Finite Group Databasebeta

← #12010043

Subgroups — #12010043

Order 1280.

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

normal subgroups only

This group has 388 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×#2396C211yesmaximal, characteristic
2#10602382C211yesmaximal, characteristic
3#10602365C211yesmaximal, characteristic
4#10598143C212yesmaximal, complemented
5#10598139C212yesmaximal, complemented
6C2×C4×C5×Q8C411yescharacteristic
7C2×C5×#117C2211yescharacteristic
8C5×#400C2211yescharacteristic
9C5×#399C411yescharacteristic
10C5×#398—21no—
11C5×#396—21no—
12C5×#395C2211yescharacteristic
13C5×#393C411yescharacteristic
14C4×C5×C4⋊C4—21no—
15C4×C5×C4⋊C4C411yescharacteristic
16C4×C5×C22⋊C4—21no—
17#65496—21no—
18#65215—21no—
19#64750—21no—
20#64750C2212yes—
21#64748C2212yes—
22#64745—21no—
23#11816—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C4×C5×Q8—41no—
27C4×C5×Q8—41no—
28C5×#117—41no—
29C5×#117—41no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4C2×C411yescharacteristic
35C2×C5×C4⋊C4D411yescharacteristic
36C2×C5×C4⋊C4C2311yescharacteristic
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4C2×C411yescharacteristic
40C2×C5×C22⋊C4D411yescharacteristic
41C2×C42×C5—21no—
42C2×C42×C5—21no—
43C2×C42×C5C2×C411yescharacteristic
44C2×C42×C5D411yescharacteristic
45C2×#681—22no—
46C5×#95—21no—
47C5×#95—21no—
48C5×#95—21no—
49C5×#95—21no—
50C5×#95C2×C411yescharacteristic
51C5×#95C2×C411yescharacteristic
52C22⋊#168—22no—
53C22⋊#168—41no—
54C22⋊#168—41no—
55C4⋊#168—41no—
56#3935—41no—
57#2396D511yescharacteristic
58#1822—51no—
59#1464—51no—
60#1354—52no—
61#1353—52no—
62C24×C5C22⋊C411yescharacteristic
63C2×C5×Q8—42no—
64C2×C5×Q8—41no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—22no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—41no—
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22×C411yescharacteristic
77C22×C4×C5C2×D411yescharacteristic
78C22×C4×C5C2×D411yescharacteristic
79C5×C4⋊C4—41no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—42no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—42no—
85C5×C4⋊C4—42no—
86C5×C22⋊C4—41no—
87C5×C22⋊C4—41no—
88C5×C22⋊C4—42no—
89C5×C22⋊C4—42no—
90C42×C5—21no—
91C42×C5—21no—
92C42×C5—21no—
93C42×C5—21no—
94C42×C5—42no—
95C42×C5—41no—
96C42×C5—41no—
97#681—42no—
98#681—42no—
99C2×#168—42no—
100C2×#168—42no—
101C2×C4×Q8Dic511yescharacteristic
102C2×#117C2×D511yescharacteristic
103#432—101no—
104#407—101no—
105#400C2×D511yescharacteristic
106#399Dic511yescharacteristic
107#398—21no—
108#396—21no—
109#395C2×D511yescharacteristic
110#393Dic511yescharacteristic
111C4×C4⋊C4—21no—
112C4×C4⋊C4Dic511yescharacteristic
113C4×C22⋊C4—21no—
114#343—52no—
115#343—101no—
116#324—52no—
117#323—101no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5C2×C22⋊C411yescharacteristic
124C5×Q8—42no—
125C5×Q8—42no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
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—41no—
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—41no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5#9911yescharacteristic
153C2×C4×C5#9911yescharacteristic
154C2×C4×C5#13712yes—
155#168—82no—
156#168—82no—
157C22×Q8C2×Dic511yescharacteristic
158C23×C4C22⋊D511yescharacteristic
159C2×M16—102no—
160C4×Q8—41no—
161C4×Q8—41no—
162#117—41no—
163#117—41no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C2×Dic511yescharacteristic
169C2×C4⋊C4C22⋊D511yescharacteristic
170C2×C4⋊C4C22×D511yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4C2×Dic511yescharacteristic
174C2×C22⋊C4C22⋊D511yescharacteristic
175C2×C42—21no—
176C2×C42—21no—
177C2×C42C2×Dic511yescharacteristic
178C2×C42C22⋊D511yescharacteristic
179C4⋊C8—201no—
180C22⋊C8—102no—
181C22⋊C8—201no—
182C22⋊C8—201no—
183#97—201no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95C2×Dic511yescharacteristic
189#95C2×Dic511yescharacteristic
190C22×C5—21no—
191C22×C5—21no—
192C22×C5—42no—
193C22×C5—41no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5—81no—
197C22×C5C2×#9911yescharacteristic
198C22×C5#41911yescharacteristic
199C22×C5#42111yescharacteristic
200C4×C5—22no—
201C4×C5—21no—
202C4×C5—21no—
203C4×C5—42no—
204C4×C5—41no—
205C4×C5—42no—
206C4×C5—41no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—81no—
210C4×C5—81no—
211C24C22⋊Dic511yescharacteristic
212C2×Q8—42no—
213C2×Q8—41no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4C22⋊Dic511yescharacteristic
223C22×C4C22⋊Dic511yescharacteristic
224C22×C4C22⋊Dic511yescharacteristic
225C22×C4C22×Dic511yescharacteristic
226C22×C4C2×C22⋊D511yescharacteristic
227C22×C4C2×C22⋊D511yescharacteristic
228M16—202no—
229M16—202no—
230C2×C8—202no—
231C2×C8—202no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—42no—
248C42—41no—
249C42—41no—
250C2×C5—21no—
251C2×C5—21no—
252C2×C5—81no—
253C2×C5#151111yescharacteristic
254C2×C5#212011yescharacteristic
255C2×C5#213211yescharacteristic
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23C2×C22⋊Dic511yescharacteristic
262Q8—42no—
263Q8—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4#396611yescharacteristic
291C2×C4#396611yescharacteristic
292C2×C4#408812yes—
293C8—402no—
294C8—402no—
295C5#1181611yescharacteristic, complemented, Sylow
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#396611yescharacteristic
304C22#6551411yescharacteristic
305C22#6552511yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#1060248211yescharacteristic
321C2#1060250911yescharacteristic
322C2#1060261311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2396C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2396D5
hypocenterC5#11816
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
4. #10598143C2semidirect
5. #10598139C2semidirect
295. C5#11816semidirect

Maximal subgroups

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

subgroupindex
C5×#23962
#106023822
#106023652
#105981432
#105981392
#118165