Finite Group Databasebeta

← #13085782

Subgroups — #13085782

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#10624226C211yesmaximal, characteristic, complemented
2C3×#6768C211yesmaximal, characteristic, complemented
3#10623798C211yesmaximal, characteristic, complemented
4#10623795C211yesmaximal, characteristic, complemented
5C5×#3714C211yesmaximal, characteristic, complemented
6#10623761C211yesmaximal, characteristic, complemented
7C3×#6658C211yesmaximal, characteristic, complemented
8#91478—31nomaximal
9#91257—31nomaximal
10C2×(C3⋊D3)⋊D5—21no—
11C5×C4⋊(C3⋊D3)C2211yescharacteristic
12C3×D60C2211yescharacteristic
13C3×C4×D15C2211yescharacteristic
14C3×C5×D12C2211yescharacteristic, complemented
15C32×Dic10C2211yescharacteristic, complemented
16#67156—21no—
17C3×#1203—21no—
18C3×D3×Dic5—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63939—51nomaximal, Hall
22C3×#711—31no—
23#6777—31no—
24D5×D12—31no—
25#6768D311yescharacteristic, complemented
26#6715—31no—
27#6705—31no—
28C5×#213—31no—
29C5×D24—31no—
30C3×#677—31no—
31#6658D311yescharacteristic, complemented
32#6656—31no—
33#6656—31no—
34#6656—61no—
35#6652—31no—
36#6649—31no—
37C2×C5×C3⋊D3—21no—
38C2×C3×D15—21no—
39C2×C3×D15D411yescharacteristic
40C2×C3×C5×D3—21no—
41(C3⋊D3)⋊D5—42no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15D411yescharacteristic
44C32×Dic5—21no—
45#4057—91noHall
46C3×#239—51no—
47D3×D12—51no—
48#3755—51no—
49C3×#215—51no—
50#3714D511yescharacteristic, complemented
51#3713—51no—
52#3711—51no—
53C2×D3×D5—61no—
54C2×D3×D5—61no—
55C3×C5×D4—31no—
56D60C2×D311yescharacteristic
57C4×D15C2×D311yescharacteristic
58C5×D12—31no—
59C5×D12—31no—
60C5×D12—61no—
61C5×D12C2×D311yescharacteristic, complemented
62C3×C22⋊D5—61no—
63C2×C3×Dic5—61no—
64C3×D20—31no—
65C3×C4×D5—31no—
66C3×Dic10—21no—
67C3×Dic10C2×D311yescharacteristic, complemented
68C3×Dic10C2×D311yescharacteristic, complemented
69#1203—21no—
70#1203—61no—
71#1201—61no—
72D3×Dic5—21no—
73C3×C5×C8—31no—
74C3⋊#168—31no—
75C3⋊#168—31no—
76C3⋊#168—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#949—151no—
79#914—151no—
80C2×C32×C5C2×D411yescharacteristic
81C5×C3⋊D3—41no—
82C3×D15—21no—
83C3×D15—41no—
84C3×C5×D3—41no—
85#711—31no—
86D4×D5—91no—
87C5×D8—91no—
88#687—91no—
89#686—91no—
90#677—31no—
91#676—91no—
92C2×D32—101no—
93C32×Q8—51no—
94C4⋊(C3⋊D3)C2×D511yescharacteristic
95C3×D12—51no—
96C3×D12—101no—
97C3×D12C2×D511yescharacteristic, complemented
98C3×C4×D3—51no—
99C3×C4×D3—101no—
100#626—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×C5×D3—61no—
108C2×C5×D3—61no—
109C2×C5×D3—121no—
110C2×C3×D5—31no—
111C2×C3×D5—61no—
112D3×D5—122no—
113D3×D5—122no—
114C3×C4×C5—21no—
115C3×C4×C5C22×D311yescharacteristic
116C3×C4×C5C22×D311yescharacteristic
117Dic15#6711yescharacteristic
118C3×Dic5—21no—
119C3×Dic5—21no—
120C3×Dic5—31no—
121C3×Dic5—41no—
122C3×#41—151no—
123#239—51no—
124D3×D4—151no—
125C2×D12—151no—
126C3×SD16—151no—
127#215—51no—
128#215—151no—
129#215—151no—
130#215—301no—
131#213—31no—
132#213—151no—
133#213—151no—
134#208—151no—
135D24—31no—
136ΓL(1,𝔽25)—151no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—181no—
139C5×D4—31no—
140C5×D4—91no—
141C22⋊D5—61no—
142C22⋊D5—181no—
143C2×Dic5—61no—
144D20—31no—
145C4×D5—31no—
146Dic10D3211yescharacteristic, complemented
147C5×C8—31no—
148#168—91no—
149C2×C3⋊D3—21no—
150C2×C3×D3—21no—
151C2×C3×D3—51no—
152C2×C3×D3—101no—
153D32—202no—
154C32×C4—101no—
155C32×C4C22×D511yescharacteristic
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—61no—
165C3×D5—121no—
166C5×D3—41no—
167C5×D3—121no—
168C5×D3—121no—
169C5×D3—241no—
170C22×D3—301no—
171C22×D3—301no—
172C3×Q8—51no—
173C3×Q8—51no—
174C3×Q8—101no—
175C3×D4—31no—
176C3×D4—151no—
177C3×D4—301no—
178C2×C3×C4—151no—
179C2×C3×C4—301no—
180#67—301no—
181D12—31no—
182D12—31no—
183D12—51no—
184D12—61no—
185D12—101no—
186D12—301no—
187D12D3×D511yescharacteristic, complemented
188C4×D3—51no—
189C4×D3—101no—
190C3×C8—31no—
191C3⋊C8—151no—
192C3⋊C8—151no—
193C3⋊C8—301no—
194C3⋊C8D3×D511yescharacteristic, complemented
195C22×C5—61no—
196C22×C5—181no—
197C2×D5—31no—
198C2×D5—61no—
199C2×D5—362no—
200C4×C5C2×D3211yescharacteristic
201Dic5—21no—
202Dic5—31no—
203C2×C32D4×D511yescharacteristic
204C3⋊D3—41no—
205C3×D3—41no—
206C3×D3—101no—
207C3×D3—201no—
208#41—151no—
209C2×D4—451no—
210SD16—151no—
211SD16—451no—
212D8—91no—
213D8—451no—
214M16—451no—
215C3×C5—21no—
216C3×C5#91411yescharacteristic, complemented
217C3×C5#94911yescharacteristic, complemented
218C22×C3—61no—
219C22×C3—151no—
220C22×C3—301no—
221C2×D3—21no—
222C2×D3—51no—
223C2×D3—61no—
224C2×D3—61no—
225C2×D3—101no—
226C2×D3—121no—
227C2×D3—602no—
228C2×D3—602no—
229C3×C4—21no—
230C3×C4—101no—
231C3×C4—101no—
232C3×C4—151no—
233C3×C4—201no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C2×D3×D511yescharacteristic
236Dic3—51no—
237C2×C5—121no—
238C2×C5—361no—
239C2×C5D3×#6711yescharacteristic
240D5—61no—
241D5—121no—
242C32#405711yescharacteristic, complemented, Sylow
243C23—901no—
244Q8—51no—
245D4—31no—
246D4—91no—
247D4—151no—
248D4—301no—
249D4—901no—
250C2×C4—151no—
251C2×C4—301no—
252C8—31no—
253C8—451no—
254C2×C3—21no—
255C2×C3—121no—
256C2×C3—301no—
257C2×C3—601no—
258C2×C3#677711yescharacteristic
259C2×C3D5×#6711yescharacteristic
260D3—41no—
261D3—101no—
262D3—121no—
263D3—121no—
264D3—201no—
265D3—241no—
266C5#6393911yescharacteristic, complemented, Sylow
267C22—61no—
268C22—151no—
269C22—181no—
270C22—301no—
271C22—1802no—
272C4—101no—
273C4—151no—
274C4D32×D511yescharacteristic
275C3—21no—
276C3#9125711yescharacteristic, complemented
277C3#9147811yescharacteristic, complemented
278C2—121no—
279C2—301no—
280C2—361no—
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. #10624226C2semidirect
2. C3×#6768C2semidirect
3. #10623798C2semidirect
4. #10623795C2semidirect
5. C5×#3714C2semidirect
6. #10623761C2semidirect
7. C3×#6658C2semidirect
14. C3×C5×D12C22semidirect
15. C32×Dic10C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
25. #6768D3semidirect
31. #6658D3semidirect
50. #3714D5semidirect
61. C5×D12C2×D3semidirect
67. C3×Dic10C2×D3semidirect
68. C3×Dic10C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
97. C3×D12C2×D5semidirect
102. C3×C3⋊C8C2×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
146. Dic10D32semidirect
187. D12D3×D5semidirect
194. C3⋊C8D3×D5semidirect
216. C3×C5#914semidirect
217. C3×C5#949semidirect
242. C32#4057semidirect
266. C5#63939semidirect
276. C3#91257semidirect
277. C3#91478semidirect

Maximal subgroups

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

subgroupindex
#106242262
C3×#67682
#106237982
#106237952
C5×#37142
#106237612
C3×#66582
#914783
#912573
#639395