Finite Group Databasebeta

← #13016896

Subgroups — #13016896

Order 1280.

412 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 118 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#10610801C211yesmaximal, characteristic
2#10604689C211yesmaximal, characteristic, complemented
3C2×#64879C211yesmaximal, characteristic, complemented
4#10604359C212yesmaximal
5#10598787C212yesmaximal, complemented
6#65856C2211yescharacteristic
7#65748C2212yes—
8C2×C4×#170C2212yes—
9C5×#424C411yescharacteristic
10#65418C411yescharacteristic
11M16×Dic5C2211yescharacteristic
12#65399C411yescharacteristic
13C2×C8×Dic5C2211yescharacteristic
14#65288C411yescharacteristic
15#65129—22no—
16#65122—22no—
17#170⋊C8—22no—
18#64879—22no—
19#64872C414yes—
20#64871C414yes—
21#8188—51nomaximal, Sylow
22C2×#709C2×C411yescharacteristic
23C22×#170—22no—
24C2×C5×M16C2×C411yescharacteristic
25C22×C5×C8C2×C411yescharacteristic
26C5×#117C2×C411yescharacteristic
27C2×C4×Dic5C2311yescharacteristic
28C2×#681C2×C411yescharacteristic
29C22×#168C2×C411yescharacteristic
30#4039C2×C412yes—
31#4033C2×C412yes—
32#4027—22no—
33#4018—21no—
34C42×D5—21no—
35C22⋊#170—22no—
36C4⋊#170C2×C414yes—
37C4×#170—22no—
38C4×#170—22no—
39C4×#170C2×C414yes—
40C5×C4⋊C8—22no—
41#3952—22no—
42#3952—22no—
43#3947—21no—
44C8×Dic5—21no—
45C8×Dic5D412yes—
46C8×Dic5D412yes—
47C4⋊#168—22no—
48#2922—51no—
49#1476—52no—
50#1452—51no—
51C2×#322—51no—
52#1278—52no—
53C22×C4×C5C22×C411yescharacteristic
54C2×C22⋊D5—21no—
55C22×Dic5#4112yes—
56#709—22no—
57#709—22no—
58C2×D20C4211yescharacteristic
59C2×C4×D5—21no—
60C2×C4×D5—22no—
61C2×Dic10C4211yescharacteristic
62C2×#170—24no—
63C2×#170—24no—
64C2×#170—44no—
65C5×M16—22no—
66C2×C5×C8—22no—
67C2×C5×C8—21no—
68C2×C5×C8C22⋊C412yes—
69C5×C4⋊C4C4212yes—
70C5×C22⋊C4—21no—
71C42×C5—21no—
72#685—22no—
73#683—22no—
74C4×Dic5—22no—
75C4×Dic5C22×C412yes—
76C4×Dic5C2×D411yescharacteristic
77C4×Dic5C2×D411yescharacteristic
78#681—22no—
79C2×#168—22no—
80C2×#168—21no—
81C2×#168C22⋊C412yes—
82C4×#41—51no—
83#433—102no—
84#431—52no—
85#431—102no—
86#424—51no—
87#424AGL(1,𝔽5)11yescharacteristic
88#408—51no—
89#408—51no—
90C4×M16—51no—
91C2×C4×C8—51no—
92C2×C4×C8—52no—
93Q8⋊C8—54no—
94D4⋊C8—54no—
95#322—102no—
96#322—102no—
97C23×C5#11711yescharacteristic
98C22×D5—21no—
99C2×C4×C5—22no—
100C2×C4×C5C2×C4211yescharacteristic
101C2×C4×C5C2×C22⋊C411yescharacteristic
102C2×C4×C5#13112yes—
103C2×C4×C5#13112yes—
104C22⋊D5—42no—
105C2×Dic5—24no—
106C2×Dic5—21no—
107C2×Dic5C4×D412yes—
108C2×Dic5C4×D412yes—
109D20—22no—
110C4×D5—41no—
111C4×D5—41no—
112C4×D5—44no—
113Dic10—22no—
114#170—44no—
115#170—44no—
116C5×C8—22no—
117C5×C8—41no—
118#168—22no—
119#168—41no—
120C2×#41—51no—
121C2×M16—51no—
122C2×M16C2×AGL(1,𝔽5)11yescharacteristic
123C22×C8—51no—
124C22×C8—102no—
125C22×C8C2×AGL(1,𝔽5)11yescharacteristic
126C4×Q8—52no—
127C4×D4—52no—
128C4×D4—102no—
129#117—101no—
130#117C2×AGL(1,𝔽5)11yescharacteristic
131C2×C42—51no—
132C2×C42—101no—
133C4⋊C8—22no—
134C4⋊C8—54no—
135C4⋊C8—102no—
136C22⋊C8—102no—
137C22⋊C8—102no—
138C22⋊C8—102no—
139#97—101no—
140C4×C8—52no—
141C4×C8—52no—
142C4×C8—54no—
143C4×C8—101no—
144C4×C8—102no—
145C4×C8—102no—
146C22×C5—22no—
147C22×C5C4×C22⋊C411yescharacteristic
148C22×C5#43312yes—
149C2×D5—41no—
150C2×D5—42no—
151C4×C5—42no—
152C4×C5#40512yes—
153C4×C5#40811yescharacteristic
154C4×C5#40811yescharacteristic
155Dic5—22no—
156Dic5—41no—
157Dic5#44414yes—
158#41—102no—
159#41—102no—
160C2×Q8—51no—
161C2×D4—51no—
162C2×D4—101no—
163C22×C4—52no—
164C22×C4—102no—
165C22×C4—101no—
166C22×C4C22×AGL(1,𝔽5)11yescharacteristic
167M16—22no—
168M16—102no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—52no—
172C2×C8—102no—
173C2×C8—104no—
174C2×C8—104no—
175C2×C8—101no—
176C2×C8—204no—
177C2×C8C22⋊AGL(1,𝔽5)12yes—
178C4⋊C4—102no—
179C4⋊C4C4×AGL(1,𝔽5)12yes—
180C22⋊C4—21no—
181C22⋊C4—102no—
182C42—21no—
183C42—52no—
184C42—51no—
185C42—51no—
186C42—102no—
187C2×C5—22no—
188C2×C5#175011yescharacteristic
189C2×C5#176512yes—
190D5—81no—
191C23—101no—
192C23#413011yescharacteristic
193Q8—102no—
194D4—102no—
195D4—202no—
196C2×C4—22no—
197C2×C4—52no—
198C2×C4—52no—
199C2×C4—104no—
200C2×C4—101no—
201C2×C4—204no—
202C2×C4—201no—
203C2×C4—201no—
204C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
205C2×C4#413112yes—
206C2×C4#413312yes—
207C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
208C8—22no—
209C8—41no—
210C8—102no—
211C8—204no—
212C8—204no—
213C8—201no—
214C5#818811yescharacteristic, complemented, Sylow
215C22—22no—
216C22—201no—
217C22—202no—
218C22#6570612yes—
219C22C4×C22⋊AGL(1,𝔽5)11yescharacteristic
220C4—42no—
221C4—54no—
222C4—102no—
223C4—201no—
224C4#6570112yes—
225C4#6577511yescharacteristic
226C4#6578711yescharacteristic
227C2—22no—
228C2—401no—
229C2#1060431311yescharacteristic
230C2#1060477812yes—
subgroupG/H
centerC2×C4C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#424C4
socleC22×C5C4×C22⋊C4
prime radicalC2×#709C2×C4
cosocleC2×C4×Dic5C23
hypercenter#424AGL(1,𝔽5)
hypocenterC5#8188
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #10604689C2semidirect
3. C2×#64879C2semidirect
5. #10598787C2semidirect
214. C5#8188semidirect

Maximal subgroups

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

subgroupindex
#106108012
#106046892
C2×#648792
#106043592
#105987872
#81885