Finite Group Databasebeta

← #12017627

Subgroups — #12017627

Order 1280.

472 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2546C211yesmaximal, characteristic
2C2×#64768C214yesmaximal
3#10602430C211yesmaximal, characteristic
4C4×#3968C211yesmaximal, characteristic
5C2×C5×C4⋊Q8C411yescharacteristic
6C2×C5×C4⋊Q8C2212yes—
7C2×C4×C5×Q8C2211yescharacteristic
8C5×#398—21no—
9C5×#398—21no—
10C5×#391C412yes—
11C5×#389—21no—
12C5×#389—21no—
13C4×C5×C4⋊C4C411yescharacteristic
14C2×#3968C2212yes—
15C2×#3968C2212yes—
16#65213—21no—
17C4×#681—21no—
18#64768—28no—
19#64750—21no—
20#64750—21no—
21#13227—51nomaximal, Sylow
22C22×C5×Q8C2×C412yes—
23C22×C5×Q8C2311yescharacteristic
24C5×C4⋊Q8—24no—
25C5×C4⋊Q8—24no—
26C5×C4⋊Q8—41no—
27C5×C4⋊Q8—41no—
28C4×C5×Q8—41no—
29C4×C5×Q8—41no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—22no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C4⋊C4D411yescharacteristic
38C2×C42×C5C2×C412yes—
39C2×C42×C5C2×C411yescharacteristic
40C2×C42×C5D411yescharacteristic
41C2×C42×C5D411yescharacteristic
42C2×#681—22no—
43C2×#681—22no—
44C5×#95—21no—
45C5×#95—22no—
46C5×#95—22no—
47#3968—22no—
48#3968—24no—
49#3968D414yes—
50C4⋊#168—41no—
51C4⋊#168—41no—
52#3935—41no—
53C4×#168—41no—
54#2546D511yescharacteristic
55C2×#356—54no—
56#1801—51no—
57C4×#101—51no—
58C2×C5×Q8—24no—
59C2×C5×Q8—41no—
60C2×C5×Q8—41no—
61C2×C5×Q8—42no—
62C2×C5×Q8—42no—
63C2×C5×Q8C2×D412yes—
64C2×C5×Q8#4112yes—
65C22×C4×C5—22no—
66C22×C4×C5—22no—
67C22×C4×C5C22⋊C411yescharacteristic
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22×C411yescharacteristic
72C22×C4×C5C2×D411yescharacteristic
73C22×C4×C5C2×D411yescharacteristic
74C5×C4⋊C4—24no—
75C5×C4⋊C4—41no—
76C5×C4⋊C4—41no—
77C5×C4⋊C4—41no—
78C5×C4⋊C4—42no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—44no—
82C5×C4⋊C4—42no—
83C5×C4⋊C4—42no—
84C42×C5—21no—
85C42×C5—21no—
86C42×C5—21no—
87C42×C5—21no—
88C42×C5—21no—
89C42×C5—21no—
90C42×C5—24no—
91#681—24no—
92#681—42no—
93#681—42no—
94#681—42no—
95C2×#168—42no—
96C2×#168—42no—
97C2×C4⋊Q8Dic511yescharacteristic
98C2×C4⋊Q8C2×D512yes—
99C2×C4×Q8C2×D511yescharacteristic
100#423—101no—
101C2×#101—52no—
102C2×#101—52no—
103C4×M16—101no—
104#398—21no—
105#398—21no—
106#391Dic512yes—
107#389—21no—
108#389—21no—
109C4×C4⋊C4Dic511yescharacteristic
110#356—108no—
111#343—101no—
112#343—101no—
113C23×C5C2×C22⋊C411yescharacteristic
114C5×Q8—42no—
115C5×Q8—42no—
116C5×Q8—84no—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—22no—
124C2×C4×C5—24no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—42no—
129C2×C4×C5—42no—
130C2×C4×C5—41no—
131C2×C4×C5—41no—
132C2×C4×C5—41no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5#9911yescharacteristic
137C2×C4×C5#9911yescharacteristic
138C2×C4×C5C4×D412yes—
139C2×C4×C5#12112yes—
140C2×C4×C5#12411yescharacteristic
141C2×C4×C5#12711yescharacteristic
142#168—44no—
143#168—82no—
144C22×Q8C2×Dic512yes—
145C22×Q8C22×D511yescharacteristic
146C2×M16—102no—
147C2×M16—102no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C2×Dic511yescharacteristic
160C2×C4⋊C4C22⋊D511yescharacteristic
161C2×C4⋊C4C22⋊D511yescharacteristic
162C2×C42C2×Dic512yes—
163C2×C42C2×Dic511yescharacteristic
164C2×C42C22⋊D511yescharacteristic
165C2×C42C22⋊D511yescharacteristic
166C4⋊C8—201no—
167C4⋊C8—201no—
168#101—54no—
169#101—102no—
170#101—104no—
171#97—201no—
172C4×C8—201no—
173#95—21no—
174#95—22no—
175#95—22no—
176C22×C5—21no—
177C22×C5—21no—
178C22×C5#39011yescharacteristic
179C22×C5C2×#9911yescharacteristic
180C22×C5#41011yescharacteristic
181C4×C5—21no—
182C4×C5—22no—
183C4×C5—21no—
184C4×C5—22no—
185C4×C5—21no—
186C4×C5—44no—
187C4×C5—41no—
188C4×C5—41no—
189C4×C5—41no—
190C4×C5—82no—
191C4×C5#35612yescomplemented
192C2×Q8—24no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8D4×D512yes—
198C2×Q8#71112yes—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4C22⋊Dic511yescharacteristic
202C22×C4C22⋊Dic511yescharacteristic
203C22×C4C22⋊Dic511yescharacteristic
204C22×C4C22⋊Dic511yescharacteristic
205C22×C4C22×Dic511yescharacteristic
206C22×C4C2×C22⋊D511yescharacteristic
207C22×C4C2×C22⋊D511yescharacteristic
208M16—104no—
209M16—202no—
210M16—202no—
211M16—202no—
212C2×C8—202no—
213C2×C8—202no—
214C4⋊C4—24no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—24no—
231C2×C5—21no—
232C2×C5—21no—
233C2×C5#196411yescharacteristic
234C2×C5C2×#35611yescharacteristic
235C2×C5#213511yescharacteristic
236C23C2×C22⋊Dic511yescharacteristic
237Q8—42no—
238Q8—42no—
239Q8—84no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4#396611yescharacteristic
260C2×C4#396611yescharacteristic
261C2×C4D4×Dic512yes—
262C2×C4#408211yescharacteristic
263C2×C4#408412yes—
264C2×C4#408611yescharacteristic
265C8—204no—
266C8—402no—
267C5#1322711yescharacteristic, complemented, Sylow
268C22—21no—
269C22—21no—
270C22C2×#396611yescharacteristic
271C22#6551011yescharacteristic
272C22#6552911yescharacteristic
273C4—21no—
274C4—22no—
275C4—21no—
276C4—22no—
277C4—21no—
278C4—44no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—82no—
283C4#6476812yescomplemented
284C2—21no—
285C2—21no—
286C2#1060247911yescharacteristic
287C2C2×#6476811yescharacteristic
288C2#1060266611yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC22×Q8C22×D5
FittingC5×#2546C2
socleC22×C5C2×#99
prime radicalC23×C5C2×C22⋊C4
cosocleC22×C5×Q8C23
hypercenter#2546D5
hypocenterC5#13227
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
191. C4×C5#356semidirect
267. C5#13227semidirect
283. C4#64768semidirect

Maximal subgroups

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

subgroupindex
C5×#25462
C2×#647682
#106024302
C4×#39682
#132275