Finite Group Databasebeta

← #13087525

Subgroups — #13087525

Order 1440.

280 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 60 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
1#10624287C211yesmaximal, characteristic
2#10624198C211yesmaximal, characteristic, complemented
3C3×#6718C211yesmaximal, characteristic, complemented
4C3×C5×#216C211yesmaximal, characteristic, complemented
5#10623806C211yesmaximal, characteristic, complemented
6#10623804C211yesmaximal, characteristic, complemented
7#10623797C211yesmaximal, characteristic, complemented
8#91807—31nomaximal
9#91512—31nomaximal
10C32×C5×Q8C2211yescharacteristic
11C4⋊(C3⋊D15)—21no—
12C4⋊(C3⋊D15)C2211yescharacteristic
13C4×C3⋊D15—21no—
14C4×C3⋊D15C2211yescharacteristic
15C3×Dic30C2211yescharacteristic
16C3×C5×Dic6C2211yescharacteristic
17#67153—21no—
18#67149—21no—
19C3×C3⋊#168C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63945—51nomaximal, Hall
22#6821—31no—
23#6821—31no—
24#6821—61no—
25Q8×D15—31no—
26D15⋊Q8—31no—
27C3×C5×Q16—31no—
28#6718D311yescharacteristic, complemented
29#6717—31no—
30#6706—31no—
31#6705—31no—
32C5×#216D311yescharacteristic, complemented
33C3×#689—31no—
34#6660—31no—
35#6658—31no—
36#6651—31no—
37C2×C3⋊D15—21no—
38C2×C3⋊D15D411yescharacteristic
39C32×C4×C5—21no—
40C32×C4×C5C2311yescharacteristic
41C3⋊Dic15D411yescharacteristic
42C3×Dic15—21no—
43C3×C5×Dic3—21no—
44#4064—91noHall
45#3832—51no—
46#3797—51no—
47C3×#216—51no—
48C3×#216D511yescharacteristic, complemented
49#3717—51no—
50#3717—51no—
51#3712—51no—
52C3×C5×Q8—21no—
53C3×C5×Q8—31no—
54C3×C5×Q8C2×D311yescharacteristic
55C3×C5×Q8C2×D311yescharacteristic
56D60—31no—
57D60—31no—
58D60—61no—
59D60—61no—
60D60—61no—
61D60—62no—
62C4×D15—31no—
63C4×D15—31no—
64C4×D15—61no—
65C4×D15—61no—
66C4×D15—61no—
67C4×D15—61no—
68C4×D15—62no—
69Dic30—61no—
70Dic30C2×D311yescharacteristic
71C5×Dic6C2×D311yescharacteristic
72C3×Dic10—31no—
73#1204—61no—
74#1200—61no—
75C3×C5×C8—31no—
76C3⋊#168C2×D311yescharacteristic
77C3×#168—31no—
78C5×C3⋊C8C2×D311yescharacteristic
79#918—151no—
80#918—151no—
81C2×C32×C5C2×D411yescharacteristic
82C3⋊D15—21no—
83C3⋊D15—41no—
84#713—91no—
85Q8×D5—91no—
86C5×Q16—31no—
87#689—31no—
88#688—91no—
89#677—91no—
90#676—91no—
91C32×Q8C2×D511yescharacteristic
92C4⋊(C3⋊D3)—51no—
93C4⋊(C3⋊D3)—101no—
94C4×C3⋊D3—51no—
95C4×C3⋊D3—101no—
96C3×Dic6—51no—
97C3×Dic6C2×D511yescharacteristic
98#627—101no—
99#624—101no—
100C3×C3⋊C8—51no—
101C3×C3⋊C8C2×D511yescharacteristic
102C2×D15—31no—
103C2×D15—31no—
104C2×D15—61no—
105C2×D15—62no—
106C2×D15—61no—
107C2×D15—61no—
108C3×C4×C5—22no—
109C3×C4×C5—21no—
110C3×C4×C5—21no—
111C3×C4×C5—21no—
112C3×C4×C5—61no—
113C3×C4×C5C22×D311yescharacteristic
114C3×C4×C5C22×D311yescharacteristic
115Dic15—21no—
116Dic15—31no—
117Dic15—31no—
118Dic15—61no—
119C3×Dic5—61no—
120C5×Dic3—21no—
121#239—151no—
122#239—151no—
123#239—301no—
124D3×Q8—151no—
125D3×Q8—151no—
126C3×Q16—31no—
127C3×Q16—151no—
128#216—51no—
129#216D1511yescharacteristic, complemented
130#215—151no—
131#215—151no—
132#204—151no—
133#204—151no—
134ΓL(1,𝔽25)—151no—
135ΓL(1,𝔽25)—151no—
136C32×C5#13711yescharacteristic, complemented, Hall
137C5×Q8—31no—
138C5×Q8D3211yescharacteristic
139D20—91no—
140D20—181no—
141C4×D5—91no—
142C4×D5—181no—
143C4×D5—181no—
144Dic10—31no—
145Dic10—181no—
146C5×C8—31no—
147#168—31no—
148C2×C3⋊D3—51no—
149C2×C3⋊D3—101no—
150C32×C4—21no—
151C32×C4C22×D511yescharacteristic
152C3⋊Dic3—51no—
153C3×Dic3—21no—
154C3×Dic3—101no—
155#137—451noSylow
156C2×C3×C5—21no—
157C2×C3×C5D3×D411yescharacteristic
158C2×C3×C5D3×D411yescharacteristic
159D15—61no—
160D15—61no—
161D15—121no—
162D15—122no—
163D15—121no—
164D15—121no—
165C3×Q8—21no—
166C3×Q8—31no—
167C3×Q8—151no—
168C3×Q8D3×D511yescharacteristic
169C3×Q8C2×D1511yescharacteristic
170D12—151no—
171D12—151no—
172D12—301no—
173D12—301no—
174D12—301no—
175D12—302no—
176C4×D3—151no—
177C4×D3—151no—
178C4×D3—301no—
179C4×D3—301no—
180C4×D3—301no—
181C4×D3—301no—
182C4×D3—301no—
183C4×D3—302no—
184Dic6—51no—
185Dic6—301no—
186Dic6—301no—
187Dic6C2×D1511yescharacteristic
188C3×C8—31no—
189C3×C8—151no—
190C3⋊C8—51no—
191C3⋊C8C2×D1511yescharacteristic
192C2×D5—91no—
193C2×D5—181no—
194C4×C5—21no—
195C4×C5—61no—
196C4×C5C2×D3211yescharacteristic
197Dic5—61no—
198Dic5—91no—
199C2×C32D4×D511yescharacteristic
200C3⋊D3—101no—
201C3⋊D3—201no—
202#41—451no—
203C2×Q8—451no—
204Q16—31no—
205Q16—151no—
206SD16—451no—
207SD16—451no—
208M16—451no—
209C3×C5—21no—
210C3×C5#91811yescharacteristic, complemented
211C3×C5#91811yescharacteristic, complemented
212C2×D3—151no—
213C2×D3—151no—
214C2×D3—301no—
215C2×D3—302no—
216C2×D3—301no—
217C2×D3—301no—
218C3×C4—22no—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—61no—
223C3×C4—301no—
224C3×C4C2×D3×D511yescharacteristic
225C3×C4C22×D1511yescharacteristic
226Dic3—21no—
227Dic3—101no—
228Dic3—151no—
229Dic3—151no—
230Dic3—301no—
231C2×C5#381111yescharacteristic
232D5—181no—
233D5—361no—
234C32#406411yescharacteristic, complemented, Sylow
235Q8—31no—
236Q8—151no—
237Q8—901no—
238Q8D3×D1511yescharacteristic
239D4—451no—
240D4—901no—
241C2×C4—451no—
242C2×C4—901no—
243C2×C4—901no—
244C8—31no—
245C8—151no—
246C2×C3—21no—
247C2×C3#679011yescharacteristic
248C2×C3D4×D1511yescharacteristic
249D3—301no—
250D3—301no—
251D3—601no—
252D3—602no—
253D3—601no—
254D3—601no—
255C5#6394511yescharacteristic, complemented, Sylow
256C22—451no—
257C22—901no—
258C4—21no—
259C4—61no—
260C4—301no—
261C4—451no—
262C4C2×D3×D1511yescharacteristic
263C3—21no—
264C3#9151211yescharacteristic, complemented
265C3#9180711yescharacteristic, complemented
266C2—901no—
267C2—1801no—
268C2#1062421811yescharacteristic
subgroupG/H
centerC2#10624218
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×Q8C22
socleC2×C32×C5C2×D4
prime radical#10624287C2
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. #10624198C2semidirect
3. C3×#6718C2semidirect
4. C3×C5×#216C2semidirect
5. #10623806C2semidirect
6. #10623804C2semidirect
7. #10623797C2semidirect
28. #6718D3semidirect
32. C5×#216D3semidirect
48. C3×#216D5semidirect
129. #216D15semidirect
136. C32×C5#137semidirect
210. C3×C5#918semidirect
211. C3×C5#918semidirect
234. C32#4064semidirect
255. C5#63945semidirect
264. C3#91512semidirect
265. C3#91807semidirect

Maximal subgroups

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

subgroupindex
#106242872
#106241982
C3×#67182
C3×C5×#2162
#106238062
#106238042
#106237972
#918073
#915123
#639455