Finite Group Databasebeta

← #12003698

Subgroups — #12003698

Order 1280.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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#10610348C211yesmaximal, characteristic
2C5×#1850C211yesmaximal, characteristic
3#10602765C212yesmaximal
4#10600250C212yesmaximal
5#10598167C211yesmaximal, characteristic
6C23×C5×C8C2211yescharacteristic
7C2×#4070C2212yes—
8C2×C5×C4⋊C8—21no—
9C2×C5×C22⋊C8C2211yescharacteristic
10C4×C5×C22⋊C4C2211yescharacteristic
11#65501C2212yes—
12#65405—21no—
13#65405—21no—
14C2×#3948—22no—
15#65028—22no—
16#65011—22no—
17C5×#336—21no—
18#64771—22no—
19#64747—22no—
20#64747—22no—
21#10566—51nomaximal, Sylow
22C23×C4×C5C2311yescharacteristic
23C22×Dic10D412yes—
24C22×C5×C8—22no—
25C22×C5×C8—21no—
26C22×C5×C8—42no—
27C22×C5×C8D411yescharacteristic
28C22×C5×C8D411yescharacteristic
29C2×C5×C22⋊C4D411yescharacteristic
30C2×C5×C22⋊C4D411yescharacteristic
31C2×C42×C5—21no—
32C2×C22⋊Dic5—22no—
33C2×C22⋊Dic5—22no—
34C2×C4⋊Dic5—22no—
35C2×C4⋊Dic5D412yes—
36#4070—22no—
37#4070—22no—
38#4070—22no—
39#4070—22no—
40C2×#683—22no—
41C2×Dic20—21no—
42C2×Dic20—21no—
43C5×C4⋊C8—42no—
44C5×C22⋊C8—22no—
45C5×#95—21no—
46#3963—22no—
47#3950—21no—
48#3950—21no—
49#3948—22no—
50#3948—22no—
51#3948—44no—
52#3189—51no—
53#1996—52no—
54#1892—52no—
55#1850D511yescharacteristic
56#1289—51no—
57C24×C5C2×D411yescharacteristic
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—41no—
61C22×C4×C5—41no—
62C22×C4×C5C2×D412yes—
63C22×C4×C5C2×D411yescharacteristic
64C22×C4×C5#4111yescharacteristic
65C22×C4×C5#4112yes—
66C22×Dic5—22no—
67C22×Dic5—22no—
68C22×Dic5—42no—
69C2×Dic10—22no—
70C2×Dic10—22no—
71C2×Dic10—42no—
72C2×Dic10—42no—
73C2×C5×C8—24no—
74C2×C5×C8—22no—
75C2×C5×C8—21no—
76C2×C5×C8—21no—
77C2×C5×C8—42no—
78C2×C5×C8—41no—
79C2×C5×C8—41no—
80C2×C5×C8—41no—
81C2×C5×C8—41no—
82C5×C22⋊C4—21no—
83C5×C22⋊C4—22no—
84C5×C22⋊C4Q1612yescomplemented
85C42×C5—41no—
86C42×C5—41no—
87C22⋊Dic5—42no—
88C22⋊Dic5—42no—
89C22⋊Dic5—42no—
90C22⋊Dic5—42no—
91C4⋊Dic5—22no—
92C4⋊Dic5—22no—
93C4⋊Dic5—42no—
94C4⋊Dic5—42no—
95#683—42no—
96#683—42no—
97Dic20—81no—
98Dic20—81no—
99C23×C8C2×D511yescharacteristic
100C2×C22⋊Q8—52no—
101#484—102no—
102#467—101no—
103#467—101no—
104#451—102no—
105C2×C4⋊C8—21no—
106C2×#103—102no—
107C2×C22⋊C8C2×D511yescharacteristic
108#380—52no—
109C4×C22⋊C4C2×D511yescharacteristic
110#342—102no—
111#342—102no—
112#340—102no—
113#336—21no—
114C23×C5—41no—
115C23×C5—41no—
116C23×C5#12112yes—
117C23×C5#12111yescharacteristic
118C23×C5C2×Q1611yescharacteristic
119C23×C5#13511yescharacteristic
120C23×C5#13712yes—
121C2×C4×C5—22no—
122C2×C4×C5—21no—
123C2×C4×C5—22no—
124C2×C4×C5—21no—
125C2×C4×C5—22no—
126C2×C4×C5—41no—
127C2×C4×C5—42no—
128C2×C4×C5—41no—
129C2×C4×C5—41no—
130C2×C4×C5—41no—
131C2×C4×C5—41no—
132C2×C4×C5—41no—
133C2×C4×C5#12011yescharacteristic
134C2×C4×C5#12312yes—
135C2×C4×C5#12411yescharacteristic
136C2×Dic5—42no—
137C2×Dic5—42no—
138C2×Dic5—82no—
139C2×Dic5—82no—
140C2×Dic5—82no—
141C2×Dic5—82no—
142C2×Dic5—82no—
143Dic10—44no—
144Dic10—42no—
145Dic10—42no—
146C5×C8—41no—
147C5×C8—42no—
148C5×C8—41no—
149C22×Q8—52no—
150C23×C4C22×D511yescharacteristic
151C2×Q16—101no—
152C2×Q16—101no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—42no—
156C22×C8C22⋊D511yescharacteristic
157C22×C8C22⋊D511yescharacteristic
158C22⋊Q8—102no—
159C22⋊Q8—102no—
160C22⋊Q8—102no—
161C22⋊Q8—102no—
162C2×C4⋊C4—52no—
163C2×C4⋊C4—102no—
164C2×C4⋊C4—102no—
165C2×C22⋊C4—102no—
166C2×C22⋊C4—102no—
167C2×C22⋊C4D2011yescharacteristic
168C2×C22⋊C4D2011yescharacteristic
169C2×C42—21no—
170#107—101no—
171#107—101no—
172C4⋊C8—42no—
173#103—102no—
174#103—102no—
175#103—204no—
176C22⋊C8—22no—
177#95—21no—
178#95—102no—
179C22×C5—22no—
180C22×C5—21no—
181C22×C5—22no—
182C22×C5—21no—
183C22×C5—41no—
184C22×C5—41no—
185C22×C5—42no—
186C22×C5#39411yescharacteristic
187C22×C5#46212yes—
188C22×C5#46412yes—
189C22×C5#46711yescharacteristic
190C22×C5#47011yescharacteristic
191C4×C5—22no—
192C4×C5—21no—
193C4×C5—42no—
194C4×C5—81no—
195C4×C5B(4,𝔽2)11yescharacteristic
196C4×C5#45811yescharacteristic
197Dic5—82no—
198Dic5—82no—
199Dic5—162no—
200C24C2×D2011yescharacteristic
201C2×Q8—102no—
202C2×Q8—102no—
203C2×Q8—202no—
204C2×Q8—202no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—102no—
210C22×C4—102no—
211C22×C4—202no—
212C22×C4#70911yescharacteristic
213C22×C4D4×D512yes—
214C22×C4#71312yes—
215C22×C4C2×C22⋊D511yescharacteristic
216Q16—401no—
217Q16—401no—
218C2×C8—24no—
219C2×C8—22no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C4⋊C4—102no—
228C4⋊C4—102no—
229C4⋊C4—202no—
230C4⋊C4—202no—
231C4⋊C4—202no—
232C4⋊C4—202no—
233C22⋊C4—21no—
234C22⋊C4—22no—
235C22⋊C4—202no—
236C22⋊C4—202no—
237C22⋊C4—202no—
238C22⋊C4—202no—
239C22⋊C4Dic2012yescomplemented
240C42—41no—
241C42—41no—
242C2×C5—22no—
243C2×C5—21no—
244C2×C5—21no—
245C2×C5—22no—
246C2×C5#204411yescharacteristic
247C2×C5#205311yescharacteristic
248C2×C5#205611yescharacteristic
249C23—41no—
250C23—41no—
251C23#404112yes—
252C23#405011yescharacteristic
253C23C2×Dic2011yescharacteristic
254C23#405512yes—
255C23#407611yescharacteristic
256Q8—204no—
257Q8—202no—
258Q8—202no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—202no—
272C2×C4—202no—
273C2×C4—402no—
274C2×C4—402no—
275C2×C4—402no—
276C2×C4—402no—
277C2×C4—402no—
278C2×C4#404012yes—
279C2×C4#408311yescharacteristic
280C2×C4#409311yescharacteristic
281C8—42no—
282C8—41no—
283C8—41no—
284C5#1056611yescharacteristic, complemented, Sylow
285C22—21no—
286C22—22no—
287C22—21no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22#6513412yes—
293C22#6513812yes—
294C22#6528011yescharacteristic
295C22#6540511yescharacteristic
296C22#6542611yescharacteristic
297C4—22no—
298C4—21no—
299C4—42no—
300C4—81no—
301C4—402no—
302C4—402no—
303C4—802no—
304C4#6532111yescharacteristic
305C4#6534211yescharacteristic
306C2—21no—
307C2—21no—
308C2—22no—
309C2—22no—
310C2#1060038911yescharacteristic
311C2#1060128011yescharacteristic
312C2#1060134011yescharacteristic
subgroupG/H
centerC22#65280
commutator, cosocleC23×C4×C5C23
FrattiniC23×C4C22×D5
FittingC5×#1850C2
socleC22×C5#394
prime radicalC24×C5C2×D4
hypercenter#1850D5
hypocenterC5#10566
LCSG ▷ C23×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
84. C5×C22⋊C4Q16semidirect
239. C22⋊C4Dic20semidirect
284. C5#10566semidirect

Maximal subgroups

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

subgroupindex
#106103482
C5×#18502
#106027652
#106002502
#105981672
#105665