Finite Group Databasebeta

← #13087586

Subgroups — #13087586

Order 1440.

330 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C22×C3×C5×Dic3C211yesmaximal, characteristic
2C2×#67153C211yesmaximal, characteristic, complemented
3#10623913C211yesmaximal, characteristic
4C3×#6719C211yesmaximal, characteristic
5Dic3⋊Dic15C211yesmaximal, characteristic, complemented
6#10623815C211yesmaximal, characteristic, complemented
7#10623814C211yesmaximal, characteristic, complemented
8#91813—31nomaximal
9#91575—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C2×C3⋊Dic15C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3—22no—
16C2×C3×C5×Dic3C2211yescharacteristic
17C2×C3×C5×Dic3C2211yescharacteristic
18#67153—22no—
19#63964—51nomaximal, Hall
20C22×C3×C4×C5—31no—
21C2×Dic30—31no—
22C22×C5×Dic3D311yescharacteristic
23C2×#1204—31no—
24#6719—31no—
25#6719—31no—
26#6719—61no—
27#6719D311yescharacteristic
28C4⋊Dic15—31no—
29#6712—31no—
30#6712—31no—
31C3×C22⋊Dic5—31no—
32Dic3⋊Dic5—31no—
33#6669—31no—
34#6668—31no—
35C22×C32×C5—21no—
36C22×C32×C5—21no—
37C22×C32×C5Q812yescomplemented
38C22×C32×C5C2311yescharacteristic
39C3⋊Dic15—21no—
40C3⋊Dic15—21no—
41C3×Dic15—21no—
42C3×Dic15—21no—
43C3×C5×Dic3—21no—
44C3×C5×Dic3D412yes—
45#4070—91noHall
46C22×C3×Dic3D511yescharacteristic
47C2×#627—51no—
48#3757—51no—
49C3×#217—51no—
50#3724—51no—
51Dic3⋊Dic3—51no—
52Dic3⋊Dic3—51no—
53C23×C3×C5—21no—
54C23×C3×C5C2×D311yescharacteristic
55C23×C3×C5C2×D311yescharacteristic
56C2×C3×C4×C5—31no—
57C2×C3×C4×C5—31no—
58C2×C3×C4×C5—62no—
59C2×Dic15—31no—
60C2×Dic15—31no—
61C2×Dic15—31no—
62C2×Dic15—31no—
63C2×Dic15—61no—
64C2×Dic15—61no—
65C2×Dic15C2×D311yescharacteristic
66C2×Dic15C2×D311yescharacteristic
67Dic30—62no—
68C2×C5×Dic3—22no—
69C2×C5×Dic3C2×D311yescharacteristic
70C2×C5×Dic3C2×D311yescharacteristic
71C2×C3×Dic5—31no—
72C2×C3×Dic5—31no—
73#1204—62no—
74#924—151no—
75#878—151no—
76C2×C32×C5—22no—
77C2×C32×C5C2×D411yescharacteristic
78C2×C32×C5C2×Q811yescharacteristic
79C2×C32×C5#4111yescharacteristic
80C22×C4×C5—31no—
81C2×Dic10—91no—
82C22⋊Dic5—31no—
83C22⋊Dic5—91no—
84C4⋊Dic5—91no—
85#683—91no—
86#683—91no—
87C23×C32C2×D511yescharacteristic
88C2×C3⋊Dic3—51no—
89C2×C3⋊Dic3—51no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—51no—
92C2×C3×Dic3—51no—
93C2×C3×Dic3C2×D511yescharacteristic
94C2×C3×Dic3C2×D511yescharacteristic
95#627—102no—
96C22×C3×C5—21no—
97C22×C3×C5—21no—
98C22×C3×C5—21no—
99C22×C3×C5—21no—
100C22×C3×C5—22no—
101C22×C3×C5—21no—
102C22×C3×C5—41no—
103C22×C3×C5—41no—
104C22×C3×C5Dic612yescomplemented
105C22×C3×C5Dic612yescomplemented
106C22×C3×C5C22×D311yescharacteristic
107C22×C3×C5C22×D311yescharacteristic
108C3×C4×C5—32no—
109C3×C4×C5—61no—
110Dic15—21no—
111Dic15—21no—
112Dic15—61no—
113Dic15—61no—
114Dic15—61no—
115Dic15—61no—
116Dic15—121no—
117Dic15—121no—
118C3×Dic5—61no—
119C3×Dic5—61no—
120C5×Dic3—21no—
121C5×Dic3#6712yes—
122C22×C3×C4—31no—
123C22×Dic3D1511yescharacteristic
124C2×Dic6—151no—
125C2×Dic6—151no—
126C3×C22⋊C4—151no—
127#217—51no—
128#217—151no—
129#217—151no—
130#217—301no—
131C4⋊Dic3—151no—
132C4⋊Dic3—151no—
133Dic3⋊C4—151no—
134Dic3⋊C4—151no—
135Dic3⋊C4—151no—
136Dic3⋊C4—151no—
137C32×C5C22⋊Q811yescharacteristic, complemented, Hall
138C23×C5D3211yescharacteristic
139C2×C4×C5—31no—
140C2×C4×C5—31no—
141C2×C4×C5—62no—
142C2×Dic5—31no—
143C2×Dic5—31no—
144C2×Dic5—91no—
145C2×Dic5—91no—
146Dic10—182no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32Dic1012yescomplemented
150C22×C32C22×D511yescharacteristic
151C3⋊Dic3—101no—
152C3⋊Dic3—101no—
153C3×Dic3—21no—
154C3×Dic3—101no—
155C3×Dic3—101no—
156C3×Dic3C22⋊D512yes—
157C22⋊Q8—451noSylow
158C2×C3×C5—22no—
159C2×C3×C5—22no—
160C2×C3×C5—21no—
161C2×C3×C5—21no—
162C2×C3×C5—21no—
163C2×C3×C5—42no—
164C2×C3×C5C2×Dic611yescharacteristic
165C2×C3×C5C2×Dic611yescharacteristic
166C2×C3×C5#23511yescharacteristic
167C2×C3×C5D3×D411yescharacteristic
168C2×C3×C5#23711yescharacteristic
169C2×C3×C5C2×#6711yescharacteristic
170C23×C3—21no—
171C23×C3D3×D511yescharacteristic
172C23×C3C2×D1511yescharacteristic
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—62no—
176C2×C3×C4—151no—
177C2×C3×C4—151no—
178C2×Dic3—22no—
179C2×Dic3—51no—
180C2×Dic3—51no—
181C2×Dic3—151no—
182C2×Dic3—151no—
183C2×Dic3—151no—
184C2×Dic3—151no—
185C2×Dic3—301no—
186C2×Dic3—301no—
187C2×Dic3C2×D1511yescharacteristic
188C2×Dic3C2×D1511yescharacteristic
189Dic6—302no—
190Dic6—302no—
191C22×C5—21no—
192C22×C5—21no—
193C22×C5#62712yescomplemented
194C22×C5C2×D3211yescharacteristic
195C4×C5—32no—
196C4×C5—61no—
197Dic5—61no—
198Dic5—61no—
199Dic5—181no—
200Dic5—181no—
201C2×C32—22no—
202C2×C32C2×Dic1011yescharacteristic
203C2×C32#70911yescharacteristic
204C2×C32C2×C22⋊D511yescharacteristic
205C2×Q8—451no—
206C22×C4—31no—
207C4⋊C4—451no—
208C4⋊C4—451no—
209C4⋊C4—451no—
210C22⋊C4—151no—
211C22⋊C4—451no—
212C3×C5—21no—
213C3×C5#87811yescharacteristic, complemented
214C3×C5#92411yescharacteristic, complemented
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—22no—
220C22×C3—21no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3#120412yescomplemented
224C22×C3Dic3012yescomplemented
225C22×C3C2×D3×D511yescharacteristic
226C22×C3C22×D1511yescharacteristic
227C3×C4—32no—
228C3×C4—61no—
229C3×C4—301no—
230C3×C4—301no—
231Dic3—21no—
232Dic3—101no—
233Dic3—101no—
234Dic3—301no—
235Dic3—301no—
236Dic3—301no—
237Dic3—301no—
238Dic3—601no—
239Dic3—601no—
240Dic3#122012yes—
241C2×C5—22no—
242C2×C5#380411yescharacteristic
243C2×C5C2×#62711yescharacteristic
244C2×C5D3×#6711yescharacteristic
245C32#407011yescharacteristic, complemented, Sylow
246C23D3×D1511yescharacteristic
247Q8—902no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—62no—
251C2×C4—151no—
252C2×C4—151no—
253C2×C4—451no—
254C2×C4—451no—
255C2×C3—22no—
256C2×C3—22no—
257C2×C3—21no—
258C2×C3—21no—
259C2×C3—21no—
260C2×C3—42no—
261C2×C3#677911yescharacteristic
262C2×C3C2×#120411yescharacteristic
263C2×C3D3×C22⋊D511yescharacteristic
264C2×C3C2×Dic3011yescharacteristic
265C2×C3#681711yescharacteristic
266C2×C3C2×#122011yescharacteristic
267C5#6396411yescharacteristic, complemented, Sylow
268C22—21no—
269C22—21no—
270C22#6715312yescomplemented
271C22C2×D3×D1511yescharacteristic
272C4—32no—
273C4—61no—
274C4—301no—
275C4—301no—
276C4—901no—
277C4—901no—
278C3—21no—
279C3#9157511yescharacteristic, complemented
280C3#9181311yescharacteristic, complemented
281C2—22no—
282C2#1062420711yescharacteristic
283C2C2×#6715311yescharacteristic
284C2D3×#122011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
Fitting, prime radicalC23×C32×C5C22
hypercenterC23D3×D15
hypocenterC32×C5C22⋊Q8
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#67153C2semidirect
5. Dic3⋊Dic15C2semidirect
6. #10623815C2semidirect
7. #10623814C2semidirect
37. C22×C32×C5Q8semidirect
104. C22×C3×C5Dic6semidirect
105. C22×C3×C5Dic6semidirect
137. C32×C5C22⋊Q8semidirect
149. C22×C32Dic10semidirect
193. C22×C5#627semidirect
213. C3×C5#878semidirect
214. C3×C5#924semidirect
223. C22×C3#1204semidirect
224. C22×C3Dic30semidirect
245. C32#4070semidirect
267. C5#63964semidirect
270. C22#67153semidirect
279. C3#91575semidirect
280. C3#91813semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C5×Dic32
C2×#671532
#106239132
C3×#67192
Dic3⋊Dic152
#106238152
#106238142
#918133
#915753
#639645