Finite Group Databasebeta

← #13017481

Subgroups — #13017481

Order 1280.

930 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 164 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×#66146C211yesmaximal, characteristic
2#10604967C214yesmaximal, complemented
3#10604877C212yesmaximal
4C23×C5×Q8C411yescharacteristic
5#66146—28no—
6C2×Q8×Dic5—22no—
7C2×Q8×Dic5C412yes—
8C2×Q8×Dic5C2212yes—
9C2×#4072—22no—
10C2×#4072C411yescharacteristic
11C2×#4072C2211yescharacteristic
12#65748C2214yes—
13#64933—22no—
14Q8⋊#170—28no—
15#64913—22no—
16#8323—51nomaximal, Sylow
17C22×C5×Q8—24no—
18C22×C5×Q8—24no—
19C22×C5×Q8—21no—
20C22×C5×Q8—22no—
21C22×C5×Q8C814yescomplemented
22C22×C5×Q8C2×C412yes—
23C23×C4×C5—21no—
24C23×C4×C5C2×C411yescharacteristic
25C2×C22⋊Dic5—21no—
26C2×C22⋊Dic5D412yes—
27Q8×Dic5—216no—
28Q8×Dic5—24no—
29Q8×Dic5—24no—
30Q8×Dic5—24no—
31Q8×Dic5D418yes—
32#4072—216no—
33#4072—24no—
34#4072—24no—
35C2×C4⋊Dic5—22no—
36C2×C4⋊Dic5—22no—
37C2×C4⋊Dic5C2×C412yes—
38C2×C4⋊Dic5D412yes—
39C2×C4×Dic5—21no—
40C2×C4×Dic5—22no—
41C2×C4×Dic5—22no—
42C2×C4×Dic5C2×C411yescharacteristic
43C2×C4×Dic5C2311yescharacteristic
44C22⋊#170—24no—
45C4⋊#170—24no—
46C4×#170—24no—
47C2×#519—51no—
48#1479—54no—
49#1458—52no—
50C24×C5C22⋊C411yescharacteristic
51C2×C5×Q8—216no—
52C2×C5×Q8—24no—
53C2×C5×Q8—24no—
54C2×C5×Q8—216no—
55C2×C5×Q8—24no—
56C2×C5×Q8—22no—
57C2×C5×Q8—22no—
58C2×C5×Q8—22no—
59C2×C5×Q8C22⋊C414yes—
60C2×C5×Q8C2×C814yes—
61C2×C5×Q8M1614yes—
62C22×C4×C5—24no—
63C22×C4×C5—24no—
64C22×C4×C5—24no—
65C22×C4×C5—22no—
66C22×C4×C5—22no—
67C22×C4×C5—22no—
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C2×C812yes—
70C22×C4×C5M1612yes—
71C22×C4×C5C22×C411yescharacteristic
72C22×Dic5—22no—
73C22×Dic5C22⋊C412yes—
74C22×Dic5C2×D412yes—
75C2×#170—44no—
76C22⋊Dic5—24no—
77C22⋊Dic5—24no—
78C4⋊Dic5—28no—
79C4⋊Dic5—22no—
80C4⋊Dic5—28no—
81C4⋊Dic5—24no—
82C4⋊Dic5C22⋊C414yes—
83C4×Dic5—28no—
84C4×Dic5—24no—
85C4×Dic5—22no—
86C4×Dic5—28no—
87C4×Dic5C2×D414yes—
88C23×Q8AGL(1,𝔽5)11yescharacteristic
89#519—108no—
90C2×C4×Q8—52no—
91C2×C4×Q8—52no—
92C2×C4×Q8—102no—
93C2×#117—51no—
94C2×#117—51no—
95C2×#117—102no—
96#431—54no—
97Q8⋊C8—108no—
98#324—102no—
99#323—102no—
100C23×C5—21no—
101C23×C5—21no—
102C23×C5—42no—
103C23×C5C22⋊C812yes—
104C23×C5C2×C22⋊C411yescharacteristic
105C5×Q8—216no—
106C5×Q8—24no—
107C5×Q8—24no—
108C5×Q8—24no—
109C5×Q8C22⋊C818yes—
110C2×C4×C5—216no—
111C2×C4×C5—24no—
112C2×C4×C5—21no—
113C2×C4×C5—24no—
114C2×C4×C5—24no—
115C2×C4×C5—21no—
116C2×C4×C5—24no—
117C2×C4×C5—24no—
118C2×C4×C5C22⋊C812yes—
119C2×C4×C5C2×C22⋊C412yes—
120C2×C4×C5C22×C811yescharacteristic
121C2×C4×C5C2×M1612yes—
122C2×C4×C5C2×M1611yescharacteristic
123C2×Dic5—22no—
124C2×Dic5—24no—
125C2×Dic5—42no—
126C2×Dic5—44no—
127C2×Dic5—42no—
128C2×Dic5#12014yes—
129#170—84no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8#17014yescomplemented
135C22×Q8C2×AGL(1,𝔽5)12yes—
136C23×C4—21no—
137C23×C4C2×AGL(1,𝔽5)11yescharacteristic
138C4×Q8—58no—
139C4×Q8—104no—
140C4×Q8—104no—
141C4×Q8—1016no—
142C4×Q8—104no—
143#117—104no—
144#117—104no—
145#117—1016no—
146C2×C4⋊C4—52no—
147C2×C4⋊C4—52no—
148C2×C4⋊C4—102no—
149C2×C4⋊C4—102no—
150C2×C22⋊C4—52no—
151C2×C22⋊C4—101no—
152C2×C42—51no—
153C2×C42—51no—
154C2×C42—102no—
155C2×C42—102no—
156C2×C42—101no—
157C4⋊C8—104no—
158C22⋊C8—104no—
159C4×C8—104no—
160C22×C5—22no—
161C22×C5—22no—
162C22×C5—42no—
163C22×C5—42no—
164C22×C5—42no—
165C22×C5#37911yescharacteristic
166C22×C5C2×C22⋊C811yescharacteristic
167C22×C5#40711yescharacteristic
168C4×C5—28no—
169C4×C5—22no—
170C4×C5C2×C22⋊C812yes—
171C4×C5#40712yes—
172Dic5—24no—
173Dic5—42no—
174Dic5—44no—
175C24C22⋊AGL(1,𝔽5)11yescharacteristic
176C2×Q8—24no—
177C2×Q8—216no—
178C2×Q8—24no—
179C2×Q8—216no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—24no—
184C2×Q8C2×#17014yes—
185C2×Q8#70414yes—
186C2×Q8C22⋊AGL(1,𝔽5)14yes—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—52no—
194C22×C4—52no—
195C22×C4—102no—
196C22×C4C2×#17012yes—
197C22×C4#70412yes—
198C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
199C22×C4C22×AGL(1,𝔽5)11yescharacteristic
200C2×C8—204no—
201C4⋊C4—54no—
202C4⋊C4—104no—
203C4⋊C4—108no—
204C4⋊C4—108no—
205C4⋊C4—102no—
206C22⋊C4—104no—
207C22⋊C4—104no—
208C42—54no—
209C42—108no—
210C42—108no—
211C42—104no—
212C42—102no—
213C2×C5—21no—
214C2×C5—21no—
215C2×C5—42no—
216C2×C5#177911yescharacteristic
217C2×C5#179612yes—
218C23—21no—
219C23—21no—
220C23—42no—
221C23C22⋊#17012yes—
222C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
223Q8—216no—
224Q8—24no—
225Q8—24no—
226Q8—24no—
227Q8C22⋊#17018yes—
228C2×C4—216no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—54no—
237C2×C4—104no—
238C2×C4—102no—
239C2×C4—202no—
240C2×C4—204no—
241C2×C4—202no—
242C2×C4C22⋊#17012yes—
243C2×C4C22×#17011yescharacteristic
244C2×C4C2×#70412yes—
245C2×C4C2×#70411yescharacteristic
246C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
247C8—404no—
248C5#832311yescharacteristic, complemented, Sylow
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22C2×C22⋊#17011yescharacteristic
255C22#6579811yescharacteristic
256C22#6580011yescharacteristic
257C4—28no—
258C4—22no—
259C4—104no—
260C4—204no—
261C4—202no—
262C4C2×C22⋊#17012yes—
263C4#6579812yes—
264C2—21no—
265C2—21no—
266C2—42no—
267C2#1060497412yes—
268C2#1060502511yescharacteristic
subgroupG/H
centerC22#65800
commutatorC2×C4×C5C22×C8
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC23×C5×Q8C4
socleC22×C5#379
prime radicalC24×C5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenterC23×Q8AGL(1,𝔽5)
hypocenterC5#8323
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #10604967C2semidirect
21. C22×C5×Q8C8semidirect
134. C22×Q8#170semidirect
248. C5#8323semidirect

Maximal subgroups

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

subgroupindex
C2×#661462
#106049672
#106048772
#83235