Finite Group Databasebeta

← #11994339

Subgroups — #11994339

Order 1280.

616 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 147 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×#2316C211yesmaximal, characteristic
2C2×#65275C211yesmaximal, characteristic
3C2×#64748C211yesmaximal, characteristic
4#10600295C212yesmaximal
5#10598115C212yesmaximal
6C2×C5×#117—21no—
7C22×C5×C4⋊C4C2211yescharacteristic
8C22×C4⋊Dic5C2211yescharacteristic
9C5×#389—24no—
10C5×#380—21no—
11C5×#380C2212yes—
12C2×#3963—21no—
13C2×#3963—21no—
14C2×#3963C2211yescharacteristic
15C2×C22⋊#168C2212yes—
16#65275—22no—
17#65275C414yescomplemented
18#64749—24no—
19#64748—22no—
20#9023—51nomaximal, Sylow
21C23×C4×C5D411yescharacteristic
22C23×C4×C5D411yescharacteristic
23C23×C4×C5C2311yescharacteristic
24C23×Dic5—21no—
25C23×Dic5—21no—
26C5×#117—42no—
27C5×#117—42no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4—22no—
32C2×C5×C4⋊C4C2×C412yes—
33C2×C5×C4⋊C4D412yes—
34C2×C5×C4⋊C4D414yes—
35C2×C5×C22⋊C4—22no—
36C2×C42×C5—22no—
37C2×C4⋊Dic5—22no—
38C2×C4⋊Dic5—22no—
39C2×C4⋊Dic5—21no—
40C2×C4⋊Dic5C2×C412yes—
41C22×#168—22no—
42C5×#95—22no—
43C5×#95—22no—
44#3963—24no—
45#3963—24no—
46#3963—21no—
47#3963C2×C412yes—
48C22⋊#168—22no—
49C22⋊#168—22no—
50C2×#400—51no—
51#2316D511yescharacteristic
52#1824—52no—
53C2×#324—51no—
54#1294—52no—
55C24×C5C2×D411yescharacteristic
56C22×C4×C5—21no—
57C22×C4×C5—21no—
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—24no—
63C22×C4×C5—42no—
64C22×C4×C5—42no—
65C22×C4×C5C22⋊C412yes—
66C22×C4×C5C22⋊C412yes—
67C22×C4×C5SD1612yes—
68C22×C4×C5Q1612yes—
69C22×C4×C5C22×C411yescharacteristic
70C22×C4×C5C2×D411yescharacteristic
71C22×C4×C5C2×D412yes—
72C22×C4×C5#4112yes—
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5—42no—
76C22×Dic5—42no—
77C22×Dic5—44no—
78C22×Dic5—41no—
79C22×Dic5—41no—
80C22×Dic5—41no—
81C22×Dic5—41no—
82C5×C4⋊C4—24no—
83C5×C4⋊C4—24no—
84C5×C4⋊C4—44no—
85C5×C4⋊C4—44no—
86C5×C4⋊C4—82no—
87C5×C22⋊C4—42no—
88C5×C22⋊C4—42no—
89C42×C5—44no—
90C4⋊Dic5—24no—
91C4⋊Dic5—42no—
92C2×#168—44no—
93C2×#168—42no—
94C2×#117—21no—
95C22×C4⋊C4—51no—
96C22×C4⋊C4C2×D511yescharacteristic
97C2×C22⋊C8—52no—
98#400—54no—
99#400—102no—
100#389—24no—
101#380—21no—
102#380C2×D512yes—
103C2×#95—51no—
104C2×#95—101no—
105C2×#95—101no—
106#340—104no—
107#324—102no—
108C23×C5—22no—
109C23×C5—22no—
110C23×C5C2×C22⋊C411yescharacteristic
111C23×C5#12011yescharacteristic
112C23×C5#12311yescharacteristic
113C23×C5C2×SD1611yescharacteristic
114C23×C5C2×Q1611yescharacteristic
115C23×C5AGL(1,ℤ/8)11yescharacteristic
116C23×C5#13711yescharacteristic
117C2×C4×C5—22no—
118C2×C4×C5—22no—
119C2×C4×C5—22no—
120C2×C4×C5—22no—
121C2×C4×C5—24no—
122C2×C4×C5—22no—
123C2×C4×C5—22no—
124C2×C4×C5—22no—
125C2×C4×C5—41no—
126C2×C4×C5—41no—
127C2×C4×C5—42no—
128C2×C4×C5—42no—
129C2×C4×C5—42no—
130C2×C4×C5—44no—
131C2×C4×C5—44no—
132C2×C4×C5—44no—
133C2×C4×C5—81no—
134C2×C4×C5#10314yes—
135C2×C4×C5C4×D412yes—
136C2×C4×C5#12112yes—
137C2×Dic5—42no—
138C2×Dic5—42no—
139C2×Dic5—44no—
140C2×Dic5—42no—
141C2×Dic5—42no—
142C2×Dic5—82no—
143C2×Dic5—82no—
144C2×Dic5—82no—
145C2×Dic5—82no—
146#168—82no—
147C23×C4—101no—
148C23×C4—101no—
149C23×C4C22⋊D511yescharacteristic
150C23×C4C22⋊D511yescharacteristic
151C23×C4C22×D511yescharacteristic
152C22×C8—102no—
153#117—42no—
154#117—42no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—52no—
160C2×C4⋊C4—102no—
161C2×C4⋊C4—102no—
162C2×C4⋊C4—101no—
163C2×C4⋊C4C4×D512yes—
164C2×C4⋊C4D2012yes—
165C2×C4⋊C4C22⋊D514yes—
166C2×C22⋊C4—22no—
167C2×C42—22no—
168C22⋊C8—102no—
169C22⋊C8—102no—
170#95—22no—
171#95—22no—
172#95—52no—
173#95—104no—
174#95—104no—
175#95—101no—
176C22×C5—22no—
177C22×C5—22no—
178C22×C5—22no—
179C22×C5—22no—
180C22×C5—21no—
181C22×C5—21no—
182C22×C5—21no—
183C22×C5—21no—
184C22×C5#38611yescharacteristic
185C22×C5C2×#10311yescharacteristic
186C22×C5#41711yescharacteristic
187C22×C5#45011yescharacteristic
188C22×C5#45111yescharacteristic
189C22×C5#45412yes—
190C22×C5#45812yes—
191C22×C5#48311yescharacteristic
192C22×C5#48411yescharacteristic
193C4×C5—24no—
194C4×C5—22no—
195C4×C5—22no—
196C4×C5—42no—
197C4×C5—84no—
198Dic5—82no—
199Dic5—82no—
200C24C2×C22⋊D511yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—102no—
211C22×C4—102no—
212C22×C4—202no—
213C22×C4—202no—
214C22×C4—204no—
215C22×C4—201no—
216C22×C4—201no—
217C22×C4—201no—
218C22×C4—201no—
219C22×C4#68512yes—
220C22×C4#68512yes—
221C22×C4#68712yes—
222C22×C4#68912yes—
223C22×C4C2×C4×D511yescharacteristic
224C22×C4C2×D2011yescharacteristic
225C22×C4#70912yes—
226C22×C4C2×C22⋊D512yes—
227C2×C8—204no—
228C2×C8—202no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—82no—
234C4⋊C4—104no—
235C4⋊C4—202no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C42—44no—
239C2×C5—21no—
240C2×C5—21no—
241C2×C5—21no—
242C2×C5—21no—
243C2×C5#161612yes—
244C2×C5#161712yes—
245C2×C5#187211yescharacteristic
246C2×C5#188811yescharacteristic
247C2×C5#189011yescharacteristic
248C23—22no—
249C23—22no—
250C23C2×#68511yescharacteristic
251C23#407511yescharacteristic
252C23#407811yescharacteristic
253C23C2×#68711yescharacteristic
254C23#408811yescharacteristic
255C23C2×#68911yescharacteristic
256C23#409911yescharacteristic
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—81no—
274C2×C4—202no—
275C2×C4—202no—
276C2×C4—204no—
277C2×C4—202no—
278C2×C4—202no—
279C2×C4—402no—
280C2×C4—402no—
281C2×C4—402no—
282C2×C4—402no—
283C2×C4#394214yes—
284C2×C4C4×C22⋊D512yes—
285C2×C4#407612yes—
286C8—402no—
287C5#902311yescharacteristic, complemented, Sylow
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22#6525511yescharacteristic
297C22C2×#394211yescharacteristic
298C22#6525911yescharacteristic
299C22#6526011yescharacteristic
300C22#6550211yescharacteristic
301C22#6550711yescharacteristic
302C22#6551711yescharacteristic
303C22#6553212yes—
304C22#6553412yes—
305C4—24no—
306C4—22no—
307C4—22no—
308C4—42no—
309C4—84no—
310C4—402no—
311C4—402no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#1060235211yescharacteristic
317C2#1060244811yescharacteristic
318C2#1060245411yescharacteristic
319C2#1060266812yes—
320C2#1060269212yes—
subgroupG/H
centerC23#4099
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2316C2
socleC23×C5#120
prime radicalC24×C5C2×D4
cosocleC23×C4×C5C23
hypercenter#2316D5
hypocenterC5#9023
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
17. #65275C4semidirect
287. C5#9023semidirect

Maximal subgroups

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

subgroupindex
C5×#23162
C2×#652752
C2×#647482
#106002952
#105981152
#90235