Finite Group Databasebeta

← #13016189

Subgroups — #13016189

Order 1280.

650 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C2×#65678C212yesmaximal
2#10605879C211yesmaximal, characteristic
3#10598725C214yesmaximal, complemented
4C2×#4038C412yes—
5C2×C42×D5C2211yescharacteristic
6C2×#4004C2214yes—
7C2×#4003—22no—
8C2×C4×#170—22no—
9#65678—24no—
10#65678C418yescomplemented
11C5×#382C411yescharacteristic
12#65265C2212yes—
13#65225C411yescharacteristic
14#64956—22no—
15#64938—22no—
16#8038—51nomaximal, Sylow
17C22×C4×D5C2×C411yescharacteristic
18C22×C4×D5D412yes—
19C22×#170—24no—
20C2×C5×C4⋊C4C2×C412yes—
21C2×C42×C5C2×C411yescharacteristic
22C2×#685—22no—
23C2×C4⋊Dic5C2×C412yes—
24C2×#683—24no—
25C2×C4×Dic5—22no—
26C2×C4×Dic5D412yes—
27C2×C4×Dic5C2311yescharacteristic
28#4038—44no—
29C42×D5C2×C414yes—
30C42×D5D412yes—
31C42×D5Q812yes—
32#4004—24no—
33#4004C2×C418yes—
34#4003—28no—
35C4×#170—28no—
36C5×#95—22no—
37#3963—22no—
38C2×#431—52no—
39#2285—51no—
40#1296—54no—
41C23×D5C22⋊C411yescharacteristic
42C22×C4×C5—22no—
43C22×C4×C5C22⋊C412yes—
44C22×C4×C5C22×C411yescharacteristic
45C22×Dic5—22no—
46C22×Dic5C22⋊C411yescharacteristic
47C22×Dic5C2×D412yes—
48C2×C4×D5—22no—
49C2×C4×D5—22no—
50C2×C4×D5—28no—
51C2×C4×D5C4212yes—
52C2×C4×D5C22⋊C414yes—
53C2×C4×D5C4⋊C412yes—
54C2×C4×D5C4⋊C414yes—
55C2×#170—28no—
56C2×#170—48no—
57C5×C4⋊C4—42no—
58C42×C5C4212yes—
59C42×C5C4⋊C412yes—
60#685—44no—
61C4⋊Dic5—42no—
62#683—48no—
63C4×Dic5—44no—
64C4×Dic5C22⋊C414yes—
65C4×Dic5C4⋊C414yes—
66C4×Dic5C22×C412yes—
67C4×Dic5C2×D411yescharacteristic
68C4×Dic5C2×Q811yescharacteristic
69C2×#117—52no—
70C22×C42—51no—
71#431—58no—
72#431—104no—
73C2×C4⋊C8—54no—
74C2×C22⋊C8—102no—
75C2×C4×C8—102no—
76#384—52no—
77#382—51no—
78#382AGL(1,𝔽5)11yescharacteristic
79#380—102no—
80C4×C4⋊C4—102no—
81C23×C5C2×C22⋊C411yescharacteristic
82C22×D5—21no—
83C22×D5—21no—
84C22×D5—24no—
85C22×D5#9512yes—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—24no—
89C2×C4×C5—42no—
90C2×C4×C5—44no—
91C2×C4×C5#9514yes—
92C2×C4×C5C2×C4211yescharacteristic
93C2×C4×C5C2×C4⋊C411yescharacteristic
94C2×Dic5—42no—
95C2×Dic5—44no—
96C2×Dic5#9512yes—
97C2×Dic5#10012yes—
98C2×Dic5#10112yes—
99C2×Dic5C2×C22⋊C412yes—
100C2×Dic5C2×C4⋊C412yes—
101C2×Dic5#13518yes—
102C4×D5—24no—
103C4×D5—24no—
104C4×D5—28no—
105C4×D5—28no—
106#170—48no—
107C23×C4—52no—
108C23×C4—51no—
109C22×C8—104no—
110#117—204no—
111C2×C4⋊C4—52no—
112C2×C4⋊C4—104no—
113C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
114C2×C22⋊C4—102no—
115C2×C42—52no—
116C2×C42—51no—
117C2×C42—54no—
118C2×C42—52no—
119C2×C42—52no—
120C2×C42—102no—
121C2×C42C2×AGL(1,𝔽5)11yescharacteristic
122C4⋊C8—58no—
123C4⋊C8—104no—
124C22⋊C8—108no—
125C4×C8—108no—
126#95—22no—
127#95—102no—
128C22×C5C2×#9511yescharacteristic
129C22×C5C2×#10011yescharacteristic
130C22×C5C2×#10111yescharacteristic
131C22×C5#41614yes—
132C2×D5—24no—
133C2×D5—22no—
134C2×D5—42no—
135C2×D5—42no—
136C4×C5—24no—
137C4×C5—21no—
138C4×C5—21no—
139C4×C5—82no—
140Dic5—82no—
141Dic5#34314yes—
142Dic5#41614yes—
143Dic5#42714yes—
144C24—51no—
145C22×C4—22no—
146C22×C4—51no—
147C22×C4—52no—
148C22×C4—52no—
149C22×C4—52no—
150C22×C4—54no—
151C22×C4—54no—
152C22×C4—102no—
153C22×C4—108no—
154C22×C4—102no—
155C22×C4—102no—
156C22×C4C22⋊AGL(1,𝔽5)12yes—
157C22×C4C22×AGL(1,𝔽5)11yescharacteristic
158C2×C8—108no—
159C2×C8—208no—
160C4⋊C4—42no—
161C4⋊C4—202no—
162C4⋊C4—208no—
163C22⋊C4—204no—
164C42—52no—
165C42—51no—
166C42—51no—
167C42—54no—
168C42—54no—
169C42—204no—
170C42C4×AGL(1,𝔽5)12yes—
171C42C4⋊AGL(1,𝔽5)12yes—
172C2×C5#154312yes—
173C2×C5#154412yes—
174C2×C5#173512yes—
175C2×C5C2×#34311yescharacteristic
176D5—42no—
177C23—52no—
178C23—104no—
179C23—101no—
180C23—101no—
181C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
182C2×C4—24no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—44no—
186C2×C4—42no—
187C2×C4—58no—
188C2×C4—52no—
189C2×C4—52no—
190C2×C4—52no—
191C2×C4—52no—
192C2×C4—52no—
193C2×C4—104no—
194C2×C4—104no—
195C2×C4—108no—
196C2×C4—108no—
197C2×C4—204no—
198C2×C4—202no—
199C2×C4(C2×AGL(1,𝔽5))⋊C414yes—
200C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
201C2×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
202C8—208no—
203C5#803811yescharacteristic, complemented, Sylow
204C22—102no—
205C22—104no—
206C22—202no—
207C22—202no—
208C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
209C22D5⋊#10011yescharacteristic
210C22D5⋊#10111yescharacteristic
211C22#6579414yes—
212C4—21no—
213C4—21no—
214C4—24no—
215C4—54no—
216C4—54no—
217C4—54no—
218C4—82no—
219C4—402no—
220C2—202no—
221C2#1060433411yescharacteristic
222C2#1060437812yes—
223C2#1060455612yes—
224C2#1060458412yes—
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#382C4
socleC23×C5C2×C22⋊C4
prime radicalC23×D5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenter#382AGL(1,𝔽5)
hypocenterC5#8038
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
3. #10598725C2semidirect
10. #65678C4semidirect
203. C5#8038semidirect

Maximal subgroups

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

subgroupindex
C2×#656782
#106058792
#105987252
#80385