Finite Group Databasebeta

← #11756429

Subgroups — #11756429

Order 960.

472 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 135 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×#124C211yesmaximal, characteristic, complemented
2#91836C211yesmaximal, characteristic, complemented
3#91833C211yesmaximal, characteristic, complemented
4#91828C212yesmaximal, complemented
5#91774C212yesmaximal, complemented
6#91773C212yesmaximal, complemented
7#91770C212yesmaximal, complemented
8#91766C212yesmaximal, complemented
9#91761C212yesmaximal, complemented
10#66004—31nomaximal, Hall
11C2×C3×C5×Q8C2211yescharacteristic
12C2×C3×C5×D4C2211yescharacteristic
13C2×#1220C2212yes—
14C2×#1220C2212yes—
15C22×Dic15C2212yes—
16C2×D60C2211yescharacteristic
17C2×C4×D15C2212yes—
18C2×Dic30C2211yescharacteristic
19C3×C5×C22⋊C4C2212yes—
20C3×C5×C22⋊C4C2212yes—
21C3×C42×C5C2211yescharacteristic
22#6719C2212yes—
23#6714C2212yes—
24#6714C2212yes—
25#6714C2212yes—
26#6712C2212yes—
27#6712C2212yes—
28#6712C2212yes—
29C4×Dic15C2212yes—
30C4×Dic15C2211yescharacteristic
31C4×Dic15C2211yescharacteristic
32#5670—51nomaximal, Hall
33C5×#124D311yescharacteristic, complemented
34#4093—31no—
35#4090—31no—
36#4085—32no—
37#4031—32no—
38#4030—32no—
39#4027—32no—
40#4023—32no—
41#4018—32no—
42C23×C3×C5C2312yes—
43C22×D15C2312yes—
44C3×C5×Q8—41no—
45C3×C5×D4—41no—
46C2×C3×C4×C5C2311yescharacteristic
47C2×C3×C4×C5C2312yes—
48C2×C3×C4×C5C2311yescharacteristic
49C2×C3×C4×C5C2311yescharacteristic
50#1220—24no—
51#1220—42no—
52C2×Dic15—24no—
53C2×Dic15C2312yes—
54C2×Dic15C2312yes—
55C2×Dic15C2312yes—
56D60—41no—
57C4×D15—24no—
58Dic30—41no—
59C3×#124D511yescharacteristic, complemented
60#947—51no—
61#944—51no—
62#939—52no—
63#885—52no—
64#884—52no—
65#67⋊C4—52no—
66#877—52no—
67#872—52no—
68C2×C5×Q8C2×D311yescharacteristic
69C2×C5×D4C2×D311yescharacteristic
70C2×C22⋊D5—32no—
71C2×C22⋊D5—32no—
72C22×Dic5—32no—
73C2×D20—31no—
74C2×C4×D5—32no—
75C2×Dic10—31no—
76C5×C22⋊C4C2×D312yes—
77C5×C22⋊C4C2×D312yes—
78C42×C5C2×D311yescharacteristic
79C22⋊Dic5—32no—
80#685—32no—
81#685—32no—
82#685—32no—
83#683—32no—
84#683—32no—
85#683—32no—
86C4×Dic5—32no—
87C4×Dic5—31no—
88C4×Dic5—31no—
89#555—151noSylow
90C22×C3×C5—22no—
91C22×C3×C5—22no—
92C22×C3×C5—22no—
93C22×C3×C5C2411yescharacteristic
94C2×D15—22no—
95C2×D15—22no—
96C2×D15—22no—
97C3×C4×C5—22no—
98C3×C4×C5—21no—
99C3×C4×C5—21no—
100C3×C4×C5—21no—
101Dic15—22no—
102Dic15—22no—
103Dic15#4114yes—
104C2×C3×Q8C2×D511yescharacteristic
105C2×C3×D4C2×D511yescharacteristic
106C2×#67—52no—
107C2×#67—52no—
108C22×Dic3—52no—
109C2×D12—51no—
110C2×C4×D3—52no—
111C2×Dic6—51no—
112C3×C22⋊C4C2×D512yes—
113C3×C22⋊C4C2×D512yes—
114C3×C42C2×D511yescharacteristic
115#217—52no—
116(C2×D3)⋊C4—52no—
117(C2×D3)⋊C4—52no—
118(C2×D3)⋊C4—52no—
119Dic3⋊C4—52no—
120Dic3⋊C4—52no—
121Dic3⋊C4—52no—
122C4×Dic3—52no—
123C4×Dic3—51no—
124C4×Dic3—51no—
125C23×C5C22×D312yes—
126C22×D5—32no—
127C5×Q8—41no—
128C5×D4—41no—
129C2×C4×C5C22×D311yescharacteristic
130C2×C4×C5C22×D312yes—
131C2×C4×C5C22×D311yescharacteristic
132C2×C4×C5C22×D311yescharacteristic
133C22⋊D5—64no—
134C22⋊D5—122no—
135C2×Dic5—32no—
136C2×Dic5—32no—
137C2×Dic5—32no—
138C2×Dic5—64no—
139D20—121no—
140C4×D5—64no—
141Dic10—121no—
142C4⋊Q8—151no—
143#124—151no—
144#124—152no—
145#124D1511yescharacteristic, complemented
146#121—152no—
147#121—152no—
148C4×D4—152no—
149#117—152no—
150#117—152no—
151C2×C3×C5—42no—
152C2×C3×C5C2×#4112yes—
153C2×C3×C5H(5,𝔽2)11yescharacteristic
154D15—42no—
155C23×C3C22×D512yes—
156C22×D3—52no—
157C3×Q8—41no—
158C3×D4—41no—
159C2×C3×C4C22×D511yescharacteristic
160C2×C3×C4C22×D512yes—
161C2×C3×C4C22×D511yescharacteristic
162C2×C3×C4C22×D511yescharacteristic
163#67—104no—
164#67—202no—
165C2×Dic3—52no—
166C2×Dic3—52no—
167C2×Dic3—52no—
168C2×Dic3—104no—
169D12—201no—
170C4×D3—104no—
171Dic6—201no—
172C22×C5—22no—
173C22×C5—22no—
174C22×C5—22no—
175C22×C5C23×D311yescharacteristic
176C2×D5—62no—
177C2×D5—62no—
178C2×D5—62no—
179C4×C5—21no—
180C4×C5—21no—
181C4×C5—21no—
182C4×C5—22no—
183Dic5—34no—
184Dic5—62no—
185Dic5—62no—
186C2×Q8—151no—
187C2×Q8C2×D1511yescharacteristic
188C2×D4—152no—
189C2×D4—152no—
190C2×D4—151no—
191C2×D4C2×D1511yescharacteristic
192C22×C4—152no—
193C22×C4—152no—
194C4⋊C4—152no—
195C4⋊C4—152no—
196C4⋊C4—152no—
197C22⋊C4—152no—
198C22⋊C4—152no—
199C22⋊C4—152no—
200C22⋊C4—152no—
201C22⋊C4C2×D1512yes—
202C22⋊C4C2×D1512yes—
203C42—151no—
204C42—152no—
205C42—151no—
206C42C2×D1511yescharacteristic
207C3×C5#55511yescharacteristic, complemented, Hall
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3C23×D511yescharacteristic
212C2×D3—102no—
213C2×D3—102no—
214C2×D3—102no—
215C3×C4—21no—
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—22no—
219Dic3—54no—
220Dic3—102no—
221Dic3—102no—
222C2×C5—42no—
223C2×C5#100411yescharacteristic
224C2×C5D3×#4112yes—
225D5—122no—
226C23—152no—
227C23C22×D1512yes—
228Q8—41no—
229Q8—601no—
230D4—41no—
231D4—304no—
232D4—601no—
233D4—602no—
234C2×C4—152no—
235C2×C4—152no—
236C2×C4—152no—
237C2×C4—304no—
238C2×C4—304no—
239C2×C4C22×D1512yes—
240C2×C4C22×D1511yescharacteristic
241C2×C4C22×D1511yescharacteristic
242C2×C4C22×D1511yescharacteristic
243C2×C3—42no—
244C2×C3#414411yescharacteristic
245C2×C3D5×#4112yes—
246D3—202no—
247C5#567011yescharacteristic, complemented, Sylow
248C22—22no—
249C22—22no—
250C22—22no—
251C22—302no—
252C22—302no—
253C22—302no—
254C22C23×D1511yescharacteristic
255C4—21no—
256C4—21no—
257C4—21no—
258C4—22no—
259C4—154no—
260C4—302no—
261C4—302no—
262C3#6600411yescharacteristic, complemented, Sylow
263C2—42no—
264C2—602no—
265C2#9209511yescharacteristic
266C2#41×D1512yes—
subgroupG/H
center, FrattiniC22C23×D15
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×#124C2
hypercenter#124D15
hypocenterC3×C5#555
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C3×C5×#124C2semidirect
2. #91836C2semidirect
3. #91833C2semidirect
4. #91828C2semidirect
5. #91774C2semidirect
6. #91773C2semidirect
7. #91770C2semidirect
8. #91766C2semidirect
9. #91761C2semidirect
33. C5×#124D3semidirect
59. C3×#124D5semidirect
145. #124D15semidirect
207. C3×C5#555semidirect
247. C5#5670semidirect
262. C3#66004semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#1242
#918362
#918332
#918282
#917742
#917732
#917702
#917662
#917612
#660043
#56705