Finite Group Databasebeta

← #12012148

Subgroups — #12012148

Order 1280.

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

normal subgroups only

This group has 336 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×#2809C211yesmaximal, characteristic
2#10600512C212yesmaximal
3#10600466C211yesmaximal, characteristic
4#10598143C212yesmaximal
5#10598134C211yesmaximal, characteristic
6#65834C2211yescharacteristic
7C5×#400C2212yes—
8C5×#398—22no—
9C5×#391—21no—
10C5×#391C2211yescharacteristic
11C5×#389—21no—
12C5×#387—22no—
13#65514—21no—
14#65259—22no—
15#64750C2211yescharacteristic
16#64748C2212yes—
17#64743—21no—
18#64672—22no—
19#12225—51nomaximal, Sylow
20C22×C5×Q8D411yescharacteristic
21C2×C5×C4⋊C4—22no—
22C2×C5×C4⋊C4—21no—
23C2×C5×C4⋊C4—21no—
24C2×C5×C4⋊C4—21no—
25C2×C5×C4⋊C4D412yes—
26C2×C5×C4⋊C4C2311yescharacteristic
27C2×C42×C5—21no—
28C2×C42×C5D411yescharacteristic
29#4072—21no—
30#4070—22no—
31C2×#681—21no—
32#4036—41no—
33#4034—41no—
34C5×#95—22no—
35C5×#95—22no—
36C5×#95—22no—
37C5×#95—22no—
38C5×#95D412yes—
39#3967—41no—
40#3967—41no—
41C22⋊#168—22no—
42#3942—42no—
43#3939—42no—
44#2809D511yescharacteristic
45#2028—51no—
46#1617—52no—
47#1353—52no—
48#1319—51no—
49C2×C5×Q8—21no—
50C2×C5×Q8—41no—
51C2×C5×Q8—41no—
52C2×C5×Q8Q1612yes—
53C22×C4×C5—21no—
54C22×C4×C5—21no—
55C22×C4×C5—22no—
56C22×C4×C5—22no—
57C22×C4×C5C2×D411yescharacteristic
58C22×C4×C5C2×D412yes—
59C2×Dic10—22no—
60C5×C4⋊C4—41no—
61C5×C4⋊C4—42no—
62C5×C4⋊C4—44no—
63C5×C4⋊C4—42no—
64C5×C4⋊C4—81no—
65C5×C4⋊C4—81no—
66C42×C5—41no—
67C42×C5—81no—
68C22⋊Dic5—41no—
69C4⋊Dic5—22no—
70C4⋊Dic5—21no—
71C4⋊Dic5—21no—
72#683—42no—
73C4×Dic5—41no—
74#681—41no—
75#681—41no—
76C2×#168—42no—
77C2×#168—41no—
78#544—51no—
79#484—102no—
80#419—101no—
81#400C2×D512yes—
82#398—22no—
83#391—21no—
84#391C2×D511yescharacteristic
85#389—21no—
86#387—22no—
87#343—51no—
88#339—101no—
89#328—102no—
90#324—52no—
91C23×C5#12011yescharacteristic
92C5×Q8—42no—
93C5×Q8—81no—
94C2×C4×C5—21no—
95C2×C4×C5—21no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—41no—
100C2×C4×C5—41no—
101C2×C4×C5—41no—
102C2×C4×C5—41no—
103C2×C4×C5—42no—
104C2×C4×C5—42no—
105C2×C4×C5—42no—
106C2×C4×C5—42no—
107C2×C4×C5—42no—
108C2×C4×C5—42no—
109C2×C4×C5—42no—
110C2×C4×C5C2×Q1611yescharacteristic
111C2×C4×C5#13711yescharacteristic
112C2×Dic5—41no—
113C2×Dic5—41no—
114Dic10—82no—
115#168—81no—
116#168—82no—
117C22×Q8C22⋊D511yescharacteristic
118C2×M16—101no—
119C4⋊Q8—201no—
120#125—201no—
121C22⋊Q8—102no—
122#117—101no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C22⋊D512yes—
128C2×C4⋊C4C22×D511yescharacteristic
129C2×C42—21no—
130C2×C42C22⋊D511yescharacteristic
131#107—202no—
132#103—202no—
133#103—201no—
134#103—201no—
135C22⋊C8—102no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95C22⋊D512yes—
141C22×C5—21no—
142C22×C5—21no—
143C22×C5#45111yescharacteristic
144C22×C5#45511yescharacteristic
145C22×C5B(4,𝔽2)11yescharacteristic
146C4×C5—21no—
147C4×C5—21no—
148C4×C5—42no—
149C4×C5—42no—
150C4×C5—42no—
151C4×C5—82no—
152C4×C5—82no—
153Dic5—81no—
154Dic5—81no—
155C2×Q8—21no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—102no—
159C2×Q8#68912yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4C2×C22⋊D511yescharacteristic
165C22×C4C2×C22⋊D512yes—
166M16—201no—
167M16—201no—
168C2×C8—202no—
169C2×C8—201no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—44no—
173C4⋊C4—42no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—101no—
177C4⋊C4—101no—
178C4⋊C4—102no—
179C4⋊C4—202no—
180C22⋊C4—201no—
181C42—41no—
182C42—81no—
183C42—201no—
184C2×C5—21no—
185C2×C5—21no—
186C2×C5#160211yescharacteristic
187C2×C5#220111yescharacteristic
188C2×C5#220411yescharacteristic
189C23#409911yescharacteristic
190Q8—42no—
191Q8—81no—
192Q8—402no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—201no—
210C2×C4—201no—
211C2×C4#408811yescharacteristic
212C2×C4C2×#68911yescharacteristic
213C8—401no—
214C8—402no—
215C5#1222511yescharacteristic, complemented, Sylow
216C22—21no—
217C22—21no—
218C22#6551711yescharacteristic
219C22#6553111yescharacteristic
220C22#6553311yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—82no—
228C4—401no—
229C4—401no—
230C2—21no—
231C2—21no—
232C2#1060268611yescharacteristic
233C2#1060272311yescharacteristic
234C2#1060275111yescharacteristic
subgroupG/H
centerC22#65533
commutator, cosocleC2×C5×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2809C2
socleC22×C5B(4,𝔽2)
prime radicalC23×C5#120
hypercenter#2809D5
hypocenterC5#12225
LCSG ▷ C2×C5×C4⋊C4 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
215. C5#12225semidirect

Maximal subgroups

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

subgroupindex
C5×#28092
#106005122
#106004662
#105981432
#105981342
#122255