Finite Group Databasebeta

← #13024618

Subgroups — #13024618

Order 1280.

334 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 88 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#10613665C211yesmaximal, characteristic
2#10603845C211yesmaximal, characteristic
3#10603828C211yesmaximal, characteristic
4#10598612C214yesmaximal
5#66062—21no—
6#65688—21no—
7C5×#500C411yescharacteristic, complemented
8#65382—21no—
9#65176—21no—
10#65176C2212yes—
11#65162—21no—
12#65162C412yescomplemented
13#64988C411yescharacteristic, complemented
14C4×C8×D5C2211yescharacteristic
15#64893—21no—
16#64893C2212yes—
17#64888—21no—
18#64887C2212yes—
19C4×#674—21no—
20#11857—51nomaximal, Sylow
21C2×C4⋊AGL(1,𝔽5)—22no—
22C2×C4⋊AGL(1,𝔽5)—41no—
23C2×C4⋊AGL(1,𝔽5)—41no—
24C5×C4⋊Q8—21no—
25#4092—41no—
26Q8×Dic5—41no—
27C2×Dic20—21no—
28C2×Dic20C2×C412yes—
29C2×C8×D5D411yescharacteristic
30C2×C8×D5C2311yescharacteristic
31#4043—22no—
32#4038—41no—
33#4036—41no—
34#4033—22no—
35C42×D5D411yescharacteristic
36#4015—21no—
37C2×#674—22no—
38C5×#107—21no—
39C5×#107C2×C412yes—
40C4×C5×C8C2×C411yescharacteristic
41C8×Dic5C2×C411yescharacteristic
42C8×Dic5D411yescharacteristic
43#3942—22no—
44#3942—22no—
45C4×#168D411yescharacteristic
46#3170—51no—
47#2162—51no—
48#1834—51no—
49#1348—54no—
50C22×AGL(1,𝔽5)—42no—
51C2×C5×Q8—41no—
52C2×C4×D5D811yescharacteristic
53C2×C4×D5D811yescharacteristic
54C2×C4×D5C2×D411yescharacteristic
55C2×Dic10—22no—
56C4⋊AGL(1,𝔽5)—42no—
57C4⋊AGL(1,𝔽5)—42no—
58C4⋊AGL(1,𝔽5)—82no—
59C4⋊AGL(1,𝔽5)—82no—
60C2×C5×C8C22⋊C411yescharacteristic
61C2×C5×C8C22×C411yescharacteristic
62C5×C4⋊C4—22no—
63C5×C4⋊C4—41no—
64C42×C5C22⋊C411yescharacteristic
65#685—42no—
66C4⋊Dic5—41no—
67C4⋊Dic5—41no—
68C4⋊Dic5—42no—
69#683—42no—
70C4×Dic5—42no—
71C4×Dic5C22⋊C411yescharacteristic
72C4×Dic5SD1611yescharacteristic
73C4×Dic5SD1611yescharacteristic
74C2×#168C22⋊C411yescharacteristic
75C2×#168C2×D411yescharacteristic
76Dic20—42no—
77Dic20—42no—
78C8×D5—41no—
79C8×D5—41no—
80#674—42no—
81#533—101no—
82#500AGL(1,𝔽5)11yescharacteristic, complemented
83#494—51no—
84#487—101no—
85#467—52no—
86#467—101no—
87C4×Q16—52no—
88C4×Q16—101no—
89C2×C4×C8—51no—
90#384—101no—
91C4⋊C16—101no—
92C22⋊C16—52no—
93C4×C16—101no—
94#340—52no—
95#340—101no—
96C22×D5C2×D811yescharacteristic
97C2×AGL(1,𝔽5)—82no—
98C2×AGL(1,𝔽5)—82no—
99C2×AGL(1,𝔽5)—82no—
100C5×Q8—82no—
101C2×C4×C5—42no—
102C2×C4×C5#10211yescharacteristic
103C2×C4×C5#10211yescharacteristic
104C2×C4×C5C2×C22⋊C411yescharacteristic
105C2×Dic5—42no—
106C2×Dic5#10211yescharacteristic
107C2×Dic5#10211yescharacteristic
108C2×Dic5C2×SD1611yescharacteristic
109C4×D5—22no—
110C4×D5—21no—
111C4×D5—21no—
112C4×D5SD3212yes—
113C4×D5Q3212yes—
114Dic10—42no—
115Dic10—42no—
116C5×C8—21no—
117C5×C8—21no—
118#168—21no—
119#168—21no—
120C2×Q16—52no—
121C2×Q16—101no—
122C22×C8—51no—
123C22×C8—51no—
124C4⋊Q8—21no—
125C4⋊Q8—101no—
126#125—201no—
127C22⋊Q8—102no—
128C22⋊Q8—201no—
129C4×Q8—102no—
130C4×Q8—201no—
131#117—201no—
132C2×C4⋊C4—102no—
133C2×C4⋊C4—201no—
134C2×C4⋊C4—201no—
135C2×C42—51no—
136C2×C16—102no—
137#107—21no—
138#107C2×AGL(1,𝔽5)12yes—
139#103—102no—
140#103—102no—
141C4×C8—51no—
142C4×C8—51no—
143C4×C8—51no—
144C4×C8C2×AGL(1,𝔽5)11yescharacteristic
145C22×C5C2×#10211yescharacteristic
146C2×D5—21no—
147C2×D5—21no—
148C2×D5C2×SD3211yescharacteristic
149C2×D5C2×Q3211yescharacteristic
150AGL(1,𝔽5)—162no—
151C4×C5—82no—
152C4×C5#35812yes—
153C4×C5#36012yes—
154C4×C5#41011yescharacteristic
155C4×C5#42111yescharacteristic
156Dic5—21no—
157Dic5—21no—
158Dic5—82no—
159Dic5#35812yes—
160C2×Q8—41no—
161C2×Q8—102no—
162C22×C4—51no—
163C22×C4—51no—
164C22×C4—51no—
165C22×C4—202no—
166Q16—202no—
167Q16—202no—
168C2×C8—51no—
169C2×C8—51no—
170C2×C8—201no—
171C2×C8—201no—
172C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
173C2×C8C22×AGL(1,𝔽5)11yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4—202no—
177C4⋊C4—202no—
178C4⋊C4—202no—
179C4⋊C4—202no—
180C4⋊C4—201no—
181C4⋊C4—201no—
182C4⋊C4—402no—
183C4⋊C4—402no—
184C22⋊C4—202no—
185C42—51no—
186C42—51no—
187C42—51no—
188C42—202no—
189C42C22⋊AGL(1,𝔽5)11yescharacteristic
190C16—202no—
191C2×C5#151711yescharacteristic
192C2×C5C2×#35811yescharacteristic
193C2×C5C2×#36011yescharacteristic
194D5—21no—
195D5—21no—
196C23—51no—
197Q8—82no—
198Q8—202no—
199Q8—202no—
200C2×C4—42no—
201C2×C4—51no—
202C2×C4—52no—
203C2×C4—52no—
204C2×C4—51no—
205C2×C4—51no—
206C2×C4—102no—
207C2×C4—101no—
208C2×C4—101no—
209C2×C4—202no—
210C2×C4—402no—
211C2×C4—402no—
212C2×C4—402no—
213C2×C4#400711yescharacteristic
214C2×C4#400711yescharacteristic
215C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
216C8—21no—
217C8—21no—
218C8—101no—
219C8—101no—
220C5#1185711yescharacteristic, complemented, Sylow
221C22—51no—
222C22—51no—
223C22—101no—
224C22—101no—
225C22C2×#400711yescharacteristic
226C4—52no—
227C4—82no—
228C4—101no—
229C4—101no—
230C4—402no—
231C4—802no—
232C4#6490312yes—
233C4#6490812yes—
234C4#6578411yescharacteristic
235C4#6578511yescharacteristic
236C2—101no—
237C2—101no—
238C2#1060463011yescharacteristic
239C2C2×#6490311yescharacteristic
240C2C2×#6490811yescharacteristic
subgroupG/H
centerC22C2×#4007
commutatorC2×C5×C8C22×C4
FrattiniC2×C8C22×AGL(1,𝔽5)
FittingC5×#500C4
socleC22×C5C2×#102
prime radicalC22×D5C2×D8
cosocleC2×C8×D5C23
hypercenter#500AGL(1,𝔽5)
hypocenterC5#11857
LCSG ▷ C2×C5×C8 ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #500

Complements

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

NH ≅ G/Ntype
7. C5×#500C4semidirect
12. #65162C4semidirect
13. #64988C4semidirect
82. #500AGL(1,𝔽5)semidirect
220. C5#11857semidirect

Maximal subgroups

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

subgroupindex
#106136652
#106038452
#106038282
#105986122
#118575