Finite Group Databasebeta

← #13085733

Subgroups — #13085733

Order 1440.

270 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 58 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#10624201C211yesmaximal, characteristic, complemented
2C3×#6766C211yesmaximal, characteristic, complemented
3#10623817C211yesmaximal, characteristic, complemented
4#10623806C211yesmaximal, characteristic, complemented
5C5×#3718C211yesmaximal, characteristic, complemented
6#10623758C211yesmaximal, characteristic, complemented
7C3×#6655C211yesmaximal, characteristic, complemented
8#91521—31nomaximal
9#91516—31nomaximal
10C4⋊(C3⋊D15)C2211yescharacteristic
11C3×C4×D15C2211yescharacteristic
12C3×C5×Dic6C2211yescharacteristic
13C3×C5×Dic6C2211yescharacteristic
14C32×D20C2211yescharacteristic
15#67151—21no—
16#67149—21no—
17C3×#1202—21no—
18C3×D5×Dic3—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×C3⋊#168C2211yescharacteristic
21#63935—51nomaximal, Hall
22#6821—31no—
23C3×#713—31no—
24#6773—31no—
25#6766D311yescharacteristic, complemented
26#6717—31no—
27C5×#216—31no—
28C5×#216—31no—
29C3×#688—31no—
30#6660—31no—
31#6655D311yescharacteristic, complemented
32#6653—31no—
33#6653—31no—
34#6653—61no—
35#6648—31no—
36D5×C3⋊C8—31no—
37C2×C3⋊D15—21no—
38C2×C3×D15D411yescharacteristic
39C2×C32×D5—21no—
40C32×C4×C5C2311yescharacteristic
41C3×Dic15D411yescharacteristic
42C3×C5×Dic3—21no—
43C3×C5×Dic3—21no—
44#4065—91noHall
45C3×#237—51no—
46#3799—51no—
47#3753—51no—
48C3×#214—51no—
49#3718D511yescharacteristic, complemented
50#3717—51no—
51D3×C3⋊C8—51no—
52C3×C5×Q8—31no—
53C3×C5×Q8—31no—
54D60—31no—
55D60—31no—
56D60—61no—
57D60—61no—
58C4×D15—61no—
59C4×D15C2×D311yescharacteristic
60C5×Dic6C2×D311yescharacteristic
61C5×Dic6C2×D311yescharacteristic
62C3×D20—21no—
63C3×D20—61no—
64C3×D20C2×D311yescharacteristic
65C3×D20C2×D311yescharacteristic
66C3×C4×D5—31no—
67C3×C4×D5—61no—
68#1202—21no—
69#1202—61no—
70#1200—61no—
71D5×Dic3—21no—
72C3⋊#168C2×D311yescharacteristic
73C3×#168—31no—
74C5×C3⋊C8—31no—
75C5×C3⋊C8—31no—
76C5×C3⋊C8—61no—
77#950—151no—
78#916—151no—
79C2×C32×C5C2×D411yescharacteristic
80C3⋊D15—41no—
81C3×D15—21no—
82C32×D5—41no—
83#713—31no—
84#713—91no—
85C5×Q16—91no—
86#688—31no—
87#688—91no—
88D40—91no—
89C8×D5—91no—
90C32×D4—51no—
91C4⋊(C3⋊D3)—51no—
92C3×#67—101no—
93C2×C3×Dic3—101no—
94C3×C4×D3—51no—
95C3×Dic6C2×D511yescharacteristic
96C3×Dic6C2×D511yescharacteristic
97#626—101no—
98#624—101no—
99C3⋊(C3⋊C8)C2×D511yescharacteristic
100C3×C3⋊C8—51no—
101C2×D15—61no—
102C2×D15—61no—
103C2×D15—121no—
104C2×D15#6711yescharacteristic
105C2×C3×D5—21no—
106C2×C3×D5—21no—
107C2×C3×D5—31no—
108C2×C3×D5—41no—
109C3×C4×C5—21no—
110C3×C4×C5—61no—
111C3×C4×C5—61no—
112C3×C4×C5C22×D311yescharacteristic
113C3×C4×C5C22×D311yescharacteristic
114Dic15#6711yescharacteristic
115C3×Dic5—31no—
116C5×Dic3—21no—
117C5×Dic3—21no—
118C3×#41—151no—
119#239—151no—
120#237—51no—
121#235—151no—
122C3×SD16—151no—
123#216—31no—
124#216—31no—
125#215—151no—
126#214—51no—
127#213—151no—
128#213—151no—
129#213—301no—
130C2×C3⋊C8—151no—
131#204—151no—
132D3×C8—151no—
133C32×C5#13511yescharacteristic, complemented, Hall
134C5×Q8—31no—
135C5×Q8—31no—
136D20—61no—
137D20—91no—
138D20—181no—
139D20D3211yescharacteristic
140C4×D5—31no—
141C4×D5—61no—
142C4×D5—181no—
143C5×C8—91no—
144#168—31no—
145C22×C32—101no—
146C2×C3⋊D3—101no—
147C2×C3×D3—51no—
148C32×C4C22×D511yescharacteristic
149C3×Dic3—21no—
150C3×Dic3—21no—
151C3×Dic3—51no—
152#135—451noSylow
153C2×C3×C5—21no—
154C2×C3×C5D3×D411yescharacteristic
155C2×C3×C5C2×#6711yescharacteristic
156D15—21no—
157D15—121no—
158D15—121no—
159D15—241no—
160C3×D5—41no—
161C3×D5—41no—
162C3×D5—61no—
163C3×D5—81no—
164C3×Q8—31no—
165C3×Q8—31no—
166C3×D4—51no—
167C3×D4—51no—
168C3×D4—101no—
169C3×D4—301no—
170C2×C3×C4—151no—
171C2×C3×C4—301no—
172#67—101no—
173#67—301no—
174C2×Dic3—101no—
175D12—151no—
176D12—151no—
177D12—301no—
178D12—301no—
179C4×D3—51no—
180C4×D3—301no—
181C4×D3—301no—
182Dic6D3×D511yescharacteristic
183Dic6D3×D511yescharacteristic
184C3×C8—151no—
185C3⋊C8—31no—
186C3⋊C8—31no—
187C3⋊C8—51no—
188C3⋊C8—61no—
189C2×D5—21no—
190C2×D5—31no—
191C2×D5—181no—
192C4×C5—61no—
193C4×C5—61no—
194C4×C5C2×D3211yescharacteristic
195Dic5—31no—
196C2×C32—201no—
197C2×C32D4×D511yescharacteristic
198C3⋊D3—201no—
199C3×D3—101no—
200#41—151no—
201#41—451no—
202Q16—91no—
203SD16—151no—
204SD16—451no—
205D8—451no—
206C2×C8—451no—
207C3×C5—21no—
208C3×C5#91611yescharacteristic, complemented
209C3×C5#95011yescharacteristic, complemented
210C22×C3—101no—
211C22×C3—101no—
212C22×C3—151no—
213C22×C3—201no—
214C2×D3—51no—
215C2×D3—301no—
216C2×D3—301no—
217C2×D3—601no—
218C3×C4—21no—
219C3×C4—61no—
220C3×C4—61no—
221C3×C4—151no—
222C3×C4C2×D3×D511yescharacteristic
223C3×C4C2×D3×D511yescharacteristic
224Dic3—21no—
225Dic3—21no—
226Dic3—51no—
227C2×C5D3×#6711yescharacteristic
228D5—41no—
229D5—61no—
230D5—361no—
231C32#406511yescharacteristic, complemented, Sylow
232Q8—31no—
233Q8—31no—
234D4—51no—
235D4—301no—
236D4—451no—
237D4—901no—
238C2×C4—151no—
239C2×C4—301no—
240C2×C4—901no—
241C8—91no—
242C8—151no—
243C2×C3—21no—
244C2×C3—201no—
245C2×C3—201no—
246C2×C3—301no—
247C2×C3—401no—
248C2×C3D5×#6711yescharacteristic
249C2×C3#679011yescharacteristic
250D3—101no—
251D3—601no—
252D3—601no—
253D3—1201no—
254C5#6393511yescharacteristic, complemented, Sylow
255C22—101no—
256C22—151no—
257C22—901no—
258C4—61no—
259C4—61no—
260C4—151no—
261C4D32×D511yescharacteristic
262C3—21no—
263C3#9151611yescharacteristic, complemented
264C3#9152111yescharacteristic, complemented
265C2—201no—
266C2—301no—
267C2—1801no—
268C2#1062402711yescharacteristic
subgroupG/H
centerC2#10624027
commutator, Fitting, cosocleC32×C4×C5C23
Frattini, hypercenterC4D32×D5
socleC2×C32×C5C2×D4
hypocenterC32×C5#135
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624201C2semidirect
2. C3×#6766C2semidirect
3. #10623817C2semidirect
4. #10623806C2semidirect
5. C5×#3718C2semidirect
6. #10623758C2semidirect
7. C3×#6655C2semidirect
25. #6766D3semidirect
31. #6655D3semidirect
49. #3718D5semidirect
133. C32×C5#135semidirect
208. C3×C5#916semidirect
209. C3×C5#950semidirect
231. C32#4065semidirect
254. C5#63935semidirect
263. C3#91516semidirect
264. C3#91521semidirect

Maximal subgroups

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

subgroupindex
#106242012
C3×#67662
#106238172
#106238062
C5×#37182
#106237582
C3×#66552
#915213
#915163
#639355