Finite Group Databasebeta

← #11756474

Subgroups — #11756474

Order 960.

668 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 147 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×#6819C211yesmaximal, characteristic, complemented
2C2×D4×D15C211yesmaximal, characteristic, complemented
3C3×C5×#127C211yesmaximal, characteristic, complemented
4#91829C211yesmaximal, characteristic, complemented
5#91828C214yesmaximal, complemented
6#91826C214yesmaximal, complemented
7#91825C211yesmaximal, characteristic, complemented
8#91764C211yesmaximal, characteristic, complemented
9#91761C211yesmaximal, characteristic, complemented
10#66049—31nomaximal, Hall
11C23×D15C2212yes—
12C2×C3×C5×D4C2211yescharacteristic
13C2×C3×C5×D4C2211yescharacteristic, complemented
14C2×C3×C5×D4C2214yescomplemented
15C2×#1220C2212yes—
16C2×#1220C2212yes—
17C2×#1220C2214yescomplemented
18C22×Dic15C2212yescomplemented
19#6819—24no—
20D4×D15—24no—
21C2×D60C2211yescharacteristic
22C2×C4×D15C2211yescharacteristic
23C2×Dic30C2211yescharacteristic, complemented
24C3×C42×C5C2211yescharacteristic, complemented
25#6719C2214yescomplemented
26#6714C2212yes—
27#6714C2214yescomplemented
28#6712C2212yescomplemented
29C4×Dic15C2211yescharacteristic, complemented
30#5715—51nomaximal, Hall
31C2×#711—31no—
32C2×D4×D5—31no—
33C5×#127D311yescharacteristic, complemented
34#4086—31no—
35#4085—34no—
36#4083—34no—
37#4082—31no—
38#4021—31no—
39#4018—31no—
40C23×C3×C5C2312yes—
41C23×C3×C5C2312yes—
42C22×D15—24no—
43C22×D15—44no—
44C22×D15C2312yes—
45C22×D15C2311yescharacteristic
46C3×C5×D4—22no—
47C3×C5×D4—22no—
48C3×C5×D4—44no—
49C2×C3×C4×C5C2312yes—
50C2×C3×C4×C5C2311yescharacteristic
51#1220—24no—
52#1220—24no—
53#1220—44no—
54C2×Dic15—24no—
55C2×Dic15C2312yes—
56C2×Dic15C2312yes—
57C2×Dic15C2311yescharacteristic
58D60—22no—
59C4×D15D414yescomplemented
60Dic30—22no—
61C2×#237—51no—
62C2×D3×D4—51no—
63C3×#127D511yescharacteristic, complemented
64#940—51no—
65#939—54no—
66#937—54no—
67#936—51no—
68#875—51no—
69#872—51no—
70C23×D5—32no—
71C2×C5×D4C2×D311yescharacteristic
72C2×C5×D4C2×D311yescharacteristic, complemented
73C2×C5×D4C2×D314yescomplemented
74C2×C22⋊D5—32no—
75C2×C22⋊D5—32no—
76C2×C22⋊D5—34no—
77C22×Dic5—32no—
78#711—64no—
79D4×D5—64no—
80C2×D20—31no—
81C2×C4×D5—31no—
82C2×Dic10—31no—
83C42×C5C2×D311yescharacteristic, complemented
84C22⋊Dic5—34no—
85#685—32no—
86#685—34no—
87#683—32no—
88C4×Dic5—31no—
89#535—151noSylow
90C22×C3×C5—22no—
91C22×C3×C5—22no—
92C22×C3×C5—24no—
93C22×C3×C5—24no—
94C22×C3×C5C2411yescharacteristic
95C2×D15—22no—
96C2×D15—22no—
97C2×D15—24no—
98C2×D15—48no—
99C2×D15C2×D412yes—
100C3×C4×C5—22no—
101C3×C4×C5C2×D412yes—
102Dic15—22no—
103Dic15—22no—
104Dic15C2×D412yes—
105C23×D3—52no—
106C2×C3×D4C2×D511yescharacteristic
107C2×C3×D4C2×D511yescharacteristic, complemented
108C2×C3×D4C2×D514yescomplemented
109C2×#67—52no—
110C2×#67—52no—
111C2×#67—54no—
112C22×Dic3—52no—
113#237—104no—
114D3×D4—104no—
115C2×D12—51no—
116C2×C4×D3—51no—
117C2×Dic6—51no—
118C3×C42C2×D511yescharacteristic, complemented
119#217—54no—
120(C2×D3)⋊C4—52no—
121(C2×D3)⋊C4—54no—
122Dic3⋊C4—52no—
123C4×Dic3—51no—
124C23×C5C22×D312yes—
125C23×C5C22×D312yes—
126C22×D5—31no—
127C22×D5—32no—
128C22×D5—64no—
129C22×D5—124no—
130C5×D4—22no—
131C5×D4—22no—
132C5×D4—44no—
133C2×C4×C5C22×D312yes—
134C2×C4×C5C22×D311yescharacteristic
135C22⋊D5—64no—
136C22⋊D5—64no—
137C22⋊D5—124no—
138C2×Dic5—31no—
139C2×Dic5—32no—
140C2×Dic5—32no—
141C2×Dic5—64no—
142D20—62no—
143C4×D5—34no—
144Dic10—62no—
145C2×#41—151no—
146C22×D4—151no—
147#127—151no—
148#127D1511yescharacteristic, complemented
149#124—151no—
150#124—151no—
151#121—154no—
152#120—154no—
153#117—151no—
154C2×C3×C5—42no—
155C2×C3×C5—42no—
156C2×C3×C5C22×D411yescharacteristic
157C2×C3×C5H(5,𝔽2)12yes—
158D15—22no—
159D15—42no—
160C23×C3C22×D512yes—
161C23×C3C22×D512yes—
162C22×D3—51no—
163C22×D3—52no—
164C22×D3—104no—
165C22×D3—204no—
166C3×D4—22no—
167C3×D4—22no—
168C3×D4—44no—
169C2×C3×C4C22×D512yes—
170C2×C3×C4C22×D511yescharacteristic
171#67—104no—
172#67—104no—
173#67—204no—
174C2×Dic3—51no—
175C2×Dic3—52no—
176C2×Dic3—52no—
177C2×Dic3—104no—
178D12—102no—
179C4×D3—54no—
180Dic6—102no—
181C22×C5—22no—
182C22×C5—22no—
183C22×C5—24no—
184C22×C5—24no—
185C22×C5C23×D311yescharacteristic
186C2×D5—32no—
187C2×D5—62no—
188C2×D5—62no—
189C2×D5—64no—
190C2×D5—128no—
191C4×C5—22no—
192C4×C5D3×D412yes—
193Dic5—32no—
194Dic5—62no—
195Dic5—62no—
196C24—152no—
197#41—304no—
198C2×Q8—151no—
199C2×D4—152no—
200C2×D4—152no—
201C2×D4—154no—
202C2×D4—151no—
203C2×D4—304no—
204C2×D4C2×D1511yescharacteristic
205C2×D4C2×D1511yescharacteristic, complemented
206C2×D4C2×D1514yescomplemented
207C22×C4—151no—
208C22×C4—152no—
209C4⋊C4—152no—
210C22⋊C4—154no—
211C22⋊C4—152no—
212C22⋊C4—154no—
213C42—151no—
214C42C2×D1511yescharacteristic, complemented
215C3×C5#53511yescharacteristic, complemented, Hall
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—24no—
219C22×C3—24no—
220C22×C3C23×D511yescharacteristic
221C2×D3—52no—
222C2×D3—102no—
223C2×D3—102no—
224C2×D3—104no—
225C2×D3—208no—
226C3×C4—22no—
227C3×C4D4×D512yes—
228Dic3—52no—
229Dic3—102no—
230Dic3—102no—
231C2×C5—42no—
232C2×C5—42no—
233C2×C5C2×D3×D411yescharacteristic
234C2×C5#100412yes—
235D5—62no—
236D5—122no—
237C23—151no—
238C23—152no—
239C23—304no—
240C23—604no—
241C23C22×D1512yes—
242C23C22×D1512yes—
243Q8—302no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—304no—
248D4—304no—
249D4—302no—
250D4—604no—
251C2×C4—151no—
252C2×C4—154no—
253C2×C4—152no—
254C2×C4—152no—
255C2×C4—304no—
256C2×C4C22×D1512yes—
257C2×C4C22×D1511yescharacteristic
258C2×C3—42no—
259C2×C3—42no—
260C2×C3C2×D4×D511yescharacteristic
261C2×C3#414412yes—
262D3—102no—
263D3—202no—
264C5#571511yescharacteristic, complemented, Sylow
265C22—22no—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—152no—
270C22—302no—
271C22—302no—
272C22—304no—
273C22—608no—
274C22C23×D1511yescharacteristic
275C4—22no—
276C4—152no—
277C4—302no—
278C4—302no—
279C4D4×D1512yes—
280C3#6604911yescharacteristic, complemented, Sylow
281C2—42no—
282C2—42no—
283C2—302no—
284C2—602no—
285C2C2×D4×D1511yescharacteristic
286C2#9209512yes—
subgroupG/H
center, FrattiniC22C23×D15
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×#127C2
hypercenter#127D15
hypocenterC3×C5#535
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#6819C2semidirect
2. C2×D4×D15C2semidirect
3. C3×C5×#127C2semidirect
4. #91829C2semidirect
5. #91828C2semidirect
6. #91826C2semidirect
7. #91825C2semidirect
8. #91764C2semidirect
9. #91761C2semidirect
13. C2×C3×C5×D4C22semidirect
14. C2×C3×C5×D4C22semidirect
17. C2×#1220C22semidirect
18. C22×Dic15C22semidirect
23. C2×Dic30C22semidirect
24. C3×C42×C5C22semidirect
25. #6719C22semidirect
27. #6714C22semidirect
28. #6712C22semidirect
29. C4×Dic15C22semidirect
33. C5×#127D3semidirect
59. C4×D15D4semidirect
63. C3×#127D5semidirect
72. C2×C5×D4C2×D3semidirect
73. C2×C5×D4C2×D3semidirect
83. C42×C5C2×D3semidirect
107. C2×C3×D4C2×D5semidirect
108. C2×C3×D4C2×D5semidirect
118. C3×C42C2×D5semidirect
148. #127D15semidirect
205. C2×D4C2×D15semidirect
206. C2×D4C2×D15semidirect
214. C42C2×D15semidirect
215. C3×C5#535semidirect
264. C5#5715semidirect
280. C3#66049semidirect

Maximal subgroups

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

subgroupindex
C2×#68192
C2×D4×D152
C3×C5×#1272
#918292
#918282
#918262
#918252
#917642
#917612
#660493
#57155