Finite Group Databasebeta

← #13085765

Subgroups — #13085765

Order 1440.

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

normal subgroups only

This group has 326 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
1D5×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#6772C211yesmaximal, characteristic, complemented
3#10623824C211yesmaximal, characteristic, complemented
4#10623804C211yesmaximal, characteristic, complemented
5#10623800C211yesmaximal, characteristic, complemented
6C5×#3714C211yesmaximal, characteristic, complemented
7C3×#6649C211yesmaximal, characteristic, complemented
8#91478—31nomaximal
9#91250—31nomaximal
10C2×D5×C3⋊D3—21no—
11C4⋊(C3⋊D15)C2211yescharacteristic
12C5×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×Dic30C2211yescharacteristic, complemented
14C3×C5×D12C2211yescharacteristic, complemented
15C32×C4×D5C2211yescharacteristic
16#67140—21no—
17C3×#1201—21no—
18C3×D3×Dic5—21no—
19C3×C3⋊#168C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63825—51nomaximal, Hall
22C3×#711—31no—
23D5×D12—31no—
24D5×D12—31no—
25D5×D12—61no—
26#6772D311yescharacteristic, complemented
27#6715—31no—
28#6706—31no—
29C5×#213—31no—
30C5×D24—31no—
31C3×#676—31no—
32#6659—31no—
33#6658—31no—
34#6656—31no—
35#6654—31no—
36#6649D311yescharacteristic, complemented
37C2×C3⋊D15—21no—
38C2×C5×C3⋊D3—21no—
39C2×C3×C5×D3—21no—
40C2×C32×D5D411yescharacteristic
41D5×C3⋊D3—42no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15—21no—
44C32×Dic5D411yescharacteristic
45#4057—91noHall
46C2×C4⋊(C3⋊D3)—51no—
47C3×#235—51no—
48C3×#208—51no—
49#3717—51no—
50#3717—51no—
51#3714—51no—
52#3714D511yescharacteristic, complemented
53C2×D3×D5—62no—
54C2×D3×D5—61no—
55C2×D3×D5—61no—
56C3×C5×D4—31no—
57D60—31no—
58D60—31no—
59D60—61no—
60Dic30C2×D311yescharacteristic, complemented
61C5×D12—31no—
62C5×D12—31no—
63C5×D12—61no—
64C5×D12C2×D311yescharacteristic, complemented
65C3×C22⋊D5—61no—
66C2×C3×Dic5—61no—
67C3×C4×D5—21no—
68C3×C4×D5C2×D311yescharacteristic
69C3×C4×D5C2×D311yescharacteristic
70C3×Dic10—31no—
71#1203—62no—
72#1203—61no—
73#1203—61no—
74#1201—21no—
75D3×Dic5—21no—
76C3×C5×C8—31no—
77C3⋊#168C2×D311yescharacteristic, complemented
78C3×#168—31no—
79C5×C3⋊C8C2×D311yescharacteristic, complemented
80#949—151no—
81#908—151no—
82C2×C32×C5C2×D411yescharacteristic
83C3⋊D15—41no—
84C5×C3⋊D3—41no—
85C3×C5×D3—41no—
86C32×D5—21no—
87#711—31no—
88D4×D5—91no—
89C5×D8—91no—
90#687—91no—
91#686—91no—
92#677—91no—
93#676—31no—
94C22×C3⋊D3—101no—
95C2×C32×C4—51no—
96C4⋊(C3⋊D3)—51no—
97C4⋊(C3⋊D3)—101no—
98C4⋊(C3⋊D3)C2×D511yescharacteristic
99C3×#67—101no—
100C3×D12C2×D511yescharacteristic, complemented
101C3×C4×D3—101no—
102C3×Dic6—51no—
103C3×C3⋊C8—51no—
104C3×C3⋊C8C2×D511yescharacteristic, complemented
105C22×C3×C5—61no—
106C2×D15—62no—
107C2×D15—61no—
108C2×D15—61no—
109C2×C5×D3—21no—
110C2×C5×D3—62no—
111C2×C5×D3—61no—
112C2×C5×D3—61no—
113C2×C3×D5—21no—
114C2×C3×D5D1211yescharacteristic
115C2×C3×D5#6711yescharacteristic
116D3×D5—124no—
117D3×D5—122no—
118D3×D5—122no—
119C3×C4×C5—21no—
120C3×C4×C5C22×D311yescharacteristic
121C3×C4×C5C22×D311yescharacteristic
122Dic15—21no—
123C3×Dic5—21no—
124C3×Dic5—61no—
125C3×Dic5D1211yescharacteristic
126C3×Dic5#6711yescharacteristic
127C3×#41—151no—
128#235—51no—
129C2×D12—151no—
130C2×D12—151no—
131C2×D12—301no—
132C3×M16—151no—
133#215—151no—
134#215—151no—
135#213—31no—
136#213—151no—
137#208—51no—
138D24—31no—
139D24—151no—
140#204—151no—
141#204—151no—
142C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
143C22×D5—181no—
144C5×D4—31no—
145C5×D4—91no—
146C22⋊D5—61no—
147C22⋊D5—181no—
148C2×Dic5—61no—
149D20—91no—
150C4×D5D3211yescharacteristic
151Dic10—31no—
152C5×C8—31no—
153#168—31no—
154C22×C32—51no—
155C2×C3⋊D3—21no—
156C2×C3⋊D3—101no—
157C2×C3⋊D3—202no—
158C2×C3×D3—21no—
159C32×C4—51no—
160C32×C4C22×D511yescharacteristic
161C3×Dic3—101no—
162AGL(1,ℤ/8)—451noSylow
163C2×C3×C5—21no—
164C2×C3×C5—121no—
165C2×C3×C5C2×D1211yescharacteristic
166C2×C3×C5C2×#6711yescharacteristic
167D15—122no—
168D15—121no—
169D15—121no—
170C3×D5—22no—
171C3×D5—21no—
172C3×D5—21no—
173C5×D3—41no—
174C5×D3—122no—
175C5×D3—121no—
176C5×D3—121no—
177C22×D3—301no—
178C22×D3—301no—
179C22×D3—302no—
180C3×Q8—151no—
181C3×D4—31no—
182C3×D4—301no—
183C2×C3×C4—51no—
184C2×C3×C4—51no—
185C2×C3×C4—101no—
186C2×C3×C4—301no—
187#67—101no—
188D12—31no—
189D12—31no—
190D12—61no—
191D12—151no—
192D12—151no—
193D12—301no—
194D12—301no—
195D12—301no—
196D12—302no—
197D12D3×D511yescharacteristic, complemented
198C4×D3—101no—
199Dic6—51no—
200C3×C8—31no—
201C3×C8—151no—
202C3⋊C8—51no—
203C3⋊C8D3×D511yescharacteristic, complemented
204C22×C5—61no—
205C22×C5—181no—
206C2×D5—181no—
207C2×D5—362no—
208C2×D5#62611yescharacteristic
209C4×C5C2×D3211yescharacteristic
210Dic5—61no—
211Dic5#62611yescharacteristic
212C2×C32—101no—
213C2×C32D4×D511yescharacteristic
214C3⋊D3—41no—
215C3⋊D3—201no—
216C3×D3—41no—
217#41—151no—
218C2×D4—451no—
219SD16—451no—
220SD16—451no—
221D8—91no—
222D8—451no—
223M16—151no—
224C3×C5—21no—
225C3×C5#90811yescharacteristic, complemented
226C3×C5#94911yescharacteristic, complemented
227C22×C3—51no—
228C22×C3—51no—
229C22×C3—61no—
230C22×C3—101no—
231C2×D3—21no—
232C2×D3—62no—
233C2×D3—61no—
234C2×D3—61no—
235C2×D3—302no—
236C2×D3—301no—
237C2×D3—301no—
238C2×D3—604no—
239C2×D3—602no—
240C2×D3—602no—
241C3×C4—21no—
242C3×C4—51no—
243C3×C4—51no—
244C3×C4—101no—
245C3×C4—301no—
246C3×C4C2×D3×D511yescharacteristic
247C3×C4C2×D3×D511yescharacteristic
248Dic3—101no—
249C2×C5—121no—
250C2×C5—361no—
251C2×C5C2×#62611yescharacteristic
252D5—21no—
253D5—361no—
254C32#405711yescharacteristic, complemented, Sylow
255C23—901no—
256Q8—151no—
257D4—31no—
258D4—91no—
259D4—301no—
260D4—451no—
261D4—901no—
262C2×C4—51no—
263C2×C4—301no—
264C8—31no—
265C8—151no—
266C2×C3—21no—
267C2×C3—102no—
268C2×C3—101no—
269C2×C3—101no—
270C2×C3—121no—
271C2×C3D5×D1211yescharacteristic
272C2×C3D5×#6711yescharacteristic
273D3—41no—
274D3—122no—
275D3—121no—
276D3—121no—
277D3—602no—
278D3—601no—
279D3—601no—
280C5#6382511yescharacteristic, complemented, Sylow
281C22—51no—
282C22—61no—
283C22—181no—
284C22—901no—
285C22—1802no—
286C4—51no—
287C4—301no—
288C4D32×D511yescharacteristic
289C3—21no—
290C3#9125011yescharacteristic, complemented
291C3#9147811yescharacteristic, complemented
292C2—101no—
293C2—121no—
294C2—361no—
295C2—1801no—
296C2D5×#62611yescharacteristic
subgroupG/H
centerC2D5×#626
commutator, Fitting, cosocleC32×C4×C5C23
Frattini, hypercenterC4D32×D5
socleC2×C32×C5C2×D4
hypocenterC32×C5AGL(1,ℤ/8)
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. D5×C4⋊(C3⋊D3)C2semidirect
2. C3×#6772C2semidirect
3. #10623824C2semidirect
4. #10623804C2semidirect
5. #10623800C2semidirect
6. C5×#3714C2semidirect
7. C3×#6649C2semidirect
13. C3×Dic30C22semidirect
14. C3×C5×D12C22semidirect
19. C3×C3⋊#168C22semidirect
20. C3×C5×C3⋊C8C22semidirect
26. #6772D3semidirect
36. #6649D3semidirect
52. #3714D5semidirect
60. Dic30C2×D3semidirect
64. C5×D12C2×D3semidirect
77. C3⋊#168C2×D3semidirect
79. C5×C3⋊C8C2×D3semidirect
100. C3×D12C2×D5semidirect
104. C3×C3⋊C8C2×D5semidirect
142. C32×C5AGL(1,ℤ/8)semidirect
197. D12D3×D5semidirect
203. C3⋊C8D3×D5semidirect
225. C3×C5#908semidirect
226. C3×C5#949semidirect
254. C32#4057semidirect
280. C5#63825semidirect
290. C3#91250semidirect
291. C3#91478semidirect

Maximal subgroups

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

subgroupindex
D5×C4⋊(C3⋊D3)2
C3×#67722
#106238242
#106238042
#106238002
C5×#37142
C3×#66492
#914783
#912503
#638255