Finite Group Databasebeta

← #12003440

Subgroups — #12003440

Order 1280.

370 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C5×#3253C211yesmaximal, characteristic
2#10602602C212yesmaximal, complemented
3#10602591C212yesmaximal, complemented
4#10601310C211yesmaximal, characteristic, complemented
5#10598207C211yesmaximal, characteristic, complemented
6C5×#532C2212yes—
7C5×#476—22no—
8C5×#469—22no—
9C5×#459—22no—
10C5×#423C2211yescharacteristic
11#65215C2212yes—
12#65213C2211yescharacteristic
13C2×#3935C2211yescharacteristic
14#64778—22no—
15#64767—22no—
16#64764—22no—
17#64760—24no—
18#10525—51nomaximal, Sylow
19C2×C5×#41—22no—
20C2×C5×D8—22no—
21C2×C5×M16D412yes—
22C5×C4⋊Q8D412yes—
23C5×#127D412yes—
24C5×#124—22no—
25C5×#121—22no—
26C5×#121—22no—
27C4×C5×D4—22no—
28C4×C5×D4—22no—
29C2×C42×C5C2311yescharacteristic
30C2×#681—21no—
31C22×#168—21no—
32C5×#106—22no—
33C5×C4⋊C8—22no—
34C5×#102—24no—
35C22⋊#168—22no—
36C4⋊#168—22no—
37C4⋊#168—22no—
38#3935—21no—
39#3935—22no—
40#3935D412yes—
41#3253D511yescharacteristic
42#1838—51no—
43#1557—52no—
44#1530—52no—
45#1307—51no—
46C5×#41—84no—
47C2×C5×Q8—22no—
48C2×C5×D4—22no—
49C2×C5×D4—22no—
50C2×C5×D4—22no—
51C2×C5×D4—42no—
52C22×C4×C5—42no—
53C22×C4×C5C2×D411yescharacteristic
54C22×C4×C5#4112yes—
55C5×D8—82no—
56C5×M16—42no—
57C2×C5×C8—22no—
58C5×C4⋊C4—22no—
59C5×C4⋊C4—22no—
60C5×C22⋊C4—42no—
61C5×C22⋊C4—42no—
62C42×C5C2×D411yescharacteristic
63C42×C5C2×D412yes—
64C42×C5#4111yescharacteristic
65#681—42no—
66C2×#168—22no—
67C2×#168—41no—
68C2×#168—41no—
69C2×#168—41no—
70C2×#168—42no—
71#532C2×D512yes—
72#476—22no—
73#469—22no—
74#459—22no—
75#432—52no—
76#423—51no—
77#423C2×D511yescharacteristic
78C2×#97—51no—
79#341—102no—
80#332—102no—
81#331—102no—
82#329—104no—
83C23×C5—42no—
84C23×C5#12411yescharacteristic
85C5×Q8—82no—
86C5×D4—42no—
87C5×D4—42no—
88C5×D4—82no—
89C5×D4—82no—
90C5×D4—84no—
91C2×C4×C5—21no—
92C2×C4×C5—21no—
93C2×C4×C5—42no—
94C2×C4×C5—42no—
95C2×C4×C5—84no—
96C2×C4×C5#12011yescharacteristic
97C2×C4×C5#12111yescharacteristic
98C2×C4×C5#12112yes—
99C2×C4×C5#12312yes—
100C5×C8—42no—
101#168—42no—
102#168—82no—
103#168—81no—
104C2×#41—22no—
105C2×D8—22no—
106C2×M16—101no—
107C2×M16C22⋊D512yes—
108C22×C8—101no—
109C4⋊Q8C22⋊D512yes—
110#127C22⋊D512yes—
111#124—22no—
112#121—22no—
113#121—22no—
114C4×D4—22no—
115C4×D4—22no—
116C2×C42C22×D511yescharacteristic
117#106—22no—
118C4⋊C8—22no—
119C4⋊C8—102no—
120C4⋊C8—102no—
121#102—24no—
122C22⋊C8—102no—
123#97—52no—
124#97—102no—
125#97—101no—
126C22×C5—22no—
127C22×C5—21no—
128C22×C5—82no—
129C22×C5—82no—
130C22×C5—82no—
131C22×C5#39411yescharacteristic
132C4×C5—22no—
133C4×C5—22no—
134C4×C5—21no—
135C4×C5—82no—
136C4×C5#45411yescharacteristic
137C4×C5#45511yescharacteristic
138#41—84no—
139C2×Q8—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—42no—
144C22×C4—42no—
145C22×C4#71112yes—
146C22×C4C2×C22⋊D511yescharacteristic
147D8—82no—
148M16—42no—
149M16—202no—
150C2×C8—22no—
151C2×C8—102no—
152C2×C8—201no—
153C2×C8—201no—
154C2×C8—201no—
155C2×C8—202no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C42D4×D511yescharacteristic
161C42#71111yescharacteristic
162C42C2×C22⋊D512yes—
163C2×C5—41no—
164C2×C5—162no—
165C2×C5#204011yescharacteristic
166C2×C5#204812yes—
167C23—42no—
168C23#408211yescharacteristic
169Q8—82no—
170D4—42no—
171D4—42no—
172D4—82no—
173D4—82no—
174D4—84no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—84no—
180C2×C4#408112yes—
181C2×C4#408411yescharacteristic
182C2×C4#408512yes—
183C2×C4#409911yescharacteristic
184C8—42no—
185C8—202no—
186C8—402no—
187C8—401no—
188C5#1052511yescharacteristic, complemented, Sylow
189C22—22no—
190C22—21no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22#6551111yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—82no—
199C4#6553111yescharacteristic
200C4#6553411yescharacteristic
201C2—41no—
202C2—162no—
203C2#1060267011yescharacteristic
204C2#1060288512yes—
subgroupG/H
centerC22#65511
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
Fitting, prime radicalC5×#3253C2
socleC22×C5#394
hypercenter#3253D5
hypocenterC5#10525
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
2. #10602602C2semidirect
3. #10602591C2semidirect
4. #10601310C2semidirect
5. #10598207C2semidirect
188. C5#10525semidirect

Maximal subgroups

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

subgroupindex
C5×#32532
#106026022
#106025912
#106013102
#105982072
#105255