Finite Group Databasebeta

← C3×#91564 #13086708

Subgroups — C3×#91564 #13086708

Order 1440.

654 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C23×C32×D5C211yesmaximal, characteristic, complemented
2C2×C3×#1220C211yesmaximal, characteristic, complemented
3C2×C3×#1202C212yesmaximal, complemented
4C3×C5×#217C211yesmaximal, characteristic, complemented
5C3×#6665C212yesmaximal, complemented
6C3×#4028—31nomaximal
7#91564C311yesmaximal, characteristic, direct product factor
8C23×C32×C5C2211yescharacteristic
9C22×C3×D15C2211yescharacteristic, complemented
10C22×C32×D5—22no—
11C22×C32×D5—24no—
12C22×C32×D5C2212yes—
13C3×#1220—22no—
14C2×C3×Dic15C2211yescharacteristic, complemented
15C2×C3×C5×Dic3C2212yescomplemented
16C3×#1202—24no—
17C3×#953—51nomaximal, Hall
18C23×C3×D5—21no—
19C23×C3×D5D311yescharacteristic, complemented
20C23×C3×D5C2×C311yescharacteristic, complemented
21C2×#1220C2×C311yescharacteristic, complemented
22C2×C3×C22⋊D5—31no—
23C2×C3×D20—32no—
24C2×#1202C2×C312yescomplemented
25C3×C5×C22⋊C4—31no—
26C5×#217C2×C311yescharacteristic, complemented
27C3×#685—32no—
28#6665C2×C312yescomplemented
29C22×C32×C5—22no—
30C22×C32×C5D412yescomplemented
31C22×C32×C5C2311yescharacteristic
32C2×C3×D15—22no—
33C2×C3×D15—21no—
34C2×C32×D5—28no—
35C2×C32×D5—22no—
36C2×C32×D5—22no—
37C2×C32×D5D414yescomplemented
38C3×Dic15—21no—
39C3×C5×Dic3—22no—
40#4028—31noHall
41C24×C32—51no—
42C2×C3×#67—51no—
43C2×C3×#67—52no—
44C3×#217—52no—
45C3×#217D511yescharacteristic, complemented
46C23×C3×C5—21no—
47C23×C3×C5C2×D311yescharacteristic
48C23×C3×C5C22×C311yescharacteristic
49C22×D15C22×C311yescharacteristic, complemented
50C22×C3×D5—24no—
51C22×C3×D5—28no—
52C22×C3×D5—22no—
53C22×C3×D5—22no—
54C22×C3×D5—22no—
55C22×C3×D5—24no—
56C22×C3×D5—24no—
57C22×C3×D5—31no—
58C22×C3×D5C2×D312yes—
59C22×C3×D5C22×C312yes—
60C2×C3×C4×C5—32no—
61#1220—22no—
62C2×Dic15C22×C311yescharacteristic, complemented
63C2×C5×Dic3C22×C312yescomplemented
64C3×C22⋊D5—62no—
65C2×C3×Dic5—31no—
66C3×D20—64no—
67#1202—24no—
68C3×#120—151no—
69#953—51no—
70C2×C32×C5—22no—
71C2×C32×C5C2×D411yescharacteristic
72C2×C32×C5C2×D412yes—
73C3×D15—41no—
74C32×D5—24no—
75C23×D5C3×D311yescharacteristic, complemented
76C2×C22⋊D5—31no—
77C2×D20—32no—
78C5×C22⋊C4—31no—
79#685—32no—
80C23×C32—52no—
81C23×C32—104no—
82C23×C32—102no—
83C23×C32C2×D511yescharacteristic
84C22×C3×D3—51no—
85C3×#67—104no—
86C3×#67—102no—
87C2×C3×Dic3—51no—
88C2×C3×Dic3C2×D512yescomplemented
89C22×C3×C5—24no—
90C22×C3×C5—22no—
91C22×C3×C5—22no—
92C22×C3×C5—22no—
93C22×C3×C5—21no—
94C22×C3×C5#6712yescomplemented
95C22×C3×C5C3×D412yescomplemented
96C22×C3×C5C22×D311yescharacteristic
97C22×C3×C5C23×C311yescharacteristic
98C2×D15—22no—
99C2×D15—21no—
100C2×C3×D5—216no—
101C2×C3×D5—28no—
102C2×C3×D5—28no—
103C2×C3×D5—24no—
104C2×C3×D5—24no—
105C2×C3×D5—22no—
106C2×C3×D5—22no—
107C2×C3×D5—22no—
108C2×C3×D5—22no—
109C2×C3×D5—24no—
110C2×C3×D5—62no—
111C2×C3×D5—61no—
112C2×C3×D5#6714yescomplemented
113C2×C3×D5C3×D414yescomplemented
114C3×C4×C5—62no—
115Dic15—21no—
116C3×Dic5—61no—
117C5×Dic3—22no—
118C24×C3—51no—
119C24×C3—51no—
120C24×C3—101no—
121C2×C3×D4—152no—
122C2×C3×D4—151no—
123C2×#67—51no—
124C2×#67—52no—
125C3×C22⋊C4—31no—
126C3×C22⋊C4—152no—
127#217—52no—
128#217C3×D511yescharacteristic, complemented
129C32×C5#12011yescharacteristic, complemented, Hall
130C23×C5C2×C3×D311yescharacteristic
131C22×D5—22no—
132C22×D5—24no—
133C22×D5—31no—
134C22×D5C2×C3×D312yes—
135C2×C4×C5—32no—
136C22⋊D5—62no—
137C2×Dic5—31no—
138D20—64no—
139C22×C32—22no—
140C22×C32—54no—
141C22×C32—108no—
142C22×C32—102no—
143C22×C32—102no—
144C22×C32D2012yescomplemented
145C22×C32C22×D511yescharacteristic
146C2×C3×D3—101no—
147C2×C3×D3—102no—
148C3×Dic3—22no—
149C3×Dic3—101no—
150#120—151noSylow
151C2×C3×C5—24no—
152C2×C3×C5—22no—
153C2×C3×C5—22no—
154C2×C3×C5—21no—
155C2×C3×C5—22no—
156C2×C3×C5C2×#6711yescharacteristic
157C2×C3×C5C2×#6712yes—
158C2×C3×C5C2×C3×D411yescharacteristic
159C2×C3×C5C2×C3×D412yes—
160D15—41no—
161C3×D5—28no—
162C3×D5—24no—
163C3×D5—24no—
164C3×D5—121no—
165C23×C3—21no—
166C23×C3—52no—
167C23×C3—52no—
168C23×C3—102no—
169C23×C3—104no—
170C23×C3—104no—
171C23×C3—108no—
172C23×C3—104no—
173C23×C3—102no—
174C23×C3—102no—
175C23×C3—151no—
176C23×C3D3×D511yescharacteristic
177C23×C3C2×C3×D511yescharacteristic
178C22×D3—51no—
179C3×D4—304no—
180C3×D4—302no—
181C2×C3×C4—32no—
182C2×C3×C4—151no—
183#67—104no—
184#67—102no—
185C2×Dic3—51no—
186C2×Dic3C2×C3×D512yescomplemented
187C22×C5—22no—
188C22×C5C3×#6712yescomplemented
189C22×C5C22×C3×D311yescharacteristic
190C2×D5—28no—
191C2×D5—22no—
192C2×D5—22no—
193C2×D5—62no—
194C2×D5—61no—
195C2×D5C3×#6714yescomplemented
196C4×C5—62no—
197Dic5—61no—
198C2×C32—22no—
199C2×C32—104no—
200C2×C32C2×D2011yescharacteristic
201C2×C32D4×D512yes—
202C3×D3—201no—
203C24—51no—
204C2×D4—152no—
205C2×D4—151no—
206C22⋊C4—31no—
207C22⋊C4—152no—
208C3×C5—21no—
209C3×C5#95311yescharacteristic, complemented
210C3×C5C3×#12011yescharacteristic, complemented
211C22×C3—24no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—54no—
217C22×C3—54no—
218C22×C3—104no—
219C22×C3—1016no—
220C22×C3—108no—
221C22×C3—108no—
222C22×C3—104no—
223C22×C3—104no—
224C22×C3—102no—
225C22×C3—102no—
226C22×C3—102no—
227C22×C3—102no—
228C22×C3—301no—
229C22×C3—302no—
230C22×C3#120212yescomplemented
231C22×C3C3×D2012yescomplemented
232C22×C3C2×D3×D511yescharacteristic
233C22×C3C22×C3×D511yescharacteristic
234C2×D3—101no—
235C2×D3—102no—
236C3×C4—62no—
237C3×C4—301no—
238Dic3—22no—
239Dic3—101no—
240C2×C5—22no—
241C2×C5C2×C3×#6711yescharacteristic
242C2×C5C2×C3×#6712yes—
243D5—24no—
244D5—121no—
245C32#402811yescharacteristic, complemented, Sylow
246C23—52no—
247C23—104no—
248C23—102no—
249C23—151no—
250C23C3×D3×D511yescharacteristic
251D4—304no—
252D4—302no—
253C2×C4—32no—
254C2×C4—151no—
255C2×C3—24no—
256C2×C3—22no—
257C2×C3—22no—
258C2×C3—21no—
259C2×C3—22no—
260C2×C3—108no—
261C2×C3—104no—
262C2×C3—104no—
263C2×C3—601no—
264C2×C3C2×#120211yescharacteristic
265C2×C3D5×#6712yes—
266C2×C3C2×C3×D2011yescharacteristic
267C2×C3C3×D4×D512yes—
268D3—201no—
269C5C3×#95311yescharacteristic, complemented, Sylow
270C22—22no—
271C22—54no—
272C22—108no—
273C22—102no—
274C22—102no—
275C22—301no—
276C22—302no—
277C22C3×#120212yescomplemented
278C22C2×C3×D3×D511yescharacteristic
279C4—62no—
280C4—301no—
281C3—21no—
282C3#9156411yescharacteristic, direct product factor
283C3C3×#402811yescharacteristic, complemented
284C2—22no—
285C2—104no—
286C2—601no—
287C2C2×C3×#120211yescharacteristic
288C2C3×D5×#6712yes—
subgroupG/H
centerC22×C3C2×D3×D5
commutator, cosocleC22×C3×C5C23×C3
FrattiniC22C2×C3×D3×D5
FittingC23×C32×C5C22
socleC22×C32×C5C23
hypercenterC23×C3D3×D5
hypocenterC3×C5C3×#120
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C32×D5C2semidirect
2. C2×C3×#1220C2semidirect
3. C2×C3×#1202C2semidirect
4. C3×C5×#217C2semidirect
5. C3×#6665C2semidirect
7. #91564C3direct (N × H)
9. C22×C3×D15C22semidirect
14. C2×C3×Dic15C22semidirect
15. C2×C3×C5×Dic3C22semidirect
19. C23×C3×D5D3semidirect
20. C23×C3×D5C2×C3semidirect
21. C2×#1220C2×C3semidirect
24. C2×#1202C2×C3semidirect
26. C5×#217C2×C3semidirect
28. #6665C2×C3semidirect
30. C22×C32×C5D4semidirect
37. C2×C32×D5D4semidirect
45. C3×#217D5semidirect
49. C22×D15C22×C3semidirect
62. C2×Dic15C22×C3semidirect
63. C2×C5×Dic3C22×C3semidirect
75. C23×D5C3×D3semidirect
88. C2×C3×Dic3C2×D5semidirect
94. C22×C3×C5#67semidirect
95. C22×C3×C5C3×D4semidirect
112. C2×C3×D5#67semidirect
113. C2×C3×D5C3×D4semidirect
128. #217C3×D5semidirect
129. C32×C5#120semidirect
144. C22×C32D20semidirect
186. C2×Dic3C2×C3×D5semidirect
188. C22×C5C3×#67semidirect
195. C2×D5C3×#67semidirect
209. C3×C5#953semidirect
210. C3×C5C3×#120semidirect
230. C22×C3#1202semidirect
231. C22×C3C3×D20semidirect
245. C32#4028semidirect
269. C5C3×#953semidirect
277. C22C3×#1202semidirect
282. C3#91564direct (N × H)
283. C3C3×#4028semidirect

Maximal subgroups

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

subgroupindex
C23×C32×D52
C2×C3×#12202
C2×C3×#12022
C3×C5×#2172
C3×#66652
C3×#40283
#915643
C3×#9535