Finite Group Databasebeta

← #12012379

Subgroups — #12012379

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×#2844C211yesmaximal, characteristic
2#10600500C212yesmaximal
3#10600336C211yesmaximal, characteristic
4#10598139C212yesmaximal
5#10598123C211yesmaximal, characteristic
6#65834C2211yescharacteristic
7C5×#400—22no—
8C5×#400—21no—
9C5×#395—22no—
10C5×#395C2212yes—
11C5×#382—21no—
12C5×#382C2211yescharacteristic
13#65517—22no—
14#65253—21no—
15#64750C2211yescharacteristic
16#64748C2212yes—
17#64743—21no—
18#64672—22no—
19#12260—51nomaximal, Sylow
20C22×C5×Q8D412yes—
21C2×C5×C4⋊C4—21no—
22C2×C5×C4⋊C4—21no—
23C2×C5×C4⋊C4—21no—
24C2×C5×C4⋊C4—22no—
25C2×C5×C4⋊C4D411yescharacteristic
26C2×C5×C4⋊C4C2311yescharacteristic
27C2×C42×C5D411yescharacteristic
28C2×#689—42no—
29#4072—21no—
30#4070—22no—
31C2×#681—21no—
32#4036—41no—
33#4034—41no—
34C5×#95—22no—
35C5×#95—22no—
36C5×#95—21no—
37C5×#95—22no—
38C5×#95—22no—
39C5×#95D412yes—
40#3967—42no—
41C22⋊#168—22no—
42#3940—41no—
43#3939—41no—
44#2844D511yescharacteristic
45#2061—51no—
46#1605—52no—
47#1354—52no—
48#1324—51no—
49C2×C5×Q8—42no—
50C2×C5×Q8—82no—
51C22×C4×C5—21no—
52C22×C4×C5—22no—
53C22×C4×C5—22no—
54C22×C4×C5—21no—
55C22×C4×C5C2×D411yescharacteristic
56C22×C4×C5C2×D412yes—
57C2×Dic10—22no—
58C5×C4⋊C4—41no—
59C5×C4⋊C4—41no—
60C5×C4⋊C4—81no—
61C5×C4⋊C4—81no—
62C5×C4⋊C4—82no—
63C5×C4⋊C4—81no—
64C42×C5—21no—
65C42×C5Q1612yescomplemented
66C22⋊Dic5—41no—
67#689—84no—
68C4⋊Dic5—21no—
69C4⋊Dic5—21no—
70C4⋊Dic5—22no—
71#683—42no—
72C4×Dic5—41no—
73#681—22no—
74#681—41no—
75C2×#168—41no—
76C2×#168—42no—
77#544—51no—
78#451—102no—
79#428—101no—
80#400—21no—
81#400—22no—
82#395—22no—
83#395C2×D512yes—
84#382—21no—
85#382C2×D511yescharacteristic
86#343—51no—
87#339—101no—
88#328—102no—
89#324—52no—
90C23×C5#12011yescharacteristic
91C5×Q8—44no—
92C5×Q8—82no—
93C2×C4×C5—21no—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—41no—
98C2×C4×C5—41no—
99C2×C4×C5—42no—
100C2×C4×C5—42no—
101C2×C4×C5—42no—
102C2×C4×C5—42no—
103C2×C4×C5—42no—
104C2×C4×C5—42no—
105C2×C4×C5—42no—
106C2×C4×C5—41no—
107C2×C4×C5—42no—
108C2×C4×C5C2×Q1611yescharacteristic
109C2×C4×C5#13711yescharacteristic
110C2×Dic5—41no—
111C2×Dic5—41no—
112Dic10—82no—
113#168—42no—
114#168—82no—
115C22×Q8C22⋊D512yes—
116C2×Q16—202no—
117C2×M16—101no—
118C4⋊Q8—201no—
119#125—201no—
120C22⋊Q8—102no—
121#117—101no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4C22⋊D511yescharacteristic
127C2×C4⋊C4C22×D511yescharacteristic
128C2×C42C22⋊D511yescharacteristic
129#107—201no—
130#106—201no—
131#103—202no—
132C22⋊C8—102no—
133#95—22no—
134#95—22no—
135#95—21no—
136#95—22no—
137#95—22no—
138#95C22⋊D512yes—
139C22×C5—21no—
140C22×C5—21no—
141C22×C5#45111yescharacteristic
142C22×C5#45611yescharacteristic
143C22×C5B(4,𝔽2)11yescharacteristic
144C4×C5—21no—
145C4×C5—21no—
146C4×C5—42no—
147C4×C5—42no—
148C4×C5—82no—
149C4×C5—82no—
150C4×C5—81no—
151Dic5—81no—
152Dic5—81no—
153C2×Q8—42no—
154C2×Q8—82no—
155C2×Q8—102no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4C2×C22⋊D511yescharacteristic
161C22×C4C2×C22⋊D512yes—
162Q16—404no—
163M16—102no—
164M16—201no—
165C2×C8—201no—
166C2×C8—202no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—82no—
172C4⋊C4—81no—
173C4⋊C4—102no—
174C4⋊C4—101no—
175C4⋊C4—101no—
176C4⋊C4—202no—
177C22⋊C4—201no—
178C42—21no—
179C42—201no—
180C42#68912yescomplemented
181C2×C5—21no—
182C2×C5—21no—
183C2×C5#160511yescharacteristic
184C2×C5#219911yescharacteristic
185C2×C5#220311yescharacteristic
186C23#409911yescharacteristic
187Q8—44no—
188Q8—82no—
189Q8—402no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—201no—
206C2×C4—201no—
207C2×C4#408811yescharacteristic
208C2×C4C2×#68911yescharacteristic
209C8—202no—
210C8—402no—
211C5#1226011yescharacteristic, complemented, Sylow
212C22—21no—
213C22—21no—
214C22#6551711yescharacteristic
215C22#6553311yescharacteristic
216C22#6553511yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—81no—
223C4—82no—
224C4—401no—
225C4—401no—
226C2—21no—
227C2—21no—
228C2#1060269711yescharacteristic
229C2#1060274311yescharacteristic
230C2#1060275211yescharacteristic
subgroupG/H
centerC22#65533
commutator, cosocleC2×C5×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2844C2
socleC22×C5B(4,𝔽2)
prime radicalC23×C5#120
hypercenter#2844D5
hypocenterC5#12260
LCSG ▷ C2×C5×C4⋊C4 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
65. C42×C5Q16semidirect
180. C42#689semidirect
211. C5#12260semidirect

Maximal subgroups

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

subgroupindex
C5×#28442
#106005002
#106003362
#105981392
#105981232
#122605