Finite Group Databasebeta

← C3×#10602121 #14290938

Subgroups — C3×#10602121 #14290938

Order 1920.

412 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 90 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×#66006C211yesmaximal, characteristic, complemented
2C3×C5×#496C211yesmaximal, characteristic
3C3×#65360C212yesmaximal, complemented
4C3×#64760C211yesmaximal, characteristic, complemented
5C3×#64685C212yesmaximal, complemented
6#10602121C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×SD16—21no—
8C2×C3×C5×D8—21no—
9C3×C5×#127C2211yescharacteristic
10C3×C5×#124C2211yescharacteristic
11C3×#4090—21no—
12C3×#4081—21no—
13C3×#4025—22no—
14C3×#4024—22no—
15C3×#4016C2212yes—
16C3×C5×#97C2211yescharacteristic
17C3×#3964—22no—
18C3×#3964—22no—
19C3×#3935C2212yes—
20C3×#1657—51nomaximal, Hall
21#66006C2×C311yescharacteristic, complemented
22C5×#496C2×C311yescharacteristic
23#65360C2×C312yescomplemented
24#64760C2×C311yescharacteristic, complemented
25#64685C2×C312yescomplemented
26C2×C3×C5×Q8D411yescharacteristic
27C2×C3×C5×D4—22no—
28C2×C3×C5×D4—21no—
29C2×C3×C5×D4D411yescharacteristic
30C22×C3×Dic5—21no—
31C2×C3×Dic10—21no—
32C3×C5×SD16—42no—
33C3×C5×D8—42no—
34C2×C3×C5×C8—21no—
35C3×C5×C22⋊C4—22no—
36C3×C42×C5C2311yescharacteristic
37C3×C22⋊Dic5—22no—
38C3×C22⋊Dic5—21no—
39C3×C4⋊Dic5D412yes—
40C3×C4⋊Dic5D412yes—
41C3×#683—22no—
42C3×#683—22no—
43C3×#683—22no—
44C3×C4×Dic5—21no—
45C2×C3×#168—22no—
46C3×#563—51no—
47C3×#496D511yescharacteristic
48C3×#490—52no—
49C3×#330—52no—
50C3×#329—51no—
51C2×C5×SD16—21no—
52C2×C5×D8—21no—
53C5×#127C22×C311yescharacteristic
54C5×#124C22×C311yescharacteristic
55#4090—21no—
56#4081—21no—
57#4025—22no—
58#4024—22no—
59#4016C22×C312yes—
60C5×#97C22×C311yescharacteristic
61#3964—22no—
62#3964—22no—
63#3935C22×C312yes—
64#1657—51noSylow
65C23×C3×C5—21no—
66C23×C3×C5—41no—
67C3×C5×Q8—41no—
68C3×C5×D4—41no—
69C3×C5×D4—41no—
70C3×C5×D4—41no—
71C3×C5×D4—44no—
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5C2×D412yes—
74C2×C3×C4×C5C2×D411yescharacteristic
75C2×C3×Dic5—22no—
76C2×C3×Dic5—22no—
77C2×C3×Dic5—81no—
78C3×Dic10—81no—
79C3×C5×C8—22no—
80C3×#168—42no—
81C2×C3×SD16—21no—
82C2×C3×D8—21no—
83C3×C4⋊Q8—101no—
84C3×#127C2×D511yescharacteristic
85C3×#126—102no—
86C3×#125—52no—
87C3×#124C2×D511yescharacteristic
88C3×#123—101no—
89C3×C22⋊Q8—102no—
90C3×#102—102no—
91C3×#102—102no—
92C3×#97—52no—
93C3×#97C2×D511yescharacteristic
94C2×C5×Q8C3×D411yescharacteristic
95C2×C5×D4—21no—
96C2×C5×D4—22no—
97C2×C5×D4C3×D411yescharacteristic
98C22×Dic5—21no—
99C2×Dic10—21no—
100C5×SD16—42no—
101C5×D8—42no—
102C2×C5×C8—21no—
103C5×C22⋊C4—22no—
104C42×C5C23×C311yescharacteristic
105C22⋊Dic5—22no—
106C22⋊Dic5—21no—
107C4⋊Dic5C3×D412yes—
108C4⋊Dic5C3×D412yes—
109#683—22no—
110#683—22no—
111#683—22no—
112C4×Dic5—21no—
113C2×#168—22no—
114#563—51no—
115#496C3×D511yescharacteristic
116#490—52no—
117#330—52no—
118#329—51no—
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—41no—
127C3×Dic5—42no—
128C3×Dic5—42no—
129C2×C3×Q8—101no—
130C2×C3×Q8C22⋊D511yescharacteristic
131C2×C3×D4—22no—
132C2×C3×D4—21no—
133C2×C3×D4C22⋊D511yescharacteristic
134C22×C3×C4—101no—
135C3×SD16—42no—
136C3×D8—42no—
137C2×C3×C8—21no—
138C2×C3×C8—102no—
139C3×C4⋊C4—52no—
140C3×C4⋊C4—52no—
141C3×C4⋊C4—102no—
142C3×C4⋊C4—102no—
143C3×C4⋊C4—102no—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—102no—
146C3×C22⋊C4—101no—
147C3×C42—101no—
148C3×C42C22×D511yescharacteristic
149C23×C5—21no—
150C23×C5—41no—
151C5×Q8—41no—
152C5×D4—41no—
153C5×D4—41no—
154C5×D4—41no—
155C5×D4—44no—
156C2×C4×C5—21no—
157C2×C4×C5C2×C3×D412yes—
158C2×C4×C5C2×C3×D411yescharacteristic
159C2×Dic5—22no—
160C2×Dic5—22no—
161C2×Dic5—81no—
162Dic10—81no—
163C5×C8—22no—
164#168—42no—
165C2×SD16—21no—
166C2×D8—21no—
167C4⋊Q8—101no—
168#127C2×C3×D511yescharacteristic
169#126—102no—
170#125—52no—
171#124C2×C3×D511yescharacteristic
172#123—101no—
173C22⋊Q8—102no—
174#102—102no—
175#102—102no—
176#97—52no—
177#97C2×C3×D511yescharacteristic
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—401no—
186C3×D4—41no—
187C3×D4—41no—
188C3×D4—41no—
189C3×D4—44no—
190C2×C3×C4—21no—
191C2×C3×C4—102no—
192C2×C3×C4—102no—
193C2×C3×C4—401no—
194C2×C3×C4D4×D512yes—
195C2×C3×C4C2×C22⋊D511yescharacteristic
196C3×C8—22no—
197C3×C8—202no—
198C22×C5—41no—
199C22×C5—42no—
200C22×C5—81no—
201C22×C5—82no—
202C22×C5C3×#12011yescharacteristic
203C4×C5—21no—
204C4×C5—22no—
205C4×C5—41no—
206Dic5—42no—
207Dic5—42no—
208C2×Q8—101no—
209C2×Q8C3×C22⋊D511yescharacteristic
210C2×D4—22no—
211C2×D4—21no—
212C2×D4C3×C22⋊D511yescharacteristic
213C22×C4—101no—
214SD16—42no—
215D8—42no—
216C2×C8—21no—
217C2×C8—102no—
218C4⋊C4—52no—
219C4⋊C4—52no—
220C4⋊C4—102no—
221C4⋊C4—102no—
222C4⋊C4—102no—
223C22⋊C4—22no—
224C22⋊C4—102no—
225C22⋊C4—101no—
226C42—101no—
227C42C22×C3×D511yescharacteristic
228C3×C5#165711yescharacteristic, complemented, Hall
229C22×C3—41no—
230C22×C3—42no—
231C22×C3—81no—
232C22×C3—82no—
233C22×C3#408311yescharacteristic
234C3×C4—21no—
235C3×C4—22no—
236C3×C4—41no—
237C3×C4—202no—
238C3×C4—202no—
239C2×C5—81no—
240C2×C5—161no—
241C2×C5C3×#45311yescharacteristic
242C2×C5C3×#45412yes—
243C23—21no—
244C23—41no—
245Q8—41no—
246Q8—401no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—44no—
251C2×C4—21no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—401no—
255C2×C4C3×D4×D512yes—
256C2×C4C2×C3×C22⋊D511yescharacteristic
257C8—22no—
258C8—202no—
259C2×C3—81no—
260C2×C3—161no—
261C2×C3#6536611yescharacteristic
262C2×C3#6549012yes—
263C5C3×#165711yescharacteristic, complemented, Sylow
264C22—41no—
265C22—42no—
266C22—81no—
267C22—82no—
268C22C3×#408311yescharacteristic
269C4—21no—
270C4—22no—
271C4—41no—
272C4—202no—
273C4—202no—
274C3#1060212111yescharacteristic, direct product factor, Sylow
275C2—81no—
276C2—161no—
277C2C3×#6536611yescharacteristic
278C2C3×#6549012yes—
subgroupG/H
centerC22×C3#4083
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
Fitting, prime radicalC3×C5×#496C2
socleC22×C3×C5#120
hypercenterC3×#496D5
hypocenterC5C3×#1657
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#496

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#66006C2semidirect
3. C3×#65360C2semidirect
4. C3×#64760C2semidirect
5. C3×#64685C2semidirect
6. #10602121C3direct (N × H)
21. #66006C2×C3semidirect
23. #65360C2×C3semidirect
24. #64760C2×C3semidirect
25. #64685C2×C3semidirect
228. C3×C5#1657semidirect
263. C5C3×#1657semidirect
274. C3#10602121direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660062
C3×C5×#4962
C3×#653602
C3×#647602
C3×#646852
#106021213
C3×#16575