Finite Group Databasebeta

← #12010017

Subgroups — #12010017

Order 1280.

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

normal subgroups only

This group has 402 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#10602387C211yesmaximal, characteristic
3#10602365C211yesmaximal, characteristic
4#10598142C212yesmaximal
5#10598137C212yesmaximal
6C2×C4×C5×Q8C411yescharacteristic, complemented
7C2×C5×#117C2211yescharacteristic
8C5×#400C411yescharacteristic, complemented
9C5×#399C2211yescharacteristic
10C5×#398—21no—
11C5×#396—21no—
12C5×#395C411yescharacteristic, complemented
13C5×#393C2211yescharacteristic
14C4×C5×C4⋊C4—21no—
15C4×C5×C4⋊C4C411yescharacteristic, complemented
16C4×C5×C22⋊C4—21no—
17#65500—21no—
18#65215—21no—
19#64748—21no—
20#64746—21no—
21#64746C2212yes—
22#64745C2212yes—
23#11810—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⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4D411yescharacteristic
40C2×C5×C22⋊C4C2311yescharacteristic
41C2×C42×C5—21no—
42C2×C42×C5—21no—
43C2×C42×C5C2×C411yescharacteristic
44C2×C42×C5D411yescharacteristic
45C2×C22⋊Dic5—22no—
46C5×#95—21no—
47C5×#95—21no—
48C5×#95—21no—
49C5×#95—21no—
50C5×#95C2×C411yescharacteristic
51C5×#95C2×C411yescharacteristic
52#3963—41no—
53#3963—41no—
54C22⋊#168—22no—
55C4⋊#168—41no—
56#3935—41no—
57#2396D511yescharacteristic
58#1820—51no—
59#1464—51no—
60#1352—52no—
61#1351—52no—
62C24×C5C2×D411yescharacteristic
63C2×C5×Q8—42no—
64C2×C5×Q8—41no—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—22no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
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×C5C22×C411yescharacteristic
78C22×C4×C5C2×D411yescharacteristic
79C22×Dic5—42no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—42no—
85C5×C4⋊C4—42no—
86C5×C4⋊C4—42no—
87C5×C22⋊C4—41no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—42no—
90C5×C22⋊C4—42no—
91C42×C5—21no—
92C42×C5—21no—
93C42×C5—21no—
94C42×C5—21no—
95C42×C5—42no—
96C42×C5—41no—
97C42×C5—41no—
98C22⋊Dic5—42no—
99C22⋊Dic5—42no—
100C2×#168—42no—
101C2×C4×Q8Dic511yescharacteristic, complemented
102C2×#117C2×D511yescharacteristic
103#432—101no—
104#400Dic511yescharacteristic, complemented
105#399C2×D511yescharacteristic
106#398—21no—
107#396—21no—
108#395Dic511yescharacteristic, complemented
109#393C2×D511yescharacteristic
110#381—101no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4Dic511yescharacteristic, complemented
113C4×C22⋊C4—21no—
114#342—52no—
115#342—101no—
116#324—101no—
117#323—52no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—41no—
121C23×C5—41no—
122C23×C5C2×C22⋊C411yescharacteristic
123C23×C5#13712yes—
124C5×Q8—42no—
125C5×Q8—42no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—22no—
129C2×C4×C5—22no—
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—21no—
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—41no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5#9911yescharacteristic
154C2×C4×C5#9911yescharacteristic
155C2×Dic5—82no—
156C2×Dic5—82no—
157C2×Dic5—82no—
158#168—82no—
159C22×Q8C2×Dic511yescharacteristic
160C23×C4C22⋊D511yescharacteristic
161C4×Q8—41no—
162C4×Q8—41no—
163#117—41no—
164#117—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4C2×Dic511yescharacteristic
170C2×C4⋊C4C2×Dic511yescharacteristic
171C2×C4⋊C4C22⋊D511yescharacteristic
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—102no—
175C2×C22⋊C4C22⋊D511yescharacteristic
176C2×C22⋊C4C22×D511yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C2×Dic511yescharacteristic
180C2×C42C22⋊D511yescharacteristic
181C4⋊C8—201no—
182C22⋊C8—102no—
183#97—201no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—201no—
189#95—201no—
190#95C2×Dic511yescharacteristic
191#95C2×Dic511yescharacteristic
192C22×C5—21no—
193C22×C5—21no—
194C22×C5—22no—
195C22×C5—42no—
196C22×C5—41no—
197C22×C5—41no—
198C22×C5—81no—
199C22×C5C2×#9911yescharacteristic
200C22×C5#41911yescharacteristic
201C22×C5#42111yescharacteristic
202C4×C5—21no—
203C4×C5—21no—
204C4×C5—42no—
205C4×C5—41no—
206C4×C5—42no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5—81no—
212C4×C5—81no—
213Dic5—162no—
214C24C2×C22⋊D511yescharacteristic
215C2×Q8—42no—
216C2×Q8—41no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—202no—
226C22×C4C22⋊Dic511yescharacteristic
227C22×C4C22⋊Dic511yescharacteristic
228C22×C4C22⋊Dic511yescharacteristic
229C22×C4C22⋊Dic511yescharacteristic
230C22×C4C22×Dic511yescharacteristic
231C22×C4C2×C22⋊D511yescharacteristic
232C2×C8—202no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—202no—
245C22⋊C4—202no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C42—41no—
252C42—41no—
253C2×C5—21no—
254C2×C5—21no—
255C2×C5—42no—
256C2×C5#151111yescharacteristic
257C2×C5#212011yescharacteristic
258C2×C5#212611yescharacteristic
259C23—21no—
260C23—21no—
261C23—41no—
262C23—41no—
263C23#408812yes—
264C23C2×C22⋊Dic511yescharacteristic
265Q8—42no—
266Q8—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—42no—
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—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—402no—
295C2×C4—402no—
296C2×C4—402no—
297C2×C4#396611yescharacteristic
298C2×C4#396611yescharacteristic
299C8—402no—
300C5#1181011yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#396611yescharacteristic
309C22#6551411yescharacteristic
310C22#6552511yescharacteristic
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—802no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2#1060248211yescharacteristic
327C2#1060250911yescharacteristic
328C2#1060261211yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2396C2
socleC22×C5C2×#99
prime radicalC24×C5C2×D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2396D5
hypocenterC5#11810
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
6. C2×C4×C5×Q8C4semidirect
8. C5×#400C4semidirect
12. C5×#395C4semidirect
15. C4×C5×C4⋊C4C4semidirect
101. C2×C4×Q8Dic5semidirect
104. #400Dic5semidirect
108. #395Dic5semidirect
112. C4×C4⋊C4Dic5semidirect
300. C5#11810semidirect

Maximal subgroups

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

subgroupindex
C5×#23962
#106023872
#106023652
#105981422
#105981372
#118105