Finite Group Databasebeta

← #12012360

Subgroups — #12012360

Order 1280.

342 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2571C211yesmaximal, characteristic
2#10600512C212yesmaximal
3#10600366C211yesmaximal, characteristic
4#10598143C212yesmaximal
5#10598126C211yesmaximal, characteristic
6#65834C2211yescharacteristic
7C5×#400—21no—
8C5×#400C2212yes—
9C5×#389—22no—
10C5×#384C2211yescharacteristic
11C5×#382—22no—
12C4×C5×C4⋊C4—21no—
13#65259—22no—
14#65253—21no—
15#64750C2211yescharacteristic
16#64748C2212yes—
17#64743—21no—
18#64672—22no—
19#12257—51nomaximal, Sylow
20C2×C5×C4⋊C4—21no—
21C2×C5×C4⋊C4—21no—
22C2×C5×C4⋊C4—22no—
23C2×C5×C4⋊C4—21no—
24C2×C5×C4⋊C4D411yescharacteristic
25C2×C5×C4⋊C4D412yes—
26C2×C5×C4⋊C4C2311yescharacteristic
27C2×C42×C5—22no—
28C2×C42×C5D411yescharacteristic
29#4072—21no—
30#4070—22no—
31C2×#681—21no—
32#4036—41no—
33#4034—41no—
34C5×#95—21no—
35C5×#95—22no—
36C5×#95—22no—
37C5×#95—22no—
38C5×#95D412yes—
39C22⋊#168—22no—
40#3942—42no—
41#3940—41no—
42#3939—41no—
43#3939—42no—
44#2571D511yescharacteristic
45#2060—51no—
46#1617—52no—
47#1353—52no—
48#1325—51no—
49C22×C4×C5—21no—
50C22×C4×C5—22no—
51C22×C4×C5—22no—
52C22×C4×C5—21no—
53C22×C4×C5C2×D411yescharacteristic
54C22×C4×C5C2×D412yes—
55C2×Dic10—22no—
56C5×C4⋊C4—41no—
57C5×C4⋊C4—41no—
58C5×C4⋊C4—42no—
59C5×C4⋊C4—42no—
60C5×C4⋊C4—44no—
61C5×C4⋊C4—42no—
62C5×C4⋊C4—81no—
63C42×C5—21no—
64C42×C5—44no—
65C42×C5Q1612yes—
66C22⋊Dic5—41no—
67C4⋊Dic5—21no—
68C4⋊Dic5—22no—
69C4⋊Dic5—21no—
70#683—42no—
71C4×Dic5—41no—
72#681—22no—
73#681—41no—
74C2×#168—41no—
75C2×#168—42no—
76#544—51no—
77#484—102no—
78#428—101no—
79#400—21no—
80#400C2×D512yes—
81#389—22no—
82#384C2×D511yescharacteristic
83#382—22no—
84C4×C4⋊C4—21no—
85#343—51no—
86#339—101no—
87#328—102no—
88#324—52no—
89C23×C5#12011yescharacteristic
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—21no—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5—42no—
97C2×C4×C5—42no—
98C2×C4×C5—42no—
99C2×C4×C5—42no—
100C2×C4×C5—42no—
101C2×C4×C5—42no—
102C2×C4×C5—42no—
103C2×C4×C5—41no—
104C2×C4×C5—42no—
105C2×C4×C5C2×Q1611yescharacteristic
106C2×C4×C5#13711yescharacteristic
107C2×Dic5—41no—
108C2×Dic5—41no—
109Dic10—82no—
110#168—42no—
111#168—82no—
112C2×M16—101no—
113C4⋊Q8—201no—
114#125—201no—
115C22⋊Q8—102no—
116#117—101no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C22⋊D511yescharacteristic
122C2×C4⋊C4C22⋊D512yes—
123C2×C4⋊C4C22×D511yescharacteristic
124C2×C42—22no—
125C2×C42C22⋊D511yescharacteristic
126#107—201no—
127#107—202no—
128#106—201no—
129#103—202no—
130C22⋊C8—102no—
131#95—21no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95C22⋊D512yes—
136C22×C5—21no—
137C22×C5—21no—
138C22×C5#45111yescharacteristic
139C22×C5#45611yescharacteristic
140C22×C5B(4,𝔽2)11yescharacteristic
141C4×C5—24no—
142C4×C5—21no—
143C4×C5—21no—
144C4×C5—42no—
145C4×C5—82no—
146C4×C5—81no—
147C4×C5—82no—
148Dic5—81no—
149Dic5—81no—
150C2×Q8—102no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4C2×C22⋊D511yescharacteristic
156C22×C4C2×C22⋊D512yes—
157M16—102no—
158M16—201no—
159C2×C8—201no—
160C2×C8—202no—
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—44no—
166C4⋊C4—42no—
167C4⋊C4—81no—
168C4⋊C4—101no—
169C4⋊C4—101no—
170C4⋊C4—102no—
171C4⋊C4—202no—
172C22⋊C4—201no—
173C42—21no—
174C42—44no—
175C42—201no—
176C42#68912yes—
177C2×C5—21no—
178C2×C5—21no—
179C2×C5#160511yescharacteristic
180C2×C5#219811yescharacteristic
181C2×C5#220411yescharacteristic
182C23#409911yescharacteristic
183Q8—402no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—201no—
200C2×C4—201no—
201C2×C4#408811yescharacteristic
202C2×C4C2×#68911yescharacteristic
203C8—202no—
204C8—402no—
205C5#1225711yescharacteristic, complemented, Sylow
206C22—21no—
207C22—21no—
208C22#6551711yescharacteristic
209C22#6553311yescharacteristic
210C22#6553511yescharacteristic
211C4—24no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—82no—
216C4—82no—
217C4—81no—
218C4—401no—
219C4—401no—
220C2—21no—
221C2—21no—
222C2#1060269711yescharacteristic
223C2#1060272311yescharacteristic
224C2#1060273911yescharacteristic
subgroupG/H
centerC22#65533
commutator, cosocleC2×C5×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2571C2
socleC22×C5B(4,𝔽2)
prime radicalC23×C5#120
hypercenter#2571D5
hypocenterC5#12257
LCSG ▷ C2×C5×C4⋊C4 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
205. C5#12257semidirect

Maximal subgroups

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

subgroupindex
C5×#25712
#106005122
#106003662
#105981432
#105981262
#122575