Finite Group Databasebeta

← C3×#10602426 #14301084

Subgroups — C3×#10602426 #14301084

Order 1920.

748 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C22×C3×#688C211yesmaximal, characteristic
3C2×C3×#3967C211yesmaximal, characteristic, complemented
4C2×C3×#3941C211yesmaximal, characteristic
5C3×#65228C211yesmaximal, characteristic
6C3×#64749C211yesmaximal, characteristic, complemented
7C3×#64744C211yesmaximal, characteristic, complemented
8#10602426C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11C3×C4×C5×Q8—22no—
12C3×C4×C5×Q8—22no—
13C2×C3×C5×C4⋊C4—21no—
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C42×C5—21no—
16C2×C3×C42×C5C2211yescharacteristic
17C2×C3×#688—22no—
18C2×C3×#688—22no—
19C2×C3×#688C414yescomplemented
20C2×C3×#685—21no—
21C2×C3×#685—21no—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C22×C3×#168C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C3×#3967—22no—
26C3×#3941—22no—
27C3×#1880—51nomaximal, Hall
28C2×C4×C5×Q8C2×C311yescharacteristic, complemented
29C22×#688C2×C311yescharacteristic
30C2×#3967C2×C311yescharacteristic, complemented
31C2×#3941C2×C311yescharacteristic
32#65228C2×C311yescharacteristic
33#64749C2×C311yescharacteristic, complemented
34#64744C2×C311yescharacteristic, complemented
35C23×C3×D5—21no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8C2×C412yes—
40C2×C3×C5×Q8D412yes—
41C2×C3×C5×Q8D412yes—
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5D411yescharacteristic
45C22×C3×C4×C5D411yescharacteristic
46C22×C3×C4×C5C2311yescharacteristic
47C22×C3×Dic5—21no—
48C2×C3×D20—21no—
49C2×C3×D20—21no—
50C2×C3×D20—21no—
51C2×C3×D20—21no—
52C2×C3×D20—21no—
53C2×C3×D20C2×C412yes—
54C3×C5×C4⋊C4—24no—
55C3×C5×C4⋊C4—21no—
56C3×C5×C4⋊C4D412yes—
57C3×C42×C5—24no—
58C3×C42×C5—21no—
59C3×C42×C5—21no—
60C3×#688—28no—
61C3×#685—42no—
62C3×#685—42no—
63C3×C4⋊Dic5—21no—
64C3×C4⋊Dic5—21no—
65C2×C3×#168—21no—
66C2×C3×#168—21no—
67C2×C3×#168—21no—
68C2×C3×#168—21no—
69C2×C3×#168—21no—
70C2×C3×#168C2×C412yes—
71C22×C3×SD16—51no—
72C2×C3×C4×Q8D511yescharacteristic, complemented
73C2×C3×#103—51no—
74C2×C3×#102—51no—
75C3×#390—51no—
76C3×#340—51no—
77C3×#336—51no—
78C22×C5×Q8C22×C311yescharacteristic
79C22×D20C22×C311yescharacteristic
80C4×C5×Q8—22no—
81C4×C5×Q8—22no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4C22×C311yescharacteristic
84C2×C42×C5—21no—
85C2×C42×C5C22×C311yescharacteristic
86C2×#688—22no—
87C2×#688—22no—
88C2×#688C3×C414yescomplemented
89C2×#685—21no—
90C2×#685—21no—
91C2×C4⋊Dic5C22×C311yescharacteristic
92C22×#168C22×C311yescharacteristic
93C22×#168C22×C311yescharacteristic
94#3967—22no—
95#3941—22no—
96#1880—51noSylow
97C23×C3×C5C2×D411yescharacteristic
98C22×C3×D5—22no—
99C22×C3×D5—41no—
100C22×C3×D5—41no—
101C22×C3×D5—41no—
102C22×C3×D5—41no—
103C22×C3×D5—41no—
104C22×C3×D5—41no—
105C3×C5×Q8—22no—
106C3×C5×Q8—22no—
107C3×C5×Q8—22no—
108C3×C5×Q8C22⋊C414yescomplemented
109C2×C3×C4×C5—22no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—41no—
119C2×C3×C4×C5—41no—
120C2×C3×C4×C5SD1612yescomplemented
121C2×C3×C4×C5C22×C411yescharacteristic
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5C2×D411yescharacteristic
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5#4111yescharacteristic
126C2×C3×C4×C5#4111yescharacteristic
127C2×C3×Dic5—41no—
128C2×C3×Dic5—41no—
129C2×C3×Dic5—41no—
130C3×D20—22no—
131C3×D20—22no—
132C3×D20—41no—
133C3×D20—41no—
134C3×#168—22no—
135C3×#168—41no—
136C22×C3×Q8C2×D511yescharacteristic
137C22×C3×D4—51no—
138C2×C3×SD16—54no—
139C2×C3×SD16—102no—
140C2×C3×SD16—102no—
141C22×C3×C8—51no—
142C22×C3×C8—51no—
143C3×C4×Q8—22no—
144C3×C4×Q8—22no—
145C2×C3×C4⋊C4—21no—
146C2×C3×C4⋊C4—51no—
147C2×C3×C4⋊C4C2×D511yescharacteristic
148C2×C3×C22⋊C4—101no—
149C2×C3×C22⋊C4—101no—
150C2×C3×C42—21no—
151C2×C3×C42C2×D511yescharacteristic
152C3×#103—102no—
153C3×#102—102no—
154C23×D5—21no—
155C2×C5×Q8—21no—
156C2×C5×Q8—21no—
157C2×C5×Q8—21no—
158C2×C5×Q8C2×C3×C412yes—
159C2×C5×Q8C3×D412yes—
160C2×C5×Q8C3×D412yes—
161C22×C4×C5—21no—
162C22×C4×C5—21no—
163C22×C4×C5C3×D411yescharacteristic
164C22×C4×C5C3×D411yescharacteristic
165C22×C4×C5C23×C311yescharacteristic
166C22×Dic5—21no—
167C2×D20—21no—
168C2×D20—21no—
169C2×D20—21no—
170C2×D20—21no—
171C2×D20—21no—
172C2×D20C2×C3×C412yes—
173C5×C4⋊C4—24no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4C3×D412yes—
176C42×C5—24no—
177C42×C5—21no—
178C42×C5—21no—
179#688—28no—
180#685—42no—
181#685—42no—
182C4⋊Dic5—21no—
183C4⋊Dic5—21no—
184C2×#168—21no—
185C2×#168—21no—
186C2×#168—21no—
187C2×#168—21no—
188C2×#168—21no—
189C2×#168C2×C3×C412yes—
190C22×SD16—51no—
191C2×C4×Q8C3×D511yescharacteristic, complemented
192C2×#103—51no—
193C2×#102—51no—
194#390—51no—
195#340—51no—
196#336—51no—
197C22×C3×C5C4×D411yescharacteristic
198C22×C3×C5#12011yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5C2×SD1611yescharacteristic
201C22×C3×C5#13511yescharacteristic
202C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
203C22×C3×C5#13711yescharacteristic
204C2×C3×D5—42no—
205C2×C3×D5—42no—
206C2×C3×D5—42no—
207C2×C3×D5—81no—
208C2×C3×D5—81no—
209C2×C3×D5—81no—
210C2×C3×D5—81no—
211C3×C4×C5—24no—
212C3×C4×C5—22no—
213C3×C4×C5—41no—
214C3×C4×C5—42no—
215C3×C4×C5C2×C22⋊C411yescharacteristic
216C3×C4×C5C4×D411yescharacteristic
217C3×C4×C5#12111yescharacteristic
218C3×C4×C5#12311yescharacteristic
219C3×Dic5—81no—
220C24×C3—101no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C2×C3×Q8C4×D512yes—
225C2×C3×Q8D2012yes—
226C2×C3×Q8C22⋊D512yes—
227C2×C3×D4—52no—
228C2×C3×D4—101no—
229C2×C3×D4—101no—
230C2×C3×D4—101no—
231C2×C3×D4—101no—
232C2×C3×D4—101no—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—101no—
236C22×C3×C4C22⋊D511yescharacteristic
237C22×C3×C4C22⋊D511yescharacteristic
238C22×C3×C4C22×D511yescharacteristic
239C3×SD16—108no—
240C2×C3×C8—52no—
241C2×C3×C8—101no—
242C2×C3×C8—101no—
243C2×C3×C8—101no—
244C2×C3×C8—101no—
245C2×C3×C8—101no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—101no—
249C3×C4⋊C4—101no—
250C3×C4⋊C4C22⋊D512yes—
251C3×C22⋊C4—202no—
252C3×C22⋊C4—202no—
253C3×C42—24no—
254C3×C42—21no—
255C3×C42—21no—
256C23×C5C2×C3×D411yescharacteristic
257C22×D5—22no—
258C22×D5—41no—
259C22×D5—41no—
260C22×D5—41no—
261C22×D5—41no—
262C22×D5—41no—
263C22×D5—41no—
264C5×Q8—22no—
265C5×Q8—22no—
266C5×Q8—22no—
267C5×Q8C3×C22⋊C414yescomplemented
268C2×C4×C5—22no—
269C2×C4×C5—21no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—22no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5C3×SD1612yescomplemented
280C2×C4×C5C22×C3×C411yescharacteristic
281C2×C4×C5C2×C3×D411yescharacteristic
282C2×C4×C5C2×C3×D411yescharacteristic
283C2×C4×C5C2×C3×D411yescharacteristic
284C2×C4×C5C3×#4111yescharacteristic
285C2×C4×C5C3×#4111yescharacteristic
286C2×Dic5—41no—
287C2×Dic5—41no—
288C2×Dic5—41no—
289D20—22no—
290D20—22no—
291D20—41no—
292D20—41no—
293#168—22no—
294#168—41no—
295C22×Q8C2×C3×D511yescharacteristic
296C22×D4—51no—
297C2×SD16—54no—
298C2×SD16—102no—
299C2×SD16—102no—
300C22×C8—51no—
301C22×C8—51no—
302C4×Q8—22no—
303C4×Q8—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—51no—
306C2×C4⋊C4C2×C3×D511yescharacteristic
307C2×C22⋊C4—101no—
308C2×C22⋊C4—101no—
309C2×C42—21no—
310C2×C42C2×C3×D511yescharacteristic
311#103—102no—
312#102—102no—
313C2×C3×C5#38611yescharacteristic
314C2×C3×C5C4×SD1611yescharacteristic
315C2×C3×C5#44011yescharacteristic
316C2×C3×C5#44811yescharacteristic
317C2×C3×C5#44911yescharacteristic
318C2×C3×C5#46011yescharacteristic
319C2×C3×C5#46411yescharacteristic
320C3×D5—82no—
321C23×C3—102no—
322C23×C3—201no—
323C23×C3—201no—
324C23×C3—201no—
325C23×C3—201no—
326C23×C3—201no—
327C23×C3—201no—
328C23×C3C2×C22⋊D511yescharacteristic
329C3×Q8—22no—
330C3×Q8—22no—
331C3×Q8—22no—
332C3×Q8#68514yescomplemented
333C3×D4—102no—
334C3×D4—102no—
335C3×D4—201no—
336C3×D4—201no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—201no—
349C2×C3×C4—201no—
350C2×C3×C4—201no—
351C2×C3×C4#68812yescomplemented
352C2×C3×C4C2×C4×D511yescharacteristic
353C2×C3×C4C2×D2011yescharacteristic
354C2×C3×C4#70911yescharacteristic
355C2×C3×C4#70911yescharacteristic
356C2×C3×C4C2×C22⋊D511yescharacteristic
357C2×C3×C4C2×C22⋊D511yescharacteristic
358C3×C8—102no—
359C3×C8—201no—
360C22×C5C3×C4×D411yescharacteristic
361C22×C5C3×#12011yescharacteristic
362C22×C5C3×#12111yescharacteristic
363C22×C5C2×C3×SD1611yescharacteristic
364C22×C5C3×#13511yescharacteristic
365C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
366C22×C5C3×#13711yescharacteristic
367C2×D5—42no—
368C2×D5—42no—
369C2×D5—42no—
370C2×D5—81no—
371C2×D5—81no—
372C2×D5—81no—
373C2×D5—81no—
374C4×C5—24no—
375C4×C5—22no—
376C4×C5—41no—
377C4×C5—42no—
378C4×C5C2×C3×C22⋊C411yescharacteristic
379C4×C5C3×C4×D411yescharacteristic
380C4×C5C3×#12111yescharacteristic
381C4×C5C3×#12311yescharacteristic
382Dic5—81no—
383C24—101no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8C3×C4×D512yes—
388C2×Q8C3×D2012yes—
389C2×Q8C3×C22⋊D512yes—
390C2×D4—52no—
391C2×D4—101no—
392C2×D4—101no—
393C2×D4—101no—
394C2×D4—101no—
395C2×D4—101no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—101no—
399C22×C4C3×C22⋊D511yescharacteristic
400C22×C4C3×C22⋊D511yescharacteristic
401C22×C4C22×C3×D511yescharacteristic
402SD16—108no—
403C2×C8—52no—
404C2×C8—101no—
405C2×C8—101no—
406C2×C8—101no—
407C2×C8—101no—
408C2×C8—101no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—101no—
412C4⋊C4—101no—
413C4⋊C4C3×C22⋊D512yes—
414C22⋊C4—202no—
415C22⋊C4—202no—
416C42—24no—
417C42—21no—
418C42—21no—
419C3×C5#188011yescharacteristic, complemented, Hall
420C22×C3—202no—
421C22×C3—202no—
422C22×C3—202no—
423C22×C3—401no—
424C22×C3—401no—
425C22×C3—401no—
426C22×C3—401no—
427C22×C3C4×C22⋊D511yescharacteristic
428C22×C3#407611yescharacteristic
429C22×C3C2×#68811yescharacteristic
430C22×C3#408811yescharacteristic
431C22×C3#409511yescharacteristic
432C22×C3#409611yescharacteristic
433C22×C3#409911yescharacteristic
434C3×C4—24no—
435C3×C4—22no—
436C3×C4—41no—
437C3×C4—42no—
438C3×C4—401no—
439C3×C4C2×#68511yescharacteristic
440C3×C4C4×C22⋊D511yescharacteristic
441C3×C4#407511yescharacteristic
442C3×C4#407611yescharacteristic
443C2×C5C3×#38611yescharacteristic
444C2×C5C3×C4×SD1611yescharacteristic
445C2×C5C3×#44011yescharacteristic
446C2×C5C3×#44811yescharacteristic
447C2×C5C3×#44911yescharacteristic
448C2×C5C3×#46011yescharacteristic
449C2×C5C3×#46411yescharacteristic
450D5—82no—
451C23—102no—
452C23—201no—
453C23—201no—
454C23—201no—
455C23—201no—
456C23—201no—
457C23—201no—
458C23C2×C3×C22⋊D511yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8C3×#68514yescomplemented
463D4—102no—
464D4—102no—
465D4—201no—
466D4—201no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—201no—
479C2×C4—201no—
480C2×C4—201no—
481C2×C4C3×#68812yescomplemented
482C2×C4C2×C3×C4×D511yescharacteristic
483C2×C4C2×C3×D2011yescharacteristic
484C2×C4C3×#70911yescharacteristic
485C2×C4C3×#70911yescharacteristic
486C2×C4C2×C3×C22⋊D511yescharacteristic
487C2×C4C2×C3×C22⋊D511yescharacteristic
488C8—102no—
489C8—201no—
490C2×C3—402no—
491C2×C3C4×#68811yescharacteristic
492C2×C3#6531511yescharacteristic
493C2×C3#6531611yescharacteristic
494C2×C3#6531711yescharacteristic
495C2×C3#6550211yescharacteristic
496C2×C3#6551611yescharacteristic
497C2×C3#6552711yescharacteristic
498C5C3×#188011yescharacteristic, complemented, Sylow
499C22—202no—
500C22—202no—
501C22—202no—
502C22—401no—
503C22—401no—
504C22—401no—
505C22—401no—
506C22C3×C4×C22⋊D511yescharacteristic
507C22C3×#407611yescharacteristic
508C22C2×C3×#68811yescharacteristic
509C22C3×#408811yescharacteristic
510C22C3×#409511yescharacteristic
511C22C3×#409611yescharacteristic
512C22C3×#409911yescharacteristic
513C4—24no—
514C4—22no—
515C4—41no—
516C4—42no—
517C4—401no—
518C4C2×C3×#68511yescharacteristic
519C4C3×C4×C22⋊D511yescharacteristic
520C4C3×#407511yescharacteristic
521C4C3×#407611yescharacteristic
522C3#1060242611yescharacteristic, direct product factor, Sylow
523C2—402no—
524C2C3×C4×#68811yescharacteristic
525C2C3×#6531511yescharacteristic
526C2C3×#6531611yescharacteristic
527C2C3×#6531711yescharacteristic
528C2C3×#6550211yescharacteristic
529C2C3×#6551611yescharacteristic
530C2C3×#6552711yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC2×C3×C4×C5×Q8C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
cosocleC22×C4×C5C23×C3
hypercenterC2×C3×C4×Q8D5
hypocenterC5C3×#1880
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×Q8C2semidirect
3. C2×C3×#3967C2semidirect
6. C3×#64749C2semidirect
7. C3×#64744C2semidirect
8. #10602426C3direct (N × H)
19. C2×C3×#688C4semidirect
28. C2×C4×C5×Q8C2×C3semidirect
30. C2×#3967C2×C3semidirect
33. #64749C2×C3semidirect
34. #64744C2×C3semidirect
72. C2×C3×C4×Q8D5semidirect
88. C2×#688C3×C4semidirect
108. C3×C5×Q8C22⋊C4semidirect
120. C2×C3×C4×C5SD16semidirect
191. C2×C4×Q8C3×D5semidirect
267. C5×Q8C3×C22⋊C4semidirect
279. C2×C4×C5C3×SD16semidirect
332. C3×Q8#685semidirect
351. C2×C3×C4#688semidirect
419. C3×C5#1880semidirect
462. Q8C3×#685semidirect
481. C2×C4C3×#688semidirect
498. C5C3×#1880semidirect
522. C3#10602426direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×Q82
C22×C3×#6882
C2×C3×#39672
C2×C3×#39412
C3×#652282
C3×#647492
C3×#647442
#106024263
C3×#18805