Finite Group Databasebeta

← #12074206

Subgroups — #12074206

Order 1280.

870 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 207 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×#3502C211yesmaximal, characteristic
2#10615055C212yesmaximal
3#10614210C211yesmaximal, characteristic
4#10602692C218yesmaximal
5#10602628C212yesmaximal
6#10602417C211yesmaximal, characteristic
7C5×Q82—22no—
8C5×#557—22no—
9C5×#557—24no—
10C5×#544—21no—
11C5×#544C2214yes—
12C5×#533—22no—
13C5×#533C2212yes—
14C5×#519—21no—
15C2×C4×C5×Q8C2211yescharacteristic
16C2×#4015C2211yescharacteristic
17#65519—21no—
18#65370—22no—
19#65367—22no—
20#65333C2218yes—
21#65310—24no—
22#65275C2214yes—
23#65266—21no—
24C4×C4⋊Dic5C2211yescharacteristic
25#65218—22no—
26#65214C2212yes—
27#64758C2218yes—
28#64748C2214yes—
29#18411—51nomaximal, Sylow
30C22×C5×Q8D411yescharacteristic
31C22×Dic10—21no—
32C5×C4⋊Q8—22no—
33C5×C4⋊Q8—22no—
34C5×C4⋊Q8—24no—
35C5×C4⋊Q8—28no—
36C5×C4⋊Q8D412yes—
37C5×C4⋊Q8D412yes—
38C5×#125—24no—
39C5×#125—24no—
40C5×#125—28no—
41C5×#125—22no—
42C5×C22⋊Q8—22no—
43C5×C22⋊Q8—24no—
44C5×C22⋊Q8C2314yes—
45C4×C5×Q8—24no—
46C4×C5×Q8—24no—
47C4×C5×Q8—24no—
48C4×C5×Q8—24no—
49C4×C5×Q8—22no—
50C4×C5×Q8—22no—
51C4×C5×Q8—22no—
52C4×C5×Q8D414yes—
53C5×#117—24no—
54C5×#117—22no—
55C2×C5×C4⋊C4D411yescharacteristic
56C2×C5×C4⋊C4C2312yes—
57C2×C42×C5D412yes—
58C2×C42×C5C2311yescharacteristic
59C2×C4⋊Dic5—21no—
60C2×C4⋊Dic5—21no—
61C2×C4⋊Dic5C2312yes—
62C2×C4×Dic5—21no—
63#4015—22no—
64#4015—22no—
65#3967—28no—
66#3963—24no—
67#3963—24no—
68#3963C2312yes—
69C22⋊#168C2314yes—
70#3939—28no—
71C4⋊#168—22no—
72C4×#168—22no—
73#3502D511yescharacteristic
74#3230—52no—
75#2568—51no—
76#1617—58no—
77#1503—52no—
78#1461—51no—
79C2×C5×Q8—24no—
80C2×C5×Q8—22no—
81C2×C5×Q8—22no—
82C2×C5×Q8—22no—
83C2×C5×Q8Q1614yes—
84C2×C5×Q8C2×D414yes—
85C22×C4×C5C2×D411yescharacteristic
86C22×C4×C5C2×D412yes—
87C22×C4×C5C2×D412yes—
88C22×C4×C5C2×D411yescharacteristic
89C22×C4×C5C2411yescharacteristic
90C22×Dic5—22no—
91C22×Dic5—22no—
92C2×Dic10—44no—
93C2×Dic10—41no—
94C2×Dic10—41no—
95C5×C4⋊C4—24no—
96C5×C4⋊C4—24no—
97C5×C4⋊C4—24no—
98C5×C4⋊C4—28no—
99C5×C4⋊C4—22no—
100C5×C4⋊C4—22no—
101C5×C4⋊C4—22no—
102C5×C4⋊C4—22no—
103C5×C4⋊C4—24no—
104C5×C4⋊C4—24no—
105C5×C4⋊C4—24no—
106C5×C4⋊C4C2×D414yes—
107C5×C22⋊C4—24no—
108C42×C5—28no—
109C42×C5—22no—
110C42×C5—22no—
111C42×C5—24no—
112C42×C5C2×D412yes—
113C42×C5C2×D412yes—
114C4⋊Dic5—24no—
115C4⋊Dic5—42no—
116C4⋊Dic5—42no—
117C4×Dic5—42no—
118C2×#168—24no—
119Q82—22no—
120#557—22no—
121#557—24no—
122#544—21no—
123#544C2×D514yes—
124#533—22no—
125#533C2×D512yes—
126C2×C4⋊Q8—51no—
127#519—21no—
128C2×C4×Q8C2×D511yescharacteristic
129#500—102no—
130#487—102no—
131#484—58no—
132#477—104no—
133#431—52no—
134#400—54no—
135#391—101no—
136#391—102no—
137#382—101no—
138C4×C4⋊C4—51no—
139#328—58no—
140#324—54no—
141C23×C5C22×D411yescharacteristic
142C5×Q8—28no—
143C5×Q8—44no—
144C5×Q8—44no—
145C5×Q8—84no—
146C2×C4×C5—22no—
147C2×C4×C5—22no—
148C2×C4×C5—22no—
149C2×C4×C5—24no—
150C2×C4×C5—24no—
151C2×C4×C5—22no—
152C2×C4×C5—41no—
153C2×C4×C5—42no—
154C2×C4×C5#12011yescharacteristic
155C2×C4×C5#12012yes—
156C2×C4×C5#12011yescharacteristic
157C2×C4×C5C2×Q1614yes—
158C2×C4×C5C2×Q1612yes—
159C2×C4×C5#13712yes—
160C2×C4×C5C22×D412yes—
161C2×Dic5—42no—
162C2×Dic5—44no—
163C2×Dic5—42no—
164C2×Dic5—42no—
165C2×Dic5—42no—
166Dic10—82no—
167Dic10—82no—
168#168—44no—
169C22×Q8—101no—
170C22×Q8C22⋊D511yescharacteristic
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—24no—
174C4⋊Q8—28no—
175C4⋊Q8—102no—
176C4⋊Q8—102no—
177C4⋊Q8C22⋊D512yes—
178C4⋊Q8C22⋊D512yes—
179#125—24no—
180#125—24no—
181#125—28no—
182#125—22no—
183C22⋊Q8—22no—
184C22⋊Q8—24no—
185C22⋊Q8C22×D514yes—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8C22⋊D514yes—
194#117—24no—
195#117—22no—
196C2×C4⋊C4—52no—
197C2×C4⋊C4—101no—
198C2×C4⋊C4—101no—
199C2×C4⋊C4C22⋊D511yescharacteristic
200C2×C4⋊C4C22×D512yes—
201C2×C42—101no—
202C2×C42C22⋊D512yes—
203C2×C42C22×D511yescharacteristic
204#107—108no—
205C4⋊C8—102no—
206#103—108no—
207C22⋊C8—54no—
208C4×C8—102no—
209#95—52no—
210#95—104no—
211#95—104no—
212C22×C5—21no—
213C22×C5—21no—
214C22×C5C2×#12011yescharacteristic
215C22×C5C22×Q1611yescharacteristic
216C22×C5C2×#13711yescharacteristic
217C4×C5—24no—
218C4×C5—21no—
219C4×C5—22no—
220C4×C5—44no—
221C4×C5—42no—
222C4×C5—44no—
223C4×C5#45114yes—
224C4×C5#45412yes—
225Dic5—82no—
226Dic5—82no—
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—204no—
232C2×Q8—201no—
233C2×Q8—201no—
234C2×Q8#68914yes—
235C2×Q8C2×C22⋊D514yes—
236C22×C4—102no—
237C22×C4—102no—
238C22×C4C2×C22⋊D511yescharacteristic
239C22×C4C2×C22⋊D512yes—
240C22×C4C2×C22⋊D512yes—
241C22×C4C2×C22⋊D511yescharacteristic
242C22×C4C23×D511yescharacteristic
243C2×C8—104no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—28no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—104no—
256C4⋊C4—202no—
257C4⋊C4—202no—
258C4⋊C4C2×C22⋊D514yes—
259C22⋊C4—24no—
260C42—28no—
261C42—22no—
262C42—22no—
263C42—24no—
264C42—202no—
265C42C2×C22⋊D512yes—
266C42C2×C22⋊D512yes—
267C2×C5—21no—
268C2×C5—21no—
269C2×C5C2×#45111yescharacteristic
270C2×C5C2×#45411yescharacteristic
271C2×C5#302811yescharacteristic
272C23C22×C22⋊D511yescharacteristic
273Q8—28no—
274Q8—44no—
275Q8—44no—
276Q8—84no—
277Q8—402no—
278Q8—402no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—202no—
288C2×C4—204no—
289C2×C4—202no—
290C2×C4—202no—
291C2×C4—202no—
292C2×C4#408812yes—
293C2×C4C2×#68914yes—
294C2×C4C2×#68912yes—
295C2×C4#409911yescharacteristic
296C2×C4#409912yes—
297C2×C4#409911yescharacteristic
298C2×C4C22×C22⋊D512yes—
299C8—204no—
300C5#1841111yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22C2×#408811yescharacteristic
304C22C22×#68911yescharacteristic
305C22C2×#409911yescharacteristic
306C4—21no—
307C4—22no—
308C4—24no—
309C4—44no—
310C4—42no—
311C4—44no—
312C4—402no—
313C4—402no—
314C4#6551714yes—
315C4#6553412yes—
316C2—21no—
317C2—21no—
318C2C2×#6551711yescharacteristic
319C2C2×#6553411yescharacteristic
320C2#1061585511yescharacteristic
subgroupG/H
centerC22C2×#4099
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#3502C2
socleC22×C5C2×#120
prime radicalC23×C5C22×D4
hypercenter#3502D5
hypocenterC5#18411
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
300. C5#18411semidirect

Maximal subgroups

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

subgroupindex
C5×#35022
#106150552
#106142102
#106026922
#106026282
#106024172
#184115