Finite Group Databasebeta

← C3×#10600506 #14288423

Subgroups — C3×#10600506 #14288423

Order 1920.

420 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 106 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×#65834C211yesmaximal, characteristic
2C3×C5×#398C211yesmaximal, characteristic
3C3×#65260C212yesmaximal
4C3×#64748C211yesmaximal, characteristic
5C3×#64672C212yesmaximal
6#10600506C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×C4⋊C4—21no—
8C2×C3×C5×C4⋊C4C2211yescharacteristic
9C2×C3×C5×C4⋊C4C2211yescharacteristic
10C3×#4072—21no—
11C3×#4070—21no—
12C3×#4070C2212yes—
13C3×#4036—22no—
14C3×#4034—22no—
15C3×C5×#95—21no—
16C3×C5×#95C2211yescharacteristic
17C3×C22⋊#168C2212yes—
18C3×#3942—22no—
19C3×#3940—22no—
20C3×#1611—51nomaximal, Hall
21#65834C2×C311yescharacteristic
22C5×#398C2×C311yescharacteristic
23#65260C2×C312yes—
24#64748C2×C311yescharacteristic
25#64672C2×C312yes—
26C22×C3×C4×C5—21no—
27C22×C3×C4×C5—21no—
28C22×C3×C4×C5D411yescharacteristic
29C22×C3×C4×C5D411yescharacteristic
30C22×C3×C4×C5C2311yescharacteristic
31C2×C3×Dic10—21no—
32C2×C3×Dic10D412yes—
33C3×C5×C4⋊C4—22no—
34C3×C5×C4⋊C4—22no—
35C3×C5×C4⋊C4—42no—
36C3×C22⋊Dic5—22no—
37C3×C4⋊Dic5—21no—
38C3×C4⋊Dic5—21no—
39C3×C4⋊Dic5—21no—
40C3×C4⋊Dic5D412yes—
41C3×#683—22no—
42C3×#683—22no—
43C3×C4×Dic5—22no—
44C2×C3×#168—22no—
45C3×#544—51no—
46C3×#483—52no—
47C3×#398D511yescharacteristic
48C3×#328—52no—
49C3×#324—51no—
50C2×C5×C4⋊C4—21no—
51C2×C5×C4⋊C4C22×C311yescharacteristic
52C2×C5×C4⋊C4C22×C311yescharacteristic
53#4072—21no—
54#4070—21no—
55#4070C22×C312yes—
56#4036—22no—
57#4034—22no—
58C5×#95—21no—
59C5×#95C22×C311yescharacteristic
60C22⋊#168C22×C312yes—
61#3942—22no—
62#3940—22no—
63#1611—51noSylow
64C23×C3×C5C2×D411yescharacteristic
65C2×C3×C4×C5—21no—
66C2×C3×C4×C5—21no—
67C2×C3×C4×C5—22no—
68C2×C3×C4×C5—21no—
69C2×C3×C4×C5—41no—
70C2×C3×C4×C5—41no—
71C2×C3×C4×C5—41no—
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5—41no—
76C2×C3×C4×C5SD1612yes—
77C2×C3×C4×C5C2×D412yes—
78C2×C3×Dic5—22no—
79C2×C3×Dic5—22no—
80C3×Dic10—42no—
81C3×Dic10—81no—
82C3×#168—42no—
83C3×C4⋊Q8—102no—
84C3×#125—102no—
85C3×C22⋊Q8—52no—
86C3×C22⋊Q8—101no—
87C3×#117—101no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2×D511yescharacteristic
90C2×C3×C4⋊C4C2×D511yescharacteristic
91C3×#106—102no—
92C3×#103—102no—
93C3×C22⋊C8—52no—
94C3×#95—21no—
95C3×#95C2×D511yescharacteristic
96C22×C4×C5—21no—
97C22×C4×C5—21no—
98C22×C4×C5C3×D411yescharacteristic
99C22×C4×C5C3×D411yescharacteristic
100C22×C4×C5C23×C311yescharacteristic
101C2×Dic10—21no—
102C2×Dic10C3×D412yes—
103C5×C4⋊C4—22no—
104C5×C4⋊C4—22no—
105C5×C4⋊C4—42no—
106C22⋊Dic5—22no—
107C4⋊Dic5—21no—
108C4⋊Dic5—21no—
109C4⋊Dic5—21no—
110C4⋊Dic5C3×D412yes—
111#683—22no—
112#683—22no—
113C4×Dic5—22no—
114C2×#168—22no—
115#544—51no—
116#483—52no—
117#398C3×D511yescharacteristic
118#328—52no—
119#324—51no—
120C22×C3×C5—21no—
121C22×C3×C5—21no—
122C22×C3×C5#12011yescharacteristic
123C22×C3×C5C2×SD1611yescharacteristic
124C22×C3×C5#13711yescharacteristic
125C3×C4×C5—22no—
126C3×C4×C5—22no—
127C3×C4×C5—42no—
128C3×C4×C5—41no—
129C3×C4×C5—81no—
130C3×Dic5—42no—
131C3×Dic5—42no—
132C2×C3×Q8—52no—
133C2×C3×Q8—101no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4C22⋊D511yescharacteristic
137C22×C3×C4C22⋊D511yescharacteristic
138C22×C3×C4C22×D511yescharacteristic
139C2×C3×C8—102no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—42no—
143C3×C4⋊C4—52no—
144C3×C4⋊C4—102no—
145C3×C4⋊C4—101no—
146C3×C4⋊C4—101no—
147C3×C4⋊C4—101no—
148C3×C4⋊C4—102no—
149C3×C22⋊C4—102no—
150C3×C42—102no—
151C23×C5C2×C3×D411yescharacteristic
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—22no—
155C2×C4×C5—21no—
156C2×C4×C5—41no—
157C2×C4×C5—41no—
158C2×C4×C5—41no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5C3×SD1612yes—
164C2×C4×C5C2×C3×D412yes—
165C2×Dic5—22no—
166C2×Dic5—22no—
167Dic10—42no—
168Dic10—81no—
169#168—42no—
170C4⋊Q8—102no—
171#125—102no—
172C22⋊Q8—52no—
173C22⋊Q8—101no—
174#117—101no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×C3×D511yescharacteristic
177C2×C4⋊C4C2×C3×D511yescharacteristic
178#106—102no—
179#103—102no—
180C22⋊C8—52no—
181#95—21no—
182#95C2×C3×D511yescharacteristic
183C2×C3×C5—21no—
184C2×C3×C5—21no—
185C2×C3×C5#44811yescharacteristic
186C2×C3×C5#45611yescharacteristic
187C2×C3×C5#45811yescharacteristic
188C23×C3C2×C22⋊D511yescharacteristic
189C3×Q8—202no—
190C3×Q8—401no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—102no—
203C2×C3×C4—102no—
204C2×C3×C4#68712yes—
205C2×C3×C4D4×D512yes—
206C3×C8—202no—
207C22×C5—21no—
208C22×C5—21no—
209C22×C5C3×#12011yescharacteristic
210C22×C5C2×C3×SD1611yescharacteristic
211C22×C5C3×#13711yescharacteristic
212C4×C5—22no—
213C4×C5—22no—
214C4×C5—42no—
215C4×C5—41no—
216C4×C5—81no—
217Dic5—42no—
218Dic5—42no—
219C2×Q8—52no—
220C2×Q8—101no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4C3×C22⋊D511yescharacteristic
224C22×C4C3×C22⋊D511yescharacteristic
225C22×C4C22×C3×D511yescharacteristic
226C2×C8—102no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—42no—
230C4⋊C4—52no—
231C4⋊C4—101no—
232C4⋊C4—101no—
233C4⋊C4—101no—
234C4⋊C4—102no—
235C4⋊C4—102no—
236C22⋊C4—102no—
237C42—102no—
238C3×C5#161111yescharacteristic, complemented, Hall
239C22×C3—21no—
240C22×C3—21no—
241C22×C3C2×#68711yescharacteristic
242C22×C3#408311yescharacteristic
243C22×C3#409711yescharacteristic
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—42no—
247C3×C4—41no—
248C3×C4—81no—
249C3×C4—202no—
250C3×C4—202no—
251C2×C5—21no—
252C2×C5—21no—
253C2×C5C3×#44811yescharacteristic
254C2×C5C3×#45611yescharacteristic
255C2×C5C3×#45811yescharacteristic
256C23C2×C3×C22⋊D511yescharacteristic
257Q8—202no—
258Q8—401no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4C3×#68712yes—
273C2×C4C3×D4×D512yes—
274C8—202no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3#6532911yescharacteristic
278C2×C3#6534211yescharacteristic
279C2×C3#6538411yescharacteristic
280C5C3×#161111yescharacteristic, complemented, Sylow
281C22—21no—
282C22—21no—
283C22C2×C3×#68711yescharacteristic
284C22C3×#408311yescharacteristic
285C22C3×#409711yescharacteristic
286C4—22no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—81no—
291C4—202no—
292C4—202no—
293C3#1060050611yescharacteristic, direct product factor, Sylow
294C2—21no—
295C2—21no—
296C2C3×#6532911yescharacteristic
297C2C3×#6534211yescharacteristic
298C2C3×#6538411yescharacteristic
subgroupG/H
centerC22×C3#4083
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#398C2
socleC22×C3×C5#120
prime radicalC23×C3×C5C2×D4
hypercenterC3×#398D5
hypocenterC5C3×#1611
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#398

Complements

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

NH ≅ G/Ntype
6. #10600506C3direct (N × H)
238. C3×C5#1611semidirect
280. C5C3×#1611semidirect
293. C3#10600506direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658342
C3×C5×#3982
C3×#652602
C3×#647482
C3×#646722
#106005063
C3×#16115