Finite Group Databasebeta

← #13087503

Subgroups — #13087503

Order 1440.

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

normal subgroups only

This group has 268 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×#6819C211yesmaximal, characteristic, complemented
2#10624200C211yesmaximal, characteristic, complemented
3#10623910C211yesmaximal, characteristic, complemented
4C3×C5×#213C211yesmaximal, characteristic, complemented
5#10623808C211yesmaximal, characteristic, complemented
6#10623802C211yesmaximal, characteristic, complemented
7C3⋊C8×D15C211yesmaximal, characteristic, complemented
8#91801—31nomaximal
9#91490—31nomaximal
10C32×C5×D4C2211yescharacteristic
11#67180C2211yescharacteristic
12C3×#1220—21no—
13C2×C3×Dic15—21no—
14C3×C4×D15C2211yescharacteristic
15C3×Dic30C2211yescharacteristic
16C3×C5×D12C2211yescharacteristic
17#67150—21no—
18D3×Dic15—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63933—51nomaximal, Hall
22#6819—31no—
23#6819D311yescharacteristic, complemented
24C3×#711—31no—
25#6772—31no—
26C3×C5×D8—31no—
27#6716—31no—
28#6716—31no—
29#6716—31no—
30#6716—61no—
31Dic60—31no—
32C8×D15—31no—
33C5×#213D311yescharacteristic, complemented
34#6662—31no—
35#6656—31no—
36D5×C3⋊C8—31no—
37C22×C32×C5—21no—
38C2×C3×D15D411yescharacteristic
39C2×C3×C5×D3—21no—
40C32×C4×C5C2311yescharacteristic
41C3⋊Dic15—21no—
42C3×Dic15—21no—
43C3×Dic15D411yescharacteristic
44#4058—91noHall
45C3×#237—51no—
46#3795—51no—
47#3754—51no—
48C3×#213D511yescharacteristic, complemented
49#3719—51no—
50#3715—51no—
51D3×C3⋊C8—51no—
52C3×C5×D4—21no—
53C3×C5×D4—31no—
54C3×C5×D4C2×D311yescharacteristic
55C3×C5×D4C2×D311yescharacteristic
56#1220—21no—
57#1220—61no—
58C2×Dic15—21no—
59C2×Dic15—61no—
60C4×D15C2×D311yescharacteristic
61Dic30—31no—
62Dic30—31no—
63Dic30—61no—
64Dic30C2×D311yescharacteristic
65C5×D12C2×D311yescharacteristic
66C3×C22⋊D5—61no—
67C2×C3×Dic5—61no—
68C3×C4×D5—31no—
69C3×Dic10—31no—
70#1201—61no—
71D3×Dic5—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×C5—41no—
80C2×C32×C5C2×D411yescharacteristic
81C3×D15—21no—
82C3×C5×D3—41no—
83#711—31no—
84#711—91no—
85C5×D8—31no—
86#687—91no—
87#687—91no—
88Dic20—91no—
89C8×D5—91no—
90C32×D4C2×D511yescharacteristic
91#634—51no—
92C3×#67—101no—
93C2×C3×Dic3—101no—
94C3×D12C2×D511yescharacteristic
95C3×C4×D3—51no—
96C3×Dic6—51no—
97#625—101no—
98D3×Dic3—101no—
99C3⋊(C3⋊C8)—51no—
100C3×C3⋊C8C2×D511yescharacteristic
101C22×C3×C5—21no—
102C22×C3×C5—21no—
103C22×C3×C5—41no—
104C22×C3×C5—61no—
105C2×D15#6711yescharacteristic
106C2×C5×D3—21no—
107C2×C3×D5—31no—
108C3×C4×C5—21no—
109C3×C4×C5C22×D311yescharacteristic
110C3×C4×C5C22×D311yescharacteristic
111Dic15—21no—
112Dic15—61no—
113Dic15—61no—
114Dic15—121no—
115Dic15#6711yescharacteristic
116C3×Dic5—31no—
117C3×Dic5—61no—
118C3×#41—151no—
119#237—51no—
120#237—151no—
121#235—151no—
122C3×D8—31no—
123#216—151no—
124#215—151no—
125#214—151no—
126#214—151no—
127#214—151no—
128#214—301no—
129#213D1511yescharacteristic, complemented
130C2×C3⋊C8—151no—
131Dic12—151no—
132D3×C8—151no—
133C32×C5#13511yescharacteristic, complemented, Hall
134C5×D4—31no—
135C5×D4D3211yescharacteristic
136C22⋊D5—61no—
137C22⋊D5—181no—
138C2×Dic5—61no—
139C2×Dic5—181no—
140C4×D5—31no—
141Dic10—31no—
142Dic10—91no—
143C5×C8—31no—
144#168—91no—
145C22×C32—21no—
146C2×C3×D3—21no—
147C2×C3×D3—51no—
148C32×C4C22×D511yescharacteristic
149C3⋊Dic3—101no—
150C3×Dic3—51no—
151C3×Dic3—101no—
152#135—451noSylow
153C2×C3×C5—21no—
154C2×C3×C5—41no—
155C2×C3×C5—41no—
156C2×C3×C5—81no—
157C2×C3×C5—121no—
158C2×C3×C5D3×D411yescharacteristic
159C2×C3×C5C2×#6711yescharacteristic
160D15—21no—
161C3×D5—61no—
162C5×D3—41no—
163C3×Q8—151no—
164C3×D4—21no—
165C3×D4—31no—
166C3×D4—301no—
167C3×D4D3×D511yescharacteristic
168C3×D4C2×D1511yescharacteristic
169C2×C3×C4—151no—
170C2×C3×C4—301no—
171#67—101no—
172#67—301no—
173#67—301no—
174C2×Dic3—101no—
175C2×Dic3—301no—
176D12C2×D1511yescharacteristic
177C4×D3—51no—
178C4×D3—301no—
179Dic6—51no—
180Dic6—151no—
181Dic6—151no—
182Dic6—301no—
183C3×C8—31no—
184C3⋊C8—151no—
185C3⋊C8—151no—
186C3⋊C8—301no—
187C3⋊C8C2×D1511yescharacteristic
188C22×C5—21no—
189C22×C5—61no—
190C2×D5—31no—
191C4×C5C2×D3211yescharacteristic
192Dic5—31no—
193Dic5—61no—
194Dic5—181no—
195C2×C32—41no—
196C2×C32D4×D511yescharacteristic
197C3×D3—41no—
198C3×D3—101no—
199#41—151no—
200#41—451no—
201Q16—451no—
202SD16—451no—
203SD16—451no—
204D8—31no—
205C2×C8—451no—
206C3×C5—21no—
207C3×C5#91211yescharacteristic, complemented
208C3×C5#95011yescharacteristic, complemented
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—41no—
212C22×C3—61no—
213C22×C3—151no—
214C2×D3—21no—
215C2×D3—51no—
216C3×C4—21no—
217C3×C4—151no—
218C3×C4—301no—
219C3×C4C2×D3×D511yescharacteristic
220C3×C4C22×D1511yescharacteristic
221Dic3—51no—
222Dic3—101no—
223Dic3—301no—
224Dic3—301no—
225Dic3—601no—
226C2×C5—41no—
227C2×C5—121no—
228C2×C5D3×#6711yescharacteristic
229D5—61no—
230C32#405811yescharacteristic, complemented, Sylow
231Q8—151no—
232Q8—451no—
233D4—31no—
234D4—301no—
235D4—901no—
236D4D3×D1511yescharacteristic
237C2×C4—151no—
238C2×C4—301no—
239C2×C4—901no—
240C8—31no—
241C8—451no—
242C2×C3—21no—
243C2×C3—41no—
244C2×C3—41no—
245C2×C3—81no—
246C2×C3—121no—
247C2×C3—301no—
248C2×C3D5×#6711yescharacteristic
249C2×C3D4×D1511yescharacteristic
250D3—41no—
251D3—101no—
252C5#6393311yescharacteristic, complemented, Sylow
253C22—21no—
254C22—61no—
255C22—151no—
256C4—151no—
257C4—301no—
258C4—901no—
259C4C2×D3×D1511yescharacteristic
260C3—21no—
261C3#9149011yescharacteristic, complemented
262C3#9180111yescharacteristic, complemented
263C2—41no—
264C2—121no—
265C2—301no—
266C2#67×D1511yescharacteristic
subgroupG/H
centerC2#67×D15
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterD4D3×D15
hypocenterC32×C5#135
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C3×#6819C2semidirect
2. #10624200C2semidirect
3. #10623910C2semidirect
4. C3×C5×#213C2semidirect
5. #10623808C2semidirect
6. #10623802C2semidirect
7. C3⋊C8×D15C2semidirect
23. #6819D3semidirect
33. C5×#213D3semidirect
48. C3×#213D5semidirect
129. #213D15semidirect
133. C32×C5#135semidirect
207. C3×C5#912semidirect
208. C3×C5#950semidirect
230. C32#4058semidirect
252. C5#63933semidirect
261. C3#91490semidirect
262. C3#91801semidirect

Maximal subgroups

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

subgroupindex
C3×#68192
#106242002
#106239102
C3×C5×#2132
#106238082
#106238022
C3⋊C8×D152
#918013
#914903
#639335