Finite Group Databasebeta

← #11756456

Subgroups — #11756456

Order 960.

480 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 139 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×#125C211yesmaximal, characteristic, complemented
2#91784C214yesmaximal, complemented
3#91784C212yesmaximal, complemented
4#91783C214yesmaximal, complemented
5#91782C212yesmaximal, complemented
6#91763C211yesmaximal, characteristic, complemented
7#91761C211yesmaximal, characteristic, complemented
8#66032—31nomaximal, Hall
9C2×D60C2212yes—
10C2×D60C2214yes—
11C2×D60C2214yes—
12C2×C4×D15C2211yescharacteristic
13C2×C4×D15C2214yes—
14C3×C5×C4⋊C4C2212yes—
15C3×C5×C4⋊C4C2214yes—
16C3×C42×C5C2211yescharacteristic
17#6714C2212yes—
18#6714C2214yes—
19#6714C2214yes—
20#6712C2212yes—
21C4×Dic15C2211yescharacteristic
22#5697—51nomaximal, Hall
23C5×#125D311yescharacteristic, complemented
24#4041—34no—
25#4041—32no—
26#4040—34no—
27#4039—32no—
28#4020—31no—
29#4018—31no—
30C22×D15C2311yescharacteristic
31C22×D15C2314yes—
32C2×C3×C4×C5C2312yes—
33C2×C3×C4×C5C2314yes—
34C2×C3×C4×C5C2311yescharacteristic
35C2×Dic15C2312yes—
36C2×Dic15C2311yescharacteristic
37D60—24no—
38D60—44no—
39D60—44no—
40C4×D15—22no—
41C4×D15—44no—
42C3×#125D511yescharacteristic, complemented
43#895—54no—
44#895—52no—
45#894—54no—
46#893—52no—
47#874—51no—
48#872—51no—
49C2×D20—32no—
50C2×D20—34no—
51C2×D20—34no—
52C2×C4×D5—31no—
53C2×C4×D5—34no—
54C5×C4⋊C4C2×D312yes—
55C5×C4⋊C4C2×D314yes—
56C42×C5C2×D311yescharacteristic
57#685—32no—
58#685—34no—
59#685—34no—
60#683—32no—
61C4×Dic5—31no—
62#540—151noSylow
63C22×C3×C5C2411yescharacteristic
64C2×D15—21no—
65C2×D15—24no—
66C2×D15—24no—
67C2×D15—22no—
68C2×D15—24no—
69C3×C4×C5—24no—
70C3×C4×C5—22no—
71C3×C4×C5#4112yes—
72Dic15—21no—
73Dic15—22no—
74C2×D12—52no—
75C2×D12—54no—
76C2×D12—54no—
77C2×C4×D3—51no—
78C2×C4×D3—54no—
79C3×C4⋊C4C2×D512yes—
80C3×C4⋊C4C2×D514yes—
81C3×C42C2×D511yescharacteristic
82(C2×D3)⋊C4—52no—
83(C2×D3)⋊C4—54no—
84(C2×D3)⋊C4—54no—
85Dic3⋊C4—52no—
86C4×Dic3—51no—
87C22×D5—31no—
88C22×D5—34no—
89C2×C4×C5C22×D312yes—
90C2×C4×C5C22×D314yes—
91C2×C4×C5C22×D311yescharacteristic
92C2×Dic5—32no—
93C2×Dic5—31no—
94D20—64no—
95D20—124no—
96D20—124no—
97C4×D5—62no—
98C4×D5—124no—
99#127—151no—
100#125D1511yescharacteristic, complemented
101#123—154no—
102#121—154no—
103#121—152no—
104C4×D4—152no—
105#117—151no—
106C2×C3×C5C2×#4111yescharacteristic
107C2×C3×C5H(5,𝔽2)12yes—
108D15—41no—
109D15—44no—
110C22×D3—51no—
111C22×D3—54no—
112C2×C3×C4C22×D512yes—
113C2×C3×C4C22×D514yes—
114C2×C3×C4C22×D511yescharacteristic
115C2×Dic3—52no—
116C2×Dic3—51no—
117D12—104no—
118D12—204no—
119D12—204no—
120C4×D3—102no—
121C4×D3—204no—
122C22×C5C23×D311yescharacteristic
123C2×D5—61no—
124C2×D5—64no—
125C2×D5—64no—
126C2×D5—62no—
127C2×D5—64no—
128C4×C5—24no—
129C4×C5—22no—
130C4×C5#23912yes—
131Dic5—61no—
132Dic5—62no—
133C2×D4—152no—
134C2×D4—154no—
135C2×D4—154no—
136C22×C4—151no—
137C22×C4—154no—
138C4⋊C4—152no—
139C4⋊C4C2×D1512yes—
140C4⋊C4C2×D1514yes—
141C22⋊C4—152no—
142C22⋊C4—154no—
143C22⋊C4—154no—
144C42—151no—
145C42C2×D1511yescharacteristic
146C3×C5#54011yescharacteristic, complemented, Hall
147C22×C3C23×D511yescharacteristic
148C2×D3—101no—
149C2×D3—104no—
150C2×D3—104no—
151C2×D3—102no—
152C2×D3—104no—
153C3×C4—24no—
154C3×C4—22no—
155C3×C4#71312yes—
156Dic3—101no—
157Dic3—102no—
158C2×C5C2×#23911yescharacteristic
159C2×C5#100912yes—
160D5—121no—
161D5—124no—
162C23—151no—
163C23—154no—
164D4—304no—
165D4—604no—
166D4—604no—
167C2×C4—152no—
168C2×C4—151no—
169C2×C4—302no—
170C2×C4—604no—
171C2×C4C22×D1512yes—
172C2×C4C22×D1514yes—
173C2×C4C22×D1511yescharacteristic
174C2×C3C2×#71311yescharacteristic
175C2×C3#414912yes—
176D3—201no—
177D3—204no—
178C5#569711yescharacteristic, complemented, Sylow
179C22—301no—
180C22—304no—
181C22—304no—
182C22—302no—
183C22—304no—
184C22C23×D1511yescharacteristic
185C4—24no—
186C4—22no—
187C4—301no—
188C4—302no—
189C4#682112yes—
190C3#6603211yescharacteristic, complemented, Sylow
191C2—601no—
192C2—604no—
193C2C2×#682111yescharacteristic
194C2#9210012yes—
subgroupG/H
center, FrattiniC22C23×D15
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×#125C2
hypercenter#125D15
hypocenterC3×C5#540
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C3×C5×#125C2semidirect
2. #91784C2semidirect
3. #91784C2semidirect
4. #91783C2semidirect
5. #91782C2semidirect
6. #91763C2semidirect
7. #91761C2semidirect
23. C5×#125D3semidirect
42. C3×#125D5semidirect
100. #125D15semidirect
146. C3×C5#540semidirect
178. C5#5697semidirect
190. C3#66032semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#1252
#917842
#917832
#917822
#917632
#917612
#660323
#56975