Finite Group Databasebeta

← C3×#10601782 #14316933

Subgroups — C3×#10601782 #14316933

Order 1920.

432 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 82 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×#454C211yesmaximal, characteristic
3C3×#65535C212yesmaximal, complemented
4C3×#64762C211yesmaximal, characteristic, complemented
5C3×#64698C212yesmaximal, complemented
6#10601782C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×2−1+4C2211yescharacteristic
8C3×C5×AGL(1,ℤ/8)—21no—
9C3×C5×#124C2211yescharacteristic
10C3×#4097—22no—
11C3×#4090C2212yes—
12C3×#4081—21no—
13C3×#4025—22no—
14C3×#4024—22no—
15C3×#4016—21no—
16C3×C5×#104—21no—
17C3×C5×#101C2211yescharacteristic
18C3×#3969—22no—
19C3×#3968C2212yes—
20C3×#2203—51nomaximal, Hall
21#66006C2×C311yescharacteristic, complemented
22C5×#454C2×C311yescharacteristic
23#65535C2×C312yescomplemented
24#64762C2×C311yescharacteristic, complemented
25#64698C2×C312yescomplemented
26C3×C5×#41—22no—
27C3×C5×#41—21no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8—41no—
30C2×C3×C5×Q8C2311yescharacteristic
31C2×C3×C5×D4D411yescharacteristic
32C22×C3×Dic5—21no—
33C2×C3×Dic10D412yes—
34C3×C5×SD16—41no—
35C3×C5×D8—41no—
36C3×C5×M16—21no—
37C3×C5×C22⋊C4—22no—
38C3×C42×C5D411yescharacteristic
39C3×C22⋊Dic5—22no—
40C3×C22⋊Dic5—21no—
41C3×#689—42no—
42C3×#687—42no—
43C3×C4⋊Dic5—21no—
44C3×C4⋊Dic5—21no—
45C3×#683—22no—
46C3×#683—22no—
47C3×#683—22no—
48C3×C4×Dic5D412yes—
49C3×#681—22no—
50C3×#563—51no—
51C3×#456—52no—
52C3×#454D511yescharacteristic
53C3×#356—52no—
54C3×#355—51no—
55C5×2−1+4C22×C311yescharacteristic
56C5×AGL(1,ℤ/8)—21no—
57C5×#124C22×C311yescharacteristic
58#4097—22no—
59#4090C22×C312yes—
60#4081—21no—
61#4025—22no—
62#4024—22no—
63#4016—21no—
64C5×#104—21no—
65C5×#101C22×C311yescharacteristic
66#3969—22no—
67#3968C22×C312yes—
68#2203—51noSylow
69C23×C3×C5—21no—
70C3×C5×Q8—22no—
71C3×C5×Q8—21no—
72C3×C5×Q8—41no—
73C3×C5×D4—22no—
74C3×C5×D4—21no—
75C3×C5×D4—41no—
76C3×C5×D4—41no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—42no—
79C2×C3×C4×C5—41no—
80C2×C3×C4×C5C2×D412yes—
81C2×C3×C4×C5C2×D411yescharacteristic
82C2×C3×Dic5—22no—
83C2×C3×Dic5—22no—
84C2×C3×Dic5—81no—
85C3×Dic10—42no—
86C3×C5×C8—41no—
87C3×#168—42no—
88C3×2−1+4C2×D511yescharacteristic
89C3×#137—102no—
90C3×AGL(1,ℤ/8)—21no—
91C3×C4⋊Q8—52no—
92C3×#126—102no—
93C3×#125—101no—
94C3×#124C2×D511yescharacteristic
95C3×#123—101no—
96C3×C22⋊Q8—102no—
97C3×#104—21no—
98C3×#104—102no—
99C3×#101—52no—
100C3×#101C2×D511yescharacteristic
101C5×#41—22no—
102C5×#41—21no—
103C5×#41—41no—
104C2×C5×Q8—41no—
105C2×C5×Q8C23×C311yescharacteristic
106C2×C5×D4C3×D411yescharacteristic
107C22×Dic5—21no—
108C2×Dic10C3×D412yes—
109C5×SD16—41no—
110C5×D8—41no—
111C5×M16—21no—
112C5×C22⋊C4—22no—
113C42×C5C3×D411yescharacteristic
114C22⋊Dic5—22no—
115C22⋊Dic5—21no—
116#689—42no—
117#687—42no—
118C4⋊Dic5—21no—
119C4⋊Dic5—21no—
120#683—22no—
121#683—22no—
122#683—22no—
123C4×Dic5C3×D412yes—
124#681—22no—
125#563—51no—
126#456—52no—
127#454C3×D511yescharacteristic
128#356—52no—
129#355—51no—
130C22×C3×C5—42no—
131C22×C3×C5—41no—
132C22×C3×C5—41no—
133C22×C3×C5#12011yescharacteristic
134C3×C4×C5—22no—
135C3×C4×C5—21no—
136C3×C4×C5—41no—
137C3×C4×C5—41no—
138C3×Dic5—42no—
139C3×Dic5—42no—
140C3×#41—22no—
141C3×#41—21no—
142C3×#41—41no—
143C2×C3×Q8—41no—
144C2×C3×Q8—52no—
145C2×C3×Q8C22×D511yescharacteristic
146C2×C3×D4C22⋊D511yescharacteristic
147C22×C3×C4—101no—
148C3×Q16—202no—
149C3×SD16—41no—
150C3×SD16—202no—
151C3×D8—41no—
152C3×M16—21no—
153C3×M16—102no—
154C3×C4⋊C4—102no—
155C3×C4⋊C4—101no—
156C3×C4⋊C4—101no—
157C3×C4⋊C4—102no—
158C3×C4⋊C4—102no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—102no—
161C3×C22⋊C4—101no—
162C3×C42—52no—
163C3×C42C22⋊D511yescharacteristic
164C23×C5—21no—
165C5×Q8—22no—
166C5×Q8—21no—
167C5×Q8—41no—
168C5×D4—22no—
169C5×D4—21no—
170C5×D4—41no—
171C5×D4—41no—
172C2×C4×C5—21no—
173C2×C4×C5—42no—
174C2×C4×C5—41no—
175C2×C4×C5C2×C3×D412yes—
176C2×C4×C5C2×C3×D411yescharacteristic
177C2×Dic5—22no—
178C2×Dic5—22no—
179C2×Dic5—81no—
180Dic10—42no—
181C5×C8—41no—
182#168—42no—
1832−1+4C2×C3×D511yescharacteristic
184#137—102no—
185AGL(1,ℤ/8)—21no—
186C4⋊Q8—52no—
187#126—102no—
188#125—101no—
189#124C2×C3×D511yescharacteristic
190#123—101no—
191C22⋊Q8—102no—
192#104—21no—
193#104—102no—
194#101—52no—
195#101C2×C3×D511yescharacteristic
196C2×C3×C5—21no—
197C2×C3×C5—81no—
198C2×C3×C5—81no—
199C2×C3×C5B(4,𝔽2)11yescharacteristic
200C23×C3—21no—
201C3×Q8—21no—
202C3×Q8—22no—
203C3×Q8—41no—
204C3×Q8—202no—
205C3×D4—21no—
206C3×D4—22no—
207C3×D4—41no—
208C3×D4—41no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—102no—
213C2×C3×C4—102no—
214C2×C3×C4—401no—
215C2×C3×C4D4×D512yes—
216C2×C3×C4C2×C22⋊D511yescharacteristic
217C3×C8—41no—
218C3×C8—202no—
219C22×C5—42no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5C3×#12011yescharacteristic
223C4×C5—22no—
224C4×C5—21no—
225C4×C5—41no—
226C4×C5—41no—
227Dic5—42no—
228Dic5—42no—
229#41—22no—
230#41—21no—
231#41—41no—
232C2×Q8—41no—
233C2×Q8—52no—
234C2×Q8C22×C3×D511yescharacteristic
235C2×D4C3×C22⋊D511yescharacteristic
236C22×C4—101no—
237Q16—202no—
238SD16—41no—
239SD16—202no—
240D8—41no—
241M16—21no—
242M16—102no—
243C4⋊C4—101no—
244C4⋊C4—102no—
245C4⋊C4—101no—
246C4⋊C4—102no—
247C4⋊C4—102no—
248C22⋊C4—22no—
249C22⋊C4—102no—
250C22⋊C4—101no—
251C42—52no—
252C42C3×C22⋊D511yescharacteristic
253C3×C5#220311yescharacteristic, complemented, Hall
254C22×C3—42no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3#408311yescharacteristic
258C3×C4—21no—
259C3×C4—22no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—202no—
263C3×C4—202no—
264C2×C5—21no—
265C2×C5—81no—
266C2×C5—81no—
267C2×C5C3×B(4,𝔽2)11yescharacteristic
268C23—21no—
269Q8—21no—
270Q8—22no—
271Q8—41no—
272Q8—202no—
273D4—21no—
274D4—22no—
275D4—41no—
276D4—41no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—102no—
281C2×C4—102no—
282C2×C4—401no—
283C2×C4C3×D4×D512yes—
284C2×C4C2×C3×C22⋊D511yescharacteristic
285C8—41no—
286C8—202no—
287C2×C3—21no—
288C2×C3—81no—
289C2×C3—81no—
290C2×C3#6532111yescharacteristic
291C5C3×#220311yescharacteristic, complemented, Sylow
292C22—42no—
293C22—41no—
294C22—41no—
295C22C3×#408311yescharacteristic
296C4—21no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—202no—
301C4—202no—
302C3#1060178211yescharacteristic, direct product factor, Sylow
303C2—21no—
304C2—81no—
305C2—81no—
306C2C3×#6532111yescharacteristic
subgroupG/H
centerC2×C3#65321
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC2×Q8C22×C3×D5
Fitting, prime radicalC3×C5×#454C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC3×#454D5
hypocenterC5C3×#2203
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#454

Complements

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

NH ≅ G/Ntype
1. C3×#66006C2semidirect
3. C3×#65535C2semidirect
4. C3×#64762C2semidirect
5. C3×#64698C2semidirect
6. #10601782C3direct (N × H)
21. #66006C2×C3semidirect
23. #65535C2×C3semidirect
24. #64762C2×C3semidirect
25. #64698C2×C3semidirect
253. C3×C5#2203semidirect
291. C5C3×#2203semidirect
302. C3#10601782direct (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×#4542
C3×#655352
C3×#647622
C3×#646982
#106017823
C3×#22035