Finite Group Databasebeta

← #13087528

Subgroups — #13087528

Order 1440.

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

normal subgroups only

This group has 260 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×Q8×D15C211yesmaximal, characteristic, complemented
2#10624201C211yesmaximal, characteristic
3#10623911C211yesmaximal, characteristic, complemented
4C3×C5×#216C211yesmaximal, characteristic, complemented
5#10623807C211yesmaximal, characteristic, complemented
6#10623804C211yesmaximal, characteristic, complemented
7#10623795C211yesmaximal, characteristic, complemented
8#91807—31nomaximal
9#91515—31nomaximal
10C32×C5×Q8C2211yescharacteristic
11C4⋊(C3⋊D15)C2211yescharacteristic
12C3×C4×D15—21no—
13C3×C4×D15C2211yescharacteristic
14C3×Dic30—21no—
15C3×Dic30C2211yescharacteristic
16C3×C5×Dic6C2211yescharacteristic
17#67151—21no—
18#67149—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63948—51nomaximal, Hall
22#6821—31no—
23Q8×D15D311yescharacteristic, complemented
24C3×Q8×D5—31no—
25#6773—31no—
26C3×C5×Q16—31no—
27#6718—31no—
28#6717—31no—
29#6717—31no—
30#6717—61no—
31#6706—31no—
32#6705—31no—
33C5×#216D311yescharacteristic, complemented
34#6661—31no—
35#6658—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×D3×Q8—51no—
46#3799—51no—
47#3755—51no—
48C3×#216D511yescharacteristic, complemented
49#3718—51no—
50#3717—51no—
51#3711—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—
60C4×D15—21no—
61C4×D15—61no—
62C4×D15C2×D311yescharacteristic
63Dic30—21no—
64Dic30C2×D311yescharacteristic
65C5×Dic6C2×D311yescharacteristic
66C3×C4×D5—31no—
67C3×C4×D5—61no—
68C3×Dic10—31no—
69C3×Dic10—61no—
70#1202—61no—
71#1200—61no—
72C3×C5×C8—31no—
73C3⋊#168—31no—
74C3⋊#168—31no—
75C3⋊#168—61no—
76C5×C3⋊C8C2×D311yescharacteristic
77#942—151no—
78#918—151no—
79C2×C32×C5C2×D411yescharacteristic
80C3⋊D15—41no—
81C3×D15—21no—
82#713—91no—
83Q8×D5—31no—
84C5×Q16—31no—
85#689—91no—
86#688—91no—
87#677—91no—
88#676—91no—
89C32×Q8C2×D511yescharacteristic
90C4⋊(C3⋊D3)—51no—
91C3×C4×D3—51no—
92C3×C4×D3—101no—
93C3×Dic6—51no—
94C3×Dic6—101no—
95C3×Dic6C2×D511yescharacteristic
96#626—101no—
97#624—101no—
98C3⋊(C3⋊C8)—51no—
99C3×C3⋊C8C2×D511yescharacteristic
100C2×D15—61no—
101C2×D15—61no—
102C2×D15—121no—
103C2×D15#6711yescharacteristic
104C2×C3×D5—31no—
105C3×C4×C5—21no—
106C3×C4×C5—21no—
107C3×C4×C5—21no—
108C3×C4×C5—41no—
109C3×C4×C5—61no—
110C3×C4×C5C22×D311yescharacteristic
111C3×C4×C5C22×D311yescharacteristic
112Dic15—21no—
113Dic15#6711yescharacteristic
114C3×Dic5—31no—
115C3×Dic5—61no—
116C5×Dic3—21no—
117C2×C3×Q8—151no—
118#239—151no—
119D3×Q8—51no—
120#235—151no—
121C3×Q16—31no—
122#216—151no—
123#216—151no—
124#216D1511yescharacteristic, complemented
125#215—151no—
126#215—151no—
127#215—151no—
128#215—301no—
129#208—151no—
130#204—151no—
131ΓL(1,𝔽25)—151no—
132C32×C5#13711yescharacteristic, complemented, Hall
133C5×Q8—31no—
134C5×Q8D3211yescharacteristic
135D20—91no—
136D20—181no—
137C4×D5—31no—
138C4×D5—61no—
139C4×D5—181no—
140Dic10—31no—
141Dic10—61no—
142C5×C8—31no—
143#168—91no—
144C2×C3⋊D3—101no—
145C2×C3×D3—51no—
146C32×C4—21no—
147C32×C4C22×D511yescharacteristic
148C3×Dic3—21no—
149C3×Dic3—51no—
150C3×Dic3—101no—
151#137—451noSylow
152C2×C3×C5—21no—
153C2×C3×C5D3×D411yescharacteristic
154C2×C3×C5C2×#6711yescharacteristic
155D15—21no—
156D15—121no—
157D15—121no—
158D15—241no—
159C3×D5—61no—
160C3×Q8—21no—
161C3×Q8—31no—
162C3×Q8—151no—
163C3×Q8—301no—
164C3×Q8D3×D511yescharacteristic
165C3×Q8C2×D1511yescharacteristic
166C2×C3×C4—151no—
167C2×C3×C4—301no—
168#67—301no—
169D12—151no—
170D12—151no—
171D12—301no—
172D12—301no—
173C4×D3—51no—
174C4×D3—101no—
175C4×D3—301no—
176C4×D3—301no—
177Dic6—51no—
178Dic6—101no—
179Dic6C2×D1511yescharacteristic
180C3×C8—31no—
181C3⋊C8—151no—
182C3⋊C8—151no—
183C3⋊C8—301no—
184C3⋊C8C2×D1511yescharacteristic
185C2×D5—31no—
186C2×D5—181no—
187C4×C5—21no—
188C4×C5—61no—
189C4×C5C2×D3211yescharacteristic
190Dic5—31no—
191Dic5—61no—
192C2×C32D4×D511yescharacteristic
193C3⋊D3—201no—
194C3×D3—101no—
195#41—451no—
196C2×Q8—151no—
197Q16—31no—
198Q16—451no—
199SD16—451no—
200SD16—451no—
201M16—451no—
202C3×C5—21no—
203C3×C5#91811yescharacteristic, complemented
204C3×C5#94211yescharacteristic, complemented
205C22×C3—151no—
206C2×D3—51no—
207C2×D3—301no—
208C2×D3—301no—
209C2×D3—601no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—41no—
214C3×C4—61no—
215C3×C4—151no—
216C3×C4—301no—
217C3×C4C2×D3×D511yescharacteristic
218C3×C4C22×D1511yescharacteristic
219Dic3—21no—
220Dic3—51no—
221Dic3—101no—
222C2×C5D3×#6711yescharacteristic
223D5—61no—
224D5—361no—
225C32#406411yescharacteristic, complemented, Sylow
226Q8—31no—
227Q8—151no—
228Q8—301no—
229Q8D3×D1511yescharacteristic
230D4—451no—
231D4—901no—
232C2×C4—151no—
233C2×C4—301no—
234C2×C4—901no—
235C8—31no—
236C8—451no—
237C2×C3—21no—
238C2×C3—301no—
239C2×C3D5×#6711yescharacteristic
240C2×C3D4×D1511yescharacteristic
241D3—101no—
242D3—601no—
243D3—601no—
244D3—1201no—
245C5#6394811yescharacteristic, complemented, Sylow
246C22—151no—
247C22—901no—
248C4—21no—
249C4—61no—
250C4—151no—
251C4—301no—
252C4C2×D3×D1511yescharacteristic
253C3—21no—
254C3#9151511yescharacteristic, complemented
255C3#9180711yescharacteristic, complemented
256C2—301no—
257C2—1801no—
258C2#67×D1511yescharacteristic
subgroupG/H
centerC2#67×D15
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×Q8C22
socleC2×C32×C5C2×D4
prime radical#10624201C2
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
1. C3×Q8×D15C2semidirect
3. #10623911C2semidirect
4. C3×C5×#216C2semidirect
5. #10623807C2semidirect
6. #10623804C2semidirect
7. #10623795C2semidirect
23. Q8×D15D3semidirect
33. C5×#216D3semidirect
48. C3×#216D5semidirect
124. #216D15semidirect
132. C32×C5#137semidirect
203. C3×C5#918semidirect
204. C3×C5#942semidirect
225. C32#4064semidirect
245. C5#63948semidirect
254. C3#91515semidirect
255. C3#91807semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×D152
#106242012
#106239112
C3×C5×#2162
#106238072
#106238042
#106237952
#918073
#915153
#639485