Finite Group Databasebeta

← #11754935

Subgroups — #11754935

Order 960.

668 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 144 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×C3×D4×D5C211yesmaximal, characteristic, complemented
2C2×#6772C211yesmaximal, characteristic, complemented
3#91825C211yesmaximal, characteristic, complemented
4C5×#940C211yesmaximal, characteristic, complemented
5#91565C214yesmaximal, complemented
6#91563C214yesmaximal, complemented
7#91463C211yesmaximal, characteristic, complemented
8#91364C211yesmaximal, characteristic, complemented
9#91357C211yesmaximal, characteristic, complemented
10#66049—31nomaximal, Hall
11C23×C3×D5C2212yes—
12C2×C3×C5×D4C2211yescharacteristic
13C2×Dic30C2211yescharacteristic, complemented
14C2×C5×#67C2214yescomplemented
15C2×C5×D12C2211yescharacteristic, complemented
16C2×C3×C22⋊D5C2212yes—
17C3×D4×D5—24no—
18C2×C3×D20C2211yescharacteristic
19C2×C3×C4×D5C2211yescharacteristic
20C2×#1201C2212yes—
21C2×#1201C2214yescomplemented
22C2×D3×Dic5C2212yescomplemented
23#6772—24no—
24#6719C2214yescomplemented
25C4×Dic15C2211yescharacteristic, complemented
26C4×C5×Dic3C2211yescharacteristic, complemented
27#6668C2212yescomplemented
28#6665C2212yes—
29#6665C2214yescomplemented
30#5805—51nomaximal, Hall
31C2×#711—31no—
32C2×D4×D5D311yescharacteristic, complemented
33C5×#127—31no—
34#4086—31no—
35#4085—34no—
36#4083—34no—
37#4082—31no—
38#4021—31no—
39#4018—31no—
40C23×C3×C5C2312yes—
41C22×C5×D3C2312yes—
42C22×C3×D5—24no—
43C22×C3×D5—44no—
44C22×C3×D5C2312yes—
45C22×C3×D5C2311yescharacteristic
46C3×C5×D4—22no—
47C2×C3×C4×C5C2311yescharacteristic
48C2×Dic15C2312yes—
49C2×Dic15C2312yes—
50Dic30—22no—
51C5×#67—44no—
52C2×C5×Dic3C2312yes—
53C5×D12—22no—
54C3×C22⋊D5—24no—
55C2×C3×Dic5C2311yescharacteristic
56C3×D20—22no—
57C3×C4×D5D414yescomplemented
58#1201—24no—
59#1201—44no—
60D3×Dic5—24no—
61C22×C3×D4—51no—
62C2×#235—51no—
63#953—54no—
64#940—51no—
65#940D511yescharacteristic, complemented
66#938—54no—
67#936—51no—
68#936—51no—
69#926—51no—
70C23×D5C2×D312yes—
71C2×C5×D4—34no—
72C2×C5×D4—31no—
73C2×C5×D4C2×D311yescharacteristic
74C2×C22⋊D5—32no—
75C2×C22⋊D5—34no—
76C2×C22⋊D5C2×D312yes—
77C22×Dic5—32no—
78#711—64no—
79D4×D5—24no—
80C2×D20C2×D311yescharacteristic
81C2×C4×D5C2×D311yescharacteristic
82C2×Dic10—31no—
83C42×C5—31no—
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—24no—
92C22×C3×C5C2411yescharacteristic
93C2×C5×D3—22no—
94C2×C5×D3—24no—
95C2×C3×D5—22no—
96C2×C3×D5—22no—
97C2×C3×D5—24no—
98C2×C3×D5—48no—
99C2×C3×D5C2×D412yes—
100C3×C4×C5C2×D412yes—
101Dic15—22no—
102Dic15—22no—
103C3×Dic5C2×D412yes—
104C5×Dic3—22no—
105C24×C3—52no—
106C2×C3×D4—52no—
107C2×C3×D4—51no—
108C2×C3×D4—104no—
109C2×C3×D4C2×D511yescharacteristic
110C22×C3×C4—51no—
111C2×#67—52no—
112C2×#67—54no—
113C2×#67C2×D514yescomplemented
114#235—104no—
115C2×D12C2×D511yescharacteristic, complemented
116C2×C4×D3—52no—
117C2×Dic6—51no—
118#217—54no—
119#217—52no—
120#217—54no—
121C4⋊Dic3—52no—
122C4×Dic3—51no—
123C4×Dic3C2×D511yescharacteristic, complemented
124C23×C5—32no—
125C23×C5C22×D312yes—
126C22×D5—24no—
127C22×D5—44no—
128C22×D5C22×D312yes—
129C22×D5C22×D311yescharacteristic
130C5×D4—22no—
131C5×D4—62no—
132C5×D4—124no—
133C2×C4×C5—32no—
134C2×C4×C5C22×D311yescharacteristic
135C22⋊D5—24no—
136C22⋊D5—64no—
137C22⋊D5—124no—
138C2×Dic5—32no—
139C2×Dic5—32no—
140C2×Dic5—64no—
141C2×Dic5C22×D311yescharacteristic
142D20—22no—
143C4×D5#6714yescomplemented
144Dic10—62no—
145C2×#41—151no—
146C22×D4—51no—
147#127—31no—
148#127—151no—
149#124—151no—
150#124—151no—
151#121—154no—
152#120—154no—
153#117—151no—
154C2×C3×C5—42no—
155C2×C3×C5C22×D411yescharacteristic
156C2×C3×C5H(5,𝔽2)12yes—
157C3×D5—22no—
158C3×D5—42no—
159C5×D3—42no—
160C23×C3—51no—
161C23×C3—52no—
162C23×C3—104no—
163C23×C3—204no—
164C23×C3C22×D512yes—
165C22×D3C22×D512yes—
166C3×D4—22no—
167C3×D4—104no—
168C3×D4—102no—
169C2×C3×C4—51no—
170C2×C3×C4—54no—
171C2×C3×C4C22×D511yescharacteristic
172#67—44no—
173#67—104no—
174#67—204no—
175C2×Dic3—52no—
176C2×Dic3—52no—
177C2×Dic3C22×D512yes—
178D12—22no—
179C4×D3—104no—
180Dic6—102no—
181C22×C5—24no—
182C22×C5—22no—
183C22×C5—64no—
184C22×C5—62no—
185C22×C5C23×D311yescharacteristic
186C2×D5—22no—
187C2×D5—24no—
188C2×D5—22no—
189C2×D5—48no—
190C2×D5C2×#6712yes—
191C4×C5—62no—
192C4×C5C2×#6712yes—
193Dic5—62no—
194Dic5—62no—
195Dic5C2×#6712yes—
196C24—52no—
197#41—304no—
198C2×Q8—151no—
199C2×D4—34no—
200C2×D4—31no—
201C2×D4—52no—
202C2×D4—51no—
203C2×D4—104no—
204C2×D4—152no—
205C2×D4—154no—
206C2×D4D3×D511yescharacteristic
207C22×C4—51no—
208C22×C4—152no—
209C4⋊C4—152no—
210C22⋊C4—152no—
211C22⋊C4—154no—
212C22⋊C4—154no—
213C42—31no—
214C42—151no—
215C3×C5#53511yescharacteristic, complemented, Hall
216C22×C3—22no—
217C22×C3—24no—
218C22×C3—52no—
219C22×C3—102no—
220C22×C3—102no—
221C22×C3—104no—
222C22×C3—208no—
223C22×C3C23×D511yescharacteristic
224C2×D3—22no—
225C2×D3—24no—
226C3×C4—52no—
227C3×C4D4×D512yes—
228Dic3—22no—
229Dic3—102no—
230Dic3—102no—
231C2×C5—42no—
232C2×C5—122no—
233C2×C5#100412yes—
234C2×C5C22×#6711yescharacteristic
235D5—22no—
236D5—42no—
237C23—32no—
238C23—51no—
239C23—52no—
240C23—104no—
241C23—204no—
242C23C2×D3×D512yes—
243Q8—302no—
244D4—22no—
245D4—62no—
246D4—104no—
247D4—102no—
248D4—124no—
249D4—304no—
250D4—604no—
251C2×C4—32no—
252C2×C4—51no—
253C2×C4—54no—
254C2×C4—152no—
255C2×C4—152no—
256C2×C4—304no—
257C2×C4C2×D3×D511yescharacteristic
258C2×C3—42no—
259C2×C3—102no—
260C2×C3—202no—
261C2×C3C2×D4×D511yescharacteristic
262C2×C3#414412yes—
263D3—42no—
264C5#580511yescharacteristic, complemented, Sylow
265C22—24no—
266C22—22no—
267C22—52no—
268C22—64no—
269C22—62no—
270C22—102no—
271C22—104no—
272C22—102no—
273C22—208no—
274C22C22×D3×D511yescharacteristic
275C4—52no—
276C4—62no—
277C4—302no—
278C4—302no—
279C4D5×#6712yes—
280C3#6604911yescharacteristic, complemented, Sylow
281C2—42no—
282C2—102no—
283C2—122no—
284C2—202no—
285C2#9202512yes—
286C2C2×D5×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#535
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×D5C2semidirect
2. C2×#6772C2semidirect
3. #91825C2semidirect
4. C5×#940C2semidirect
5. #91565C2semidirect
6. #91563C2semidirect
7. #91463C2semidirect
8. #91364C2semidirect
9. #91357C2semidirect
13. C2×Dic30C22semidirect
14. C2×C5×#67C22semidirect
15. C2×C5×D12C22semidirect
21. C2×#1201C22semidirect
22. C2×D3×Dic5C22semidirect
24. #6719C22semidirect
25. C4×Dic15C22semidirect
26. C4×C5×Dic3C22semidirect
27. #6668C22semidirect
29. #6665C22semidirect
32. C2×D4×D5D3semidirect
57. C3×C4×D5D4semidirect
65. #940D5semidirect
113. C2×#67C2×D5semidirect
115. C2×D12C2×D5semidirect
123. C4×Dic3C2×D5semidirect
143. C4×D5#67semidirect
215. C3×C5#535semidirect
264. C5#5805semidirect
280. C3#66049semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D52
C2×#67722
#918252
C5×#9402
#915652
#915632
#914632
#913642
#913572
#660493
#58055