Finite Group Databasebeta

← C3×C5×#1640 #14274674

Subgroups — C3×C5×#1640 #14274674

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#559C211yesmaximal, characteristic
2C3×C5×#486C211yesmaximal, characteristic
3C3×C5×#475C211yesmaximal, characteristic, complemented
4C3×C5×#462C211yesmaximal, characteristic, complemented
5C3×C5×#331C211yesmaximal, characteristic
6C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
8C5×#1640C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×Q16—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×#127C2211yescharacteristic
12C3×C5×#124—21no—
13C3×C5×#124—21no—
14C3×C5×#123—22no—
15C3×C4×C5×Q8C2211yescharacteristic
16C3×C4×C5×Q8C2211yescharacteristic
17C3×C4×C5×D4—21no—
18C3×C4×C5×D4—21no—
19C3×C5×#106—21no—
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C5×#103—21no—
23C3×C5×#103—21no—
24C3×C5×#102—22no—
25C3×C4×C5×C8C2211yescharacteristic
26C3×#1640C511yesmaximal, characteristic, direct product factor, Hall
27C5×#559C2×C311yescharacteristic
28C5×#486C2×C311yescharacteristic
29C5×#475C2×C311yescharacteristic, complemented
30C5×#462C2×C311yescharacteristic, complemented
31C5×#331C2×C311yescharacteristic
32C5×Q8⋊C8C2×C311yescharacteristic, complemented
33C5×Q8⋊C8C2×C311yescharacteristic, complemented
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C3×C5×Q8D411yescharacteristic
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4D412yes—
40C22×C3×C4×C5—22no—
41C3×C5×Q16—42no—
42C2×C3×C5×C8—21no—
43C2×C3×C5×C8—21no—
44C2×C3×C5×C8—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4D411yescharacteristic
50C3×C5×C4⋊C4D411yescharacteristic
51C3×C5×C22⋊C4—22no—
52C3×C5×C22⋊C4—22no—
53C3×C5×C22⋊C4—22no—
54C3×C42×C5—21no—
55C3×C42×C5—21no—
56C3×C42×C5C2311yescharacteristic
57C3×#559C2×C511yescharacteristic
58C3×#486C2×C511yescharacteristic
59C3×#475C2×C511yescharacteristic, complemented
60C3×#462C2×C511yescharacteristic, complemented
61C3×#331C2×C511yescharacteristic
62C3×Q8⋊C8C2×C511yescharacteristic, complemented
63C3×Q8⋊C8C2×C511yescharacteristic, complemented
64C2×C5×Q16—21no—
65C5×C4⋊Q8C22×C311yescharacteristic
66C5×#127C22×C311yescharacteristic
67C5×#124—21no—
68C5×#124—21no—
69C5×#123—22no—
70C4×C5×Q8C22×C311yescharacteristic
71C4×C5×Q8C22×C311yescharacteristic
72C4×C5×D4—21no—
73C4×C5×D4—21no—
74C5×#106—21no—
75C5×C4⋊C8C22×C311yescharacteristic
76C5×C4⋊C8C22×C311yescharacteristic
77C5×#103—21no—
78C5×#103—21no—
79C5×#102—22no—
80C4×C5×C8C22×C311yescharacteristic
81#1640C3×C511yescharacteristic, direct product factor, Sylow
82C23×C3×C5—22no—
83C3×C5×Q8—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—21no—
86C3×C5×Q8—21no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×D4—42no—
90C3×C5×D4—41no—
91C3×C5×D4—41no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—82no—
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C3×C5×C8—41no—
104C3×C5×C8—41no—
105C3×C5×C8—41no—
106C2×C3×Q16—21no—
107C3×C4⋊Q8C22×C511yescharacteristic
108C3×#127C22×C511yescharacteristic
109C3×#124—21no—
110C3×#124—21no—
111C3×#123—22no—
112C3×C4×Q8C22×C511yescharacteristic
113C3×C4×Q8C22×C511yescharacteristic
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×#106—21no—
117C3×C4⋊C8C22×C511yescharacteristic
118C3×C4⋊C8C22×C511yescharacteristic
119C3×#103—21no—
120C3×#103—21no—
121C3×#102—22no—
122C3×C4×C8C22×C511yescharacteristic
123C2×C5×Q8—21no—
124C2×C5×Q8C3×D411yescharacteristic
125C2×C5×Q8C3×D411yescharacteristic
126C2×C5×D4—21no—
127C2×C5×D4—21no—
128C2×C5×D4C3×D412yes—
129C22×C4×C5—22no—
130C5×Q16—42no—
131C2×C5×C8—21no—
132C2×C5×C8—21no—
133C2×C5×C8—21no—
134C5×C4⋊C4—21no—
135C5×C4⋊C4—21no—
136C5×C4⋊C4—21no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4C3×D411yescharacteristic
139C5×C4⋊C4C3×D411yescharacteristic
140C5×C22⋊C4—22no—
141C5×C22⋊C4—22no—
142C5×C22⋊C4—22no—
143C42×C5—21no—
144C42×C5—21no—
145C42×C5C23×C311yescharacteristic
146#559C2×C3×C511yescharacteristic
147#486C2×C3×C511yescharacteristic
148#475C2×C3×C511yescharacteristic, complemented
149#462C2×C3×C511yescharacteristic, complemented
150#331C2×C3×C511yescharacteristic
151Q8⋊C8C2×C3×C511yescharacteristic, complemented
152Q8⋊C8C2×C3×C511yescharacteristic, complemented
153C22×C3×C5—42no—
154C22×C3×C5—42no—
155C22×C3×C5—42no—
156C22×C3×C5#12011yescharacteristic
157C3×C4×C5—21no—
158C3×C4×C5—41no—
159C3×C4×C5—41no—
160C3×C4×C5—41no—
161C3×C4×C5—41no—
162C3×C4×C5—81no—
163C3×C4×C5#13511yescharacteristic
164C3×C4×C5#13511yescharacteristic
165C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
166C3×C4×C5#13711yescharacteristic
167C2×C3×Q8—21no—
168C2×C3×Q8C5×D411yescharacteristic
169C2×C3×Q8C5×D411yescharacteristic
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4C5×D412yes—
173C22×C3×C4—22no—
174C3×Q16—42no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4C5×D411yescharacteristic
183C3×C4⋊C4C5×D411yescharacteristic
184C3×C22⋊C4—22no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C42—21no—
188C3×C42—21no—
189C3×C42C23×C511yescharacteristic
190C23×C5—22no—
191C5×Q8—21no—
192C5×Q8—21no—
193C5×Q8—21no—
194C5×Q8—21no—
195C5×Q8—41no—
196C5×Q8—41no—
197C5×D4—42no—
198C5×D4—41no—
199C5×D4—41no—
200C5×D4—41no—
201C5×D4—41no—
202C2×C4×C5—21no—
203C2×C4×C5—21no—
204C2×C4×C5—21no—
205C2×C4×C5—21no—
206C2×C4×C5—41no—
207C2×C4×C5—82no—
208C2×C4×C5C2×C3×D411yescharacteristic
209C2×C4×C5C2×C3×D411yescharacteristic
210C2×C4×C5C2×C3×D411yescharacteristic
211C5×C8—41no—
212C5×C8—41no—
213C5×C8—41no—
214C2×Q16—21no—
215C4⋊Q8C22×C3×C511yescharacteristic
216#127C22×C3×C511yescharacteristic
217#124—21no—
218#124—21no—
219#123—22no—
220C4×Q8C22×C3×C511yescharacteristic
221C4×Q8C22×C3×C511yescharacteristic
222C4×D4—21no—
223C4×D4—21no—
224#106—21no—
225C4⋊C8C22×C3×C511yescharacteristic
226C4⋊C8C22×C3×C511yescharacteristic
227#103—21no—
228#103—21no—
229#102—22no—
230C4×C8C22×C3×C511yescharacteristic
231C2×C3×C5—82no—
232C2×C3×C5#44911yescharacteristic
233C2×C3×C5#45211yescharacteristic
234C2×C3×C5#45511yescharacteristic
235C23×C3—22no—
236C3×Q8—21no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×D4—42no—
243C3×D4—41no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—41no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—82no—
253C2×C3×C4C2×C5×D411yescharacteristic
254C2×C3×C4C2×C5×D411yescharacteristic
255C2×C3×C4C2×C5×D411yescharacteristic
256C3×C8—41no—
257C3×C8—41no—
258C3×C8—41no—
259C22×C5—42no—
260C22×C5—42no—
261C22×C5—42no—
262C22×C5C3×#12011yescharacteristic
263C4×C5—21no—
264C4×C5—41no—
265C4×C5—41no—
266C4×C5—41no—
267C4×C5—41no—
268C4×C5—81no—
269C4×C5C3×#13511yescharacteristic
270C4×C5C3×#13511yescharacteristic
271C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
272C4×C5C3×#13711yescharacteristic
273C2×Q8—21no—
274C2×Q8C3×C5×D411yescharacteristic
275C2×Q8C3×C5×D411yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4C3×C5×D412yes—
279C22×C4—22no—
280Q16—42no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4C3×C5×D411yescharacteristic
289C4⋊C4C3×C5×D411yescharacteristic
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C42—21no—
294C42—21no—
295C42C23×C3×C511yescharacteristic
296C3×C5#164011yescharacteristic, direct product factor, Hall
297C22×C3—42no—
298C22×C3—42no—
299C22×C3—42no—
300C22×C3C5×#12011yescharacteristic
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—81no—
307C3×C4C5×#13511yescharacteristic
308C3×C4C5×#13511yescharacteristic
309C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
310C3×C4C5×#13711yescharacteristic
311C2×C5—82no—
312C2×C5C3×#44911yescharacteristic
313C2×C5C3×#45211yescharacteristic
314C2×C5C3×#45511yescharacteristic
315C23—22no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—41no—
321Q8—41no—
322D4—42no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—41no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—82no—
333C2×C4C2×C3×C5×D411yescharacteristic
334C2×C4C2×C3×C5×D411yescharacteristic
335C2×C4C2×C3×C5×D411yescharacteristic
336C8—41no—
337C8—41no—
338C8—41no—
339C2×C3—82no—
340C2×C3C5×#44911yescharacteristic
341C2×C3C5×#45211yescharacteristic
342C2×C3C5×#45511yescharacteristic
343C5C3×#164011yescharacteristic, direct product factor, Sylow
344C22—42no—
345C22—42no—
346C22—42no—
347C22C3×C5×#12011yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—81no—
354C4C3×C5×#13511yescharacteristic
355C4C3×C5×#13511yescharacteristic
356C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
357C4C3×C5×#13711yescharacteristic
358C3C5×#164011yescharacteristic, direct product factor, Sylow
359C2—82no—
360C2C3×C5×#44911yescharacteristic
361C2C3×C5×#45211yescharacteristic
362C2C3×C5×#45511yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC3×C5×#127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C3×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×C5×#475C2semidirect
4. C3×C5×#462C2semidirect
6. C3×C5×Q8⋊C8C2semidirect
7. C3×C5×Q8⋊C8C2semidirect
8. C5×#1640C3direct (N × H)
26. C3×#1640C5direct (N × H)
29. C5×#475C2×C3semidirect
30. C5×#462C2×C3semidirect
32. C5×Q8⋊C8C2×C3semidirect
33. C5×Q8⋊C8C2×C3semidirect
59. C3×#475C2×C5semidirect
60. C3×#462C2×C5semidirect
62. C3×Q8⋊C8C2×C5semidirect
63. C3×Q8⋊C8C2×C5semidirect
81. #1640C3×C5direct (N × H)
148. #475C2×C3×C5semidirect
149. #462C2×C3×C5semidirect
151. Q8⋊C8C2×C3×C5semidirect
152. Q8⋊C8C2×C3×C5semidirect
296. C3×C5#1640direct (N × H)
343. C5C3×#1640direct (N × H)
358. C3C5×#1640direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5592
C3×C5×#4862
C3×C5×#4752
C3×C5×#4622
C3×C5×#3312
C3×C5×Q8⋊C82
C5×#16403
C3×#16405