Finite Group Databasebeta

← C3×C5×#1906 #14274940

Subgroups — C3×C5×#1906 #14274940

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#575C211yesmaximal, characteristic
2C3×C4×C5×#41C211yesmaximal, characteristic, complemented
3C3×C5×#421C211yesmaximal, characteristic, complemented
4C2×C3×C5×#104C211yesmaximal, characteristic, complemented
5C3×C5×#410C211yesmaximal, characteristic, complemented
6C3×C5×#344C211yesmaximal, characteristic, complemented
7C3×C5×#339C211yesmaximal, characteristic, complemented
8C5×#1906C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×#41C2211yescharacteristic
11C3×C5×#137—21no—
12C3×C5×#137C412yescomplemented
13C3×C5×AGL(1,ℤ/8)—21no—
14C3×C5×AGL(1,ℤ/8)C412yescomplemented
15C3×C5×#135—22no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C3×C5×M16C2211yescharacteristic
18C3×C4×C5×Q8—21no—
19C3×C4×C5×D4—21no—
20C3×C4×C5×D4—22no—
21C3×C5×#117—21no—
22C3×C5×#117C2211yescharacteristic
23C3×C5×#117C2211yescharacteristic
24C2×C3×C42×C5—21no—
25C2×C3×C42×C5C2211yescharacteristic
26C3×C5×#104—22no—
27C3×C5×#104—21no—
28C3×C5×#104—21no—
29C3×C5×#99—22no—
30C3×#1906C511yesmaximal, characteristic, direct product factor, Hall
31C5×#575C2×C311yescharacteristic
32C4×C5×#41C2×C311yescharacteristic, complemented
33C5×#421C2×C311yescharacteristic, complemented
34C2×C5×#104C2×C311yescharacteristic, complemented
35C5×#410C2×C311yescharacteristic, complemented
36C5×#344C2×C311yescharacteristic, complemented
37C5×#339C2×C311yescharacteristic, complemented
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C3×C5×#41C2×C412yes—
45C3×C5×#41D412yes—
46C2×C3×C5×Q8C2×C411yescharacteristic, complemented
47C2×C3×C5×Q8D411yescharacteristic
48C2×C3×C5×D4—21no—
49C2×C3×C5×D4—21no—
50C2×C3×C5×D4C2×C411yescharacteristic, complemented
51C2×C3×C5×D4D411yescharacteristic
52C22×C3×C4×C5—21no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—21no—
55C22×C3×C4×C5—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C3×C5×Q16—22no—
58C3×C5×SD16—22no—
59C3×C5×SD16—22no—
60C3×C5×D8—22no—
61C3×C5×M16—21no—
62C3×C5×M16—21no—
63C3×C5×M16—21no—
64C3×C5×M16C2×C412yescomplemented
65C2×C3×C5×C8—21no—
66C2×C3×C5×C8—21no—
67C3×C5×C4⋊C4—22no—
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4—21no—
70C3×C5×C22⋊C4—22no—
71C3×C5×C22⋊C4—21no—
72C3×C5×C22⋊C4D412yes—
73C3×C42×C5—22no—
74C3×C42×C5—21no—
75C3×C42×C5—21no—
76C3×C42×C5—21no—
77C3×C42×C5D411yescharacteristic
78C3×C42×C5D411yescharacteristic
79C3×#575C2×C511yescharacteristic
80C3×C4×#41C2×C511yescharacteristic, complemented
81C3×#421C2×C511yescharacteristic, complemented
82C2×C3×#104C2×C511yescharacteristic, complemented
83C3×#410C2×C511yescharacteristic, complemented
84C3×#344C2×C511yescharacteristic, complemented
85C3×#339C2×C511yescharacteristic, complemented
86C2×C5×#41C22×C311yescharacteristic
87C2×C5×#41C22×C311yescharacteristic
88C5×#137—21no—
89C5×#137C3×C412yescomplemented
90C5×AGL(1,ℤ/8)—21no—
91C5×AGL(1,ℤ/8)C3×C412yescomplemented
92C5×#135—22no—
93C2×C5×M16C22×C311yescharacteristic
94C2×C5×M16C22×C311yescharacteristic
95C4×C5×Q8—21no—
96C4×C5×D4—21no—
97C4×C5×D4—22no—
98C5×#117—21no—
99C5×#117C22×C311yescharacteristic
100C5×#117C22×C311yescharacteristic
101C2×C42×C5—21no—
102C2×C42×C5C22×C311yescharacteristic
103C5×#104—22no—
104C5×#104—21no—
105C5×#104—21no—
106C5×#99—22no—
107#1906C3×C511yescharacteristic, direct product factor, Sylow
108C23×C3×C5—21no—
109C23×C3×C5—21no—
110C23×C3×C5C2×D411yescharacteristic
111C3×C5×Q8—21no—
112C3×C5×Q8—21no—
113C3×C5×Q8—21no—
114C3×C5×Q8C22⋊C412yes—
115C3×C5×D4—22no—
116C3×C5×D4—22no—
117C3×C5×D4—21no—
118C3×C5×D4—21no—
119C3×C5×D4—21no—
120C3×C5×D4—41no—
121C3×C5×D4—41no—
122C3×C5×D4C22⋊C412yes—
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—42no—
131C2×C3×C4×C5—44no—
132C2×C3×C4×C5—41no—
133C2×C3×C4×C5—41no—
134C2×C3×C4×C5—41no—
135C2×C3×C4×C5C22×C411yescharacteristic
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5#4111yescharacteristic
140C2×C3×C4×C5#4111yescharacteristic
141C3×C5×C8—22no—
142C3×C5×C8—41no—
143C2×C3×#41C22×C511yescharacteristic
144C2×C3×#41C22×C511yescharacteristic
145C3×#137—21no—
146C3×#137C4×C512yescomplemented
147C3×AGL(1,ℤ/8)—21no—
148C3×AGL(1,ℤ/8)C4×C512yescomplemented
149C3×#135—22no—
150C2×C3×M16C22×C511yescharacteristic
151C2×C3×M16C22×C511yescharacteristic
152C3×C4×Q8—21no—
153C3×C4×D4—21no—
154C3×C4×D4—22no—
155C3×#117—21no—
156C3×#117C22×C511yescharacteristic
157C3×#117C22×C511yescharacteristic
158C2×C3×C42—21no—
159C2×C3×C42C22×C511yescharacteristic
160C3×#104—22no—
161C3×#104—21no—
162C3×#104—21no—
163C3×#99—22no—
164C5×#41—21no—
165C5×#41—21no—
166C5×#41—21no—
167C5×#41—21no—
168C5×#41—21no—
169C5×#41—21no—
170C5×#41C2×C3×C412yes—
171C5×#41C3×D412yes—
172C2×C5×Q8C2×C3×C411yescharacteristic, complemented
173C2×C5×Q8C3×D411yescharacteristic
174C2×C5×D4—21no—
175C2×C5×D4—21no—
176C2×C5×D4C2×C3×C411yescharacteristic, complemented
177C2×C5×D4C3×D411yescharacteristic
178C22×C4×C5—21no—
179C22×C4×C5—22no—
180C22×C4×C5—21no—
181C22×C4×C5—21no—
182C22×C4×C5C23×C311yescharacteristic
183C5×Q16—22no—
184C5×SD16—22no—
185C5×SD16—22no—
186C5×D8—22no—
187C5×M16—21no—
188C5×M16—21no—
189C5×M16—21no—
190C5×M16C2×C3×C412yescomplemented
191C2×C5×C8—21no—
192C2×C5×C8—21no—
193C5×C4⋊C4—22no—
194C5×C4⋊C4—21no—
195C5×C4⋊C4—21no—
196C5×C22⋊C4—22no—
197C5×C22⋊C4—21no—
198C5×C22⋊C4C3×D412yes—
199C42×C5—22no—
200C42×C5—21no—
201C42×C5—21no—
202C42×C5—21no—
203C42×C5C3×D411yescharacteristic
204C42×C5C3×D411yescharacteristic
205#575C2×C3×C511yescharacteristic
206C4×#41C2×C3×C511yescharacteristic, complemented
207#421C2×C3×C511yescharacteristic, complemented
208C2×#104C2×C3×C511yescharacteristic, complemented
209#410C2×C3×C511yescharacteristic, complemented
210#344C2×C3×C511yescharacteristic, complemented
211#339C2×C3×C511yescharacteristic, complemented
212C22×C3×C5—22no—
213C22×C3×C5—41no—
214C22×C3×C5—42no—
215C22×C3×C5—41no—
216C22×C3×C5—81no—
217C22×C3×C5C4×D411yescharacteristic
218C22×C3×C5#12011yescharacteristic
219C22×C3×C5#12111yescharacteristic
220C3×C4×C5—22no—
221C3×C4×C5—22no—
222C3×C4×C5—42no—
223C3×C4×C5—41no—
224C3×C4×C5—41no—
225C3×C4×C5—81no—
226C3×C4×C5C2×C22⋊C411yescharacteristic
227C3×C4×C5C4×D411yescharacteristic
228C3×C4×C5#12111yescharacteristic
229C3×C4×C5#12311yescharacteristic
230C3×#41—21no—
231C3×#41—21no—
232C3×#41—21no—
233C3×#41—21no—
234C3×#41—21no—
235C3×#41—21no—
236C3×#41C2×C4×C512yes—
237C3×#41C5×D412yes—
238C2×C3×Q8C2×C4×C511yescharacteristic, complemented
239C2×C3×Q8C5×D411yescharacteristic
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4C2×C4×C511yescharacteristic, complemented
243C2×C3×D4C5×D411yescharacteristic
244C22×C3×C4—21no—
245C22×C3×C4—22no—
246C22×C3×C4—21no—
247C22×C3×C4—21no—
248C22×C3×C4C23×C511yescharacteristic
249C3×Q16—22no—
250C3×SD16—22no—
251C3×SD16—22no—
252C3×D8—22no—
253C3×M16—21no—
254C3×M16—21no—
255C3×M16—21no—
256C3×M16C2×C4×C512yescomplemented
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4C5×D412yes—
265C3×C42—22no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42C5×D411yescharacteristic
270C3×C42C5×D411yescharacteristic
271C23×C5—21no—
272C23×C5—21no—
273C23×C5C2×C3×D411yescharacteristic
274C5×Q8—21no—
275C5×Q8—21no—
276C5×Q8—21no—
277C5×Q8C3×C22⋊C412yes—
278C5×D4—22no—
279C5×D4—22no—
280C5×D4—21no—
281C5×D4—21no—
282C5×D4—21no—
283C5×D4—41no—
284C5×D4—41no—
285C5×D4C3×C22⋊C412yes—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—21no—
291C2×C4×C5—22no—
292C2×C4×C5—21no—
293C2×C4×C5—42no—
294C2×C4×C5—44no—
295C2×C4×C5—41no—
296C2×C4×C5—41no—
297C2×C4×C5—41no—
298C2×C4×C5C22×C3×C411yescharacteristic
299C2×C4×C5C2×C3×D411yescharacteristic
300C2×C4×C5C2×C3×D411yescharacteristic
301C2×C4×C5C2×C3×D411yescharacteristic
302C2×C4×C5C3×#4111yescharacteristic
303C2×C4×C5C3×#4111yescharacteristic
304C5×C8—22no—
305C5×C8—41no—
306C2×#41C22×C3×C511yescharacteristic
307C2×#41C22×C3×C511yescharacteristic
308#137—21no—
309#137C3×C4×C512yescomplemented
310AGL(1,ℤ/8)—21no—
311AGL(1,ℤ/8)C3×C4×C512yescomplemented
312#135—22no—
313C2×M16C22×C3×C511yescharacteristic
314C2×M16C22×C3×C511yescharacteristic
315C4×Q8—21no—
316C4×D4—21no—
317C4×D4—22no—
318#117—21no—
319#117C22×C3×C511yescharacteristic
320#117C22×C3×C511yescharacteristic
321C2×C42—21no—
322C2×C42C22×C3×C511yescharacteristic
323#104—22no—
324#104—21no—
325#104—21no—
326#99—22no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5—42no—
331C2×C3×C5—81no—
332C2×C3×C5#38611yescharacteristic
333C23×C3—21no—
334C23×C3—21no—
335C23×C3C2×C5×D411yescharacteristic
336C3×Q8—21no—
337C3×Q8—21no—
338C3×Q8—21no—
339C3×Q8C5×C22⋊C412yes—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—21no—
343C3×D4—21no—
344C3×D4—21no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4C5×C22⋊C412yes—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4C22×C4×C511yescharacteristic
361C2×C3×C4C2×C5×D411yescharacteristic
362C2×C3×C4C2×C5×D411yescharacteristic
363C2×C3×C4C2×C5×D411yescharacteristic
364C2×C3×C4C5×#4111yescharacteristic
365C2×C3×C4C5×#4111yescharacteristic
366C3×C8—22no—
367C3×C8—41no—
368C22×C5—22no—
369C22×C5—41no—
370C22×C5—42no—
371C22×C5—41no—
372C22×C5—81no—
373C22×C5C3×C4×D411yescharacteristic
374C22×C5C3×#12011yescharacteristic
375C22×C5C3×#12111yescharacteristic
376C4×C5—22no—
377C4×C5—22no—
378C4×C5—42no—
379C4×C5—41no—
380C4×C5—41no—
381C4×C5—81no—
382C4×C5C2×C3×C22⋊C411yescharacteristic
383C4×C5C3×C4×D411yescharacteristic
384C4×C5C3×#12111yescharacteristic
385C4×C5C3×#12311yescharacteristic
386#41—21no—
387#41—21no—
388#41—21no—
389#41—21no—
390#41—21no—
391#41—21no—
392#41C2×C3×C4×C512yes—
393#41C3×C5×D412yes—
394C2×Q8C2×C3×C4×C511yescharacteristic, complemented
395C2×Q8C3×C5×D411yescharacteristic
396C2×D4—21no—
397C2×D4—21no—
398C2×D4C2×C3×C4×C511yescharacteristic, complemented
399C2×D4C3×C5×D411yescharacteristic
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4C23×C3×C511yescharacteristic
405Q16—22no—
406SD16—22no—
407SD16—22no—
408D8—22no—
409M16—21no—
410M16—21no—
411M16—21no—
412M16C2×C3×C4×C512yescomplemented
413C2×C8—21no—
414C2×C8—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C22⋊C4—22no—
419C22⋊C4—21no—
420C22⋊C4C3×C5×D412yes—
421C42—22no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42C3×C5×D411yescharacteristic
426C42C3×C5×D411yescharacteristic
427C3×C5#190611yescharacteristic, direct product factor, Hall
428C22×C3—22no—
429C22×C3—41no—
430C22×C3—42no—
431C22×C3—41no—
432C22×C3—81no—
433C22×C3C4×C5×D411yescharacteristic
434C22×C3C5×#12011yescharacteristic
435C22×C3C5×#12111yescharacteristic
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—42no—
439C3×C4—41no—
440C3×C4—41no—
441C3×C4—81no—
442C3×C4C2×C5×C22⋊C411yescharacteristic
443C3×C4C4×C5×D411yescharacteristic
444C3×C4C5×#12111yescharacteristic
445C3×C4C5×#12311yescharacteristic
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—42no—
450C2×C5—81no—
451C2×C5C3×#38611yescharacteristic
452C23—21no—
453C23—21no—
454C23C2×C3×C5×D411yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—21no—
458Q8C3×C5×C22⋊C412yes—
459D4—22no—
460D4—22no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—41no—
465D4—41no—
466D4C3×C5×C22⋊C412yes—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4C22×C3×C4×C511yescharacteristic
480C2×C4C2×C3×C5×D411yescharacteristic
481C2×C4C2×C3×C5×D411yescharacteristic
482C2×C4C2×C3×C5×D411yescharacteristic
483C2×C4C3×C5×#4111yescharacteristic
484C2×C4C3×C5×#4111yescharacteristic
485C8—22no—
486C8—41no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—42no—
491C2×C3—81no—
492C2×C3C5×#38611yescharacteristic
493C5C3×#190611yescharacteristic, direct product factor, Sylow
494C22—22no—
495C22—41no—
496C22—42no—
497C22—41no—
498C22—81no—
499C22C3×C4×C5×D411yescharacteristic
500C22C3×C5×#12011yescharacteristic
501C22C3×C5×#12111yescharacteristic
502C4—22no—
503C4—22no—
504C4—42no—
505C4—41no—
506C4—41no—
507C4—81no—
508C4C2×C3×C5×C22⋊C411yescharacteristic
509C4C3×C4×C5×D411yescharacteristic
510C4C3×C5×#12111yescharacteristic
511C4C3×C5×#12311yescharacteristic
512C3C5×#190611yescharacteristic, direct product factor, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—42no—
517C2—81no—
518C2C3×C5×#38611yescharacteristic
subgroupG/H
centerC3×C4×C5#121
commutatorC2×C4C22×C3×C4×C5
Frattini, cosocleC22×C4C23×C3×C5
socleC2×C3×C5#386
prime radicalC3×C5×#575C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4×C5 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C4×C5×#41C2semidirect
3. C3×C5×#421C2semidirect
4. C2×C3×C5×#104C2semidirect
5. C3×C5×#410C2semidirect
6. C3×C5×#344C2semidirect
7. C3×C5×#339C2semidirect
8. C5×#1906C3direct (N × H)
12. C3×C5×#137C4semidirect
14. C3×C5×AGL(1,ℤ/8)C4semidirect
30. C3×#1906C5direct (N × H)
32. C4×C5×#41C2×C3semidirect
33. C5×#421C2×C3semidirect
34. C2×C5×#104C2×C3semidirect
35. C5×#410C2×C3semidirect
36. C5×#344C2×C3semidirect
37. C5×#339C2×C3semidirect
46. C2×C3×C5×Q8C2×C4semidirect
50. C2×C3×C5×D4C2×C4semidirect
64. C3×C5×M16C2×C4semidirect
80. C3×C4×#41C2×C5semidirect
81. C3×#421C2×C5semidirect
82. C2×C3×#104C2×C5semidirect
83. C3×#410C2×C5semidirect
84. C3×#344C2×C5semidirect
85. C3×#339C2×C5semidirect
89. C5×#137C3×C4semidirect
91. C5×AGL(1,ℤ/8)C3×C4semidirect
107. #1906C3×C5direct (N × H)
146. C3×#137C4×C5semidirect
148. C3×AGL(1,ℤ/8)C4×C5semidirect
172. C2×C5×Q8C2×C3×C4semidirect
176. C2×C5×D4C2×C3×C4semidirect
190. C5×M16C2×C3×C4semidirect
206. C4×#41C2×C3×C5semidirect
207. #421C2×C3×C5semidirect
208. C2×#104C2×C3×C5semidirect
209. #410C2×C3×C5semidirect
210. #344C2×C3×C5semidirect
211. #339C2×C3×C5semidirect
238. C2×C3×Q8C2×C4×C5semidirect
242. C2×C3×D4C2×C4×C5semidirect
256. C3×M16C2×C4×C5semidirect
309. #137C3×C4×C5semidirect
311. AGL(1,ℤ/8)C3×C4×C5semidirect
394. C2×Q8C2×C3×C4×C5semidirect
398. C2×D4C2×C3×C4×C5semidirect
412. M16C2×C3×C4×C5semidirect
427. C3×C5#1906direct (N × H)
493. C5C3×#1906direct (N × H)
512. C3C5×#1906direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5752
C3×C4×C5×#412
C3×C5×#4212
C2×C3×C5×#1042
C3×C5×#4102
C3×C5×#3442
C3×C5×#3392
C5×#19063
C3×#19065