Finite Group Databasebeta

← C3×C5×#1657 #14274691

Subgroups — C3×C5×#1657 #14274691

Order 1920.

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

normal subgroups only

This group has 412 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, complemented
2C3×C5×#496C211yesmaximal, characteristic
3C3×C5×#490C212yesmaximal, complemented
4C3×C5×#330C212yesmaximal, complemented
5C3×C5×#329C211yesmaximal, characteristic, complemented
6C5×#1657C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×SD16—21no—
8C2×C3×C5×D8—21no—
9C3×C5×C4⋊Q8—21no—
10C3×C5×#127C2211yescharacteristic
11C3×C5×#126—22no—
12C3×C5×#125C2212yes—
13C3×C5×#124C2211yescharacteristic
14C3×C5×#123—21no—
15C3×C5×C22⋊Q8—22no—
16C3×C5×#102—22no—
17C3×C5×#102—22no—
18C3×C5×#97C2212yes—
19C3×C5×#97C2211yescharacteristic
20C3×#1657C511yesmaximal, characteristic, direct product factor, Hall
21C5×#563C2×C311yescharacteristic, complemented
22C5×#496C2×C311yescharacteristic
23C5×#490C2×C312yescomplemented
24C5×#330C2×C312yescomplemented
25C5×#329C2×C311yescharacteristic, complemented
26C2×C3×C5×Q8—21no—
27C2×C3×C5×Q8D411yescharacteristic
28C2×C3×C5×D4—21no—
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4D411yescharacteristic
31C22×C3×C4×C5—21no—
32C3×C5×SD16—42no—
33C3×C5×D8—42no—
34C2×C3×C5×C8—21no—
35C2×C3×C5×C8—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, complemented
47C3×#496C2×C511yescharacteristic
48C3×#490C2×C512yescomplemented
49C3×#330C2×C512yescomplemented
50C3×#329C2×C511yescharacteristic, complemented
51C2×C5×SD16—21no—
52C2×C5×D8—21no—
53C5×C4⋊Q8—21no—
54C5×#127C22×C311yescharacteristic
55C5×#126—22no—
56C5×#125C22×C312yes—
57C5×#124C22×C311yescharacteristic
58C5×#123—21no—
59C5×C22⋊Q8—22no—
60C5×#102—22no—
61C5×#102—22no—
62C5×#97C22×C312yes—
63C5×#97C22×C311yescharacteristic
64#1657C3×C511yescharacteristic, direct product factor, Sylow
65C23×C3×C5—21no—
66C23×C3×C5—41no—
67C3×C5×Q8—41no—
68C3×C5×Q8—81no—
69C3×C5×D4—41no—
70C3×C5×D4—41no—
71C3×C5×D4—41no—
72C3×C5×D4—44no—
73C2×C3×C4×C5—22no—
74C2×C3×C4×C5—21no—
75C2×C3×C4×C5—22no—
76C2×C3×C4×C5—81no—
77C2×C3×C4×C5C2×D412yes—
78C2×C3×C4×C5C2×D411yescharacteristic
79C3×C5×C8—22no—
80C3×C5×C8—42no—
81C2×C3×SD16—21no—
82C2×C3×D8—21no—
83C3×C4⋊Q8—21no—
84C3×#127C22×C511yescharacteristic
85C3×#126—22no—
86C3×#125C22×C512yes—
87C3×#124C22×C511yescharacteristic
88C3×#123—21no—
89C3×C22⋊Q8—22no—
90C3×#102—22no—
91C3×#102—22no—
92C3×#97C22×C512yes—
93C3×#97C22×C511yescharacteristic
94C2×C5×Q8—21no—
95C2×C5×Q8C3×D411yescharacteristic
96C2×C5×D4—21no—
97C2×C5×D4—22no—
98C2×C5×D4C3×D411yescharacteristic
99C22×C4×C5—21no—
100C5×SD16—42no—
101C5×D8—42no—
102C2×C5×C8—21no—
103C2×C5×C8—22no—
104C5×C4⋊C4—22no—
105C5×C4⋊C4—22no—
106C5×C4⋊C4—22no—
107C5×C4⋊C4C3×D412yes—
108C5×C4⋊C4C3×D412yes—
109C5×C22⋊C4—22no—
110C5×C22⋊C4—21no—
111C5×C22⋊C4—22no—
112C42×C5—21no—
113C42×C5C23×C311yescharacteristic
114#563C2×C3×C511yescharacteristic, complemented
115#496C2×C3×C511yescharacteristic
116#490C2×C3×C512yescomplemented
117#330C2×C3×C512yescomplemented
118#329C2×C3×C511yescharacteristic, complemented
119C22×C3×C5—41no—
120C22×C3×C5—42no—
121C22×C3×C5—81no—
122C22×C3×C5—82no—
123C22×C3×C5#12011yescharacteristic
124C3×C4×C5—21no—
125C3×C4×C5—22no—
126C3×C4×C5—42no—
127C3×C4×C5—41no—
128C3×C4×C5—42no—
129C2×C3×Q8—21no—
130C2×C3×Q8C5×D411yescharacteristic
131C2×C3×D4—22no—
132C2×C3×D4—21no—
133C2×C3×D4C5×D411yescharacteristic
134C22×C3×C4—21no—
135C3×SD16—42no—
136C3×D8—42no—
137C2×C3×C8—21no—
138C2×C3×C8—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4C5×D412yes—
143C3×C4⋊C4C5×D412yes—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—22no—
147C3×C42—21no—
148C3×C42C23×C511yescharacteristic
149C23×C5—21no—
150C23×C5—41no—
151C5×Q8—41no—
152C5×Q8—81no—
153C5×D4—41no—
154C5×D4—41no—
155C5×D4—41no—
156C5×D4—44no—
157C2×C4×C5—22no—
158C2×C4×C5—21no—
159C2×C4×C5—22no—
160C2×C4×C5—81no—
161C2×C4×C5C2×C3×D412yes—
162C2×C4×C5C2×C3×D411yescharacteristic
163C5×C8—22no—
164C5×C8—42no—
165C2×SD16—21no—
166C2×D8—21no—
167C4⋊Q8—21no—
168#127C22×C3×C511yescharacteristic
169#126—22no—
170#125C22×C3×C512yes—
171#124C22×C3×C511yescharacteristic
172#123—21no—
173C22⋊Q8—22no—
174#102—22no—
175#102—22no—
176#97C22×C3×C512yes—
177#97C22×C3×C511yescharacteristic
178C2×C3×C5—81no—
179C2×C3×C5—161no—
180C2×C3×C5#45311yescharacteristic
181C2×C3×C5#45412yes—
182C23×C3—21no—
183C23×C3—41no—
184C3×Q8—41no—
185C3×Q8—81no—
186C3×D4—41no—
187C3×D4—41no—
188C3×D4—41no—
189C3×D4—44no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—22no—
193C2×C3×C4—81no—
194C2×C3×C4C2×C5×D412yes—
195C2×C3×C4C2×C5×D411yescharacteristic
196C3×C8—22no—
197C3×C8—42no—
198C22×C5—41no—
199C22×C5—42no—
200C22×C5—81no—
201C22×C5—82no—
202C22×C5C3×#12011yescharacteristic
203C4×C5—21no—
204C4×C5—22no—
205C4×C5—42no—
206C4×C5—41no—
207C4×C5—42no—
208C2×Q8—21no—
209C2×Q8C3×C5×D411yescharacteristic
210C2×D4—22no—
211C2×D4—21no—
212C2×D4C3×C5×D411yescharacteristic
213C22×C4—21no—
214SD16—42no—
215D8—42no—
216C2×C8—21no—
217C2×C8—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4C3×C5×D412yes—
222C4⋊C4C3×C5×D412yes—
223C22⋊C4—22no—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C42—21no—
227C42C23×C3×C511yescharacteristic
228C3×C5#165711yescharacteristic, direct product factor, Hall
229C22×C3—41no—
230C22×C3—42no—
231C22×C3—81no—
232C22×C3—82no—
233C22×C3C5×#12011yescharacteristic
234C3×C4—21no—
235C3×C4—22no—
236C3×C4—42no—
237C3×C4—41no—
238C3×C4—42no—
239C2×C5—81no—
240C2×C5—161no—
241C2×C5C3×#45311yescharacteristic
242C2×C5C3×#45412yes—
243C23—21no—
244C23—41no—
245Q8—41no—
246Q8—81no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—44no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—81no—
255C2×C4C2×C3×C5×D412yes—
256C2×C4C2×C3×C5×D411yescharacteristic
257C8—22no—
258C8—42no—
259C2×C3—81no—
260C2×C3—161no—
261C2×C3C5×#45311yescharacteristic
262C2×C3C5×#45412yes—
263C5C3×#165711yescharacteristic, direct product factor, Sylow
264C22—41no—
265C22—42no—
266C22—81no—
267C22—82no—
268C22C3×C5×#12011yescharacteristic
269C4—21no—
270C4—22no—
271C4—42no—
272C4—41no—
273C4—42no—
274C3C5×#165711yescharacteristic, direct product factor, Sylow
275C2—81no—
276C2—161no—
277C2C3×C5×#45311yescharacteristic
278C2C3×C5×#45412yes—
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC3×C5×#496C2
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
1. C3×C5×#563C2semidirect
3. C3×C5×#490C2semidirect
4. C3×C5×#330C2semidirect
5. C3×C5×#329C2semidirect
6. C5×#1657C3direct (N × H)
20. C3×#1657C5direct (N × H)
21. C5×#563C2×C3semidirect
23. C5×#490C2×C3semidirect
24. C5×#330C2×C3semidirect
25. C5×#329C2×C3semidirect
46. C3×#563C2×C5semidirect
48. C3×#490C2×C5semidirect
49. C3×#330C2×C5semidirect
50. C3×#329C2×C5semidirect
64. #1657C3×C5direct (N × H)
114. #563C2×C3×C5semidirect
116. #490C2×C3×C5semidirect
117. #330C2×C3×C5semidirect
118. #329C2×C3×C5semidirect
228. C3×C5#1657direct (N × H)
263. C5C3×#1657direct (N × H)
274. C3C5×#1657direct (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×#4962
C3×C5×#4902
C3×C5×#3302
C3×C5×#3292
C5×#16573
C3×#16575