102no—
283C2×C3×Q8—54no—
284C2×C3×Q8—52no—
285C2×C3×Q8—51no—
286C2×C3×Q8—101no—
287C2×C3×Q8—102no—
288C2×C3×D4—51no—
289C2×C3×D4—101no—
290C22×C3×C4—21no—
291C22×C3×C4—51no—
292C22×C3×C4—51no—
293C22×C3×C4—101no—
294C22×C3×C4—101no—
295C22×C3×C4—101no—
296C22×C3×C4—101no—
297C22×C3×C4C22×D511yescharacteristic
298C3×Q16—58no—
299C3×SD16—58no—
300C3×M16—102no—
301C3×M16C4×D514yescomplemented
302C2×C3×C8—52no—
303C2×C3×C8C22×D512yes—
304C3×C4⋊C4—21no—
305C3×C4⋊C4—102no—
306C3×C4⋊C4—102no—
307C3×C4⋊C4—101no—
308C3×C4⋊C4—101no—
309C3×C4⋊C4C22×D512yes—
310C3×C4⋊C4C22×D511yescharacteristic
311C3×C4⋊C4C22×D511yescharacteristic
312C3×C22⋊C4—21no—
313C3×C22⋊C4—101no—
314C3×C22⋊C4—101no—
315C3×C42—21no—
316C3×C42—51no—
317C3×C42—51no—
318C3×C42—51no—
319C3×C42—51no—
320C3×C42—101no—
321C3×C42—101no—
322C3×C42—102no—
323C3×C42—102no—
324C23×C5C2×C3×D411yescharacteristic
325C22×D5—21no—
326C2×C4×C5—22no—
327C2×C4×C5—21no—
328C2×C4×C5—21no—
329C2×C4×C5—42no—
330C2×C4×C5C22×C3×C412yes—
331C2×C4×C5C2×C3×D411yescharacteristic
332C2×C4×C5C3×#4112yes—
333C2×C4×C5C24×C311yescharacteristic
334C22⋊D5—24no—
335C2×Dic5—24no—
336C2×Dic5—22no—
337C2×Dic5—21no—
338C2×Dic5—22no—
339C2×Dic5—41no—
340C2×Dic5C2×C3×D411yescharacteristic
341C2×Dic5C2×C3×D411yescharacteristic
342C2×Dic5C2×C3×D411yescharacteristic
343C2×Dic5C2×C3×D411yescharacteristic
344D20—21no—
345D20C22×C3×C412yes—
346C4×D5—24no—
347C4×D5—42no—
348C4×D5—42no—
349Dic10—24no—
350Dic10—22no—
351Dic10—21no—
352Dic10C22×C3×C414yes—
353Dic10C22×C3×C412yes—
354C5×C8C22×C3×C414yes—
355#168—22no—
356C2×#41—51no—
357C22×Q8—51no—
358#137—58no—
359C2×Q16—52no—
360C2×SD16—52no—
361C2×M16—51no—
362C2×M16C2×C3×D511yescharacteristic
363C4×Q8—52no—
364C4×Q8—52no—
365C4×Q8—51no—
366C4×Q8—51no—
367C4×Q8—101no—
368C4×Q8—101no—
369C4×D4—51no—
370C4×D4—51no—
371C4×D4—101no—
372C4×D4—101no—
373#117—101no—
374#117C2×C3×D511yescharacteristic
375C2×C4⋊C4—101no—
376C2×C4⋊C4C2×C3×D511yescharacteristic
377C2×C42—51no—
378C2×C42—101no—
379C2×C42—101no—
380#107C2×C3×D512yes—
381#106C2×C3×D512yes—
382#103—52no—
383#103—52no—
384#103—52no—
385#102—52no—
386#97—52no—
387C4×C8—52no—
388C2×C3×C5—22no—
389C2×C3×C5C2×C4×D411yescharacteristic
390C2×C3×C5C2×#13711yescharacteristic
391C2×C3×C5#57511yescharacteristic
392C3×D5—42no—
393C23×C3—101no—
394C23×C3D4×D511yescharacteristic
395C3×Q8—54no—
396C3×Q8—52no—
397C3×Q8—102no—
398C3×Q8—101no—
399C3×Q8—104no—
400C3×D4—52no—
401C3×D4—104no—
402C3×D4—101no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
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—101no—
413C2×C3×C4—102no—
414C2×C3×C4—104no—
415C2×C3×C4—104no—
416C2×C3×C4—202no—
417C2×C3×C4—202no—
418C2×C3×C4—201no—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4D4×D511yescharacteristic
421C2×C3×C4#71312yes—
422C2×C3×C4C23×D511yescharacteristic
423C3×C8—102no—
424C3×C8C2×C4×D514yes—
425C22×C5—21no—
426C22×C5—21no—
427C22×C5C3×C4×D412yes—
428C22×C5C22×C3×D411yescharacteristic
429C2×D5—22no—
430C2×D5—41no—
431C2×D5—41no—
432C4×C5—41no—
433C4×C5—41no—
434C4×C5—42no—
435C4×C5C3×C4×D412yes—
436C4×C5C23×C3×C411yescharacteristic
437C4×C5C2×C3×#4111yescharacteristic
438Dic5—21no—
439Dic5—21no—
440Dic5—21no—
441Dic5—24no—
442Dic5—22no—
443Dic5C3×#13712yescomplemented
444#41—54no—
445#41—102no—
446C2×Q8—54no—
447C2×Q8—52no—
448C2×Q8—51no—
449C2×Q8—101no—
450C2×Q8—102no—
451C2×D4—51no—
452C2×D4—101no—
453C22×C4—21no—
454C22×C4—51no—
455C22×C4—51no—
456C22×C4—101no—
457C22×C4—101no—
458C22×C4—101no—
459C22×C4—101no—
460C22×C4C22×C3×D511yescharacteristic
461Q16—58no—
462SD16—58no—
463M16—102no—
464M16C3×C4×D514yescomplemented
465C2×C8—52no—
466C2×C8C22×C3×D512yes—
467C4⋊C4—21no—
468C4⋊C4—102no—
469C4⋊C4—102no—
470C4⋊C4—101no—
471C4⋊C4—101no—
472C4⋊C4C22×C3×D512yes—
473C4⋊C4C22×C3×D511yescharacteristic
474C4⋊C4C22×C3×D511yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—101no—
477C22⋊C4—101no—
478C42—21no—
479C42—51no—
480C42—51no—
481C42—51no—
482C42—51no—
483C42—101no—
484C42—101no—
485C42—102no—
486C42—102no—
487C3×C5C4×#13711yescharacteristic, complemented, Hall
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—102no—
491C22×C3—201no—
492C22×C3—201no—
493C22×C3#403912yes—
494C22×C3C2×D4×D511yescharacteristic
495C3×C4—42no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—52no—
499C3×C4—102no—
500C3×C4—104no—
501C3×C4—101no—
502C3×C4—101no—
503C3×C4—101no—
504C3×C4#403912yes—
505C3×C4C22×C4×D511yescharacteristic
506C3×C4C2×#71311yescharacteristic
507C2×C5—22no—
508C2×C5C2×C3×C4×D411yescharacteristic
509C2×C5C2×C3×#13711yescharacteristic
510C2×C5C3×#57511yescharacteristic
511D5—42no—
512C23—101no—
513C23C3×D4×D511yescharacteristic
514Q8—54no—
515Q8—52no—
516Q8—102no—
517Q8—101no—
518Q8—104no—
519D4—52no—
520D4—104no—
521D4—101no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—42no—
526C2×C4—51no—
527C2×C4—51no—
528C2×C4—51no—
529C2×C4—51no—
530C2×C4—102no—
531C2×C4—101no—
532C2×C4—102no—
533C2×C4—104no—
534C2×C4—104no—
535C2×C4—202no—
536C2×C4—202no—
537C2×C4—201no—
538C2×C4C2×C3×C4×D512yes—
539C2×C4C3×D4×D511yescharacteristic
540C2×C4C3×#71312yes—
541C2×C4C23×C3×D511yescharacteristic
542C8—102no—
543C8C2×C3×C4×D514yes—
544C2×C3—22no—
545C2×C3—202no—
546C2×C3C2×#403911yescharacteristic
547C2×C3#6610711yescharacteristic
548C2×C3D5×#13711yescharacteristic
549C5C3×C4×#13711yescharacteristic, complemented, Sylow
550C22—21no—
551C22—21no—
552C22—102no—
553C22—201no—
554C22—201no—
555C22C3×#403912yes—
556C22C2×C3×D4×D511yescharacteristic
557C4—42no—
558C4—41no—
559C4—41no—
560C4—52no—
561C4—101no—
562C4—101no—
563C4—101no—
564C4—102no—
565C4—104no—
566C4C3×#403912yes—
567C4C22×C3×C4×D511yescharacteristic
568C4C2×C3×#71311yescharacteristic
569C3#1061093111yescharacteristic, direct product factor, Sylow
570C2—22no—
571C2—202no—
572C2C2×C3×#403911yescharacteristic
573C2C3×#6610711yescharacteristic
574C2C3×D5×#13711yescharacteristic
subgroupG/H
centerC22×C3C2×D4×D5
commutatorC4×C5C23×C3×C4
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#428C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×#709C22
cosocleC2×C4×C5C24×C3
hypercenterC3×#428D5
hypocenterC5C3×C4×#137
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#428

Complements

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

NH ≅ G/Ntype
3. C2×C3×#4033C2semidirect
4. C3×C5×#428C2semidirect
5. C3×M16×Dic5C2semidirect
6. C3×#65293C2semidirect
8. C3×#65178C2semidirect
9. C3×#65162C2semidirect
10. C3×#65142C2semidirect
11. C3×#65141C2semidirect
12. #10610931C3direct (N × H)
22. C3×#4055C4semidirect
48. C2×#4033C2×C3semidirect
49. C5×#428C2×C3semidirect
50. M16×Dic5C2×C3semidirect
51. #65293C2×C3semidirect
53. #65178C2×C3semidirect
54. #65162C2×C3semidirect
55. #65142C2×C3semidirect
56. #65141C2×C3semidirect
71. C2×C3×Dic10C2×C4semidirect
74. C3×C5×M16C2×C4semidirect
98. C3×Dic20C2×C4semidirect
99. C3×#677C2×C4semidirect
107. C3×#428D5semidirect
120. #4055C3×C4semidirect
222. C2×Dic10C2×C3×C4semidirect
225. C5×M16C2×C3×C4semidirect
249. Dic20C2×C3×C4semidirect
250. #677C2×C3×C4semidirect
258. #428C3×D5semidirect
280. C3×Dic5#137semidirect
301. C3×M16C4×D5semidirect
443. Dic5C3×#137semidirect
464. M16C3×C4×D5semidirect
487. C3×C5C4×#137semidirect
549. C5C3×C4×#137semidirect
569. C3#10610931direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40552
C3×#658562
C2×C3×#40332
C3×C5×#4282
C3×M16×Dic52
C3×#652932
C3×#652552
C3×#651782
C3×#651622
C3×#651422
C3×#651412
#106109313
C3×C4×#1375