Finite Group Databasebeta

← C3×#91258 #13086405

Subgroups — C3×#91258 #13086405

Order 1440.

294 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 76 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×#6777C211yesmaximal, characteristic, complemented
2C3×#6766C211yesmaximal, characteristic, complemented
3C3×#6705C211yesmaximal, characteristic, complemented
4C3×C5×#204C211yesmaximal, characteristic, complemented
5C32×D40C211yesmaximal, characteristic, complemented
6C3×#6655C211yesmaximal, characteristic, complemented
7C3×#6652C211yesmaximal, characteristic, complemented
8C3×#4060—31nomaximal
9#91258C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×D5—21no—
11C3×C4×D15C2211yescharacteristic
12C3×C5×D12C2211yescharacteristic
13C3×C5×Dic6C2211yescharacteristic, complemented
14C32×D20C2211yescharacteristic
15C32×D20C2211yescharacteristic, complemented
16C3×#1202—21no—
17C3×#1201—21no—
18C3×D5×Dic3—21no—
19C32×C5×C8C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21C3×#911—51nomaximal, Hall
22C3×#713—31no—
23C3×D4×D5—31no—
24#6777C2×C311yescharacteristic, complemented
25#6766C2×C311yescharacteristic, complemented
26C3×C5×SD16—31no—
27#6705C2×C311yescharacteristic, complemented
28C5×#204C2×C311yescharacteristic, complemented
29C3×#688—31no—
30C3×#686—31no—
31C3×D40—21no—
32C3×D40D311yescharacteristic, complemented
33C3×D40C2×C311yescharacteristic, complemented
34C3×#676—31no—
35#6655C2×C311yescharacteristic, complemented
36#6652C2×C311yescharacteristic, complemented
37C2×C3×D15D411yescharacteristic
38C2×C3×C5×D3—21no—
39C2×C32×D5—21no—
40C2×C32×D5—21no—
41C3×D3×D5—42no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15D411yescharacteristic
44C3×C5×Dic3—21no—
45#4060—31noHall
46C3×#237—51no—
47C3×D3×D4—51no—
48C32×D8—51no—
49C3×#214—51no—
50C3×#213—51no—
51C3×#204D511yescharacteristic, complemented
52C3×ΓL(1,𝔽25)—51no—
53C22×C3×D5—61no—
54C2×D3×D5—21no—
55C3×C5×Q8—31no—
56C3×C5×D4—31no—
57C4×D15C22×C311yescharacteristic
58C5×D12C22×C311yescharacteristic
59C5×Dic6C22×C311yescharacteristic, complemented
60C3×C22⋊D5—61no—
61C3×D20—21no—
62C3×D20—21no—
63C3×D20—61no—
64C3×D20C2×D311yescharacteristic
65C3×D20C2×D311yescharacteristic, complemented
66C3×D20C22×C311yescharacteristic
67C3×D20C22×C311yescharacteristic, complemented
68C3×C4×D5—31no—
69C3×C4×D5—61no—
70#1202—21no—
71#1201—21no—
72D5×Dic3—21no—
73C3×C5×C8—21no—
74C3×C5×C8C2×D311yescharacteristic, complemented
75C3×C5×C8C22×C311yescharacteristic, complemented
76C3⋊#168C22×C311yescharacteristic, complemented
77C3×#168—31no—
78C3×AGL(1,ℤ/8)—151no—
79#911—51no—
80C2×C32×C5C2×D411yescharacteristic
81C3×D15—21no—
82C3×C5×D3—41no—
83C32×D5—41no—
84C32×D5—41no—
85#713—31no—
86D4×D5—31no—
87C5×SD16—31no—
88#688—31no—
89#686—31no—
90D40C3×D311yescharacteristic, complemented
91#676—31no—
92C22×C3×D3—101no—
93C32×D4—51no—
94C32×D4—51no—
95C3×#67—101no—
96C3×#67—101no—
97C2×C3×Dic3—101no—
98C3×D12C2×D511yescharacteristic
99C3×C4×D3—51no—
100C3×Dic6C2×D511yescharacteristic, complemented
101C32×C8C2×D511yescharacteristic, complemented
102C3×C3⋊C8—51no—
103C22×C3×C5—61no—
104C2×D15C3×D411yescharacteristic
105C2×C5×D3—21no—
106C2×C3×D5—21no—
107C2×C3×D5—21no—
108C2×C3×D5—21no—
109C2×C3×D5—21no—
110C2×C3×D5—31no—
111C2×C3×D5—41no—
112C2×C3×D5—41no—
113C2×C3×D5—122no—
114D3×D5—42no—
115C3×C4×C5—21no—
116C3×C4×C5—61no—
117C3×C4×C5C22×D311yescharacteristic
118C3×C4×C5C23×C311yescharacteristic
119Dic15C3×D411yescharacteristic
120C3×Dic5—31no—
121C5×Dic3—21no—
122C3×#41—151no—
123C2×C3×D4—151no—
124#237—51no—
125D3×D4—51no—
126C3×SD16—31no—
127C3×SD16—151no—
128C3×D8—51no—
129C3×D8—51no—
130C3×D8—101no—
131C3×D8—151no—
132C3×M16—151no—
133#214—51no—
134#213—51no—
135#204C3×D511yescharacteristic, complemented
136ΓL(1,𝔽25)—51no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—61no—
139C5×Q8—31no—
140C5×D4—31no—
141C22⋊D5—61no—
142D20—61no—
143D20C2×C3×D311yescharacteristic
144D20C2×C3×D311yescharacteristic, complemented
145C4×D5—31no—
146C4×D5—61no—
147C5×C8C2×C3×D311yescharacteristic, complemented
148#168—31no—
149C22×C32—101no—
150C22×C32—101no—
151C2×C3×D3—21no—
152C2×C3×D3—51no—
153C2×C3×D3—202no—
154C32×C4C22×D511yescharacteristic
155C3×Dic3—21no—
156C3×Dic3—51no—
157AGL(1,ℤ/8)—151noSylow
158C2×C3×C5—21no—
159C2×C3×C5—121no—
160C2×C3×C5D3×D411yescharacteristic
161C2×C3×C5C2×C3×D411yescharacteristic
162D15—21no—
163C3×D5—41no—
164C3×D5—41no—
165C3×D5—41no—
166C3×D5—41no—
167C3×D5—61no—
168C3×D5—81no—
169C3×D5—81no—
170C5×D3—41no—
171C23×C3—301no—
172C22×D3—101no—
173C3×Q8—31no—
174C3×D4—31no—
175C3×D4—51no—
176C3×D4—51no—
177C3×D4—51no—
178C3×D4—51no—
179C3×D4—101no—
180C3×D4—101no—
181C3×D4—301no—
182C3×D4—301no—
183C2×C3×C4—151no—
184C2×C3×C4—301no—
185#67—101no—
186#67—101no—
187C2×Dic3—101no—
188D12C2×C3×D511yescharacteristic
189C4×D3—51no—
190Dic6C2×C3×D511yescharacteristic, complemented
191C3×C8—21no—
192C3×C8—151no—
193C3×C8D3×D511yescharacteristic, complemented
194C3×C8C2×C3×D511yescharacteristic, complemented
195C3⋊C8—51no—
196C22×C5—61no—
197C2×D5—21no—
198C2×D5—21no—
199C2×D5—31no—
200C2×D5—122no—
201C4×C5—61no—
202C4×C5C22×C3×D311yescharacteristic
203Dic5—31no—
204C2×C32—201no—
205C2×C32—201no—
206C2×C32D4×D511yescharacteristic
207C3×D3—41no—
208C3×D3—101no—
209#41—151no—
210C2×D4—151no—
211SD16—31no—
212SD16—151no—
213D8—51no—
214D8—151no—
215M16—151no—
216C3×C5—21no—
217C3×C5#91111yescharacteristic, complemented
218C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
219C22×C3—61no—
220C22×C3—101no—
221C22×C3—101no—
222C22×C3—101no—
223C22×C3—101no—
224C22×C3—151no—
225C22×C3—201no—
226C22×C3—201no—
227C22×C3—602no—
228C2×D3—21no—
229C2×D3—51no—
230C2×D3—202no—
231C3×C4—21no—
232C3×C4—61no—
233C3×C4—151no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C22×C3×D511yescharacteristic
236Dic3—21no—
237Dic3—51no—
238C2×C5—121no—
239C2×C5C3×D3×D411yescharacteristic
240D5—41no—
241D5—41no—
242D5—61no—
243C32#406011yescharacteristic, complemented, Sylow
244C23—301no—
245Q8—31no—
246D4—31no—
247D4—51no—
248D4—51no—
249D4—301no—
250D4—301no—
251C2×C4—151no—
252C2×C4—301no—
253C8—151no—
254C8C3×D3×D511yescharacteristic, complemented
255C2×C3—21no—
256C2×C3—121no—
257C2×C3—201no—
258C2×C3—201no—
259C2×C3—201no—
260C2×C3—201no—
261C2×C3—301no—
262C2×C3—401no—
263C2×C3—401no—
264C2×C3#677711yescharacteristic
265C2×C3C3×D4×D511yescharacteristic
266D3—41no—
267D3—101no—
268C5C3×#91111yescharacteristic, complemented, Sylow
269C22—61no—
270C22—101no—
271C22—101no—
272C22—151no—
273C22—602no—
274C4—61no—
275C4—151no—
276C4C2×C3×D3×D511yescharacteristic
277C3—21no—
278C3#9125811yescharacteristic, direct product factor
279C3C3×#406011yescharacteristic, complemented
280C2—121no—
281C2—201no—
282C2—201no—
283C2—301no—
284C2C3×#677711yescharacteristic
subgroupG/H
centerC2×C3#6777
commutator, cosocleC3×C4×C5C23×C3
FrattiniC4C2×C3×D3×D5
FittingC32×C5×C8C22
socleC2×C32×C5C2×D4
hypercenterC3×C8D3×D5
hypocenterC3×C5C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×#6777C2semidirect
2. C3×#6766C2semidirect
3. C3×#6705C2semidirect
4. C3×C5×#204C2semidirect
5. C32×D40C2semidirect
6. C3×#6655C2semidirect
7. C3×#6652C2semidirect
9. #91258C3direct (N × H)
13. C3×C5×Dic6C22semidirect
15. C32×D20C22semidirect
19. C32×C5×C8C22semidirect
20. C3×C3⋊#168C22semidirect
24. #6777C2×C3semidirect
25. #6766C2×C3semidirect
27. #6705C2×C3semidirect
28. C5×#204C2×C3semidirect
32. C3×D40D3semidirect
33. C3×D40C2×C3semidirect
35. #6655C2×C3semidirect
36. #6652C2×C3semidirect
51. C3×#204D5semidirect
59. C5×Dic6C22×C3semidirect
65. C3×D20C2×D3semidirect
67. C3×D20C22×C3semidirect
74. C3×C5×C8C2×D3semidirect
75. C3×C5×C8C22×C3semidirect
76. C3⋊#168C22×C3semidirect
90. D40C3×D3semidirect
100. C3×Dic6C2×D5semidirect
101. C32×C8C2×D5semidirect
135. #204C3×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
144. D20C2×C3×D3semidirect
147. C5×C8C2×C3×D3semidirect
190. Dic6C2×C3×D5semidirect
193. C3×C8D3×D5semidirect
194. C3×C8C2×C3×D5semidirect
217. C3×C5#911semidirect
218. C3×C5C3×AGL(1,ℤ/8)semidirect
243. C32#4060semidirect
254. C8C3×D3×D5semidirect
268. C5C3×#911semidirect
278. C3#91258direct (N × H)
279. C3C3×#4060semidirect

Maximal subgroups

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

subgroupindex
C3×#67772
C3×#67662
C3×#67052
C3×C5×#2042
C32×D402
C3×#66552
C3×#66522
C3×#40603
#912583
C3×#9115