Finite Group Databasebeta

← #13087521

Subgroups — #13087521

Order 1440.

258 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 62 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×#6821C211yesmaximal, characteristic
2Dic6×D15C211yesmaximal, characteristic, complemented
3#10623912C211yesmaximal, characteristic, complemented
4C3×C5×#216C211yesmaximal, characteristic, complemented
5#10623803C211yesmaximal, characteristic, complemented
6#10623801C211yesmaximal, characteristic, complemented
7#10623795C211yesmaximal, characteristic, complemented
8#91807—31nomaximal
9#91508—31nomaximal
10C32×C5×Q8C2211yescharacteristic
11#67180C2211yescharacteristic
12C3×D60—21no—
13C3×D60C2211yescharacteristic
14C3×C4×D15—21no—
15C3×C4×D15C2211yescharacteristic
16C3×C5×Dic6C2211yescharacteristic
17#67153—21no—
18Dic3×D15—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63943—51nomaximal, Hall
22#6821D311yescharacteristic
23Q8×D15—31no—
24C3×#713—31no—
25D5×Dic6—31no—
26C3×C5×Q16—31no—
27#6718—31no—
28#6718—31no—
29#6718—61no—
30#6717—31no—
31#6706—31no—
32#6705—31no—
33C5×#216D311yescharacteristic, complemented
34#6657—31no—
35#6655—31no—
36#6649—31no—
37C2×C3×D15—21no—
38C2×C3×D15D411yescharacteristic
39C32×C4×C5—21no—
40C32×C4×C5C2311yescharacteristic
41C3⋊Dic15—21no—
42C3×Dic15D411yescharacteristic
43C3×C5×Dic3—21no—
44#4064—91noHall
45C3×#239—51no—
46D3×Dic6—51no—
47#3756—51no—
48C3×#216D511yescharacteristic, complemented
49#3716—51no—
50#3715—51no—
51#3711—51no—
52C3×C5×Q8—21no—
53C3×C5×Q8—31no—
54C3×C5×Q8C2×D311yescharacteristic
55C3×C5×Q8C2×D311yescharacteristic
56D60—21no—
57D60C2×D311yescharacteristic
58C4×D15—21no—
59C4×D15—61no—
60C4×D15C2×D311yescharacteristic
61Dic30—31no—
62Dic30—31no—
63Dic30—61no—
64Dic30—61no—
65C5×Dic6C2×D311yescharacteristic
66C3×D20—31no—
67C3×D20—61no—
68C3×C4×D5—31no—
69C3×C4×D5—61no—
70#1204—61no—
71D5×Dic3—61no—
72C3×C5×C8—31no—
73C3⋊#168—31no—
74C3⋊#168—31no—
75C3⋊#168—61no—
76C5×C3⋊C8C2×D311yescharacteristic
77#951—151no—
78#918—151no—
79C2×C32×C5C2×D411yescharacteristic
80C3×D15—21no—
81C3×D15—41no—
82#713—31no—
83Q8×D5—91no—
84C5×Q16—31no—
85#689—91no—
86#688—91no—
87#677—91no—
88#676—91no—
89C32×Q8C2×D511yescharacteristic
90#634—51no—
91C3×D12—51no—
92C3×D12—101no—
93C3×C4×D3—51no—
94C3×C4×D3—101no—
95C3×Dic6C2×D511yescharacteristic
96#627—101no—
97D3×Dic3—101no—
98C3⋊(C3⋊C8)—51no—
99C3×C3⋊C8C2×D511yescharacteristic
100C2×D15—21no—
101C2×D15#6711yescharacteristic
102C2×C3×D5—31no—
103C2×C3×D5—61no—
104C3×C4×C5—21no—
105C3×C4×C5—21no—
106C3×C4×C5—21no—
107C3×C4×C5—41no—
108C3×C4×C5—61no—
109C3×C4×C5C22×D311yescharacteristic
110C3×C4×C5C22×D311yescharacteristic
111Dic15—61no—
112Dic15—61no—
113Dic15—121no—
114Dic15#6711yescharacteristic
115C3×Dic5—31no—
116C5×Dic3—21no—
117C3×#41—151no—
118#239—51no—
119D3×Q8—151no—
120C2×Dic6—151no—
121C3×Q16—31no—
122#216—151no—
123#216—151no—
124#216—301no—
125#216D1511yescharacteristic, complemented
126#215—151no—
127#214—151no—
128#214—151no—
129#208—151no—
130#204—151no—
131ΓL(1,𝔽25)—151no—
132C32×C5#13711yescharacteristic, complemented, Hall
133C5×Q8—31no—
134C5×Q8D3211yescharacteristic
135D20—31no—
136D20—61no—
137C4×D5—31no—
138C4×D5—61no—
139C4×D5—181no—
140Dic10—91no—
141Dic10—181no—
142C5×C8—31no—
143#168—91no—
144C2×C3×D3—51no—
145C2×C3×D3—101no—
146C32×C4—21no—
147C32×C4C22×D511yescharacteristic
148C3⋊Dic3—101no—
149C3×Dic3—21no—
150C3×Dic3—51no—
151#137—451noSylow
152C2×C3×C5—21no—
153C2×C3×C5D3×D411yescharacteristic
154C2×C3×C5C2×#6711yescharacteristic
155D15—21no—
156D15—41no—
157C3×D5—61no—
158C3×D5—121no—
159C3×Q8—21no—
160C3×Q8—31no—
161C3×Q8D3×D511yescharacteristic
162C3×Q8C2×D1511yescharacteristic
163C3×D4—151no—
164C3×D4—301no—
165C2×C3×C4—151no—
166C2×C3×C4—301no—
167C2×Dic3—301no—
168D12—51no—
169D12—101no—
170C4×D3—51no—
171C4×D3—101no—
172C4×D3—301no—
173Dic6—151no—
174Dic6—151no—
175Dic6—301no—
176Dic6—301no—
177Dic6—301no—
178Dic6C2×D1511yescharacteristic
179C3×C8—31no—
180C3⋊C8—151no—
181C3⋊C8—151no—
182C3⋊C8—301no—
183C3⋊C8C2×D1511yescharacteristic
184C2×D5—31no—
185C2×D5—61no—
186C4×C5—21no—
187C4×C5—61no—
188C4×C5C2×D3211yescharacteristic
189Dic5—31no—
190Dic5—181no—
191C2×C32D4×D511yescharacteristic
192C3×D3—101no—
193C3×D3—201no—
194#41—151no—
195C2×Q8—451no—
196Q16—31no—
197Q16—451no—
198SD16—451no—
199SD16—451no—
200M16—451no—
201C3×C5—21no—
202C3×C5#91811yescharacteristic, complemented
203C3×C5#95111yescharacteristic, complemented
204C22×C3—151no—
205C22×C3—301no—
206C2×D3—51no—
207C2×D3—101no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—41no—
212C3×C4—61no—
213C3×C4—151no—
214C3×C4C2×D3×D511yescharacteristic
215C3×C4C22×D1511yescharacteristic
216Dic3—21no—
217Dic3—51no—
218Dic3—301no—
219Dic3—301no—
220Dic3—601no—
221C2×C5D3×#6711yescharacteristic
222D5—61no—
223D5—121no—
224C32#406411yescharacteristic, complemented, Sylow
225Q8—31no—
226Q8—451no—
227Q8—901no—
228Q8D3×D1511yescharacteristic
229D4—151no—
230D4—301no—
231C2×C4—151no—
232C2×C4—301no—
233C2×C4—901no—
234C8—31no—
235C8—451no—
236C2×C3—21no—
237C2×C3—301no—
238C2×C3—601no—
239C2×C3D5×#6711yescharacteristic
240C2×C3D4×D1511yescharacteristic
241D3—101no—
242D3—201no—
243C5#6394311yescharacteristic, complemented, Sylow
244C22—151no—
245C22—301no—
246C4—21no—
247C4—61no—
248C4—151no—
249C4—901no—
250C4C2×D3×D1511yescharacteristic
251C3—21no—
252C3#9150811yescharacteristic, complemented
253C3#9180711yescharacteristic, complemented
254C2—301no—
255C2—601no—
256C2#67×D1511yescharacteristic
subgroupG/H
centerC2#67×D15
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×Q8C22
socleC2×C32×C5C2×D4
prime radicalC3×#6821C2
hypercenterQ8D3×D15
hypocenterC32×C5#137
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
2. Dic6×D15C2semidirect
3. #10623912C2semidirect
4. C3×C5×#216C2semidirect
5. #10623803C2semidirect
6. #10623801C2semidirect
7. #10623795C2semidirect
33. C5×#216D3semidirect
48. C3×#216D5semidirect
125. #216D15semidirect
132. C32×C5#137semidirect
202. C3×C5#918semidirect
203. C3×C5#951semidirect
224. C32#4064semidirect
243. C5#63943semidirect
252. C3#91508semidirect
253. C3#91807semidirect

Maximal subgroups

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

subgroupindex
C3×#68212
Dic6×D152
#106239122
C3×C5×#2162
#106238032
#106238012
#106237952
#918073
#915083
#639435