Finite Group Databasebeta

← C2×C3×#65430 #14429293

Subgroups — C2×C3×#65430 #14429293

Order 1920.

936 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#131C211yesmaximal, characteristic, complemented
2C2×C3×#4097C211yesmaximal, characteristic, complemented
3C2×C3×#4095C211yesmaximal, characteristic
4C22×C3×#677C211yesmaximal, characteristic, complemented
5C3×#65430C218yesmaximal, direct product factor
6C2×C3×#3956C211yesmaximal, characteristic, complemented
7C2×C3×#3955C211yesmaximal, characteristic, complemented
8C2×C3×#3951C211yesmaximal, characteristic, complemented
9C2×#65430C311yesmaximal, characteristic, direct product factor, Hall
10C2×C3×C5×#41C2211yescharacteristic
11C22×C3×D20C2211yescharacteristic
12C22×C3×Dic10C2211yescharacteristic
13C3×C5×#131—22no—
14C3×C5×#131C2214yescomplemented
15C2×C3×C5×M16—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C22×C3×C5×C8—21no—
18C22×C3×C5×C8C2211yescharacteristic
19C3×#4097—22no—
20C3×#4097C2214yescomplemented
21C3×#4095—22no—
22C3×#4095C2214yes—
23C2×C3×#689—21no—
24C2×C3×#688—21no—
25C2×C3×#687—21no—
26C2×C3×#686—21no—
27C2×C3×#681C2211yescharacteristic
28C2×C3×#681C2211yescharacteristic
29C2×C3×#677—21no—
30C2×C3×#677—21no—
31C2×C3×#677—22no—
32C2×C3×#677C2214yescomplemented
33C3×#3956C2214yescomplemented
34C3×#3955C2214yescomplemented
35C3×#3951C2214yescomplemented
36C2×C3×#471—51nomaximal, Hall
37C2×C5×#131C2×C311yescharacteristic, complemented
38C2×#4097C2×C311yescharacteristic, complemented
39C2×#4095C2×C311yescharacteristic
40C22×#677C2×C311yescharacteristic, complemented
41#65430C2×C318yesdirect product factor
42C2×#3956C2×C311yescharacteristic, complemented
43C2×#3955C2×C311yescharacteristic, complemented
44C2×#3951C2×C311yescharacteristic, complemented
45C23×C3×D5—21no—
46C3×C5×#41—21no—
47C3×C5×#41—21no—
48C3×C5×#41D412yes—
49C3×C5×#41C2312yes—
50C2×C3×C5×Q8D411yescharacteristic
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4D411yescharacteristic
53C22×C3×C4×C5—21no—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×Dic5—21no—
56C2×C3×D20—21no—
57C2×C3×D20—21no—
58C2×C3×D20—21no—
59C2×C3×D20—21no—
60C2×C3×D20—21no—
61C2×C3×D20C2312yes—
62C2×C3×Dic10—21no—
63C2×C3×Dic10—21no—
64C2×C3×Dic10—21no—
65C2×C3×Dic10—21no—
66C2×C3×Dic10—21no—
67C2×C3×Dic10C2312yes—
68C3×C5×M16—21no—
69C3×C5×M16—24no—
70C3×C5×M16C2312yes—
71C2×C3×C5×C8—21no—
72C2×C3×C5×C8—24no—
73C2×C3×C5×C8D411yescharacteristic
74C2×C3×C5×C8D411yescharacteristic
75C2×C3×C5×C8D412yes—
76C2×C3×C5×C8C2312yes—
77C3×#689—24no—
78C3×#688—24no—
79C3×#687—24no—
80C3×#686—24no—
81C3×#681—21no—
82C3×#681—21no—
83C3×#681C2312yes—
84C3×#681C2312yes—
85C2×C3×#168—21no—
86C2×C3×#168—21no—
87C3×#677—24no—
88C3×#677—24no—
89C2×C3×#137—51no—
90C2×C3×AGL(1,ℤ/8)—51no—
91C22×C3×SD16—51no—
92C2×C3×#131D511yescharacteristic, complemented
93C3×#471—58no—
94C2×C3×#108—51no—
95C2×C3×#101—51no—
96C2×C3×#100—51no—
97C2×C5×#41C22×C311yescharacteristic
98C22×D20C22×C311yescharacteristic
99C22×Dic10C22×C311yescharacteristic
100C5×#131—22no—
101C5×#131C22×C314yescomplemented
102C2×C5×M16—21no—
103C2×C5×M16C22×C311yescharacteristic
104C22×C5×C8—21no—
105C22×C5×C8C22×C311yescharacteristic
106#4097—22no—
107#4097C22×C314yescomplemented
108#4095—22no—
109#4095C22×C314yes—
110C2×#689—21no—
111C2×#688—21no—
112C2×#687—21no—
113C2×#686—21no—
114C2×#681C22×C311yescharacteristic
115C2×#681C22×C311yescharacteristic
116C2×#677—21no—
117C2×#677—21no—
118C2×#677—22no—
119C2×#677C22×C314yescomplemented
120#3956C22×C314yescomplemented
121#3955C22×C314yescomplemented
122#3951C22×C314yescomplemented
123C2×#471—51noSylow
124C23×C3×C5—21no—
125C23×C3×C5#4111yescharacteristic
126C22×C3×D5—22no—
127C22×C3×D5—41no—
128C22×C3×D5—41no—
129C22×C3×D5—82no—
130C3×C5×Q8—21no—
131C3×C5×Q8C2×D412yes—
132C3×C5×D4—22no—
133C3×C5×D4—22no—
134C3×C5×D4—21no—
135C3×C5×D4C2×D412yes—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5#4111yescharacteristic
144C2×C3×C4×C5C2411yescharacteristic
145C2×C3×Dic5—22no—
146C2×C3×Dic5—41no—
147C2×C3×Dic5—41no—
148C3×D20—22no—
149C3×D20—22no—
150C3×D20—41no—
151C3×D20—41no—
152C3×Dic10—22no—
153C3×Dic10—22no—
154C3×Dic10—41no—
155C3×Dic10—41no—
156C3×C5×C8—22no—
157C3×C5×C8C2×D412yes—
158C3×C5×C8C2×D412yes—
159C3×#168—22no—
160C3×#168—22no—
161C2×C3×#41C2×D511yescharacteristic
162C22×C3×Q8—51no—
163C22×C3×D4—51no—
164C3×#137—54no—
165C3×#137—102no—
166C3×AGL(1,ℤ/8)—54no—
167C3×AGL(1,ℤ/8)—102no—
168C2×C3×Q16—101no—
169C2×C3×SD16—54no—
170C2×C3×SD16—101no—
171C2×C3×SD16—101no—
172C2×C3×SD16—101no—
173C2×C3×SD16—101no—
174C2×C3×SD16—102no—
175C2×C3×D8—101no—
176C3×#131—22no—
177C3×#131C2×D514yescomplemented
178C2×C3×M16—21no—
179C2×C3×M16—51no—
180C2×C3×M16—51no—
181C2×C3×M16C2×D511yescharacteristic
182C22×C3×C8—21no—
183C22×C3×C8C2×D511yescharacteristic
184C3×#108—54no—
185C3×#101—54no—
186C3×#100—54no—
187C23×D5—21no—
188C5×#41—21no—
189C5×#41—21no—
190C5×#41C3×D412yes—
191C5×#41C23×C312yes—
192C2×C5×Q8C3×D411yescharacteristic
193C2×C5×D4—21no—
194C2×C5×D4C3×D411yescharacteristic
195C22×C4×C5—21no—
196C22×C4×C5C23×C311yescharacteristic
197C22×Dic5—21no—
198C2×D20—21no—
199C2×D20—21no—
200C2×D20—21no—
201C2×D20—21no—
202C2×D20—21no—
203C2×D20C23×C312yes—
204C2×Dic10—21no—
205C2×Dic10—21no—
206C2×Dic10—21no—
207C2×Dic10—21no—
208C2×Dic10—21no—
209C2×Dic10C23×C312yes—
210C5×M16—21no—
211C5×M16—24no—
212C5×M16C23×C312yes—
213C2×C5×C8—21no—
214C2×C5×C8—24no—
215C2×C5×C8C3×D411yescharacteristic
216C2×C5×C8C3×D411yescharacteristic
217C2×C5×C8C3×D412yes—
218C2×C5×C8C23×C312yes—
219#689—24no—
220#688—24no—
221#687—24no—
222#686—24no—
223#681—21no—
224#681—21no—
225#681C23×C312yes—
226#681C23×C312yes—
227C2×#168—21no—
228C2×#168—21no—
229#677—24no—
230#677—24no—
231C2×#137—51no—
232C2×AGL(1,ℤ/8)—51no—
233C22×SD16—51no—
234C2×#131C3×D511yescharacteristic, complemented
235#471—58no—
236C2×#108—51no—
237C2×#101—51no—
238C2×#100—51no—
239C22×C3×C5—22no—
240C22×C3×C5—22no—
241C22×C3×C5—42no—
242C22×C3×C5#12111yescharacteristic
243C22×C3×C5#12111yescharacteristic
244C22×C3×C5C2×#4111yescharacteristic
245C2×C3×D5—42no—
246C2×C3×D5—82no—
247C2×C3×D5—82no—
248C2×C3×D5—82no—
249C3×C4×C5—22no—
250C3×C4×C5#12111yescharacteristic
251C3×C4×C5#12111yescharacteristic
252C3×C4×C5C22×D411yescharacteristic
253C3×C4×C5C22×D411yescharacteristic
254C3×Dic5—42no—
255C24×C3—101no—
256C3×#41—21no—
257C3×#41—21no—
258C3×#41D2012yes—
259C3×#41C22×D512yes—
260C2×C3×Q8—52no—
261C2×C3×Q8—101no—
262C2×C3×Q8—101no—
263C2×C3×Q8—101no—
264C2×C3×Q8—101no—
265C2×C3×Q8—101no—
266C2×C3×Q8D2011yescharacteristic
267C2×C3×D4—21no—
268C2×C3×D4—52no—
269C2×C3×D4—101no—
270C2×C3×D4—101no—
271C2×C3×D4—101no—
272C2×C3×D4—101no—
273C2×C3×D4—101no—
274C2×C3×D4D2011yescharacteristic
275C22×C3×C4—21no—
276C22×C3×C4—101no—
277C22×C3×C4C22×D511yescharacteristic
278C3×Q16—104no—
279C3×SD16—104no—
280C3×SD16—104no—
281C3×SD16—104no—
282C3×SD16—104no—
283C3×D8—104no—
284C3×M16—21no—
285C3×M16—24no—
286C3×M16—52no—
287C3×M16—52no—
288C3×M16—101no—
289C3×M16—101no—
290C3×M16C22×D512yes—
291C2×C3×C8—21no—
292C2×C3×C8—24no—
293C2×C3×C8—101no—
294C2×C3×C8—101no—
295C2×C3×C8C22⋊D511yescharacteristic
296C2×C3×C8C22⋊D511yescharacteristic
297C2×C3×C8C22⋊D512yes—
298C2×C3×C8C22×D512yes—
299C23×C5—21no—
300C23×C5C3×#4111yescharacteristic
301C22×D5—22no—
302C22×D5—41no—
303C22×D5—41no—
304C22×D5—82no—
305C5×Q8—21no—
306C5×Q8C2×C3×D412yes—
307C5×D4—22no—
308C5×D4—22no—
309C5×D4—21no—
310C5×D4C2×C3×D412yes—
311C2×C4×C5—21no—
312C2×C4×C5—22no—
313C2×C4×C5—22no—
314C2×C4×C5C2×C3×D411yescharacteristic
315C2×C4×C5C2×C3×D411yescharacteristic
316C2×C4×C5C2×C3×D411yescharacteristic
317C2×C4×C5C2×C3×D411yescharacteristic
318C2×C4×C5C3×#4111yescharacteristic
319C2×C4×C5C24×C311yescharacteristic
320C2×Dic5—22no—
321C2×Dic5—41no—
322C2×Dic5—41no—
323D20—22no—
324D20—22no—
325D20—41no—
326D20—41no—
327Dic10—22no—
328Dic10—22no—
329Dic10—41no—
330Dic10—41no—
331C5×C8—22no—
332C5×C8C2×C3×D412yes—
333C5×C8C2×C3×D412yes—
334#168—22no—
335#168—22no—
336C2×#41C2×C3×D511yescharacteristic
337C22×Q8—51no—
338C22×D4—51no—
339#137—54no—
340#137—102no—
341AGL(1,ℤ/8)—54no—
342AGL(1,ℤ/8)—102no—
343C2×Q16—101no—
344C2×SD16—54no—
345C2×SD16—101no—
346C2×SD16—101no—
347C2×SD16—101no—
348C2×SD16—101no—
349C2×SD16—102no—
350C2×D8—101no—
351#131—22no—
352#131C2×C3×D514yescomplemented
353C2×M16—21no—
354C2×M16—51no—
355C2×M16—51no—
356C2×M16C2×C3×D511yescharacteristic
357C22×C8—21no—
358C22×C8C2×C3×D511yescharacteristic
359#108—54no—
360#101—54no—
361#100—54no—
362C2×C3×C5—21no—
363C2×C3×C5—21no—
364C2×C3×C5—42no—
365C2×C3×C5#47112yescomplemented
366C2×C3×C5C2×#12111yescharacteristic
367C3×D5—82no—
368C23×C3—21no—
369C23×C3—102no—
370C23×C3—201no—
371C23×C3—201no—
372C23×C3—402no—
373C23×C3#70911yescharacteristic
374C3×Q8—21no—
375C3×Q8—102no—
376C3×Q8—102no—
377C3×Q8—201no—
378C3×Q8—201no—
379C3×Q8C2×D2012yes—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—21no—
383C3×D4—102no—
384C3×D4—102no—
385C3×D4—201no—
386C3×D4—201no—
387C3×D4C2×D2012yes—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—102no—
392C2×C3×C4—201no—
393C2×C3×C4—201no—
394C2×C3×C4C2×D2011yescharacteristic
395C2×C3×C4C2×D2011yescharacteristic
396C2×C3×C4#70911yescharacteristic
397C2×C3×C4C2×C22⋊D511yescharacteristic
398C2×C3×C4C2×C22⋊D511yescharacteristic
399C2×C3×C4C23×D511yescharacteristic
400C3×C8—22no—
401C3×C8—102no—
402C3×C8—102no—
403C3×C8C2×C22⋊D512yes—
404C3×C8C2×C22⋊D512yes—
405C22×C5—22no—
406C22×C5—22no—
407C22×C5—42no—
408C22×C5C3×#12111yescharacteristic
409C22×C5C3×#12111yescharacteristic
410C22×C5C2×C3×#4111yescharacteristic
411C2×D5—42no—
412C2×D5—82no—
413C2×D5—82no—
414C2×D5—82no—
415C4×C5—22no—
416C4×C5C3×#12111yescharacteristic
417C4×C5C3×#12111yescharacteristic
418C4×C5C22×C3×D411yescharacteristic
419C4×C5C22×C3×D411yescharacteristic
420Dic5—42no—
421C24—101no—
422#41—21no—
423#41—21no—
424#41C3×D2012yes—
425#41C22×C3×D512yes—
426C2×Q8—52no—
427C2×Q8—101no—
428C2×Q8—101no—
429C2×Q8—101no—
430C2×Q8—101no—
431C2×Q8—101no—
432C2×Q8C3×D2011yescharacteristic
433C2×D4—21no—
434C2×D4—52no—
435C2×D4—101no—
436C2×D4—101no—
437C2×D4—101no—
438C2×D4—101no—
439C2×D4—101no—
440C2×D4C3×D2011yescharacteristic
441C22×C4—21no—
442C22×C4—101no—
443C22×C4C22×C3×D511yescharacteristic
444Q16—104no—
445SD16—104no—
446SD16—104no—
447SD16—104no—
448SD16—104no—
449D8—104no—
450M16—21no—
451M16—24no—
452M16—52no—
453M16—52no—
454M16—101no—
455M16—101no—
456M16C22×C3×D512yes—
457C2×C8—21no—
458C2×C8—24no—
459C2×C8—101no—
460C2×C8—101no—
461C2×C8C3×C22⋊D511yescharacteristic
462C2×C8C3×C22⋊D511yescharacteristic
463C2×C8C3×C22⋊D512yes—
464C2×C8C22×C3×D512yes—
465C3×C5C2×#47111yescharacteristic, complemented, Hall
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—42no—
469C22×C3—202no—
470C22×C3—402no—
471C22×C3—402no—
472C22×C3—402no—
473C22×C3#407611yescharacteristic
474C22×C3#407611yescharacteristic
475C22×C3C2×#70911yescharacteristic
476C3×C4—22no—
477C3×C4—202no—
478C3×C4#407611yescharacteristic
479C3×C4#407611yescharacteristic
480C3×C4C22×D2011yescharacteristic
481C3×C4C22×C22⋊D511yescharacteristic
482C2×C5—21no—
483C2×C5—21no—
484C2×C5—42no—
485C2×C5C3×#47112yescomplemented
486C2×C5C2×C3×#12111yescharacteristic
487D5—82no—
488C23—21no—
489C23—102no—
490C23—201no—
491C23—201no—
492C23—402no—
493C23C3×#70911yescharacteristic
494Q8—21no—
495Q8—102no—
496Q8—102no—
497Q8—201no—
498Q8—201no—
499Q8C2×C3×D2012yes—
500D4—22no—
501D4—22no—
502D4—21no—
503D4—102no—
504D4—102no—
505D4—201no—
506D4—201no—
507D4C2×C3×D2012yes—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—102no—
512C2×C4—201no—
513C2×C4—201no—
514C2×C4C2×C3×D2011yescharacteristic
515C2×C4C2×C3×D2011yescharacteristic
516C2×C4C3×#70911yescharacteristic
517C2×C4C2×C3×C22⋊D511yescharacteristic
518C2×C4C2×C3×C22⋊D511yescharacteristic
519C2×C4C23×C3×D511yescharacteristic
520C8—22no—
521C8—102no—
522C8—102no—
523C8C2×C3×C22⋊D512yes—
524C8C2×C3×C22⋊D512yes—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—42no—
528C2×C3—402no—
529C2×C3#6543012yesdirect product factor
530C2×C3C2×#407611yescharacteristic
531C5C2×C3×#47111yescharacteristic, complemented, Sylow
532C22—22no—
533C22—22no—
534C22—42no—
535C22—202no—
536C22—402no—
537C22—402no—
538C22—402no—
539C22C3×#407611yescharacteristic
540C22C3×#407611yescharacteristic
541C22C2×C3×#70911yescharacteristic
542C4—22no—
543C4—202no—
544C4C3×#407611yescharacteristic
545C4C3×#407611yescharacteristic
546C4C22×C3×D2011yescharacteristic
547C4C22×C3×C22⋊D511yescharacteristic
548C3C2×#6543011yescharacteristic, direct product factor, Sylow
549C2—21no—
550C2—21no—
551C2—42no—
552C2—402no—
553C2C3×#6543012yesdirect product factor
554C2C2×C3×#407611yescharacteristic
subgroupG/H
centerC22×C3#4076
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#131C2
socleC22×C3×C5#121
prime radicalC2×C3×#4095C2
hypercenterC2×C3×#131D5
hypocenterC5C2×C3×#471
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#131C2semidirect
2. C2×C3×#4097C2semidirect
4. C22×C3×#677C2semidirect
5. C3×#65430C2direct (N × H)
6. C2×C3×#3956C2semidirect
7. C2×C3×#3955C2semidirect
8. C2×C3×#3951C2semidirect
9. C2×#65430C3direct (N × H)
14. C3×C5×#131C22semidirect
20. C3×#4097C22semidirect
32. C2×C3×#677C22semidirect
33. C3×#3956C22semidirect
34. C3×#3955C22semidirect
35. C3×#3951C22semidirect
37. C2×C5×#131C2×C3semidirect
38. C2×#4097C2×C3semidirect
40. C22×#677C2×C3semidirect
41. #65430C2×C3direct (N × H)
42. C2×#3956C2×C3semidirect
43. C2×#3955C2×C3semidirect
44. C2×#3951C2×C3semidirect
92. C2×C3×#131D5semidirect
101. C5×#131C22×C3semidirect
107. #4097C22×C3semidirect
119. C2×#677C22×C3semidirect
120. #3956C22×C3semidirect
121. #3955C22×C3semidirect
122. #3951C22×C3semidirect
177. C3×#131C2×D5semidirect
234. C2×#131C3×D5semidirect
352. #131C2×C3×D5semidirect
365. C2×C3×C5#471semidirect
465. C3×C5C2×#471semidirect
485. C2×C5C3×#471semidirect
529. C2×C3#65430direct (N × H)
531. C5C2×C3×#471semidirect
548. C3C2×#65430direct (N × H)
553. C2C3×#65430direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1312
C2×C3×#40972
C2×C3×#40952
C22×C3×#6772
C3×#654302
C2×C3×#39562
C2×C3×#39552
C2×C3×#39512
C2×#654303
C2×C3×#4715