Finite Group Databasebeta

← C4×#91307 #14416902

Subgroups — C4×#91307 #14416902

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×C4×C5×Dic6C211yesmaximal, characteristic, complemented
2C3×C4×#709C211yesmaximal, characteristic
3C2×#91307C211yesmaximal, characteristic
4C4×#6710C211yesmaximal, characteristic, complemented
5#11748472C211yesmaximal, characteristic, complemented
6#11748407C211yesmaximal, characteristic, complemented
7#11748368C211yesmaximal, characteristic
8#11747089C212yesmaximal, complemented
9C4×#6661C212yesmaximal, complemented
10#11747048C212yesmaximal, complemented
11C4×#6655C212yesmaximal, complemented
12C4×#4088—31nomaximal, Hall
13C22×C5×Dic6C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×C3×C42×C5C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17C2×#6710C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C5×Dic3⋊C4—21no—
20C2×C4×C5×Dic3—21no—
21C4×C5×Dic6—21no—
22C4×C5×Dic6—21no—
23C4×C5×Dic6C2212yes—
24C4×C5×Dic6C2212yes—
25C3×C4×C22⋊D5—21no—
26C3×C4×C22⋊D5—21no—
27C3×#4072C2211yescharacteristic
28C3×C4×D20C2211yescharacteristic
29C3×C4×D20C2211yescharacteristic
30C3×#4018—21no—
31C3×C42×D5—21no—
32C3×C4×Dic10C2211yescharacteristic
33C3×C4×Dic10C2211yescharacteristic
34C2×#6661C2212yes—
35#91307C418yesdirect product factor
36C2×#6655C2212yes—
37#91087C2212yes—
38C4×C3⋊#168C2212yes—
39#90989C2212yes—
40#90986C2212yes—
41#90972C2212yes—
42#90972C2212yes—
43#90970C2212yes—
44#90964C2212yes—
45C4×#951—51nomaximal, Hall
46C2×C4×C5×Q8—31no—
47C2×#4088—31no—
48C4×#709D311yescharacteristic
49#65514—31no—
50#65319—32no—
51C4×#689—32no—
52#65315—32no—
53C4×#688—32no—
54#65255—31no—
55#65253—31no—
56C4×#681—31no—
57C22×C3×C4×C5D411yescharacteristic
58C22×C3×C4×C5D411yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C22×C5×Dic3—21no—
61C22×C5×Dic3—21no—
62C2×C5×Dic6—21no—
63C2×C5×Dic6—22no—
64C2×C5×Dic6C2×C414yescomplemented
65C2×C5×Dic6C2312yes—
66C2×C3×C22⋊D5—21no—
67C3×#709—22no—
68C3×#709C2×C414yes—
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×C4×D5—21no—
72C2×C3×C4×D5—21no—
73C2×C3×Dic10C2311yescharacteristic
74C3×C42×C5D411yescharacteristic
75C3×C42×C5D411yescharacteristic
76C3×C42×C5C2311yescharacteristic
77C3×C42×C5C2311yescharacteristic
78#6710—22no—
79#6710C2×C414yescomplemented
80C2×C3⋊#168C2312yes—
81C2×C3⋊#168C2312yes—
82C5×C4⋊Dic3—21no—
83C5×C4⋊Dic3C2312yes—
84C5×Dic3⋊C4—22no—
85C5×Dic3⋊C4—22no—
86C4×C5×Dic3—22no—
87C4×C5×Dic3—22no—
88C3×C22⋊Dic5—21no—
89C3×#685—21no—
90C3×#685—21no—
91C3×C4⋊Dic5C2311yescharacteristic
92C3×C4⋊Dic5C2311yescharacteristic
93C3×#683—21no—
94C3×#683—21no—
95C3×C4×Dic5—21no—
96C3×C4×Dic5—21no—
97C3×C4×Dic5—21no—
98#6661C2×C418yescomplemented
99#6655C2×C418yescomplemented
100C3×C4×#41—51no—
101C2×#951—51no—
102C2×C4×Dic6D511yescharacteristic, complemented
103#5232—51no—
104#5030—52no—
105C4×#216—52no—
106#5018—52no—
107C4×#214—52no—
108#5004—51no—
109#4995—51no—
110C4×#208—51no—
111C22×C5×Q8—31no—
112C2×#709C2×D311yescharacteristic
113C4×C5×Q8—32no—
114C4×C5×Q8—32no—
115C4×C5×Q8—61no—
116C4×C5×Q8—61no—
117C2×C5×C4⋊C4—31no—
118C2×C5×C4⋊C4—61no—
119C2×C42×C5—61no—
120C2×C42×C5C2×D311yescharacteristic
121C2×#689—32no—
122#4088—38no—
123C2×#688—32no—
124C4×C22⋊D5—21no—
125C4×C22⋊D5—21no—
126#4072C2×D311yescharacteristic
127C2×#681—31no—
128C2×#681—31no—
129C4×D20C2×D311yescharacteristic
130C4×D20C2×D311yescharacteristic
131#4018—21no—
132C42×D5—21no—
133C4×Dic10C2×D311yescharacteristic
134C4×Dic10C2×D311yescharacteristic
135#3967—32no—
136#3967—32no—
137#3942—32no—
138#3941—32no—
139#3940—32no—
140#3939—32no—
141#3935—32no—
142C4×#168—32no—
143C4×#137—151noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×C3×D5—21no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—41no—
148C2×C3×C4×C5C22×C412yes—
149C2×C3×C4×C5C2×D411yescharacteristic
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5C2×D411yescharacteristic
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2×D411yescharacteristic
154C2×C3×C4×C5#4112yes—
155C2×C3×C4×C5C2411yescharacteristic
156C2×C5×Dic3—24no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—42no—
160C5×Dic6—24no—
161C5×Dic6—22no—
162C5×Dic6C22×C414yes—
163C3×C22⋊D5—24no—
164C2×C3×Dic5—21no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—21no—
167C3×D20—21no—
168C3×D20C22×C412yes—
169C3×C4×D5—24no—
170C3×C4×D5—42no—
171C3×C4×D5—42no—
172C3×Dic10—21no—
173C3×Dic10C22×C412yes—
174C3⋊#168—22no—
175C3⋊#168C22×C414yes—
176C2×C3×#41—51no—
177C22×Dic6C2×D511yescharacteristic
178C3×C4×Q8—51no—
179C3×C4×Q8—51no—
180C3×C4×D4—51no—
181C3×C4×D4—51no—
182C3×C4×D4—101no—
183C3×C4×D4—101no—
184C3×#117—51no—
185C3×#117—101no—
186C2×C3×C42—101no—
187C2×C3×C42C2×D511yescharacteristic
188#951—58no—
189C2×#216—52no—
190C2×#214—52no—
191C2×C4⋊Dic3C2×D511yescharacteristic
192C2×Dic3⋊C4—21no—
193C2×C4×Dic3—21no—
194C2×#208—51no—
195C2×#208—51no—
196C4×Dic6—21no—
197C4×Dic6—21no—
198C4×Dic6C2×D512yes—
199C4×Dic6C2×D512yes—
200#835—52no—
201#832—52no—
202#810—52no—
203#810—52no—
204#808—52no—
205#807—52no—
206#803—52no—
207C4×C3⋊C8—52no—
208C2×C5×Q8—34no—
209C2×C5×Q8—32no—
210C2×C5×Q8—62no—
211C2×C5×Q8—61no—
212C22×C4×C5—61no—
213C22×C4×C5—61no—
214C22×C4×C5#6711yescharacteristic
215C22×C4×C5#6711yescharacteristic
216C22×C4×C5C22×D311yescharacteristic
217C2×C22⋊D5—21no—
218#709—22no—
219#709C4×D314yes—
220C2×D20C22×D311yescharacteristic
221C2×C4×D5—21no—
222C2×C4×D5—21no—
223C2×C4×D5—21no—
224C2×Dic10C22×D311yescharacteristic
225C5×C4⋊C4—32no—
226C5×C4⋊C4—61no—
227C5×C4⋊C4—62no—
228C5×C4⋊C4—62no—
229C42×C5—62no—
230C42×C5—62no—
231C42×C5#6711yescharacteristic
232C42×C5#6711yescharacteristic
233C42×C5C22×D311yescharacteristic
234C42×C5C22×D311yescharacteristic
235C22⋊Dic5—21no—
236#689—38no—
237#688—38no—
238#685—21no—
239#685—21no—
240C4⋊Dic5C22×D311yescharacteristic
241C4⋊Dic5C22×D311yescharacteristic
242#683—21no—
243#683—21no—
244C4×Dic5—21no—
245C4×Dic5—21no—
246C4×Dic5—21no—
247#681—34no—
248#681—62no—
249C2×#168—32no—
250C2×#168—32no—
251C2×#137—151no—
252C4×#41—51no—
253C2×C4×Q8—31no—
254#441—152no—
255#440—152no—
256C4×Q16—152no—
257C4×SD16—152no—
258#428—151no—
259#419—151no—
260#417—151no—
261C4×M16—151no—
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—21no—
270C3×C4×C5—21no—
271C3×C4×C5—21no—
272C3×C4×C5C4×D412yes—
273C3×C4×C5#13712yescomplemented
274C3×C4×C5C23×C411yescharacteristic
275C3×C4×C5C2×#4111yescharacteristic
276C3×Dic5—22no—
277C3×Dic5—41no—
278C3×Dic5—41no—
279C5×Dic3—24no—
280C5×Dic3—42no—
281C3×#41—54no—
282C3×#41—102no—
283C2×C3×Q8—51no—
284C2×C3×D4—51no—
285C2×C3×D4—101no—
286C22×C3×C4—101no—
287C22×C3×C4—101no—
288C22×C3×C4—101no—
289C22×C3×C4C22⋊D511yescharacteristic
290C22×C3×C4C22⋊D511yescharacteristic
291C22×C3×C4C22×D511yescharacteristic
292C22×Dic3—21no—
293C22×Dic3—21no—
294C2×Dic6—22no—
295C2×Dic6—21no—
296C2×Dic6C4×D514yescomplemented
297C2×Dic6C22×D512yes—
298C3×C4⋊C4—51no—
299C3×C4⋊C4—51no—
300C3×C4⋊C4—101no—
301C3×C4⋊C4—101no—
302C3×C22⋊C4—101no—
303C3×C22⋊C4—101no—
304C3×C22⋊C4—101no—
305C3×C42—101no—
306C3×C42—101no—
307C3×C42—101no—
308C3×C42C22⋊D511yescharacteristic
309C3×C42C22⋊D511yescharacteristic
310C3×C42C22×D511yescharacteristic
311C3×C42C22×D511yescharacteristic
312#216—58no—
313#214—58no—
314C4⋊Dic3—21no—
315C4⋊Dic3C22×D512yes—
316Dic3⋊C4—22no—
317Dic3⋊C4—22no—
318C4×Dic3—22no—
319C4×Dic3—22no—
320#208—54no—
321#208—102no—
322C2×C3⋊C8—52no—
323C2×C3⋊C8—52no—
324C23×C5C2×#6711yescharacteristic
325C22×D5—21no—
326C5×Q8—34no—
327C5×Q8—64no—
328C5×Q8—62no—
329C2×C4×C5—22no—
330C2×C4×C5—41no—
331C2×C4×C5—64no—
332C2×C4×C5—62no—
333C2×C4×C5—62no—
334C2×C4×C5—122no—
335C2×C4×C5C2×C4×D312yes—
336C2×C4×C5#23512yes—
337C2×C4×C5C2×#6711yescharacteristic
338C2×C4×C5C2×#6711yescharacteristic
339C2×C4×C5C2×#6711yescharacteristic
340C2×C4×C5C2×#6711yescharacteristic
341C2×C4×C5C2×#6711yescharacteristic
342C2×C4×C5C23×D311yescharacteristic
343C22⋊D5—24no—
344C2×Dic5—21no—
345C2×Dic5—21no—
346C2×Dic5—21no—
347D20—21no—
348D20C2×C4×D312yes—
349C4×D5—24no—
350C4×D5—42no—
351C4×D5—42no—
352Dic10—21no—
353Dic10C2×C4×D312yes—
354#168—34no—
355#168—62no—
356C2×#41—51no—
357C22×Q8—31no—
358#137—158no—
359C2×Q16—152no—
360C2×SD16—152no—
361C2×M16—151no—
362C2×M16—151no—
363C4×Q8—32no—
364C4×Q8—32no—
365C4×Q8—51no—
366C4×Q8—51no—
367C4×Q8—61no—
368C4×Q8—61no—
369C4×D4—51no—
370C4×D4—51no—
371C4×D4—101no—
372C4×D4—101no—
373#117—51no—
374#117—101no—
375C2×C4⋊C4—31no—
376C2×C4⋊C4—61no—
377C2×C42—61no—
378C2×C42—101no—
379C2×C42D3×D511yescharacteristic
380#107—152no—
381#106—152no—
382#103—152no—
383#103—152no—
384#103—152no—
385#102—152no—
386#97—152no—
387C4×C8—152no—
388C2×C3×C5—22no—
389C2×C3×C5C2×C4×D411yescharacteristic
390C2×C3×C5C2×#13711yescharacteristic
391C2×C3×C5#57511yescharacteristic
392C3×D5—42no—
393C23×C3—101no—
394C23×C3C2×C22⋊D511yescharacteristic
395C3×Q8—52no—
396C3×Q8—101no—
397C3×D4—52no—
398C3×D4—104no—
399C3×D4—101no—
400C2×C3×C4—22no—
401C2×C3×C4—41no—
402C2×C3×C4—101no—
403C2×C3×C4—104no—
404C2×C3×C4—101no—
405C2×C3×C4—101no—
406C2×C3×C4—202no—
407C2×C3×C4—202no—
408C2×C3×C4C2×C4×D512yes—
409C2×C3×C4#70912yes—
410C2×C3×C4C2×C22⋊D511yescharacteristic
411C2×C3×C4C2×C22⋊D511yescharacteristic
412C2×C3×C4C2×C22⋊D511yescharacteristic
413C2×C3×C4C2×C22⋊D511yescharacteristic
414C2×C3×C4C2×C22⋊D511yescharacteristic
415C2×C3×C4C23×D511yescharacteristic
416C2×Dic3—22no—
417C2×Dic3—24no—
418C2×Dic3—22no—
419C2×Dic3—42no—
420Dic6—22no—
421Dic6—24no—
422Dic6C2×C4×D514yes—
423C3⋊C8—54no—
424C3⋊C8—102no—
425C22×C5—21no—
426C22×C5—21no—
427C22×C5C4×#6712yes—
428C22×C5C22×#6711yescharacteristic
429C2×D5—22no—
430C2×D5—41no—
431C2×D5—41no—
432C4×C5—21no—
433C4×C5—21no—
434C4×C5—21no—
435C4×C5—64no—
436C4×C5—122no—
437C4×C5C4×#6712yes—
438C4×C5#95112yescomplemented
439C4×C5C22×C4×D311yescharacteristic
440C4×C5C2×#23511yescharacteristic
441Dic5—22no—
442Dic5—41no—
443Dic5—41no—
444#41—54no—
445#41—102no—
446C2×Q8—34no—
447C2×Q8—32no—
448C2×Q8—51no—
449C2×Q8—62no—
450C2×Q8—61no—
451C2×D4—51no—
452C2×D4—101no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—101no—
456C22×C4—101no—
457C22×C4—101no—
458C22×C4#120111yescharacteristic
459C22×C4#120111yescharacteristic
460C22×C4C2×D3×D511yescharacteristic
461Q16—158no—
462SD16—158no—
463M16—154no—
464M16—302no—
465C2×C8—152no—
466C2×C8—152no—
467C4⋊C4—32no—
468C4⋊C4—51no—
469C4⋊C4—51no—
470C4⋊C4—61no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C4⋊C4—101no—
474C4⋊C4—101no—
475C22⋊C4—101no—
476C22⋊C4—101no—
477C22⋊C4—101no—
478C42—62no—
479C42—62no—
480C42—101no—
481C42—101no—
482C42—101no—
483C42#120111yescharacteristic
484C42#120111yescharacteristic
485C42C2×D3×D511yescharacteristic
486C42C2×D3×D511yescharacteristic
487C3×C5C4×#13711yescharacteristic, complemented, Hall
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—102no—
491C22×C3—201no—
492C22×C3—201no—
493C22×C3C4×C22⋊D512yes—
494C22×C3C22×C22⋊D511yescharacteristic
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—102no—
499C3×C4—201no—
500C3×C4—201no—
501C3×C4C4×C22⋊D512yes—
502C3×C4#408812yescomplemented
503C3×C4C22×C4×D511yescharacteristic
504C3×C4C2×#70911yescharacteristic
505Dic3—24no—
506Dic3—42no—
507C2×C5—22no—
508C2×C5C2×C4×#6711yescharacteristic
509C2×C5#582211yescharacteristic
510C2×C5C2×#95111yescharacteristic
511D5—42no—
512C23—101no—
513C23C2×#120111yescharacteristic
514Q8—34no—
515Q8—52no—
516Q8—64no—
517Q8—62no—
518Q8—101no—
519D4—52no—
520D4—104no—
521D4—101no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—64no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—101no—
528C2×C4—104no—
529C2×C4—101no—
530C2×C4—101no—
531C2×C4—122no—
532C2×C4—202no—
533C2×C4—202no—
534C2×C4#677112yes—
535C2×C4C4×D3×D512yes—
536C2×C4C2×#120111yescharacteristic
537C2×C4C2×#120111yescharacteristic
538C2×C4C2×#120111yescharacteristic
539C2×C4C2×#120111yescharacteristic
540C2×C4C2×#120111yescharacteristic
541C2×C4C22×D3×D511yescharacteristic
542C8—154no—
543C8—302no—
544C2×C3—22no—
545C2×C3—202no—
546C2×C3C2×C4×C22⋊D511yescharacteristic
547C2×C3C2×#408811yescharacteristic
548C2×C3#6615611yescharacteristic
549C5C4×#95111yescharacteristic, complemented, Sylow
550C22—21no—
551C22—21no—
552C22—102no—
553C22—201no—
554C22—201no—
555C22C4×#120112yes—
556C22C22×#120111yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—64no—
561C4—102no—
562C4—122no—
563C4—201no—
564C4—201no—
565C4#9130712yesdirect product factor
566C4C4×#120112yes—
567C4C2×#677111yescharacteristic
568C4C2×C4×D3×D511yescharacteristic
569C3C4×#408811yescharacteristic, complemented, Sylow
570C2—22no—
571C2—202no—
572C2#1175413811yescharacteristic
573C2C2×#9130711yescharacteristic
574C2C2×C4×#120111yescharacteristic
subgroupG/H
centerC2×C4C2×#1201
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C42×C5C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#709C22
cosocleC2×C3×C4×C5C24
hypercenterC2×C42D3×D5
hypocenterC3×C5C4×#137
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×Dic6C2semidirect
4. C4×#6710C2semidirect
5. #11748472C2semidirect
6. #11748407C2semidirect
8. #11747089C2semidirect
9. C4×#6661C2semidirect
10. #11747048C2semidirect
11. C4×#6655C2semidirect
35. #91307C4direct (N × H)
64. C2×C5×Dic6C2×C4semidirect
79. #6710C2×C4semidirect
98. #6661C2×C4semidirect
99. #6655C2×C4semidirect
102. C2×C4×Dic6D5semidirect
273. C3×C4×C5#137semidirect
296. C2×Dic6C4×D5semidirect
438. C4×C5#951semidirect
487. C3×C5C4×#137semidirect
502. C3×C4#4088semidirect
549. C5C4×#951semidirect
565. C4#91307direct (N × H)
569. C3C4×#4088semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×Dic62
C3×C4×#7092
C2×#913072
C4×#67102
#117484722
#117484072
#117483682
#117470892
C4×#66612
#117470482
C4×#66552
C4×#40883
C4×#9515