Finite Group Databasebeta

← #11997155

Subgroups — #11997155

Order 1280.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2536C211yesmaximal, characteristic
2#10614166C211yesmaximal, characteristic, complemented
3#10602430C211yesmaximal, characteristic, complemented
4#10602391C212yesmaximal
5#10598122C212yesmaximal, complemented
6C2×C5×C22⋊Q8—21no—
7C2×C4×C5×Q8C2211yescharacteristic
8C2×C4×C5×D4C2211yescharacteristic
9C5×#398C412yes—
10C5×#394C412yes—
11C5×#388—22no—
12C5×#382C2211yescharacteristic
13C5×#380—21no—
14C2×#3968—21no—
15Q8×#168—24no—
16#65214—22no—
17#65214C2212yes—
18#65213C2212yes—
19D4⋊#168—24no—
20#64750—21no—
21#9486—51nomaximal, Sylow
22C22×C5×Q8D411yescharacteristic
23C22×C5×D4C2×C411yescharacteristic
24C23×C4×C5—21no—
25C5×C22⋊Q8—42no—
26C5×C22⋊Q8—42no—
27C4×C5×Q8—22no—
28C4×C5×Q8—22no—
29C4×C5×D4—22no—
30C4×C5×D4—22no—
31C2×C5×C4⋊C4—22no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C4⋊C4C2×C412yes—
35C2×C5×C4⋊C4D411yescharacteristic
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—22no—
38C2×C5×C22⋊C4—21no—
39C2×C42×C5D412yes—
40C2×C42×C5C2311yescharacteristic
41C2×#681—22no—
42C5×#95—22no—
43C5×#95—21no—
44C5×#95C2×C412yes—
45#3968—42no—
46C22⋊#168—22no—
47C4⋊#168—24no—
48C4⋊#168—22no—
49C4⋊#168—22no—
50C4⋊#168—22no—
51C4×#168—24no—
52C4×#168D414yes—
53#2982—51no—
54#2536D511yescharacteristic
55#1801—51no—
56#1486—52no—
57#1286—52no—
58C24×C5—21no—
59C2×C5×Q8—22no—
60C2×C5×Q8—22no—
61C2×C5×Q8D812yes—
62C2×C5×Q8SD1612yes—
63C2×C5×D4—22no—
64C2×C5×D4—42no—
65C22×C4×C5—22no—
66C22×C4×C5—21no—
67C22×C4×C5—42no—
68C22×C4×C5—41no—
69C22×C4×C5—41no—
70C22×C4×C5—42no—
71C22×C4×C5C22⋊C412yes—
72C22×C4×C5C22⋊C412yes—
73C22×C4×C5C22×C411yescharacteristic
74C22×C4×C5C2×D411yescharacteristic
75C22×C4×C5C2×D411yescharacteristic
76C5×C4⋊C4—24no—
77C5×C4⋊C4—22no—
78C5×C4⋊C4—22no—
79C5×C4⋊C4—44no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C22⋊C4—44no—
83C5×C22⋊C4—44no—
84C5×C22⋊C4—41no—
85C5×C22⋊C4—41no—
86C42×C5—22no—
87C42×C5—22no—
88C42×C5C2×D412yes—
89C42×C5#4112yes—
90#681—42no—
91#681—42no—
92C2×#168—24no—
93C2×#168—42no—
94C2×C22⋊Q8—21no—
95C2×C4×Q8C2×D511yescharacteristic
96C2×C4×D4C2×D511yescharacteristic
97C8×Q8—104no—
98#431—52no—
99#431—102no—
100#423—52no—
101C2×#101—101no—
102#398Dic512yes—
103#394Dic512yes—
104#388—22no—
105#382C2×D511yescharacteristic
106#380—21no—
107#343—101no—
108D4⋊C8—104no—
109C23×C5—22no—
110C23×C5—41no—
111C23×C5—41no—
112C23×C5—41no—
113C23×C5—41no—
114C23×C5—41no—
115C23×C5—41no—
116C23×C5C2×C22⋊C411yescharacteristic
117C5×Q8—24no—
118C5×Q8—24no—
119C5×D4—24no—
120C5×D4—42no—
121C2×C4×C5—22no—
122C2×C4×C5—22no—
123C2×C4×C5—22no—
124C2×C4×C5—41no—
125C2×C4×C5—41no—
126C2×C4×C5—42no—
127C2×C4×C5—42no—
128C2×C4×C5—42no—
129C2×C4×C5—41no—
130C2×C4×C5—42no—
131C2×C4×C5—41no—
132C2×C4×C5—41no—
133C2×C4×C5—82no—
134C2×C4×C5—82no—
135C2×C4×C5#10214yes—
136C2×C4×C5C4×D412yes—
137C2×C4×C5#12011yescharacteristic
138C2×C4×C5#12112yes—
139C2×C4×C5#12311yescharacteristic
140C2×C4×C5C2×D811yescharacteristic
141C2×C4×C5C2×SD1611yescharacteristic
142C2×C4×C5#13512yes—
143#168—44no—
144#168—82no—
145C22×Q8C22⋊D511yescharacteristic
146C22×D4C2×Dic511yescharacteristic
147C23×C4—21no—
148C2×M16—102no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×D4—22no—
154C4×D4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×Dic511yescharacteristic
158C2×C4⋊C4C2×Dic512yes—
159C2×C4⋊C4C22⋊D511yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C42C22⋊D512yes—
164C2×C42C22×D511yescharacteristic
165C4⋊C8—104no—
166C4⋊C8—102no—
167C4⋊C8—102no—
168C4⋊C8—102no—
169#101—202no—
170C22⋊C8—102no—
171C4×C8—54no—
172C4×C8—104no—
173#95—22no—
174#95—21no—
175#95C2×Dic512yes—
176C22×C5—21no—
177C22×C5—21no—
178C22×C5—42no—
179C22×C5—42no—
180C22×C5—42no—
181C22×C5—81no—
182C22×C5—81no—
183C22×C5—81no—
184C22×C5—81no—
185C22×C5#38611yescharacteristic
186C22×C5C2×#10211yescharacteristic
187C22×C5#41611yescharacteristic
188C4×C5—22no—
189C4×C5—24no—
190C4×C5—21no—
191C4×C5—21no—
192C4×C5—42no—
193C4×C5—82no—
194C4×C5—81no—
195C4×C5#46512yes—
196C4×C5#46612yes—
197C24—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8#68612yes—
201C2×Q8#68712yes—
202C2×D4—22no—
203C2×D4—42no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4C22⋊Dic512yes—
211C22×C4C22⋊Dic512yes—
212C22×C4C22×Dic511yescharacteristic
213C22×C4C2×C22⋊D511yescharacteristic
214C22×C4C2×C22⋊D511yescharacteristic
215M16—202no—
216M16—202no—
217C2×C8—104no—
218C2×C8—202no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—44no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C42—22no—
230C42—22no—
231C42D4×D512yes—
232C42#71112yes—
233C2×C5—21no—
234C2×C5—21no—
235C2×C5—82no—
236C2×C5#189111yescharacteristic
237C2×C5#189311yescharacteristic
238C2×C5#191211yescharacteristic
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23C2×C22⋊Dic511yescharacteristic
247Q8—24no—
248Q8—24no—
249D4—24no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4#396414yes—
266C2×C4C2×#68611yescharacteristic
267C2×C4C2×#68711yescharacteristic
268C2×C4D4×Dic512yes—
269C2×C4#408111yescharacteristic
270C2×C4#408311yescharacteristic
271C2×C4#408512yes—
272C2×C4#409612yes—
273C8—204no—
274C8—402no—
275C5#948611yescharacteristic, complemented, Sylow
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22C2×#396411yescharacteristic
286C22#6550911yescharacteristic
287C22#6552211yescharacteristic
288C4—22no—
289C4—21no—
290C4—21no—
291C4—24no—
292C4—42no—
293C4—82no—
294C4—81no—
295C4#6532712yes—
296C4#6533012yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#1060247411yescharacteristic
301C2#1060250711yescharacteristic
302C2#1060265711yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#2536C2
socleC22×C5#386
prime radicalC22×C5×D4C2×C4
cosocleC2×C42×C5C23
hypercenter#2536D5
hypocenterC5#9486
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. #10614166C2semidirect
3. #10602430C2semidirect
5. #10598122C2semidirect
275. C5#9486semidirect

Maximal subgroups

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

subgroupindex
C5×#25362
#106141662
#106024302
#106023912
#105981222
#94865