Finite Group Databasebeta

← C2×D3×#686 #11754693

Subgroups — C2×D3×#686 #11754693

Order 960.

676 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 160 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
1C2×C5×D3×D4C211yesmaximal, characteristic, complemented
2C2×D3×D20C211yesmaximal, characteristic, complemented
3C2×#6715C211yesmaximal, characteristic, complemented
4C2×C3×#686C211yesmaximal, characteristic, complemented
5D3×#686C218yesmaximal, direct product factor
6C2×#6654C211yesmaximal, characteristic, complemented
7C2×#6652C211yesmaximal, characteristic, complemented
8C2×D3×#168C211yesmaximal, characteristic, complemented
9C22×#686—31nomaximal, Hall
10C23×C5×D3—21no—
11C22×D3×D5—21no—
12C2×C3×C5×D4C2211yescharacteristic, complemented
13C2×D60C2211yescharacteristic, complemented
14C2×C5×#67—21no—
15C5×D3×D4—22no—
16C5×D3×D4C2214yescomplemented
17C2×C5×D12C2211yescharacteristic, complemented
18C2×C4×C5×D3C2211yescharacteristic
19C2×C3×D20C2211yescharacteristic, complemented
20C2×#1202—21no—
21D3×D20—22no—
22D3×D20C2214yescomplemented
23#6715C2214yescomplemented
24C2×C3⋊#168C2211yescharacteristic, complemented
25C3×#686C2214yescomplemented
26C2×C3×#168C2211yescharacteristic, complemented
27#6654C2214yescomplemented
28#6652C2214yescomplemented
29D3×#168C2214yescomplemented
30C2×D3×D8—51nomaximal, Hall
31C22×C5×D4—31no—
32C22×D20—31no—
33C2×#686—31no—
34C2×#686—31no—
35C2×#686—38no—
36C2×#686—31no—
37C2×#686D311yescharacteristic, direct product factor
38C22×#168—31no—
39C23×C3×C5—21no—
40C22×D15—21no—
41C22×C5×D3—21no—
42C22×C5×D3—24no—
43C22×C5×D3—44no—
44C22×C5×D3D411yescharacteristic
45C22×C3×D5—21no—
46C2×D3×D5—24no—
47C2×D3×D5—44no—
48C3×C5×D4—21no—
49C3×C5×D4C2312yescomplemented
50C2×C3×C4×C5C2311yescharacteristic
51D60—21no—
52D60C2312yescomplemented
53C5×#67—24no—
54C2×C5×Dic3D411yescharacteristic
55C5×D12—21no—
56C5×D12C2312yescomplemented
57C4×C5×D3D412yes—
58C4×C5×D3C2312yes—
59C3×D20—21no—
60C3×D20C2312yescomplemented
61#1202—24no—
62C3⋊#168C2312yescomplemented
63C3×#168C2312yescomplemented
64C2×D3×D4—51no—
65C2×D3×D4D511yescharacteristic, complemented
66C2×C3×D8—51no—
67C2×#213—51no—
68C2×#213—51no—
69D3×D8—58no—
70C2×D24—51no—
71C2×D3×C8—51no—
72C24×C5—61no—
73C23×D5—61no—
74C2×C5×D4—34no—
75C2×C5×D4—31no—
76C2×C5×D4—62no—
77C2×C5×D4—61no—
78C2×C5×D4C2×D311yescharacteristic, complemented
79C22×C4×C5—31no—
80C2×D20—31no—
81C2×D20—34no—
82C2×D20—62no—
83C2×D20—61no—
84C2×D20C2×D311yescharacteristic, complemented
85#686—34no—
86#686—34no—
87#686—34no—
88#686C2×D314yesdirect product factor
89C2×#168—31no—
90C2×#168—34no—
91C2×#168C2×D311yescharacteristic, complemented
92C22×D8—151noSylow
93C22×C3×C5—22no—
94C22×C3×C5—42no—
95C22×C3×C5C2×D411yescharacteristic
96C2×D15—22no—
97C2×D15—42no—
98C2×C5×D3—22no—
99C2×C5×D3—42no—
100C2×C5×D3—48no—
101C2×C5×D3D814yescomplemented
102C2×C5×D3C2×D412yes—
103C2×C3×D5—22no—
104C2×C3×D5—42no—
105D3×D5—48no—
106C3×C4×C5C2×D411yescharacteristic
107C3×C4×C5C2411yescharacteristic
108C5×Dic3C2×D412yes—
109C23×D3—21no—
110C23×D3—101no—
111C2×C3×D4—51no—
112C2×C3×D4C2×D511yescharacteristic, complemented
113C2×#67—21no—
114C2×#67—101no—
115D3×D4—22no—
116D3×D4—54no—
117D3×D4—102no—
118D3×D4C2×D514yescomplemented
119C2×D12—51no—
120C2×D12C2×D511yescharacteristic, complemented
121C2×C4×D3C2×D511yescharacteristic
122C3×D8—54no—
123C2×C3×C8—51no—
124#213—54no—
125#213—54no—
126C2×C3⋊C8—51no—
127D24—54no—
128D3×C8—54no—
129C23×C5—21no—
130C23×C5—31no—
131C23×C5—64no—
132C23×C5—61no—
133C23×C5—124no—
134C22×D5—21no—
135C22×D5—64no—
136C22×D5—61no—
137C22×D5—124no—
138C5×D4—21no—
139C5×D4—32no—
140C5×D4—64no—
141C5×D4—61no—
142C5×D4C22×D312yescomplemented
143C2×C4×C5—31no—
144C2×C4×C5—32no—
145C2×C4×C5—32no—
146C2×C4×C5C22×D311yescharacteristic
147D20—21no—
148D20—32no—
149D20—64no—
150D20—61no—
151D20C22×D312yescomplemented
152#168—32no—
153#168C22×D312yescomplemented
154C22×D4—31no—
155C22×D4—151no—
156C2×D8—51no—
157C2×D8—151no—
158C2×D8—158no—
159C2×D8—151no—
160C2×D8—151no—
161C22×C8—151no—
162C2×C3×C5—42no—
163C2×C3×C5C2×D812yescomplemented
164C2×C3×C5C22×D411yescharacteristic
165D15—42no—
166C3×D5—42no—
167C5×D3—42no—
168C5×D3C2×D814yescomplemented
169C23×C3—21no—
170C23×C3—101no—
171C22×D3—21no—
172C22×D3—24no—
173C22×D3—44no—
174C22×D3—101no—
175C22×D3—104no—
176C22×D3—204no—
177C22×D3C22⋊D511yescharacteristic
178C3×D4—21no—
179C3×D4—52no—
180C3×D4—101no—
181C3×D4C22×D512yescomplemented
182C2×C3×C4C22×D511yescharacteristic
183#67—24no—
184#67—104no—
185C2×Dic3C22⋊D511yescharacteristic
186D12—21no—
187D12—52no—
188D12—101no—
189D12C22×D512yescomplemented
190C4×D3C22⋊D512yes—
191C4×D3C22×D512yes—
192C3×C8—52no—
193C3⋊C8—52no—
194C22×C5—22no—
195C22×C5—34no—
196C22×C5—32no—
197C22×C5—42no—
198C22×C5—62no—
199C22×C5—128no—
200C22×C5—122no—
201C22×C5D3×D411yescharacteristic
202C2×D5—22no—
203C2×D5—42no—
204C2×D5—62no—
205C2×D5—128no—
206C2×D5—122no—
207C4×C5—32no—
208C4×C5D3×D411yescharacteristic
209C4×C5C23×D311yescharacteristic
210C24—61no—
211C24—301no—
212C2×D4—34no—
213C2×D4—31no—
214C2×D4—51no—
215C2×D4—62no—
216C2×D4—61no—
217C2×D4—154no—
218C2×D4—151no—
219C2×D4—302no—
220C2×D4—301no—
221C2×D4D3×D511yescharacteristic, complemented
222C22×C4—31no—
223D8—54no—
224D8—154no—
225D8—154no—
226D8—154no—
227C2×C8—51no—
228C2×C8—154no—
229C2×C8—151no—
230C3×C5C22×D811yescharacteristic, complemented, Hall
231C22×C3—22no—
232C22×C3—42no—
233C22×C3—102no—
234C22×C3—202no—
235C22×C3C2×C22⋊D511yescharacteristic
236C2×D3—22no—
237C2×D3—42no—
238C2×D3—48no—
239C2×D3—102no—
240C2×D3—202no—
241C2×D3—208no—
242C2×D3#68614yesdirect product factor
243C2×D3C2×C22⋊D512yes—
244C3×C4C2×C22⋊D511yescharacteristic
245C3×C4C23×D511yescharacteristic
246Dic3C2×C22⋊D512yes—
247C2×C5—34no—
248C2×C5—42no—
249C2×C5—122no—
250C2×C5D3×D812yescomplemented
251C2×C5C2×D3×D411yescharacteristic
252D5—42no—
253D5—122no—
254C23—21no—
255C23—31no—
256C23—64no—
257C23—61no—
258C23—101no—
259C23—124no—
260C23—304no—
261C23—301no—
262C23—604no—
263D4—21no—
264D4—32no—
265D4—52no—
266D4—64no—
267D4—61no—
268D4—101no—
269D4—152no—
270D4—304no—
271D4—301no—
272D4C2×D3×D512yescomplemented
273C2×C4—31no—
274C2×C4—32no—
275C2×C4—32no—
276C2×C4C2×D3×D511yescharacteristic
277C8—52no—
278C8—152no—
279C2×C3—42no—
280C2×C3—202no—
281C2×C3C2×#68612yescomplemented
282C2×C3C22×C22⋊D511yescharacteristic
283D3—42no—
284D3—202no—
285D3C2×#68614yesdirect product factor
286C5C2×D3×D811yescharacteristic, complemented, Sylow
287C22—22no—
288C22—34no—
289C22—32no—
290C22—42no—
291C22—62no—
292C22—102no—
293C22—128no—
294C22—122no—
295C22—202no—
296C22—302no—
297C22—608no—
298C22—602no—
299C22D3×C22⋊D511yescharacteristic
300C4—32no—
301C4D3×C22⋊D511yescharacteristic
302C4C22×D3×D511yescharacteristic
303C3C22×#68611yescharacteristic, complemented, Sylow
304C2—34no—
305C2—42no—
306C2—122no—
307C2—202no—
308C2—602no—
309C2D3×#68612yesdirect product factor
310C2C2×D3×C22⋊D511yescharacteristic
subgroupG/H
centerC22D3×C22⋊D5
commutator, cosocleC3×C4×C5C24
FrattiniC4C22×D3×D5
FittingC2×C3×C5×D4C22
socleC22×C3×C5C2×D4
hypercenterC2×D4D3×D5
hypocenterC3×C5C22×D8
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×D3×D4C2semidirect
2. C2×D3×D20C2semidirect
3. C2×#6715C2semidirect
4. C2×C3×#686C2semidirect
5. D3×#686C2direct (N × H)
6. C2×#6654C2semidirect
7. C2×#6652C2semidirect
8. C2×D3×#168C2semidirect
12. C2×C3×C5×D4C22semidirect
13. C2×D60C22semidirect
16. C5×D3×D4C22semidirect
17. C2×C5×D12C22semidirect
19. C2×C3×D20C22semidirect
22. D3×D20C22semidirect
23. #6715C22semidirect
24. C2×C3⋊#168C22semidirect
25. C3×#686C22semidirect
26. C2×C3×#168C22semidirect
27. #6654C22semidirect
28. #6652C22semidirect
29. D3×#168C22semidirect
37. C2×#686D3direct (N × H)
49. C3×C5×D4C23semidirect
52. D60C23semidirect
56. C5×D12C23semidirect
60. C3×D20C23semidirect
62. C3⋊#168C23semidirect
63. C3×#168C23semidirect
65. C2×D3×D4D5semidirect
78. C2×C5×D4C2×D3semidirect
84. C2×D20C2×D3semidirect
88. #686C2×D3direct (N × H)
91. C2×#168C2×D3semidirect
101. C2×C5×D3D8semidirect
112. C2×C3×D4C2×D5semidirect
118. D3×D4C2×D5semidirect
120. C2×D12C2×D5semidirect
142. C5×D4C22×D3semidirect
151. D20C22×D3semidirect
153. #168C22×D3semidirect
163. C2×C3×C5C2×D8semidirect
168. C5×D3C2×D8semidirect
181. C3×D4C22×D5semidirect
189. D12C22×D5semidirect
221. C2×D4D3×D5semidirect
230. C3×C5C22×D8semidirect
242. C2×D3#686direct (N × H)
250. C2×C5D3×D8semidirect
272. D4C2×D3×D5semidirect
281. C2×C3C2×#686semidirect
285. D3C2×#686direct (N × H)
286. C5C2×D3×D8semidirect
303. C3C22×#686semidirect
309. C2D3×#686direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×D3×D42
C2×D3×D202
C2×#67152
C2×C3×#6862
D3×#6862
C2×#66542
C2×#66522
C2×D3×#1682
C22×#6863
C2×D3×D85