Finite Group Databasebeta

← #13023103

Subgroups — #13023103

Order 1280.

678 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#10615758C211yesmaximal, characteristic
2#10604877C212yesmaximal, complemented
3#10598837C212yesmaximal
4#10598819C212yesmaximal, complemented
5C24×C4×C5C411yescharacteristic
6C2×#4072C411yescharacteristic
7C2×#4072C2211yescharacteristic
8#65748—22no—
9#65536C411yescharacteristic
10#65536C2211yescharacteristic
11#65501—22no—
12#65500C411yescharacteristic
13#65500C2211yescharacteristic
14C4×C22⋊Dic5—22no—
15#64933C2214yes—
16#64913—22no—
17#11425—51nomaximal, Sylow
18C25×C5C2×C411yescharacteristic
19C23×C4×C5—21no—
20C23×C4×C5—21no—
21C23×C4×C5—22no—
22C23×C4×C5—41no—
23C23×C4×C5—41no—
24C23×C4×C5—42no—
25C23×C4×C5C814yescomplemented
26C23×C4×C5C2×C411yescharacteristic
27C23×C4×C5C2×C411yescharacteristic
28C2×C22⋊Dic5—22no—
29C2×C22⋊Dic5—22no—
30C2×C22⋊Dic5C2×C411yescharacteristic
31C2×C22⋊Dic5C2×C411yescharacteristic
32C2×C22⋊Dic5C2×C411yescharacteristic
33C2×C22⋊Dic5C2311yescharacteristic
34#4072—41no—
35#4072—41no—
36#4072—42no—
37C2×C4⋊Dic5—21no—
38C2×C4⋊Dic5D412yes—
39C2×C4×Dic5—21no—
40C2×C4×Dic5D412yes—
41C22⋊#170—24no—
42C4⋊#170—42no—
43C4×#170—42no—
44#3963—22no—
45#3963—22no—
46#2280—51no—
47#1458—52no—
48#1317—52no—
49#1316—52no—
50C24×C5—21no—
51C24×C5—21no—
52C24×C5—41no—
53C24×C5—41no—
54C24×C5—44no—
55C24×C5C2×C812yes—
56C24×C5C22×C411yescharacteristic
57C22×C4×C5—21no—
58C22×C4×C5—21no—
59C22×C4×C5—22no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—24no—
63C22×C4×C5—24no—
64C22×C4×C5—42no—
65C22×C4×C5—42no—
66C22×C4×C5—44no—
67C22×C4×C5—44no—
68C22×C4×C5—44no—
69C22×C4×C5—44no—
70C22×C4×C5—41no—
71C22×C4×C5—41no—
72C22×C4×C5—42no—
73C22×C4×C5—41no—
74C22×C4×C5—41no—
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C2×C812yes—
78C22×C4×C5C2×C812yes—
79C22×C4×C5M1612yes—
80C22×Dic5—21no—
81C22×Dic5—21no—
82C22×Dic5C22⋊C412yes—
83C22×Dic5C2×D412yes—
84C2×#170—44no—
85C22⋊Dic5—41no—
86C22⋊Dic5—41no—
87C22⋊Dic5—44no—
88C22⋊Dic5—44no—
89C22⋊Dic5—41no—
90C22⋊Dic5—41no—
91C4⋊Dic5—42no—
92C4⋊Dic5—42no—
93C4×Dic5—22no—
94C4×Dic5—22no—
95C4×Dic5—42no—
96C24×C4AGL(1,𝔽5)11yescharacteristic
97C2×#117—51no—
98C2×#117—51no—
99#431—102no—
100#381—51no—
101#381—51no—
102#380—102no—
103#379—51no—
104#379—51no—
105C4×C22⋊C4—102no—
106#324—102no—
107#323—54no—
108C23×C5—24no—
109C23×C5—22no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—42no—
113C23×C5—44no—
114C23×C5—42no—
115C23×C5—42no—
116C23×C5—42no—
117C23×C5—44no—
118C23×C5—44no—
119C23×C5—48no—
120C23×C5—44no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5C2×C22⋊C411yescharacteristic
124C23×C5C22×C811yescharacteristic
125C23×C5C2×M1611yescharacteristic
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—22no—
129C2×C4×C5—22no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—48no—
134C2×C4×C5—42no—
135C2×C4×C5—41no—
136C2×C4×C5—41no—
137C2×C4×C5—44no—
138C2×C4×C5—44no—
139C2×C4×C5—42no—
140C2×C4×C5C22⋊C812yes—
141C2×C4×C5C22⋊C812yes—
142C2×C4×C5#9911yescharacteristic
143C2×C4×C5#9911yescharacteristic
144C2×C4×C5#10012yes—
145C2×Dic5—22no—
146C2×Dic5—22no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149C2×Dic5—42no—
150C2×Dic5—42no—
151C2×Dic5—41no—
152C2×Dic5—41no—
153#170—84no—
154C25C2×AGL(1,𝔽5)11yescharacteristic
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—42no—
161C23×C4#17014yescomplemented
162C23×C4C2×AGL(1,𝔽5)11yescharacteristic
163C23×C4C2×AGL(1,𝔽5)11yescharacteristic
164#117—201no—
165#117—201no—
166#117—202no—
167C2×C4⋊C4—52no—
168C2×C4⋊C4—101no—
169C2×C22⋊C4—51no—
170C2×C22⋊C4—51no—
171C2×C22⋊C4—51no—
172C2×C22⋊C4—51no—
173C2×C22⋊C4—102no—
174C2×C22⋊C4—102no—
175C2×C42—52no—
176C2×C42—101no—
177C4⋊C8—202no—
178C22⋊C8—104no—
179C4×C8—202no—
180#95—102no—
181#95—102no—
182C22×C5—22no—
183C22×C5—24no—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—41no—
187C22×C5—41no—
188C22×C5—44no—
189C22×C5—42no—
190C22×C5—42no—
191C22×C5—44no—
192C22×C5—44no—
193C22×C5—48no—
194C22×C5—44no—
195C22×C5—42no—
196C22×C5—42no—
197C22×C5C2×C22⋊C811yescharacteristic
198C22×C5C2×#9911yescharacteristic
199C22×C5C2×#10011yescharacteristic
200C4×C5—22no—
201C4×C5—42no—
202C4×C5#32312yes—
203C4×C5#32312yes—
204Dic5—42no—
205Dic5—42no—
206Dic5—81no—
207Dic5—81no—
208C24—21no—
209C24—21no—
210C24—41no—
211C24—41no—
212C24—44no—
213C24C2×#17012yes—
214C24C22×AGL(1,𝔽5)11yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—52no—
234C22×C4—52no—
235C22×C4—101no—
236C22×C4—101no—
237C22×C4C2×#17012yes—
238C22×C4C2×#17012yes—
239C22×C4#70412yes—
240C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
241C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
242C2×C8—204no—
243C4⋊C4—202no—
244C4⋊C4—202no—
245C22⋊C4—201no—
246C22⋊C4—201no—
247C22⋊C4—204no—
248C22⋊C4—204no—
249C22⋊C4—201no—
250C22⋊C4—201no—
251C42—102no—
252C42—102no—
253C42—202no—
254C2×C5—21no—
255C2×C5—21no—
256C2×C5—44no—
257C2×C5—41no—
258C2×C5—41no—
259C2×C5C2×#32311yescharacteristic
260C2×C5#212012yes—
261C23—24no—
262C23—21no—
263C23—22no—
264C23—21no—
265C23—42no—
266C23—44no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—44no—
271C23—44no—
272C23—48no—
273C23—44no—
274C23—41no—
275C23—41no—
276C23C22×#17011yescharacteristic
277C23C2×#70411yescharacteristic
278C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—48no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—102no—
294C2×C4—102no—
295C2×C4—202no—
296C2×C4—202no—
297C2×C4—202no—
298C2×C4—202no—
299C2×C4—201no—
300C2×C4—201no—
301C2×C4#401111yescharacteristic
302C2×C4#401111yescharacteristic
303C2×C4C22⋊#17012yes—
304C2×C4C22⋊#17012yes—
305C2×C4#401312yes—
306C8—404no—
307C5#1142511yescharacteristic, complemented, Sylow
308C22—22no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22—48no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22C2×#401111yescharacteristic
324C22C2×C22⋊#17011yescharacteristic
325C22C2×#401311yescharacteristic
326C4—22no—
327C4—42no—
328C4—202no—
329C4—202no—
330C4—401no—
331C4—401no—
332C4#6493312yes—
333C4#6493312yes—
334C2—21no—
335C2—21no—
336C2—44no—
337C2—41no—
338C2—41no—
339C2#1060488712yes—
340C2C2×#6493311yescharacteristic
subgroupG/H
centerC2×C4#4011
commutatorC23×C5C22×C8
FrattiniC24C22×AGL(1,𝔽5)
FittingC24×C4×C5C4
socleC22×C5C2×#99
prime radicalC25×C5C2×C4
cosocleC2×C22⋊Dic5C23
hypercenterC24×C4AGL(1,𝔽5)
hypocenterC5#11425
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
2. #10604877C2semidirect
4. #10598819C2semidirect
25. C23×C4×C5C8semidirect
161. C23×C4#170semidirect
307. C5#11425semidirect

Maximal subgroups

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

subgroupindex
#106157582
#106048772
#105988372
#105988192
#114255