Finite Group Databasebeta

← #12010048

Subgroups — #12010048

Order 1280.

386 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C5×#2627C211yesmaximal, characteristic
2#10602355C211yesmaximal, characteristic
3#10602345C211yesmaximal, characteristic
4#10598143C212yesmaximal, complemented
5#10598139C212yesmaximal, complemented
6C2×C5×C22⋊Q8C411yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C5×#400C411yescharacteristic
9C5×#400C2211yescharacteristic
10C5×#398—21no—
11C5×#398—21no—
12C5×#395C411yescharacteristic
13C5×#395C2211yescharacteristic
14C5×#385—21no—
15C5×#381—21no—
16C5×#380C411yescharacteristic
17#65496—21no—
18#65496—21no—
19#64750—21no—
20#64750C2212yes—
21#64748—21no—
22#64748C2212yes—
23#11817—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C23×C4×C5C2×C411yescharacteristic
26C23×C4×C5D411yescharacteristic
27C23×C4×C5D411yescharacteristic
28C5×C22⋊Q8—41no—
29C5×C22⋊Q8—41no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—41no—
35C2×C5×C4⋊C4—41no—
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C4⋊C4C2×C411yescharacteristic
38C2×C5×C4⋊C4D411yescharacteristic
39C2×C5×C4⋊C4D411yescharacteristic
40C2×C5×C4⋊C4C2311yescharacteristic
41C2×C5×C22⋊C4—21no—
42C2×C5×C22⋊C4—21no—
43C2×#681—22no—
44C2×#681—41no—
45C5×#95—21no—
46C5×#95—21no—
47C5×#95—21no—
48C5×#95—21no—
49C5×#95—21no—
50C5×#95—21no—
51C5×#95C2×C411yescharacteristic
52C5×#95C2×C411yescharacteristic
53C22⋊#168—22no—
54C22⋊#168—41no—
55C22⋊#168—41no—
56C22⋊#168—41no—
57#2627D511yescharacteristic
58#1821—51no—
59#1463—51no—
60#1354—52no—
61#1353—52no—
62C24×C5C22⋊C411yescharacteristic
63C2×C5×Q8—41no—
64C2×C5×Q8—81no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—22no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—41no—
77C22×C4×C5—81no—
78C22×C4×C5—81no—
79C22×C4×C5C22⋊C411yescharacteristic
80C22×C4×C5C22⋊C411yescharacteristic
81C22×C4×C5C22⋊C411yescharacteristic
82C22×C4×C5C22×C411yescharacteristic
83C22×C4×C5C2×D411yescharacteristic
84C22×C4×C5C2×D411yescharacteristic
85C5×C4⋊C4—41no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—41no—
89C5×C4⋊C4—41no—
90C5×C4⋊C4—42no—
91C5×C4⋊C4—81no—
92C5×C4⋊C4—81no—
93C5×C4⋊C4—81no—
94C5×C22⋊C4—42no—
95C5×C22⋊C4—41no—
96C5×C22⋊C4—41no—
97#681—42no—
98#681—42no—
99#681—82no—
100C2×#168—42no—
101C2×#168—42no—
102C2×C22⋊Q8Dic511yescharacteristic
103C22×C4⋊C4C2×D511yescharacteristic
104#407—101no—
105#407—101no—
106#400Dic511yescharacteristic
107#400C2×D511yescharacteristic
108#398—21no—
109#398—21no—
110#395Dic511yescharacteristic
111#395C2×D511yescharacteristic
112#385—21no—
113#381—21no—
114#380Dic511yescharacteristic
115#343—52no—
116#343—101no—
117#324—52no—
118#324—101no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5#9911yescharacteristic
124C23×C5#9911yescharacteristic
125C23×C5C2×C22⋊C411yescharacteristic
126C5×Q8—82no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5—42no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—81no—
155C2×C4×C5—81no—
156C2×C4×C5—81no—
157C2×C4×C5#13712yes—
158#168—82no—
159#168—82no—
160C22×Q8C2×Dic511yescharacteristic
161C23×C4C2×Dic511yescharacteristic
162C23×C4C22⋊D511yescharacteristic
163C23×C4C22⋊D511yescharacteristic
164C2×M16—102no—
165C2×M16—201no—
166C22⋊Q8—41no—
167C22⋊Q8—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4C2×Dic511yescharacteristic
175C2×C4⋊C4C2×Dic511yescharacteristic
176C2×C4⋊C4C22⋊D511yescharacteristic
177C2×C4⋊C4C22⋊D511yescharacteristic
178C2×C4⋊C4C22×D511yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C22⋊C8—102no—
182C22⋊C8—201no—
183C22⋊C8—201no—
184C22⋊C8—201no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95C2×Dic511yescharacteristic
192#95C2×Dic511yescharacteristic
193C22×C5—21no—
194C22×C5—21no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—42no—
202C22×C5—42no—
203C22×C5C2×#9911yescharacteristic
204C22×C5#41911yescharacteristic
205C22×C5#42111yescharacteristic
206C4×C5—22no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5—41no—
212C4×C5—81no—
213C4×C5—81no—
214C4×C5—81no—
215C4×C5—81no—
216C24C22⋊Dic511yescharacteristic
217C2×Q8—41no—
218C2×Q8—81no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4C22⋊Dic511yescharacteristic
234C22×C4C22⋊Dic511yescharacteristic
235C22×C4C22⋊Dic511yescharacteristic
236C22×C4C22×Dic511yescharacteristic
237C22×C4C2×C22⋊D511yescharacteristic
238C22×C4C2×C22⋊D511yescharacteristic
239M16—202no—
240M16—202no—
241M16—402no—
242C2×C8—202no—
243C2×C8—202no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C22⋊C4—42no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C2×C5—21no—
257C2×C5—21no—
258C2×C5—41no—
259C2×C5—41no—
260C2×C5#151111yescharacteristic
261C2×C5#212111yescharacteristic
262C2×C5#213311yescharacteristic
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23#396611yescharacteristic
268C23#396611yescharacteristic
269C23C2×C22⋊Dic511yescharacteristic
270Q8—82no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4#408812yes—
302C8—402no—
303C8—402no—
304C5#1181711yescharacteristic, complemented, Sylow
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#396611yescharacteristic
316C22#6551411yescharacteristic
317C22#6552511yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#1060248211yescharacteristic
333C2#1060264011yescharacteristic
334C2#1060266511yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2627C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2627D5
hypocenterC5#11817
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
4. #10598143C2semidirect
5. #10598139C2semidirect
304. C5#11817semidirect

Maximal subgroups

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

subgroupindex
C5×#26272
#106023552
#106023452
#105981432
#105981392
#118175