Finite Group Databasebeta

← #11756563

Subgroups — #11756563

Order 960.

692 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×C3×C5×D4C211yesmaximal, characteristic, complemented
2C22×#1220C211yesmaximal, characteristic, complemented
3#91842C212yesmaximal, complemented
4C2×#6719C211yesmaximal, characteristic, complemented
5#91828C214yesmaximal, complemented
6#91826C212yesmaximal, complemented
7#91824C212yesmaximal, complemented
8#91818C212yesmaximal, complemented
9#66138—31nomaximal, Hall
10C24×C3×C5C2211yescharacteristic
11C24×C3×C5C2211yescharacteristic, complemented
12C23×D15C2211yescharacteristic, complemented
13C2×C3×C5×D4—24no—
14C2×C3×C5×D4C2212yes—
15C2×C3×C5×D4C2212yescomplemented
16C22×C3×C4×C5C2211yescharacteristic, complemented
17C2×#1220—24no—
18C2×#1220C2212yes—
19C2×#1220C2212yescomplemented
20C2×#1220C2214yescomplemented
21C22×Dic15C2211yescharacteristic, complemented
22C22×Dic15C2212yescomplemented
23#6719C2212yescomplemented
24#6719C2212yescomplemented
25#6719C2214yescomplemented
26#6714C2214yescomplemented
27#6712C2214yescomplemented
28#5804—51nomaximal, Hall
29C22×C5×D4D311yescharacteristic, complemented
30C22×C22⋊D5—31no—
31#4099—32no—
32C2×C22⋊Dic5—31no—
33#4085—34no—
34#4083—32no—
35#4081—32no—
36#4075—32no—
37C23×C3×C5—22no—
38C23×C3×C5—42no—
39C23×C3×C5—42no—
40C23×C3×C5D414yescomplemented
41C23×C3×C5C2312yes—
42C23×C3×C5C2311yescharacteristic
43C23×C3×C5C2311yescharacteristic
44C23×C3×C5C2311yescharacteristic
45C22×D15—22no—
46C22×D15—42no—
47C22×D15C2312yes—
48C3×C5×D4—24no—
49C3×C5×D4—24no—
50C2×C3×C4×C5—22no—
51C2×C3×C4×C5C2312yes—
52#1220—24no—
53#1220—24no—
54#1220—44no—
55C2×Dic15—22no—
56C2×Dic15—42no—
57C2×Dic15C2312yes—
58C2×Dic15C2314yes—
59C22×C3×D4D511yescharacteristic, complemented
60C22×#67—51no—
61#953—52no—
62C2×#217—51no—
63#939—54no—
64#937—52no—
65#935—52no—
66#929—52no—
67C24×C5C2×D311yescharacteristic
68C24×C5C2×D311yescharacteristic, complemented
69C23×D5—31no—
70C2×C5×D4—24no—
71C2×C5×D4C2×D312yes—
72C2×C5×D4C2×D312yescomplemented
73C22×C4×C5C2×D311yescharacteristic, complemented
74C2×C22⋊D5—32no—
75C2×C22⋊D5—32no—
76C2×C22⋊D5—34no—
77C2×C22⋊D5—64no—
78C22×Dic5—31no—
79C22×Dic5—32no—
80C22⋊Dic5—32no—
81C22⋊Dic5—32no—
82C22⋊Dic5—34no—
83#685—34no—
84#683—34no—
85#534—151noSylow
86C22×C3×C5—22no—
87C22×C3×C5—22no—
88C22×C3×C5—22no—
89C22×C3×C5—22no—
90C22×C3×C5—24no—
91C22×C3×C5—44no—
92C22×C3×C5—44no—
93C22×C3×C5C2×D412yes—
94C22×C3×C5C2×D412yes—
95C22×C3×C5C2×D412yes—
96C22×C3×C5C2411yescharacteristic
97C2×D15—22no—
98C2×D15—24no—
99C2×D15—44no—
100C3×C4×C5—22no—
101Dic15—22no—
102Dic15—24no—
103C24×C3C2×D511yescharacteristic
104C24×C3C2×D511yescharacteristic, complemented
105C23×D3—51no—
106C2×C3×D4—24no—
107C2×C3×D4C2×D512yes—
108C2×C3×D4C2×D512yescomplemented
109C22×C3×C4C2×D511yescharacteristic, complemented
110C2×#67—52no—
111C2×#67—52no—
112C2×#67—54no—
113C2×#67—104no—
114C22×Dic3—51no—
115C22×Dic3—52no—
116#217—52no—
117#217—52no—
118#217—54no—
119(C2×D3)⋊C4—54no—
120Dic3⋊C4—54no—
121C23×C5—22no—
122C23×C5—42no—
123C23×C5—42no—
124C23×C5#6714yescomplemented
125C23×C5C22×D312yes—
126C23×C5C22×D311yescharacteristic
127C23×C5C22×D311yescharacteristic
128C23×C5C22×D311yescharacteristic
129C22×D5—32no—
130C22×D5—62no—
131C22×D5—122no—
132C5×D4—24no—
133C5×D4—24no—
134C2×C4×C5—22no—
135C2×C4×C5C22×D312yes—
136C22⋊D5—64no—
137C22⋊D5—64no—
138C22⋊D5—124no—
139C2×Dic5—32no—
140C2×Dic5—34no—
141C2×Dic5—62no—
142C2×Dic5—122no—
143C22×D4—151no—
144C22×D4D1511yescharacteristic, complemented
145#123—152no—
146#123—152no—
147#121—154no—
148#120—152no—
149#120—152no—
150C2×C22⋊C4—151no—
151C2×C3×C5—22no—
152C2×C3×C5—22no—
153C2×C3×C5—22no—
154C2×C3×C5—42no—
155C2×C3×C5C22×D411yescharacteristic
156C2×C3×C5H(5,𝔽2)12yes—
157D15—42no—
158C23×C3—22no—
159C23×C3—42no—
160C23×C3—42no—
161C23×C3C22⋊D514yescomplemented
162C23×C3C22×D512yes—
163C23×C3C22×D511yescharacteristic
164C23×C3C22×D511yescharacteristic
165C23×C3C22×D511yescharacteristic
166C22×D3—52no—
167C22×D3—102no—
168C22×D3—202no—
169C3×D4—24no—
170C3×D4—24no—
171C2×C3×C4—22no—
172C2×C3×C4C22×D512yes—
173#67—104no—
174#67—104no—
175#67—204no—
176C2×Dic3—52no—
177C2×Dic3—54no—
178C2×Dic3—102no—
179C2×Dic3—202no—
180C22×C5—22no—
181C22×C5—22no—
182C22×C5—22no—
183C22×C5—24no—
184C22×C5—22no—
185C22×C5—44no—
186C22×C5—44no—
187C22×C5C2×#6712yes—
188C22×C5C2×#6712yes—
189C22×C5C2×#6712yes—
190C22×C5C23×D311yescharacteristic
191C2×D5—64no—
192C2×D5—62no—
193C2×D5—124no—
194C4×C5—22no—
195Dic5—64no—
196Dic5—62no—
197C24—151no—
198C24C2×D1511yescharacteristic
199C24C2×D1511yescharacteristic, complemented
200C2×D4—24no—
201C2×D4—152no—
202C2×D4—152no—
203C2×D4—154no—
204C2×D4—304no—
205C2×D4C2×D1512yes—
206C2×D4C2×D1512yescomplemented
207C22×C4—151no—
208C22×C4—152no—
209C22×C4C2×D1511yescharacteristic, complemented
210C4⋊C4—154no—
211C22⋊C4—152no—
212C22⋊C4—152no—
213C22⋊C4—154no—
214C22⋊C4—154no—
215C3×C5#53411yescharacteristic, complemented, Hall
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—22no—
219C22×C3—24no—
220C22×C3—22no—
221C22×C3—44no—
222C22×C3—44no—
223C22×C3C2×C22⋊D512yes—
224C22×C3C2×C22⋊D512yes—
225C22×C3C2×C22⋊D512yes—
226C22×C3C23×D511yescharacteristic
227C2×D3—104no—
228C2×D3—102no—
229C2×D3—204no—
230C3×C4—22no—
231Dic3—104no—
232Dic3—102no—
233C2×C5—22no—
234C2×C5—22no—
235C2×C5—22no—
236C2×C5—42no—
237C2×C5#100412yes—
238C2×C5C22×#6711yescharacteristic
239D5—122no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—152no—
244C23—302no—
245C23—602no—
246C23#122014yescomplemented
247C23C22×D1512yes—
248C23C22×D1511yescharacteristic
249C23C22×D1511yescharacteristic
250C23C22×D1511yescharacteristic
251D4—24no—
252D4—24no—
253D4—304no—
254D4—304no—
255D4—604no—
256C2×C4—22no—
257C2×C4—154no—
258C2×C4—152no—
259C2×C4—302no—
260C2×C4—602no—
261C2×C4C22×D1512yes—
262C2×C3—22no—
263C2×C3—22no—
264C2×C3—22no—
265C2×C3—42no—
266C2×C3#414412yes—
267C2×C3C22×C22⋊D511yescharacteristic
268D3—202no—
269C5#580411yescharacteristic, complemented, Sylow
270C22—22no—
271C22—22no—
272C22—22no—
273C22—24no—
274C22—22no—
275C22—44no—
276C22—44no—
277C22—304no—
278C22—302no—
279C22—604no—
280C22C2×#122012yes—
281C22C2×#122012yes—
282C22C2×#122012yes—
283C22C23×D1511yescharacteristic
284C4—22no—
285C4—304no—
286C4—302no—
287C3#6613811yescharacteristic, complemented, Sylow
288C2—22no—
289C2—22no—
290C2—22no—
291C2—42no—
292C2—602no—
293C2#9209512yes—
294C2C22×#122011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D15
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C5×D4C2
hypercenterC22×D4D15
hypocenterC3×C5#534
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×D4C2semidirect
2. C22×#1220C2semidirect
3. #91842C2semidirect
4. C2×#6719C2semidirect
5. #91828C2semidirect
6. #91826C2semidirect
7. #91824C2semidirect
8. #91818C2semidirect
11. C24×C3×C5C22semidirect
12. C23×D15C22semidirect
15. C2×C3×C5×D4C22semidirect
16. C22×C3×C4×C5C22semidirect
19. C2×#1220C22semidirect
20. C2×#1220C22semidirect
21. C22×Dic15C22semidirect
22. C22×Dic15C22semidirect
23. #6719C22semidirect
24. #6719C22semidirect
25. #6719C22semidirect
26. #6714C22semidirect
27. #6712C22semidirect
29. C22×C5×D4D3semidirect
40. C23×C3×C5D4semidirect
59. C22×C3×D4D5semidirect
68. C24×C5C2×D3semidirect
72. C2×C5×D4C2×D3semidirect
73. C22×C4×C5C2×D3semidirect
104. C24×C3C2×D5semidirect
108. C2×C3×D4C2×D5semidirect
109. C22×C3×C4C2×D5semidirect
124. C23×C5#67semidirect
144. C22×D4D15semidirect
161. C23×C3C22⋊D5semidirect
199. C24C2×D15semidirect
206. C2×D4C2×D15semidirect
209. C22×C4C2×D15semidirect
215. C3×C5#534semidirect
246. C23#1220semidirect
269. C5#5804semidirect
287. C3#66138semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C5×D42
C22×#12202
#918422
C2×#67192
#918282
#918262
#918242
#918182
#661383
#58045