Finite Group Databasebeta

← #11749052

Subgroups — #11749052

Order 960.

240 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 62 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
1#91827C211yesmaximal, characteristic
2C2×C3×#687C211yesmaximal, characteristic, complemented
3#91390C211yesmaximal, characteristic, complemented
4C2×#6651C211yesmaximal, characteristic, complemented
5C5×#832C211yesmaximal, characteristic, complemented
6#90988C211yesmaximal, characteristic, complemented
7#90971C211yesmaximal, characteristic, complemented
8#65069—31nomaximal, Hall
9C2×C3×C5×D4C2211yescharacteristic
10C2×#1220—21no—
11C2×C4×D15C2211yescharacteristic
12C2×C3×Dic10C2211yescharacteristic
13#6719—21no—
14C4⋊Dic15C2211yescharacteristic
15C5×C4⋊Dic3C2211yescharacteristic
16C2×C5×C3⋊C8C2211yescharacteristic
17C3×#687—22no—
18C2×C3×#168C2211yescharacteristic
19Dic3⋊Dic5—21no—
20#6667—21no—
21#6651—22no—
22#5174—51nomaximal, Hall
23#4084—31no—
24C2×#687D311yescharacteristic, complemented
25C2×#676—31no—
26#4043—31no—
27C5×#102—31no—
28#3948—31no—
29#3939—31no—
30C23×C3×C5—21no—
31C22×D15D411yescharacteristic
32C3×C5×D4—21no—
33C3×C5×D4—21no—
34C2×C3×C4×C5C2311yescharacteristic
35#1220—42no—
36C2×Dic15—21no—
37C2×Dic15D411yescharacteristic
38C4×D15—21no—
39C4×D15—21no—
40C2×C5×Dic3—21no—
41C2×C3×Dic5—21no—
42C3×Dic10—21no—
43C3×Dic10—21no—
44C3×#168D412yescomplemented
45C5×C3⋊C8—21no—
46C2×C3×SD16—51no—
47#946—51no—
48#938—51no—
49C2×ΓL(1,𝔽25)—51no—
50#835—51no—
51#832D511yescharacteristic, complemented
52#818—51no—
53C2×C5×D4C2×D311yescharacteristic
54C2×C22⋊D5—61no—
55C2×C4×D5—31no—
56C2×Dic10C2×D311yescharacteristic
57C2×C5×C8—31no—
58C5×C4⋊C4—31no—
59C22⋊Dic5—61no—
60#687—22no—
61#685—61no—
62C4⋊Dic5—31no—
63C4⋊Dic5—61no—
64C2×#168C2×D311yescharacteristic
65#676—62no—
66#468—151noSylow
67C22×C3×C5—41no—
68C22×C3×C5—41no—
69C22×C3×C5—41no—
70C22×C3×C5C2×D411yescharacteristic
71C2×D15—21no—
72C2×D15—21no—
73C2×D15—21no—
74C3×C4×C5C2×D411yescharacteristic
75C3×C4×C5#4111yescharacteristic
76Dic15—21no—
77Dic15—41no—
78C3×Dic5—41no—
79C5×Dic3—41no—
80C2×C3×Q8—51no—
81C2×C3×D4C2×D511yescharacteristic
82C2×#67—101no—
83C2×C4×D3—51no—
84C3×SD16—102no—
85C2×C3×C8—51no—
86#217—101no—
87(C2×D3)⋊C4—101no—
88C4⋊Dic3—51no—
89C4⋊Dic3C2×D511yescharacteristic
90Dic3⋊C4—101no—
91C2×C3⋊C8C2×D511yescharacteristic
92ΓL(1,𝔽25)—102no—
93C23×C5—21no—
94C22×D5—31no—
95C5×D4—21no—
96C5×D4—21no—
97C2×C4×C5—61no—
98C2×C4×C5C22×D311yescharacteristic
99C22⋊D5—122no—
100C2×Dic5—21no—
101C2×Dic5—31no—
102C2×Dic5—61no—
103C4×D5—61no—
104C4×D5—61no—
105Dic10—21no—
106Dic10—21no—
107C5×C8—61no—
108#168#6712yescomplemented
109C2×SD16—51no—
110C2×M16—151no—
111C22⋊Q8—151no—
112#121—151no—
113#107—151no—
114#103—151no—
115#102—31no—
116C2×C3×C5—81no—
117C2×C3×C5#12111yescharacteristic
118C2×C3×C5AGL(1,ℤ/8)11yescharacteristic
119C2×C3×C5#13711yescharacteristic
120D15—41no—
121C23×C3—21no—
122C22×D3—51no—
123C3×Q8—101no—
124C3×Q8—101no—
125C3×D4—21no—
126C3×D4—21no—
127C2×C3×C4—101no—
128C2×C3×C4C22×D511yescharacteristic
129#67—202no—
130C2×Dic3—21no—
131C2×Dic3—51no—
132C2×Dic3—101no—
133C4×D3—101no—
134C4×D3—101no—
135C3×C8—52no—
136C3⋊C8—21no—
137C22×C5—41no—
138C22×C5—41no—
139C22×C5—41no—
140C22×C5D3×D411yescharacteristic
141C2×D5—61no—
142C2×D5—61no—
143C2×D5—61no—
144C4×C5—121no—
145C4×C5#23711yescharacteristic
146C4×C5C2×#6711yescharacteristic
147Dic5—41no—
148Dic5—61no—
149Dic5—121no—
150C2×Q8—51no—
151C2×D4—301no—
152C2×D4D3×D511yescharacteristic
153C22×C4—151no—
154SD16—102no—
155M16—302no—
156C2×C8—31no—
157C2×C8—51no—
158C4⋊C4—31no—
159C4⋊C4—151no—
160C4⋊C4—301no—
161C22⋊C4—301no—
162C22⋊C4—301no—
163C3×C5#46811yescharacteristic, complemented, Hall
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3D4×D511yescharacteristic
168C2×D3—101no—
169C2×D3—101no—
170C2×D3—101no—
171C3×C4—201no—
172C3×C4#70911yescharacteristic
173C3×C4D4×D511yescharacteristic
174Dic3—41no—
175Dic3—101no—
176Dic3—201no—
177C2×C5—81no—
178C2×C5#91411yescharacteristic
179C2×C5#91511yescharacteristic
180C2×C5#93811yescharacteristic
181D5—121no—
182C23—21no—
183C23—151no—
184Q8—101no—
185Q8—101no—
186D4—21no—
187D4—21no—
188D4—602no—
189C2×C4—61no—
190C2×C4—101no—
191C2×C4—151no—
192C2×C4—301no—
193C2×C4—301no—
194C2×C4—301no—
195C2×C4C2×D3×D511yescharacteristic
196C8—52no—
197C8—61no—
198C2×C3—81no—
199C2×C3#403011yescharacteristic
200C2×C3#405711yescharacteristic
201C2×C3#406111yescharacteristic
202D3—201no—
203C5#517411yescharacteristic, complemented, Sylow
204C22—41no—
205C22—41no—
206C22—41no—
207C22—301no—
208C22—301no—
209C22—301no—
210C22#679011yescharacteristic
211C4—121no—
212C4—201no—
213C4—301no—
214C4—601no—
215C4#677911yescharacteristic
216C4D5×#6711yescharacteristic
217C3#6506911yescharacteristic, complemented, Sylow
218C2—81no—
219C2—601no—
220C2#9148811yescharacteristic
221C2#9150011yescharacteristic
222C2#9153311yescharacteristic
subgroupG/H
centerC22#6790
commutator, cosocleC2×C3×C4×C5C23
FrattiniC2×C4C2×D3×D5
FittingC2×C3×C5×D4C22
socleC22×C3×C5C2×D4
prime radical#91827C2
hypercenterC2×D4D3×D5
hypocenterC3×C5#468
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#687C2semidirect
3. #91390C2semidirect
4. C2×#6651C2semidirect
5. C5×#832C2semidirect
6. #90988C2semidirect
7. #90971C2semidirect
24. C2×#687D3semidirect
44. C3×#168D4semidirect
51. #832D5semidirect
108. #168#67semidirect
163. C3×C5#468semidirect
203. C5#5174semidirect
217. C3#65069semidirect

Maximal subgroups

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

subgroupindex
#918272
C2×C3×#6872
#913902
C2×#66512
C5×#8322
#909882
#909712
#650693
#51745