Finite Group Databasebeta

← C3×C5×#1650 #14274684

Subgroups — C3×C5×#1650 #14274684

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#556C211yesmaximal, characteristic
2C3×C5×#500C211yesmaximal, characteristic, complemented
3C3×C5×#475C211yesmaximal, characteristic, complemented
4C3×C5×#461C211yesmaximal, characteristic
5C3×C5×#332C211yesmaximal, characteristic, complemented
6C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×C5×D4⋊C8C211yesmaximal, characteristic
8C5×#1650C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×SD16—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×C4⋊Q8C2211yescharacteristic
12C3×C5×#126—22no—
13C3×C5×#124—21no—
14C3×C5×C22⋊Q8—21no—
15C3×C4×C5×Q8—21no—
16C3×C4×C5×Q8C2211yescharacteristic
17C3×C4×C5×D4C2211yescharacteristic
18C3×C5×#117—21no—
19C3×C5×#107—22no—
20C3×C5×#106—21no—
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C5×C4⋊C8C2211yescharacteristic
23C3×C5×#103—21no—
24C3×C5×#103—21no—
25C3×C4×C5×C8C2211yescharacteristic
26C3×#1650C511yesmaximal, characteristic, direct product factor, Hall
27C5×#556C2×C311yescharacteristic
28C5×#500C2×C311yescharacteristic, complemented
29C5×#475C2×C311yescharacteristic, complemented
30C5×#461C2×C311yescharacteristic
31C5×#332C2×C311yescharacteristic, complemented
32C5×Q8⋊C8C2×C311yescharacteristic, complemented
33C5×D4⋊C8C2×C311yescharacteristic
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8—21no—
36C2×C3×C5×Q8D411yescharacteristic
37C2×C3×C5×D4D411yescharacteristic
38C22×C3×C4×C5—21no—
39C3×C5×SD16—42no—
40C2×C3×C5×C8—21no—
41C2×C3×C5×C8—21no—
42C2×C3×C5×C8—21no—
43C3×C5×C4⋊C4—22no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4D411yescharacteristic
49C3×C5×C4⋊C4D411yescharacteristic
50C3×C5×C4⋊C4D412yes—
51C3×C5×C22⋊C4—22no—
52C3×C5×C22⋊C4—22no—
53C3×C5×C22⋊C4—21no—
54C3×C42×C5—22no—
55C3×C42×C5—21no—
56C3×C42×C5C2311yescharacteristic
57C3×#556C2×C511yescharacteristic
58C3×#500C2×C511yescharacteristic, complemented
59C3×#475C2×C511yescharacteristic, complemented
60C3×#461C2×C511yescharacteristic
61C3×#332C2×C511yescharacteristic, complemented
62C3×Q8⋊C8C2×C511yescharacteristic, complemented
63C3×D4⋊C8C2×C511yescharacteristic
64C2×C5×SD16—21no—
65C5×C4⋊Q8C22×C311yescharacteristic
66C5×C4⋊Q8C22×C311yescharacteristic
67C5×#126—22no—
68C5×#124—21no—
69C5×C22⋊Q8—21no—
70C4×C5×Q8—21no—
71C4×C5×Q8C22×C311yescharacteristic
72C4×C5×D4C22×C311yescharacteristic
73C5×#117—21no—
74C5×#107—22no—
75C5×#106—21no—
76C5×C4⋊C8C22×C311yescharacteristic
77C5×C4⋊C8C22×C311yescharacteristic
78C5×#103—21no—
79C5×#103—21no—
80C4×C5×C8C22×C311yescharacteristic
81#1650C3×C511yescharacteristic, direct product factor, Sylow
82C23×C3×C5—21no—
83C3×C5×Q8—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—41no—
86C3×C5×Q8—41no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×D4—21no—
90C3×C5×D4—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—41no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C3×C5×C8—22no—
102C3×C5×C8—41no—
103C3×C5×C8—41no—
104C2×C3×SD16—21no—
105C3×C4⋊Q8C22×C511yescharacteristic
106C3×C4⋊Q8C22×C511yescharacteristic
107C3×#126—22no—
108C3×#124—21no—
109C3×C22⋊Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8C22×C511yescharacteristic
112C3×C4×D4C22×C511yescharacteristic
113C3×#117—21no—
114C3×#107—22no—
115C3×#106—21no—
116C3×C4⋊C8C22×C511yescharacteristic
117C3×C4⋊C8C22×C511yescharacteristic
118C3×#103—21no—
119C3×#103—21no—
120C3×C4×C8C22×C511yescharacteristic
121C2×C5×Q8—21no—
122C2×C5×Q8—21no—
123C2×C5×Q8C3×D411yescharacteristic
124C2×C5×D4C3×D411yescharacteristic
125C22×C4×C5—21no—
126C5×SD16—42no—
127C2×C5×C8—21no—
128C2×C5×C8—21no—
129C2×C5×C8—21no—
130C5×C4⋊C4—22no—
131C5×C4⋊C4—21no—
132C5×C4⋊C4—21no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4—21no—
135C5×C4⋊C4C3×D411yescharacteristic
136C5×C4⋊C4C3×D411yescharacteristic
137C5×C4⋊C4C3×D412yes—
138C5×C22⋊C4—22no—
139C5×C22⋊C4—22no—
140C5×C22⋊C4—21no—
141C42×C5—22no—
142C42×C5—21no—
143C42×C5C23×C311yescharacteristic
144#556C2×C3×C511yescharacteristic
145#500C2×C3×C511yescharacteristic, complemented
146#475C2×C3×C511yescharacteristic, complemented
147#461C2×C3×C511yescharacteristic
148#332C2×C3×C511yescharacteristic, complemented
149Q8⋊C8C2×C3×C511yescharacteristic, complemented
150D4⋊C8C2×C3×C511yescharacteristic
151C22×C3×C5—41no—
152C22×C3×C5—41no—
153C22×C3×C5—41no—
154C22×C3×C5#12011yescharacteristic
155C3×C4×C5—21no—
156C3×C4×C5—42no—
157C3×C4×C5—41no—
158C3×C4×C5—41no—
159C3×C4×C5—41no—
160C3×C4×C5—81no—
161C3×C4×C5#13511yescharacteristic
162C3×C4×C5#13511yescharacteristic
163C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
164C3×C4×C5#13711yescharacteristic
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8C5×D411yescharacteristic
168C2×C3×D4C5×D411yescharacteristic
169C22×C3×C4—21no—
170C3×SD16—42no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4C5×D411yescharacteristic
180C3×C4⋊C4C5×D411yescharacteristic
181C3×C4⋊C4C5×D412yes—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—21no—
185C3×C42—22no—
186C3×C42—21no—
187C3×C42C23×C511yescharacteristic
188C23×C5—21no—
189C5×Q8—21no—
190C5×Q8—21no—
191C5×Q8—41no—
192C5×Q8—41no—
193C5×Q8—41no—
194C5×Q8—41no—
195C5×D4—21no—
196C5×D4—21no—
197C2×C4×C5—21no—
198C2×C4×C5—21no—
199C2×C4×C5—21no—
200C2×C4×C5—22no—
201C2×C4×C5—41no—
202C2×C4×C5—41no—
203C2×C4×C5—41no—
204C2×C4×C5C2×C3×D411yescharacteristic
205C2×C4×C5C2×C3×D411yescharacteristic
206C2×C4×C5C2×C3×D411yescharacteristic
207C5×C8—22no—
208C5×C8—41no—
209C5×C8—41no—
210C2×SD16—21no—
211C4⋊Q8C22×C3×C511yescharacteristic
212C4⋊Q8C22×C3×C511yescharacteristic
213#126—22no—
214#124—21no—
215C22⋊Q8—21no—
216C4×Q8—21no—
217C4×Q8C22×C3×C511yescharacteristic
218C4×D4C22×C3×C511yescharacteristic
219#117—21no—
220#107—22no—
221#106—21no—
222C4⋊C8C22×C3×C511yescharacteristic
223C4⋊C8C22×C3×C511yescharacteristic
224#103—21no—
225#103—21no—
226C4×C8C22×C3×C511yescharacteristic
227C2×C3×C5—81no—
228C2×C3×C5#44911yescharacteristic
229C2×C3×C5#45211yescharacteristic
230C2×C3×C5#45611yescharacteristic
231C23×C3—21no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×D4—21no—
239C3×D4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4C2×C5×D411yescharacteristic
248C2×C3×C4C2×C5×D411yescharacteristic
249C2×C3×C4C2×C5×D411yescharacteristic
250C3×C8—22no—
251C3×C8—41no—
252C3×C8—41no—
253C22×C5—41no—
254C22×C5—41no—
255C22×C5—41no—
256C22×C5C3×#12011yescharacteristic
257C4×C5—21no—
258C4×C5—42no—
259C4×C5—41no—
260C4×C5—41no—
261C4×C5—41no—
262C4×C5—81no—
263C4×C5C3×#13511yescharacteristic
264C4×C5C3×#13511yescharacteristic
265C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
266C4×C5C3×#13711yescharacteristic
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8C3×C5×D411yescharacteristic
270C2×D4C3×C5×D411yescharacteristic
271C22×C4—21no—
272SD16—42no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4C3×C5×D411yescharacteristic
282C4⋊C4C3×C5×D411yescharacteristic
283C4⋊C4C3×C5×D412yes—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—21no—
287C42—22no—
288C42—21no—
289C42C23×C3×C511yescharacteristic
290C3×C5#165011yescharacteristic, direct product factor, Hall
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3C5×#12011yescharacteristic
295C3×C4—21no—
296C3×C4—42no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—81no—
301C3×C4C5×#13511yescharacteristic
302C3×C4C5×#13511yescharacteristic
303C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
304C3×C4C5×#13711yescharacteristic
305C2×C5—81no—
306C2×C5C3×#44911yescharacteristic
307C2×C5C3×#45211yescharacteristic
308C2×C5C3×#45611yescharacteristic
309C23—21no—
310Q8—21no—
311Q8—21no—
312Q8—41no—
313Q8—41no—
314Q8—41no—
315Q8—41no—
316D4—21no—
317D4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4C2×C3×C5×D411yescharacteristic
326C2×C4C2×C3×C5×D411yescharacteristic
327C2×C4C2×C3×C5×D411yescharacteristic
328C8—22no—
329C8—41no—
330C8—41no—
331C2×C3—81no—
332C2×C3C5×#44911yescharacteristic
333C2×C3C5×#45211yescharacteristic
334C2×C3C5×#45611yescharacteristic
335C5C3×#165011yescharacteristic, direct product factor, Sylow
336C22—41no—
337C22—41no—
338C22—41no—
339C22C3×C5×#12011yescharacteristic
340C4—21no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4C3×C5×#13511yescharacteristic
347C4C3×C5×#13511yescharacteristic
348C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
349C4C3×C5×#13711yescharacteristic
350C3C5×#165011yescharacteristic, direct product factor, Sylow
351C2—81no—
352C2C3×C5×#44911yescharacteristic
353C2C3×C5×#45211yescharacteristic
354C2C3×C5×#45611yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC2×C3×C5×D4D4
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
2. C3×C5×#500C2semidirect
3. C3×C5×#475C2semidirect
5. C3×C5×#332C2semidirect
6. C3×C5×Q8⋊C8C2semidirect
8. C5×#1650C3direct (N × H)
26. C3×#1650C5direct (N × H)
28. C5×#500C2×C3semidirect
29. C5×#475C2×C3semidirect
31. C5×#332C2×C3semidirect
32. C5×Q8⋊C8C2×C3semidirect
58. C3×#500C2×C5semidirect
59. C3×#475C2×C5semidirect
61. C3×#332C2×C5semidirect
62. C3×Q8⋊C8C2×C5semidirect
81. #1650C3×C5direct (N × H)
145. #500C2×C3×C5semidirect
146. #475C2×C3×C5semidirect
148. #332C2×C3×C5semidirect
149. Q8⋊C8C2×C3×C5semidirect
290. C3×C5#1650direct (N × H)
335. C5C3×#1650direct (N × H)
350. C3C5×#1650direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5562
C3×C5×#5002
C3×C5×#4752
C3×C5×#4612
C3×C5×#3322
C3×C5×Q8⋊C82
C3×C5×D4⋊C82
C5×#16503
C3×#16505