Finite Group Databasebeta

← C3×D3×S5 #72247568

Subgroups — C3×D3×S5 #72247568

Order 2160.

281 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 20 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×D3×A5C211yesmaximal, characteristic, complemented
2C32×S5C211yesmaximal, characteristic, complemented
3C3×ΓL(2,𝔽4)C211yesmaximal, characteristic, complemented
4C2×C3×S5—31nomaximal
5D3×S5C311yesmaximal, characteristic, direct product factor
6C32×A5C2211yescharacteristic, complemented
7C3×D3×S4—51nomaximal, Hall
8C3×D3×AGL(1,𝔽5)—61nomaximal
9C2×C3×A5—31no—
10D3×A5C2×C311yescharacteristic, complemented
11ΓL(2,𝔽4)C2×C311yescharacteristic, complemented
12C3×S5—21no—
13C3×S5—31no—
14C3×S5D311yescharacteristic, direct product factor
15C3×S5C2×C311yescharacteristic, complemented
16C2×S5—31no—
17C2×C3×D32—101nomaximal
18C3×D3×A4—51no—
19C3×C3⋊S4—51no—
20C32×S4—51no—
21C3×D3×D5—61no—
22C3×ΓL(1,𝔽16)—61no—
23C32×AGL(1,𝔽5)—61no—
24C3×A5—21no—
25C3×A5C2×D311yescharacteristic, complemented
26C3×A5C22×C311yescharacteristic, complemented
27C2×C3×S4—151no—
28D3×S4—51no—
29C3×D3×D4—151no—
30C2×C3×AGL(1,𝔽5)—181no—
31D3×AGL(1,𝔽5)—61no—
32C2×A5—31no—
33S5—31no—
34S5C3×D311yescharacteristic, direct product factor
35C2×C3×C3⋊D3—101no—
36C2×C32×D3—101no—
37C2×C32×D3—101no—
38C32×A4—51no—
39C3×D32—101no—
40C3×D32—101no—
41C3×D32—101no—
42C3×D32—101no—
43C3×D15—61no—
44C3×C5×D3—61no—
45C32×D5—61no—
46C22×C3×D3—51no—
47C22×C3×D3—151no—
48C22×C3×D3—301no—
49C2×C3×A4—151no—
50C2×D32—101no—
51D3×A4—51no—
52D3×A4—101no—
53C3⋊S4—51no—
54C3×S4—51no—
55C3×S4—51no—
56C3×S4—101no—
57C3×S4—151no—
58C3×S4—301no—
59C32×D4—151no—
60C3×#67—151no—
61C3×#67—151no—
62C3×D12—151no—
63C3×C4×D3—151no—
64C2×C3×D5—181no—
65D3×D5—61no—
66ΓL(1,𝔽16)—61no—
67C3×AGL(1,𝔽5)—61no—
68C3×AGL(1,𝔽5)—61no—
69C3×AGL(1,𝔽5)—121no—
70C3×AGL(1,𝔽5)—181no—
71A5C2×C3×D311yescharacteristic, complemented
72C2×C33—101no—
73C3×C3⋊D3—101no—
74C3×C3⋊D3—101no—
75C32×D3—101no—
76C32×D3—101no—
77C32×D3—101no—
78C32×D3—101no—
79C2×S4—151no—
80C2×C3×D4—451no—
81D3×D4—151no—
82C32×C5—61no—
83C2×AGL(1,𝔽5)—181no—
84C22×C32—51no—
85C22×C32—151no—
86C22×C32—301no—
87C2×C3⋊D3—101no—
88C2×C3×D3—101no—
89C2×C3×D3—101no—
90C2×C3×D3—101no—
91C2×C3×D3—101no—
92C2×C3×D3—151no—
93C2×C3×D3—151no—
94C2×C3×D3—151no—
95C2×C3×D3—201no—
96C2×C3×D3—201no—
97C2×C3×D3—301no—
98C2×C3×D3—301no—
99C2×C3×D3—301no—
100C2×C3×D3—301no—
101C2×C3×D3—301no—
102C2×C3×D3—301no—
103C2×C3×D3—601no—
104C3×A4—51no—
105C3×A4—51no—
106C3×A4—101no—
107C3×A4—101no—
108C3×A4—101no—
109C3×A4—201no—
110D32—101no—
111D32—101no—
112D32—101no—
113D32—101no—
114C32×C4—151no—
115C3×Dic3—151no—
116C2×C3×C5—181no—
117D15—61no—
118C3×D5—61no—
119C3×D5—61no—
120C3×D5—121no—
121C3×D5—181no—
122C5×D3—61no—
123C33—101noSylow
124C23×C3—151no—
125C23×C3—451no—
126C22×D3—51no—
127C22×D3—151no—
128C22×D3—301no—
129C2×A4—151no—
130C2×A4—301no—
131S4—51no—
132S4—151no—
133S4—301no—
134C3×D4—151no—
135C3×D4—151no—
136C3×D4—301no—
137C3×D4—451no—
138C3×D4—451no—
139C3×D4—451no—
140C2×C3×C4—451no—
141#67—151no—
142#67—151no—
143D12—151no—
144C4×D3—151no—
145C2×D5—181no—
146AGL(1,𝔽5)—61no—
147AGL(1,𝔽5)—181no—
148C2×C32—101no—
149C2×C32—101no—
150C2×C32—101no—
151C2×C32—151no—
152C2×C32—201no—
153C2×C32—201no—
154C2×C32—201no—
155C2×C32—301no—
156C2×C32—301no—
157C2×C32—401no—
158C3⋊D3—101no—
159C3⋊D3—101no—
160C3×D3—101no—
161C3×D3—101no—
162C3×D3—101no—
163C3×D3—101no—
164C3×D3—101no—
165C3×D3—101no—
166C3×D3—101no—
167C3×D3—151no—
168C3×D3—201no—
169C3×D3—201no—
170C3×D3—201no—
171C3×D3—201no—
172C3×D3—301no—
173C3×D3—301no—
174C3×D3—601no—
175C3×D3—601no—
176C3×D3S511yescharacteristic, direct product factor
177C2×D4—451noSylow
178C3×C5—61no—
179C3×C5—61no—
180C3×C5—121no—
181C22×C3—51no—
182C22×C3—51no—
183C22×C3—101no—
184C22×C3—151no—
185C22×C3—151no—
186C22×C3—301no—
187C22×C3—301no—
188C22×C3—301no—
189C22×C3—451no—
190C22×C3—451no—
191C22×C3—451no—
192C22×C3—601no—
193C22×C3—901no—
194C2×D3—101no—
195C2×D3—101no—
196C2×D3—151no—
197C2×D3—151no—
198C2×D3—151no—
199C2×D3—301no—
200C2×D3—301no—
201C2×D3—301no—
202C2×D3—301no—
203C2×D3—301no—
204C2×D3—301no—
205C2×D3—601no—
206A4—51no—
207A4—101no—
208A4—101no—
209A4—201no—
210C3×C4—151no—
211C3×C4—151no—
212C3×C4—301no—
213C3×C4—451no—
214Dic3—151no—
215C2×C5—181no—
216D5—61no—
217D5—181no—
218C32—101no—
219C32—101no—
220C32—201no—
221C32—201no—
222C32—201no—
223C32—401no—
224C32C2×S511yescharacteristic, complemented
225C23—151no—
226C23—451no—
227D4—151no—
228D4—451no—
229D4—451no—
230D4—451no—
231C2×C4—451no—
232C2×C3—31no—
233C2×C3—101no—
234C2×C3—101no—
235C2×C3—101no—
236C2×C3—151no—
237C2×C3—151no—
238C2×C3—201no—
239C2×C3—201no—
240C2×C3—201no—
241C2×C3—301no—
242C2×C3—301no—
243C2×C3—301no—
244C2×C3—301no—
245C2×C3—401no—
246C2×C3—451no—
247C2×C3—601no—
248C2×C3—601no—
249D3—101no—
250D3—101no—
251D3—101no—
252D3—151no—
253D3—301no—
254D3—301no—
255D3—601no—
256D3—601no—
257D3C3×S511yescharacteristic, direct product factor
258C5—61noSylow
259C22—51no—
260C22—151no—
261C22—301no—
262C22—451no—
263C22—451no—
264C22—451no—
265C22—901no—
266C4—151no—
267C4—451no—
268C3—21no—
269C3—101no—
270C3—201no—
271C3—201no—
272C3—401no—
273C3D3×S511yescharacteristic, direct product factor
274C3C2×C3×S511yescharacteristic, complemented
275C2—31no—
276C2—101no—
277C2—151no—
278C2—301no—
279C2—451no—
subgroupG/H
center, hypercenterC3D3×S5
commutator, cosocle, hypocenterC3×A5C22×C3
FrattiniC1G
FittingC32C2×S5
socleC32×A5C22
LCSG ▷ C3×A5
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×D3×A5C2semidirect
2. C32×S5C2semidirect
3. C3×ΓL(2,𝔽4)C2semidirect
5. D3×S5C3direct (N × H)
6. C32×A5C22semidirect
10. D3×A5C2×C3semidirect
11. ΓL(2,𝔽4)C2×C3semidirect
14. C3×S5D3direct (N × H)
15. C3×S5C2×C3semidirect
25. C3×A5C2×D3semidirect
26. C3×A5C22×C3semidirect
34. S5C3×D3direct (N × H)
71. A5C2×C3×D3semidirect
176. C3×D3S5direct (N × H)
224. C32C2×S5semidirect
257. D3C3×S5direct (N × H)
273. C3D3×S5direct (N × H)
274. C3C2×C3×S5semidirect

Maximal subgroups

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

subgroupindex
C3×D3×A52
C32×S52
C3×ΓL(2,𝔽4)2
C2×C3×S53
D3×S53
C3×D3×S45
C3×D3×AGL(1,𝔽5)6
C2×C3×D3210