Finite Group Databasebeta

← #13018082

Subgroups — #13018082

Order 1280.

1026 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 116 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×#4034C211yesmaximal, characteristic
2#10604905C212yesmaximal
3#10604882C212yesmaximal
4#10604881C212yesmaximal
5C23×Dic10C411yescharacteristic
6C23×Dic10C2211yescharacteristic
7C22×C5×C4⋊C4C411yescharacteristic
8C22×C4⋊Dic5C2211yescharacteristic
9C22×#683C411yescharacteristic
10C22×#683C2211yescharacteristic
11C22×C4×Dic5C411yescharacteristic
12C2×#4034—24no—
13C2×#4034—24no—
14C2×#4034—24no—
15#65798C2214yes—
16C2×#4005—24no—
17#64914—24no—
18#64913—24no—
19#8551—51nomaximal, Sylow
20C23×C4×C5C2×C411yescharacteristic
21C23×C4×C5C2×C411yescharacteristic
22C23×C4×C5C2×C411yescharacteristic
23C23×Dic5C2×C411yescharacteristic
24C23×Dic5C2×C411yescharacteristic
25C23×Dic5C2×C411yescharacteristic
26C23×Dic5C2311yescharacteristic
27C22×Dic10—22no—
28C22×Dic10—22no—
29C22×Dic10—22no—
30C22×Dic10—22no—
31C22×Dic10—22no—
32C22×Dic10—44no—
33C22×Dic10—44no—
34C22×Dic10D414yes—
35C2×#704—24no—
36C2×C5×C4⋊C4—22no—
37C2×C5×C4⋊C4—22no—
38C2×C5×C4⋊C4—22no—
39C2×C4⋊Dic5—22no—
40C2×C4⋊Dic5—22no—
41C2×C4⋊Dic5D414yes—
42C2×#683—22no—
43C2×#683—22no—
44C2×#683—22no—
45C2×#683—22no—
46C2×#683—22no—
47C2×#683D414yes—
48C2×C4×Dic5—22no—
49C2×C4×Dic5—22no—
50C2×C4×Dic5—22no—
51#4034—232no—
52C22⋊#170—24no—
53C22⋊#170—24no—
54#4005—48no—
55C22×C4⋊Q8—51no—
56#1784—52no—
57#1463—52no—
58#1463—52no—
59C24×C5C22×C411yescharacteristic
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—24no—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5C22⋊C412yes—
70C22×C4×C5C22⋊C412yes—
71C22×C4×C5C22⋊C412yes—
72C22×Dic5—21no—
73C22×Dic5—21no—
74C22×Dic5—21no—
75C22×Dic5—21no—
76C22×Dic5—24no—
77C22×Dic5—24no—
78C22×Dic5—21no—
79C22×Dic5—21no—
80C22×Dic5—42no—
81C22×Dic5—42no—
82C22×Dic5C22⋊C412yes—
83C22×Dic5C22⋊C412yes—
84C22×Dic5C22⋊C412yes—
85C22×Dic5C2×D412yes—
86C22×Dic5C2×D412yes—
87C22×Dic5C2×D412yes—
88C2×Dic10—28no—
89C2×Dic10—28no—
90C2×Dic10—28no—
91C2×Dic10—44no—
92C2×Dic10—48no—
93C2×Dic10—48no—
94C2×Dic10—48no—
95C2×Dic10—48no—
96C2×Dic10—44no—
97C2×Dic10—44no—
98#704—48no—
99C2×#170—44no—
100C5×C4⋊C4—28no—
101C4⋊Dic5—28no—
102#683—28no—
103#683—28no—
104C4×Dic5—28no—
105C23×Q8—51no—
106C23×Q8—51no—
107C2×C4⋊Q8—104no—
108C2×C4⋊Q8—104no—
109C2×C4⋊Q8—104no—
110C22×C4⋊C4—51no—
111C22×C4⋊C4—51no—
112C22×C4⋊C4—51no—
113C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic
114C22×C42—51no—
115C2×#101—104no—
116#407—54no—
117#324—104no—
118#324—104no—
119C23×C5—22no—
120C23×C5—22no—
121C23×C5—22no—
122C23×C5C2×C22⋊C411yescharacteristic
123C23×C5C2×C22⋊C411yescharacteristic
124C23×C5C2×C22⋊C411yescharacteristic
125C2×C4×C5—24no—
126C2×C4×C5—24no—
127C2×C4×C5—22no—
128C2×C4×C5—22no—
129C2×C4×C5—22no—
130C2×C4×C5—22no—
131C2×C4×C5—24no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5#10114yescomplemented
138C2×Dic5—24no—
139C2×Dic5—24no—
140C2×Dic5—24no—
141C2×Dic5—24no—
142C2×Dic5—22no—
143C2×Dic5—22no—
144C2×Dic5—24no—
145C2×Dic5—22no—
146C2×Dic5—22no—
147C2×Dic5—22no—
148C2×Dic5—42no—
149C2×Dic5—42no—
150C2×Dic5—42no—
151C2×Dic5—42no—
152C2×Dic5—42no—
153C2×Dic5#9914yes—
154C2×Dic5#10114yescomplemented
155C2×Dic5#12014yes—
156Dic10—416no—
157Dic10—416no—
158#170—84no—
159C22×Q8—54no—
160C22×Q8—102no—
161C22×Q8—102no—
162C22×Q8—102no—
163C22×Q8—102no—
164C22×Q8—102no—
165C22×Q8—204no—
166C22×Q8—204no—
167C23×C4—51no—
168C23×C4—51no—
169C23×C4—51no—
170C23×C4—51no—
171C23×C4C2×AGL(1,𝔽5)11yescharacteristic
172C23×C4C2×AGL(1,𝔽5)11yescharacteristic
173C23×C4C2×AGL(1,𝔽5)11yescharacteristic
174C2×M16—104no—
175C4⋊Q8—1032no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—54no—
180C2×C4⋊C4—54no—
181C2×C4⋊C4—102no—
182C2×C4⋊C4—102no—
183C2×C4⋊C4—102no—
184C2×C4⋊C4—102no—
185C2×C4⋊C4—102no—
186C2×C4⋊C4—102no—
187C2×C4⋊C4—102no—
188C2×C42—102no—
189C2×C42—102no—
190C2×C42—102no—
191#101—208no—
192C22⋊C8—104no—
193C22⋊C8—104no—
194C22×C5—28no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5#37911yescharacteristic
202C22×C5C2×#9912yes—
203C22×C5C2×#10112yes—
204C22×C5C2×#10112yes—
205C4×C5—24no—
206C4×C5—42no—
207C4×C5—42no—
208Dic5—24no—
209Dic5—24no—
210Dic5—24no—
211Dic5—42no—
212C24C22×AGL(1,𝔽5)11yescharacteristic
213C2×Q8—108no—
214C2×Q8—108no—
215C2×Q8—108no—
216C2×Q8—204no—
217C2×Q8—204no—
218C2×Q8—208no—
219C2×Q8—208no—
220C2×Q8—208no—
221C2×Q8—204no—
222C2×Q8—208no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—52no—
233C22×C4—52no—
234C22×C4—52no—
235C22×C4—52no—
236C22×C4—52no—
237C22×C4—52no—
238C22×C4—104no—
239C22×C4—104no—
240C22×C4—101no—
241C22×C4—101no—
242C22×C4—101no—
243C22×C4—101no—
244C22×C4—101no—
245C22×C4—101no—
246C22×C4—202no—
247C22×C4—202no—
248C22×C4C22⋊AGL(1,𝔽5)12yes—
249C22×C4C22⋊AGL(1,𝔽5)12yes—
250C22×C4C22⋊AGL(1,𝔽5)12yes—
251M16—208no—
252C2×C8—204no—
253C4⋊C4—28no—
254C4⋊C4—108no—
255C4⋊C4—108no—
256C4⋊C4—108no—
257C42—108no—
258C2×C5—22no—
259C2×C5—22no—
260C2×C5—22no—
261C2×C5#178111yescharacteristic
262C2×C5#178411yescharacteristic
263C2×C5#178411yescharacteristic
264C23—22no—
265C23—22no—
266C23—22no—
267C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
268C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
269C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
270Q8—2016no—
271Q8—2016no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—54no—
285C2×C4—54no—
286C2×C4—54no—
287C2×C4—104no—
288C2×C4—104no—
289C2×C4—104no—
290C2×C4—104no—
291C2×C4—102no—
292C2×C4—102no—
293C2×C4—104no—
294C2×C4—102no—
295C2×C4—102no—
296C2×C4—102no—
297C2×C4—202no—
298C2×C4—202no—
299C2×C4—202no—
300C2×C4—202no—
301C2×C4—202no—
302C2×C4#400514yescomplemented
303C8—404no—
304C5#855111yescharacteristic, complemented, Sylow
305C22—28no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22C2×#400512yes—
313C22D5⋊#9912yes—
314C22D5⋊#10112yes—
315C22#6580011yescharacteristic
316C4—24no—
317C4—42no—
318C4—42no—
319C4—104no—
320C4—104no—
321C4—104no—
322C4—202no—
323C2—22no—
324C2—22no—
325C2—22no—
326C2#1060490511yescharacteristic
327C2#1060494111yescharacteristic
328C2#1060498611yescharacteristic
subgroupG/H
centerC22#65800
commutator, prime radicalC24×C5C22×C4
FrattiniC24C22×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC22×C5#379
cosocleC23×Dic5C23
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5#8551
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
137. C2×C4×C5#101semidirect
154. C2×Dic5#101semidirect
302. C2×C4#4005semidirect
304. C5#8551semidirect

Maximal subgroups

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

subgroupindex
C22×#40342
#106049052
#106048822
#106048812
#85515