Finite Group Databasebeta

← #13085777

Subgroups — #13085777

Order 1440.

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

normal subgroups only

This group has 294 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
1D12×D15C211yesmaximal, characteristic, complemented
2C3×#6766C211yesmaximal, characteristic, complemented
3#10623819C211yesmaximal, characteristic, complemented
4#10623806C211yesmaximal, characteristic, complemented
5C5×#3715C211yesmaximal, characteristic, complemented
6#10623758C211yesmaximal, characteristic, complemented
7C3×#6652C211yesmaximal, characteristic, complemented
8#91502—31nomaximal
9#91498—31nomaximal
10C2×D3×D15—21no—
11C4⋊(C3⋊D15)C2211yescharacteristic
12C3×C4×D15C2211yescharacteristic
13C3×C5×D12C2211yescharacteristic
14C3×C5×Dic6C2211yescharacteristic, complemented
15C32×D20C2211yescharacteristic, complemented
16#67152—21no—
17C3×#1202—21no—
18C3×D5×Dic3—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21#63925—51nomaximal, Hall
22D4×D15—31no—
23C3×#713—31no—
24D5×D12—31no—
25#6766D311yescharacteristic, complemented
26#6717—31no—
27C5×#215—31no—
28C5×#214—31no—
29C3×#686—31no—
30#6660—31no—
31#6653—31no—
32#6653—31no—
33#6653—61no—
34#6652D311yescharacteristic, complemented
35#6651—31no—
36#6649—31no—
37C2×C3⋊D15—21no—
38C2×C3×D15D411yescharacteristic
39C2×C3×C5×D3—21no—
40C2×C32×D5—21no—
41D3×D15—42no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15D411yescharacteristic
44C3×C5×Dic3—21no—
45#4060—91noHall
46C3×#237—51no—
47D3×D12—51no—
48#3753—51no—
49C3×#213—51no—
50#3717—51no—
51#3715D511yescharacteristic, complemented
52#3711—51no—
53C22×D15—61no—
54C2×D3×D5—61no—
55C3×C5×Q8—31no—
56C3×C5×D4—31no—
57#1220—61no—
58D60—31no—
59D60—31no—
60D60—61no—
61C4×D15C2×D311yescharacteristic
62C5×D12C2×D311yescharacteristic
63C5×Dic6C2×D311yescharacteristic, complemented
64C3×D20—21no—
65C3×D20—61no—
66C3×D20C2×D311yescharacteristic, complemented
67C3×D20C2×D311yescharacteristic, complemented
68C3×C4×D5—31no—
69C3×C4×D5—61no—
70#1203—61no—
71#1202—21no—
72D5×Dic3—21no—
73C3⋊#168C2×D311yescharacteristic, complemented
74C3×#168—31no—
75C5×C3⋊C8—31no—
76C5×C3⋊C8—31no—
77C5×C3⋊C8—61no—
78#949—151no—
79#911—151no—
80C2×C32×C5C2×D411yescharacteristic
81C3⋊D15—41no—
82C3×D15—21no—
83C3×C5×D3—41no—
84C32×D5—41no—
85#713—31no—
86D4×D5—91no—
87C5×SD16—91no—
88#688—91no—
89#686—31no—
90D40—91no—
91#676—91no—
92C2×D32—101no—
93C32×D4—51no—
94C4⋊(C3⋊D3)—51no—
95C3×#67—101no—
96C2×C3×Dic3—101no—
97C3×D12C2×D511yescharacteristic
98C3×C4×D3—51no—
99C3×Dic6C2×D511yescharacteristic, complemented
100#626—101no—
101C3⋊(C3⋊C8)C2×D511yescharacteristic, complemented
102C3×C3⋊C8—51no—
103C22×C3×C5—61no—
104C2×D15—61no—
105C2×D15—61no—
106C2×D15—121no—
107C2×D15—122no—
108C2×D15#6711yescharacteristic
109C2×C5×D3—21no—
110C2×C3×D5—21no—
111C2×C3×D5—21no—
112C2×C3×D5—31no—
113C2×C3×D5—41no—
114D3×D5—122no—
115C3×C4×C5—21no—
116C3×C4×C5—61no—
117C3×C4×C5C22×D311yescharacteristic
118C3×C4×C5C22×D311yescharacteristic
119Dic15#6711yescharacteristic
120C3×Dic5—31no—
121C5×Dic3—21no—
122C3×#41—151no—
123#237—51no—
124D3×D4—151no—
125C2×D12—151no—
126C3×D8—151no—
127#215—31no—
128#215—151no—
129#214—31no—
130#213—51no—
131#213—151no—
132#213—151no—
133#213—301no—
134#208—151no—
135#204—151no—
136ΓL(1,𝔽25)—151no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—181no—
139C5×Q8—31no—
140C5×D4—31no—
141C22⋊D5—181no—
142D20—61no—
143D20—91no—
144D20D3211yescharacteristic, complemented
145C4×D5—31no—
146C4×D5—61no—
147C5×C8—91no—
148#168—31no—
149C22×C32—101no—
150C2×C3⋊D3—101no—
151C2×C3×D3—21no—
152C2×C3×D3—51no—
153D32—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—121no—
164D15—121no—
165D15—241no—
166C3×D5—41no—
167C3×D5—41no—
168C3×D5—61no—
169C3×D5—81no—
170C5×D3—41no—
171C22×D3—301no—
172C22×D3—301no—
173C3×Q8—31no—
174C3×D4—31no—
175C3×D4—51no—
176C3×D4—51no—
177C3×D4—101no—
178C3×D4—301no—
179C2×C3×C4—151no—
180C2×C3×C4—301no—
181#67—101no—
182#67—301no—
183C2×Dic3—101no—
184D12—151no—
185D12—151no—
186D12—301no—
187D12—301no—
188D12D3×D511yescharacteristic
189C4×D3—51no—
190Dic6D3×D511yescharacteristic, complemented
191C3×C8—151no—
192C3⋊C8—31no—
193C3⋊C8—31no—
194C3⋊C8—51no—
195C3⋊C8—61no—
196C22×C5—61no—
197C2×D5—21no—
198C2×D5—31no—
199C2×D5—181no—
200C2×D5—362no—
201C4×C5—61no—
202C4×C5C2×D3211yescharacteristic
203Dic5—31no—
204C2×C32—201no—
205C2×C32D4×D511yescharacteristic
206C3⋊D3—201no—
207C3×D3—41no—
208C3×D3—101no—
209#41—151no—
210C2×D4—451no—
211SD16—91no—
212SD16—451no—
213D8—151no—
214D8—451no—
215M16—451no—
216C3×C5—21no—
217C3×C5#91111yescharacteristic, complemented
218C3×C5#94911yescharacteristic, complemented
219C22×C3—61no—
220C22×C3—101no—
221C22×C3—101no—
222C22×C3—151no—
223C22×C3—201no—
224C2×D3—21no—
225C2×D3—51no—
226C2×D3—301no—
227C2×D3—301no—
228C2×D3—601no—
229C2×D3—602no—
230C2×D3—602no—
231C3×C4—21no—
232C3×C4—61no—
233C3×C4—151no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C2×D3×D511yescharacteristic
236Dic3—21no—
237Dic3—51no—
238C2×C5—121no—
239C2×C5D3×#6711yescharacteristic
240D5—41no—
241D5—61no—
242D5—361no—
243C32#406011yescharacteristic, complemented, Sylow
244C23—901no—
245Q8—31no—
246D4—31no—
247D4—51no—
248D4—301no—
249D4—451no—
250D4—901no—
251C2×C4—151no—
252C2×C4—301no—
253C8—91no—
254C8—151no—
255C2×C3—21no—
256C2×C3—121no—
257C2×C3—201no—
258C2×C3—201no—
259C2×C3—301no—
260C2×C3—401no—
261C2×C3D5×#6711yescharacteristic
262C2×C3#679011yescharacteristic
263D3—41no—
264D3—101no—
265D3—601no—
266D3—601no—
267D3—1201no—
268C5#6392511yescharacteristic, complemented, Sylow
269C22—61no—
270C22—101no—
271C22—151no—
272C22—901no—
273C22—1802no—
274C4—61no—
275C4—151no—
276C4D32×D511yescharacteristic
277C3—21no—
278C3#9149811yescharacteristic, complemented
279C3#9150211yescharacteristic, complemented
280C2—121no—
281C2—201no—
282C2—301no—
283C2—1801no—
284C2#1062402711yescharacteristic
subgroupG/H
centerC2#10624027
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. D12×D15C2semidirect
2. C3×#6766C2semidirect
3. #10623819C2semidirect
4. #10623806C2semidirect
5. C5×#3715C2semidirect
6. #10623758C2semidirect
7. C3×#6652C2semidirect
14. C3×C5×Dic6C22semidirect
15. C32×D20C22semidirect
19. C5×C3⋊(C3⋊C8)C22semidirect
20. C3×C3⋊#168C22semidirect
25. #6766D3semidirect
34. #6652D3semidirect
51. #3715D5semidirect
63. C5×Dic6C2×D3semidirect
66. C3×D20C2×D3semidirect
67. C3×D20C2×D3semidirect
73. C3⋊#168C2×D3semidirect
99. C3×Dic6C2×D5semidirect
101. C3⋊(C3⋊C8)C2×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
144. D20D32semidirect
190. Dic6D3×D5semidirect
217. C3×C5#911semidirect
218. C3×C5#949semidirect
243. C32#4060semidirect
268. C5#63925semidirect
278. C3#91498semidirect
279. C3#91502semidirect

Maximal subgroups

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

subgroupindex
D12×D152
C3×#67662
#106238192
#106238062
C5×#37152
#106237582
C3×#66522
#915023
#914983
#639255