Finite Group Databasebeta

← C3×C5×#2200 #14275234

Subgroups — C3×C5×#2200 #14275234

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#560C211yesmaximal, characteristic, complemented
2C3×C5×#458C211yesmaximal, characteristic, complemented
3C3×C5×B(4,𝔽2)C211yesmaximal, characteristic, complemented
4C3×C5×#453C211yesmaximal, characteristic, complemented
5C3×C5×#353C211yesmaximal, characteristic, complemented
6C3×C5×#352C211yesmaximal, characteristic, complemented
7C3×C5×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8C5×#2200C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×H(5,𝔽2)C2211yescharacteristic
10C3×C5×AGL(1,ℤ/8)—21no—
11C3×C5×#127C2211yescharacteristic
12C3×C5×#126—21no—
13C3×C5×#123—21no—
14C3×C5×#123—21no—
15C3×C5×#123C2211yescharacteristic
16C3×C5×#121—21no—
17C3×C5×#121—21no—
18C3×C5×#121—21no—
19C3×C5×#120—21no—
20C3×C5×#120—21no—
21C3×C5×#120C2211yescharacteristic
22C3×C5×#104—21no—
23C3×C5×#100C2211yescharacteristic
24C3×C5×#99—21no—
25C3×C5×#99—21no—
26C3×C5×#99C2211yescharacteristic
27C3×C5×#99C2211yescharacteristic
28C3×#2200C511yesmaximal, characteristic, direct product factor, Hall
29C5×#560C2×C311yescharacteristic, complemented
30C5×#458C2×C311yescharacteristic, complemented
31C5×B(4,𝔽2)C2×C311yescharacteristic, complemented
32C5×#453C2×C311yescharacteristic, complemented
33C5×#353C2×C311yescharacteristic, complemented
34C5×#352C2×C311yescharacteristic, complemented
35C5×AΣL(1,𝔽16)C2×C311yescharacteristic, complemented
36C24×C3×C5D411yescharacteristic
37C3×C5×#41—21no—
38C3×C5×#41—41no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4—21no—
43C2×C3×C5×D4—21no—
44C2×C3×C5×D4—21no—
45C2×C3×C5×D4—21no—
46C2×C3×C5×D4—41no—
47C2×C3×C5×D4—41no—
48C2×C3×C5×D4D411yescharacteristic, complemented
49C2×C3×C5×D4C2311yescharacteristic
50C22×C3×C4×C5—21no—
51C22×C3×C4×C5D411yescharacteristic, complemented
52C3×C5×SD16—41no—
53C3×C5×D8—41no—
54C3×C5×M16—21no—
55C3×C5×C4⋊C4—21no—
56C3×C5×C4⋊C4—21no—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4—41no—
59C3×C5×C22⋊C4—21no—
60C3×C5×C22⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C5×C22⋊C4—21no—
63C3×C5×C22⋊C4—21no—
64C3×C5×C22⋊C4—21no—
65C3×C5×C22⋊C4—21no—
66C3×C5×C22⋊C4—41no—
67C3×C5×C22⋊C4—41no—
68C3×C5×C22⋊C4D411yescharacteristic
69C3×C5×C22⋊C4D411yescharacteristic, complemented
70C3×C42×C5D411yescharacteristic, complemented
71C3×#560C2×C511yescharacteristic, complemented
72C3×#458C2×C511yescharacteristic, complemented
73C3×B(4,𝔽2)C2×C511yescharacteristic, complemented
74C3×#453C2×C511yescharacteristic, complemented
75C3×#353C2×C511yescharacteristic, complemented
76C3×#352C2×C511yescharacteristic, complemented
77C3×AΣL(1,𝔽16)C2×C511yescharacteristic, complemented
78C5×H(5,𝔽2)C22×C311yescharacteristic
79C5×AGL(1,ℤ/8)—21no—
80C5×#127C22×C311yescharacteristic
81C5×#126—21no—
82C5×#123—21no—
83C5×#123—21no—
84C5×#123C22×C311yescharacteristic
85C5×#121—21no—
86C5×#121—21no—
87C5×#121—21no—
88C5×#120—21no—
89C5×#120—21no—
90C5×#120C22×C311yescharacteristic
91C5×#104—21no—
92C5×#100C22×C311yescharacteristic
93C5×#99—21no—
94C5×#99—21no—
95C5×#99C22×C311yescharacteristic
96C5×#99C22×C311yescharacteristic
97#2200C3×C511yescharacteristic, direct product factor, Sylow
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5—21no—
101C23×C3×C5—21no—
102C23×C3×C5—41no—
103C23×C3×C5—81no—
104C23×C3×C5C2×D411yescharacteristic
105C23×C3×C5C2×D411yescharacteristic
106C3×C5×Q8—21no—
107C3×C5×D4—21no—
108C3×C5×D4—41no—
109C3×C5×D4—41no—
110C3×C5×D4—41no—
111C3×C5×D4—41no—
112C3×C5×D4—41no—
113C3×C5×D4—81no—
114C3×C5×D4—81no—
115C3×C5×D4—81no—
116C3×C5×D4—81no—
117C3×C5×D4—81no—
118C3×C5×D4—82no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—21no—
122C2×C3×C4×C5—21no—
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—41no—
126C2×C3×C4×C5—41no—
127C2×C3×C4×C5—41no—
128C2×C3×C4×C5—41no—
129C2×C3×C4×C5—81no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C3×C5×C8—41no—
132C3×H(5,𝔽2)C22×C511yescharacteristic
133C3×AGL(1,ℤ/8)—21no—
134C3×#127C22×C511yescharacteristic
135C3×#126—21no—
136C3×#123—21no—
137C3×#123—21no—
138C3×#123C22×C511yescharacteristic
139C3×#121—21no—
140C3×#121—21no—
141C3×#121—21no—
142C3×#120—21no—
143C3×#120—21no—
144C3×#120C22×C511yescharacteristic
145C3×#104—21no—
146C3×#100C22×C511yescharacteristic
147C3×#99—21no—
148C3×#99—21no—
149C3×#99C22×C511yescharacteristic
150C3×#99C22×C511yescharacteristic
151C24×C5C3×D411yescharacteristic
152C5×#41—21no—
153C5×#41—41no—
154C2×C5×D4—21no—
155C2×C5×D4—21no—
156C2×C5×D4—21no—
157C2×C5×D4—21no—
158C2×C5×D4—21no—
159C2×C5×D4—21no—
160C2×C5×D4—21no—
161C2×C5×D4—41no—
162C2×C5×D4—41no—
163C2×C5×D4C3×D411yescharacteristic, complemented
164C2×C5×D4C23×C311yescharacteristic
165C22×C4×C5—21no—
166C22×C4×C5C3×D411yescharacteristic, complemented
167C5×SD16—41no—
168C5×D8—41no—
169C5×M16—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—41no—
174C5×C22⋊C4—21no—
175C5×C22⋊C4—21no—
176C5×C22⋊C4—21no—
177C5×C22⋊C4—21no—
178C5×C22⋊C4—21no—
179C5×C22⋊C4—21no—
180C5×C22⋊C4—21no—
181C5×C22⋊C4—41no—
182C5×C22⋊C4—41no—
183C5×C22⋊C4C3×D411yescharacteristic
184C5×C22⋊C4C3×D411yescharacteristic, complemented
185C42×C5C3×D411yescharacteristic, complemented
186#560C2×C3×C511yescharacteristic, complemented
187#458C2×C3×C511yescharacteristic, complemented
188B(4,𝔽2)C2×C3×C511yescharacteristic, complemented
189#453C2×C3×C511yescharacteristic, complemented
190#353C2×C3×C511yescharacteristic, complemented
191#352C2×C3×C511yescharacteristic, complemented
192AΣL(1,𝔽16)C2×C3×C511yescharacteristic, complemented
193C22×C3×C5—21no—
194C22×C3×C5—21no—
195C22×C3×C5—41no—
196C22×C3×C5—41no—
197C22×C3×C5—41no—
198C22×C3×C5—41no—
199C22×C3×C5—41no—
200C22×C3×C5—41no—
201C22×C3×C5—42no—
202C22×C3×C5—41no—
203C22×C3×C5—41no—
204C22×C3×C5—41no—
205C22×C3×C5—81no—
206C22×C3×C5—81no—
207C22×C3×C5—81no—
208C22×C3×C5#12011yescharacteristic
209C3×C4×C5—21no—
210C3×C4×C5—41no—
211C3×C4×C5—41no—
212C3×C4×C5—41no—
213C3×C4×C5—41no—
214C3×C4×C5—41no—
215C3×C4×C5—81no—
216C3×C4×C5—81no—
217C24×C3C5×D411yescharacteristic
218C3×#41—21no—
219C3×#41—41no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—41no—
228C2×C3×D4—41no—
229C2×C3×D4C5×D411yescharacteristic, complemented
230C2×C3×D4C23×C511yescharacteristic
231C22×C3×C4—21no—
232C22×C3×C4C5×D411yescharacteristic, complemented
233C3×SD16—41no—
234C3×D8—41no—
235C3×M16—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—41no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—21no—
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—41no—
248C3×C22⋊C4—41no—
249C3×C22⋊C4C5×D411yescharacteristic
250C3×C22⋊C4C5×D411yescharacteristic, complemented
251C3×C42C5×D411yescharacteristic, complemented
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—41no—
257C23×C5—81no—
258C23×C5C2×C3×D411yescharacteristic
259C23×C5C2×C3×D411yescharacteristic
260C5×Q8—21no—
261C5×D4—21no—
262C5×D4—41no—
263C5×D4—41no—
264C5×D4—41no—
265C5×D4—41no—
266C5×D4—41no—
267C5×D4—81no—
268C5×D4—81no—
269C5×D4—81no—
270C5×D4—81no—
271C5×D4—81no—
272C5×D4—82no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—81no—
284C2×C4×C5C2×C3×D411yescharacteristic
285C5×C8—41no—
286H(5,𝔽2)C22×C3×C511yescharacteristic
287AGL(1,ℤ/8)—21no—
288#127C22×C3×C511yescharacteristic
289#126—21no—
290#123—21no—
291#123—21no—
292#123C22×C3×C511yescharacteristic
293#121—21no—
294#121—21no—
295#121—21no—
296#120—21no—
297#120—21no—
298#120C22×C3×C511yescharacteristic
299#104—21no—
300#100C22×C3×C511yescharacteristic
301#99—21no—
302#99—21no—
303#99C22×C3×C511yescharacteristic
304#99C22×C3×C511yescharacteristic
305C2×C3×C5—21no—
306C2×C3×C5—41no—
307C2×C3×C5—41no—
308C2×C3×C5—81no—
309C2×C3×C5—81no—
310C2×C3×C5—81no—
311C2×C3×C5B(4,𝔽2)11yescharacteristic
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—41no—
317C23×C3—81no—
318C23×C3C2×C5×D411yescharacteristic
319C23×C3C2×C5×D411yescharacteristic
320C3×Q8—21no—
321C3×D4—21no—
322C3×D4—41no—
323C3×D4—41no—
324C3×D4—41no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—81no—
328C3×D4—81no—
329C3×D4—81no—
330C3×D4—81no—
331C3×D4—81no—
332C3×D4—82no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—81no—
344C2×C3×C4C2×C5×D411yescharacteristic
345C3×C8—41no—
346C22×C5—21no—
347C22×C5—21no—
348C22×C5—41no—
349C22×C5—41no—
350C22×C5—41no—
351C22×C5—41no—
352C22×C5—41no—
353C22×C5—41no—
354C22×C5—42no—
355C22×C5—41no—
356C22×C5—41no—
357C22×C5—41no—
358C22×C5—81no—
359C22×C5—81no—
360C22×C5—81no—
361C22×C5C3×#12011yescharacteristic
362C4×C5—21no—
363C4×C5—41no—
364C4×C5—41no—
365C4×C5—41no—
366C4×C5—41no—
367C4×C5—41no—
368C4×C5—81no—
369C4×C5—81no—
370C24C3×C5×D411yescharacteristic
371#41—21no—
372#41—41no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4C3×C5×D411yescharacteristic, complemented
383C2×D4C23×C3×C511yescharacteristic
384C22×C4—21no—
385C22×C4C3×C5×D411yescharacteristic, complemented
386SD16—41no—
387D8—41no—
388M16—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—41no—
393C22⋊C4—21no—
394C22⋊C4—21no—
395C22⋊C4—21no—
396C22⋊C4—21no—
397C22⋊C4—21no—
398C22⋊C4—21no—
399C22⋊C4—21no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4C3×C5×D411yescharacteristic
403C22⋊C4C3×C5×D411yescharacteristic, complemented
404C42C3×C5×D411yescharacteristic, complemented
405C3×C5#220011yescharacteristic, direct product factor, Hall
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—42no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3—81no—
421C22×C3C5×#12011yescharacteristic
422C3×C4—21no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—81no—
429C3×C4—81no—
430C2×C5—21no—
431C2×C5—41no—
432C2×C5—41no—
433C2×C5—81no—
434C2×C5—81no—
435C2×C5—81no—
436C2×C5C3×B(4,𝔽2)11yescharacteristic
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—41no—
442C23—81no—
443C23C2×C3×C5×D411yescharacteristic
444C23C2×C3×C5×D411yescharacteristic
445Q8—21no—
446D4—21no—
447D4—41no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—82no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—81no—
469C2×C4C2×C3×C5×D411yescharacteristic
470C8—41no—
471C2×C3—21no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—81no—
475C2×C3—81no—
476C2×C3—81no—
477C2×C3C5×B(4,𝔽2)11yescharacteristic
478C5C3×#220011yescharacteristic, direct product factor, Sylow
479C22—21no—
480C22—21no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—42no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22C3×C5×#12011yescharacteristic
495C4—21no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—81no—
502C4—81no—
503C3C5×#220011yescharacteristic, direct product factor, Sylow
504C2—21no—
505C2—41no—
506C2—41no—
507C2—81no—
508C2—81no—
509C2—81no—
510C2C3×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C3×C5
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C5 ◁ C22×C3×C5 ◁ C2×C3×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#560C2semidirect
2. C3×C5×#458C2semidirect
3. C3×C5×B(4,𝔽2)C2semidirect
4. C3×C5×#453C2semidirect
5. C3×C5×#353C2semidirect
6. C3×C5×#352C2semidirect
7. C3×C5×AΣL(1,𝔽16)C2semidirect
8. C5×#2200C3direct (N × H)
28. C3×#2200C5direct (N × H)
29. C5×#560C2×C3semidirect
30. C5×#458C2×C3semidirect
31. C5×B(4,𝔽2)C2×C3semidirect
32. C5×#453C2×C3semidirect
33. C5×#353C2×C3semidirect
34. C5×#352C2×C3semidirect
35. C5×AΣL(1,𝔽16)C2×C3semidirect
48. C2×C3×C5×D4D4semidirect
51. C22×C3×C4×C5D4semidirect
69. C3×C5×C22⋊C4D4semidirect
70. C3×C42×C5D4semidirect
71. C3×#560C2×C5semidirect
72. C3×#458C2×C5semidirect
73. C3×B(4,𝔽2)C2×C5semidirect
74. C3×#453C2×C5semidirect
75. C3×#353C2×C5semidirect
76. C3×#352C2×C5semidirect
77. C3×AΣL(1,𝔽16)C2×C5semidirect
97. #2200C3×C5direct (N × H)
163. C2×C5×D4C3×D4semidirect
166. C22×C4×C5C3×D4semidirect
184. C5×C22⋊C4C3×D4semidirect
185. C42×C5C3×D4semidirect
186. #560C2×C3×C5semidirect
187. #458C2×C3×C5semidirect
188. B(4,𝔽2)C2×C3×C5semidirect
189. #453C2×C3×C5semidirect
190. #353C2×C3×C5semidirect
191. #352C2×C3×C5semidirect
192. AΣL(1,𝔽16)C2×C3×C5semidirect
229. C2×C3×D4C5×D4semidirect
232. C22×C3×C4C5×D4semidirect
250. C3×C22⋊C4C5×D4semidirect
251. C3×C42C5×D4semidirect
382. C2×D4C3×C5×D4semidirect
385. C22×C4C3×C5×D4semidirect
403. C22⋊C4C3×C5×D4semidirect
404. C42C3×C5×D4semidirect
405. C3×C5#2200direct (N × H)
478. C5C3×#2200direct (N × H)
503. C3C5×#2200direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5602
C3×C5×#4582
C3×C5×B(4,𝔽2)2
C3×C5×#4532
C3×C5×#3532
C3×C5×#3522
C3×C5×AΣL(1,𝔽16)2
C5×#22003
C3×#22005