Finite Group Databasebeta

← #13034410

Subgroups — #13034410

Order 1280.

1276 conjugacy classes of subgroups; only the 260 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 260 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#10617757C211yesmaximal, characteristic
2C2×#65771C212yesmaximal
3#10604384C214yesmaximal, complemented
4#10604219C218yesmaximal, complemented
5C22×#711C2211yescharacteristic
6C2×#4130C2214yes—
7C5×#518C411yescharacteristic, complemented
8#66132C411yescharacteristic, complemented
9#65880—24no—
10C4×#711—24no—
11D5×#117C2212yes—
12#65850C412yescomplemented
13#65820C2214yes—
14C2×#4023C414yescomplemented
15C2×#4011C2218yes—
16#65771—24no—
17#65771—24no—
18#64866C22116yes—
19#20710—51nomaximal, Sylow
20C22×C5×D4C2×C411yescharacteristic
21C22×C22⋊D5C2312yes—
22C23×Dic5C2×C412yes—
23C2×#711—24no—
24C2×#711—24no—
25C2×#711D414yes—
26C2×#711D414yes—
27C22×C4×D5C2311yescharacteristic
28C22×Dic10C2×C411yescharacteristic
29C2×C22⋊AGL(1,𝔽5)C2318yes—
30#4130—24no—
31#4130—24no—
32#4130—24no—
33#4130—24no—
34C2×C4⋊AGL(1,𝔽5)—24no—
35C2×C4×AGL(1,𝔽5)—24no—
36C4×C5×D4—24no—
37C5×#117C2×C412yes—
38C2×C5×C22⋊C4C2×C414yes—
39C2×C22⋊Dic5C2×C414yes—
40D4×Dic5—24no—
41C4×C22⋊D5—28no—
42#4072C2×C412yes—
43C2×#683—24no—
44C2×C4×Dic5—24no—
45#4038—22no—
46D5×C4⋊C4—22no—
47#4033—24no—
48#4027—28no—
49D5×C22⋊C4C2314yes—
50#4023—24no—
51#4023—24no—
52#4023C2×C414yes—
53#4023C2×C418yes—
54#4018—22no—
55C42×D5—22no—
56C4×Dic10—24no—
57#4011—216no—
58#3428—51no—
59C2×#410—52no—
60#1828—58no—
61#1740—54no—
62C24×C5C22×C412yes—
63C23×D5C2411yescharacteristic
64C22×AGL(1,𝔽5)—28no—
65C2×C5×D4—24no—
66C2×C5×D4C22⋊C414yes—
67C22×C4×C5—24no—
68C22×C4×C5C22×C411yescharacteristic
69C2×C22⋊D5—28no—
70C2×C22⋊D5C2×D418yes—
71C22×Dic5—24no—
72C22×Dic5—24no—
73C22×Dic5—24no—
74C22×Dic5C22⋊C414yes—
75C22×Dic5C22⋊C414yes—
76C22×Dic5C22×C412yes—
77C22×Dic5C22×C411yescharacteristic
78#711—28no—
79#711—28no—
80#711—28no—
81#711—28no—
82C2×C4×D5—24no—
83C2×C4×D5C2×D412yes—
84C2×C4×D5C2×D412yes—
85C2×C4×D5#4114yes—
86C2×C4×D5#4114yes—
87C2×Dic10—24no—
88C2×Dic10C22⋊C414yes—
89C22⋊AGL(1,𝔽5)—28no—
90C22⋊AGL(1,𝔽5)—28no—
91C4⋊AGL(1,𝔽5)—44no—
92C4⋊AGL(1,𝔽5)—44no—
93C4×AGL(1,𝔽5)—44no—
94C4×AGL(1,𝔽5)—44no—
95C5×C4⋊C4—22no—
96C5×C22⋊C4—24no—
97C5×C22⋊C4C22×C414yes—
98C42×C5—22no—
99C22⋊Dic5—24no—
100C22⋊Dic5C22×C414yes—
101#685—24no—
102C4⋊Dic5—22no—
103#683—28no—
104#683—24no—
105C4×Dic5—22no—
106C4×Dic5—28no—
107C4×Dic5—24no—
108C22×#41—51no—
109#520—104no—
110#519—52no—
111#518—51no—
112#518—54no—
113#518AGL(1,𝔽5)11yescharacteristic, complemented
114C4×#41—104no—
115C2×#117—52no—
116C2×#117—54no—
117C2×#117—54no—
118#410—104no—
119#410—104no—
120C2×#99—58no—
121#342—516no—
122C23×C5—24no—
123C23×C5—82no—
124C23×C5C2×C22⋊C414yes—
125C23×C5C23×C411yescharacteristic
126C22×D5—81no—
127C22×D5C22×D412yes—
128C22×D5C2×#4114yes—
129C2×AGL(1,𝔽5)—48no—
130C2×AGL(1,𝔽5)—88no—
131C5×D4—28no—
132C2×C4×C5—24no—
133C2×C4×C5—84no—
134C2×C4×C5C2×C22⋊C412yes—
135C2×C4×C5#11714yes—
136C22⋊D5—216no—
137C2×Dic5—24no—
138C2×Dic5—24no—
139C2×Dic5—24no—
140C2×Dic5—28no—
141C2×Dic5—24no—
142C2×Dic5—84no—
143C2×Dic5C2×C22⋊C414yes—
144C2×Dic5C2×C22⋊C411yescharacteristic
145C2×Dic5C2×C22⋊C411yescharacteristic
146C2×Dic5#11712yes—
147C2×Dic5#11712yes—
148C4×D5—84no—
149C4×D5#12314yes—
150C4×D5#12314yes—
151C4×D5#12314yes—
152C4×D5#12314yes—
153Dic10—28no—
154C2×#41—54no—
155C2×#41—54no—
156C2×#41—104no—
157C2×#41—104no—
158C22×Q8—51no—
159C22×D4—52no—
160C22×D4C2×AGL(1,𝔽5)11yescharacteristic
161C23×C4—51no—
162C23×C4—52no—
163C4×Q8—104no—
164C4×Q8—104no—
165C4×D4—24no—
166C4×D4—108no—
167C4×D4—108no—
168C4×D4—104no—
169#117—52no—
170#117—54no—
171#117—58no—
172#117—102no—
173#117—104no—
174#117—104no—
175#117—102no—
176#117—104no—
177#117—104no—
178#117—104no—
179#117—104no—
180#117C2×AGL(1,𝔽5)12yes—
181C2×C4⋊C4—102no—
182C2×C4⋊C4—104no—
183C2×C4⋊C4—104no—
184C2×C22⋊C4—54no—
185C2×C22⋊C4—54no—
186C2×C22⋊C4—58no—
187C2×C22⋊C4C2×AGL(1,𝔽5)14yes—
188C2×C42—102no—
189C2×C42—104no—
190C2×C42—104no—
191#99—1016no—
192C22×C5—24no—
193C22×C5—41no—
194C22×C5—44no—
195C22×C5—84no—
196C22×C5C22×C22⋊C411yescharacteristic
197C22×C5C2×#11712yes—
198C2×D5—42no—
199C2×D5—44no—
200C2×D5C2×#12314yes—
201AGL(1,𝔽5)—88no—
202C4×C5—44no—
203C4×C5#38114yes—
204Dic5—24no—
205Dic5—44no—
206Dic5#38111yescharacteristic
207Dic5#38112yes—
208Dic5#38111yescharacteristic
209C24—51no—
210C24C22×AGL(1,𝔽5)12yes—
211#41—108no—
212#41—108no—
213#41—108no—
214#41—108no—
215C2×Q8—54no—
216C2×Q8—104no—
217C2×D4—24no—
218C2×D4—58no—
219C2×D4—108no—
220C2×D4C22⋊AGL(1,𝔽5)14yes—
221C22×C4—24no—
222C22×C4—51no—
223C22×C4—54no—
224C22×C4—54no—
225C22×C4—52no—
226C22×C4—52no—
227C22×C4—54no—
228C22×C4—54no—
229C22×C4—52no—
230C22×C4—104no—
231C22×C4—104no—
232C22×C4—104no—
233C22×C4—104no—
234C22×C4—108no—
235C22×C4C22×AGL(1,𝔽5)11yescharacteristic
236C4⋊C4—22no—
237C4⋊C4—104no—
238C4⋊C4—108no—
239C4⋊C4—102no—
240C4⋊C4—204no—
241C4⋊C4—204no—
242C22⋊C4—24no—
243C22⋊C4—54no—
244C22⋊C4—104no—
245C22⋊C4—104no—
246C22⋊C4—108no—
247C22⋊C4—108no—
248C22⋊C4C22×AGL(1,𝔽5)14yes—
249C42—22no—
250C42—104no—
251C42—108no—
252C42—102no—
253C42—204no—
254C42—204no—
255C2×C5—21no—
256C2×C5—22no—
257C2×C5—44no—
258C2×C5C2×#38111yescharacteristic
259D5—24no—
260C23—24no—
261C23—54no—
262C23—52no—
263C23—82no—
264C23—401no—
265C23C2×C22⋊AGL(1,𝔽5)14yes—
266C23C23×AGL(1,𝔽5)11yescharacteristic
267Q8—108no—
268D4—28no—
269D4—1016no—
270C2×C4—24no—
271C2×C4—52no—
272C2×C4—52no—
273C2×C4—51no—
274C2×C4—51no—
275C2×C4—54no—
276C2×C4—54no—
277C2×C4—54no—
278C2×C4—54no—
279C2×C4—54no—
280C2×C4—84no—
281C2×C4—104no—
282C2×C4—104no—
283C2×C4—108no—
284C2×C4—104no—
285C2×C4—104no—
286C2×C4—208no—
287C2×C4—404no—
288C2×C4—404no—
289C2×C4—408no—
290C2×C4#413014yes—
291C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
292C5#2071011yescharacteristic, complemented, Sylow
293C22—24no—
294C22—41no—
295C22—44no—
296C22—54no—
297C22—84no—
298C22—202no—
299C22—204no—
300C22C2×#413012yes—
301C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
302C4—44no—
303C4—51no—
304C4—51no—
305C4—52no—
306C4—104no—
307C4—204no—
308C4—408no—
309C4#6576414yes—
310C2—21no—
311C2—22no—
312C2—44no—
313C2—104no—
314C2C2×#6576411yescharacteristic
subgroupG/H
centerC2C2×#65764
commutatorC23×C5C23×C4
FrattiniC23C23×AGL(1,𝔽5)
FittingC5×#518C4
socleC2×C5C2×#381
prime radicalC22×#711C22
cosocleC23×D5C24
hypercenter#518AGL(1,𝔽5)
hypocenterC5#20710
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
3. #10604384C2semidirect
4. #10604219C2semidirect
7. C5×#518C4semidirect
8. #66132C4semidirect
12. #65850C4semidirect
14. C2×#4023C4semidirect
113. #518AGL(1,𝔽5)semidirect
292. C5#20710semidirect

Maximal subgroups

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

subgroupindex
#106177572
C2×#657712
#106043842
#106042192
#207105