Finite Group Databasebeta

← #11751266

Subgroups — #11751266

Order 960.

248 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 69 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×C5×SD16C211yesmaximal, characteristic, complemented
2#91836C211yesmaximal, characteristic, complemented
3D4×Dic15C211yesmaximal, characteristic, complemented
4C2×#6715C211yesmaximal, characteristic
5#91119C211yesmaximal, characteristic, complemented
6#91105C211yesmaximal, characteristic, complemented
7#91098C211yesmaximal, characteristic, complemented
8#65454—31nomaximal, Hall
9C2×C3×C5×Q8C2211yescharacteristic
10C2×C3×C5×D4C2211yescharacteristic
11C22×Dic15—21no—
12C2×D60C2211yescharacteristic
13C3×C5×SD16—22no—
14C2×C3×C5×C8C2211yescharacteristic
15#6719—21no—
16#6715—22no—
17#6714—21no—
18#6714—21no—
19C4⋊Dic15C2211yescharacteristic
20C4×Dic15C2211yescharacteristic
21C2×C3⋊#168C2211yescharacteristic
22#5165—51nomaximal, Hall
23C2×C5×SD16D311yescharacteristic, complemented
24#4093—31no—
25D4×Dic5—31no—
26C2×#686—31no—
27#3967—31no—
28#3953—31no—
29#3946—31no—
30C23×C3×C5—21no—
31C22×D15—21no—
32C3×C5×Q8—21no—
33C3×C5×Q8—21no—
34C3×C5×D4—21no—
35C3×C5×D4D412yes—
36C2×C3×C4×C5—21no—
37C2×C3×C4×C5C2311yescharacteristic
38C2×Dic15—21no—
39C2×Dic15—42no—
40C2×Dic15D411yescharacteristic
41C2×Dic15D411yescharacteristic
42D60—21no—
43D60—21no—
44C3×C5×C8—21no—
45C3⋊#168—21no—
46C2×C3×SD16D511yescharacteristic, complemented
47#947—51no—
48D4×Dic3—51no—
49C2×#213—51no—
50#835—51no—
51#821—51no—
52Dic3⋊C8—51no—
53C2×C5×Q8C2×D311yescharacteristic
54C2×C5×D4C2×D311yescharacteristic
55C22×Dic5—61no—
56C2×D20—31no—
57C5×SD16—22no—
58C2×C5×C8C2×D311yescharacteristic
59C22⋊Dic5—61no—
60#686—62no—
61#685—61no—
62#685—61no—
63C4⋊Dic5—31no—
64C4×Dic5—31no—
65C2×#168—31no—
66#463—151noSylow
67C22×C3×C5—22no—
68C22×C3×C5—41no—
69C22×C3×C5—41no—
70C22×C3×C5C2×D411yescharacteristic
71C2×D15—41no—
72C2×D15—41no—
73C2×D15—41no—
74C3×C4×C5—41no—
75C3×C4×C5C2×D411yescharacteristic
76C3×C4×C5#4111yescharacteristic
77Dic15—21no—
78Dic15—21no—
79Dic15—41no—
80C2×C3×Q8C2×D511yescharacteristic
81C2×C3×D4C2×D511yescharacteristic
82C22×Dic3—101no—
83C2×D12—51no—
84C3×SD16—22no—
85C2×C3×C8C2×D511yescharacteristic
86#217—101no—
87#213—102no—
88(C2×D3)⋊C4—101no—
89(C2×D3)⋊C4—101no—
90C4⋊Dic3—51no—
91C4×Dic3—51no—
92C2×C3⋊C8—51no—
93C23×C5—21no—
94C22×D5—61no—
95C5×Q8—21no—
96C5×Q8—21no—
97C5×D4—21no—
98C5×D4#6712yes—
99C2×C4×C5—21no—
100C2×C4×C5C22×D311yescharacteristic
101C2×Dic5—31no—
102C2×Dic5—31no—
103C2×Dic5—61no—
104C2×Dic5—122no—
105D20—61no—
106D20—61no—
107C5×C8—21no—
108#168—61no—
109C2×SD16D1511yescharacteristic, complemented
110C2×D8—151no—
111#124—151no—
112C4×D4—151no—
113C4⋊C8—151no—
114#103—151no—
115#102—151no—
116C2×C3×C5—42no—
117C2×C3×C5#12111yescharacteristic
118C2×C3×C5#13511yescharacteristic
119C2×C3×C5AGL(1,ℤ/8)11yescharacteristic
120D15—81no—
121C23×C3—21no—
122C22×D3—101no—
123C3×Q8—21no—
124C3×Q8—21no—
125C3×D4—21no—
126C3×D4C22⋊D512yes—
127C2×C3×C4—21no—
128C2×C3×C4C22×D511yescharacteristic
129C2×Dic3—51no—
130C2×Dic3—51no—
131C2×Dic3—101no—
132C2×Dic3—202no—
133D12—101no—
134D12—101no—
135C3×C8—21no—
136C3⋊C8—101no—
137C22×C5—22no—
138C22×C5—41no—
139C22×C5—41no—
140C22×C5D3×D411yescharacteristic
141C2×D5—121no—
142C2×D5—121no—
143C2×D5—121no—
144C4×C5—41no—
145C4×C5#23711yescharacteristic
146C4×C5C2×#6711yescharacteristic
147Dic5—61no—
148Dic5—61no—
149Dic5—121no—
150C2×Q8C2×D1511yescharacteristic
151C2×D4—151no—
152C2×D4C2×D1511yescharacteristic
153C22×C4—301no—
154SD16—22no—
155D8—302no—
156C2×C8—151no—
157C2×C8C2×D1511yescharacteristic
158C4⋊C4—151no—
159C22⋊C4—301no—
160C22⋊C4—301no—
161C22⋊C4—301no—
162C42—151no—
163C3×C5#46311yescharacteristic, complemented, Hall
164C22×C3—22no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3D4×D511yescharacteristic
168C2×D3—201no—
169C2×D3—201no—
170C2×D3—201no—
171C3×C4—41no—
172C3×C4#71111yescharacteristic
173C3×C4C2×C22⋊D511yescharacteristic
174Dic3—101no—
175Dic3—101no—
176Dic3—201no—
177C2×C5—42no—
178C2×C5#91411yescharacteristic
179C2×C5#91611yescharacteristic
180C2×C5#93911yescharacteristic
181D5—241no—
182C23—21no—
183C23—301no—
184Q8—21no—
185Q8—21no—
186D4—21no—
187D4—301no—
188D4—301no—
189D4#122012yes—
190C2×C4—21no—
191C2×C4—151no—
192C2×C4—151no—
193C2×C4—301no—
194C2×C4—602no—
195C2×C4C22×D1511yescharacteristic
196C8—21no—
197C8—301no—
198C2×C3—42no—
199C2×C3#406011yescharacteristic
200C2×C3#406211yescharacteristic
201C2×C3#408511yescharacteristic
202D3—401no—
203C5#516511yescharacteristic, complemented, Sylow
204C22—22no—
205C22—41no—
206C22—41no—
207C22—601no—
208C22—601no—
209C22—601no—
210C22D4×D1511yescharacteristic
211C4—41no—
212C4—301no—
213C4—301no—
214C4—601no—
215C4#681911yescharacteristic
216C4C2×#122011yescharacteristic
217C3#6545411yescharacteristic, complemented, Sylow
218C2—42no—
219C2—1201no—
220C2#9180311yescharacteristic
221C2#9180511yescharacteristic
222C2#9182811yescharacteristic
subgroupG/H
centerC22D4×D15
commutator, cosocleC2×C3×C4×C5C23
FrattiniC2×C4C22×D15
FittingC2×C3×C5×SD16C2
socleC22×C3×C5C2×D4
prime radicalC2×#6715C2
hypercenterC2×SD16D15
hypocenterC3×C5#463
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×SD16C2semidirect
2. #91836C2semidirect
3. D4×Dic15C2semidirect
5. #91119C2semidirect
6. #91105C2semidirect
7. #91098C2semidirect
23. C2×C5×SD16D3semidirect
46. C2×C3×SD16D5semidirect
109. C2×SD16D15semidirect
163. C3×C5#463semidirect
203. C5#5165semidirect
217. C3#65454semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×SD162
#918362
D4×Dic152
C2×#67152
#911192
#911052
#910982
#654543
#51655