Finite Group Databasebeta

← #12003464

Subgroups — #12003464

Order 1280.

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

normal subgroups only

This group has 326 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#10601310C211yesmaximal, characteristic
5#10598207C211yesmaximal, characteristic
6C5×#533C2212yes—
7C5×#475—22no—
8C5×#470—22no—
9C5×#462—22no—
10C5×#423C2211yescharacteristic
11#65215C2212yes—
12#65213C2211yescharacteristic
13C2×#3935C2211yescharacteristic
14#64778—22no—
15#64767—22no—
16#64766—22no—
17#64763—24no—
18#10529—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—
36C4⋊#168—22no—
37C4⋊#168—22no—
38#3935—21no—
39#3935—22no—
40#3935D412yes—
41#3255D511yescharacteristic
42#1838—51no—
43#1559—52no—
44#1531—52no—
45#1307—51no—
46C2×C5×Q8—22no—
47C2×C5×Q8—22no—
48C22×C4×C5C2×D411yescharacteristic
49C22×C4×C5#4112yes—
50C5×Q16—82no—
51C5×M16—42no—
52C2×C5×C8—22no—
53C5×C4⋊C4—22no—
54C5×C4⋊C4—22no—
55C5×C4⋊C4—22no—
56C5×C4⋊C4—22no—
57C5×C4⋊C4—42no—
58C5×C4⋊C4—42no—
59C5×C22⋊C4—42no—
60C42×C5—42no—
61C42×C5C2×D411yescharacteristic
62C42×C5C2×D412yes—
63C42×C5#4111yescharacteristic
64#681—42no—
65C2×#168—22no—
66C2×#168—41no—
67C2×#168—41no—
68C2×#168—41no—
69C2×#168—42no—
70#533C2×D512yes—
71#475—22no—
72#470—22no—
73#462—22no—
74#432—52no—
75#423—51no—
76#423C2×D511yescharacteristic
77C2×#97—51no—
78#341—102no—
79#333—102no—
80#332—102no—
81#330—104no—
82C23×C5#12411yescharacteristic
83C5×Q8—42no—
84C5×Q8—42no—
85C5×Q8—82no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—42no—
89C2×C4×C5—42no—
90C2×C4×C5—42no—
91C2×C4×C5#12011yescharacteristic
92C2×C4×C5#12111yescharacteristic
93C2×C4×C5#12112yes—
94C2×C4×C5#12312yes—
95C5×C8—42no—
96#168—42no—
97#168—82no—
98#168—81no—
99C2×Q16—22no—
100C2×M16—101no—
101C2×M16C22⋊D512yes—
102C22×C8—101no—
103C4⋊Q8C22⋊D512yes—
104C4⋊Q8C22⋊D512yes—
105#125—22no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C4×Q8—22no—
109C4×Q8—22no—
110#117—22no—
111C2×C42C22×D511yescharacteristic
112#106—22no—
113C4⋊C8—22no—
114C4⋊C8—102no—
115C4⋊C8—102no—
116#103—24no—
117C22⋊C8—102no—
118#97—52no—
119#97—102no—
120#97—101no—
121C22×C5—22no—
122C22×C5—21no—
123C22×C5#39411yescharacteristic
124C4×C5—22no—
125C4×C5—22no—
126C4×C5—21no—
127C4×C5—82no—
128C4×C5—82no—
129C4×C5#45411yescharacteristic
130C4×C5#45511yescharacteristic
131C2×Q8—22no—
132C2×Q8—22no—
133C22×C4#71112yes—
134C22×C4C2×C22⋊D511yescharacteristic
135Q16—82no—
136M16—42no—
137M16—202no—
138C2×C8—22no—
139C2×C8—102no—
140C2×C8—201no—
141C2×C8—201no—
142C2×C8—201no—
143C2×C8—202no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—42no—
151C42—42no—
152C42D4×D511yescharacteristic
153C42#71111yescharacteristic
154C42C2×C22⋊D512yes—
155C2×C5—41no—
156C2×C5#204011yescharacteristic
157C2×C5#204912yes—
158C23#408211yescharacteristic
159Q8—42no—
160Q8—42no—
161Q8—82no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4#408112yes—
168C2×C4#408411yescharacteristic
169C2×C4#408512yes—
170C2×C4#409911yescharacteristic
171C8—42no—
172C8—202no—
173C8—402no—
174C8—401no—
175C5#1052911yescharacteristic, complemented, Sylow
176C22—22no—
177C22—21no—
178C22#6551111yescharacteristic
179C4—22no—
180C4—22no—
181C4—21no—
182C4—82no—
183C4—82no—
184C4#6553111yescharacteristic
185C4#6553411yescharacteristic
186C2—41no—
187C2#1060267011yescharacteristic
188C2#1060289712yes—
subgroupG/H
centerC22#65511
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#3255C2
socleC22×C5#394
prime radicalC23×C5#124
hypercenter#3255D5
hypocenterC5#10529
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
175. C5#10529semidirect

Maximal subgroups

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

subgroupindex
C5×#32552
#106026262
#106026152
#106013102
#105982072
#105295