Finite Group Databasebeta

← C3×C5×#1612 #14274646

Subgroups — C3×C5×#1612 #14274646

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#538C211yesmaximal, characteristic
2C3×C5×#483C211yesmaximal, characteristic, complemented
3C3×C5×#480C211yesmaximal, characteristic, complemented
4C3×C5×#397C211yesmaximal, characteristic, complemented
5C3×C5×#328C211yesmaximal, characteristic, complemented
6C3×C5×#327C211yesmaximal, characteristic, complemented
7C3×C5×#323C211yesmaximal, characteristic, complemented
8C5×#1612C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×#126—21no—
10C3×C5×#124—21no—
11C3×C5×#123—21no—
12C3×C5×C22⋊Q8C2211yescharacteristic
13C3×C5×#121—21no—
14C3×C5×#121C2211yescharacteristic
15C3×C5×#120—21no—
16C3×C4×C5×D4—21no—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4—21no—
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C3×C5×#107—21no—
21C3×C5×#106—21no—
22C3×C5×#103—21no—
23C3×C5×#102—21no—
24C3×C5×C22⋊C8C2211yescharacteristic
25C3×C5×C22⋊C8C2211yescharacteristic
26C3×C5×#95—21no—
27C3×C5×#95C2211yescharacteristic
28C3×#1612C511yesmaximal, characteristic, direct product factor, Hall
29C5×#538C2×C311yescharacteristic
30C5×#483C2×C311yescharacteristic, complemented
31C5×#480C2×C311yescharacteristic, complemented
32C5×#397C2×C311yescharacteristic, complemented
33C5×#328C2×C311yescharacteristic, complemented
34C5×#327C2×C311yescharacteristic, complemented
35C5×#323C2×C311yescharacteristic, complemented
36C24×C3×C5D411yescharacteristic
37C2×C3×C5×Q8D411yescharacteristic
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4D411yescharacteristic
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5—21no—
45C22×C3×C4×C5D411yescharacteristic
46C22×C3×C4×C5C2311yescharacteristic
47C2×C3×C5×C8—21no—
48C2×C3×C5×C8—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4D411yescharacteristic
54C3×C5×C4⋊C4D411yescharacteristic
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C5×C22⋊C4—21no—
59C3×C5×C22⋊C4—21no—
60C3×C5×C22⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C5×C22⋊C4—42no—
63C3×C42×C5—21no—
64C3×#538C2×C511yescharacteristic
65C3×#483C2×C511yescharacteristic, complemented
66C3×#480C2×C511yescharacteristic, complemented
67C3×#397C2×C511yescharacteristic, complemented
68C3×#328C2×C511yescharacteristic, complemented
69C3×#327C2×C511yescharacteristic, complemented
70C3×#323C2×C511yescharacteristic, complemented
71C5×#126—21no—
72C5×#124—21no—
73C5×#123—21no—
74C5×C22⋊Q8C22×C311yescharacteristic
75C5×#121—21no—
76C5×#121C22×C311yescharacteristic
77C5×#120—21no—
78C4×C5×D4—21no—
79C2×C5×C4⋊C4C22×C311yescharacteristic
80C2×C5×C22⋊C4—21no—
81C2×C5×C22⋊C4C22×C311yescharacteristic
82C5×#107—21no—
83C5×#106—21no—
84C5×#103—21no—
85C5×#102—21no—
86C5×C22⋊C8C22×C311yescharacteristic
87C5×C22⋊C8C22×C311yescharacteristic
88C5×#95—21no—
89C5×#95C22×C311yescharacteristic
90#1612C3×C511yescharacteristic, direct product factor, Sylow
91C23×C3×C5—21no—
92C23×C3×C5—21no—
93C23×C3×C5—21no—
94C23×C3×C5—21no—
95C23×C3×C5—41no—
96C23×C3×C5—41no—
97C23×C3×C5C2×D411yescharacteristic
98C3×C5×Q8—41no—
99C3×C5×D4—41no—
100C3×C5×D4—41no—
101C3×C5×D4—41no—
102C3×C5×D4—81no—
103C3×C5×D4—81no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—21no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5—41no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5—41no—
114C2×C3×C4×C5—41no—
115C2×C3×C4×C5—41no—
116C2×C3×C4×C5—81no—
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5C2×D411yescharacteristic
119C3×C5×C8—41no—
120C3×C5×C8—41no—
121C3×#126—21no—
122C3×#124—21no—
123C3×#123—21no—
124C3×C22⋊Q8C22×C511yescharacteristic
125C3×#121—21no—
126C3×#121C22×C511yescharacteristic
127C3×#120—21no—
128C3×C4×D4—21no—
129C2×C3×C4⋊C4C22×C511yescharacteristic
130C2×C3×C22⋊C4—21no—
131C2×C3×C22⋊C4C22×C511yescharacteristic
132C3×#107—21no—
133C3×#106—21no—
134C3×#103—21no—
135C3×#102—21no—
136C3×C22⋊C8C22×C511yescharacteristic
137C3×C22⋊C8C22×C511yescharacteristic
138C3×#95—21no—
139C3×#95C22×C511yescharacteristic
140C24×C5C3×D411yescharacteristic
141C2×C5×Q8C3×D411yescharacteristic
142C2×C5×D4—21no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4C3×D411yescharacteristic
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5C3×D411yescharacteristic
150C22×C4×C5C23×C311yescharacteristic
151C2×C5×C8—21no—
152C2×C5×C8—21no—
153C5×C4⋊C4—21no—
154C5×C4⋊C4—21no—
155C5×C4⋊C4—21no—
156C5×C4⋊C4—21no—
157C5×C4⋊C4C3×D411yescharacteristic
158C5×C4⋊C4C3×D411yescharacteristic
159C5×C22⋊C4—21no—
160C5×C22⋊C4—21no—
161C5×C22⋊C4—21no—
162C5×C22⋊C4—21no—
163C5×C22⋊C4—21no—
164C5×C22⋊C4—21no—
165C5×C22⋊C4—21no—
166C5×C22⋊C4—42no—
167C42×C5—21no—
168#538C2×C3×C511yescharacteristic
169#483C2×C3×C511yescharacteristic, complemented
170#480C2×C3×C511yescharacteristic, complemented
171#397C2×C3×C511yescharacteristic, complemented
172#328C2×C3×C511yescharacteristic, complemented
173#327C2×C3×C511yescharacteristic, complemented
174#323C2×C3×C511yescharacteristic, complemented
175C22×C3×C5—21no—
176C22×C3×C5—21no—
177C22×C3×C5—42no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5—41no—
181C22×C3×C5—41no—
182C22×C3×C5—41no—
183C22×C3×C5—41no—
184C22×C3×C5—81no—
185C22×C3×C5#12011yescharacteristic
186C22×C3×C5#13511yescharacteristic
187C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
188C3×C4×C5—21no—
189C3×C4×C5—21no—
190C3×C4×C5—41no—
191C3×C4×C5—41no—
192C3×C4×C5—41no—
193C3×C4×C5—41no—
194C3×C4×C5—81no—
195C3×C4×C5—81no—
196C24×C3C5×D411yescharacteristic
197C2×C3×Q8C5×D411yescharacteristic
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4C5×D411yescharacteristic
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C5×D411yescharacteristic
206C22×C3×C4C23×C511yescharacteristic
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4C5×D411yescharacteristic
214C3×C4⋊C4C5×D411yescharacteristic
215C3×C22⋊C4—21no—
216C3×C22⋊C4—21no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—42no—
223C3×C42—21no—
224C23×C5—21no—
225C23×C5—21no—
226C23×C5—21no—
227C23×C5—21no—
228C23×C5—41no—
229C23×C5—41no—
230C23×C5C2×C3×D411yescharacteristic
231C5×Q8—41no—
232C5×D4—41no—
233C5×D4—41no—
234C5×D4—41no—
235C5×D4—81no—
236C5×D4—81no—
237C2×C4×C5—21no—
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—41no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—41no—
249C2×C4×C5—81no—
250C2×C4×C5C2×C3×D411yescharacteristic
251C2×C4×C5C2×C3×D411yescharacteristic
252C5×C8—41no—
253C5×C8—41no—
254#126—21no—
255#124—21no—
256#123—21no—
257C22⋊Q8C22×C3×C511yescharacteristic
258#121—21no—
259#121C22×C3×C511yescharacteristic
260#120—21no—
261C4×D4—21no—
262C2×C4⋊C4C22×C3×C511yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4C22×C3×C511yescharacteristic
265#107—21no—
266#106—21no—
267#103—21no—
268#102—21no—
269C22⋊C8C22×C3×C511yescharacteristic
270C22⋊C8C22×C3×C511yescharacteristic
271#95—21no—
272#95C22×C3×C511yescharacteristic
273C2×C3×C5—21no—
274C2×C3×C5—21no—
275C2×C3×C5—81no—
276C2×C3×C5—81no—
277C2×C3×C5#44911yescharacteristic
278C2×C3×C5#45511yescharacteristic
279C2×C3×C5#45811yescharacteristic
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3C2×C5×D411yescharacteristic
287C3×Q8—41no—
288C3×D4—41no—
289C3×D4—41no—
290C3×D4—41no—
291C3×D4—81no—
292C3×D4—81no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—81no—
306C2×C3×C4C2×C5×D411yescharacteristic
307C2×C3×C4C2×C5×D411yescharacteristic
308C3×C8—41no—
309C3×C8—41no—
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—42no—
313C22×C5—41no—
314C22×C5—41no—
315C22×C5—41no—
316C22×C5—41no—
317C22×C5—41no—
318C22×C5—41no—
319C22×C5—81no—
320C22×C5C3×#12011yescharacteristic
321C22×C5C3×#13511yescharacteristic
322C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
323C4×C5—21no—
324C4×C5—21no—
325C4×C5—41no—
326C4×C5—41no—
327C4×C5—41no—
328C4×C5—41no—
329C4×C5—81no—
330C4×C5—81no—
331C24C3×C5×D411yescharacteristic
332C2×Q8C3×C5×D411yescharacteristic
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4C3×C5×D411yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4C3×C5×D411yescharacteristic
341C22×C4C23×C3×C511yescharacteristic
342C2×C8—21no—
343C2×C8—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4C3×C5×D411yescharacteristic
349C4⋊C4C3×C5×D411yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—42no—
358C42—21no—
359C3×C5#161211yescharacteristic, direct product factor, Hall
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—42no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—81no—
370C22×C3C5×#12011yescharacteristic
371C22×C3C5×#13511yescharacteristic
372C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—81no—
380C3×C4—81no—
381C2×C5—21no—
382C2×C5—21no—
383C2×C5—81no—
384C2×C5—81no—
385C2×C5C3×#44911yescharacteristic
386C2×C5C3×#45511yescharacteristic
387C2×C5C3×#45811yescharacteristic
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23C2×C3×C5×D411yescharacteristic
395Q8—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—81no—
400D4—81no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4C2×C3×C5×D411yescharacteristic
415C2×C4C2×C3×C5×D411yescharacteristic
416C8—41no—
417C8—41no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—81no—
421C2×C3—81no—
422C2×C3C5×#44911yescharacteristic
423C2×C3C5×#45511yescharacteristic
424C2×C3C5×#45811yescharacteristic
425C5C3×#161211yescharacteristic, direct product factor, Sylow
426C22—21no—
427C22—21no—
428C22—42no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22C3×C5×#12011yescharacteristic
437C22C3×C5×#13511yescharacteristic
438C22C3×C5×AGL(1,ℤ/8)11yescharacteristic
439C4—21no—
440C4—21no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—81no—
446C4—81no—
447C3C5×#161211yescharacteristic, direct product factor, Sylow
448C2—21no—
449C2—21no—
450C2—81no—
451C2—81no—
452C2C3×C5×#44911yescharacteristic
453C2C3×C5×#45511yescharacteristic
454C2C3×C5×#45811yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC3×C5×#538C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C5×#483C2semidirect
3. C3×C5×#480C2semidirect
4. C3×C5×#397C2semidirect
5. C3×C5×#328C2semidirect
6. C3×C5×#327C2semidirect
7. C3×C5×#323C2semidirect
8. C5×#1612C3direct (N × H)
28. C3×#1612C5direct (N × H)
30. C5×#483C2×C3semidirect
31. C5×#480C2×C3semidirect
32. C5×#397C2×C3semidirect
33. C5×#328C2×C3semidirect
34. C5×#327C2×C3semidirect
35. C5×#323C2×C3semidirect
65. C3×#483C2×C5semidirect
66. C3×#480C2×C5semidirect
67. C3×#397C2×C5semidirect
68. C3×#328C2×C5semidirect
69. C3×#327C2×C5semidirect
70. C3×#323C2×C5semidirect
90. #1612C3×C5direct (N × H)
169. #483C2×C3×C5semidirect
170. #480C2×C3×C5semidirect
171. #397C2×C3×C5semidirect
172. #328C2×C3×C5semidirect
173. #327C2×C3×C5semidirect
174. #323C2×C3×C5semidirect
359. C3×C5#1612direct (N × H)
425. C5C3×#1612direct (N × H)
447. C3C5×#1612direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5382
C3×C5×#4832
C3×C5×#4802
C3×C5×#3972
C3×C5×#3282
C3×C5×#3272
C3×C5×#3232
C5×#16123
C3×#16125