Finite Group Databasebeta

← C3×#91257 #13086404

Subgroups — C3×#91257 #13086404

Order 1440.

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

normal subgroups only

This group has 292 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×#6768C211yesmaximal, characteristic, complemented
3C3×#6705C211yesmaximal, characteristic, complemented
4C3×C5×D24C211yesmaximal, characteristic, complemented
5C32×#677C211yesmaximal, characteristic, complemented
6C3×#6656C211yesmaximal, characteristic, complemented
7C3×#6652C211yesmaximal, characteristic, complemented
8C3×#4057—31nomaximal
9#91257C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×D5—21no—
11C3×C4×D15C2211yescharacteristic
12C3×C5×D12C2211yescharacteristic
13C3×C5×D12C2211yescharacteristic, complemented
14C32×D20C2211yescharacteristic
15C32×Dic10C2211yescharacteristic, complemented
16C3×#1203—21no—
17C3×#1201—21no—
18C3×D3×Dic5—21no—
19C32×C5×C8C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21C3×#914—51nomaximal, Hall
22C3×#711—31no—
23C3×D4×D5—31no—
24#6777C2×C311yescharacteristic, complemented
25#6768C2×C311yescharacteristic, complemented
26C3×C5×D8—31no—
27#6705C2×C311yescharacteristic, complemented
28C5×D24C2×C311yescharacteristic, complemented
29C3×#687—31no—
30C3×#686—31no—
31C3×#677—21no—
32C3×#677D311yescharacteristic, complemented
33C3×#677C2×C311yescharacteristic, complemented
34C3×#676—31no—
35#6656C2×C311yescharacteristic, complemented
36#6652C2×C311yescharacteristic, complemented
37C2×C3×D15D411yescharacteristic
38C2×C3×C5×D3—21no—
39C2×C3×C5×D3—21no—
40C2×C32×D5—21no—
41C3×D3×D5—42no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15D411yescharacteristic
44C32×Dic5—21no—
45#4057—31noHall
46C3×#239—51no—
47C3×D3×D4—51no—
48C32×SD16—51no—
49C3×#215—51no—
50C3×#213—51no—
51C3×D24D511yescharacteristic, complemented
52C3×ΓL(1,𝔽25)—51no—
53C22×C3×D5—61no—
54C2×D3×D5—21no—
55C3×C5×D4—31no—
56C3×C5×D4—31no—
57C4×D15C22×C311yescharacteristic
58C5×D12C22×C311yescharacteristic
59C5×D12C22×C311yescharacteristic, complemented
60C3×C22⋊D5—61no—
61C3×C22⋊D5—61no—
62C2×C3×Dic5—61no—
63C3×D20—21no—
64C3×D20C2×D311yescharacteristic
65C3×D20C22×C311yescharacteristic
66C3×C4×D5—31no—
67C3×Dic10—21no—
68C3×Dic10C2×D311yescharacteristic, complemented
69C3×Dic10C22×C311yescharacteristic, complemented
70#1203—21no—
71#1201—21no—
72D3×Dic5—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#914—51no—
80C2×C32×C5C2×D411yescharacteristic
81C3×D15—21no—
82C3×C5×D3—41no—
83C3×C5×D3—41no—
84C32×D5—41no—
85#711—31no—
86D4×D5—31no—
87C5×D8—31no—
88#687—31no—
89#686—31no—
90#677C3×D311yescharacteristic, complemented
91#676—31no—
92C22×C3×D3—101no—
93C32×Q8—51no—
94C32×D4—51no—
95C3×#67—101no—
96C3×D12—101no—
97C3×D12C2×D511yescharacteristic
98C3×D12C2×D511yescharacteristic, complemented
99C3×C4×D3—51no—
100C3×C4×D3—101no—
101C32×C8C2×D511yescharacteristic, complemented
102C3×C3⋊C8—51no—
103C22×C3×C5—61no—
104C22×C3×C5—61no—
105C2×D15C3×D411yescharacteristic
106C2×C5×D3—21no—
107C2×C5×D3—21no—
108C2×C3×D5—21no—
109C2×C3×D5—21no—
110C2×C3×D5—31no—
111C2×C3×D5—41no—
112C2×C3×D5—122no—
113D3×D5—42no—
114C3×C4×C5—21no—
115C3×C4×C5C22×D311yescharacteristic
116C3×C4×C5C23×C311yescharacteristic
117Dic15C3×D411yescharacteristic
118C3×Dic5—21no—
119C3×Dic5—21no—
120C3×Dic5—31no—
121C3×Dic5—41no—
122C3×#41—151no—
123C2×C3×D4—151no—
124#239—51no—
125D3×D4—51no—
126C3×SD16—51no—
127C3×SD16—51no—
128C3×SD16—101no—
129C3×SD16—151no—
130C3×D8—31no—
131C3×D8—151no—
132C3×M16—151no—
133#215—51no—
134#213—51no—
135D24C3×D511yescharacteristic, complemented
136ΓL(1,𝔽25)—51no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—61no—
139C5×D4—31no—
140C5×D4—31no—
141C22⋊D5—61no—
142C22⋊D5—61no—
143C2×Dic5—61no—
144D20C2×C3×D311yescharacteristic
145C4×D5—31no—
146Dic10C2×C3×D311yescharacteristic, complemented
147C5×C8C2×C3×D311yescharacteristic, complemented
148#168—31no—
149C22×C32—101no—
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—51no—
153C2×C3×D3—202no—
154C32×C4—101no—
155C32×C4C22×D511yescharacteristic
156C3×Dic3—51no—
157AGL(1,ℤ/8)—151noSylow
158C2×C3×C5—21no—
159C2×C3×C5—121no—
160C2×C3×C5—121no—
161C2×C3×C5D3×D411yescharacteristic
162C2×C3×C5C2×C3×D411yescharacteristic
163D15—21no—
164C3×D5—41no—
165C3×D5—41no—
166C3×D5—61no—
167C3×D5—81no—
168C5×D3—41no—
169C5×D3—41no—
170C23×C3—301no—
171C22×D3—101no—
172C3×Q8—51no—
173C3×Q8—51no—
174C3×Q8—101no—
175C3×D4—31no—
176C3×D4—31no—
177C3×D4—51no—
178C3×D4—51no—
179C3×D4—101no—
180C3×D4—301no—
181C3×D4—301no—
182C2×C3×C4—151no—
183C2×C3×C4—301no—
184#67—101no—
185D12—101no—
186D12C2×C3×D511yescharacteristic
187D12C2×C3×D511yescharacteristic, complemented
188C4×D3—51no—
189C4×D3—101no—
190C3×C8—21no—
191C3×C8—151no—
192C3×C8D3×D511yescharacteristic, complemented
193C3×C8C2×C3×D511yescharacteristic, complemented
194C3⋊C8—51no—
195C22×C5—61no—
196C22×C5—61no—
197C2×D5—21no—
198C2×D5—31no—
199C2×D5—122no—
200C4×C5C22×C3×D311yescharacteristic
201Dic5—21no—
202Dic5—31no—
203C2×C32—201no—
204C2×C32D4×D511yescharacteristic
205C3×D3—41no—
206C3×D3—41no—
207C3×D3—101no—
208#41—151no—
209C2×D4—151no—
210SD16—51no—
211SD16—151no—
212D8—31no—
213D8—151no—
214M16—151no—
215C3×C5—21no—
216C3×C5#91411yescharacteristic, complemented
217C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
218C22×C3—61no—
219C22×C3—61no—
220C22×C3—101no—
221C22×C3—101no—
222C22×C3—151no—
223C22×C3—201no—
224C22×C3—602no—
225C2×D3—21no—
226C2×D3—21no—
227C2×D3—51no—
228C2×D3—202no—
229C3×C4—21no—
230C3×C4—101no—
231C3×C4—101no—
232C3×C4—151no—
233C3×C4—201no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C22×C3×D511yescharacteristic
236Dic3—51no—
237C2×C5—121no—
238C2×C5—121no—
239C2×C5C3×D3×D411yescharacteristic
240D5—41no—
241D5—61no—
242C32#405711yescharacteristic, complemented, Sylow
243C23—301no—
244Q8—51no—
245D4—31no—
246D4—31no—
247D4—51no—
248D4—301no—
249D4—301no—
250C2×C4—151no—
251C2×C4—301no—
252C8—151no—
253C8C3×D3×D511yescharacteristic, complemented
254C2×C3—21no—
255C2×C3—121no—
256C2×C3—121no—
257C2×C3—201no—
258C2×C3—201no—
259C2×C3—301no—
260C2×C3—401no—
261C2×C3#677711yescharacteristic
262C2×C3C3×D4×D511yescharacteristic
263D3—41no—
264D3—41no—
265D3—101no—
266C5C3×#91411yescharacteristic, complemented, Sylow
267C22—61no—
268C22—61no—
269C22—101no—
270C22—151no—
271C22—602no—
272C4—101no—
273C4—151no—
274C4C2×C3×D3×D511yescharacteristic
275C3—21no—
276C3#9125711yescharacteristic, direct product factor
277C3C3×#405711yescharacteristic, complemented
278C2—121no—
279C2—121no—
280C2—201no—
281C2—301no—
282C2C3×#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×#6768C2semidirect
3. C3×#6705C2semidirect
4. C3×C5×D24C2semidirect
5. C32×#677C2semidirect
6. C3×#6656C2semidirect
7. C3×#6652C2semidirect
9. #91257C3direct (N × H)
13. C3×C5×D12C22semidirect
15. C32×Dic10C22semidirect
19. C32×C5×C8C22semidirect
20. C3×C3⋊#168C22semidirect
24. #6777C2×C3semidirect
25. #6768C2×C3semidirect
27. #6705C2×C3semidirect
28. C5×D24C2×C3semidirect
32. C3×#677D3semidirect
33. C3×#677C2×C3semidirect
35. #6656C2×C3semidirect
36. #6652C2×C3semidirect
51. C3×D24D5semidirect
59. C5×D12C22×C3semidirect
68. C3×Dic10C2×D3semidirect
69. C3×Dic10C22×C3semidirect
74. C3×C5×C8C2×D3semidirect
75. C3×C5×C8C22×C3semidirect
76. C3⋊#168C22×C3semidirect
90. #677C3×D3semidirect
98. C3×D12C2×D5semidirect
101. C32×C8C2×D5semidirect
135. D24C3×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
146. Dic10C2×C3×D3semidirect
147. C5×C8C2×C3×D3semidirect
187. D12C2×C3×D5semidirect
192. C3×C8D3×D5semidirect
193. C3×C8C2×C3×D5semidirect
216. C3×C5#914semidirect
217. C3×C5C3×AGL(1,ℤ/8)semidirect
242. C32#4057semidirect
253. C8C3×D3×D5semidirect
266. C5C3×#914semidirect
276. C3#91257direct (N × H)
277. C3C3×#4057semidirect

Maximal subgroups

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

subgroupindex
C3×#67772
C3×#67682
C3×#67052
C3×C5×D242
C32×#6772
C3×#66562
C3×#66522
C3×#40573
#912573
C3×#9145