Finite Group Databasebeta

← #13087491

Subgroups — #13087491

Order 1440.

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

normal subgroups only

This group has 294 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
1C3×#6819C211yesmaximal, characteristic, complemented
2D12×D15C211yesmaximal, characteristic, complemented
3#10623909C211yesmaximal, characteristic, complemented
4C3×C5×#213C211yesmaximal, characteristic, complemented
5#10623804C211yesmaximal, characteristic, complemented
6#10623802C211yesmaximal, characteristic, complemented
7#10623795C211yesmaximal, characteristic, complemented
8#91800—31nomaximal
9#91478—31nomaximal
10C2×D3×D15—21no—
11C32×C5×D4C2211yescharacteristic, complemented
12C4⋊(C3⋊D15)C2211yescharacteristic
13C3×#1220—21no—
14C2×C3×Dic15—21no—
15C3×C4×D15C2211yescharacteristic
16C3×Dic30C2211yescharacteristic, complemented
17C3×C5×D12C2211yescharacteristic
18#67152—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63925—51nomaximal, Hall
22#6819D311yescharacteristic, complemented
23D4×D15—31no—
24C3×#711—31no—
25D5×D12—31no—
26C3×C5×D8—31no—
27#6716—31no—
28#6715—31no—
29#6715—31no—
30#6715—61no—
31#6706—31no—
32#6705—31no—
33C5×#213D311yescharacteristic, complemented
34#6658—31no—
35#6656—31no—
36#6649—31no—
37C22×C32×C5—21no—
38C2×C3⋊D15—21no—
39C2×C3×D15D411yescharacteristic
40C2×C3×C5×D3—21no—
41D3×D15—42no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15—21no—
44C3×Dic15D411yescharacteristic
45#4057—91noHall
46C3×#237—51no—
47D3×D12—51no—
48#3753—51no—
49C3×#213D511yescharacteristic, complemented
50#3717—51no—
51#3715—51no—
52#3711—51no—
53C22×D15—61no—
54C2×D3×D5—61no—
55C3×C5×D4—21no—
56C3×C5×D4—31no—
57C3×C5×D4C2×D311yescharacteristic, complemented
58C3×C5×D4C2×D311yescharacteristic, complemented
59#1220—21no—
60#1220—61no—
61C2×Dic15—21no—
62D60—31no—
63D60—31no—
64D60—61no—
65C4×D15C2×D311yescharacteristic
66Dic30C2×D311yescharacteristic, complemented
67C5×D12C2×D311yescharacteristic
68C3×C22⋊D5—61no—
69C2×C3×Dic5—61no—
70C3×C4×D5—31no—
71C3×Dic10—31no—
72#1203—61no—
73C3×C5×C8—31no—
74C3⋊#168—31no—
75C3⋊#168—31no—
76C3⋊#168—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#949—151no—
79#911—151no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C3⋊D15—41no—
83C3×D15—21no—
84C3×C5×D3—41no—
85#711—31no—
86D4×D5—91no—
87C5×D8—31no—
88#687—91no—
89#686—91no—
90#677—91no—
91#676—91no—
92C2×D32—101no—
93C32×D4C2×D511yescharacteristic, complemented
94C4⋊(C3⋊D3)—51no—
95C3×#67—101no—
96C2×C3×Dic3—101no—
97C3×D12C2×D511yescharacteristic
98C3×C4×D3—51no—
99C3×Dic6—51no—
100#626—101no—
101C3⋊(C3⋊C8)—51no—
102C3×C3⋊C8C2×D511yescharacteristic, complemented
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—41no—
106C22×C3×C5—61no—
107C2×D15—61no—
108C2×D15—61no—
109C2×D15—121no—
110C2×D15—122no—
111C2×D15#6711yescharacteristic
112C2×C5×D3—21no—
113C2×C3×D5—31no—
114D3×D5—122no—
115C3×C4×C5—21no—
116C3×C4×C5C22×D311yescharacteristic
117C3×C4×C5C22×D311yescharacteristic
118Dic15—21no—
119Dic15#6711yescharacteristic
120C3×Dic5—31no—
121C3×Dic5—61no—
122C3×#41—151no—
123#237—51no—
124D3×D4—151no—
125C2×D12—151no—
126C3×D8—31no—
127#215—151no—
128#215—151no—
129#214—151no—
130#213—151no—
131#213—151no—
132#213—301no—
133#213D1511yescharacteristic, complemented
134#208—151no—
135#204—151no—
136ΓL(1,𝔽25)—151no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—181no—
139C5×D4—31no—
140C5×D4D3211yescharacteristic, complemented
141C22⋊D5—61no—
142C22⋊D5—181no—
143C2×Dic5—61no—
144D20—91no—
145C4×D5—31no—
146Dic10—31no—
147C5×C8—31no—
148#168—91no—
149C22×C32—21no—
150C2×C3⋊D3—101no—
151C2×C3×D3—21no—
152C2×C3×D3—51no—
153D32—202no—
154C32×C4C22×D511yescharacteristic
155C3×Dic3—51no—
156C3×Dic3—101no—
157AGL(1,ℤ/8)—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5—81no—
162C2×C3×C5—121no—
163C2×C3×C5D3×D411yescharacteristic
164C2×C3×C5C2×#6711yescharacteristic
165D15—21no—
166D15—121no—
167D15—121no—
168D15—241no—
169C3×D5—61no—
170C5×D3—41no—
171C22×D3—301no—
172C22×D3—301no—
173C3×Q8—151no—
174C3×D4—21no—
175C3×D4—31no—
176C3×D4—301no—
177C3×D4D3×D511yescharacteristic, complemented
178C3×D4C2×D1511yescharacteristic, complemented
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—
188D12C2×D1511yescharacteristic
189C4×D3—51no—
190Dic6—51no—
191C3×C8—31no—
192C3⋊C8—151no—
193C3⋊C8—151no—
194C3⋊C8—301no—
195C3⋊C8C2×D1511yescharacteristic, complemented
196C22×C5—21no—
197C22×C5—61no—
198C2×D5—31no—
199C2×D5—181no—
200C2×D5—362no—
201C4×C5C2×D3211yescharacteristic
202Dic5—31no—
203Dic5—61no—
204C2×C32—41no—
205C2×C32D4×D511yescharacteristic
206C3⋊D3—201no—
207C3×D3—41no—
208C3×D3—101no—
209#41—151no—
210C2×D4—451no—
211SD16—451no—
212SD16—451no—
213D8—31no—
214D8—451no—
215M16—451no—
216C3×C5—21no—
217C3×C5#91111yescharacteristic, complemented
218C3×C5#94911yescharacteristic, complemented
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—41no—
222C22×C3—61no—
223C22×C3—151no—
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—151no—
233C3×C4—301no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C22×D1511yescharacteristic
236Dic3—51no—
237Dic3—101no—
238C2×C5—41no—
239C2×C5—121no—
240C2×C5D3×#6711yescharacteristic
241D5—61no—
242D5—361no—
243C32#405711yescharacteristic, complemented, Sylow
244C23—901no—
245Q8—151no—
246D4—31no—
247D4—301no—
248D4—451no—
249D4—901no—
250D4D3×D1511yescharacteristic, complemented
251C2×C4—151no—
252C2×C4—301no—
253C8—31no—
254C8—451no—
255C2×C3—21no—
256C2×C3—41no—
257C2×C3—41no—
258C2×C3—81no—
259C2×C3—121no—
260C2×C3—301no—
261C2×C3D5×#6711yescharacteristic
262C2×C3D4×D1511yescharacteristic
263D3—41no—
264D3—101no—
265D3—601no—
266D3—601no—
267D3—1201no—
268C5#6392511yescharacteristic, complemented, Sylow
269C22—21no—
270C22—61no—
271C22—151no—
272C22—901no—
273C22—1802no—
274C4—151no—
275C4—301no—
276C4C2×D3×D1511yescharacteristic
277C3—21no—
278C3#9147811yescharacteristic, complemented
279C3#9180011yescharacteristic, complemented
280C2—41no—
281C2—121no—
282C2—301no—
283C2—1801no—
284C2#67×D1511yescharacteristic
subgroupG/H
centerC2#67×D15
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterD4D3×D15
hypocenterC32×C5AGL(1,ℤ/8)
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#6819C2semidirect
2. D12×D15C2semidirect
3. #10623909C2semidirect
4. C3×C5×#213C2semidirect
5. #10623804C2semidirect
6. #10623802C2semidirect
7. #10623795C2semidirect
11. C32×C5×D4C22semidirect
16. C3×Dic30C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
22. #6819D3semidirect
33. C5×#213D3semidirect
49. C3×#213D5semidirect
57. C3×C5×D4C2×D3semidirect
58. C3×C5×D4C2×D3semidirect
66. Dic30C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
93. C32×D4C2×D5semidirect
102. C3×C3⋊C8C2×D5semidirect
133. #213D15semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
140. C5×D4D32semidirect
177. C3×D4D3×D5semidirect
178. C3×D4C2×D15semidirect
195. C3⋊C8C2×D15semidirect
217. C3×C5#911semidirect
218. C3×C5#949semidirect
243. C32#4057semidirect
250. D4D3×D15semidirect
268. C5#63925semidirect
278. C3#91478semidirect
279. C3#91800semidirect

Maximal subgroups

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

subgroupindex
C3×#68192
D12×D152
#106239092
C3×C5×#2132
#106238042
#106238022
#106237952
#918003
#914783
#639255