Finite Group Databasebeta

← C3×C5×#1605 #14274639

Subgroups — C3×C5×#1605 #14274639

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#544C211yesmaximal, characteristic
2C3×C5×#451C212yesmaximal
3C3×C5×#395C211yesmaximal, characteristic
4C3×C5×#328C212yesmaximal
5C3×C5×#324C211yesmaximal, characteristic
6C5×#1605C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×Q8C2211yescharacteristic
8C2×C3×C5×Q16—22no—
9C3×C5×C4⋊Q8—22no—
10C3×C5×#125—22no—
11C3×C5×C22⋊Q8—21no—
12C3×C5×C22⋊Q8C2212yes—
13C3×C5×#117—21no—
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×#1605C511yesmaximal, characteristic, direct product factor, Hall
21C5×#544C2×C311yescharacteristic
22C5×#451C2×C312yes—
23C5×#395C2×C311yescharacteristic
24C5×#328C2×C312yes—
25C5×#324C2×C311yescharacteristic
26C2×C3×C5×Q8—21no—
27C2×C3×C5×Q8—22no—
28C2×C3×C5×Q8—42no—
29C2×C3×C5×Q8D412yes—
30C22×C3×C4×C5—21no—
31C22×C3×C4×C5—21no—
32C22×C3×C4×C5D411yescharacteristic
33C22×C3×C4×C5D411yescharacteristic
34C22×C3×C4×C5C2311yescharacteristic
35C3×C5×Q16—44no—
36C2×C3×C5×C8—22no—
37C3×C5×C4⋊C4—21no—
38C3×C5×C4⋊C4—21no—
39C3×C5×C4⋊C4—21no—
40C3×C5×C4⋊C4—22no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C4⋊C4—22no—
43C3×C5×C4⋊C4—42no—
44C3×C5×C4⋊C4D412yes—
45C3×C5×C22⋊C4—22no—
46C3×C42×C5—22no—
47C3×#544C2×C511yescharacteristic
48C3×#451C2×C512yes—
49C3×#395C2×C511yescharacteristic
50C3×#328C2×C512yes—
51C3×#324C2×C511yescharacteristic
52C22×C5×Q8C22×C311yescharacteristic
53C2×C5×Q16—22no—
54C5×C4⋊Q8—22no—
55C5×#125—22no—
56C5×C22⋊Q8—21no—
57C5×C22⋊Q8C22×C312yes—
58C5×#117—21no—
59C2×C5×C4⋊C4—21no—
60C2×C5×C4⋊C4C22×C311yescharacteristic
61C5×#103—22no—
62C5×C22⋊C8C22×C312yes—
63C5×#95—21no—
64C5×#95C22×C311yescharacteristic
65#1605C3×C511yescharacteristic, direct product factor, Sylow
66C23×C3×C5C2×D411yescharacteristic
67C3×C5×Q8—24no—
68C3×C5×Q8—42no—
69C3×C5×Q8—42no—
70C3×C5×Q8—81no—
71C2×C3×C4×C5—21no—
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5—22no—
74C2×C3×C4×C5—22no—
75C2×C3×C4×C5—22no—
76C2×C3×C4×C5—22no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—41no—
79C2×C3×C4×C5—41no—
80C2×C3×C4×C5—41no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5Q1612yescomplemented
85C2×C3×C4×C5C2×D412yes—
86C3×C5×C8—42no—
87C22×C3×Q8C22×C511yescharacteristic
88C2×C3×Q16—22no—
89C3×C4⋊Q8—22no—
90C3×#125—22no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8C22×C512yes—
93C3×#117—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C22×C511yescharacteristic
96C3×#103—22no—
97C3×C22⋊C8C22×C512yes—
98C3×#95—21no—
99C3×#95C22×C511yescharacteristic
100C2×C5×Q8—21no—
101C2×C5×Q8—22no—
102C2×C5×Q8—42no—
103C2×C5×Q8C3×D412yes—
104C22×C4×C5—21no—
105C22×C4×C5—21no—
106C22×C4×C5C3×D411yescharacteristic
107C22×C4×C5C3×D411yescharacteristic
108C22×C4×C5C23×C311yescharacteristic
109C5×Q16—44no—
110C2×C5×C8—22no—
111C5×C4⋊C4—21no—
112C5×C4⋊C4—21no—
113C5×C4⋊C4—21no—
114C5×C4⋊C4—22no—
115C5×C4⋊C4—22no—
116C5×C4⋊C4—22no—
117C5×C4⋊C4—42no—
118C5×C4⋊C4C3×D412yes—
119C5×C22⋊C4—22no—
120C42×C5—22no—
121#544C2×C3×C511yescharacteristic
122#451C2×C3×C512yes—
123#395C2×C3×C511yescharacteristic
124#328C2×C3×C512yes—
125#324C2×C3×C511yescharacteristic
126C22×C3×C5—21no—
127C22×C3×C5—21no—
128C22×C3×C5#12011yescharacteristic
129C22×C3×C5C2×Q1611yescharacteristic
130C22×C3×C5#13711yescharacteristic
131C3×C4×C5—22no—
132C3×C4×C5—22no—
133C3×C4×C5—42no—
134C3×C4×C5—42no—
135C3×C4×C5—42no—
136C3×C4×C5—41no—
137C3×C4×C5—42no—
138C2×C3×Q8—21no—
139C2×C3×Q8—22no—
140C2×C3×Q8—42no—
141C2×C3×Q8C5×D412yes—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4C5×D411yescharacteristic
145C22×C3×C4C5×D411yescharacteristic
146C22×C3×C4C23×C511yescharacteristic
147C3×Q16—44no—
148C2×C3×C8—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4C5×D412yes—
157C3×C22⋊C4—22no—
158C3×C42—22no—
159C23×C5C2×C3×D411yescharacteristic
160C5×Q8—24no—
161C5×Q8—42no—
162C5×Q8—42no—
163C5×Q8—81no—
164C2×C4×C5—21no—
165C2×C4×C5—21no—
166C2×C4×C5—22no—
167C2×C4×C5—22no—
168C2×C4×C5—22no—
169C2×C4×C5—22no—
170C2×C4×C5—21no—
171C2×C4×C5—41no—
172C2×C4×C5—41no—
173C2×C4×C5—41no—
174C2×C4×C5—41no—
175C2×C4×C5—41no—
176C2×C4×C5—41no—
177C2×C4×C5C3×Q1612yescomplemented
178C2×C4×C5C2×C3×D412yes—
179C5×C8—42no—
180C22×Q8C22×C3×C511yescharacteristic
181C2×Q16—22no—
182C4⋊Q8—22no—
183#125—22no—
184C22⋊Q8—21no—
185C22⋊Q8C22×C3×C512yes—
186#117—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C22×C3×C511yescharacteristic
189#103—22no—
190C22⋊C8C22×C3×C512yes—
191#95—21no—
192#95C22×C3×C511yescharacteristic
193C2×C3×C5—21no—
194C2×C3×C5—21no—
195C2×C3×C5#45111yescharacteristic
196C2×C3×C5#45611yescharacteristic
197C2×C3×C5B(4,𝔽2)11yescharacteristic
198C23×C3C2×C5×D411yescharacteristic
199C3×Q8—24no—
200C3×Q8—42no—
201C3×Q8—42no—
202C3×Q8—81no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4C5×Q1612yescomplemented
217C2×C3×C4C2×C5×D412yes—
218C3×C8—42no—
219C22×C5—21no—
220C22×C5—21no—
221C22×C5C3×#12011yescharacteristic
222C22×C5C2×C3×Q1611yescharacteristic
223C22×C5C3×#13711yescharacteristic
224C4×C5—22no—
225C4×C5—22no—
226C4×C5—42no—
227C4×C5—42no—
228C4×C5—42no—
229C4×C5—41no—
230C4×C5—42no—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8C3×C5×D412yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4C3×C5×D411yescharacteristic
238C22×C4C3×C5×D411yescharacteristic
239C22×C4C23×C3×C511yescharacteristic
240Q16—44no—
241C2×C8—22no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4C3×C5×D412yes—
250C22⋊C4—22no—
251C42—22no—
252C3×C5#160511yescharacteristic, direct product factor, Hall
253C22×C3—21no—
254C22×C3—21no—
255C22×C3C5×#12011yescharacteristic
256C22×C3C2×C5×Q1611yescharacteristic
257C22×C3C5×#13711yescharacteristic
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—42no—
261C3×C4—42no—
262C3×C4—42no—
263C3×C4—41no—
264C3×C4—42no—
265C2×C5—21no—
266C2×C5—21no—
267C2×C5C3×#45111yescharacteristic
268C2×C5C3×#45611yescharacteristic
269C2×C5C3×B(4,𝔽2)11yescharacteristic
270C23C2×C3×C5×D411yescharacteristic
271Q8—24no—
272Q8—42no—
273Q8—42no—
274Q8—81no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4C3×C5×Q1612yescomplemented
289C2×C4C2×C3×C5×D412yes—
290C8—42no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3C5×#45111yescharacteristic
294C2×C3C5×#45611yescharacteristic
295C2×C3C5×B(4,𝔽2)11yescharacteristic
296C5C3×#160511yescharacteristic, direct product factor, Sylow
297C22—21no—
298C22—21no—
299C22C3×C5×#12011yescharacteristic
300C22C2×C3×C5×Q1611yescharacteristic
301C22C3×C5×#13711yescharacteristic
302C4—22no—
303C4—22no—
304C4—42no—
305C4—42no—
306C4—42no—
307C4—41no—
308C4—42no—
309C3C5×#160511yescharacteristic, direct product factor, Sylow
310C2—21no—
311C2—21no—
312C2C3×C5×#45111yescharacteristic
313C2C3×C5×#45611yescharacteristic
314C2C3×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC23×C3×C5C2×D4
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
6. C5×#1605C3direct (N × H)
20. C3×#1605C5direct (N × H)
65. #1605C3×C5direct (N × H)
84. C2×C3×C4×C5Q16semidirect
177. C2×C4×C5C3×Q16semidirect
216. C2×C3×C4C5×Q16semidirect
252. C3×C5#1605direct (N × H)
288. C2×C4C3×C5×Q16semidirect
296. C5C3×#1605direct (N × H)
309. C3C5×#1605direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5442
C3×C5×#4512
C3×C5×#3952
C3×C5×#3282
C3×C5×#3242
C5×#16053
C3×#16055