Finite Group Databasebeta

← C3×#10601788 #14317008

Subgroups — C3×#10601788 #14317008

Order 1920.

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

normal subgroups only

This group has 448 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×#66056C211yesmaximal, characteristic, complemented
2C3×C5×#456C211yesmaximal, characteristic
3C3×#65535C212yesmaximal
4C3×#64768C211yesmaximal, characteristic, complemented
5C3×#64698C212yesmaximal, complemented
6#10601788C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×2−1+4C2211yescharacteristic
8C3×C5×#137—21no—
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×#4097—22no—
11C3×#4090—21no—
12C3×#4090C2212yes—
13C3×#4034—22no—
14C3×#4033—22no—
15C3×C4×Dic10—21no—
16C3×C5×#104—21no—
17C3×C5×#101C2211yescharacteristic
18C3×#3969—22no—
19C3×#3968C2212yes—
20C3×#2205—51nomaximal, Hall
21#66056C2×C311yescharacteristic, complemented
22C5×#456C2×C311yescharacteristic
23#65535C2×C312yes—
24#64768C2×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×Q8D411yescharacteristic
31C2×C3×C5×Q8C2311yescharacteristic
32C2×C3×Dic10—21no—
33C2×C3×Dic10—21no—
34C2×C3×Dic10D412yes—
35C3×C5×Q16—41no—
36C3×C5×SD16—41no—
37C3×C5×M16—21no—
38C3×C5×C4⋊C4—22no—
39C3×C42×C5D411yescharacteristic
40C3×#689—42no—
41C3×#687—42no—
42C3×C4⋊Dic5—21no—
43C3×#683—22no—
44C3×#683—22no—
45C3×#683—22no—
46C3×C4×Dic5—21no—
47C3×C4×Dic5—22no—
48C3×C4×Dic5D412yes—
49C3×#681—22no—
50C3×Q82—51no—
51C3×#456—52no—
52C3×#456D511yescharacteristic
53C3×#356—51no—
54C3×#356—52no—
55C5×2−1+4C22×C311yescharacteristic
56C5×#137—21no—
57C5×C4⋊Q8C22×C311yescharacteristic
58#4097—22no—
59#4090—21no—
60#4090C22×C312yes—
61#4034—22no—
62#4033—22no—
63C4×Dic10—21no—
64C5×#104—21no—
65C5×#101C22×C311yescharacteristic
66#3969—22no—
67#3968C22×C312yes—
68#2205—51noSylow
69C3×C5×Q8—22no—
70C3×C5×Q8—21no—
71C3×C5×Q8—41no—
72C3×C5×Q8—41no—
73C3×C5×D4—22no—
74C3×C5×D4—21no—
75C3×C5×D4—41no—
76C2×C3×C4×C5—21no—
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—
84C3×Dic10—24no—
85C3×Dic10—42no—
86C3×Dic10—81no—
87C3×C5×C8—41no—
88C3×#168—42no—
89C3×2−1+4C2×D511yescharacteristic
90C3×#137—21no—
91C3×#137—102no—
92C3×C4⋊Q8—52no—
93C3×C4⋊Q8—102no—
94C3×C4⋊Q8—101no—
95C3×C4⋊Q8C2×D511yescharacteristic
96C3×C4×Q8—102no—
97C3×C4×Q8—101no—
98C3×#104—21no—
99C3×#104—102no—
100C3×#101—52no—
101C3×#101C2×D511yescharacteristic
102C5×#41—22no—
103C5×#41—21no—
104C5×#41—41no—
105C2×C5×Q8—41no—
106C2×C5×Q8C3×D411yescharacteristic
107C2×C5×Q8C23×C311yescharacteristic
108C2×Dic10—21no—
109C2×Dic10—21no—
110C2×Dic10C3×D412yes—
111C5×Q16—41no—
112C5×SD16—41no—
113C5×M16—21no—
114C5×C4⋊C4—22no—
115C42×C5C3×D411yescharacteristic
116#689—42no—
117#687—42no—
118C4⋊Dic5—21no—
119#683—22no—
120#683—22no—
121#683—22no—
122C4×Dic5—22no—
123C4×Dic5—21no—
124C4×Dic5C3×D412yes—
125#681—22no—
126Q82—51no—
127#456—52no—
128#456C3×D511yescharacteristic
129#356—51no—
130#356—52no—
131C22×C3×C5—41no—
132C22×C3×C5#12011yescharacteristic
133C3×C4×C5—22no—
134C3×C4×C5—22no—
135C3×C4×C5—21no—
136C3×C4×C5—41no—
137C3×C4×C5—41no—
138C3×Dic5—24no—
139C3×Dic5—42no—
140C3×#41—22no—
141C3×#41—21no—
142C3×#41—41no—
143C2×C3×Q8—41no—
144C2×C3×Q8—52no—
145C2×C3×Q8—101no—
146C2×C3×Q8—101no—
147C2×C3×Q8C22⋊D511yescharacteristic
148C2×C3×Q8C22×D511yescharacteristic
149C3×Q16—41no—
150C3×Q16—202no—
151C3×SD16—41no—
152C3×SD16—202no—
153C3×M16—21no—
154C3×M16—102no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—102no—
157C3×C4⋊C4—101no—
158C3×C4⋊C4—102no—
159C3×C4⋊C4—102no—
160C3×C42—52no—
161C3×C42—102no—
162C3×C42—101no—
163C3×C42C22⋊D511yescharacteristic
164C5×Q8—22no—
165C5×Q8—21no—
166C5×Q8—41no—
167C5×Q8—41no—
168C5×D4—22no—
169C5×D4—21no—
170C5×D4—41no—
171C2×C4×C5—21no—
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—
179Dic10—24no—
180Dic10—42no—
181Dic10—81no—
182C5×C8—41no—
183#168—42no—
1842−1+4C2×C3×D511yescharacteristic
185#137—21no—
186#137—102no—
187C4⋊Q8—52no—
188C4⋊Q8—102no—
189C4⋊Q8—101no—
190C4⋊Q8C2×C3×D511yescharacteristic
191C4×Q8—102no—
192C4×Q8—101no—
193#104—21no—
194#104—102no—
195#101—52no—
196#101C2×C3×D511yescharacteristic
197C2×C3×C5—21no—
198C2×C3×C5—81no—
199C2×C3×C5B(4,𝔽2)11yescharacteristic
200C3×Q8—21no—
201C3×Q8—22no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×Q8—104no—
205C3×Q8—202no—
206C3×Q8—401no—
207C3×D4—21no—
208C3×D4—22no—
209C3×D4—41no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—102no—
215C2×C3×C4—102no—
216C2×C3×C4D4×D512yes—
217C2×C3×C4C2×C22⋊D511yescharacteristic
218C3×C8—41no—
219C3×C8—202no—
220C22×C5—41no—
221C22×C5C3×#12011yescharacteristic
222C4×C5—22no—
223C4×C5—22no—
224C4×C5—21no—
225C4×C5—41no—
226C4×C5—41no—
227Dic5—24no—
228Dic5—42no—
229#41—22no—
230#41—21no—
231#41—41no—
232C2×Q8—41no—
233C2×Q8—52no—
234C2×Q8—101no—
235C2×Q8—101no—
236C2×Q8C3×C22⋊D511yescharacteristic
237C2×Q8C22×C3×D511yescharacteristic
238Q16—41no—
239Q16—202no—
240SD16—41no—
241SD16—202no—
242M16—21no—
243M16—102no—
244C4⋊C4—22no—
245C4⋊C4—102no—
246C4⋊C4—101no—
247C4⋊C4—102no—
248C4⋊C4—102no—
249C42—52no—
250C42—102no—
251C42—101no—
252C42C3×C22⋊D511yescharacteristic
253C3×C5#220511yescharacteristic, complemented, Hall
254C22×C3—41no—
255C22×C3#408311yescharacteristic
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—104no—
262C3×C4—202no—
263C2×C5—21no—
264C2×C5—81no—
265C2×C5C3×B(4,𝔽2)11yescharacteristic
266Q8—21no—
267Q8—22no—
268Q8—41no—
269Q8—41no—
270Q8—104no—
271Q8—202no—
272Q8—401no—
273D4—21no—
274D4—22no—
275D4—41no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—102no—
281C2×C4—102no—
282C2×C4C3×D4×D512yes—
283C2×C4C2×C3×C22⋊D511yescharacteristic
284C8—41no—
285C8—202no—
286C2×C3—21no—
287C2×C3—81no—
288C2×C3#6532111yescharacteristic
289C5C3×#220511yescharacteristic, complemented, Sylow
290C22—41no—
291C22C3×#408311yescharacteristic
292C4—22no—
293C4—21no—
294C4—22no—
295C4—41no—
296C4—41no—
297C4—104no—
298C4—202no—
299C3#1060178811yescharacteristic, direct product factor, Sylow
300C2—21no—
301C2—81no—
302C2C3×#6532111yescharacteristic
subgroupG/H
centerC2×C3#65321
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC2×Q8C22×C3×D5
FittingC3×C5×#456C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC3×C5×2−1+4C22
hypercenterC3×#456D5
hypocenterC5C3×#2205
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#456

Complements

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

NH ≅ G/Ntype
1. C3×#66056C2semidirect
4. C3×#64768C2semidirect
5. C3×#64698C2semidirect
6. #10601788C3direct (N × H)
21. #66056C2×C3semidirect
24. #64768C2×C3semidirect
25. #64698C2×C3semidirect
253. C3×C5#2205semidirect
289. C5C3×#2205semidirect
299. C3#10601788direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660562
C3×C5×#4562
C3×#655352
C3×#647682
C3×#646982
#106017883
C3×#22055