Finite Group Databasebeta

← C3×C5×#1660 #14274694

Subgroups — C3×C5×#1660 #14274694

Order 1920.

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

normal subgroups only

This group has 392 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×#563C211yesmaximal, characteristic
2C3×C5×#501C212yesmaximal, complemented
3C3×C5×#497C211yesmaximal, characteristic
4C3×C5×#330C212yesmaximal, complemented
5C3×C5×#329C211yesmaximal, characteristic
6C5×#1660C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×Q16—21no—
8C2×C3×C5×SD16—21no—
9C3×C5×C4⋊Q8—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×#126—22no—
12C3×C5×#125C2212yes—
13C3×C5×#124C2211yescharacteristic
14C3×C5×#123—21no—
15C3×C5×C22⋊Q8—22no—
16C3×C5×#107—22no—
17C3×C5×#106—22no—
18C3×C5×#97C2211yescharacteristic
19C3×C5×#97C2212yes—
20C3×#1660C511yesmaximal, characteristic, direct product factor, Hall
21C5×#563C2×C311yescharacteristic
22C5×#501C2×C312yescomplemented
23C5×#497C2×C311yescharacteristic
24C5×#330C2×C312yescomplemented
25C5×#329C2×C311yescharacteristic
26C2×C3×C5×Q8—21no—
27C2×C3×C5×Q8—21no—
28C2×C3×C5×Q8D411yescharacteristic
29C2×C3×C5×D4D411yescharacteristic
30C22×C3×C4×C5—21no—
31C3×C5×Q16—42no—
32C3×C5×SD16—42no—
33C2×C3×C5×C8—21no—
34C2×C3×C5×C8—22no—
35C3×C5×C4⋊C4—22no—
36C3×C5×C4⋊C4—22no—
37C3×C5×C4⋊C4—22no—
38C3×C5×C4⋊C4—22no—
39C3×C5×C4⋊C4D412yes—
40C3×C5×C4⋊C4D412yes—
41C3×C5×C22⋊C4—22no—
42C3×C5×C22⋊C4—21no—
43C3×C5×C22⋊C4—22no—
44C3×C42×C5—21no—
45C3×C42×C5C2311yescharacteristic
46C3×#563C2×C511yescharacteristic
47C3×#501C2×C512yescomplemented
48C3×#497C2×C511yescharacteristic
49C3×#330C2×C512yescomplemented
50C3×#329C2×C511yescharacteristic
51C2×C5×Q16—21no—
52C2×C5×SD16—21no—
53C5×C4⋊Q8—21no—
54C5×C4⋊Q8C22×C311yescharacteristic
55C5×#126—22no—
56C5×#125C22×C312yes—
57C5×#124C22×C311yescharacteristic
58C5×#123—21no—
59C5×C22⋊Q8—22no—
60C5×#107—22no—
61C5×#106—22no—
62C5×#97C22×C312yes—
63C5×#97C22×C311yescharacteristic
64#1660C3×C511yescharacteristic, direct product factor, Sylow
65C23×C3×C5—21no—
66C3×C5×Q8—41no—
67C3×C5×Q8—41no—
68C3×C5×Q8—41no—
69C3×C5×Q8—81no—
70C3×C5×D4—41no—
71C2×C3×C4×C5—22no—
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5—22no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5—81no—
76C2×C3×C4×C5C2×D412yes—
77C2×C3×C4×C5C2×D411yescharacteristic
78C3×C5×C8—24no—
79C3×C5×C8—22no—
80C2×C3×Q16—21no—
81C2×C3×SD16—21no—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8C22×C511yescharacteristic
84C3×#126—22no—
85C3×#125C22×C512yes—
86C3×#124C22×C511yescharacteristic
87C3×#123—21no—
88C3×C22⋊Q8—22no—
89C3×#107—22no—
90C3×#106—22no—
91C3×#97C22×C512yes—
92C3×#97C22×C511yescharacteristic
93C2×C5×Q8—21no—
94C2×C5×Q8—21no—
95C2×C5×Q8C3×D411yescharacteristic
96C2×C5×D4C3×D411yescharacteristic
97C22×C4×C5—21no—
98C5×Q16—42no—
99C5×SD16—42no—
100C2×C5×C8—21no—
101C2×C5×C8—22no—
102C5×C4⋊C4—22no—
103C5×C4⋊C4—22no—
104C5×C4⋊C4—22no—
105C5×C4⋊C4—22no—
106C5×C4⋊C4C3×D412yes—
107C5×C4⋊C4C3×D412yes—
108C5×C22⋊C4—22no—
109C5×C22⋊C4—21no—
110C5×C22⋊C4—22no—
111C42×C5—21no—
112C42×C5C23×C311yescharacteristic
113#563C2×C3×C511yescharacteristic
114#501C2×C3×C512yescomplemented
115#497C2×C3×C511yescharacteristic
116#330C2×C3×C512yescomplemented
117#329C2×C3×C511yescharacteristic
118C22×C3×C5—41no—
119C22×C3×C5—42no—
120C22×C3×C5#12011yescharacteristic
121C3×C4×C5—21no—
122C3×C4×C5—22no—
123C3×C4×C5—42no—
124C3×C4×C5—41no—
125C3×C4×C5—42no—
126C3×C4×C5—81no—
127C2×C3×Q8—21no—
128C2×C3×Q8—21no—
129C2×C3×Q8C5×D411yescharacteristic
130C2×C3×D4C5×D411yescharacteristic
131C22×C3×C4—21no—
132C3×Q16—42no—
133C3×SD16—42no—
134C2×C3×C8—22no—
135C2×C3×C8—21no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4C5×D412yes—
141C3×C4⋊C4C5×D412yes—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—22no—
145C3×C42—21no—
146C3×C42C23×C511yescharacteristic
147C23×C5—21no—
148C5×Q8—41no—
149C5×Q8—41no—
150C5×Q8—41no—
151C5×Q8—81no—
152C5×D4—41no—
153C2×C4×C5—22no—
154C2×C4×C5—21no—
155C2×C4×C5—22no—
156C2×C4×C5—41no—
157C2×C4×C5—81no—
158C2×C4×C5C2×C3×D412yes—
159C2×C4×C5C2×C3×D411yescharacteristic
160C5×C8—24no—
161C5×C8—22no—
162C2×Q16—21no—
163C2×SD16—21no—
164C4⋊Q8—21no—
165C4⋊Q8C22×C3×C511yescharacteristic
166#126—22no—
167#125C22×C3×C512yes—
168#124C22×C3×C511yescharacteristic
169#123—21no—
170C22⋊Q8—22no—
171#107—22no—
172#106—22no—
173#97C22×C3×C512yes—
174#97C22×C3×C511yescharacteristic
175C2×C3×C5—81no—
176C2×C3×C5#45311yescharacteristic
177C2×C3×C5#45612yes—
178C23×C3—21no—
179C3×Q8—41no—
180C3×Q8—41no—
181C3×Q8—41no—
182C3×Q8—81no—
183C3×D4—41no—
184C2×C3×C4—22no—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—41no—
188C2×C3×C4—81no—
189C2×C3×C4C2×C5×D412yes—
190C2×C3×C4C2×C5×D411yescharacteristic
191C3×C8—24no—
192C3×C8—22no—
193C22×C5—41no—
194C22×C5—42no—
195C22×C5C3×#12011yescharacteristic
196C4×C5—21no—
197C4×C5—22no—
198C4×C5—42no—
199C4×C5—41no—
200C4×C5—42no—
201C4×C5—81no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8C3×C5×D411yescharacteristic
205C2×D4C3×C5×D411yescharacteristic
206C22×C4—21no—
207Q16—42no—
208SD16—42no—
209C2×C8—22no—
210C2×C8—21no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4C3×C5×D412yes—
216C4⋊C4C3×C5×D412yes—
217C22⋊C4—22no—
218C22⋊C4—21no—
219C22⋊C4—22no—
220C42—21no—
221C42C23×C3×C511yescharacteristic
222C3×C5#166011yescharacteristic, direct product factor, Hall
223C22×C3—41no—
224C22×C3—42no—
225C22×C3C5×#12011yescharacteristic
226C3×C4—21no—
227C3×C4—22no—
228C3×C4—42no—
229C3×C4—41no—
230C3×C4—42no—
231C3×C4—81no—
232C2×C5—81no—
233C2×C5C3×#45311yescharacteristic
234C2×C5C3×#45612yes—
235C23—21no—
236Q8—41no—
237Q8—41no—
238Q8—41no—
239Q8—81no—
240D4—41no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—81no—
246C2×C4C2×C3×C5×D412yes—
247C2×C4C2×C3×C5×D411yescharacteristic
248C8—22no—
249C8—24no—
250C2×C3—81no—
251C2×C3C5×#45311yescharacteristic
252C2×C3C5×#45612yes—
253C5C3×#166011yescharacteristic, direct product factor, Sylow
254C22—41no—
255C22—42no—
256C22C3×C5×#12011yescharacteristic
257C4—21no—
258C4—22no—
259C4—42no—
260C4—41no—
261C4—42no—
262C4—81no—
263C3C5×#166011yescharacteristic, direct product factor, Sylow
264C2—81no—
265C2C3×C5×#45311yescharacteristic
266C2C3×C5×#45612yes—
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×#501C2semidirect
4. C3×C5×#330C2semidirect
6. C5×#1660C3direct (N × H)
20. C3×#1660C5direct (N × H)
22. C5×#501C2×C3semidirect
24. C5×#330C2×C3semidirect
47. C3×#501C2×C5semidirect
49. C3×#330C2×C5semidirect
64. #1660C3×C5direct (N × H)
114. #501C2×C3×C5semidirect
116. #330C2×C3×C5semidirect
222. C3×C5#1660direct (N × H)
253. C5C3×#1660direct (N × H)
263. C3C5×#1660direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5632
C3×C5×#5012
C3×C5×#4972
C3×C5×#3302
C3×C5×#3292
C5×#16603
C3×#16605