Finite Group Databasebeta

← C3×C5×#1601 #14274635

Subgroups — C3×C5×#1601 #14274635

Order 1920.

616 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#534C211yesmaximal, characteristic
2C3×C5×#448C212yesmaximal, complemented
3C3×C5×#393C211yesmaximal, characteristic, complemented
4C3×C5×#327C212yesmaximal, complemented
5C3×C5×#323C211yesmaximal, characteristic, complemented
6C5×#1601C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×Q8C2211yescharacteristic, complemented
8C22×C3×C5×D4—21no—
9C2×C3×C5×SD16—22no—
10C3×C5×#123—22no—
11C3×C5×#121—21no—
12C3×C5×#121C2212yes—
13C3×C5×#120—22no—
14C2×C3×C5×C22⋊C4—21no—
15C2×C3×C5×C22⋊C4C2211yescharacteristic
16C3×C5×#103—22no—
17C3×C5×C22⋊C8C2212yescomplemented
18C3×C5×#95—21no—
19C3×C5×#95C2211yescharacteristic, complemented
20C3×#1601C511yesmaximal, characteristic, direct product factor, Hall
21C5×#534C2×C311yescharacteristic
22C5×#448C2×C312yescomplemented
23C5×#393C2×C311yescharacteristic, complemented
24C5×#327C2×C312yescomplemented
25C5×#323C2×C311yescharacteristic, complemented
26C24×C3×C5—21no—
27C24×C3×C5D411yescharacteristic
28C2×C3×C5×Q8—22no—
29C2×C3×C5×Q8—42no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4—42no—
34C2×C3×C5×D4—42no—
35C2×C3×C5×D4D412yes—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5D411yescharacteristic
40C22×C3×C4×C5C2311yescharacteristic
41C3×C5×SD16—44no—
42C2×C3×C5×C8—22no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4D412yes—
45C3×C5×C22⋊C4—22no—
46C3×C5×C22⋊C4—22no—
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—42no—
49C3×#534C2×C511yescharacteristic
50C3×#448C2×C512yescomplemented
51C3×#393C2×C511yescharacteristic, complemented
52C3×#327C2×C512yescomplemented
53C3×#323C2×C511yescharacteristic, complemented
54C22×C5×Q8C22×C311yescharacteristic, complemented
55C22×C5×D4—21no—
56C2×C5×SD16—22no—
57C5×#123—22no—
58C5×#121—21no—
59C5×#121C22×C312yes—
60C5×#120—22no—
61C2×C5×C22⋊C4—21no—
62C2×C5×C22⋊C4C22×C311yescharacteristic
63C5×#103—22no—
64C5×C22⋊C8C22×C312yescomplemented
65C5×#95—21no—
66C5×#95C22×C311yescharacteristic, complemented
67#1601C3×C511yescharacteristic, direct product factor, Sylow
68C23×C3×C5—21no—
69C23×C3×C5—21no—
70C23×C3×C5—22no—
71C23×C3×C5—41no—
72C23×C3×C5—41no—
73C23×C3×C5—42no—
74C23×C3×C5—81no—
75C23×C3×C5—81no—
76C23×C3×C5SD1612yescomplemented
77C23×C3×C5C2×D411yescharacteristic
78C3×C5×Q8—24no—
79C3×C5×Q8—42no—
80C3×C5×D4—42no—
81C3×C5×D4—44no—
82C3×C5×D4—42no—
83C3×C5×D4—42no—
84C3×C5×D4—81no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—22no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5—42no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5—41no—
92C2×C3×C4×C5—41no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—41no—
95C2×C3×C4×C5—41no—
96C2×C3×C4×C5C2×D412yes—
97C3×C5×C8—42no—
98C22×C3×Q8C22×C511yescharacteristic, complemented
99C22×C3×D4—21no—
100C2×C3×SD16—22no—
101C3×#123—22no—
102C3×#121—21no—
103C3×#121C22×C512yes—
104C3×#120—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4C22×C511yescharacteristic
107C3×#103—22no—
108C3×C22⋊C8C22×C512yescomplemented
109C3×#95—21no—
110C3×#95C22×C511yescharacteristic, complemented
111C24×C5—21no—
112C24×C5C3×D411yescharacteristic
113C2×C5×Q8—22no—
114C2×C5×Q8—42no—
115C2×C5×D4—22no—
116C2×C5×D4—22no—
117C2×C5×D4—21no—
118C2×C5×D4—42no—
119C2×C5×D4—42no—
120C2×C5×D4C3×D412yes—
121C22×C4×C5—21no—
122C22×C4×C5—21no—
123C22×C4×C5—21no—
124C22×C4×C5C3×D411yescharacteristic
125C22×C4×C5C23×C311yescharacteristic
126C5×SD16—44no—
127C2×C5×C8—22no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4C3×D412yes—
130C5×C22⋊C4—22no—
131C5×C22⋊C4—22no—
132C5×C22⋊C4—22no—
133C5×C22⋊C4—42no—
134#534C2×C3×C511yescharacteristic
135#448C2×C3×C512yescomplemented
136#393C2×C3×C511yescharacteristic, complemented
137#327C2×C3×C512yescomplemented
138#323C2×C3×C511yescharacteristic, complemented
139C22×C3×C5—21no—
140C22×C3×C5—21no—
141C22×C3×C5—22no—
142C22×C3×C5—42no—
143C22×C3×C5—41no—
144C22×C3×C5—41no—
145C22×C3×C5—42no—
146C22×C3×C5—42no—
147C22×C3×C5—42no—
148C22×C3×C5—81no—
149C22×C3×C5—84no—
150C22×C3×C5#12011yescharacteristic
151C22×C3×C5C2×SD1611yescharacteristic
152C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
153C3×C4×C5—22no—
154C3×C4×C5—42no—
155C3×C4×C5—41no—
156C3×C4×C5—42no—
157C3×C4×C5—81no—
158C24×C3—21no—
159C24×C3C5×D411yescharacteristic
160C2×C3×Q8—22no—
161C2×C3×Q8—42no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4C5×D412yes—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4C5×D411yescharacteristic
172C22×C3×C4C23×C511yescharacteristic
173C3×SD16—44no—
174C2×C3×C8—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4C5×D412yes—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—42no—
181C23×C5—21no—
182C23×C5—21no—
183C23×C5—22no—
184C23×C5—41no—
185C23×C5—41no—
186C23×C5—42no—
187C23×C5—81no—
188C23×C5—81no—
189C23×C5C3×SD1612yescomplemented
190C23×C5C2×C3×D411yescharacteristic
191C5×Q8—24no—
192C5×Q8—42no—
193C5×D4—42no—
194C5×D4—44no—
195C5×D4—42no—
196C5×D4—42no—
197C5×D4—81no—
198C2×C4×C5—22no—
199C2×C4×C5—21no—
200C2×C4×C5—22no—
201C2×C4×C5—41no—
202C2×C4×C5—42no—
203C2×C4×C5—41no—
204C2×C4×C5—41no—
205C2×C4×C5—41no—
206C2×C4×C5—41no—
207C2×C4×C5—41no—
208C2×C4×C5—41no—
209C2×C4×C5C2×C3×D412yes—
210C5×C8—42no—
211C22×Q8C22×C3×C511yescharacteristic, complemented
212C22×D4—21no—
213C2×SD16—22no—
214#123—22no—
215#121—21no—
216#121C22×C3×C512yes—
217#120—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C22×C3×C511yescharacteristic
220#103—22no—
221C22⋊C8C22×C3×C512yescomplemented
222#95—21no—
223#95C22×C3×C511yescharacteristic, complemented
224C2×C3×C5—21no—
225C2×C3×C5—21no—
226C2×C3×C5—42no—
227C2×C3×C5—82no—
228C2×C3×C5#45011yescharacteristic
229C2×C3×C5#45511yescharacteristic
230C2×C3×C5B(4,𝔽2)11yescharacteristic
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—22no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—42no—
237C23×C3—81no—
238C23×C3—81no—
239C23×C3C5×SD1612yescomplemented
240C23×C3C2×C5×D411yescharacteristic
241C3×Q8—24no—
242C3×Q8—42no—
243C3×D4—42no—
244C3×D4—44no—
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—81no—
248C2×C3×C4—22no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4C2×C5×D412yes—
260C3×C8—42no—
261C22×C5—21no—
262C22×C5—21no—
263C22×C5—22no—
264C22×C5—42no—
265C22×C5—41no—
266C22×C5—41no—
267C22×C5—42no—
268C22×C5—42no—
269C22×C5—42no—
270C22×C5—81no—
271C22×C5—84no—
272C22×C5C3×#12011yescharacteristic
273C22×C5C2×C3×SD1611yescharacteristic
274C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
275C4×C5—22no—
276C4×C5—42no—
277C4×C5—41no—
278C4×C5—42no—
279C4×C5—81no—
280C24—21no—
281C24C3×C5×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—42no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4C3×C5×D412yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4C3×C5×D411yescharacteristic
294C22×C4C23×C3×C511yescharacteristic
295SD16—44no—
296C2×C8—22no—
297C4⋊C4—21no—
298C4⋊C4C3×C5×D412yes—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—42no—
303C3×C5#160111yescharacteristic, direct product factor, Hall
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—22no—
307C22×C3—42no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—81no—
314C22×C3—84no—
315C22×C3C5×#12011yescharacteristic
316C22×C3C2×C5×SD1611yescharacteristic
317C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
318C3×C4—22no—
319C3×C4—42no—
320C3×C4—41no—
321C3×C4—42no—
322C3×C4—81no—
323C2×C5—21no—
324C2×C5—21no—
325C2×C5—42no—
326C2×C5—82no—
327C2×C5C3×#45011yescharacteristic
328C2×C5C3×#45511yescharacteristic
329C2×C5C3×B(4,𝔽2)11yescharacteristic
330C23—21no—
331C23—21no—
332C23—22no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—81no—
337C23—81no—
338C23C3×C5×SD1612yescomplemented
339C23C2×C3×C5×D411yescharacteristic
340Q8—24no—
341Q8—42no—
342D4—42no—
343D4—44no—
344D4—42no—
345D4—42no—
346D4—81no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4C2×C3×C5×D412yes—
359C8—42no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—42no—
363C2×C3—82no—
364C2×C3C5×#45011yescharacteristic
365C2×C3C5×#45511yescharacteristic
366C2×C3C5×B(4,𝔽2)11yescharacteristic
367C5C3×#160111yescharacteristic, direct product factor, Sylow
368C22—21no—
369C22—21no—
370C22—22no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—81no—
378C22—84no—
379C22C3×C5×#12011yescharacteristic
380C22C2×C3×C5×SD1611yescharacteristic
381C22C3×C5×AGL(1,ℤ/8)11yescharacteristic
382C4—22no—
383C4—42no—
384C4—41no—
385C4—42no—
386C4—81no—
387C3C5×#160111yescharacteristic, direct product factor, Sylow
388C2—21no—
389C2—21no—
390C2—42no—
391C2—82no—
392C2C3×C5×#45011yescharacteristic
393C2C3×C5×#45511yescharacteristic
394C2C3×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC3×C5×#534C2
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×#448C2semidirect
3. C3×C5×#393C2semidirect
4. C3×C5×#327C2semidirect
5. C3×C5×#323C2semidirect
6. C5×#1601C3direct (N × H)
7. C22×C3×C5×Q8C22semidirect
17. C3×C5×C22⋊C8C22semidirect
19. C3×C5×#95C22semidirect
20. C3×#1601C5direct (N × H)
22. C5×#448C2×C3semidirect
23. C5×#393C2×C3semidirect
24. C5×#327C2×C3semidirect
25. C5×#323C2×C3semidirect
50. C3×#448C2×C5semidirect
51. C3×#393C2×C5semidirect
52. C3×#327C2×C5semidirect
53. C3×#323C2×C5semidirect
54. C22×C5×Q8C22×C3semidirect
64. C5×C22⋊C8C22×C3semidirect
66. C5×#95C22×C3semidirect
67. #1601C3×C5direct (N × H)
76. C23×C3×C5SD16semidirect
98. C22×C3×Q8C22×C5semidirect
108. C3×C22⋊C8C22×C5semidirect
110. C3×#95C22×C5semidirect
135. #448C2×C3×C5semidirect
136. #393C2×C3×C5semidirect
137. #327C2×C3×C5semidirect
138. #323C2×C3×C5semidirect
189. C23×C5C3×SD16semidirect
211. C22×Q8C22×C3×C5semidirect
221. C22⋊C8C22×C3×C5semidirect
223. #95C22×C3×C5semidirect
239. C23×C3C5×SD16semidirect
303. C3×C5#1601direct (N × H)
338. C23C3×C5×SD16semidirect
367. C5C3×#1601direct (N × H)
387. C3C5×#1601direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5342
C3×C5×#4482
C3×C5×#3932
C3×C5×#3272
C3×C5×#3232
C5×#16013
C3×#16015