Finite Group Databasebeta

← #11751117

Subgroups — #11751117

Order 960.

248 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 75 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×C4×C5×D4C211yesmaximal, characteristic, complemented
2C2×#6716C211yesmaximal, characteristic, complemented
3C2×#6715C211yesmaximal, characteristic
4#91764C211yesmaximal, characteristic, complemented
5#91094C211yesmaximal, characteristic, complemented
6#91093C211yesmaximal, characteristic, complemented
7C4⋊(C3⋊#168)C211yesmaximal, characteristic, complemented
8#65305—31nomaximal, Hall
9C2×C3×C5×D4C2211yescharacteristic
10C22×C3×C4×C5—21no—
11C2×D60C2211yescharacteristic
12C2×Dic30C2211yescharacteristic
13C3×C5×C4⋊C4C2211yescharacteristic
14C3×C5×C22⋊C4—21no—
15C3×C42×C5C2211yescharacteristic
16#6716—22no—
17#6715—22no—
18#6714—21no—
19#6714—21no—
20C2×C3⋊#168C2211yescharacteristic
21C2×C3⋊#168C2211yescharacteristic
22#5016—51nomaximal, Hall
23C4×C5×D4D311yescharacteristic, complemented
24C2×#687—31no—
25C2×#686—31no—
26#4021—31no—
27#3942—31no—
28#3941—31no—
29C4⋊#168—31no—
30C23×C3×C5—21no—
31C22×D15—21no—
32C3×C5×D4—21no—
33C3×C5×D4D412yes—
34C2×C3×C4×C5—21no—
35C2×C3×C4×C5—42no—
36C2×C3×C4×C5D411yescharacteristic
37C2×C3×C4×C5D411yescharacteristic
38C2×C3×C4×C5C2311yescharacteristic
39C2×Dic15—21no—
40D60—21no—
41D60—21no—
42Dic30—21no—
43Dic30—21no—
44C3⋊#168—21no—
45C3⋊#168—21no—
46C3×C4×D4D511yescharacteristic, complemented
47C2×#214—51no—
48C2×#213—51no—
49#875—51no—
50#810—51no—
51#809—51no—
52C4⋊(C3⋊C8)—51no—
53C2×C5×D4C2×D311yescharacteristic
54C22×C4×C5—21no—
55C2×D20—31no—
56C2×Dic10—31no—
57C5×C4⋊C4C2×D311yescharacteristic
58C5×C22⋊C4—21no—
59C42×C5C2×D311yescharacteristic
60#687—62no—
61#686—62no—
62#685—61no—
63#685—61no—
64C2×#168—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—21no—
75C3×C4×C5—21no—
76C3×C4×C5—41no—
77C3×C4×C5C2×D411yescharacteristic
78C3×C4×C5#4111yescharacteristic
79Dic15—41no—
80C2×C3×D4C2×D511yescharacteristic
81C22×C3×C4—21no—
82C2×D12—51no—
83C2×Dic6—51no—
84C3×C4⋊C4C2×D511yescharacteristic
85C3×C22⋊C4—21no—
86C3×C42C2×D511yescharacteristic
87#214—102no—
88#213—102no—
89(C2×D3)⋊C4—101no—
90(C2×D3)⋊C4—101no—
91C2×C3⋊C8—51no—
92C2×C3⋊C8—51no—
93C23×C5—21no—
94C22×D5—61no—
95C5×D4—21no—
96C5×D4D1212yes—
97C2×C4×C5—21no—
98C2×C4×C5—42no—
99C2×C4×C5#6711yescharacteristic
100C2×C4×C5#6711yescharacteristic
101C2×C4×C5C22×D311yescharacteristic
102C2×Dic5—61no—
103D20—61no—
104D20—61no—
105Dic10—61no—
106Dic10—61no—
107#168—61no—
108#168—61no—
109C2×SD16—151no—
110C2×D8—151no—
111#124—151no—
112C4×D4D1511yescharacteristic, complemented
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×D4—21no—
124C3×D4D2012yes—
125C2×C3×C4—21no—
126C2×C3×C4—42no—
127C2×C3×C4C22⋊D511yescharacteristic
128C2×C3×C4C22⋊D511yescharacteristic
129C2×C3×C4C22×D511yescharacteristic
130C2×Dic3—101no—
131D12—101no—
132D12—101no—
133Dic6—101no—
134Dic6—101no—
135C3⋊C8—101no—
136C3⋊C8—101no—
137C22×C5—22no—
138C22×C5—41no—
139C22×C5—41no—
140C22×C5C2×#6711yescharacteristic
141C2×D5—121no—
142C2×D5—121no—
143C2×D5—121no—
144C4×C5—21no—
145C4×C5—21no—
146C4×C5—41no—
147C4×C5C2×D1211yescharacteristic
148C4×C5#23511yescharacteristic
149Dic5—121no—
150C2×Q8—151no—
151C2×D4—151no—
152C2×D4C2×D1511yescharacteristic
153C22×C4—21no—
154SD16—302no—
155D8—302no—
156C2×C8—151no—
157C2×C8—151no—
158C4⋊C4C2×D1511yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—301no—
161C22⋊C4—301no—
162C42C2×D1511yescharacteristic
163C3×C5#46311yescharacteristic, complemented, Hall
164C22×C3—22no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3C2×C22⋊D511yescharacteristic
168C2×D3—201no—
169C2×D3—201no—
170C2×D3—201no—
171C3×C4—21no—
172C3×C4—21no—
173C3×C4—41no—
174C3×C4C2×D2011yescharacteristic
175C3×C4#70911yescharacteristic
176Dic3—201no—
177C2×C5—42no—
178C2×C5#93011yescharacteristic
179C2×C5#93211yescharacteristic
180C2×C5#95011yescharacteristic
181D5—241no—
182C23—21no—
183C23—301no—
184Q8—301no—
185Q8—301no—
186D4—21no—
187D4—301no—
188D4—301no—
189D4D6012yes—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—301no—
193C2×C4#122011yescharacteristic
194C2×C4#122011yescharacteristic
195C2×C4C22×D1511yescharacteristic
196C8—301no—
197C8—301no—
198C2×C3—42no—
199C2×C3#407611yescharacteristic
200C2×C3#407811yescharacteristic
201C2×C3#409611yescharacteristic
202D3—401no—
203C5#501611yescharacteristic, complemented, Sylow
204C22—22no—
205C22—41no—
206C22—41no—
207C22—601no—
208C22—601no—
209C22—601no—
210C22C2×#122011yescharacteristic
211C4—21no—
212C4—21no—
213C4—41no—
214C4—601no—
215C4C2×D6011yescharacteristic
216C4#681711yescharacteristic
217C3#6530511yescharacteristic, complemented, Sylow
218C2—42no—
219C2—1201no—
220C2#9181911yescharacteristic
221C2#9182111yescharacteristic
222C2#9183911yescharacteristic
subgroupG/H
centerC22C2×#1220
commutator, cosocleC2×C3×C4×C5C23
FrattiniC2×C4C22×D15
FittingC3×C4×C5×D4C2
socleC22×C3×C5C2×D4
prime radicalC2×#6715C2
hypercenterC4×D4D15
hypocenterC3×C5#463
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×C4×C5×D4C2semidirect
2. C2×#6716C2semidirect
4. #91764C2semidirect
5. #91094C2semidirect
6. #91093C2semidirect
7. C4⋊(C3⋊#168)C2semidirect
23. C4×C5×D4D3semidirect
46. C3×C4×D4D5semidirect
112. C4×D4D15semidirect
163. C3×C5#463semidirect
203. C5#5016semidirect
217. C3#65305semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C5×D42
C2×#67162
C2×#67152
#917642
#910942
#910932
C4⋊(C3⋊#168)2
#653053
#50165