Finite Group Databasebeta

← #13020949

Subgroups — #13020949

Order 1280.

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

normal subgroups only

This group has 640 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×#65274C211yesmaximal, characteristic
2#10604900C211yesmaximal, characteristic
3#10604897C211yesmaximal, characteristic
4C2×#64913C214yesmaximal
5C22×C5×C4⋊C4C411yescharacteristic
6C22×C4⋊Dic5C411yescharacteristic
7C22×C4⋊Dic5C2212yes—
8C22×C4×Dic5C2211yescharacteristic
9C2×C22⋊#170C2212yes—
10C2×C22⋊#170C2212yes—
11C2×#4004—21no—
12C2×#4002—21no—
13C2×#3963C412yes—
14#65274—21no—
15#65274—21no—
16#65274—21no—
17#65274—21no—
18#64918—21no—
19#64918—21no—
20#64913—28no—
21#9654—51nomaximal, Sylow
22C23×C4×C5C2×C412yes—
23C23×C4×C5C2×C411yescharacteristic
24C23×Dic5C2×C411yescharacteristic
25C23×Dic5C2×C412yes—
26C23×Dic5C2311yescharacteristic
27C22×#170—22no—
28C22×#170—22no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—24no—
32C2×C4⋊Dic5—21no—
33C2×C4⋊Dic5—21no—
34C2×C4⋊Dic5—24no—
35C2×C4⋊Dic5—22no—
36C2×C4⋊Dic5—22no—
37C2×C4⋊Dic5—24no—
38C2×C4⋊Dic5—42no—
39C2×C4×Dic5—42no—
40C2×C4×Dic5D411yescharacteristic
41C2×C4×Dic5D411yescharacteristic
42C2×C4×Dic5D411yescharacteristic
43C2×C4×Dic5D411yescharacteristic
44C22⋊#170—22no—
45C22⋊#170—24no—
46C22⋊#170D414yes—
47#4004—42no—
48#4002—42no—
49#3963—22no—
50#3963—22no—
51C2×#389—51no—
52#1797—51no—
53#1749—51no—
54C2×#324—54no—
55C24×C5C22×C411yescharacteristic
56C22×C4×C5—22no—
57C22×C4×C5—22no—
58C22×C4×C5—22no—
59C22×C4×C5—22no—
60C22×C4×C5—24no—
61C22×C4×C5—22no—
62C22×C4×C5—24no—
63C22×C4×C5C22⋊C411yescharacteristic
64C22×C4×C5C22⋊C411yescharacteristic
65C22×C4×C5M1614yes—
66C22×Dic5—22no—
67C22×Dic5—22no—
68C22×Dic5—22no—
69C22×Dic5—24no—
70C22×Dic5—41no—
71C22×Dic5—41no—
72C22×Dic5—41no—
73C22×Dic5—42no—
74C22×Dic5—42no—
75C22×Dic5C22⋊C411yescharacteristic
76C22×Dic5C22⋊C411yescharacteristic
77C22×Dic5C2×D411yescharacteristic
78C22×Dic5C2×D411yescharacteristic
79C22×Dic5C2×D412yes—
80C22×Dic5#4112yes—
81C2×#170—24no—
82C2×#170—42no—
83C2×#170—42no—
84C2×#170—42no—
85C2×#170—42no—
86C2×#170—42no—
87C5×C4⋊C4—24no—
88C5×C4⋊C4—24no—
89C4⋊Dic5—42no—
90C4⋊Dic5—42no—
91C4⋊Dic5—44no—
92C4⋊Dic5—42no—
93C4⋊Dic5—42no—
94C4×Dic5—22no—
95C4×Dic5—22no—
96C4×Dic5—41no—
97C4×Dic5—41no—
98C22×C4⋊C4—52no—
99C22×C4⋊C4—51no—
100C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic
101C22×C42—51no—
102C2×C4⋊C8—101no—
103C2×C22⋊C8—52no—
104C2×C22⋊C8—52no—
105C2×#97—101no—
106#389—101no—
107#389—101no—
108#389—101no—
109#389—101no—
110C2×#95—52no—
111#336—101no—
112#336—101no—
113#324—108no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5C2×C22⋊C411yescharacteristic
119C23×C5C4×D412yes—
120C23×C5C2×M1612yes—
121C23×C5#13112yes—
122C2×C4×C5—28no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—24no—
128C2×C4×C5—28no—
129C2×C4×C5—42no—
130C2×C4×C5—42no—
131C2×C4×C5—44no—
132C2×C4×C5#9912yes—
133C2×C4×C5#10112yes—
134C2×Dic5—21no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—42no—
144C2×Dic5—42no—
145C2×Dic5—44no—
146C2×Dic5—42no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149C2×Dic5—84no—
150C2×Dic5#10112yes—
151C2×Dic5#12112yes—
152C2×Dic5#12411yescharacteristic
153C2×Dic5#12711yescharacteristic
154#170—44no—
155#170—82no—
156C23×C4—51no—
157C23×C4—51no—
158C23×C4—52no—
159C23×C4C2×AGL(1,𝔽5)12yes—
160C23×C4C2×AGL(1,𝔽5)11yescharacteristic
161C22×C8—102no—
162C22×C8—102no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—104no—
167C2×C4⋊C4—102no—
168C2×C4⋊C4—102no—
169C2×C4⋊C4—104no—
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—
185C22×C5—22no—
186C22×C5—22no—
187C22×C5—22no—
188C22×C5—22no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5#39011yescharacteristic
192C22×C5#40711yescharacteristic
193C22×C5#40811yescharacteristic
194C22×C5C2×#9911yescharacteristic
195C22×C5#41011yescharacteristic
196C22×C5C2×#10111yescharacteristic
197C22×C5C2×#10111yescharacteristic
198C22×C5#43514yes—
199C4×C5—22no—
200C4×C5—22no—
201C4×C5—44no—
202Dic5—22no—
203Dic5—21no—
204Dic5—21no—
205Dic5—22no—
206Dic5—41no—
207Dic5—82no—
208C24C22×AGL(1,𝔽5)11yescharacteristic
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—51no—
217C22×C4—51no—
218C22×C4—51no—
219C22×C4—52no—
220C22×C4—52no—
221C22×C4—51no—
222C22×C4—102no—
223C22×C4—102no—
224C22×C4—102no—
225C22×C4—104no—
226C22×C4—201no—
227C22×C4—201no—
228C22×C4—201no—
229C22×C4—202no—
230C22×C4—202no—
231C22×C4#70414yes—
232C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
233C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
234C2×C8—104no—
235C2×C8—202no—
236C2×C8—202no—
237C2×C8—202no—
238C2×C8—202no—
239C2×C8—202no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—202no—
243C4⋊C4—204no—
244C4⋊C4—202no—
245C4⋊C4—202no—
246C4⋊C4—202no—
247C42—102no—
248C42—102no—
249C42—201no—
250C42—201no—
251C2×C5—24no—
252C2×C5#146311yescharacteristic
253C2×C5#146711yescharacteristic
254C2×C5#147011yescharacteristic
255C2×C5#147211yescharacteristic
256C2×C5#195511yescharacteristic
257C2×C5#196411yescharacteristic
258C2×C5#196611yescharacteristic
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23#413112yes—
264C23D4×AGL(1,𝔽5)12yes—
265C23C2×#70412yes—
266C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
267C2×C4—28no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—28no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—52no—
278C2×C4—51no—
279C2×C4—51no—
280C2×C4—52no—
281C2×C4—101no—
282C2×C4—101no—
283C2×C4—101no—
284C2×C4—101no—
285C2×C4—101no—
286C2×C4—101no—
287C2×C4—101no—
288C2×C4—101no—
289C2×C4—101no—
290C2×C4—202no—
291C2×C4—202no—
292C2×C4—204no—
293C2×C4—202no—
294C2×C4—202no—
295C2×C4—202no—
296C2×C4—404no—
297C2×C4#399912yes—
298C2×C4#400512yes—
299C8—204no—
300C8—402no—
301C5#965411yescharacteristic, complemented, Sylow
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22C2×#399911yescharacteristic
309C22C2×#400511yescharacteristic
310C22#6577314yes—
311C22#6577611yescharacteristic
312C22#6578011yescharacteristic
313C22#6578511yescharacteristic
314C22D5⋊#10111yescharacteristic
315C22#6579811yescharacteristic
316C4—22no—
317C4—22no—
318C4—44no—
319C4—102no—
320C4—102no—
321C4—101no—
322C4—101no—
323C4—201no—
324C4—402no—
325C2—24no—
326C2#1060429811yescharacteristic
327C2#1060430111yescharacteristic
328C2#1060430911yescharacteristic
329C2#1060433911yescharacteristic
330C2#1060488111yescharacteristic
331C2#1060494911yescharacteristic
332C2#1060497711yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutator, prime radicalC24×C5C22×C4
FrattiniC24C22×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC23×C5C2×C22⋊C4
cosocleC23×Dic5C23
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5#9654
LCSG ▷ C24×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
301. C5#9654semidirect

Maximal subgroups

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

subgroupindex
C2×#652742
#106049002
#106048972
C2×#649132
#96545