Finite Group Databasebeta

← C3×#10604551 #14313758

Subgroups — C3×#10604551 #14313758

Order 1920.

460 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 104 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×#66009C211yesmaximal, characteristic
2C3×#65777C212yesmaximal
3C3×#64929C212yesmaximal, complemented
4C3×#64855C212yesmaximal, complemented
5#10604551C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×Q8×D5C411yescharacteristic
7C2×C3×#711C2211yescharacteristic
8C2×C3×#704—22no—
9C3×C5×#124C411yescharacteristic
10C3×#4090C2211yescharacteristic
11C3×#4082C411yescharacteristic
12C3×#4029—22no—
13C3×#4024—22no—
14C3×#4018C411yescharacteristic
15C3×#4015C2211yescharacteristic
16C3×#4013—22no—
17C3×#4005C2214yes—
18C3×#2135—51nomaximal, Hall
19#66009C2×C311yescharacteristic
20#65777C2×C312yes—
21#64929C2×C312yescomplemented
22#64855C2×C312yescomplemented
23C2×C3×C5×Q8C2×C411yescharacteristic
24C2×C3×C5×D4C2×C411yescharacteristic
25C2×C3×C22⋊D5D412yes—
26C22×C3×Dic5D412yes—
27C3×Q8×D5—42no—
28C3×#711—42no—
29C2×C3×C4×D5—21no—
30C2×C3×C4×D5C2×C411yescharacteristic
31C2×C3×Dic10—21no—
32C2×C3×Dic10C2×C411yescharacteristic
33C2×C3×Dic10C2311yescharacteristic
34C3×#704—24no—
35C3×#704—42no—
36C2×C3×#170—42no—
37C3×C5×C22⋊C4—22no—
38C3×C42×C5C2×C411yescharacteristic
39C3×C22⋊Dic5—22no—
40C3×#685—21no—
41C3×C4⋊Dic5—22no—
42C3×#683—21no—
43C3×#683—22no—
44C3×C4×Dic5C2×C411yescharacteristic
45C3×#536—51no—
46C3×#413—52no—
47C3×#356—52no—
48C3×#356—52no—
49C2×Q8×D5C3×C411yescharacteristic
50C2×#711C22×C311yescharacteristic
51C2×#704—22no—
52C5×#124C3×C411yescharacteristic
53#4090C22×C311yescharacteristic
54#4082C3×C411yescharacteristic
55#4029—22no—
56#4024—22no—
57#4018C3×C411yescharacteristic
58#4015C22×C311yescharacteristic
59#4013—22no—
60#4005C22×C314yes—
61#2135—51noSylow
62C23×C3×C5C22⋊C412yes—
63C22×C3×D5C22⋊C411yescharacteristic
64C3×C5×Q8—41no—
65C3×C5×D4—41no—
66C2×C3×C4×C5—21no—
67C2×C3×C4×C5—21no—
68C2×C3×C4×C5C22×C411yescharacteristic
69C3×C22⋊D5—42no—
70C2×C3×Dic5—22no—
71C2×C3×Dic5—22no—
72C2×C3×Dic5—21no—
73C2×C3×Dic5—21no—
74C2×C3×Dic5C22⋊C411yescharacteristic
75C2×C3×Dic5C2×D412yes—
76C3×C4×D5—22no—
77C3×C4×D5—42no—
78C3×Dic10—41no—
79C3×Dic10—42no—
80C3×Dic10—41no—
81C3×#170—44no—
82C2×C3×#41—51no—
83C22×C3×Q8—51no—
84C2×C3×M16—102no—
85C3×C4⋊Q8—51no—
86C3×C4⋊Q8—51no—
87C3×#124—51no—
88C3×#124AGL(1,𝔽5)11yescharacteristic
89C3×#123—102no—
90C3×C22⋊Q8—102no—
91C3×#117—51no—
92C3×#101—54no—
93C3×#100—102no—
94C2×C5×Q8C2×C3×C411yescharacteristic
95C2×C5×D4C2×C3×C411yescharacteristic
96C2×C22⋊D5C3×D412yes—
97C22×Dic5C3×D412yes—
98Q8×D5—42no—
99#711—42no—
100C2×C4×D5—21no—
101C2×C4×D5C2×C3×C411yescharacteristic
102C2×Dic10—21no—
103C2×Dic10C2×C3×C411yescharacteristic
104C2×Dic10C23×C311yescharacteristic
105#704—24no—
106#704—42no—
107C2×#170—42no—
108C5×C22⋊C4—22no—
109C42×C5C2×C3×C411yescharacteristic
110C22⋊Dic5—22no—
111#685—21no—
112C4⋊Dic5—22no—
113#683—21no—
114#683—22no—
115C4×Dic5C2×C3×C411yescharacteristic
116#536—51no—
117#413—52no—
118#356—52no—
119#356—52no—
120C22×C3×C5—22no—
121C22×C3×C5—42no—
122C22×C3×C5C2×C22⋊C411yescharacteristic
123C2×C3×D5—21no—
124C2×C3×D5—41no—
125C3×C4×C5—21no—
126C3×C4×C5—41no—
127C3×C4×C5—41no—
128C3×Dic5—22no—
129C3×Dic5—41no—
130C3×Dic5—41no—
131C3×Dic5#9912yes—
132C3×#41—202no—
133C2×C3×Q8—51no—
134C2×C3×Q8—51no—
135C2×C3×Q8—101no—
136C2×C3×Q8—202no—
137C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
138C2×C3×D4—52no—
139C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
140C22×C3×C4—52no—
141C22×C3×C4—51no—
142C22×C3×C4—101no—
143C3×M16—104no—
144C3×M16—202no—
145C2×C3×C8—202no—
146C3×C4⋊C4—101no—
147C3×C4⋊C4—102no—
148C3×C4⋊C4—102no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—101no—
151C3×C22⋊C4—102no—
152C3×C42—51no—
153C3×C42C2×AGL(1,𝔽5)11yescharacteristic
154C23×C5C3×C22⋊C412yes—
155C22×D5C3×C22⋊C411yescharacteristic
156C5×Q8—41no—
157C5×D4—41no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5C22×C3×C411yescharacteristic
161C22⋊D5—42no—
162C2×Dic5—22no—
163C2×Dic5—22no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5C3×C22⋊C411yescharacteristic
167C2×Dic5C2×C3×D412yes—
168C4×D5—22no—
169C4×D5—42no—
170Dic10—41no—
171Dic10—42no—
172Dic10—41no—
173#170—44no—
174C2×#41—51no—
175C22×Q8—51no—
176C2×M16—102no—
177C4⋊Q8—51no—
178C4⋊Q8—51no—
179#124—51no—
180#124C3×AGL(1,𝔽5)11yescharacteristic
181#123—102no—
182C22⋊Q8—102no—
183#117—51no—
184#101—54no—
185#100—102no—
186C2×C3×C5—21no—
187C2×C3×C5—42no—
188C2×C3×C5C2×#9911yescharacteristic
189C3×D5—41no—
190C23×C3—51no—
191C23×C3C22⋊AGL(1,𝔽5)12yes—
192C3×Q8—41no—
193C3×Q8—201no—
194C3×Q8—201no—
195C3×Q8—202no—
196C3×D4—41no—
197C3×D4—202no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—51no—
201C2×C3×C4—52no—
202C2×C3×C4—102no—
203C2×C3×C4—102no—
204C2×C3×C4—102no—
205C2×C3×C4—101no—
206C2×C3×C4—101no—
207C2×C3×C4—202no—
208C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
209C3×C8—204no—
210C22×C5—22no—
211C22×C5—42no—
212C22×C5C2×C3×C22⋊C411yescharacteristic
213C2×D5—21no—
214C2×D5—41no—
215C4×C5—21no—
216C4×C5—41no—
217C4×C5—41no—
218Dic5—22no—
219Dic5—41no—
220Dic5—41no—
221Dic5C3×#9912yes—
222#41—202no—
223C2×Q8—51no—
224C2×Q8—51no—
225C2×Q8—101no—
226C2×Q8—202no—
227C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
228C2×D4—52no—
229C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
230C22×C4—52no—
231C22×C4—51no—
232C22×C4—101no—
233M16—104no—
234M16—202no—
235C2×C8—202no—
236C4⋊C4—101no—
237C4⋊C4—102no—
238C4⋊C4—102no—
239C22⋊C4—22no—
240C22⋊C4—101no—
241C22⋊C4—102no—
242C42—51no—
243C42C2×C3×AGL(1,𝔽5)11yescharacteristic
244C3×C5#213511yescharacteristic, complemented, Hall
245C22×C3—22no—
246C22×C3—42no—
247C22×C3—101no—
248C22×C3—201no—
249C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
250C3×C4—21no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—52no—
254C3×C4—102no—
255C3×C4—201no—
256C3×C4—201no—
257C2×C5—21no—
258C2×C5—42no—
259C2×C5C2×C3×#9911yescharacteristic
260D5—41no—
261C23—51no—
262C23C3×C22⋊AGL(1,𝔽5)12yes—
263Q8—41no—
264Q8—201no—
265Q8—201no—
266Q8—202no—
267D4—41no—
268D4—202no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—51no—
272C2×C4—52no—
273C2×C4—102no—
274C2×C4—102no—
275C2×C4—102no—
276C2×C4—101no—
277C2×C4—101no—
278C2×C4—202no—
279C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
280C8—204no—
281C2×C3—21no—
282C2×C3—42no—
283C2×C3—201no—
284C2×C3D5⋊#9911yescharacteristic
285C5C3×#213511yescharacteristic, complemented, Sylow
286C22—22no—
287C22—42no—
288C22—101no—
289C22—201no—
290C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
291C4—21no—
292C4—41no—
293C4—41no—
294C4—52no—
295C4—102no—
296C4—201no—
297C4—201no—
298C3#1060455111yescharacteristic, direct product factor, Sylow
299C2—21no—
300C2—42no—
301C2—201no—
302C2C3×D5⋊#9911yescharacteristic
subgroupG/H
centerC2×C3D5⋊#99
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#124C4
socleC2×C3×C5C2×#99
prime radicalC2×C3×#711C22
cosocleC2×Dic10C23×C3
hypercenterC3×#124AGL(1,𝔽5)
hypocenterC5C3×#2135
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
3. C3×#64929C2semidirect
4. C3×#64855C2semidirect
5. #10604551C3direct (N × H)
21. #64929C2×C3semidirect
22. #64855C2×C3semidirect
244. C3×C5#2135semidirect
285. C5C3×#2135semidirect
298. C3#10604551direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660092
C3×#657772
C3×#649292
C3×#648552
#106045513
C3×#21355