Finite Group Databasebeta

← #13021468

Subgroups — #13021468

Order 1280.

664 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 126 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×#65519C211yesmaximal, characteristic
2#10604900C211yesmaximal, characteristic
3#10604897C211yesmaximal, characteristic
4#10598821C214yesmaximal
5C23×C5×Q8C411yescharacteristic
6C22×C4⋊Dic5C411yescharacteristic
7C22×C4×Dic5C2211yescharacteristic
8C2×C22⋊#170C2212yes—
9C2×C22⋊#170C2212yes—
10C2×#4004—21no—
11C2×#4002—21no—
12#65519—21no—
13#65519—21no—
14#65519—21no—
15#65519—21no—
16C2×#3963C412yes—
17C2×#3963C2212yes—
18#64918—21no—
19#64918—21no—
20#9882—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×C5×Q8—21no—
23C22×C5×Q8—24no—
24C22×C5×Q8—42no—
25C22×C5×Q8—42no—
26C23×C4×C5C2×C412yes—
27C23×C4×C5C2×C411yescharacteristic
28C23×Dic5C2×C411yescharacteristic
29C23×Dic5C2×C412yes—
30C23×Dic5C2311yescharacteristic
31C22×#170—22no—
32C22×#170—22no—
33C2×C4⋊Dic5—21no—
34C2×C4⋊Dic5—21no—
35C2×C4⋊Dic5—42no—
36C2×C4×Dic5—42no—
37C2×C4×Dic5D411yescharacteristic
38C2×C4×Dic5D411yescharacteristic
39C2×C4×Dic5D411yescharacteristic
40C2×C4×Dic5D411yescharacteristic
41C22⋊#170—22no—
42C22⋊#170—24no—
43C22⋊#170Q814yescomplemented
44#4004—42no—
45#4002—42no—
46#3963—22no—
47#3963—22no—
48#3963—22no—
49#3963—22no—
50C2×#391—51no—
51#1797—51no—
52#1749—51no—
53#1280—54no—
54C24×C5C22×C411yescharacteristic
55C2×C5×Q8—24no—
56C2×C5×Q8—24no—
57C2×C5×Q8—416no—
58C2×C5×Q8—42no—
59C2×C5×Q8—42no—
60C2×C5×Q8—42no—
61C2×C5×Q8—42no—
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5—22no—
65C22×C4×C5—22no—
66C22×C4×C5—22no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5M1614yes—
72C22×Dic5—22no—
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5—24no—
76C22×Dic5—41no—
77C22×Dic5—41no—
78C22×Dic5—41no—
79C22×Dic5—42no—
80C22×Dic5—42no—
81C22×Dic5C22⋊C411yescharacteristic
82C22×Dic5C22⋊C411yescharacteristic
83C22×Dic5C2×D411yescharacteristic
84C22×Dic5C2×D411yescharacteristic
85C22×Dic5C2×Q812yes—
86C22×Dic5#4112yes—
87C2×#170—24no—
88C2×#170—42no—
89C2×#170—42no—
90C2×#170—42no—
91C2×#170—42no—
92C2×#170—42no—
93C4⋊Dic5—42no—
94C4⋊Dic5—42no—
95C4×Dic5—22no—
96C4×Dic5—22no—
97C4×Dic5—41no—
98C4×Dic5—41no—
99C23×Q8AGL(1,𝔽5)11yescharacteristic
100C22×C4⋊C4—51no—
101C22×C42—51no—
102C2×C4⋊C8—101no—
103C2×C22⋊C8—52no—
104C2×C22⋊C8—52no—
105C2×#97—101no—
106#391—101no—
107#391—101no—
108#391—101no—
109#391—101no—
110C2×#95—52no—
111C2×#95—52no—
112#336—101no—
113#336—101no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5C2×C22⋊C411yescharacteristic
119C23×C5C4×Q812yes—
120C23×C5C2×M1612yes—
121C23×C5#13112yes—
122C5×Q8—48no—
123C5×Q8—48no—
124C2×C4×C5—28no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—24no—
130C2×C4×C5—44no—
131C2×C4×C5—42no—
132C2×C4×C5—42no—
133C2×C4×C5—44no—
134C2×C4×C5#9912yes—
135C2×C4×C5#10012yes—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—21no—
144C2×Dic5—21no—
145C2×Dic5—42no—
146C2×Dic5—42no—
147C2×Dic5—44no—
148C2×Dic5—42no—
149C2×Dic5—42no—
150C2×Dic5—42no—
151C2×Dic5—84no—
152C2×Dic5#10112yes—
153C2×Dic5C22⋊Q812yes—
154C2×Dic5#12411yescharacteristic
155C2×Dic5C4⋊Q811yescharacteristic
156#170—44no—
157#170—82no—
158C22×Q8—24no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—42no—
162C22×Q8—42no—
163C23×C4—51no—
164C23×C4—51no—
165C23×C4—52no—
166C23×C4C2×AGL(1,𝔽5)12yes—
167C23×C4C2×AGL(1,𝔽5)11yescharacteristic
168C22×C8—102no—
169C22×C8—102no—
170C2×C4⋊C4—101no—
171C2×C4⋊C4—101no—
172C2×C4⋊C4—202no—
173C2×C42—51no—
174C2×C42—51no—
175C2×C42—51no—
176C2×C42—51no—
177C2×C42—202no—
178C4⋊C8—202no—
179C22⋊C8—54no—
180C22⋊C8—102no—
181C22⋊C8—104no—
182#97—202no—
183#95—102no—
184#95—102no—
185#95—102no—
186#95—102no—
187C22×C5—22no—
188C22×C5—22no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—22no—
192C22×C5—22no—
193C22×C5#39111yescharacteristic
194C22×C5#40711yescharacteristic
195C22×C5#40811yescharacteristic
196C22×C5C2×#9911yescharacteristic
197C22×C5#41011yescharacteristic
198C22×C5C2×#10011yescharacteristic
199C22×C5C2×#10111yescharacteristic
200C22×C5#44614yes—
201C4×C5—22no—
202C4×C5—22no—
203C4×C5—44no—
204Dic5—22no—
205Dic5—22no—
206Dic5—21no—
207Dic5—21no—
208Dic5—41no—
209Dic5—82no—
210C24C22×AGL(1,𝔽5)11yescharacteristic
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—416no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—51no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—52no—
229C22×C4—52no—
230C22×C4—51no—
231C22×C4—102no—
232C22×C4—102no—
233C22×C4—102no—
234C22×C4—104no—
235C22×C4—201no—
236C22×C4—201no—
237C22×C4—201no—
238C22×C4—202no—
239C22×C4—202no—
240C22×C4#70414yes—
241C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
242C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
243C2×C8—104no—
244C2×C8—202no—
245C2×C8—202no—
246C2×C8—202no—
247C2×C8—202no—
248C2×C8—202no—
249C4⋊C4—202no—
250C4⋊C4—202no—
251C42—102no—
252C42—102no—
253C42—201no—
254C42—201no—
255C2×C5—24no—
256C2×C5#146211yescharacteristic
257C2×C5#146711yescharacteristic
258C2×C5#146911yescharacteristic
259C2×C5#147211yescharacteristic
260C2×C5#198511yescharacteristic
261C2×C5#199611yescharacteristic
262C2×C5#199711yescharacteristic
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23#413312yes—
268C23Q8×AGL(1,𝔽5)12yes—
269C23C2×#70412yes—
270C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
271Q8—48no—
272Q8—48no—
273C2×C4—28no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—52no—
284C2×C4—52no—
285C2×C4—51no—
286C2×C4—51no—
287C2×C4—101no—
288C2×C4—101no—
289C2×C4—101no—
290C2×C4—101no—
291C2×C4—101no—
292C2×C4—101no—
293C2×C4—101no—
294C2×C4—101no—
295C2×C4—101no—
296C2×C4—202no—
297C2×C4—202no—
298C2×C4—204no—
299C2×C4—202no—
300C2×C4—202no—
301C2×C4—202no—
302C2×C4—404no—
303C2×C4#401112yes—
304C2×C4#401312yes—
305C8—204no—
306C8—402no—
307C5#988211yescharacteristic, complemented, Sylow
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22#6578511yescharacteristic
315C22#6578711yescharacteristic
316C22#6578814yes—
317C22D5⋊#10111yescharacteristic
318C22(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
319C22C2×#401111yescharacteristic
320C22#6579811yescharacteristic
321C22C2×#401311yescharacteristic
322C4—22no—
323C4—22no—
324C4—44no—
325C4—102no—
326C4—102no—
327C4—101no—
328C4—101no—
329C4—201no—
330C4—402no—
331C2—24no—
332C2#1060497711yescharacteristic
333C2#1060497811yescharacteristic
334C2#1060497911yescharacteristic
335C2#1060498211yescharacteristic
336C2#1060498511yescharacteristic
337C2#1060498711yescharacteristic
338C2#1060502211yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutator, prime radicalC24×C5C22×C4
FrattiniC24C22×AGL(1,𝔽5)
FittingC23×C5×Q8C4
socleC23×C5C2×C22⋊C4
cosocleC23×Dic5C23
hypercenterC23×Q8AGL(1,𝔽5)
hypocenterC5#9882
LCSG ▷ C24×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
43. C22⋊#170Q8semidirect
307. C5#9882semidirect

Maximal subgroups

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

subgroupindex
C2×#655192
#106049002
#106048972
#105988212
#98825