Finite Group Databasebeta

← #12002415

Subgroups — #12002415

Order 1280.

402 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 93 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×#2864C211yesmaximal, characteristic, complemented
2#10600338C212yesmaximal
3#10600188C211yesmaximal, characteristic
4#10598122C212yesmaximal, complemented
5#10598102C211yesmaximal, characteristic, complemented
6#65811C2211yescharacteristic
7C5×#389—22no—
8C5×#387—24no—
9C5×#382C2212yes—
10C43×C5C2211yescharacteristic
11C2×#3937—22no—
12#65214C2212yes—
13#65213C2211yescharacteristic
14#64671—24no—
15#10368—51nomaximal, Sylow
16C2×#709—21no—
17C2×C5×C4⋊C4—24no—
18C2×C5×C4⋊C4—22no—
19C2×C42×C5—22no—
20C2×C42×C5D412yes—
21C2×C42×C5C2311yescharacteristic
22#4076—22no—
23C2×#681—21no—
24#4021—21no—
25#4020Q811yescharacteristic
26C4×D20—22no—
27#4015Q811yescharacteristic
28C5×#95—24no—
29C5×#95—22no—
30C5×#95D414yes—
31C22⋊#168—22no—
32#3937—44no—
33C4⋊#168—22no—
34C4⋊#168—22no—
35C4×#168—22no—
36#2864D511yescharacteristic, complemented
37#1962—51no—
38#1517—52no—
39#1286—52no—
40#1276—51no—
41C22×C4×C5—22no—
42C22×C4×C5—24no—
43C22×C4×C5C2×D412yes—
44C22×C4×C5C2×D411yescharacteristic
45C2×C22⋊D5—41no—
46#709—42no—
47#709—42no—
48C2×D20—21no—
49C2×D20—22no—
50C2×C4×D5—41no—
51C2×Dic10—21no—
52C5×C4⋊C4—48no—
53C5×C4⋊C4—42no—
54C5×C4⋊C4—42no—
55C42×C5—22no—
56C42×C5—22no—
57C42×C5—44no—
58C42×C5C2×Q811yescharacteristic
59C42×C5#4111yescharacteristic
60C42×C5#4112yes—
61#685—42no—
62C4⋊Dic5—22no—
63#681—42no—
64C2×#168—42no—
65C2×#168—41no—
66#532—51no—
67#431—52no—
68#423—51no—
69C2×#104—102no—
70#389—22no—
71#387—24no—
72#382C2×D512yes—
73C43C2×D511yescharacteristic
74#327—104no—
75C23×C5#12011yescharacteristic
76C22×D5—41no—
77C2×C4×C5—24no—
78C2×C4×C5—42no—
79C2×C4×C5—42no—
80C2×C4×C5—42no—
81C2×C4×C5—48no—
82C2×C4×C5—44no—
83C2×C4×C5C22⋊Q812yes—
84C2×C4×C5C22⋊Q811yescharacteristic
85C2×C4×C5#12412yes—
86C2×C4×C5#12411yescharacteristic
87C2×C4×C5#13512yes—
88C2×C4×C5AGL(1,ℤ/8)12yes—
89C22⋊D5—82no—
90C2×Dic5—41no—
91D20—42no—
92D20—82no—
93C4×D5—81no—
94C4×D5—81no—
95Dic10—42no—
96#168—82no—
97#168—81no—
98C2×#41—101no—
99C2×M16—101no—
100C4⋊Q8—51no—
101#127—51no—
102#124—101no—
103#121—102no—
104C4×D4—102no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C42—22no—
108C2×C42C22⋊D512yes—
109C2×C42C22×D511yescharacteristic
110C4⋊C8—102no—
111C4⋊C8—102no—
112#104—204no—
113C22⋊C8—102no—
114C4×C8—102no—
115#95—24no—
116#95—22no—
117#95C22⋊D514yes—
118C22×C5—22no—
119C22×C5#39311yescharacteristic
120C22×C5#44912yes—
121C2×D5—81no—
122C2×D5—82no—
123C4×C5—22no—
124C4×C5—22no—
125C4×C5—44no—
126C4×C5—84no—
127C4×C5#47812yes—
128C4×C5#48811yescharacteristic
129C4×C5#49011yescharacteristic
130Dic5—81no—
131#41—202no—
132#41—202no—
133C2×Q8—101no—
134C2×D4—101no—
135C2×D4—102no—
136C2×D4—201no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—201no—
140C22×C4C2×C22⋊D512yes—
141C22×C4C2×C22⋊D511yescharacteristic
142M16—202no—
143C2×C8—202no—
144C2×C8—201no—
145C4⋊C4—48no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—102no—
149C22⋊C4—202no—
150C42—22no—
151C42—22no—
152C42—44no—
153C42Q8×D511yescharacteristic
154C42#71311yescharacteristic
155C42#71312yes—
156C2×C5—22no—
157C2×C5#202511yescharacteristic
158C2×C5#203112yes—
159D5—161no—
160C23—201no—
161C23#409911yescharacteristic
162Q8—202no—
163D4—202no—
164D4—402no—
165D4—402no—
166C2×C4—24no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—48no—
171C2×C4—44no—
172C2×C4—201no—
173C2×C4—401no—
174C2×C4—401no—
175C2×C4#409212yes—
176C2×C4#409211yescharacteristic
177C2×C4#409312yes—
178C2×C4#409311yescharacteristic
179C2×C4#409512yes—
180C2×C4#409612yes—
181C8—402no—
182C8—401no—
183C5#1036811yescharacteristic, complemented, Sylow
184C22—22no—
185C22—401no—
186C22—402no—
187C22#6552011yescharacteristic
188C22#6552712yes—
189C4—22no—
190C4—22no—
191C4—44no—
192C4—84no—
193C4—401no—
194C4#6535512yes—
195C4#6535611yescharacteristic
196C4#6535711yescharacteristic
197C2—22no—
198C2—801no—
199C2#1060253611yescharacteristic
200C2#1060263512yes—
subgroupG/H
centerC22#65520
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#2864C2
socleC22×C5#393
prime radical#65811C22
hypercenter#2864D5
hypocenterC5#10368
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2864

Complements

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

NH ≅ G/Ntype
1. C5×#2864C2semidirect
4. #10598122C2semidirect
5. #10598102C2semidirect
36. #2864D5semidirect
183. C5#10368semidirect

Maximal subgroups

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

subgroupindex
C5×#28642
#106003382
#106001882
#105981222
#105981022
#103685