Finite Group Databasebeta

← C3×#10602742 #14291002

Subgroups — C3×#10602742 #14291002

Order 1920.

456 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 98 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×#65349C211yesmaximal, characteristic, complemented
3C3×#65344C212yesmaximal, complemented
4C3×#64760C211yesmaximal, characteristic, complemented
5C3×#64760C212yesmaximal, complemented
6#10602742C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×#124—21no—
8C3×C5×#124—22no—
9C3×C5×#124C2212yes—
10C3×C5×#124C2211yescharacteristic
11C3×C5×#120—21no—
12C3×C5×#120—22no—
13C2×C3×#689—21no—
14C2×C3×#687—21no—
15C3×#4015C2211yescharacteristic, complemented
16C3×#3967—22no—
17C3×#3964—22no—
18C3×#3935C2212yescomplemented
19C3×#3935C2211yescharacteristic, complemented
20C3×#1659—51nomaximal, Hall
21C5×H(3,𝔽4)C2×C311yescharacteristic
22#65349C2×C311yescharacteristic, complemented
23#65344C2×C312yescomplemented
24#64760C2×C312yescomplemented
25#64760C2×C311yescharacteristic, complemented
26C24×C3×C5—21no—
27C2×C3×C5×Q8D411yescharacteristic
28C2×C3×C5×Q8D412yes—
29C2×C3×C5×D4—21no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4D411yescharacteristic
32C2×C3×C5×D4D412yes—
33C2×C3×Dic10—21no—
34C3×C5×C22⋊C4—22no—
35C3×C5×C22⋊C4—21no—
36C3×C5×C22⋊C4—22no—
37C3×C5×C22⋊C4—22no—
38C3×C5×C22⋊C4—22no—
39C3×C42×C5—21no—
40C3×C42×C5C2311yescharacteristic
41C3×#689—42no—
42C3×#687—42no—
43C3×C4⋊Dic5—22no—
44C2×C3×#168—21no—
45C2×C3×#168—22no—
46C3×H(3,𝔽4)D511yescharacteristic
47C3×#497—51no—
48C3×#489—52no—
49C3×#329—51no—
50C3×#329—52no—
51C5×#124—22no—
52C5×#124—21no—
53C5×#124C22×C312yes—
54C5×#124C22×C311yescharacteristic
55C5×#120—22no—
56C5×#120—21no—
57C2×#689—21no—
58C2×#687—21no—
59#4015C22×C311yescharacteristic, complemented
60#3967—22no—
61#3964—22no—
62#3935C22×C312yescomplemented
63#3935C22×C311yescharacteristic, complemented
64#1659—51noSylow
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×D4—41no—
72C3×C5×D4—42no—
73C3×C5×D4—81no—
74C3×C5×D4—82no—
75C2×C3×C4×C5—21no—
76C2×C3×C4×C5—22no—
77C2×C3×C4×C5C2×D412yes—
78C2×C3×C4×C5C2×D411yescharacteristic
79C2×C3×Dic5—41no—
80C3×Dic10—41no—
81C3×Dic10—41no—
82C3×#168—22no—
83C3×#168—42no—
84C2×C3×Q16—101no—
85C2×C3×SD16—101no—
86C3×C4⋊Q8—51no—
87C3×#124—22no—
88C3×#124—21no—
89C3×#124C2×D512yes—
90C3×#124C2×D511yescharacteristic
91C3×#120—22no—
92C3×#120—21no—
93C3×#103—102no—
94C3×#102—102no—
95C3×#97—52no—
96C3×#97—51no—
97C24×C5—21no—
98C2×C5×Q8C3×D412yes—
99C2×C5×Q8C3×D411yescharacteristic
100C2×C5×D4—21no—
101C2×C5×D4—22no—
102C2×C5×D4C3×D412yes—
103C2×C5×D4C3×D411yescharacteristic
104C2×Dic10—21no—
105C5×C22⋊C4—22no—
106C5×C22⋊C4—21no—
107C5×C22⋊C4—22no—
108C5×C22⋊C4—22no—
109C5×C22⋊C4—22no—
110C42×C5—21no—
111C42×C5C23×C311yescharacteristic
112#689—42no—
113#687—42no—
114C4⋊Dic5—22no—
115C2×#168—21no—
116C2×#168—22no—
117H(3,𝔽4)C3×D511yescharacteristic
118#497—51no—
119#489—52no—
120#329—51no—
121#329—52no—
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×Dic5—81no—
134C24×C3—21no—
135C2×C3×Q8—101no—
136C2×C3×Q8C22⋊D512yes—
137C2×C3×Q8C22⋊D511yescharacteristic
138C2×C3×D4—21no—
139C2×C3×D4—22no—
140C2×C3×D4C22⋊D512yes—
141C2×C3×D4C22⋊D511yescharacteristic
142C3×Q16—202no—
143C3×SD16—202no—
144C2×C3×C8—101no—
145C2×C3×C8—102no—
146C3×C4⋊C4—102no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C42—21no—
153C3×C42C22×D511yescharacteristic
154C23×C5—21no—
155C23×C5—22no—
156C23×C5—81no—
157C23×C5—82no—
158C5×Q8—41no—
159C5×Q8—42no—
160C5×D4—41no—
161C5×D4—42no—
162C5×D4—81no—
163C5×D4—82no—
164C2×C4×C5—21no—
165C2×C4×C5—22no—
166C2×C4×C5C2×C3×D412yes—
167C2×C4×C5C2×C3×D411yescharacteristic
168C2×Dic5—41no—
169Dic10—41no—
170Dic10—41no—
171#168—22no—
172#168—42no—
173C2×Q16—101no—
174C2×SD16—101no—
175C4⋊Q8—51no—
176#124—22no—
177#124—21no—
178#124C2×C3×D512yes—
179#124C2×C3×D511yescharacteristic
180#120—22no—
181#120—21no—
182#103—102no—
183#102—102no—
184#97—52no—
185#97—51no—
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—201no—
197C3×Q8—201no—
198C3×D4—41no—
199C3×D4—42no—
200C3×D4—81no—
201C3×D4—82no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—201no—
205C2×C3×C4C2×C22⋊D512yes—
206C2×C3×C4C2×C22⋊D511yescharacteristic
207C3×C8—102no—
208C3×C8—202no—
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—
220Dic5—81no—
221C24—21no—
222C2×Q8—101no—
223C2×Q8C3×C22⋊D512yes—
224C2×Q8C3×C22⋊D511yescharacteristic
225C2×D4—21no—
226C2×D4—22no—
227C2×D4C3×C22⋊D512yes—
228C2×D4C3×C22⋊D511yescharacteristic
229Q16—202no—
230SD16—202no—
231C2×C8—101no—
232C2×C8—102no—
233C4⋊C4—102no—
234C22⋊C4—22no—
235C22⋊C4—21no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C42—21no—
240C42C22×C3×D511yescharacteristic
241C3×C5#165911yescharacteristic, complemented, Hall
242C22×C3—41no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3—42no—
246C22×C3—42no—
247C22×C3—84no—
248C22×C3#409911yescharacteristic
249C3×C4—21no—
250C3×C4—22no—
251C3×C4—41no—
252C3×C4—42no—
253C3×C4—401no—
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—201no—
265Q8—201no—
266D4—41no—
267D4—42no—
268D4—81no—
269D4—82no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—201no—
273C2×C4C2×C3×C22⋊D512yes—
274C2×C4C2×C3×C22⋊D511yescharacteristic
275C8—102no—
276C8—202no—
277C2×C3—81no—
278C2×C3—82no—
279C2×C3#6553011yescharacteristic
280C2×C3#6553112yes—
281C5C3×#165911yescharacteristic, complemented, Sylow
282C22—41no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—84no—
288C22C3×#409911yescharacteristic
289C4—21no—
290C4—22no—
291C4—41no—
292C4—42no—
293C4—401no—
294C3#1060274211yescharacteristic, direct product factor, Sylow
295C2—81no—
296C2—82no—
297C2C3×#6553011yescharacteristic
298C2C3×#6553112yes—
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
Fitting, prime radicalC3×C5×H(3,𝔽4)C2
socleC22×C3×C5#120
hypercenterC3×H(3,𝔽4)D5
hypocenterC5C3×#1659
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
2. C3×#65349C2semidirect
3. C3×#65344C2semidirect
4. C3×#64760C2semidirect
5. C3×#64760C2semidirect
6. #10602742C3direct (N × H)
15. C3×#4015C22semidirect
18. C3×#3935C22semidirect
19. C3×#3935C22semidirect
22. #65349C2×C3semidirect
23. #65344C2×C3semidirect
24. #64760C2×C3semidirect
25. #64760C2×C3semidirect
59. #4015C22×C3semidirect
62. #3935C22×C3semidirect
63. #3935C22×C3semidirect
241. C3×C5#1659semidirect
281. C5C3×#1659semidirect
294. C3#10602742direct (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×#653492
C3×#653442
C3×#647602
#106027423
C3×#16595