Finite Group Databasebeta

← #13085744

Subgroups — #13085744

Order 1440.

288 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 56 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#10624196C211yesmaximal, characteristic, complemented
2#10624151C211yesmaximal, characteristic, complemented
3#10623824C211yesmaximal, characteristic, complemented
4#10623794C211yesmaximal, characteristic, complemented
5C5×#3714C211yesmaximal, characteristic, complemented
6C3×#6662C211yesmaximal, characteristic, complemented
7C3×#6656C211yesmaximal, characteristic, complemented
8#91499—31nomaximal
9#91279—31nomaximal
10C4×C3⋊D15C2211yescharacteristic
11C5×C4⋊(C3⋊D3)C2211yescharacteristic
12C3×Dic30C2211yescharacteristic
13C3×C5×D12C2211yescharacteristic
14C32×Dic10C2211yescharacteristic
15#67152—21no—
16D3×Dic15—21no—
17#67140—21no—
18C3⋊D3×Dic5—21no—
19C3×C3⋊#168C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63950—51nomaximal, Hall
22#6819—31no—
23#6768—31no—
24#6768—31no—
25#6768—31no—
26#6768—61no—
27C8×D15—31no—
28C5×#213—31no—
29C5×D24—31no—
30C3×#687—31no—
31C3×Dic20—31no—
32#6662D311yescharacteristic, complemented
33#6659—31no—
34#6656—31no—
35#6656D311yescharacteristic, complemented
36#6648—31no—
37C2×C3⋊D15D411yescharacteristic
38C2×C5×C3⋊D3—21no—
39C2×C3×C5×D3—21no—
40C32×C4×C5C2311yescharacteristic
41C3⋊Dic15D411yescharacteristic
42C3×Dic15—21no—
43C32×Dic5—21no—
44#4058—91noHall
45#3832—51no—
46#3796—51no—
47C3×#216—51no—
48C3×#215—51no—
49#3717—51no—
50#3714D511yescharacteristic, complemented
51#3710—51no—
52C3×C5×D4—31no—
53#1220—61no—
54C2×Dic15—61no—
55C4×D15—31no—
56C4×D15—31no—
57C4×D15—61no—
58Dic30C2×D311yescharacteristic
59C5×D12—31no—
60C5×D12—31no—
61C5×D12—61no—
62C5×D12C2×D311yescharacteristic
63C3×Dic10—21no—
64C3×Dic10—31no—
65C3×Dic10C2×D311yescharacteristic
66C3×Dic10C2×D311yescharacteristic
67#1203—61no—
68#1203—61no—
69#1203—62no—
70#1203—61no—
71D3×Dic5—61no—
72D3×Dic5—61no—
73D3×Dic5—62no—
74D3×Dic5—61no—
75C3×C5×C8—31no—
76C3⋊#168C2×D311yescharacteristic
77C3×#168—31no—
78C5×C3⋊C8C2×D311yescharacteristic
79#919—151no—
80#916—151no—
81C2×C32×C5C2×D411yescharacteristic
82C3⋊D15—21no—
83C5×C3⋊D3—41no—
84C3×C5×D3—41no—
85#711—91no—
86#711—91no—
87C5×D8—91no—
88#687—31no—
89#687—91no—
90Dic20—31no—
91C8×D5—91no—
92C32×Q8—51no—
93C4⋊(C3⋊D3)—101no—
94C4⋊(C3⋊D3)C2×D511yescharacteristic
95C4×C3⋊D3—51no—
96C4×C3⋊D3—101no—
97C3×D12C2×D511yescharacteristic
98C3×Dic6—51no—
99#626—101no—
100D3×Dic3—101no—
101C3×C3⋊C8—51no—
102C3×C3⋊C8C2×D511yescharacteristic
103C22×C3×C5—61no—
104C2×D15—31no—
105C2×D15—31no—
106C2×D15—61no—
107C2×C5×D3—21no—
108C2×C5×D3—62no—
109C2×C5×D3—61no—
110C2×C5×D3—61no—
111C3×C4×C5—21no—
112C3×C4×C5C22×D311yescharacteristic
113C3×C4×C5C22×D311yescharacteristic
114Dic15—21no—
115Dic15—31no—
116Dic15—31no—
117Dic15—61no—
118C3×Dic5—22no—
119C3×Dic5—21no—
120C3×Dic5—21no—
121C3×Dic5—61no—
122#239—151no—
123#239—151no—
124#239—151no—
125#239—301no—
126#237—151no—
127C3×Q16—151no—
128C3×SD16—151no—
129#216—51no—
130#215—51no—
131#215—151no—
132#213—31no—
133D24—31no—
134#204—151no—
135D3×C8—151no—
136D3×C8—151no—
137C32×C5#13511yescharacteristic, complemented, Hall
138C5×D4—31no—
139C5×D4—91no—
140C22⋊D5—181no—
141C22⋊D5—181no—
142C2×Dic5—181no—
143C2×Dic5—181no—
144C4×D5—91no—
145Dic10—31no—
146Dic10D3211yescharacteristic
147C5×C8—31no—
148#168—31no—
149C2×C3⋊D3—21no—
150C2×C3⋊D3—51no—
151C2×C3×D3—21no—
152C32×C4—101no—
153C32×C4C22×D511yescharacteristic
154C3⋊Dic3—51no—
155C3×Dic3—101no—
156#135—451noSylow
157C2×C3×C5—21no—
158C2×C3×C5—121no—
159C2×C3×C5D3×D411yescharacteristic
160C2×C3×C5D3×D411yescharacteristic
161D15—61no—
162D15—61no—
163D15—121no—
164C5×D3—41no—
165C5×D3—122no—
166C5×D3—121no—
167C5×D3—121no—
168C3×Q8—51no—
169C3×Q8—51no—
170C3×Q8—101no—
171C3×Q8—151no—
172C3×D4—31no—
173#67—301no—
174C2×Dic3—301no—
175D12—31no—
176D12—31no—
177D12—61no—
178D12—301no—
179D12—301no—
180D12—302no—
181D12—301no—
182D12D3×D511yescharacteristic
183C4×D3—151no—
184C4×D3—151no—
185C4×D3—301no—
186C4×D3—301no—
187C4×D3—301no—
188C4×D3—301no—
189C4×D3—302no—
190Dic6—51no—
191C3×C8—31no—
192C3×C8—151no—
193C3⋊C8—51no—
194C3⋊C8D3×D511yescharacteristic
195C22×C5—61no—
196C22×C5—181no—
197C2×D5—91no—
198C4×C5C2×D3211yescharacteristic
199Dic5—21no—
200Dic5—61no—
201Dic5—91no—
202C2×C32D4×D511yescharacteristic
203C3⋊D3—41no—
204C3⋊D3—101no—
205C3×D3—41no—
206#41—451no—
207#41—451no—
208Q16—151no—
209SD16—151no—
210SD16—451no—
211D8—91no—
212C2×C8—451no—
213C3×C5—21no—
214C3×C5#91611yescharacteristic, complemented
215C3×C5#91911yescharacteristic, complemented
216C22×C3—61no—
217C2×D3—21no—
218C2×D3—62no—
219C2×D3—61no—
220C2×D3—61no—
221C2×D3—151no—
222C2×D3—151no—
223C2×D3—301no—
224C3×C4—21no—
225C3×C4—102no—
226C3×C4—101no—
227C3×C4—101no—
228C3×C4—301no—
229C3×C4C2×D3×D511yescharacteristic
230C3×C4C2×D3×D511yescharacteristic
231Dic3—101no—
232Dic3—151no—
233Dic3—151no—
234Dic3—301no—
235C2×C5—121no—
236C2×C5—361no—
237C2×C5#381111yescharacteristic
238D5—181no—
239C32#405811yescharacteristic, complemented, Sylow
240Q8—51no—
241Q8—151no—
242D4—31no—
243D4—91no—
244D4—901no—
245D4—901no—
246C2×C4—451no—
247C2×C4—901no—
248C2×C4—901no—
249C8—31no—
250C8—151no—
251C2×C3—21no—
252C2×C3—121no—
253C2×C3#677711yescharacteristic
254C2×C3#679011yescharacteristic
255D3—41no—
256D3—122no—
257D3—121no—
258D3—121no—
259D3—301no—
260D3—301no—
261D3—601no—
262C5#6395011yescharacteristic, complemented, Sylow
263C22—61no—
264C22—181no—
265C22—451no—
266C4—101no—
267C4—301no—
268C4—451no—
269C4D32×D511yescharacteristic
270C3—21no—
271C3#9127911yescharacteristic, complemented
272C3#9149911yescharacteristic, complemented
273C2—121no—
274C2—361no—
275C2—901no—
276C2#1062403011yescharacteristic
subgroupG/H
centerC2#10624030
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. #10624196C2semidirect
2. #10624151C2semidirect
3. #10623824C2semidirect
4. #10623794C2semidirect
5. C5×#3714C2semidirect
6. C3×#6662C2semidirect
7. C3×#6656C2semidirect
32. #6662D3semidirect
35. #6656D3semidirect
50. #3714D5semidirect
137. C32×C5#135semidirect
214. C3×C5#916semidirect
215. C3×C5#919semidirect
239. C32#4058semidirect
262. C5#63950semidirect
271. C3#91279semidirect
272. C3#91499semidirect

Maximal subgroups

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

subgroupindex
#106241962
#106241512
#106238242
#106237942
C5×#37142
C3×#66622
C3×#66562
#914993
#912793
#639505