Finite Group Databasebeta

← C3×C5×#2002 #14275036

Subgroups — C3×C5×#2002 #14275036

Order 1920.

956 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×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C3×C5×#535C211yesmaximal, characteristic, complemented
3C2×C3×C5×#127C211yesmaximal, characteristic, complemented
4C2×C3×C5×#104C211yesmaximal, characteristic, complemented
5C3×C5×#418C211yesmaximal, characteristic, complemented
6C2×C3×C5×#100C211yesmaximal, characteristic, complemented
7C3×C5×#339C211yesmaximal, characteristic, complemented
8C5×#2002C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×C5×D4—22no—
11C22×C3×C5×D4C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13C3×C5×AGL(1,ℤ/8)—22no—
14C3×C5×AGL(1,ℤ/8)—22no—
15C2×C3×C5×SD16—21no—
16C2×C3×C5×D8—21no—
17C2×C3×C5×M16C2211yescharacteristic
18C2×C3×C5×M16C2211yescharacteristic
19C3×C5×#127—22no—
20C3×C5×#127—21no—
21C3×C5×#127—22no—
22C3×C5×#124—21no—
23C3×C5×#121—22no—
24C3×C5×#120—22no—
25C3×C5×#117C2211yescharacteristic
26C2×C3×C42×C5C2211yescharacteristic
27C3×C5×#104—22no—
28C3×C5×#102—22no—
29C3×C5×#100—22no—
30C3×#2002C511yesmaximal, characteristic, direct product factor, Hall
31C2×C5×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
32C5×#535C2×C311yescharacteristic, complemented
33C2×C5×#127C2×C311yescharacteristic, complemented
34C2×C5×#104C2×C311yescharacteristic, complemented
35C5×#418C2×C311yescharacteristic, complemented
36C2×C5×#100C2×C311yescharacteristic, complemented
37C5×#339C2×C311yescharacteristic, complemented
38C24×C3×C5—21no—
39C24×C3×C5D412yes—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C2×C3×C5×Q8D411yescharacteristic, complemented
45C2×C3×C5×D4—28no—
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4—24no—
48C2×C3×C5×D4—21no—
49C2×C3×C5×D4—21no—
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4—21no—
53C2×C3×C5×D4—21no—
54C2×C3×C5×D4—21no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4—21no—
59C2×C3×C5×D4—48no—
60C2×C3×C5×D4D412yescomplemented
61C2×C3×C5×D4D411yescharacteristic, complemented
62C22×C3×C4×C5—21no—
63C22×C3×C4×C5—21no—
64C22×C3×C4×C5C2311yescharacteristic
65C3×C5×SD16—24no—
66C3×C5×D8—24no—
67C3×C5×M16—21no—
68C3×C5×M16—21no—
69C3×C5×M16—21no—
70C3×C5×M16D412yescomplemented
71C2×C3×C5×C8—21no—
72C2×C3×C5×C8—21no—
73C3×C5×C4⋊C4D412yescomplemented
74C3×C5×C22⋊C4—24no—
75C3×C5×C22⋊C4—21no—
76C3×C42×C5—21no—
77C3×C42×C5—21no—
78C3×C42×C5D412yescomplemented
79C2×C3×AGL(1,ℤ/8)C2×C511yescharacteristic, complemented
80C3×#535C2×C511yescharacteristic, complemented
81C2×C3×#127C2×C511yescharacteristic, complemented
82C2×C3×#104C2×C511yescharacteristic, complemented
83C3×#418C2×C511yescharacteristic, complemented
84C2×C3×#100C2×C511yescharacteristic, complemented
85C3×#339C2×C511yescharacteristic, complemented
86C2×C5×#41C22×C311yescharacteristic
87C22×C5×D4—22no—
88C22×C5×D4C22×C311yescharacteristic
89C22×C5×D4C22×C311yescharacteristic
90C5×AGL(1,ℤ/8)—22no—
91C5×AGL(1,ℤ/8)—22no—
92C2×C5×SD16—21no—
93C2×C5×D8—21no—
94C2×C5×M16C22×C311yescharacteristic
95C2×C5×M16C22×C311yescharacteristic
96C5×#127—22no—
97C5×#127—21no—
98C5×#127—22no—
99C5×#124—21no—
100C5×#121—22no—
101C5×#120—22no—
102C5×#117C22×C311yescharacteristic
103C2×C42×C5C22×C311yescharacteristic
104C5×#104—22no—
105C5×#102—22no—
106C5×#100—22no—
107#2002C3×C511yescharacteristic, direct product factor, Sylow
108C23×C3×C5—22no—
109C23×C3×C5—22no—
110C23×C3×C5—22no—
111C23×C3×C5—21no—
112C23×C3×C5—22no—
113C23×C3×C5—44no—
114C23×C3×C5—44no—
115C23×C3×C5—41no—
116C23×C3×C5—41no—
117C23×C3×C5C2×D411yescharacteristic
118C3×C5×Q8—21no—
119C3×C5×Q8—21no—
120C3×C5×D4—21no—
121C3×C5×D4—22no—
122C3×C5×D4—22no—
123C3×C5×D4—21no—
124C3×C5×D4—416no—
125C3×C5×D4—41no—
126C3×C5×D4—41no—
127C3×C5×D4—41no—
128C3×C5×D4—41no—
129C3×C5×D4—41no—
130C3×C5×D4—41no—
131C3×C5×D4—41no—
132C3×C5×D4—41no—
133C3×C5×D4—84no—
134C2×C3×C4×C5—24no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—41no—
139C2×C3×C4×C5—41no—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5#4111yescharacteristic
146C3×C5×C8—22no—
147C3×C5×C8—41no—
148C2×C3×#41C22×C511yescharacteristic
149C22×C3×D4—22no—
150C22×C3×D4C22×C511yescharacteristic
151C22×C3×D4C22×C511yescharacteristic
152C3×AGL(1,ℤ/8)—22no—
153C3×AGL(1,ℤ/8)—22no—
154C2×C3×SD16—21no—
155C2×C3×D8—21no—
156C2×C3×M16C22×C511yescharacteristic
157C2×C3×M16C22×C511yescharacteristic
158C3×#127—22no—
159C3×#127—21no—
160C3×#127—22no—
161C3×#124—21no—
162C3×#121—22no—
163C3×#120—22no—
164C3×#117C22×C511yescharacteristic
165C2×C3×C42C22×C511yescharacteristic
166C3×#104—22no—
167C3×#102—22no—
168C3×#100—22no—
169C24×C5—21no—
170C24×C5C3×D412yes—
171C5×#41—21no—
172C5×#41—21no—
173C5×#41—21no—
174C5×#41—21no—
175C2×C5×Q8C3×D411yescharacteristic, complemented
176C2×C5×D4—28no—
177C2×C5×D4—21no—
178C2×C5×D4—24no—
179C2×C5×D4—21no—
180C2×C5×D4—21no—
181C2×C5×D4—21no—
182C2×C5×D4—21no—
183C2×C5×D4—21no—
184C2×C5×D4—21no—
185C2×C5×D4—21no—
186C2×C5×D4—21no—
187C2×C5×D4—21no—
188C2×C5×D4—21no—
189C2×C5×D4—21no—
190C2×C5×D4—48no—
191C2×C5×D4C3×D411yescharacteristic, complemented
192C2×C5×D4C3×D412yescomplemented
193C22×C4×C5—21no—
194C22×C4×C5—21no—
195C22×C4×C5C23×C311yescharacteristic
196C5×SD16—24no—
197C5×D8—24no—
198C5×M16—21no—
199C5×M16—21no—
200C5×M16—21no—
201C5×M16C3×D412yescomplemented
202C2×C5×C8—21no—
203C2×C5×C8—21no—
204C5×C4⋊C4C3×D412yescomplemented
205C5×C22⋊C4—24no—
206C5×C22⋊C4—21no—
207C42×C5—21no—
208C42×C5—21no—
209C42×C5C3×D412yescomplemented
210C2×AGL(1,ℤ/8)C2×C3×C511yescharacteristic, complemented
211#535C2×C3×C511yescharacteristic, complemented
212C2×#127C2×C3×C511yescharacteristic, complemented
213C2×#104C2×C3×C511yescharacteristic, complemented
214#418C2×C3×C511yescharacteristic, complemented
215C2×#100C2×C3×C511yescharacteristic, complemented
216#339C2×C3×C511yescharacteristic, complemented
217C22×C3×C5—22no—
218C22×C3×C5—44no—
219C22×C3×C5—44no—
220C22×C3×C5—41no—
221C22×C3×C5—42no—
222C22×C3×C5—42no—
223C22×C3×C5—42no—
224C22×C3×C5—44no—
225C22×C3×C5—82no—
226C22×C3×C5—82no—
227C22×C3×C5—82no—
228C22×C3×C5#12011yescharacteristic
229C22×C3×C5#12011yescharacteristic
230C22×C3×C5#12111yescharacteristic
231C3×C4×C5—24no—
232C3×C4×C5—41no—
233C3×C4×C5—42no—
234C3×C4×C5#12011yescharacteristic
235C3×C4×C5#12111yescharacteristic
236C3×C4×C5#12111yescharacteristic
237C3×C4×C5#12711yescharacteristic
238C24×C3—21no—
239C24×C3C5×D412yes—
240C3×#41—21no—
241C3×#41—21no—
242C3×#41—21no—
243C3×#41—21no—
244C2×C3×Q8C5×D411yescharacteristic, complemented
245C2×C3×D4—28no—
246C2×C3×D4—21no—
247C2×C3×D4—24no—
248C2×C3×D4—21no—
249C2×C3×D4—21no—
250C2×C3×D4—21no—
251C2×C3×D4—21no—
252C2×C3×D4—21no—
253C2×C3×D4—21no—
254C2×C3×D4—21no—
255C2×C3×D4—21no—
256C2×C3×D4—21no—
257C2×C3×D4—21no—
258C2×C3×D4—21no—
259C2×C3×D4—48no—
260C2×C3×D4C5×D411yescharacteristic, complemented
261C2×C3×D4C5×D412yescomplemented
262C22×C3×C4—21no—
263C22×C3×C4—21no—
264C22×C3×C4C23×C511yescharacteristic
265C3×SD16—24no—
266C3×D8—24no—
267C3×M16—21no—
268C3×M16—21no—
269C3×M16—21no—
270C3×M16C5×D412yescomplemented
271C2×C3×C8—21no—
272C2×C3×C8—21no—
273C3×C4⋊C4C5×D412yescomplemented
274C3×C22⋊C4—24no—
275C3×C22⋊C4—21no—
276C3×C42—21no—
277C3×C42—21no—
278C3×C42C5×D412yescomplemented
279C23×C5—22no—
280C23×C5—21no—
281C23×C5—22no—
282C23×C5—22no—
283C23×C5—22no—
284C23×C5—44no—
285C23×C5—44no—
286C23×C5—41no—
287C23×C5—41no—
288C23×C5C2×C3×D411yescharacteristic
289C5×Q8—21no—
290C5×Q8—21no—
291C5×D4—21no—
292C5×D4—22no—
293C5×D4—22no—
294C5×D4—21no—
295C5×D4—416no—
296C5×D4—41no—
297C5×D4—41no—
298C5×D4—41no—
299C5×D4—41no—
300C5×D4—41no—
301C5×D4—41no—
302C5×D4—41no—
303C5×D4—41no—
304C5×D4—84no—
305C2×C4×C5—24no—
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—21no—
309C2×C4×C5—41no—
310C2×C4×C5—41no—
311C2×C4×C5C2×C3×D411yescharacteristic
312C2×C4×C5C2×C3×D411yescharacteristic
313C2×C4×C5C2×C3×D411yescharacteristic
314C2×C4×C5C2×C3×D411yescharacteristic
315C2×C4×C5C2×C3×D411yescharacteristic
316C2×C4×C5C3×#4111yescharacteristic
317C5×C8—22no—
318C5×C8—41no—
319C2×#41C22×C3×C511yescharacteristic
320C22×D4—22no—
321C22×D4C22×C3×C511yescharacteristic
322C22×D4C22×C3×C511yescharacteristic
323AGL(1,ℤ/8)—22no—
324AGL(1,ℤ/8)—22no—
325C2×SD16—21no—
326C2×D8—21no—
327C2×M16C22×C3×C511yescharacteristic
328C2×M16C22×C3×C511yescharacteristic
329#127—22no—
330#127—21no—
331#127—22no—
332#124—21no—
333#121—22no—
334#120—22no—
335#117C22×C3×C511yescharacteristic
336C2×C42C22×C3×C511yescharacteristic
337#104—22no—
338#102—22no—
339#100—22no—
340C2×C3×C5—21no—
341C2×C3×C5—21no—
342C2×C3×C5—81no—
343C2×C3×C5—82no—
344C2×C3×C5—82no—
345C2×C3×C5#39211yescharacteristic
346C2×C3×C5#45312yes—
347C23×C3—22no—
348C23×C3—21no—
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—44no—
353C23×C3—44no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3C2×C5×D411yescharacteristic
357C3×Q8—21no—
358C3×Q8—21no—
359C3×D4—21no—
360C3×D4—22no—
361C3×D4—22no—
362C3×D4—21no—
363C3×D4—416no—
364C3×D4—41no—
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—84no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4C2×C5×D411yescharacteristic
380C2×C3×C4C2×C5×D411yescharacteristic
381C2×C3×C4C2×C5×D411yescharacteristic
382C2×C3×C4C2×C5×D411yescharacteristic
383C2×C3×C4C2×C5×D411yescharacteristic
384C2×C3×C4C5×#4111yescharacteristic
385C3×C8—22no—
386C3×C8—41no—
387C22×C5—22no—
388C22×C5—44no—
389C22×C5—44no—
390C22×C5—41no—
391C22×C5—42no—
392C22×C5—42no—
393C22×C5—42no—
394C22×C5—44no—
395C22×C5—82no—
396C22×C5—82no—
397C22×C5—82no—
398C22×C5C3×#12011yescharacteristic
399C22×C5C3×#12011yescharacteristic
400C22×C5C3×#12111yescharacteristic
401C4×C5—24no—
402C4×C5—41no—
403C4×C5—42no—
404C4×C5C3×#12011yescharacteristic
405C4×C5C3×#12111yescharacteristic
406C4×C5C3×#12111yescharacteristic
407C4×C5C3×#12711yescharacteristic
408C24—21no—
409C24C3×C5×D412yes—
410#41—21no—
411#41—21no—
412#41—21no—
413#41—21no—
414C2×Q8C3×C5×D411yescharacteristic, complemented
415C2×D4—28no—
416C2×D4—21no—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—48no—
430C2×D4C3×C5×D411yescharacteristic, complemented
431C2×D4C3×C5×D412yescomplemented
432C22×C4—21no—
433C22×C4—21no—
434C22×C4C23×C3×C511yescharacteristic
435SD16—24no—
436D8—24no—
437M16—21no—
438M16—21no—
439M16—21no—
440M16C3×C5×D412yescomplemented
441C2×C8—21no—
442C2×C8—21no—
443C4⋊C4C3×C5×D412yescomplemented
444C22⋊C4—24no—
445C22⋊C4—21no—
446C42—21no—
447C42—21no—
448C42C3×C5×D412yescomplemented
449C3×C5#200211yescharacteristic, direct product factor, Hall
450C22×C3—22no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—41no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—44no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3C5×#12011yescharacteristic
462C22×C3C5×#12011yescharacteristic
463C22×C3C5×#12111yescharacteristic
464C3×C4—24no—
465C3×C4—41no—
466C3×C4—42no—
467C3×C4C5×#12011yescharacteristic
468C3×C4C5×#12111yescharacteristic
469C3×C4C5×#12111yescharacteristic
470C3×C4C5×#12711yescharacteristic
471C2×C5—21no—
472C2×C5—21no—
473C2×C5—82no—
474C2×C5—82no—
475C2×C5—81no—
476C2×C5C3×#39211yescharacteristic
477C2×C5C3×#45312yes—
478C23—22no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—44no—
484C23—44no—
485C23—41no—
486C23—41no—
487C23C2×C3×C5×D411yescharacteristic
488Q8—21no—
489Q8—21no—
490D4—21no—
491D4—22no—
492D4—22no—
493D4—21no—
494D4—416no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—84no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4C2×C3×C5×D411yescharacteristic
511C2×C4C2×C3×C5×D411yescharacteristic
512C2×C4C2×C3×C5×D411yescharacteristic
513C2×C4C2×C3×C5×D411yescharacteristic
514C2×C4C2×C3×C5×D411yescharacteristic
515C2×C4C3×C5×#4111yescharacteristic
516C8—22no—
517C8—41no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—82no—
521C2×C3—82no—
522C2×C3—81no—
523C2×C3C5×#39211yescharacteristic
524C2×C3C5×#45312yes—
525C5C3×#200211yescharacteristic, direct product factor, Sylow
526C22—22no—
527C22—44no—
528C22—44no—
529C22—42no—
530C22—44no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22C3×C5×#12011yescharacteristic
538C22C3×C5×#12011yescharacteristic
539C22C3×C5×#12111yescharacteristic
540C4—24no—
541C4—42no—
542C4—41no—
543C4C3×C5×#12011yescharacteristic
544C4C3×C5×#12111yescharacteristic
545C4C3×C5×#12111yescharacteristic
546C4C3×C5×#12711yescharacteristic
547C3C5×#200211yescharacteristic, direct product factor, Sylow
548C2—21no—
549C2—21no—
550C2—82no—
551C2—82no—
552C2—81no—
553C2C3×C5×#39211yescharacteristic
554C2C3×C5×#45312yes—
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×AGL(1,ℤ/8)C2semidirect
2. C3×C5×#535C2semidirect
3. C2×C3×C5×#127C2semidirect
4. C2×C3×C5×#104C2semidirect
5. C3×C5×#418C2semidirect
6. C2×C3×C5×#100C2semidirect
7. C3×C5×#339C2semidirect
8. C5×#2002C3direct (N × H)
30. C3×#2002C5direct (N × H)
31. C2×C5×AGL(1,ℤ/8)C2×C3semidirect
32. C5×#535C2×C3semidirect
33. C2×C5×#127C2×C3semidirect
34. C2×C5×#104C2×C3semidirect
35. C5×#418C2×C3semidirect
36. C2×C5×#100C2×C3semidirect
37. C5×#339C2×C3semidirect
44. C2×C3×C5×Q8D4semidirect
60. C2×C3×C5×D4D4semidirect
61. C2×C3×C5×D4D4semidirect
70. C3×C5×M16D4semidirect
73. C3×C5×C4⋊C4D4semidirect
78. C3×C42×C5D4semidirect
79. C2×C3×AGL(1,ℤ/8)C2×C5semidirect
80. C3×#535C2×C5semidirect
81. C2×C3×#127C2×C5semidirect
82. C2×C3×#104C2×C5semidirect
83. C3×#418C2×C5semidirect
84. C2×C3×#100C2×C5semidirect
85. C3×#339C2×C5semidirect
107. #2002C3×C5direct (N × H)
175. C2×C5×Q8C3×D4semidirect
191. C2×C5×D4C3×D4semidirect
192. C2×C5×D4C3×D4semidirect
201. C5×M16C3×D4semidirect
204. C5×C4⋊C4C3×D4semidirect
209. C42×C5C3×D4semidirect
210. C2×AGL(1,ℤ/8)C2×C3×C5semidirect
211. #535C2×C3×C5semidirect
212. C2×#127C2×C3×C5semidirect
213. C2×#104C2×C3×C5semidirect
214. #418C2×C3×C5semidirect
215. C2×#100C2×C3×C5semidirect
216. #339C2×C3×C5semidirect
244. C2×C3×Q8C5×D4semidirect
260. C2×C3×D4C5×D4semidirect
261. C2×C3×D4C5×D4semidirect
270. C3×M16C5×D4semidirect
273. C3×C4⋊C4C5×D4semidirect
278. C3×C42C5×D4semidirect
414. C2×Q8C3×C5×D4semidirect
430. C2×D4C3×C5×D4semidirect
431. C2×D4C3×C5×D4semidirect
440. M16C3×C5×D4semidirect
443. C4⋊C4C3×C5×D4semidirect
448. C42C3×C5×D4semidirect
449. C3×C5#2002direct (N × H)
525. C5C3×#2002direct (N × H)
547. C3C5×#2002direct (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×AGL(1,ℤ/8)2
C3×C5×#5352
C2×C3×C5×#1272
C2×C3×C5×#1042
C3×C5×#4182
C2×C3×C5×#1002
C3×C5×#3392
C5×#20023
C3×#20025