Finite Group Databasebeta

← C3×#10602122 #14290996

Subgroups — C3×#10602122 #14290996

Order 1920.

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

normal subgroups only

This group has 456 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×#66017C211yesmaximal, characteristic
2C3×C5×#497C211yesmaximal, characteristic, complemented
3C3×#65378C212yesmaximal, complemented
4C3×#64760C211yesmaximal, characteristic, complemented
5C3×#64684C212yesmaximal, complemented
6#10602122C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×Q16—21no—
8C2×C3×C5×SD16—21no—
9C3×C5×C4⋊Q8C2211yescharacteristic, complemented
10C3×C5×#124C2211yescharacteristic
11C3×#4093—21no—
12C3×#4083—21no—
13C3×#4031—22no—
14C3×#4028—22no—
15C3×#4021C2212yes—
16C3×C5×#97C2211yescharacteristic, complemented
17C3×#3942—22no—
18C3×#3941—22no—
19C3×#3935C2212yescomplemented
20C3×#1659—51nomaximal, Hall
21#66017C2×C311yescharacteristic
22C5×#497C2×C311yescharacteristic, complemented
23#65378C2×C312yescomplemented
24#64760C2×C311yescharacteristic, complemented
25#64684C2×C312yescomplemented
26C23×C3×D5—21no—
27C2×C3×C5×Q8—21no—
28C2×C3×C5×Q8D411yescharacteristic
29C2×C3×C5×D4D411yescharacteristic
30C2×C3×C22⋊D5—22no—
31C2×C3×D20—21no—
32C2×C3×D20D412yes—
33C2×C3×Dic10D412yes—
34C3×C5×Q16—42no—
35C3×C5×SD16—42no—
36C2×C3×C5×C8—21no—
37C3×C5×C4⋊C4—22no—
38C3×C5×C22⋊C4—22no—
39C3×C42×C5C2311yescharacteristic
40C3×C22⋊Dic5—21no—
41C3×#685—22no—
42C3×#685—22no—
43C3×#685—22no—
44C3×C4×Dic5—21no—
45C2×C3×#168—22no—
46C3×H(3,𝔽4)—51no—
47C3×#497D511yescharacteristic, complemented
48C3×#489—52no—
49C3×#329—51no—
50C3×#329—52no—
51C2×C5×Q16—21no—
52C2×C5×SD16—21no—
53C5×C4⋊Q8C22×C311yescharacteristic, complemented
54C5×#124C22×C311yescharacteristic
55#4093—21no—
56#4083—21no—
57#4031—22no—
58#4028—22no—
59#4021C22×C312yes—
60C5×#97C22×C311yescharacteristic, complemented
61#3942—22no—
62#3941—22no—
63#3935C22×C312yescomplemented
64#1659—51noSylow
65C23×C3×C5—21no—
66C22×C3×D5—22no—
67C22×C3×D5—81no—
68C22×C3×D5—82no—
69C3×C5×Q8—41no—
70C3×C5×Q8—41no—
71C3×C5×Q8—41no—
72C3×C5×D4—41no—
73C2×C3×C4×C5—21no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5C2×D412yes—
76C2×C3×C4×C5C2×D411yescharacteristic
77C3×C22⋊D5—82no—
78C2×C3×Dic5—22no—
79C3×D20—42no—
80C3×D20—81no—
81C3×Dic10—42no—
82C3×C5×C8—22no—
83C3×#168—42no—
84C2×C3×Q16—21no—
85C2×C3×SD16—21no—
86C3×C4⋊Q8C2×D511yescharacteristic, complemented
87C3×#124—52no—
88C3×#124—102no—
89C3×#124—101no—
90C3×#124C2×D511yescharacteristic
91C3×#120—102no—
92C3×#120—101no—
93C3×#103—102no—
94C3×#102—102no—
95C3×#97—52no—
96C3×#97C2×D511yescharacteristic, complemented
97C23×D5—21no—
98C2×C5×Q8—21no—
99C2×C5×Q8C3×D411yescharacteristic
100C2×C5×D4C3×D411yescharacteristic
101C2×C22⋊D5—22no—
102C2×D20—21no—
103C2×D20C3×D412yes—
104C2×Dic10C3×D412yes—
105C5×Q16—42no—
106C5×SD16—42no—
107C2×C5×C8—21no—
108C5×C4⋊C4—22no—
109C5×C22⋊C4—22no—
110C42×C5C23×C311yescharacteristic
111C22⋊Dic5—21no—
112#685—22no—
113#685—22no—
114#685—22no—
115C4×Dic5—21no—
116C2×#168—22no—
117H(3,𝔽4)—51no—
118#497C3×D511yescharacteristic, complemented
119#489—52no—
120#329—51no—
121#329—52no—
122C22×C3×C5—41no—
123C22×C3×C5—42no—
124C22×C3×C5#12011yescharacteristic
125C2×C3×D5—42no—
126C2×C3×D5—42no—
127C2×C3×D5—42no—
128C2×C3×D5—84no—
129C3×C4×C5—21no—
130C3×C4×C5—22no—
131C3×C4×C5—41no—
132C3×C4×C5—81no—
133C3×Dic5—42no—
134C24×C3—101no—
135C2×C3×Q8—21no—
136C2×C3×Q8—52no—
137C2×C3×Q8C22⋊D511yescharacteristic
138C2×C3×D4—52no—
139C2×C3×D4—101no—
140C2×C3×D4—102no—
141C2×C3×D4C22⋊D511yescharacteristic
142C3×Q16—42no—
143C3×SD16—42no—
144C2×C3×C8—21no—
145C2×C3×C8—102no—
146C3×C4⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—102no—
149C3×C22⋊C4—101no—
150C3×C22⋊C4—102no—
151C3×C22⋊C4—102no—
152C3×C42—101no—
153C3×C42C22×D511yescharacteristic
154C23×C5—21no—
155C22×D5—22no—
156C22×D5—81no—
157C22×D5—82no—
158C5×Q8—41no—
159C5×Q8—41no—
160C5×Q8—41no—
161C5×D4—41no—
162C2×C4×C5—21no—
163C2×C4×C5—41no—
164C2×C4×C5C2×C3×D412yes—
165C2×C4×C5C2×C3×D411yescharacteristic
166C22⋊D5—82no—
167C2×Dic5—22no—
168D20—42no—
169D20—81no—
170Dic10—42no—
171C5×C8—22no—
172#168—42no—
173C2×Q16—21no—
174C2×SD16—21no—
175C4⋊Q8C2×C3×D511yescharacteristic, complemented
176#124—52no—
177#124—102no—
178#124—101no—
179#124C2×C3×D511yescharacteristic
180#120—102no—
181#120—101no—
182#103—102no—
183#102—102no—
184#97—52no—
185#97C2×C3×D511yescharacteristic, complemented
186C2×C3×C5—81no—
187C2×C3×C5#45311yescharacteristic
188C2×C3×C5#45512yes—
189C3×D5—82no—
190C23×C3—21no—
191C23×C3—102no—
192C23×C3—401no—
193C23×C3—402no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—202no—
198C3×D4—41no—
199C3×D4—202no—
200C3×D4—401no—
201C3×D4—402no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4—102no—
205C2×C3×C4D4×D512yes—
206C2×C3×C4C2×C22⋊D511yescharacteristic
207C3×C8—22no—
208C3×C8—202no—
209C22×C5—41no—
210C22×C5—42no—
211C22×C5C3×#12011yescharacteristic
212C2×D5—42no—
213C2×D5—42no—
214C2×D5—42no—
215C2×D5—84no—
216C4×C5—21no—
217C4×C5—22no—
218C4×C5—41no—
219C4×C5—81no—
220Dic5—42no—
221C24—101no—
222C2×Q8—21no—
223C2×Q8—52no—
224C2×Q8C3×C22⋊D511yescharacteristic
225C2×D4—52no—
226C2×D4—101no—
227C2×D4—102no—
228C2×D4C3×C22⋊D511yescharacteristic
229Q16—42no—
230SD16—42no—
231C2×C8—21no—
232C2×C8—102no—
233C4⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—102no—
236C22⋊C4—101no—
237C22⋊C4—102no—
238C22⋊C4—102no—
239C42—101no—
240C42C22×C3×D511yescharacteristic
241C3×C5#165911yescharacteristic, complemented, Hall
242C22×C3—41no—
243C22×C3—42no—
244C22×C3—202no—
245C22×C3—202no—
246C22×C3—202no—
247C22×C3—404no—
248C22×C3#408311yescharacteristic
249C3×C4—21no—
250C3×C4—22no—
251C3×C4—41no—
252C3×C4—81no—
253C3×C4—202no—
254C2×C5—81no—
255C2×C5C3×#45311yescharacteristic
256C2×C5C3×#45512yes—
257D5—82no—
258C23—21no—
259C23—102no—
260C23—401no—
261C23—402no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—202no—
266D4—41no—
267D4—202no—
268D4—401no—
269D4—402no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—102no—
273C2×C4C3×D4×D512yes—
274C2×C4C2×C3×C22⋊D511yescharacteristic
275C8—22no—
276C8—202no—
277C2×C3—81no—
278C2×C3—402no—
279C2×C3#6536611yescharacteristic
280C2×C3#6549112yes—
281C5C3×#165911yescharacteristic, complemented, Sylow
282C22—41no—
283C22—42no—
284C22—202no—
285C22—202no—
286C22—202no—
287C22—404no—
288C22C3×#408311yescharacteristic
289C4—21no—
290C4—22no—
291C4—41no—
292C4—81no—
293C4—202no—
294C3#1060212211yescharacteristic, direct product factor, Sylow
295C2—81no—
296C2—402no—
297C2C3×#6536611yescharacteristic
298C2C3×#6549112yes—
subgroupG/H
centerC22×C3#4083
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#497C2
socleC22×C3×C5#120
prime radicalC3×#66017C2
hypercenterC3×#497D5
hypocenterC5C3×#1659
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#497

Complements

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

NH ≅ G/Ntype
2. C3×C5×#497C2semidirect
3. C3×#65378C2semidirect
4. C3×#64760C2semidirect
5. C3×#64684C2semidirect
6. #10602122C3direct (N × H)
9. C3×C5×C4⋊Q8C22semidirect
16. C3×C5×#97C22semidirect
19. C3×#3935C22semidirect
22. C5×#497C2×C3semidirect
23. #65378C2×C3semidirect
24. #64760C2×C3semidirect
25. #64684C2×C3semidirect
47. C3×#497D5semidirect
53. C5×C4⋊Q8C22×C3semidirect
60. C5×#97C22×C3semidirect
63. #3935C22×C3semidirect
86. C3×C4⋊Q8C2×D5semidirect
96. C3×#97C2×D5semidirect
118. #497C3×D5semidirect
175. C4⋊Q8C2×C3×D5semidirect
185. #97C2×C3×D5semidirect
241. C3×C5#1659semidirect
281. C5C3×#1659semidirect
294. C3#10602122direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660172
C3×C5×#4972
C3×#653782
C3×#647602
C3×#646842
#106021223
C3×#16595