Finite Group Databasebeta

← #13085732

Subgroups — #13085732

Order 1440.

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

normal subgroups only

This group has 268 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#10624200C211yesmaximal, characteristic, complemented
2C3×#6766C211yesmaximal, characteristic, complemented
3#10623817C211yesmaximal, characteristic, complemented
4#10623809C211yesmaximal, characteristic, complemented
5C5×#3715C211yesmaximal, characteristic, complemented
6#10623762C211yesmaximal, characteristic, complemented
7C3×#6652C211yesmaximal, characteristic, complemented
8#91514—31nomaximal
9#91497—31nomaximal
10#67180C2211yescharacteristic
11C3×C4×D15C2211yescharacteristic
12C3×C5×D12C2211yescharacteristic
13C3×C5×Dic6C2211yescharacteristic
14C32×D20C2211yescharacteristic
15#67150—21no—
16D3×Dic15—21no—
17C3×#1202—21no—
18C3×D5×Dic3—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×C3⋊#168C2211yescharacteristic
21#63933—51nomaximal, Hall
22#6819—31no—
23C3×#713—31no—
24#6772—31no—
25#6766D311yescharacteristic, complemented
26#6718—31no—
27C5×#215—31no—
28C5×#214—31no—
29C3×#686—31no—
30#6663—31no—
31#6657—31no—
32#6657—31no—
33#6657—61no—
34#6652D311yescharacteristic, complemented
35#6648—31no—
36D5×C3⋊C8—31no—
37C2×C3×D15D411yescharacteristic
38C2×C3×C5×D3—21no—
39C2×C32×D5—21no—
40C32×C4×C5C2311yescharacteristic
41C3⋊Dic15—21no—
42C3×Dic15D411yescharacteristic
43C3×C5×Dic3—21no—
44#4062—91noHall
45C3×#237—51no—
46#3795—51no—
47#3754—51no—
48C3×#213—51no—
49#3719—51no—
50#3715D511yescharacteristic, complemented
51D3×C3⋊C8—51no—
52C3×C5×Q8—31no—
53C3×C5×D4—31no—
54#1220—61no—
55C2×Dic15—61no—
56C4×D15C2×D311yescharacteristic
57Dic30—31no—
58Dic30—31no—
59Dic30—61no—
60C5×D12C2×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#1201—61no—
70D3×Dic5—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#912—151no—
79C2×C32×C5C2×D411yescharacteristic
80C3×D15—21no—
81C3×C5×D3—41no—
82C32×D5—41no—
83#713—31no—
84#711—91no—
85C5×SD16—91no—
86#689—91no—
87#686—31no—
88#677—91no—
89C8×D5—91no—
90C32×D4—51no—
91#634—51no—
92C3×#67—101no—
93C2×C3×Dic3—101no—
94C3×D12C2×D511yescharacteristic
95C3×C4×D3—51no—
96C3×Dic6C2×D511yescharacteristic
97#625—101no—
98D3×Dic3—101no—
99C3⋊(C3⋊C8)C2×D511yescharacteristic
100C3×C3⋊C8—51no—
101C22×C3×C5—61no—
102C2×D15#6711yescharacteristic
103C2×C5×D3—21no—
104C2×C3×D5—21no—
105C2×C3×D5—21no—
106C2×C3×D5—31no—
107C2×C3×D5—41no—
108C3×C4×C5—21no—
109C3×C4×C5—61no—
110C3×C4×C5C22×D311yescharacteristic
111C3×C4×C5C22×D311yescharacteristic
112Dic15—61no—
113Dic15—61no—
114Dic15—121no—
115Dic15#6711yescharacteristic
116C3×Dic5—31no—
117C5×Dic3—21no—
118C3×#41—151no—
119#237—51no—
120#237—151no—
121#235—151no—
122C3×D8—151no—
123#216—151no—
124#215—31no—
125#214—31no—
126#214—151no—
127#214—151no—
128#214—301no—
129#213—51no—
130C2×C3⋊C8—151no—
131Dic12—151no—
132D3×C8—151no—
133C32×C5#13511yescharacteristic, complemented, Hall
134C5×Q8—31no—
135C5×D4—31no—
136C22⋊D5—181no—
137C2×Dic5—181no—
138D20—61no—
139D20D3211yescharacteristic
140C4×D5—31no—
141C4×D5—61no—
142Dic10—91no—
143C5×C8—91no—
144#168—31no—
145C22×C32—101no—
146C2×C3×D3—21no—
147C2×C3×D3—51no—
148C32×C4C22×D511yescharacteristic
149C3⋊Dic3—101no—
150C3×Dic3—21no—
151C3×Dic3—51no—
152#135—451noSylow
153C2×C3×C5—21no—
154C2×C3×C5—121no—
155C2×C3×C5D3×D411yescharacteristic
156C2×C3×C5C2×#6711yescharacteristic
157D15—21no—
158C3×D5—41no—
159C3×D5—41no—
160C3×D5—61no—
161C3×D5—81no—
162C5×D3—41no—
163C3×Q8—31no—
164C3×D4—31no—
165C3×D4—51no—
166C3×D4—51no—
167C3×D4—101no—
168C3×D4—301no—
169C2×C3×C4—151no—
170C2×C3×C4—301no—
171#67—101no—
172#67—301no—
173#67—301no—
174C2×Dic3—101no—
175C2×Dic3—301no—
176D12D3×D511yescharacteristic
177C4×D3—51no—
178C4×D3—301no—
179Dic6—151no—
180Dic6—151no—
181Dic6—301no—
182Dic6D3×D511yescharacteristic
183C3×C8—151no—
184C3⋊C8—31no—
185C3⋊C8—31no—
186C3⋊C8—51no—
187C3⋊C8—61no—
188C22×C5—61no—
189C2×D5—21no—
190C2×D5—31no—
191C4×C5—61no—
192C4×C5C2×D3211yescharacteristic
193Dic5—31no—
194Dic5—181no—
195C2×C32—201no—
196C2×C32D4×D511yescharacteristic
197C3×D3—41no—
198C3×D3—101no—
199#41—151no—
200#41—451no—
201Q16—451no—
202SD16—91no—
203SD16—451no—
204D8—151no—
205C2×C8—451no—
206C3×C5—21no—
207C3×C5#91211yescharacteristic, complemented
208C3×C5#95011yescharacteristic, complemented
209C22×C3—61no—
210C22×C3—101no—
211C22×C3—101no—
212C22×C3—151no—
213C22×C3—201no—
214C2×D3—21no—
215C2×D3—51no—
216C3×C4—21no—
217C3×C4—61no—
218C3×C4—151no—
219C3×C4C2×D3×D511yescharacteristic
220C3×C4C2×D3×D511yescharacteristic
221Dic3—21no—
222Dic3—51no—
223Dic3—301no—
224Dic3—301no—
225Dic3—601no—
226C2×C5—121no—
227C2×C5D3×#6711yescharacteristic
228D5—41no—
229D5—61no—
230C32#406211yescharacteristic, complemented, Sylow
231Q8—31no—
232Q8—451no—
233D4—31no—
234D4—51no—
235D4—301no—
236D4—901no—
237C2×C4—151no—
238C2×C4—301no—
239C2×C4—901no—
240C8—91no—
241C8—151no—
242C2×C3—21no—
243C2×C3—121no—
244C2×C3—201no—
245C2×C3—201no—
246C2×C3—301no—
247C2×C3—401no—
248C2×C3D5×#6711yescharacteristic
249C2×C3#679011yescharacteristic
250D3—41no—
251D3—101no—
252C5#6393311yescharacteristic, complemented, Sylow
253C22—61no—
254C22—101no—
255C22—151no—
256C4—61no—
257C4—151no—
258C4—901no—
259C4D32×D511yescharacteristic
260C3—21no—
261C3#9149711yescharacteristic, complemented
262C3#9151411yescharacteristic, complemented
263C2—121no—
264C2—201no—
265C2—301no—
266C2#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. #10624200C2semidirect
2. C3×#6766C2semidirect
3. #10623817C2semidirect
4. #10623809C2semidirect
5. C5×#3715C2semidirect
6. #10623762C2semidirect
7. C3×#6652C2semidirect
25. #6766D3semidirect
34. #6652D3semidirect
50. #3715D5semidirect
133. C32×C5#135semidirect
207. C3×C5#912semidirect
208. C3×C5#950semidirect
230. C32#4062semidirect
252. C5#63933semidirect
261. C3#91497semidirect
262. C3#91514semidirect

Maximal subgroups

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

subgroupindex
#106242002
C3×#67662
#106238172
#106238092
C5×#37152
#106237622
C3×#66522
#915143
#914973
#639335