Finite Group Databasebeta

← #11752415

Subgroups — #11752415

Order 960.

380 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C3×C4×D5C211yesmaximal, characteristic
2C2×#6758C212yesmaximal, complemented
3#91241C212yesmaximal, complemented
4#91232C212yesmaximal, complemented
5#65756—31nomaximal, Hall
6C23×C3×D5C411yescharacteristic
7C22×C3×C4×C5C411yescharacteristic
8C22×C3×Dic5C2211yescharacteristic
9C2×C3×C4×D5—24no—
10C2×C3×C4×D5C411yescharacteristic
11C2×C3×C4×D5C411yescharacteristic
12C2×C3×C4×D5C2211yescharacteristic
13C2×C3×C4×D5C2211yescharacteristic
14C2×C3⋊#170C2214yescomplemented
15#6758—24no—
16#5207—51nomaximal, Hall
17C22×C4×D5D311yescharacteristic
18C2×#700—32no—
19C22⋊#170—32no—
20#4003—32no—
21C23×C3×C5C2×C411yescharacteristic
22C22×C3×D5—22no—
23C22×C3×D5—22no—
24C22×C3×D5—22no—
25C22×C3×D5C2×C411yescharacteristic
26C22×C3×D5C2×C411yescharacteristic
27C2×C3×C4×C5—22no—
28C2×C3×C4×C5C2×C411yescharacteristic
29C2×C3×C4×C5C2×C411yescharacteristic
30C2×C3×Dic5—22no—
31C2×C3×Dic5C2×C411yescharacteristic
32C2×C3×Dic5C2311yescharacteristic
33C3×C4×D5—22no—
34C3×C4×D5—22no—
35C3×C4×D5—22no—
36C3×C4×D5D414yes—
37C3⋊#170—24no—
38C23×C3×C4—51no—
39C2×#208—52no—
40#830—52no—
41#830—52no—
42C23×D5Dic311yescharacteristic
43C22×C4×C5Dic311yescharacteristic
44C22×Dic5C2×D311yescharacteristic
45C2×C4×D5—24no—
46C2×C4×D5Dic311yescharacteristic
47C2×C4×D5Dic311yescharacteristic
48C2×C4×D5C2×D311yescharacteristic
49C2×C4×D5C2×D311yescharacteristic
50C2×#170—34no—
51#700—64no—
52#407—151noSylow
53C22×C3×C5—21no—
54C22×C3×C5—21no—
55C22×C3×C5M1612yescomplemented
56C22×C3×C5C22×C411yescharacteristic
57C2×C3×D5—28no—
58C2×C3×D5—21no—
59C2×C3×D5—21no—
60C2×C3×D5—21no—
61C2×C3×D5—21no—
62C2×C3×D5C22⋊C412yes—
63C2×C3×D5M1612yescomplemented
64C3×C4×C5—21no—
65C3×C4×C5C22⋊C412yes—
66C3×Dic5—21no—
67C3×Dic5C2×D412yes—
68C24×C3—51no—
69C22×C3×C4—51no—
70C22×C3×C4—51no—
71C22×C3×C4—51no—
72C22×C3×C4—51no—
73C22×C3×C4—51no—
74C22×C3×C4—104no—
75C22×C3×C4AGL(1,𝔽5)11yescharacteristic
76#208—104no—
77C2×C3⋊C8—54no—
78C23×C5C2×Dic311yescharacteristic
79C22×D5—22no—
80C22×D5—22no—
81C22×D5—22no—
82C22×D5C2×Dic311yescharacteristic
83C22×D5C2×Dic311yescharacteristic
84C2×C4×C5—22no—
85C2×C4×C5C2×Dic311yescharacteristic
86C2×C4×C5C2×Dic311yescharacteristic
87C2×Dic5—22no—
88C2×Dic5C2×Dic311yescharacteristic
89C2×Dic5C22×D311yescharacteristic
90C4×D5—22no—
91C4×D5—22no—
92C4×D5—22no—
93C4×D5#6714yes—
94#170—64no—
95C23×C4—51no—
96C2×M16—152no—
97C22⋊C8—152no—
98C22⋊C8—152no—
99C2×C3×C5—22no—
100C2×C3×C5C2×C22⋊C411yescharacteristic
101C2×C3×C5C2×M1611yescharacteristic
102C2×C3×C5C2×M1611yescharacteristic
103C3×D5—22no—
104C3×D5—22no—
105C23×C3—51no—
106C23×C3—51no—
107C23×C3—102no—
108C23×C3—102no—
109C23×C3—102no—
110C23×C3C2×AGL(1,𝔽5)11yescharacteristic
111C2×C3×C4—22no—
112C2×C3×C4—51no—
113C2×C3×C4—54no—
114C2×C3×C4—51no—
115C2×C3×C4—102no—
116C2×C3×C4—102no—
117C2×C3×C4—102no—
118C2×C3×C4—102no—
119C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
120C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
121C3⋊C8—104no—
122C22×C5—21no—
123C22×C5—21no—
124C22×C5#20812yescomplemented
125C22×C5C22×Dic311yescharacteristic
126C2×D5—28no—
127C2×D5—21no—
128C2×D5—21no—
129C2×D5—21no—
130C2×D5—21no—
131C2×D5#20812yescomplemented
132C2×D5#21712yes—
133C4×C5—21no—
134C4×C5#21712yes—
135Dic5—21no—
136Dic5C2×#6712yes—
137C24—51no—
138C22×C4—51no—
139C22×C4—51no—
140C22×C4—51no—
141C22×C4—51no—
142C22×C4—51no—
143C22×C4—104no—
144C22×C4ΓL(1,𝔽16)11yescharacteristic
145M16—304no—
146C2×C8—154no—
147C3×C5#40711yescharacteristic, complemented, Hall
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—52no—
151C22×C3—52no—
152C22×C3—108no—
153C22×C3—101no—
154C22×C3—101no—
155C22×C3—101no—
156C22×C3—101no—
157C22×C3#70012yescomplemented
158C22×C3C22×AGL(1,𝔽5)11yescharacteristic
159C3×C4—21no—
160C3×C4—52no—
161C3×C4—101no—
162C3×C4C22⋊AGL(1,𝔽5)12yes—
163C2×C5—22no—
164C2×C5C2×#20811yescharacteristic
165C2×C5C2×#20811yescharacteristic
166C2×C5C2×#21711yescharacteristic
167D5—22no—
168D5—22no—
169C23—51no—
170C23—51no—
171C23—102no—
172C23—102no—
173C23—102no—
174C23C2×ΓL(1,𝔽16)11yescharacteristic
175C2×C4—22no—
176C2×C4—51no—
177C2×C4—54no—
178C2×C4—51no—
179C2×C4—102no—
180C2×C4—102no—
181C2×C4—102no—
182C2×C4—102no—
183C2×C4C2×ΓL(1,𝔽16)11yescharacteristic
184C2×C4C2×ΓL(1,𝔽16)11yescharacteristic
185C8—304no—
186C2×C3—22no—
187C2×C3—102no—
188C2×C3—102no—
189C2×C3C2×#70011yescharacteristic
190C2×C3D5⋊M1611yescharacteristic
191C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
192C5#520711yescharacteristic, complemented, Sylow
193C22—21no—
194C22—21no—
195C22—52no—
196C22—52no—
197C22—108no—
198C22—101no—
199C22—101no—
200C22—101no—
201C22—101no—
202C22#675812yescomplemented
203C22C22×ΓL(1,𝔽16)11yescharacteristic
204C4—21no—
205C4—52no—
206C4—101no—
207C4#676312yes—
208C3#6575611yescharacteristic, complemented, Sylow
209C2—22no—
210C2—102no—
211C2—102no—
212C2C2×#675811yescharacteristic
213C2D5⋊#20811yescharacteristic
214C2C2×#676311yescharacteristic
subgroupG/H
center, FrattiniC22C22×ΓL(1,𝔽16)
commutator, socleC22×C3×C5C22×C4
FittingC22×C3×C4×C5C4
prime radicalC23×C3×D5C4
cosocleC2×C3×Dic5C23
hypercenterC22×C4ΓL(1,𝔽16)
hypocenterC3×C5#407
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#6758C2semidirect
3. #91241C2semidirect
4. #91232C2semidirect
14. C2×C3⋊#170C22semidirect
55. C22×C3×C5M16semidirect
63. C2×C3×D5M16semidirect
124. C22×C5#208semidirect
131. C2×D5#208semidirect
147. C3×C5#407semidirect
157. C22×C3#700semidirect
192. C5#5207semidirect
202. C22#6758semidirect
208. C3#65756semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D52
C2×#67582
#912412
#912322
#657563
#52075