Finite Group Databasebeta

← #13087727

Subgroups — #13087727

Order 1440.

294 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 59 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
2#10624226C211yesmaximal, characteristic, complemented
3C5×#3753C211yesmaximal, characteristic, complemented
4C3×#6715C211yesmaximal, characteristic, complemented
5#10623825C211yesmaximal, characteristic, complemented
6#10623824C211yesmaximal, characteristic, complemented
7#10623819C211yesmaximal, characteristic, complemented
8#91482—31nomaximal
9#91478—31nomaximal
10C2×(C3⋊D3)⋊D5—21no—
11C32×C5×D4C2211yescharacteristic, complemented
12C5×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×#1220—21no—
14C2×C3×Dic15—21no—
15C3×D60C2211yescharacteristic
16C3×C4×D15C2211yescharacteristic
17C3×Dic30C2211yescharacteristic, complemented
18#67156—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21#63925—51nomaximal, Hall
22#6819D311yescharacteristic, complemented
23C3×#711—31no—
24#6777—31no—
25D5×D12—31no—
26#6715D311yescharacteristic, complemented
27C5×#213—31no—
28C5×#213—31no—
29C5×#213—61no—
30C3×#686—31no—
31#6659—31no—
32#6658—31no—
33#6657—31no—
34#6656—31no—
35#6651—31no—
36#6649—31no—
37C22×C32×C5—21no—
38C2×C5×C3⋊D3—21no—
39C2×C3×D15—21no—
40C2×C3×D15D411yescharacteristic
41(C3⋊D3)⋊D5—42no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15—21no—
44C3×Dic15D411yescharacteristic
45#4057—91noHall
46C3×#237—51no—
47D3×D12—51no—
48#3753D511yescharacteristic, complemented
49C3×#213—51no—
50#3717—51no—
51#3715—51no—
52#3711—51no—
53C2×D3×D5—61no—
54C2×D3×D5—61no—
55C3×C5×D4—21no—
56C3×C5×D4C2×D311yescharacteristic, complemented
57C3×C5×D4C2×D311yescharacteristic, complemented
58#1220—21no—
59C2×Dic15—21no—
60D60C2×D311yescharacteristic
61C4×D15C2×D311yescharacteristic
62Dic30C2×D311yescharacteristic, complemented
63C5×D12—31no—
64C5×D12—31no—
65C5×D12—61no—
66C3×C22⋊D5—61no—
67C2×C3×Dic5—61no—
68C3×D20—31no—
69C3×C4×D5—31no—
70C3×Dic10—31no—
71#1203—61no—
72#1201—61no—
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×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C5×C3⋊D3—41no—
83C3×D15—21no—
84C3×D15—41no—
85#711—31no—
86D4×D5—91no—
87C5×D8—91no—
88#687—91no—
89#686—31no—
90#677—91no—
91#676—91no—
92C2×D32—101no—
93C32×D4C2×D511yescharacteristic, complemented
94C4⋊(C3⋊D3)C2×D511yescharacteristic
95C3×#67—101no—
96C2×C3×Dic3—101no—
97C3×D12—51no—
98C3×C4×D3—51no—
99C3×Dic6—51no—
100#626—101no—
101C3⋊(C3⋊C8)C2×D511yescharacteristic, complemented
102C3×C3⋊C8—51no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—41no—
106C2×D15—21no—
107C2×D15#6711yescharacteristic
108C2×C5×D3—61no—
109C2×C5×D3—61no—
110C2×C5×D3—121no—
111C2×C3×D5—31no—
112C2×C3×D5—61no—
113D3×D5—122no—
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—151no—
127#215—151no—
128#215—151no—
129#214—151no—
130#213—31no—
131#213—31no—
132#213—51no—
133#213—61no—
134#208—151no—
135#204—151no—
136ΓL(1,𝔽25)—151no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—181no—
139C5×D4—91no—
140C5×D4D3211yescharacteristic, complemented
141C22⋊D5—61no—
142C22⋊D5—181no—
143C2×Dic5—61no—
144D20—31no—
145C4×D5—31no—
146Dic10—31no—
147C5×C8—91no—
148#168—31no—
149C22×C32—21no—
150C2×C3⋊D3—21no—
151C2×C3×D3—51no—
152C2×C3×D3—101no—
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×C5D3×D411yescharacteristic
163C2×C3×C5C2×#6711yescharacteristic
164D15—21no—
165D15—41no—
166C3×D5—61no—
167C3×D5—121no—
168C5×D3—121no—
169C5×D3—121no—
170C5×D3—241no—
171C22×D3—301no—
172C22×D3—301no—
173C3×Q8—151no—
174C3×D4—21no—
175C3×D4—151no—
176C3×D4—301no—
177C3×D4D3×D511yescharacteristic, complemented
178C3×D4D3×D511yescharacteristic, complemented
179C2×C3×C4—151no—
180C2×C3×C4—301no—
181#67—101no—
182#67—301no—
183C2×Dic3—101no—
184D12—31no—
185D12—31no—
186D12—51no—
187D12—61no—
188D12—301no—
189C4×D3—51no—
190Dic6—51no—
191C3×C8—151no—
192C3⋊C8—31no—
193C3⋊C8—31no—
194C3⋊C8—51no—
195C3⋊C8—61no—
196C22×C5—21no—
197C22×C5—181no—
198C2×D5—31no—
199C2×D5—61no—
200C2×D5—362no—
201C4×C5C2×D3211yescharacteristic
202Dic5—31no—
203Dic5—61no—
204C2×C32—41no—
205C2×C32D4×D511yescharacteristic
206C3⋊D3—41no—
207C3×D3—101no—
208C3×D3—201no—
209#41—151no—
210C2×D4—451no—
211SD16—451no—
212SD16—451no—
213D8—91no—
214D8—151no—
215M16—451no—
216C3×C5—21no—
217C3×C5#91111yescharacteristic, complemented
218C3×C5#94911yescharacteristic, complemented
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—41no—
222C22×C3—151no—
223C22×C3—301no—
224C2×D3—51no—
225C2×D3—61no—
226C2×D3—61no—
227C2×D3—101no—
228C2×D3—121no—
229C2×D3—602no—
230C2×D3—602no—
231C3×C4—21no—
232C3×C4—151no—
233C3×C4—301no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C2×D3×D511yescharacteristic
236Dic3—51no—
237Dic3—101no—
238C2×C5—41no—
239C2×C5—361no—
240C2×C5D3×#6711yescharacteristic
241D5—61no—
242D5—121no—
243C32#405711yescharacteristic, complemented, Sylow
244C23—901no—
245Q8—151no—
246D4—91no—
247D4—151no—
248D4—301no—
249D4—901no—
250D4(C3⋊D3)⋊D511yescharacteristic, complemented
251C2×C4—151no—
252C2×C4—301no—
253C8—91no—
254C8—151no—
255C2×C3—21no—
256C2×C3—41no—
257C2×C3—41no—
258C2×C3—81no—
259C2×C3—301no—
260C2×C3—601no—
261C2×C3D5×#6711yescharacteristic
262C2×C3#679011yescharacteristic
263D3—101no—
264D3—121no—
265D3—121no—
266D3—201no—
267D3—241no—
268C5#6392511yescharacteristic, complemented, Sylow
269C22—21no—
270C22—151no—
271C22—181no—
272C22—301no—
273C22—1802no—
274C4—151no—
275C4—301no—
276C4C2×(C3⋊D3)⋊D511yescharacteristic
277C3—21no—
278C3#9147811yescharacteristic, complemented
279C3#9148211yescharacteristic, complemented
280C2—41no—
281C2—301no—
282C2—361no—
283C2—601no—
284C2#1062423511yescharacteristic
subgroupG/H
centerC2#10624235
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×(C3⋊D3)⋊D5
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterD4(C3⋊D3)⋊D5
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. #10624226C2semidirect
3. C5×#3753C2semidirect
4. C3×#6715C2semidirect
5. #10623825C2semidirect
6. #10623824C2semidirect
7. #10623819C2semidirect
11. C32×C5×D4C22semidirect
17. C3×Dic30C22semidirect
19. C5×C3⋊(C3⋊C8)C22semidirect
20. C3×C3⋊#168C22semidirect
22. #6819D3semidirect
26. #6715D3semidirect
48. #3753D5semidirect
56. C3×C5×D4C2×D3semidirect
57. C3×C5×D4C2×D3semidirect
62. Dic30C2×D3semidirect
73. C3⋊#168C2×D3semidirect
93. C32×D4C2×D5semidirect
101. C3⋊(C3⋊C8)C2×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
140. C5×D4D32semidirect
177. C3×D4D3×D5semidirect
178. C3×D4D3×D5semidirect
217. C3×C5#911semidirect
218. C3×C5#949semidirect
243. C32#4057semidirect
250. D4(C3⋊D3)⋊D5semidirect
268. C5#63925semidirect
278. C3#91478semidirect
279. C3#91482semidirect

Maximal subgroups

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

subgroupindex
C3×#68192
#106242262
C5×#37532
C3×#67152
#106238252
#106238242
#106238192
#914823
#914783
#639255