Finite Group Databasebeta

← #13020917

Subgroups — #13020917

Order 1280.

714 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 140 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#10606050C211yesmaximal, characteristic
2C2×#64871C214yesmaximal, complemented
3#10604191C211yesmaximal, characteristic, complemented
4#10604174C211yesmaximal, characteristic, complemented
5C2×#4039—22no—
6C2×#4039C2212yes—
7C2×#4020C411yescharacteristic
8C2×#4004—22no—
9C2×C4⋊#170C2212yes—
10C2×C4×#170C2212yes—
11C5×#384C411yescharacteristic
12#65277—22no—
13#65277C412yes—
14C42×Dic5C2211yescharacteristic
15#64918—22no—
16#64871—28no—
17#9644—51nomaximal, Sylow
18C22×D20—22no—
19C22×D20C2×C412yes—
20C22×C4×D5—22no—
21C22×#170—22no—
22C22×#170—22no—
23C2×C5×C4⋊C4—22no—
24C2×C5×C4⋊C4C2×C412yes—
25C2×C42×C5C2×C411yescharacteristic
26C2×#685—24no—
27C2×#685—22no—
28C2×C4×Dic5C2×C411yescharacteristic
29C2×C4×Dic5D412yes—
30C2×C4×Dic5D412yes—
31C2×C4×Dic5C2311yescharacteristic
32#4039—24no—
33#4039—24no—
34#4039—44no—
35#4039—44no—
36#4020—24no—
37#4004—44no—
38C4⋊#170—24no—
39C4×#170—22no—
40C4×#170D414yescomplemented
41C4×#127—51no—
42#1842—51no—
43#1744—51no—
44C2×D4⋊C8—54no—
45C23×D5—22no—
46C22×C4×C5—22no—
47C22×C4×C5C22⋊C412yes—
48C22×C4×C5C22×C411yescharacteristic
49C22×Dic5C22⋊C412yes—
50C22×Dic5C2×D411yescharacteristic
51C22×Dic5C2×D411yescharacteristic
52C2×D20—24no—
53C2×D20—24no—
54C2×D20—28no—
55C2×D20—22no—
56C2×D20—44no—
57C2×D20—44no—
58C2×D20M1614yes—
59C2×C4×D5—42no—
60C2×C4×D5—42no—
61C2×C4×D5—44no—
62C2×C4×D5—44no—
63C2×#170—24no—
64C2×#170—44no—
65C2×#170—42no—
66C2×#170—44no—
67C5×C4⋊C4—24no—
68C5×C4⋊C4—44no—
69C42×C5—22no—
70#685—48no—
71#685—44no—
72C4×Dic5—22no—
73C4×Dic5—42no—
74C4×Dic5D814yes—
75C4×Dic5SD1614yes—
76C4×Dic5C2×D412yes—
77C4×Dic5#4112yes—
78C2×#127—51no—
79C2×C4×D4—52no—
80C2×C4×D4—102no—
81C2×C4⋊C8—52no—
82C2×C4⋊C8—102no—
83C2×C4×C8—52no—
84#390—52no—
85#390—102no—
86#384AGL(1,𝔽5)11yescharacteristic
87C43—51no—
88#336—102no—
89D4⋊C8—108no—
90C23×C5C2×C22⋊C411yescharacteristic
91C22×D5—24no—
92C22×D5—42no—
93C22×D5—42no—
94C22×D5—44no—
95C22×D5—44no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—42no—
99C2×C4×C5—44no—
100C2×C4×C5#10214yes—
101C2×C4×C5C4×D412yes—
102C2×C4×C5C2×M1612yes—
103C2×C4×C5#13112yes—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5#10214yes—
108C2×Dic5#12112yes—
109C2×Dic5#12411yescharacteristic
110C2×Dic5#12711yescharacteristic
111C2×Dic5C2×D812yes—
112C2×Dic5C2×SD1612yes—
113D20—28no—
114D20—416no—
115D20—44no—
116C4×D5—84no—
117C4×D5—82no—
118C4×D5—82no—
119#170—44no—
120#170—82no—
121C22×D4—52no—
122C22×D4—102no—
123C23×C4—102no—
124C22×C8—102no—
125C22×C8—102no—
126#127—104no—
127C4×D4—104no—
128C4×D4—104no—
129C4×D4—204no—
130C4×D4—204no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
133C2×C22⋊C4—102no—
134C2×C22⋊C4—104no—
135C2×C42—52no—
136C2×C42—51no—
137C2×C42—51no—
138C2×C42—52no—
139C2×C42C2×AGL(1,𝔽5)11yescharacteristic
140C4⋊C8—104no—
141C4⋊C8—204no—
142C4×C8—54no—
143C4×C8—102no—
144C22×C5#39011yescharacteristic
145C22×C5#40711yescharacteristic
146C22×C5#40811yescharacteristic
147C22×C5C2×#10212yes—
148C22×C5#42112yes—
149C2×D5—44no—
150C2×D5—48no—
151C2×D5—82no—
152C2×D5—82no—
153C2×D5—84no—
154C4×C5—24no—
155C4×C5—82no—
156C4×C5#43514yes—
157Dic5—22no—
158Dic5—24no—
159Dic5#48612yes—
160Dic5#49211yescharacteristic
161Dic5#49311yescharacteristic
162C24—102no—
163C2×D4—54no—
164C2×D4—108no—
165C2×D4—102no—
166C2×D4—104no—
167C2×D4—104no—
168C2×D4—204no—
169C2×D4—204no—
170C22×C4—22no—
171C22×C4—51no—
172C22×C4—51no—
173C22×C4—52no—
174C22×C4—202no—
175C22×C4—202no—
176C22×C4—204no—
177C22×C4—204no—
178C22×C4C22⋊AGL(1,𝔽5)12yes—
179C22×C4C22×AGL(1,𝔽5)11yescharacteristic
180C2×C8—104no—
181C2×C8—204no—
182C2×C8—202no—
183C2×C8—204no—
184C4⋊C4—24no—
185C4⋊C4—44no—
186C22⋊C4—204no—
187C22⋊C4—208no—
188C42—22no—
189C42—52no—
190C42—52no—
191C42—54no—
192C42—54no—
193C42—102no—
194C42—202no—
195C2×C5#148012yes—
196C2×C5#148612yes—
197C2×C5#195511yescharacteristic
198C2×C5#195711yescharacteristic
199C2×C5#196311yescharacteristic
200D5—84no—
201C23—104no—
202C23—204no—
203C23—202no—
204C23—204no—
205C23—202no—
206C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
207D4—108no—
208D4—2016no—
209D4—204no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—52no—
215C2×C4—52no—
216C2×C4—51no—
217C2×C4—51no—
218C2×C4—54no—
219C2×C4—52no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—102no—
223C2×C4—402no—
224C2×C4—404no—
225C2×C4—402no—
226C2×C4#400714yes—
227C2×C4D5⋊M1612yes—
228C2×C4D4×AGL(1,𝔽5)12yes—
229C2×C4#413312yes—
230C8—204no—
231C8—402no—
232C5#964411yescharacteristic, complemented, Sylow
233C22—204no—
234C22—208no—
235C22—402no—
236C22—404no—
237C22—402no—
238C22#6575511yescharacteristic
239C22C2×#400712yes—
240C22#6578011yescharacteristic
241C22#6578412yes—
242C22#6578711yescharacteristic
243C4—24no—
244C4—51no—
245C4—52no—
246C4—51no—
247C4—82no—
248C4—104no—
249C4—102no—
250C4#6570414yes—
251C2—404no—
252C2#1060427212yes—
253C2#1060432911yescharacteristic
254C2#1060461611yescharacteristic
255C2#1060464412yes—
256C2#1060466411yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#384C4
socleC23×C5C2×C22⋊C4
prime radicalC2×#4020C4
cosocleC2×C4×Dic5C23
hypercenter#384AGL(1,𝔽5)
hypocenterC5#9644
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#64871C2semidirect
3. #10604191C2semidirect
4. #10604174C2semidirect
40. C4×#170D4semidirect
232. C5#9644semidirect

Maximal subgroups

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

subgroupindex
#106060502
C2×#648712
#106041912
#106041742
#96445