Finite Group Databasebeta

← #13087505

Subgroups — #13087505

Order 1440.

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

normal subgroups only

This group has 270 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#10624201C211yesmaximal, characteristic, complemented
3#10623909C211yesmaximal, characteristic, complemented
4C3×C5×#214C211yesmaximal, characteristic, complemented
5#10623807C211yesmaximal, characteristic, complemented
6#10623804C211yesmaximal, characteristic, complemented
7C3⋊C8×D15C211yesmaximal, characteristic, complemented
8#91805—31nomaximal
9#91492—31nomaximal
10C32×C5×D4C2211yescharacteristic
11C4⋊(C3⋊D15)C2211yescharacteristic
12C3×#1220—21no—
13C2×C3×Dic15—21no—
14C3×C4×D15C2211yescharacteristic
15C3×Dic30C2211yescharacteristic
16C3×C5×Dic6C2211yescharacteristic
17#67151—21no—
18#67149—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63935—51nomaximal, Hall
22#6821—31no—
23#6819D311yescharacteristic, complemented
24C3×#711—31no—
25#6773—31no—
26C3×C5×SD16—31no—
27#6718—31no—
28#6715—31no—
29#6715—31no—
30#6715—61no—
31#6706—31no—
32C8×D15—31no—
33C5×#214D311yescharacteristic, complemented
34#6661—31no—
35#6658—31no—
36D5×C3⋊C8—31no—
37C22×C32×C5—21no—
38C2×C3⋊D15—21no—
39C2×C3×D15D411yescharacteristic
40C32×C4×C5C2311yescharacteristic
41C3×Dic15—21no—
42C3×Dic15D411yescharacteristic
43C3×C5×Dic3—21no—
44#4062—91noHall
45C3×#237—51no—
46#3799—51no—
47#3753—51no—
48C3×#214D511yescharacteristic, complemented
49#3718—51no—
50#3717—51no—
51D3×C3⋊C8—51no—
52C3×C5×Q8—31no—
53C3×C5×D4—21no—
54C3×C5×D4C2×D311yescharacteristic
55C3×C5×D4C2×D311yescharacteristic
56#1220—21no—
57C2×Dic15—21no—
58D60—31no—
59D60—31no—
60D60—61no—
61D60—61no—
62C4×D15—61no—
63C4×D15C2×D311yescharacteristic
64Dic30C2×D311yescharacteristic
65C5×Dic6C2×D311yescharacteristic
66C3×C22⋊D5—61no—
67C2×C3×Dic5—61no—
68C3×C4×D5—31no—
69C3×Dic10—31no—
70#1202—61no—
71#1200—61no—
72C3×C5×C8—31no—
73C3⋊#168—31no—
74C3⋊#168—31no—
75C3⋊#168—61no—
76C5×C3⋊C8C2×D311yescharacteristic
77#950—151no—
78#916—151no—
79C2×C32×C5—41no—
80C2×C32×C5C2×D411yescharacteristic
81C3⋊D15—41no—
82C3×D15—21no—
83#713—91no—
84#711—31no—
85C5×SD16—31no—
86#689—91no—
87#686—91no—
88#677—91no—
89C8×D5—91no—
90C32×D4C2×D511yescharacteristic
91C4⋊(C3⋊D3)—51no—
92C3×#67—101no—
93C2×C3×Dic3—101no—
94C3×C4×D3—51no—
95C3×Dic6—51no—
96C3×Dic6C2×D511yescharacteristic
97#626—101no—
98#624—101no—
99C3⋊(C3⋊C8)—51no—
100C3×C3⋊C8C2×D511yescharacteristic
101C22×C3×C5—21no—
102C22×C3×C5—21no—
103C22×C3×C5—41no—
104C2×D15—61no—
105C2×D15—61no—
106C2×D15—121no—
107C2×D15#6711yescharacteristic
108C2×C3×D5—31no—
109C3×C4×C5—21no—
110C3×C4×C5—61no—
111C3×C4×C5C22×D311yescharacteristic
112C3×C4×C5C22×D311yescharacteristic
113Dic15—21no—
114Dic15#6711yescharacteristic
115C3×Dic5—31no—
116C3×Dic5—61no—
117C5×Dic3—21no—
118C3×#41—151no—
119#239—151no—
120#237—51no—
121#235—151no—
122C3×SD16—31no—
123#216—151no—
124#216—151no—
125#215—151no—
126#214D1511yescharacteristic, complemented
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×D4D3211yescharacteristic
136C22⋊D5—61no—
137C2×Dic5—61no—
138D20—91no—
139D20—181no—
140C4×D5—31no—
141C4×D5—181no—
142Dic10—31no—
143C5×C8—31no—
144#168—91no—
145C22×C32—21no—
146C2×C3⋊D3—101no—
147C2×C3×D3—51no—
148C32×C4C22×D511yescharacteristic
149C3×Dic3—21no—
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×C5D3×D411yescharacteristic
158C2×C3×C5C2×#6711yescharacteristic
159D15—21no—
160D15—121no—
161D15—121no—
162D15—241no—
163C3×D5—61no—
164C3×Q8—31no—
165C3×Q8—151no—
166C3×D4—21no—
167C3×D4—301no—
168C3×D4D3×D511yescharacteristic
169C3×D4C2×D1511yescharacteristic
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—
182Dic6—51no—
183Dic6C2×D1511yescharacteristic
184C3×C8—31no—
185C3⋊C8—151no—
186C3⋊C8—151no—
187C3⋊C8—301no—
188C3⋊C8C2×D1511yescharacteristic
189C22×C5—21no—
190C2×D5—31no—
191C2×D5—181no—
192C4×C5—61no—
193C4×C5C2×D3211yescharacteristic
194Dic5—31no—
195Dic5—61no—
196C2×C32—41no—
197C2×C32D4×D511yescharacteristic
198C3⋊D3—201no—
199C3×D3—101no—
200#41—151no—
201#41—451no—
202Q16—451no—
203SD16—31no—
204SD16—451no—
205D8—451no—
206C2×C8—451no—
207C3×C5—21no—
208C3×C5#91611yescharacteristic, complemented
209C3×C5#95011yescharacteristic, complemented
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—41no—
213C22×C3—151no—
214C2×D3—51no—
215C2×D3—301no—
216C2×D3—301no—
217C2×D3—601no—
218C3×C4—21no—
219C3×C4—61no—
220C3×C4—151no—
221C3×C4—301no—
222C3×C4C2×D3×D511yescharacteristic
223C3×C4C22×D1511yescharacteristic
224Dic3—21no—
225Dic3—51no—
226Dic3—101no—
227C2×C5—41no—
228C2×C5D3×#6711yescharacteristic
229D5—61no—
230D5—361no—
231C32#406211yescharacteristic, complemented, Sylow
232Q8—31no—
233Q8—151no—
234D4—301no—
235D4—451no—
236D4—901no—
237D4D3×D1511yescharacteristic
238C2×C4—151no—
239C2×C4—301no—
240C2×C4—901no—
241C8—31no—
242C8—451no—
243C2×C3—21no—
244C2×C3—41no—
245C2×C3—41no—
246C2×C3—81no—
247C2×C3—301no—
248C2×C3D5×#6711yescharacteristic
249C2×C3D4×D1511yescharacteristic
250D3—101no—
251D3—601no—
252D3—601no—
253D3—1201no—
254C5#6393511yescharacteristic, complemented, Sylow
255C22—21no—
256C22—151no—
257C22—901no—
258C4—61no—
259C4—151no—
260C4—301no—
261C4C2×D3×D1511yescharacteristic
262C3—21no—
263C3#9149211yescharacteristic, complemented
264C3#9180511yescharacteristic, complemented
265C2—41no—
266C2—301no—
267C2—1801no—
268C2#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. #10624201C2semidirect
3. #10623909C2semidirect
4. C3×C5×#214C2semidirect
5. #10623807C2semidirect
6. #10623804C2semidirect
7. C3⋊C8×D15C2semidirect
23. #6819D3semidirect
33. C5×#214D3semidirect
48. C3×#214D5semidirect
126. #214D15semidirect
133. C32×C5#135semidirect
208. C3×C5#916semidirect
209. C3×C5#950semidirect
231. C32#4062semidirect
254. C5#63935semidirect
263. C3#91492semidirect
264. C3#91805semidirect

Maximal subgroups

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

subgroupindex
C3×#68192
#106242012
#106239092
C3×C5×#2142
#106238072
#106238042
C3⋊C8×D152
#918053
#914923
#639355