Finite Group Databasebeta

← C3×C5×#1604 #14274638

Subgroups — C3×C5×#1604 #14274638

Order 1920.

564 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×#537C211yesmaximal, characteristic
2C3×C5×#448C212yesmaximal, complemented
3C3×C5×#395C211yesmaximal, characteristic, complemented
4C3×C5×#327C212yesmaximal, complemented
5C3×C5×#324C211yesmaximal, characteristic, complemented
6C5×#1604C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41—21no—
8C22×C3×C5×Q8C2211yescharacteristic
9C2×C3×C5×SD16—22no—
10C3×C5×C22⋊Q8—22no—
11C3×C5×#121—22no—
12C3×C5×#121—21no—
13C3×C5×#121C2212yes—
14C2×C3×C5×C4⋊C4—21no—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C3×C5×#103—22no—
17C3×C5×C22⋊C8C2212yes—
18C3×C5×#95—21no—
19C3×C5×#95C2211yescharacteristic
20C3×#1604C511yesmaximal, characteristic, direct product factor, Hall
21C5×#537C2×C311yescharacteristic
22C5×#448C2×C312yescomplemented
23C5×#395C2×C311yescharacteristic, complemented
24C5×#327C2×C312yescomplemented
25C5×#324C2×C311yescharacteristic, complemented
26C3×C5×#41—42no—
27C3×C5×#41—42no—
28C2×C3×C5×Q8—21no—
29C2×C3×C5×Q8—22no—
30C2×C3×C5×Q8—42no—
31C2×C3×C5×D4—21no—
32C2×C3×C5×D4—22no—
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4D412yes—
35C22×C3×C4×C5—22no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5D411yescharacteristic
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⋊C4—22no—
45C3×C5×C4⋊C4—42no—
46C3×C5×C4⋊C4D412yes—
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—22no—
49C3×#537C2×C511yescharacteristic
50C3×#448C2×C512yescomplemented
51C3×#395C2×C511yescharacteristic, complemented
52C3×#327C2×C512yescomplemented
53C3×#324C2×C511yescharacteristic, complemented
54C2×C5×#41—21no—
55C22×C5×Q8C22×C311yescharacteristic
56C2×C5×SD16—22no—
57C5×C22⋊Q8—22no—
58C5×#121—22no—
59C5×#121—21no—
60C5×#121C22×C312yes—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4C22×C311yescharacteristic
63C5×#103—22no—
64C5×C22⋊C8C22×C312yes—
65C5×#95—21no—
66C5×#95C22×C311yescharacteristic
67#1604C3×C511yescharacteristic, direct product factor, Sylow
68C23×C3×C5—22no—
69C23×C3×C5C2×D411yescharacteristic
70C3×C5×Q8—24no—
71C3×C5×Q8—42no—
72C3×C5×Q8—42no—
73C3×C5×D4—42no—
74C3×C5×D4—42no—
75C3×C5×D4—42no—
76C3×C5×D4—42no—
77C3×C5×D4—81no—
78C2×C3×C4×C5—21no—
79C2×C3×C4×C5—21no—
80C2×C3×C4×C5—22no—
81C2×C3×C4×C5—22no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—22no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—44no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5SD1612yescomplemented
92C2×C3×C4×C5C2×D412yes—
93C3×C5×C8—42no—
94C2×C3×#41—21no—
95C22×C3×Q8C22×C511yescharacteristic
96C2×C3×SD16—22no—
97C3×C22⋊Q8—22no—
98C3×#121—22no—
99C3×#121—21no—
100C3×#121C22×C512yes—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4C22×C511yescharacteristic
103C3×#103—22no—
104C3×C22⋊C8C22×C512yes—
105C3×#95—21no—
106C3×#95C22×C511yescharacteristic
107C5×#41—42no—
108C5×#41—42no—
109C2×C5×Q8—21no—
110C2×C5×Q8—22no—
111C2×C5×Q8—42no—
112C2×C5×D4—21no—
113C2×C5×D4—22no—
114C2×C5×D4—21no—
115C2×C5×D4C3×D412yes—
116C22×C4×C5—22no—
117C22×C4×C5—21no—
118C22×C4×C5—21no—
119C22×C4×C5C3×D411yescharacteristic
120C22×C4×C5C3×D411yescharacteristic
121C22×C4×C5C23×C311yescharacteristic
122C5×SD16—44no—
123C2×C5×C8—22no—
124C5×C4⋊C4—21no—
125C5×C4⋊C4—22no—
126C5×C4⋊C4—42no—
127C5×C4⋊C4C3×D412yes—
128C5×C22⋊C4—22no—
129C5×C22⋊C4—22no—
130#537C2×C3×C511yescharacteristic
131#448C2×C3×C512yescomplemented
132#395C2×C3×C511yescharacteristic, complemented
133#327C2×C3×C512yescomplemented
134#324C2×C3×C511yescharacteristic, complemented
135C22×C3×C5—21no—
136C22×C3×C5—21no—
137C22×C3×C5—42no—
138C22×C3×C5—42no—
139C22×C3×C5—42no—
140C22×C3×C5#12011yescharacteristic
141C22×C3×C5C2×SD1611yescharacteristic
142C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
143C3×C4×C5—22no—
144C3×C4×C5—22no—
145C3×C4×C5—42no—
146C3×C4×C5—42no—
147C3×C4×C5—42no—
148C3×C4×C5—41no—
149C3×#41—42no—
150C3×#41—42no—
151C2×C3×Q8—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—42no—
154C2×C3×D4—21no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4C5×D412yes—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4C5×D411yescharacteristic
162C22×C3×C4C5×D411yescharacteristic
163C22×C3×C4C23×C511yescharacteristic
164C3×SD16—44no—
165C2×C3×C8—22no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4C5×D412yes—
170C3×C22⋊C4—22no—
171C3×C22⋊C4—22no—
172C23×C5—22no—
173C23×C5C2×C3×D411yescharacteristic
174C5×Q8—24no—
175C5×Q8—42no—
176C5×Q8—42no—
177C5×D4—42no—
178C5×D4—42no—
179C5×D4—42no—
180C5×D4—42no—
181C5×D4—81no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—22no—
185C2×C4×C5—22no—
186C2×C4×C5—21no—
187C2×C4×C5—22no—
188C2×C4×C5—41no—
189C2×C4×C5—44no—
190C2×C4×C5—41no—
191C2×C4×C5—41no—
192C2×C4×C5—41no—
193C2×C4×C5—41no—
194C2×C4×C5—41no—
195C2×C4×C5C3×SD1612yescomplemented
196C2×C4×C5C2×C3×D412yes—
197C5×C8—42no—
198C2×#41—21no—
199C22×Q8C22×C3×C511yescharacteristic
200C2×SD16—22no—
201C22⋊Q8—22no—
202#121—22no—
203#121—21no—
204#121C22×C3×C512yes—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C22×C3×C511yescharacteristic
207#103—22no—
208C22⋊C8C22×C3×C512yes—
209#95—21no—
210#95C22×C3×C511yescharacteristic
211C2×C3×C5—21no—
212C2×C3×C5—21no—
213C2×C3×C5—82no—
214C2×C3×C5#45011yescharacteristic
215C2×C3×C5#45611yescharacteristic
216C2×C3×C5B(4,𝔽2)11yescharacteristic
217C23×C3—22no—
218C23×C3C2×C5×D411yescharacteristic
219C3×Q8—24no—
220C3×Q8—42no—
221C3×Q8—42no—
222C3×D4—42no—
223C3×D4—42no—
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—81no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—44no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4C5×SD1612yescomplemented
241C2×C3×C4C2×C5×D412yes—
242C3×C8—42no—
243C22×C5—21no—
244C22×C5—21no—
245C22×C5—42no—
246C22×C5—42no—
247C22×C5—42no—
248C22×C5C3×#12011yescharacteristic
249C22×C5C2×C3×SD1611yescharacteristic
250C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
251C4×C5—22no—
252C4×C5—22no—
253C4×C5—42no—
254C4×C5—42no—
255C4×C5—42no—
256C4×C5—41no—
257#41—42no—
258#41—42no—
259C2×Q8—21no—
260C2×Q8—22no—
261C2×Q8—42no—
262C2×D4—21no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4C3×C5×D412yes—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4C3×C5×D411yescharacteristic
270C22×C4C3×C5×D411yescharacteristic
271C22×C4C23×C3×C511yescharacteristic
272SD16—44no—
273C2×C8—22no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4C3×C5×D412yes—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C3×C5#160411yescharacteristic, direct product factor, Hall
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3—42no—
286C22×C3C5×#12011yescharacteristic
287C22×C3C2×C5×SD1611yescharacteristic
288C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
289C3×C4—22no—
290C3×C4—22no—
291C3×C4—42no—
292C3×C4—42no—
293C3×C4—42no—
294C3×C4—41no—
295C2×C5—21no—
296C2×C5—21no—
297C2×C5—82no—
298C2×C5C3×#45011yescharacteristic
299C2×C5C3×#45611yescharacteristic
300C2×C5C3×B(4,𝔽2)11yescharacteristic
301C23—22no—
302C23C2×C3×C5×D411yescharacteristic
303Q8—24no—
304Q8—42no—
305Q8—42no—
306D4—42no—
307D4—42no—
308D4—42no—
309D4—42no—
310D4—81no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4C3×C5×SD1612yescomplemented
325C2×C4C2×C3×C5×D412yes—
326C8—42no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—82no—
330C2×C3C5×#45011yescharacteristic
331C2×C3C5×#45611yescharacteristic
332C2×C3C5×B(4,𝔽2)11yescharacteristic
333C5C3×#160411yescharacteristic, direct product factor, Sylow
334C22—21no—
335C22—21no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22C3×C5×#12011yescharacteristic
340C22C2×C3×C5×SD1611yescharacteristic
341C22C3×C5×AGL(1,ℤ/8)11yescharacteristic
342C4—22no—
343C4—22no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—41no—
348C3C5×#160411yescharacteristic, direct product factor, Sylow
349C2—21no—
350C2—21no—
351C2—82no—
352C2C3×C5×#45011yescharacteristic
353C2C3×C5×#45611yescharacteristic
354C2C3×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC3×C5×#537C2
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×#395C2semidirect
4. C3×C5×#327C2semidirect
5. C3×C5×#324C2semidirect
6. C5×#1604C3direct (N × H)
20. C3×#1604C5direct (N × H)
22. C5×#448C2×C3semidirect
23. C5×#395C2×C3semidirect
24. C5×#327C2×C3semidirect
25. C5×#324C2×C3semidirect
50. C3×#448C2×C5semidirect
51. C3×#395C2×C5semidirect
52. C3×#327C2×C5semidirect
53. C3×#324C2×C5semidirect
67. #1604C3×C5direct (N × H)
91. C2×C3×C4×C5SD16semidirect
131. #448C2×C3×C5semidirect
132. #395C2×C3×C5semidirect
133. #327C2×C3×C5semidirect
134. #324C2×C3×C5semidirect
195. C2×C4×C5C3×SD16semidirect
240. C2×C3×C4C5×SD16semidirect
280. C3×C5#1604direct (N × H)
324. C2×C4C3×C5×SD16semidirect
333. C5C3×#1604direct (N × H)
348. C3C5×#1604direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5372
C3×C5×#4482
C3×C5×#3952
C3×C5×#3272
C3×C5×#3242
C5×#16043
C3×#16045