Finite Group Databasebeta

← #13017420

Subgroups — #13017420

Order 1280.

1050 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C22×Q8×Dic5C211yesmaximal, characteristic
2#10604901C212yesmaximal
3C2×#64913C214yesmaximal
4C23×C5×Q8C411yescharacteristic
5C2×Q8×Dic5—28no—
6C2×Q8×Dic5—24no—
7C22×C4⋊Dic5C411yescharacteristic
8C22×C4⋊Dic5C2212yes—
9C22×C4×Dic5C412yes—
10C22×C4×Dic5C2211yescharacteristic
11C2×C22⋊#170C2214yes—
12#64918—24no—
13#64913—28no—
14#8303—51nomaximal, Sylow
15C22×C5×Q8—24no—
16C22×C5×Q8—22no—
17C22×C5×Q8—22no—
18C22×C5×Q8—24no—
19C22×C5×Q8C814yescomplemented
20C23×C4×C5C2×C412yes—
21C23×C4×C5C2×C411yescharacteristic
22C23×Dic5C2×C411yescharacteristic
23C23×Dic5C2×C412yes—
24C23×Dic5C2311yescharacteristic
25C22×#170—24no—
26Q8×Dic5—232no—
27C2×C4⋊Dic5—22no—
28C2×C4⋊Dic5—24no—
29C2×C4⋊Dic5—24no—
30C2×C4⋊Dic5—24no—
31C2×C4⋊Dic5D418yes—
32C2×C4×Dic5—24no—
33C2×C4×Dic5—24no—
34C2×C4×Dic5—24no—
35C2×C4×Dic5—24no—
36C2×C4×Dic5D414yes—
37C22⋊#170—28no—
38C22×C4×Q8—51no—
39#1817—52no—
40C2×#324—54no—
41C24×C5C22×C411yescharacteristic
42C2×C5×Q8—216no—
43C2×C5×Q8—28no—
44C2×C5×Q8—28no—
45C2×C5×Q8—216no—
46C2×C5×Q8—28no—
47C22×C4×C5—22no—
48C22×C4×C5—22no—
49C22×C4×C5—22no—
50C22×C4×C5—22no—
51C22×C4×C5—22no—
52C22×C4×C5—22no—
53C22×C4×C5C22⋊C414yes—
54C22×C4×C5C22⋊C412yes—
55C22×C4×C5C2×C814yes—
56C22×C4×C5C2×C812yes—
57C22×C4×C5M1614yes—
58C22×C4×C5M1612yes—
59C22×Dic5—22no—
60C22×Dic5—24no—
61C22×Dic5—22no—
62C22×Dic5—22no—
63C22×Dic5—42no—
64C22×Dic5—44no—
65C22×Dic5C22⋊C412yes—
66C22×Dic5C22⋊C414yes—
67C22×Dic5C2×D412yes—
68C22×Dic5C2×D414yes—
69C2×#170—48no—
70C2×#170—44no—
71C4⋊Dic5—28no—
72C4⋊Dic5—216no—
73C4×Dic5—216no—
74C4×Dic5—28no—
75C23×Q8AGL(1,𝔽5)11yescharacteristic
76C2×C4×Q8—104no—
77C2×C4×Q8—108no—
78C22×C4⋊C4—52no—
79C22×C4⋊C4—51no—
80C22×C42—51no—
81C22×C42—52no—
82C2×C22⋊C8—54no—
83#336—104no—
84#324—108no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5C2×C22⋊C412yes—
88C23×C5C2×C22⋊C411yescharacteristic
89C23×C5C22×C811yescharacteristic
90C23×C5C2×M1611yescharacteristic
91C23×C5C2×M1612yes—
92C5×Q8—232no—
93C2×C4×C5—24no—
94C2×C4×C5—22no—
95C2×C4×C5—24no—
96C2×C4×C5—24no—
97C2×C4×C5—24no—
98C2×C4×C5—24no—
99C2×C4×C5—22no—
100C2×C4×C5—24no—
101C2×C4×C5—24no—
102C2×C4×C5—24no—
103C2×C4×C5C22⋊C818yes—
104C2×C4×C5C22⋊C814yes—
105C2×Dic5—28no—
106C2×Dic5—22no—
107C2×Dic5—24no—
108C2×Dic5—22no—
109C2×Dic5—24no—
110C2×Dic5—44no—
111C2×Dic5—44no—
112C2×Dic5—44no—
113C2×Dic5—44no—
114C2×Dic5#10114yes—
115C2×Dic5#12014yes—
116#170—84no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—22no—
121C22×Q8#17014yescomplemented
122C23×C4—51no—
123C23×C4—51no—
124C23×C4—52no—
125C23×C4C2×AGL(1,𝔽5)12yes—
126C23×C4C2×AGL(1,𝔽5)11yescharacteristic
127C22×C8—104no—
128C4×Q8—1032no—
129C2×C4⋊C4—58no—
130C2×C4⋊C4—104no—
131C2×C4⋊C4—104no—
132C2×C4⋊C4—104no—
133C2×C4⋊C4—102no—
134C2×C42—54no—
135C2×C42—104no—
136C2×C42—104no—
137C2×C42—104no—
138C2×C42—104no—
139C22⋊C8—108no—
140C22×C5—22no—
141C22×C5—22no—
142C22×C5—24no—
143C22×C5—22no—
144C22×C5—22no—
145C22×C5#37911yescharacteristic
146C22×C5C2×C22⋊C812yes—
147C22×C5C2×C22⋊C811yescharacteristic
148C22×C5#40712yes—
149C22×C5#40711yescharacteristic
150C22×C5#41012yes—
151C22×C5C2×#10112yes—
152C4×C5—28no—
153C4×C5—24no—
154Dic5—24no—
155Dic5—24no—
156Dic5—44no—
157C24C22×AGL(1,𝔽5)11yescharacteristic
158C2×Q8—28no—
159C2×Q8—216no—
160C2×Q8—28no—
161C2×Q8—216no—
162C2×Q8—28no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—52no—
170C22×C4—52no—
171C22×C4—54no—
172C22×C4—54no—
173C22×C4—104no—
174C22×C4—102no—
175C22×C4—102no—
176C22×C4—102no—
177C22×C4—202no—
178C22×C4—204no—
179C22×C4C2×#17014yes—
180C22×C4C2×#17012yes—
181C22×C4#70414yes—
182C22×C4#70412yes—
183C22×C4C22⋊AGL(1,𝔽5)14yes—
184C22×C4C22⋊AGL(1,𝔽5)12yes—
185C2×C8—204no—
186C2×C8—208no—
187C4⋊C4—1016no—
188C4⋊C4—108no—
189C42—108no—
190C42—1016no—
191C2×C5—22no—
192C2×C5—22no—
193C2×C5#146112yes—
194C2×C5#146712yes—
195C2×C5#177911yescharacteristic
196C2×C5#178211yescharacteristic
197C2×C5#178411yescharacteristic
198C23—22no—
199C23—22no—
200C23C22×#17011yescharacteristic
201C23C2×#70411yescharacteristic
202C23C2×#70412yes—
203C23C2×C22⋊AGL(1,𝔽5)12yes—
204C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
205Q8—232no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—54no—
217C2×C4—54no—
218C2×C4—108no—
219C2×C4—102no—
220C2×C4—104no—
221C2×C4—102no—
222C2×C4—104no—
223C2×C4—204no—
224C2×C4—204no—
225C2×C4—204no—
226C2×C4—204no—
227C2×C4C22⋊#17018yes—
228C2×C4C22⋊#17014yes—
229C8—404no—
230C5#830311yescharacteristic, complemented, Sylow
231C22—22no—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—22no—
236C22#6578512yes—
237C22D5⋊#10112yes—
238C22C2×C22⋊#17012yes—
239C22C2×C22⋊#17011yescharacteristic
240C22#6579812yes—
241C22#6579811yescharacteristic
242C22#6580011yescharacteristic
243C4—28no—
244C4—24no—
245C4—104no—
246C4—104no—
247C4—204no—
248C2—22no—
249C2—22no—
250C2#1060497612yes—
251C2#1060497712yes—
252C2#1060498111yescharacteristic
253C2#1060498611yescharacteristic
254C2#1060502511yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C8
FrattiniC24C22×AGL(1,𝔽5)
FittingC23×C5×Q8C4
socleC23×C5C2×C22⋊C4
prime radicalC24×C5C22×C4
cosocleC23×Dic5C23
hypercenterC23×Q8AGL(1,𝔽5)
hypocenterC5#8303
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
19. C22×C5×Q8C8semidirect
121. C22×Q8#170semidirect
230. C5#8303semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic52
#106049012
C2×#649132
#83035