Finite Group Databasebeta

← #11996728

Subgroups — #11996728

Order 1280.

554 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#10614207C211yesmaximal, characteristic
2#10614167C211yesmaximal, characteristic, complemented
3C5×#1801C211yesmaximal, characteristic, complemented
4#10601295C212yesmaximal
5#10598206C212yesmaximal, complemented
6C2×C4×C5×Q8C2211yescharacteristic
7C2×#4093—21no—
8C2×#4020C2211yescharacteristic
9C5×#423C2212yes—
10C2×C5×#101—21no—
11#65313—24no—
12#65280C414yes—
13#65277—21no—
14#65228—22no—
15C4×C4⋊Dic5C2211yescharacteristic
16#65215—22no—
17#65214C2212yes—
18C5×#343—21no—
19#64705—24no—
20#9406—51nomaximal, Sylow
21C22×C5×Q8D411yescharacteristic
22C22×D20—21no—
23C22×D20—21no—
24C22×D20C2×C412yes—
25C2×C5×M16—22no—
26C4×C5×Q8—22no—
27C4×C5×Q8—22no—
28C2×C5×C4⋊C4C2×C412yes—
29C2×C5×C4⋊C4D411yescharacteristic
30C2×C42×C5D412yes—
31C2×C42×C5C2311yescharacteristic
32#4093—44no—
33C2×#685—24no—
34C2×#685—24no—
35C2×C4⋊Dic5—21no—
36C2×C4×Dic5—21no—
37#4020—22no—
38#4020—22no—
39C5×C4⋊C8—22no—
40C5×C4⋊C8—22no—
41C5×#101—42no—
42#3963C2×C412yes—
43C22⋊#168—22no—
44C4⋊#168—24no—
45C4⋊#168D414yes—
46#3935—24no—
47C4×#168—22no—
48#2983—51no—
49#2548—51no—
50#1801D511yescharacteristic, complemented
51#1545—52no—
52#1303—52no—
53C23×D5—22no—
54C2×C5×Q8—22no—
55C2×C5×Q8—21no—
56C2×C5×Q8—22no—
57C2×C5×Q8—21no—
58C22×C4×C5C22⋊C412yes—
59C22×C4×C5C22⋊C412yes—
60C22×C4×C5C22×C411yescharacteristic
61C22×C4×C5C2×D411yescharacteristic
62C22×C4×C5C2×D411yescharacteristic
63C22×Dic5—22no—
64C2×D20—24no—
65C2×D20—44no—
66C2×D20—41no—
67C2×D20—42no—
68C2×D20—41no—
69C2×D20—41no—
70C2×D20—42no—
71C2×D20—41no—
72C2×D20—82no—
73C5×M16—42no—
74C5×M16—42no—
75C2×C5×C8—42no—
76C5×C4⋊C4—24no—
77C5×C4⋊C4—22no—
78C42×C5—22no—
79C42×C5—22no—
80C42×C5C2×D412yes—
81C42×C5#4112yes—
82#685—48no—
83#685—84no—
84C4⋊Dic5—42no—
85C4×Dic5—42no—
86C2×#168—24no—
87C2×#127—51no—
88C2×#124—101no—
89C2×C4×Q8C2×D511yescharacteristic
90#446—104no—
91#432—102no—
92#431—52no—
93#423C2×D512yes—
94C2×#101—21no—
95#394—54no—
96#390—101no—
97#390—102no—
98C4×C4⋊C4—51no—
99#343—21no—
100#331—104no—
101C23×C5C2×C22⋊C411yescharacteristic
102C22×D5—44no—
103C22×D5—44no—
104C22×D5—42no—
105C22×D5—42no—
106C22×D5—42no—
107C5×Q8—42no—
108C5×Q8—42no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—41no—
115C2×C4×C5—42no—
116C2×C4×C5C4×D412yes—
117C2×C4×C5#12011yescharacteristic
118C2×C4×C5#12112yes—
119C2×C4×C5#12311yescharacteristic
120C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C5AGL(1,ℤ/8)12yes—
122C2×C4×C5#13711yescharacteristic
123C2×Dic5—42no—
124C2×Dic5—42no—
125C2×Dic5—42no—
126D20—84no—
127D20—82no—
128D20—82no—
129D20—82no—
130D20—82no—
131C5×C8—82no—
132#168—44no—
133C22×Q8C22⋊D511yescharacteristic
134C22×D4—52no—
135C22×D4—101no—
136C22×D4—101no—
137C2×M16—22no—
138#127—102no—
139#127—102no—
140#124—204no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—101no—
144C2×C4⋊C4C4×D512yes—
145C2×C4⋊C4C22⋊D511yescharacteristic
146C2×C22⋊C4—104no—
147C2×C22⋊C4—104no—
148C2×C42—101no—
149C2×C42D2012yes—
150C2×C42C22×D511yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—54no—
154C4⋊C8—104no—
155#101—42no—
156C22⋊C8—102no—
157#97—104no—
158C4×C8—102no—
159#95—52no—
160C22×C5—21no—
161C22×C5—21no—
162C22×C5#38611yescharacteristic
163C22×C5#41711yescharacteristic
164C22×C5#41811yescharacteristic
165C2×D5—84no—
166C2×D5—84no—
167C2×D5—82no—
168C2×D5—82no—
169C2×D5—82no—
170C4×C5—21no—
171C4×C5—22no—
172C4×C5—42no—
173C4×C5—42no—
174C4×C5#45412yes—
175C4×C5#46812yes—
176C4×C5#46912yes—
177Dic5—82no—
178C24—102no—
179C2×Q8—21no—
180C2×Q8—22no—
181C2×Q8—21no—
182C2×Q8—22no—
183C2×D4—104no—
184C2×D4—204no—
185C2×D4—201no—
186C2×D4—201no—
187C2×D4—201no—
188C2×D4—202no—
189C2×D4—201no—
190C2×D4—202no—
191C2×D4—402no—
192C22×C4—102no—
193C22×C4#68512yes—
194C22×C4#68512yes—
195C22×C4C2×C4×D511yescharacteristic
196C22×C4C2×D2011yescharacteristic
197C22×C4C2×C22⋊D511yescharacteristic
198M16—42no—
199M16—42no—
200C2×C8—42no—
201C2×C8—104no—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—202no—
205C22⋊C4—208no—
206C22⋊C4—404no—
207C42—22no—
208C42—22no—
209C42—202no—
210C42D4×D512yes—
211C42#71112yes—
212C2×C5—21no—
213C2×C5—21no—
214C2×C5#188811yescharacteristic
215C2×C5#189511yescharacteristic
216C2×C5#191411yescharacteristic
217D5—162no—
218C23—204no—
219C23—204no—
220C23—202no—
221C23—202no—
222C23—202no—
223C23C2×#68511yescharacteristic
224Q8—42no—
225Q8—42no—
226D4—404no—
227D4—402no—
228D4—402no—
229D4—402no—
230D4—402no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—202no—
239C2×C4—202no—
240C2×C4—202no—
241C2×C4#402712yes—
242C2×C4#402811yescharacteristic
243C2×C4#403211yescharacteristic
244C2×C4#407811yescharacteristic
245C2×C4#408512yes—
246C2×C4#408811yescharacteristic
247C2×C4#409512yes—
248C8—82no—
249C8—204no—
250C5#940611yescharacteristic, complemented, Sylow
251C22—21no—
252C22—21no—
253C22—404no—
254C22—404no—
255C22—402no—
256C22—402no—
257C22—402no—
258C22#6524711yescharacteristic
259C22#6525511yescharacteristic
260C22#6529011yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—402no—
266C4#6504212yes—
267C4#6532812yes—
268C4#6533812yes—
269C2—21no—
270C2—21no—
271C2—802no—
272C2#1060031011yescharacteristic
273C2#1060089711yescharacteristic
274C2#1060093011yescharacteristic
subgroupG/H
centerC22#65247
commutatorC22×C4×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#1801C2
socleC22×C5#386
prime radicalC2×#4020C22
cosocleC2×C42×C5C23
hypercenter#1801D5
hypocenterC5#9406
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #10614167C2semidirect
3. C5×#1801C2semidirect
5. #10598206C2semidirect
50. #1801D5semidirect
250. C5#9406semidirect

Maximal subgroups

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

subgroupindex
#106142072
#106141672
C5×#18012
#106012952
#105982062
#94065