Finite Group Databasebeta

← #11756346

Subgroups — #11756346

Order 960.

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

normal subgroups only

This group has 496 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
1C3×C5×#120C211yesmaximal, characteristic
2C2×#6719C212yesmaximal, complemented
3#91825C211yesmaximal, characteristic, complemented
4#91824C211yesmaximal, characteristic, complemented
5#91824C212yesmaximal, complemented
6D4×Dic15C212yesmaximal, complemented
7#91768C212yesmaximal, complemented
8#91767C212yesmaximal, complemented
9#91766C212yesmaximal, complemented
10#65921—31nomaximal, Hall
11C24×C3×C5C2211yescharacteristic
12C2×C3×C5×D4C2212yes—
13C2×C3×C5×D4C2211yescharacteristic
14C22×Dic15C2212yes—
15C22×Dic15C2212yes—
16C22×Dic15C2211yescharacteristic, complemented
17C2×Dic30C2211yescharacteristic, complemented
18C3×C5×C22⋊C4C2212yes—
19C3×C5×C22⋊C4C2211yescharacteristic
20#6719C2212yes—
21#6719C2212yes—
22#6719C2212yes—
23#6719C2212yes—
24#6719C2212yescomplemented
25#6719C2211yescharacteristic, complemented
26C4⋊Dic15C2212yescomplemented
27#6712C2212yescomplemented
28#6712C2212yescomplemented
29#6712C2212yescomplemented
30C4×Dic15C2211yescharacteristic, complemented
31C4×Dic15C2212yescomplemented
32#5587—51nomaximal, Hall
33C5×#120D311yescharacteristic
34C2×C22⋊Dic5—32no—
35#4082—31no—
36#4081—31no—
37#4081—32no—
38D4×Dic5—32no—
39#4025—32no—
40#4024—32no—
41#4023—32no—
42C23×C3×C5—24no—
43C23×C3×C5—41no—
44C23×C3×C5C2311yescharacteristic
45C23×C3×C5C2311yescharacteristic
46C23×C3×C5C2312yes—
47C3×C5×D4—24no—
48C3×C5×D4—41no—
49C2×C3×C4×C5C2312yes—
50C2×C3×C4×C5C2311yescharacteristic
51C2×Dic15—24no—
52C2×Dic15—42no—
53C2×Dic15—41no—
54C2×Dic15C2312yes—
55C2×Dic15C2312yes—
56C2×Dic15C2312yes—
57C2×Dic15C2312yes—
58Dic30—41no—
59C3×#120D511yescharacteristic
60C2×#217—52no—
61#936—51no—
62#935—51no—
63#935—52no—
64D4×Dic3—52no—
65#879—52no—
66#878—52no—
67#877—52no—
68C24×C5C2×D311yescharacteristic
69C2×C5×D4C2×D312yes—
70C2×C5×D4C2×D311yescharacteristic
71C22×Dic5—32no—
72C22×Dic5—32no—
73C22×Dic5—31no—
74C2×Dic10—31no—
75C5×C22⋊C4C2×D312yes—
76C5×C22⋊C4C2×D311yescharacteristic
77C22⋊Dic5—32no—
78C22⋊Dic5—32no—
79C22⋊Dic5—32no—
80C22⋊Dic5—32no—
81C22⋊Dic5—32no—
82C22⋊Dic5—31no—
83C4⋊Dic5—32no—
84#683—32no—
85#683—32no—
86#683—32no—
87C4×Dic5—31no—
88C4×Dic5—32no—
89#551—151noSylow
90C22×C3×C5—22no—
91C22×C3×C5—22no—
92C22×C3×C5—22no—
93C22×C3×C5—22no—
94C22×C3×C5—21no—
95C22×C3×C5—21no—
96C22×C3×C5—44no—
97C22×C3×C5#4114yes—
98C22×C3×C5C2411yescharacteristic
99C3×C4×C5—22no—
100C3×C4×C5—21no—
101Dic15—22no—
102Dic15—22no—
103Dic15—22no—
104Dic15—22no—
105C24×C3C2×D511yescharacteristic
106C2×C3×D4C2×D512yes—
107C2×C3×D4C2×D511yescharacteristic
108C22×Dic3—52no—
109C22×Dic3—52no—
110C22×Dic3—51no—
111C2×Dic6—51no—
112C3×C22⋊C4C2×D512yes—
113C3×C22⋊C4C2×D511yescharacteristic
114#217—52no—
115#217—52no—
116#217—52no—
117#217—52no—
118#217—52no—
119#217—51no—
120C4⋊Dic3—52no—
121Dic3⋊C4—52no—
122Dic3⋊C4—52no—
123Dic3⋊C4—52no—
124C4×Dic3—51no—
125C4×Dic3—52no—
126C23×C5—24no—
127C23×C5—41no—
128C23×C5C22×D311yescharacteristic
129C23×C5C22×D311yescharacteristic
130C23×C5C22×D312yes—
131C5×D4—24no—
132C5×D4—41no—
133C2×C4×C5C22×D312yes—
134C2×C4×C5C22×D311yescharacteristic
135C2×Dic5—32no—
136C2×Dic5—32no—
137C2×Dic5—32no—
138C2×Dic5—32no—
139C2×Dic5—64no—
140C2×Dic5—122no—
141C2×Dic5—121no—
142Dic10—121no—
143#126—152no—
144#124—151no—
145#123—152no—
146#123—151no—
147C22⋊Q8—152no—
148#120D1511yescharacteristic
149C4×D4—152no—
150#117—152no—
151C2×C22⋊C4—152no—
152C2×C3×C5—24no—
153C2×C3×C5—41no—
154C2×C3×C5—41no—
155C2×C3×C5C2×#4112yes—
156C2×C3×C5H(5,𝔽2)11yescharacteristic
157C23×C3—24no—
158C23×C3—41no—
159C23×C3C22×D511yescharacteristic
160C23×C3C22×D511yescharacteristic
161C23×C3C22×D512yes—
162C3×D4—24no—
163C3×D4—41no—
164C2×C3×C4C22×D512yes—
165C2×C3×C4C22×D511yescharacteristic
166C2×Dic3—52no—
167C2×Dic3—52no—
168C2×Dic3—52no—
169C2×Dic3—52no—
170C2×Dic3—104no—
171C2×Dic3—202no—
172C2×Dic3—201no—
173Dic6—201no—
174C22×C5—22no—
175C22×C5—22no—
176C22×C5—22no—
177C22×C5—22no—
178C22×C5—21no—
179C22×C5—21no—
180C22×C5—44no—
181C22×C5#23714yes—
182C22×C5C23×D311yescharacteristic
183C4×C5—22no—
184C4×C5—21no—
185Dic5—62no—
186Dic5—62no—
187Dic5—62no—
188Dic5—62no—
189C24C2×D1511yescharacteristic
190C2×Q8—151no—
191C2×D4C2×D1512yes—
192C2×D4C2×D1511yescharacteristic
193C22×C4—152no—
194C22×C4—152no—
195C22×C4—151no—
196C4⋊C4—152no—
197C4⋊C4—152no—
198C4⋊C4—152no—
199C4⋊C4—152no—
200C22⋊C4—152no—
201C22⋊C4—152no—
202C22⋊C4—152no—
203C22⋊C4—152no—
204C22⋊C4—152no—
205C22⋊C4—151no—
206C22⋊C4C2×D1512yes—
207C22⋊C4C2×D1511yescharacteristic
208C42—151no—
209C42—152no—
210C3×C5#55111yescharacteristic, complemented, Hall
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—44no—
218C22×C3#71114yes—
219C22×C3C23×D511yescharacteristic
220C3×C4—22no—
221C3×C4—21no—
222Dic3—102no—
223Dic3—102no—
224Dic3—102no—
225Dic3—102no—
226C2×C5—24no—
227C2×C5—41no—
228C2×C5—41no—
229C2×C5C2×#23712yes—
230C2×C5#100411yescharacteristic
231C23—24no—
232C23—41no—
233C23C22×D1511yescharacteristic
234C23C22×D1511yescharacteristic
235C23C22×D1512yes—
236Q8—601no—
237D4—24no—
238D4—41no—
239C2×C4—152no—
240C2×C4—152no—
241C2×C4—152no—
242C2×C4—152no—
243C2×C4—304no—
244C2×C4—601no—
245C2×C4—602no—
246C2×C4C22×D1512yes—
247C2×C4C22×D1511yescharacteristic
248C2×C3—24no—
249C2×C3—41no—
250C2×C3—41no—
251C2×C3C2×#71112yes—
252C2×C3#414411yescharacteristic
253C5#558711yescharacteristic, complemented, Sylow
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—21no—
259C22—21no—
260C22—44no—
261C22#681914yes—
262C22C23×D1511yescharacteristic
263C4—22no—
264C4—21no—
265C4—302no—
266C4—302no—
267C4—302no—
268C4—302no—
269C3#6592111yescharacteristic, complemented, Sylow
270C2—24no—
271C2—41no—
272C2—41no—
273C2C2×#681912yes—
274C2#9209511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D15
commutator, socle, cosocleC22×C3×C5C24
Fitting, prime radicalC3×C5×#120C2
hypercenter#120D15
hypocenterC3×C5#551
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
2. C2×#6719C2semidirect
3. #91825C2semidirect
4. #91824C2semidirect
5. #91824C2semidirect
6. D4×Dic15C2semidirect
7. #91768C2semidirect
8. #91767C2semidirect
9. #91766C2semidirect
16. C22×Dic15C22semidirect
17. C2×Dic30C22semidirect
24. #6719C22semidirect
25. #6719C22semidirect
26. C4⋊Dic15C22semidirect
27. #6712C22semidirect
28. #6712C22semidirect
29. #6712C22semidirect
30. C4×Dic15C22semidirect
31. C4×Dic15C22semidirect
210. C3×C5#551semidirect
253. C5#5587semidirect
269. C3#65921semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#1202
C2×#67192
#918252
#918242
D4×Dic152
#917682
#917672
#917662
#659213
#55875