Finite Group Databasebeta

← C3×C5×#1659 #14274693

Subgroups — C3×C5×#1659 #14274693

Order 1920.

456 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×H(3,𝔽4)C211yesmaximal, characteristic
2C3×C5×#497C211yesmaximal, characteristic, complemented
3C3×C5×#489C212yesmaximal, complemented
4C3×C5×#329C212yesmaximal, complemented
5C3×C5×#329C211yesmaximal, characteristic, complemented
6C5×#1659C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×Q16—21no—
8C2×C3×C5×SD16—21no—
9C3×C5×C4⋊Q8C2211yescharacteristic, complemented
10C3×C5×#124—21no—
11C3×C5×#124—22no—
12C3×C5×#124C2212yes—
13C3×C5×#124C2211yescharacteristic
14C3×C5×#120—21no—
15C3×C5×#120—22no—
16C3×C5×#103—22no—
17C3×C5×#102—22no—
18C3×C5×#97C2212yescomplemented
19C3×C5×#97C2211yescharacteristic, complemented
20C3×#1659C511yesmaximal, characteristic, direct product factor, Hall
21C5×H(3,𝔽4)C2×C311yescharacteristic
22C5×#497C2×C311yescharacteristic, complemented
23C5×#489C2×C312yescomplemented
24C5×#329C2×C312yescomplemented
25C5×#329C2×C311yescharacteristic, complemented
26C24×C3×C5—21no—
27C2×C3×C5×Q8—21no—
28C2×C3×C5×Q8D411yescharacteristic
29C2×C3×C5×Q8D412yes—
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4D411yescharacteristic
33C2×C3×C5×D4D412yes—
34C3×C5×Q16—42no—
35C3×C5×SD16—42no—
36C2×C3×C5×C8—21no—
37C2×C3×C5×C8—22no—
38C3×C5×C4⋊C4—22no—
39C3×C5×C22⋊C4—22no—
40C3×C5×C22⋊C4—21no—
41C3×C5×C22⋊C4—22no—
42C3×C5×C22⋊C4—22no—
43C3×C5×C22⋊C4—22no—
44C3×C42×C5—21no—
45C3×C42×C5C2311yescharacteristic
46C3×H(3,𝔽4)C2×C511yescharacteristic
47C3×#497C2×C511yescharacteristic, complemented
48C3×#489C2×C512yescomplemented
49C3×#329C2×C511yescharacteristic, complemented
50C3×#329C2×C512yescomplemented
51C2×C5×Q16—21no—
52C2×C5×SD16—21no—
53C5×C4⋊Q8C22×C311yescharacteristic, complemented
54C5×#124—21no—
55C5×#124—22no—
56C5×#124C22×C312yes—
57C5×#124C22×C311yescharacteristic
58C5×#120—21no—
59C5×#120—22no—
60C5×#103—22no—
61C5×#102—22no—
62C5×#97C22×C312yescomplemented
63C5×#97C22×C311yescharacteristic, complemented
64#1659C3×C511yescharacteristic, direct product factor, Sylow
65C23×C3×C5—21no—
66C23×C3×C5—22no—
67C23×C3×C5—81no—
68C23×C3×C5—82no—
69C3×C5×Q8—41no—
70C3×C5×Q8—42no—
71C3×C5×Q8—41no—
72C3×C5×Q8—41no—
73C3×C5×D4—41no—
74C3×C5×D4—42no—
75C3×C5×D4—81no—
76C3×C5×D4—82no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5—41no—
80C2×C3×C4×C5C2×D412yes—
81C2×C3×C4×C5C2×D411yescharacteristic
82C3×C5×C8—22no—
83C3×C5×C8—42no—
84C2×C3×Q16—21no—
85C2×C3×SD16—21no—
86C3×C4⋊Q8C22×C511yescharacteristic, complemented
87C3×#124—22no—
88C3×#124—21no—
89C3×#124C22×C512yes—
90C3×#124C22×C511yescharacteristic
91C3×#120—22no—
92C3×#120—21no—
93C3×#103—22no—
94C3×#102—22no—
95C3×#97C22×C512yescomplemented
96C3×#97C22×C511yescharacteristic, complemented
97C24×C5—21no—
98C2×C5×Q8—21no—
99C2×C5×Q8C3×D411yescharacteristic
100C2×C5×Q8C3×D412yes—
101C2×C5×D4—21no—
102C2×C5×D4—22no—
103C2×C5×D4C3×D411yescharacteristic
104C2×C5×D4C3×D412yes—
105C5×Q16—42no—
106C5×SD16—42no—
107C2×C5×C8—21no—
108C2×C5×C8—22no—
109C5×C4⋊C4—22no—
110C5×C22⋊C4—22no—
111C5×C22⋊C4—21no—
112C5×C22⋊C4—22no—
113C5×C22⋊C4—22no—
114C5×C22⋊C4—22no—
115C42×C5—21no—
116C42×C5C23×C311yescharacteristic
117H(3,𝔽4)C2×C3×C511yescharacteristic
118#497C2×C3×C511yescharacteristic, complemented
119#489C2×C3×C512yescomplemented
120#329C2×C3×C511yescharacteristic, complemented
121#329C2×C3×C512yescomplemented
122C22×C3×C5—41no—
123C22×C3×C5—42no—
124C22×C3×C5—42no—
125C22×C3×C5—42no—
126C22×C3×C5—42no—
127C22×C3×C5—84no—
128C22×C3×C5#12011yescharacteristic
129C3×C4×C5—21no—
130C3×C4×C5—22no—
131C3×C4×C5—41no—
132C3×C4×C5—42no—
133C3×C4×C5—81no—
134C24×C3—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8C5×D412yes—
137C2×C3×Q8C5×D411yescharacteristic
138C2×C3×D4—21no—
139C2×C3×D4—22no—
140C2×C3×D4C5×D412yes—
141C2×C3×D4C5×D411yescharacteristic
142C3×Q16—42no—
143C3×SD16—42no—
144C2×C3×C8—21no—
145C2×C3×C8—22no—
146C3×C4⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C42—21no—
153C3×C42C23×C511yescharacteristic
154C23×C5—21no—
155C23×C5—22no—
156C23×C5—81no—
157C23×C5—82no—
158C5×Q8—41no—
159C5×Q8—42no—
160C5×Q8—41no—
161C5×Q8—41no—
162C5×D4—41no—
163C5×D4—42no—
164C5×D4—81no—
165C5×D4—82no—
166C2×C4×C5—21no—
167C2×C4×C5—22no—
168C2×C4×C5—41no—
169C2×C4×C5C2×C3×D412yes—
170C2×C4×C5C2×C3×D411yescharacteristic
171C5×C8—22no—
172C5×C8—42no—
173C2×Q16—21no—
174C2×SD16—21no—
175C4⋊Q8C22×C3×C511yescharacteristic, complemented
176#124—22no—
177#124—21no—
178#124C22×C3×C512yes—
179#124C22×C3×C511yescharacteristic
180#120—22no—
181#120—21no—
182#103—22no—
183#102—22no—
184#97C22×C3×C512yescomplemented
185#97C22×C3×C511yescharacteristic, complemented
186C2×C3×C5—81no—
187C2×C3×C5—82no—
188C2×C3×C5#45311yescharacteristic
189C2×C3×C5#45512yes—
190C23×C3—21no—
191C23×C3—22no—
192C23×C3—81no—
193C23×C3—82no—
194C3×Q8—41no—
195C3×Q8—42no—
196C3×Q8—41no—
197C3×Q8—41no—
198C3×D4—41no—
199C3×D4—42no—
200C3×D4—81no—
201C3×D4—82no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4C2×C5×D412yes—
206C2×C3×C4C2×C5×D411yescharacteristic
207C3×C8—22no—
208C3×C8—42no—
209C22×C5—41no—
210C22×C5—42no—
211C22×C5—42no—
212C22×C5—42no—
213C22×C5—42no—
214C22×C5—84no—
215C22×C5C3×#12011yescharacteristic
216C4×C5—21no—
217C4×C5—22no—
218C4×C5—41no—
219C4×C5—42no—
220C4×C5—81no—
221C24—21no—
222C2×Q8—21no—
223C2×Q8C3×C5×D412yes—
224C2×Q8C3×C5×D411yescharacteristic
225C2×D4—21no—
226C2×D4—22no—
227C2×D4C3×C5×D412yes—
228C2×D4C3×C5×D411yescharacteristic
229Q16—42no—
230SD16—42no—
231C2×C8—21no—
232C2×C8—22no—
233C4⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—21no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C42—21no—
240C42C23×C3×C511yescharacteristic
241C3×C5#165911yescharacteristic, direct product factor, Hall
242C22×C3—41no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3—42no—
246C22×C3—42no—
247C22×C3—84no—
248C22×C3C5×#12011yescharacteristic
249C3×C4—21no—
250C3×C4—22no—
251C3×C4—41no—
252C3×C4—42no—
253C3×C4—81no—
254C2×C5—81no—
255C2×C5—82no—
256C2×C5C3×#45311yescharacteristic
257C2×C5C3×#45512yes—
258C23—21no—
259C23—22no—
260C23—81no—
261C23—82no—
262Q8—41no—
263Q8—42no—
264Q8—41no—
265Q8—41no—
266D4—41no—
267D4—42no—
268D4—81no—
269D4—82no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4C2×C3×C5×D412yes—
274C2×C4C2×C3×C5×D411yescharacteristic
275C8—22no—
276C8—42no—
277C2×C3—81no—
278C2×C3—82no—
279C2×C3C5×#45311yescharacteristic
280C2×C3C5×#45512yes—
281C5C3×#165911yescharacteristic, direct product factor, Sylow
282C22—41no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—84no—
288C22C3×C5×#12011yescharacteristic
289C4—21no—
290C4—22no—
291C4—41no—
292C4—42no—
293C4—81no—
294C3C5×#165911yescharacteristic, direct product factor, Sylow
295C2—81no—
296C2—82no—
297C2C3×C5×#45311yescharacteristic
298C2C3×C5×#45512yes—
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC3×C5×H(3,𝔽4)C2
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×#497C2semidirect
3. C3×C5×#489C2semidirect
4. C3×C5×#329C2semidirect
5. C3×C5×#329C2semidirect
6. C5×#1659C3direct (N × H)
9. C3×C5×C4⋊Q8C22semidirect
18. C3×C5×#97C22semidirect
19. C3×C5×#97C22semidirect
20. C3×#1659C5direct (N × H)
22. C5×#497C2×C3semidirect
23. C5×#489C2×C3semidirect
24. C5×#329C2×C3semidirect
25. C5×#329C2×C3semidirect
47. C3×#497C2×C5semidirect
48. C3×#489C2×C5semidirect
49. C3×#329C2×C5semidirect
50. C3×#329C2×C5semidirect
53. C5×C4⋊Q8C22×C3semidirect
62. C5×#97C22×C3semidirect
63. C5×#97C22×C3semidirect
64. #1659C3×C5direct (N × H)
86. C3×C4⋊Q8C22×C5semidirect
95. C3×#97C22×C5semidirect
96. C3×#97C22×C5semidirect
118. #497C2×C3×C5semidirect
119. #489C2×C3×C5semidirect
120. #329C2×C3×C5semidirect
121. #329C2×C3×C5semidirect
175. C4⋊Q8C22×C3×C5semidirect
184. #97C22×C3×C5semidirect
185. #97C22×C3×C5semidirect
241. C3×C5#1659direct (N × H)
281. C5C3×#1659direct (N × H)
294. C3C5×#1659direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×H(3,𝔽4)2
C3×C5×#4972
C3×C5×#4892
C3×C5×#3292
C5×#16593
C3×#16595