Finite Group Databasebeta

← #13085780

Subgroups — #13085780

Order 1440.

292 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 292 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×#6777C211yesmaximal, characteristic, complemented
3#10623798C211yesmaximal, characteristic, complemented
4#10623795C211yesmaximal, characteristic, complemented
5C5×#3716C211yesmaximal, characteristic, complemented
6#10623760C211yesmaximal, characteristic, complemented
7C3×#6653C211yesmaximal, characteristic, complemented
8#91491—31nomaximal
9#91258—31nomaximal
10C2×C3×D3×D5—21no—
11C5×#634C2211yescharacteristic, complemented
12C3×D60C2211yescharacteristic, complemented
13C3×C4×D15C2211yescharacteristic
14C3×C5×D12C2211yescharacteristic
15C32×D20C2211yescharacteristic
16#67157—21no—
17#67154—21no—
18C3×#1201—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63932—51nomaximal, Hall
22C3×D4×D5—31no—
23#6777D311yescharacteristic, complemented
24#6773—31no—
25#6766—31no—
26#6715—31no—
27#6705—31no—
28C5×#214—31no—
29C5×#204—31no—
30C3×D40—31no—
31#6655—31no—
32#6655—31no—
33#6655—61no—
34#6653D311yescharacteristic, complemented
35#6652—31no—
36#6649—31no—
37C2×C3×D15—21no—
38C2×C3×D15D411yescharacteristic
39C2×C3×C5×D3—21no—
40C2×C32×D5—21no—
41C3×D3×D5—42no—
42C32×C4×C5C2311yescharacteristic
43C5×C3⋊Dic3—21no—
44C3×Dic15D411yescharacteristic
45#4060—91noHall
46C3×D3×D4—51no—
47#3795—51no—
48#3754—51no—
49C3×#213—51no—
50#3716D511yescharacteristic, complemented
51#3713—51no—
52#3711—51no—
53C22×C3×D5—61no—
54C2×D3×D5—21no—
55C3×C5×D4—31no—
56D60C2×D311yescharacteristic, complemented
57C4×D15C2×D311yescharacteristic
58C5×D12C2×D311yescharacteristic
59C5×Dic6—31no—
60C5×Dic6—31no—
61C5×Dic6—61no—
62C3×C22⋊D5—61no—
63C3×D20—21no—
64C3×D20—31no—
65C3×D20C2×D311yescharacteristic
66C3×D20C2×D311yescharacteristic
67C3×C4×D5—31no—
68#1202—61no—
69#1202—61no—
70#1201—21no—
71#1200—61no—
72D5×Dic3—61no—
73C3×C5×C8—31no—
74C3⋊#168—31no—
75C3⋊#168—31no—
76C3⋊#168—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#932—151no—
79#911—151no—
80C2×C32×C5C2×D411yescharacteristic
81C3×D15—21no—
82C3×D15—41no—
83C3×C5×D3—41no—
84C32×D5—41no—
85#713—91no—
86D4×D5—31no—
87C5×SD16—91no—
88#688—91no—
89#686—91no—
90D40—31no—
91#676—91no—
92C22×C3×D3—101no—
93C32×D4—51no—
94#634C2×D511yescharacteristic, complemented
95C3×#67—101no—
96C3×D12—51no—
97C3×D12C2×D511yescharacteristic
98C3×C4×D3—51no—
99#625—101no—
100D3×Dic3—101no—
101C3⋊(C3⋊C8)—51no—
102C3×C3⋊C8C2×D511yescharacteristic, complemented
103C22×C3×C5—61no—
104C2×D15—21no—
105C2×D15#6711yescharacteristic
106C2×C5×D3—21no—
107C2×C3×D5—21no—
108C2×C3×D5—21no—
109C2×C3×D5—31no—
110C2×C3×D5—41no—
111C2×C3×D5—61no—
112C2×C3×D5—122no—
113D3×D5—42no—
114C3×C4×C5—21no—
115C3×C4×C5C22×D311yescharacteristic
116C3×C4×C5C22×D311yescharacteristic
117Dic15#6711yescharacteristic
118C3×Dic5—31no—
119C5×Dic3—61no—
120C5×Dic3—61no—
121C5×Dic3—121no—
122C2×C3×D4—151no—
123#237—151no—
124D3×D4—51no—
125#235—151no—
126C3×D8—151no—
127#214—31no—
128#214—151no—
129#214—151no—
130#214—301no—
131#213—51no—
132#213—151no—
133#213—151no—
134#208—151no—
135#204—31no—
136ΓL(1,𝔽25)—151no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—61no—
139C5×Q8—91no—
140C5×D4—31no—
141C22⋊D5—61no—
142D20—31no—
143D20—181no—
144D20D3211yescharacteristic
145C4×D5—31no—
146C4×D5—181no—
147C5×C8—31no—
148#168—91no—
149C22×C32—101no—
150C2×C3×D3—21no—
151C2×C3×D3—51no—
152C2×C3×D3—101no—
153C2×C3×D3—202no—
154C32×C4C22×D511yescharacteristic
155C3⋊Dic3—21no—
156C3×Dic3—51no—
157AGL(1,ℤ/8)—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—121no—
160C2×C3×C5D3×D411yescharacteristic
161C2×C3×C5C2×#6711yescharacteristic
162D15—21no—
163D15—41no—
164C3×D5—41no—
165C3×D5—41no—
166C3×D5—61no—
167C3×D5—81no—
168C3×D5—121no—
169C5×D3—41no—
170C23×C3—301no—
171C22×D3—101no—
172C3×D4—31no—
173C3×D4—51no—
174C3×D4—51no—
175C3×D4—101no—
176C3×D4—151no—
177C3×D4—301no—
178C2×C3×C4—151no—
179#67—101no—
180#67—301no—
181#67—301no—
182C2×Dic3—301no—
183D12—51no—
184D12D3×D511yescharacteristic
185C4×D3—51no—
186C4×D3—301no—
187Dic6—31no—
188Dic6—31no—
189Dic6—61no—
190C3×C8—31no—
191C3⋊C8—151no—
192C3⋊C8—151no—
193C3⋊C8—301no—
194C3⋊C8D3×D511yescharacteristic, complemented
195C22×C5—61no—
196C2×D5—21no—
197C2×D5—31no—
198C2×D5—61no—
199C2×D5—122no—
200C4×C5—181no—
201C4×C5C2×D3211yescharacteristic
202Dic5—31no—
203C2×C32—201no—
204C2×C32D4×D511yescharacteristic
205C3×D3—41no—
206C3×D3—101no—
207C3×D3—201no—
208#41—451no—
209C2×D4—151no—
210SD16—91no—
211SD16—451no—
212D8—151no—
213D8—451no—
214M16—451no—
215C3×C5—21no—
216C3×C5#91111yescharacteristic, complemented
217C3×C5#93211yescharacteristic, complemented
218C22×C3—61no—
219C22×C3—101no—
220C22×C3—101no—
221C22×C3—151no—
222C22×C3—201no—
223C22×C3—301no—
224C22×C3—602no—
225C2×D3—21no—
226C2×D3—51no—
227C2×D3—101no—
228C2×D3—202no—
229C3×C4—21no—
230C3×C4—151no—
231C3×C4C2×D3×D511yescharacteristic
232C3×C4C2×D3×D511yescharacteristic
233Dic3—51no—
234Dic3—61no—
235Dic3—61no—
236Dic3—121no—
237C2×C5—121no—
238C2×C5D3×#6711yescharacteristic
239D5—41no—
240D5—61no—
241D5—121no—
242C32#406011yescharacteristic, complemented, Sylow
243C23—301no—
244Q8—91no—
245D4—31no—
246D4—51no—
247D4—151no—
248D4—301no—
249D4—901no—
250C2×C4—151no—
251C2×C4—901no—
252C8—31no—
253C8—451no—
254C2×C3—21no—
255C2×C3—121no—
256C2×C3—201no—
257C2×C3—201no—
258C2×C3—301no—
259C2×C3—401no—
260C2×C3—601no—
261C2×C3#677711yescharacteristic
262C2×C3D5×#6711yescharacteristic
263D3—41no—
264D3—101no—
265D3—201no—
266C5#6393211yescharacteristic, complemented, Sylow
267C22—61no—
268C22—101no—
269C22—151no—
270C22—301no—
271C22—602no—
272C4—151no—
273C4—181no—
274C4D32×D511yescharacteristic
275C3—21no—
276C3#9125811yescharacteristic, complemented
277C3#9149111yescharacteristic, complemented
278C2—121no—
279C2—201no—
280C2—301no—
281C2—601no—
282C2#1062402811yescharacteristic
subgroupG/H
centerC2#10624028
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. #10624221C2semidirect
2. C3×#6777C2semidirect
3. #10623798C2semidirect
4. #10623795C2semidirect
5. C5×#3716C2semidirect
6. #10623760C2semidirect
7. C3×#6653C2semidirect
11. C5×#634C22semidirect
12. C3×D60C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
23. #6777D3semidirect
34. #6653D3semidirect
50. #3716D5semidirect
56. D60C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
94. #634C2×D5semidirect
102. C3×C3⋊C8C2×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
194. C3⋊C8D3×D5semidirect
216. C3×C5#911semidirect
217. C3×C5#932semidirect
242. C32#4060semidirect
266. C5#63932semidirect
276. C3#91258semidirect
277. C3#91491semidirect

Maximal subgroups

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

subgroupindex
#106242212
C3×#67772
#106237982
#106237952
C5×#37162
#106237602
C3×#66532
#914913
#912583
#639325