Finite Group Databasebeta

← #13085779

Subgroups — #13085779

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
1#10624224C211yesmaximal, characteristic, complemented
2C3×#6777C211yesmaximal, characteristic, complemented
3#10623802C211yesmaximal, characteristic, complemented
4#10623795C211yesmaximal, characteristic, complemented
5C5×#3714C211yesmaximal, characteristic, complemented
6#10623757C211yesmaximal, characteristic, complemented
7C3×#6657C211yesmaximal, characteristic, complemented
8#91489—31nomaximal
9#91257—31nomaximal
10C2×C3×D3×D5—21no—
11C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
12C3×C4×D15C2211yescharacteristic
13C3×Dic30C2211yescharacteristic, complemented
14C3×C5×D12C2211yescharacteristic
15C32×D20C2211yescharacteristic
16#67156—21no—
17#67155—21no—
18C3×#1201—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63934—51nomaximal, Hall
22C3×D4×D5—31no—
23#6777D311yescharacteristic, complemented
24#6772—31no—
25#6768—31no—
26#6716—31no—
27#6705—31no—
28C5×#213—31no—
29C5×D24—31no—
30C3×#677—31no—
31#6657D311yescharacteristic, complemented
32#6656—31no—
33#6652—31no—
34#6652—31no—
35#6652—61no—
36#6649—31no—
37C2×C5×C3⋊D3—21no—
38C2×C3×D15D411yescharacteristic
39C2×C3×C5×D3—21no—
40C2×C32×D5—21no—
41C3×D3×D5—42no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15—21no—
44C3×Dic15D411yescharacteristic
45#4057—91noHall
46C3×D3×D4—51no—
47#3799—51no—
48#3753—51no—
49C3×#214—51no—
50#3715—51no—
51#3714D511yescharacteristic, complemented
52#3711—51no—
53C22×C3×D5—61no—
54C2×D3×D5—21no—
55C3×C5×D4—31no—
56C4×D15C2×D311yescharacteristic
57Dic30C2×D311yescharacteristic, complemented
58C5×D12—31no—
59C5×D12—31no—
60C5×D12—61no—
61C5×D12C2×D311yescharacteristic
62C3×C22⋊D5—61no—
63C3×D20—21no—
64C3×D20C2×D311yescharacteristic
65C3×D20C2×D311yescharacteristic
66C3×C4×D5—31no—
67C3×Dic10—31no—
68#1203—61no—
69#1201—21no—
70#1201—61no—
71D3×Dic5—61no—
72D3×Dic5—61no—
73C3×C5×C8—31no—
74C3⋊#168—31no—
75C3⋊#168—31no—
76C3⋊#168—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#932—151no—
79#914—151no—
80C2×C32×C5C2×D411yescharacteristic
81C5×C3⋊D3—41no—
82C3×D15—21no—
83C3×C5×D3—41no—
84C32×D5—41no—
85#711—91no—
86D4×D5—31no—
87C5×D8—91no—
88#687—91no—
89#686—91no—
90#677—31no—
91#676—91no—
92C22×C3×D3—101no—
93C32×D4—51no—
94C4⋊(C3⋊D3)C2×D511yescharacteristic, complemented
95C3×#67—101no—
96C3×D12C2×D511yescharacteristic
97C3×C4×D3—51no—
98C3×Dic6—51no—
99#626—101no—
100#624—101no—
101C3⋊(C3⋊C8)—51no—
102C3×C3⋊C8C2×D511yescharacteristic, complemented
103C22×C3×C5—61no—
104C2×D15#6711yescharacteristic
105C2×C5×D3—21no—
106C2×C5×D3—61no—
107C2×C5×D3—61no—
108C2×C5×D3—121no—
109C2×C3×D5—21no—
110C2×C3×D5—21no—
111C2×C3×D5—31no—
112C2×C3×D5—41no—
113C2×C3×D5—122no—
114D3×D5—42no—
115C3×C4×C5—21no—
116C3×C4×C5C22×D311yescharacteristic
117C3×C4×C5C22×D311yescharacteristic
118Dic15—21no—
119Dic15#6711yescharacteristic
120C3×Dic5—31no—
121C3×Dic5—61no—
122C2×C3×D4—151no—
123#239—151no—
124D3×D4—51no—
125#235—151no—
126C3×SD16—151no—
127#215—151no—
128#214—51no—
129#214—151no—
130#213—31no—
131#213—151no—
132#213—151no—
133#213—301no—
134#208—151no—
135D24—31no—
136ΓL(1,𝔽25)—151no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—61no—
139C5×D4—31no—
140C5×D4—91no—
141C22⋊D5—61no—
142C22⋊D5—181no—
143C2×Dic5—181no—
144D20D3211yescharacteristic
145C4×D5—31no—
146Dic10—31no—
147C5×C8—31no—
148#168—91no—
149C22×C32—101no—
150C2×C3⋊D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—51no—
153C2×C3×D3—202no—
154C32×C4C22×D511yescharacteristic
155C3×Dic3—51no—
156C3×Dic3—101no—
157AGL(1,ℤ/8)—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—121no—
160C2×C3×C5D3×D411yescharacteristic
161C2×C3×C5C2×#6711yescharacteristic
162D15—21no—
163C3×D5—41no—
164C3×D5—41no—
165C3×D5—61no—
166C3×D5—81no—
167C5×D3—41no—
168C5×D3—121no—
169C5×D3—121no—
170C5×D3—241no—
171C23×C3—301no—
172C22×D3—101no—
173C3×Q8—151no—
174C3×D4—31no—
175C3×D4—51no—
176C3×D4—51no—
177C3×D4—101no—
178C3×D4—301no—
179C2×C3×C4—151no—
180#67—101no—
181#67—301no—
182D12—31no—
183D12—31no—
184D12—61no—
185D12—301no—
186D12D3×D511yescharacteristic
187C4×D3—51no—
188C4×D3—301no—
189C4×D3—301no—
190Dic6—51no—
191C3×C8—31no—
192C3⋊C8—151no—
193C3⋊C8—151no—
194C3⋊C8—301no—
195C3⋊C8D3×D511yescharacteristic, complemented
196C22×C5—61no—
197C22×C5—181no—
198C2×D5—21no—
199C2×D5—31no—
200C2×D5—122no—
201C4×C5C2×D3211yescharacteristic
202Dic5—31no—
203Dic5—61no—
204C2×C32—201no—
205C2×C32D4×D511yescharacteristic
206C3⋊D3—41no—
207C3×D3—41no—
208C3×D3—101no—
209#41—451no—
210C2×D4—151no—
211SD16—151no—
212SD16—451no—
213D8—91no—
214D8—451no—
215M16—451no—
216C3×C5—21no—
217C3×C5#91411yescharacteristic, complemented
218C3×C5#93211yescharacteristic, complemented
219C22×C3—61no—
220C22×C3—101no—
221C22×C3—101no—
222C22×C3—151no—
223C22×C3—201no—
224C22×C3—602no—
225C2×D3—21no—
226C2×D3—51no—
227C2×D3—61no—
228C2×D3—61no—
229C2×D3—121no—
230C2×D3—202no—
231C3×C4—21no—
232C3×C4—151no—
233C3×C4—301no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C2×D3×D511yescharacteristic
236Dic3—51no—
237Dic3—101no—
238C2×C5—121no—
239C2×C5—361no—
240C2×C5D3×#6711yescharacteristic
241D5—41no—
242D5—61no—
243C32#405711yescharacteristic, complemented, Sylow
244C23—301no—
245Q8—151no—
246D4—31no—
247D4—51no—
248D4—91no—
249D4—301no—
250D4—901no—
251C2×C4—151no—
252C2×C4—901no—
253C8—31no—
254C8—451no—
255C2×C3—21no—
256C2×C3—121no—
257C2×C3—201no—
258C2×C3—201no—
259C2×C3—301no—
260C2×C3—401no—
261C2×C3#677711yescharacteristic
262C2×C3D5×#6711yescharacteristic
263D3—41no—
264D3—101no—
265D3—121no—
266D3—121no—
267D3—241no—
268C5#6393411yescharacteristic, complemented, Sylow
269C22—61no—
270C22—101no—
271C22—151no—
272C22—181no—
273C22—602no—
274C4—151no—
275C4—301no—
276C4D32×D511yescharacteristic
277C3—21no—
278C3#9125711yescharacteristic, complemented
279C3#9148911yescharacteristic, complemented
280C2—121no—
281C2—201no—
282C2—301no—
283C2—361no—
284C2#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. #10624224C2semidirect
2. C3×#6777C2semidirect
3. #10623802C2semidirect
4. #10623795C2semidirect
5. C5×#3714C2semidirect
6. #10623757C2semidirect
7. C3×#6657C2semidirect
11. C5×C4⋊(C3⋊D3)C22semidirect
13. C3×Dic30C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
23. #6777D3semidirect
31. #6657D3semidirect
51. #3714D5semidirect
57. Dic30C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
94. C4⋊(C3⋊D3)C2×D5semidirect
102. C3×C3⋊C8C2×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
195. C3⋊C8D3×D5semidirect
217. C3×C5#914semidirect
218. C3×C5#932semidirect
243. C32#4057semidirect
268. C5#63934semidirect
278. C3#91257semidirect
279. C3#91489semidirect

Maximal subgroups

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

subgroupindex
#106242242
C3×#67772
#106238022
#106237952
C5×#37142
#106237572
C3×#66572
#914893
#912573
#639345