Finite Group Databasebeta

← #13087411

Subgroups — #13087411

Order 1440.

322 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×C3×C5×Dic6C211yesmaximal, characteristic, complemented
2C2×Dic3×D15C211yesmaximal, characteristic
3#10623906C211yesmaximal, characteristic, complemented
4C3×#6714C211yesmaximal, characteristic
5Dic3⋊Dic15C211yesmaximal, characteristic, complemented
6#10623814C211yesmaximal, characteristic, complemented
7#10623811C211yesmaximal, characteristic
8#91835—31nomaximal
9#91421—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C2×C32×C4×C5C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C3×C5×Dic6—22no—
18Dic3×D15—22no—
19#63887—51nomaximal, Hall
20C2×C3×C5×Q8—31no—
21C2×C4×D15—31no—
22C2×C5×Dic6D311yescharacteristic, complemented
23C2×D5×Dic3—31no—
24#6714—31no—
25#6714D311yescharacteristic
26C4⋊Dic15—31no—
27#6712—31no—
28#6712—31no—
29#6712—31no—
30#6712—61no—
31C3×#685—31no—
32Dic3⋊Dic5—31no—
33#6668—31no—
34#6665—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C3×D15—21no—
37C2×C3×D15—21no—
38C2×C3×D15Q812yescomplemented
39C32×C4×C5—21no—
40C3⋊Dic15—21no—
41C3⋊Dic15—21no—
42C3×Dic15—21no—
43C3×C5×Dic3—21no—
44C3×C5×Dic3D412yes—
45#4092—91noHall
46C2×C3×Dic6D511yescharacteristic, complemented
47C2×D3×Dic3—51no—
48#3750—51no—
49C3×(C2×D3)⋊C4—51no—
50Dic3⋊Dic3—51no—
51Dic3⋊Dic3—51no—
52(C2×D3)⋊Dic3—51no—
53C22×D15C2×D311yescharacteristic
54C22×C3×D5—31no—
55C3×C5×Q8—62no—
56C2×C3×C4×C5—21no—
57C2×C3×C4×C5—31no—
58C2×C3×C4×C5—31no—
59C2×C3×C4×C5C2×D311yescharacteristic
60C2×C3×C4×C5C2×D311yescharacteristic
61C2×Dic15—31no—
62C2×Dic15—31no—
63C2×Dic15—31no—
64C2×Dic15—31no—
65C2×Dic15—61no—
66C2×Dic15—61no—
67C2×Dic15C2×D311yescharacteristic
68C4×D15—62no—
69C2×C5×Dic3C2×D311yescharacteristic
70C2×C5×Dic3C2×D311yescharacteristic
71C5×Dic6—22no—
72C2×C3×Dic5—31no—
73D5×Dic3—62no—
74#946—151no—
75#878—151no—
76C2×C32×C5C2×D411yescharacteristic
77C2×C32×C5C2×Q811yescharacteristic
78C2×C32×C5#4111yescharacteristic
79C3×D15—22no—
80C2×C5×Q8—31no—
81C2×C4×D5—91no—
82#685—31no—
83#685—91no—
84C4⋊Dic5—91no—
85#683—91no—
86#683—91no—
87C22×C3×D3—51no—
88C2×C32×C4C2×D511yescharacteristic
89C2×C3⋊Dic3—51no—
90C2×C3⋊Dic3—51no—
91C2×C3×Dic3—51no—
92C2×C3×Dic3C2×D511yescharacteristic
93C2×C3×Dic3C2×D511yescharacteristic
94C3×Dic6—22no—
95D3×Dic3—102no—
96C22×C3×C5—21no—
97C22×C3×C5C22×D311yescharacteristic
98C22×C3×C5C22×D311yescharacteristic
99C2×D15—21no—
100C2×D15—21no—
101C2×D15Dic612yescomplemented
102C2×C3×D5—32no—
103C2×C3×D5—61no—
104C2×C3×D5—61no—
105C3×C4×C5—21no—
106C3×C4×C5—21no—
107C3×C4×C5—32no—
108C3×C4×C5—41no—
109C3×C4×C5—61no—
110Dic15—21no—
111Dic15—62no—
112Dic15—61no—
113Dic15—61no—
114Dic15—61no—
115Dic15—61no—
116Dic15—121no—
117C3×Dic5—61no—
118C5×Dic3—21no—
119C5×Dic3#6712yes—
120C2×C3×Q8—31no—
121C22×Dic3—151no—
122C2×C4×D3—151no—
123C2×Dic6D1511yescharacteristic, complemented
124C3×C22⋊C4—151no—
125#217—151no—
126(C2×D3)⋊C4—51no—
127(C2×D3)⋊C4—151no—
128C4⋊Dic3—151no—
129C4⋊Dic3—151no—
130Dic3⋊C4—151no—
131Dic3⋊C4—151no—
132Dic3⋊C4—151no—
133Dic3⋊C4—151no—
134Dic3⋊C4—301no—
135C32×C5C22⋊Q811yescharacteristic, complemented, Hall
136C22×D5—31no—
137C5×Q8—62no—
138C2×C4×C5—31no—
139C2×C4×C5—31no—
140C2×C4×C5D3211yescharacteristic
141C2×Dic5—31no—
142C2×Dic5—91no—
143C2×Dic5—91no—
144C4×D5—182no—
145C22×C32C22×D511yescharacteristic
146C2×C3×D3—52no—
147C2×C3×D3—101no—
148C2×C3×D3—101no—
149C32×C4—21no—
150C3⋊Dic3—101no—
151C3⋊Dic3—101no—
152C3×Dic3—21no—
153C3×Dic3—101no—
154C3×Dic3C22⋊D512yes—
155C22⋊Q8—451noSylow
156C2×C3×C5—21no—
157C2×C3×C5—21no—
158C2×C3×C5—21no—
159C2×C3×C5C2×Dic611yescharacteristic
160C2×C3×C5D3×D411yescharacteristic
161C2×C3×C5#23711yescharacteristic
162C2×C3×C5D3×Q811yescharacteristic
163C2×C3×C5#23911yescharacteristic
164C2×C3×C5C2×#6711yescharacteristic
165D15—22no—
166C3×D5—62no—
167C23×C3—151no—
168C22×D3—51no—
169C3×Q8—62no—
170C2×C3×C4—21no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—151no—
174C2×C3×C4D3×D511yescharacteristic
175C2×C3×C4C2×D1511yescharacteristic
176C2×Dic3—51no—
177C2×Dic3—151no—
178C2×Dic3—151no—
179C2×Dic3—151no—
180C2×Dic3—151no—
181C2×Dic3—301no—
182C2×Dic3—301no—
183C2×Dic3—302no—
184C2×Dic3C2×D1511yescharacteristic
185C2×Dic3C2×D1511yescharacteristic
186C4×D3—302no—
187Dic6—22no—
188C22×C5C2×D3211yescharacteristic
189C2×D5—32no—
190C2×D5—61no—
191C2×D5—61no—
192C4×C5—21no—
193C4×C5—32no—
194C4×C5—61no—
195Dic5—61no—
196Dic5—181no—
197Dic5—181no—
198C2×C32Q8×D511yescharacteristic
199C2×C32#71311yescharacteristic
200C2×C32C2×C22⋊D511yescharacteristic
201C3×D3—102no—
202C2×Q8—31no—
203C22×C4—451no—
204C4⋊C4—451no—
205C4⋊C4—451no—
206C4⋊C4—451no—
207C22⋊C4—151no—
208C22⋊C4—451no—
209C3×C5—21no—
210C3×C5#87811yescharacteristic, complemented
211C3×C5#94611yescharacteristic, complemented
212C22×C3—21no—
213C22×C3—152no—
214C22×C3—301no—
215C22×C3—301no—
216C22×C3C2×D3×D511yescharacteristic
217C22×C3C22×D1511yescharacteristic
218C2×D3—52no—
219C2×D3—101no—
220C2×D3—101no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—32no—
224C3×C4—41no—
225C3×C4—61no—
226C3×C4—301no—
227Dic3—21no—
228Dic3—101no—
229Dic3—302no—
230Dic3—301no—
231Dic3—301no—
232Dic3—301no—
233Dic3—301no—
234Dic3—601no—
235Dic3#122012yes—
236C2×C5D3×Dic611yescharacteristic
237C2×C5#379611yescharacteristic
238C2×C5D3×#6711yescharacteristic
239D5—62no—
240C32#409211yescharacteristic, complemented, Sylow
241C23—151no—
242Q8—62no—
243C2×C4—31no—
244C2×C4—31no—
245C2×C4—151no—
246C2×C4—451no—
247C2×C4—451no—
248C2×C4—902no—
249C2×C4D3×D1511yescharacteristic
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—302no—
254C2×C3D5×Dic611yescharacteristic
255C2×C3#676611yescharacteristic
256C2×C3D3×C22⋊D511yescharacteristic
257C2×C3Q8×D1511yescharacteristic
258C2×C3#682111yescharacteristic
259C2×C3C2×#122011yescharacteristic
260D3—102no—
261C5#6388711yescharacteristic, complemented, Sylow
262C22—152no—
263C22—301no—
264C22—301no—
265C22C2×D3×D1511yescharacteristic
266C4—21no—
267C4—32no—
268C4—61no—
269C4—301no—
270C4—901no—
271C4—901no—
272C3—21no—
273C3#9142111yescharacteristic, complemented
274C3#9183511yescharacteristic, complemented
275C2—302no—
276C2Dic6×D1511yescharacteristic
277C2#1062419411yescharacteristic
278C2D3×#122011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
prime radicalC22×C3×D15C22
hypercenterC2×C4D3×D15
hypocenterC32×C5C22⋊Q8
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×Dic6C2semidirect
3. #10623906C2semidirect
5. Dic3⋊Dic15C2semidirect
6. #10623814C2semidirect
22. C2×C5×Dic6D3semidirect
38. C2×C3×D15Q8semidirect
46. C2×C3×Dic6D5semidirect
101. C2×D15Dic6semidirect
123. C2×Dic6D15semidirect
135. C32×C5C22⋊Q8semidirect
210. C3×C5#878semidirect
211. C3×C5#946semidirect
240. C32#4092semidirect
261. C5#63887semidirect
273. C3#91421semidirect
274. C3#91835semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×Dic62
C2×Dic3×D152
#106239062
C3×#67142
Dic3⋊Dic152
#106238142
#106238112
#918353
#914213
#638875