Finite Group Databasebeta

← #13085737

Subgroups — #13085737

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#10624221C211yesmaximal, characteristic, complemented
2C3×#6766C211yesmaximal, characteristic, complemented
3#10623801C211yesmaximal, characteristic, complemented
4C3⋊C8×D15C211yesmaximal, characteristic, complemented
5C5×#3719C211yesmaximal, characteristic, complemented
6#10623760C211yesmaximal, characteristic, complemented
7C3×#6653C211yesmaximal, characteristic, complemented
8#91516—31nomaximal
9#91277—31nomaximal
10C5×#634C2211yescharacteristic
11C3×D60C2211yescharacteristic
12C3×C4×D15C2211yescharacteristic
13C3×C5×Dic6C2211yescharacteristic
14C32×D20C2211yescharacteristic
15#67157—21no—
16#67154—21no—
17C3×#1202—21no—
18C3×D5×Dic3—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63933—51nomaximal, Hall
22C3×#713—31no—
23#6773—31no—
24#6766—31no—
25#6766D311yescharacteristic, complemented
26#6717—31no—
27C8×D15—31no—
28C5×#216—31no—
29C5×Dic12—31no—
30C3×D40—31no—
31#6655—31no—
32#6655—31no—
33#6655—31no—
34#6655—61no—
35#6653D311yescharacteristic, complemented
36D5×C3⋊C8—31no—
37C2×C3×D15—21no—
38C2×C3×D15D411yescharacteristic
39C2×C32×D5—21no—
40C32×C4×C5C2311yescharacteristic
41C5×C3⋊Dic3—21no—
42C3×Dic15D411yescharacteristic
43C3×C5×Dic3—21no—
44#4065—91noHall
45C3×#237—51no—
46#3795—51no—
47#3754—51no—
48C3×#213—51no—
49#3719D511yescharacteristic, complemented
50#3715—51no—
51D3×C3⋊C8—51no—
52C3×C5×Q8—31no—
53D60C2×D311yescharacteristic
54C4×D15C2×D311yescharacteristic
55C5×Dic6—31no—
56C5×Dic6—31no—
57C5×Dic6—61no—
58C5×Dic6C2×D311yescharacteristic
59C3×D20—21no—
60C3×D20—31no—
61C3×D20—61no—
62C3×D20C2×D311yescharacteristic
63C3×D20C2×D311yescharacteristic
64C3×C4×D5—31no—
65C3×C4×D5—61no—
66#1202—21no—
67#1202—61no—
68#1202—61no—
69#1200—61no—
70D5×Dic3—21no—
71D5×Dic3—61no—
72C3×C5×C8—31no—
73C3⋊#168—31no—
74C3⋊#168—31no—
75C3⋊#168—61no—
76C5×C3⋊C8C2×D311yescharacteristic
77#950—151no—
78#912—151no—
79C2×C32×C5C2×D411yescharacteristic
80C3×D15—21no—
81C3×D15—41no—
82C32×D5—41no—
83#713—31no—
84#713—91no—
85C5×Q16—91no—
86#688—91no—
87#688—91no—
88D40—31no—
89C8×D5—91no—
90C32×D4—51no—
91#634C2×D511yescharacteristic
92C3×#67—101no—
93C2×C3×Dic3—101no—
94C3×D12—51no—
95C3×C4×D3—51no—
96C3×Dic6C2×D511yescharacteristic
97#625—101no—
98D3×Dic3—101no—
99C3⋊(C3⋊C8)—51no—
100C3×C3⋊C8C2×D511yescharacteristic
101C2×D15—21no—
102C2×D15#6711yescharacteristic
103C2×C3×D5—21no—
104C2×C3×D5—21no—
105C2×C3×D5—31no—
106C2×C3×D5—41no—
107C2×C3×D5—61no—
108C3×C4×C5—21no—
109C3×C4×C5—61no—
110C3×C4×C5C22×D311yescharacteristic
111C3×C4×C5C22×D311yescharacteristic
112Dic15#6711yescharacteristic
113C3×Dic5—31no—
114C5×Dic3—21no—
115C5×Dic3—61no—
116C5×Dic3—61no—
117C5×Dic3—121no—
118C3×#41—151no—
119#237—51no—
120#237—151no—
121#235—151no—
122C3×D8—151no—
123#216—31no—
124#215—151no—
125#214—151no—
126#214—151no—
127#214—151no—
128#214—301no—
129#213—51no—
130C2×C3⋊C8—151no—
131Dic12—31no—
132D3×C8—151no—
133C32×C5#13511yescharacteristic, complemented, Hall
134C5×Q8—31no—
135C5×Q8—91no—
136D20—31no—
137D20—61no—
138D20—181no—
139D20D3211yescharacteristic
140C4×D5—31no—
141C4×D5—61no—
142C4×D5—181no—
143C5×C8—31no—
144#168—91no—
145C22×C32—101no—
146C2×C3×D3—51no—
147C2×C3×D3—101no—
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—41no—
158C3×D5—41no—
159C3×D5—41no—
160C3×D5—61no—
161C3×D5—81no—
162C3×D5—121no—
163C3×Q8—31no—
164C3×D4—51no—
165C3×D4—51no—
166C3×D4—101no—
167C3×D4—151no—
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—
176D12—51no—
177C4×D3—51no—
178C4×D3—301no—
179Dic6—31no—
180Dic6—31no—
181Dic6—61no—
182Dic6D3×D511yescharacteristic
183C3×C8—31no—
184C3⋊C8—151no—
185C3⋊C8—151no—
186C3⋊C8—301no—
187C3⋊C8D3×D511yescharacteristic
188C2×D5—21no—
189C2×D5—31no—
190C2×D5—61no—
191C4×C5—61no—
192C4×C5—181no—
193C4×C5C2×D3211yescharacteristic
194Dic5—31no—
195C2×C32—201no—
196C2×C32D4×D511yescharacteristic
197C3×D3—101no—
198C3×D3—201no—
199#41—151no—
200#41—451no—
201Q16—91no—
202SD16—451no—
203SD16—451no—
204D8—151no—
205C2×C8—451no—
206C3×C5—21no—
207C3×C5#91211yescharacteristic, complemented
208C3×C5#95011yescharacteristic, complemented
209C22×C3—101no—
210C22×C3—101no—
211C22×C3—151no—
212C22×C3—201no—
213C22×C3—301no—
214C2×D3—51no—
215C2×D3—101no—
216C3×C4—21no—
217C3×C4—61no—
218C3×C4—151no—
219C3×C4C2×D3×D511yescharacteristic
220C3×C4C2×D3×D511yescharacteristic
221Dic3—21no—
222Dic3—51no—
223Dic3—61no—
224Dic3—61no—
225Dic3—121no—
226C2×C5D3×#6711yescharacteristic
227D5—41no—
228D5—61no—
229D5—121no—
230C32#406511yescharacteristic, complemented, Sylow
231Q8—31no—
232Q8—91no—
233D4—51no—
234D4—151no—
235D4—301no—
236D4—901no—
237C2×C4—151no—
238C2×C4—301no—
239C2×C4—901no—
240C8—31no—
241C8—451no—
242C2×C3—21no—
243C2×C3—201no—
244C2×C3—201no—
245C2×C3—301no—
246C2×C3—401no—
247C2×C3—601no—
248C2×C3#677711yescharacteristic
249C2×C3D5×#6711yescharacteristic
250D3—101no—
251D3—201no—
252C5#6393311yescharacteristic, complemented, Sylow
253C22—101no—
254C22—151no—
255C22—301no—
256C4—61no—
257C4—151no—
258C4—181no—
259C4D32×D511yescharacteristic
260C3—21no—
261C3#9127711yescharacteristic, complemented
262C3#9151611yescharacteristic, complemented
263C2—201no—
264C2—301no—
265C2—601no—
266C2#1062402811yescharacteristic
subgroupG/H
centerC2#10624028
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. #10624221C2semidirect
2. C3×#6766C2semidirect
3. #10623801C2semidirect
4. C3⋊C8×D15C2semidirect
5. C5×#3719C2semidirect
6. #10623760C2semidirect
7. C3×#6653C2semidirect
25. #6766D3semidirect
35. #6653D3semidirect
49. #3719D5semidirect
133. C32×C5#135semidirect
207. C3×C5#912semidirect
208. C3×C5#950semidirect
230. C32#4065semidirect
252. C5#63933semidirect
261. C3#91277semidirect
262. C3#91516semidirect

Maximal subgroups

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

subgroupindex
#106242212
C3×#67662
#106238012
C3⋊C8×D152
C5×#37192
#106237602
C3×#66532
#915163
#912773
#639335