Finite Group Databasebeta

← #13013300

Subgroups — #13013300

Order 1280.

476 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 54 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#10602919C211yesmaximal, characteristic
2#10598817C212yesmaximal, complemented
3C24×C4×C5C411yescharacteristic
4C2×C22⋊#168—22no—
5C2×C22⋊#168C411yescharacteristic
6C2×C22⋊#168C2211yescharacteristic
7C22⋊#674—22no—
8#7054—51nomaximal, Sylow
9C25×C5C811yescharacteristic
10C23×C4×C5—21no—
11C23×C4×C5—21no—
12C23×C4×C5—44no—
13C23×C4×C5—41no—
14C23×C4×C5—41no—
15C23×C4×C5C811yescharacteristic
16C23×C4×C5C2×C411yescharacteristic
17C22×#168—21no—
18C22×#168D412yes—
19C2×#674—42no—
20C22⋊#168—41no—
21C22⋊#168—44no—
22C22⋊#168—41no—
23#1779—51no—
24#1314—52no—
25C24×C5—21no—
26C24×C5—21no—
27C24×C5—42no—
28C24×C5—41no—
29C24×C5—42no—
30C24×C5—41no—
31C24×C5C1612yescomplemented
32C24×C5C2×C811yescharacteristic
33C22×C4×C5—24no—
34C22×C4×C5—21no—
35C22×C4×C5—21no—
36C22×C4×C5—21no—
37C22×C4×C5—42no—
38C22×C4×C5—42no—
39C22×C4×C5—42no—
40C22×C4×C5—42no—
41C22×C4×C5—42no—
42C22×C4×C5—42no—
43C22×C4×C5—44no—
44C22×C4×C5—44no—
45C22×C4×C5—44no—
46C22×C4×C5—41no—
47C22×C4×C5—41no—
48C22×C4×C5—41no—
49C22×C4×C5—44no—
50C22×C4×C5C22⋊C411yescharacteristic
51C22×C4×C5M1611yescharacteristic
52C2×#168—22no—
53C2×#168—42no—
54C2×#168—42no—
55C2×#168—41no—
56#674—82no—
57C24×C4AGL(1,𝔽5)11yescharacteristic
58C2×C22⋊C8—51no—
59C2×C22⋊C8—51no—
60C2×C22⋊C8—102no—
61C22⋊C16—102no—
62C23×C5—22no—
63C23×C5—22no—
64C23×C5—21no—
65C23×C5—22no—
66C23×C5—21no—
67C23×C5—41no—
68C23×C5—44no—
69C23×C5—42no—
70C23×C5—41no—
71C23×C5—41no—
72C23×C5—41no—
73C23×C5—44no—
74C23×C5—42no—
75C23×C5—42no—
76C23×C5—42no—
77C23×C5—42no—
78C23×C5—48no—
79C23×C5—42no—
80C23×C5—41no—
81C23×C5—41no—
82C23×C5C22⋊C811yescharacteristic
83C23×C5C2×C1611yescharacteristic
84C23×C5M3211yescharacteristic
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5—21no—
88C2×C4×C5—44no—
89C2×C4×C5—42no—
90C2×C4×C5—41no—
91C2×C4×C5—48no—
92C2×C4×C5—42no—
93C2×C4×C5—41no—
94C2×C4×C5—44no—
95C2×C4×C5—42no—
96C2×C4×C5—41no—
97C2×C4×C5—42no—
98C2×C4×C5#9911yescharacteristic
99C2×C4×C5#10011yescharacteristic
100#168—42no—
101#168—81no—
102C25#17011yescharacteristic
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—44no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4#17011yescharacteristic
109C23×C4C2×AGL(1,𝔽5)11yescharacteristic
110C22×C8—52no—
111C22×C8—101no—
112C2×C16—202no—
113C22⋊C8—201no—
114C22⋊C8—204no—
115C22⋊C8—201no—
116C22×C5—21no—
117C22×C5—21no—
118C22×C5—24no—
119C22×C5—21no—
120C22×C5—21no—
121C22×C5—41no—
122C22×C5—41no—
123C22×C5—42no—
124C22×C5—42no—
125C22×C5—42no—
126C22×C5—42no—
127C22×C5—44no—
128C22×C5—44no—
129C22×C5—44no—
130C22×C5—44no—
131C22×C5—42no—
132C22×C5—42no—
133C22×C5—41no—
134C22×C5—42no—
135C22×C5—41no—
136C22×C5#32311yescharacteristic
137C22×C5C22⋊C1611yescharacteristic
138C22×C5#34911yescharacteristic
139C4×C5—21no—
140C4×C5—42no—
141C4×C5—41no—
142C4×C5AΣL(1,𝔽16)12yes—
143C24—21no—
144C24—21no—
145C24—42no—
146C24—41no—
147C24—42no—
148C24—41no—
149C24#67412yescomplemented
150C24C2×#17011yescharacteristic
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—44no—
168C22×C4#70411yescharacteristic
169C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
170C2×C8—102no—
171C2×C8—202no—
172C2×C8—202no—
173C2×C8—201no—
174C16—402no—
175C2×C5—21no—
176C2×C5—21no—
177C2×C5—44no—
178C2×C5—41no—
179C2×C5—41no—
180C2×C5#131411yescharacteristic
181C2×C5#131611yescharacteristic
182C2×C5#132011yescharacteristic
183C23—22no—
184C23—22no—
185C23—22no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—44no—
190C23—42no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—44no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—48no—
200C23—42no—
201C23—41no—
202C23—41no—
203C23C2×#67411yescharacteristic
204C23#399811yescharacteristic
205C23C22⋊#17011yescharacteristic
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—48no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4#401111yescharacteristic
220C2×C4#401311yescharacteristic
221C8—202no—
222C8—401no—
223C5#705411yescharacteristic, complemented, Sylow
224C22—21no—
225C22—21no—
226C22—24no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—44no—
236C22—44no—
237C22—44no—
238C22—44no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22—42no—
243C22—41no—
244C22C22⋊#67411yescharacteristic
245C22#6491211yescharacteristic
246C22#6493311yescharacteristic
247C4—21no—
248C4—42no—
249C4—41no—
250C4#6493412yes—
251C2—21no—
252C2—21no—
253C2—44no—
254C2—41no—
255C2—41no—
256C2#1059881711yescharacteristic
257C2#1059881811yescharacteristic
258C2#1059883711yescharacteristic
subgroupG/H
centerC2×C4#4011
commutatorC23×C5C2×C16
FrattiniC23×C4C2×AGL(1,𝔽5)
FittingC24×C4×C5C4
socleC22×C5#323
prime radicalC25×C5C8
cosocleC2×C22⋊#168C22
hypercenterC24×C4AGL(1,𝔽5)
hypocenterC5#7054
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
2. #10598817C2semidirect
31. C24×C5C16semidirect
149. C24#674semidirect
223. C5#7054semidirect

Maximal subgroups

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

subgroupindex
#106029192
#105988172
#70545