Finite Group Databasebeta

← #12003459

Subgroups — #12003459

Order 1280.

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

normal subgroups only

This group has 342 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×#3255C211yesmaximal, characteristic
2#10602626C212yesmaximal
3#10602615C212yesmaximal
4#10601312C211yesmaximal, characteristic
5#10598207C211yesmaximal, characteristic
6C5×#533C2212yes—
7C5×#475—22no—
8C5×#470—22no—
9C5×#462—22no—
10C5×#423C2211yescharacteristic
11C2×#3954—22no—
12#65215C2212yes—
13#65213C2211yescharacteristic
14C2×#3935C2211yescharacteristic
15#64767—22no—
16#64766—22no—
17#64763—24no—
18#10528—51nomaximal, Sylow
19C2×C5×Q16—22no—
20C2×C5×M16D412yes—
21C5×C4⋊Q8D412yes—
22C5×C4⋊Q8D412yes—
23C5×#125—22no—
24C5×C22⋊Q8—22no—
25C5×C22⋊Q8—22no—
26C4×C5×Q8—22no—
27C4×C5×Q8—22no—
28C5×#117—22no—
29C2×C42×C5C2311yescharacteristic
30C2×#681—21no—
31C22×#168—21no—
32C5×#106—22no—
33C5×C4⋊C8—22no—
34C5×#103—24no—
35C22⋊#168—22no—
36#3954—44no—
37C4⋊#168—22no—
38C4⋊#168—22no—
39#3935—21no—
40#3935—22no—
41#3935D412yes—
42#3255D511yescharacteristic
43#1849—51no—
44#1559—52no—
45#1531—52no—
46#1307—51no—
47C2×C5×Q8—22no—
48C2×C5×Q8—22no—
49C22×C4×C5C2×D411yescharacteristic
50C22×C4×C5#4112yes—
51C5×Q16—82no—
52C5×M16—42no—
53C2×C5×C8—22no—
54C5×C4⋊C4—22no—
55C5×C4⋊C4—22no—
56C5×C4⋊C4—22no—
57C5×C4⋊C4—22no—
58C5×C4⋊C4—42no—
59C5×C4⋊C4—42no—
60C5×C22⋊C4—42no—
61C42×C5—42no—
62C42×C5C2×D411yescharacteristic
63C42×C5C2×D412yes—
64C42×C5#4111yescharacteristic
65#681—42no—
66C2×#168—22no—
67C2×#168—22no—
68C2×#168—22no—
69C2×#168—41no—
70C2×#168—42no—
71#533C2×D512yes—
72#475—22no—
73#470—22no—
74#462—22no—
75#432—52no—
76C2×#108—102no—
77#423—51no—
78#423C2×D511yescharacteristic
79C2×#97—51no—
80#333—102no—
81#332—102no—
82#330—104no—
83C23×C5#12411yescharacteristic
84C5×Q8—42no—
85C5×Q8—42no—
86C5×Q8—82no—
87C2×C4×C5—21no—
88C2×C4×C5—21no—
89C2×C4×C5—42no—
90C2×C4×C5—42no—
91C2×C4×C5—42no—
92C2×C4×C5#12011yescharacteristic
93C2×C4×C5#12111yescharacteristic
94C2×C4×C5#12112yes—
95C2×C4×C5#12312yes—
96C5×C8—42no—
97#168—24no—
98#168—82no—
99#168—81no—
100C2×Q16—22no—
101C2×M16—101no—
102C2×M16C22⋊D512yes—
103C22×C8—101no—
104C4⋊Q8C22⋊D512yes—
105C4⋊Q8C22⋊D512yes—
106#125—22no—
107C22⋊Q8—22no—
108C22⋊Q8—22no—
109C4×Q8—22no—
110C4×Q8—22no—
111#117—22no—
112C2×C42C22×D511yescharacteristic
113#108—204no—
114#106—22no—
115C4⋊C8—22no—
116C4⋊C8—102no—
117C4⋊C8—102no—
118#103—24no—
119C22⋊C8—102no—
120#97—52no—
121#97—102no—
122#97—101no—
123C22×C5—22no—
124C22×C5—21no—
125C22×C5#39411yescharacteristic
126C4×C5—22no—
127C4×C5—22no—
128C4×C5—21no—
129C4×C5—82no—
130C4×C5—82no—
131C4×C5#45311yescharacteristic
132C4×C5#45611yescharacteristic
133C2×Q8—22no—
134C2×Q8—22no—
135C22×C4#71112yes—
136C22×C4C2×C22⋊D511yescharacteristic
137Q16—82no—
138M16—42no—
139M16—202no—
140C2×C8—22no—
141C2×C8—102no—
142C2×C8—102no—
143C2×C8—102no—
144C2×C8—201no—
145C2×C8—202no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C22⋊C4—42no—
153C42—42no—
154C42D4×D511yescharacteristic
155C42#71111yescharacteristic
156C42C2×C22⋊D512yes—
157C2×C5—41no—
158C2×C5#203911yescharacteristic
159C2×C5#204912yes—
160C23#408211yescharacteristic
161Q8—42no—
162Q8—42no—
163Q8—82no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4#408112yes—
170C2×C4#408411yescharacteristic
171C2×C4#408512yes—
172C2×C4#409911yescharacteristic
173C8—42no—
174C8—104no—
175C8—402no—
176C8—401no—
177C5#1052811yescharacteristic, complemented, Sylow
178C22—22no—
179C22—21no—
180C22#6551111yescharacteristic
181C4—22no—
182C4—22no—
183C4—21no—
184C4—82no—
185C4—82no—
186C4#6553011yescharacteristic
187C4#6553511yescharacteristic
188C2—41no—
189C2#1060266911yescharacteristic
190C2#1060289712yes—
subgroupG/H
centerC22#65511
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#3255C2
socleC22×C5#394
prime radicalC23×C5#124
hypercenter#3255D5
hypocenterC5#10528
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

Complements

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

NH ≅ G/Ntype
177. C5#10528semidirect

Maximal subgroups

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

subgroupindex
C5×#32552
#106026262
#106026152
#106013122
#105982072
#105285