Finite Group Databasebeta

← #11751143

Subgroups — #11751143

Order 960.

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

normal subgroups only

This group has 240 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
1C3×C5×#121C211yesmaximal, characteristic
2C2×#6716C211yesmaximal, characteristic, complemented
3#91813C211yesmaximal, characteristic, complemented
4C2×#6710C211yesmaximal, characteristic, complemented
5#91116C211yesmaximal, characteristic, complemented
6#91094C211yesmaximal, characteristic, complemented
7#91091C211yesmaximal, characteristic, complemented
8#65331—31nomaximal, Hall
9C2×C3×C5×D4—21no—
10C2×C3×C5×D4C2211yescharacteristic
11C22×C3×C4×C5C2211yescharacteristic
12C2×Dic30C2211yescharacteristic
13C3×C5×C4⋊C4C2211yescharacteristic
14C3×C5×C22⋊C4—21no—
15#6719—21no—
16#6716—22no—
17C4⋊Dic15C2211yescharacteristic
18#6712—21no—
19#6710—22no—
20C2×C3⋊#168C2211yescharacteristic
21C2×C3⋊#168C2211yescharacteristic
22#5042—51nomaximal, Hall
23C5×#121D311yescharacteristic
24C2×#687—31no—
25#4070—31no—
26C2×#681—31no—
27#3964—31no—
28#3942—31no—
29#3939—31no—
30C23×C3×C5—21no—
31C23×C3×C5D411yescharacteristic
32C3×C5×D4—21no—
33C3×C5×D4—21no—
34C3×C5×D4—42no—
35C2×C3×C4×C5—21no—
36C2×C3×C4×C5—21no—
37C2×C3×C4×C5—21no—
38C2×C3×C4×C5D411yescharacteristic
39C2×C3×C4×C5C2311yescharacteristic
40C2×Dic15—21no—
41C2×Dic15—21no—
42Dic30—21no—
43Dic30—21no—
44C3⋊#168—21no—
45C3⋊#168D412yescomplemented
46C3×#121D511yescharacteristic
47C2×#214—51no—
48#924—51no—
49C2×#208—51no—
50#832—51no—
51#810—51no—
52#807—51no—
53C2×C5×D4—21no—
54C2×C5×D4C2×D311yescharacteristic
55C22×C4×C5C2×D311yescharacteristic
56C2×Dic10—31no—
57C5×C4⋊C4C2×D311yescharacteristic
58C5×C22⋊C4—21no—
59C22⋊Dic5—61no—
60#687—62no—
61C4⋊Dic5—31no—
62#683—61no—
63#681—62no—
64C2×#168—31no—
65C2×#168—31no—
66#468—151noSylow
67C22×C3×C5—21no—
68C22×C3×C5—21no—
69C22×C3×C5—21no—
70C22×C3×C5—41no—
71C22×C3×C5—41no—
72C22×C3×C5—41no—
73C22×C3×C5C2×D411yescharacteristic
74C3×C4×C5—21no—
75C3×C4×C5—41no—
76C3×C4×C5C2×D411yescharacteristic
77C3×C4×C5#4111yescharacteristic
78Dic15—41no—
79Dic15—41no—
80C2×C3×D4—21no—
81C2×C3×D4C2×D511yescharacteristic
82C22×C3×C4C2×D511yescharacteristic
83C2×Dic6—51no—
84C3×C4⋊C4C2×D511yescharacteristic
85C3×C22⋊C4—21no—
86#217—101no—
87#214—102no—
88C4⋊Dic3—51no—
89Dic3⋊C4—101no—
90#208—102no—
91C2×C3⋊C8—51no—
92C2×C3⋊C8—51no—
93C23×C5—21no—
94C23×C5#6711yescharacteristic
95C5×D4—21no—
96C5×D4—21no—
97C5×D4—42no—
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5—21no—
101C2×C4×C5#6711yescharacteristic
102C2×C4×C5C22×D311yescharacteristic
103C2×Dic5—61no—
104C2×Dic5—61no—
105Dic10—61no—
106Dic10—61no—
107#168—32no—
108#168—61no—
109C2×SD16—151no—
110C2×M16—151no—
111C22⋊Q8—151no—
112#121D1511yescharacteristic
113#107—151no—
114#103—151no—
115#102—151no—
116C2×C3×C5—41no—
117C2×C3×C5—81no—
118C2×C3×C5#12111yescharacteristic
119C2×C3×C5AGL(1,ℤ/8)11yescharacteristic
120C2×C3×C5#13711yescharacteristic
121C23×C3—21no—
122C23×C3C22⋊D511yescharacteristic
123C3×D4—21no—
124C3×D4—21no—
125C3×D4—42no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4C22⋊D511yescharacteristic
130C2×C3×C4C22×D511yescharacteristic
131C2×Dic3—101no—
132C2×Dic3—101no—
133Dic6—101no—
134Dic6—101no—
135C3⋊C8—52no—
136C3⋊C8—101no—
137C22×C5—21no—
138C22×C5—21no—
139C22×C5—21no—
140C22×C5—41no—
141C22×C5—41no—
142C22×C5—41no—
143C22×C5C2×#6711yescharacteristic
144C4×C5—21no—
145C4×C5—41no—
146C4×C5D3×D411yescharacteristic
147C4×C5#23711yescharacteristic
148Dic5—121no—
149Dic5—121no—
150C2×Q8—151no—
151C2×D4—21no—
152C2×D4C2×D1511yescharacteristic
153C22×C4C2×D1511yescharacteristic
154SD16—302no—
155M16—302no—
156C2×C8—151no—
157C2×C8—151no—
158C4⋊C4—151no—
159C4⋊C4—301no—
160C4⋊C4C2×D1511yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—301no—
163C3×C5#46811yescharacteristic, complemented, Hall
164C22×C3—21no—
165C22×C3—21no—
166C22×C3—21no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3C2×C22⋊D511yescharacteristic
171C3×C4—21no—
172C3×C4—41no—
173C3×C4D4×D511yescharacteristic
174C3×C4#71111yescharacteristic
175Dic3—201no—
176Dic3—201no—
177C2×C5—41no—
178C2×C5—81no—
179C2×C5#93211yescharacteristic
180C2×C5#93911yescharacteristic
181C2×C5#95111yescharacteristic
182C23—21no—
183C23#122011yescharacteristic
184Q8—301no—
185Q8—301no—
186D4—21no—
187D4—21no—
188D4—42no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—301no—
193C2×C4—301no—
194C2×C4#122011yescharacteristic
195C2×C4C22×D1511yescharacteristic
196C8—152no—
197C8—301no—
198C2×C3—41no—
199C2×C3—81no—
200C2×C3#407811yescharacteristic
201C2×C3#408511yescharacteristic
202C2×C3#409711yescharacteristic
203C5#504211yescharacteristic, complemented, Sylow
204C22—21no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22C2×#122011yescharacteristic
211C4—21no—
212C4—41no—
213C4—601no—
214C4—601no—
215C4D4×D1511yescharacteristic
216C4#681911yescharacteristic
217C3#6533111yescharacteristic, complemented, Sylow
218C2—41no—
219C2—81no—
220C2#9182111yescharacteristic
221C2#9182811yescharacteristic
222C2#9184011yescharacteristic
subgroupG/H
centerC22C2×#1220
commutator, cosocleC2×C3×C4×C5C23
FrattiniC2×C4C22×D15
Fitting, prime radicalC3×C5×#121C2
socleC22×C3×C5C2×D4
hypercenter#121D15
hypocenterC3×C5#468
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C2×#6716C2semidirect
3. #91813C2semidirect
4. C2×#6710C2semidirect
5. #91116C2semidirect
6. #91094C2semidirect
7. #91091C2semidirect
45. C3⋊#168D4semidirect
163. C3×C5#468semidirect
203. C5#5042semidirect
217. C3#65331semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#1212
C2×#67162
#918132
C2×#67102
#911162
#910942
#910912
#653313
#50425