Finite Group Databasebeta

← #13033962

Subgroups — #13033962

Order 1280.

934 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65754C211yesmaximal, characteristic
2C22×C4⋊#170C211yesmaximal, characteristic
3#10605594C211yesmaximal, characteristic
4#10604315C212yesmaximal, complemented
5C4⋊C4×#170C212yesmaximal, complemented
6#10604226C212yesmaximal
7C22⋊C4×#170C212yesmaximal
8#10603778C214yesmaximal, complemented
9C23×#170C2212yes—
10C22×C4×Dic5C2211yescharacteristic
11C2×C22⋊#170C2211yescharacteristic
12C2×C22⋊#170C2211yescharacteristic
13#65754C418yescomplemented
14C2×#4004C2212yes—
15C2×#4002C2214yes—
16C2×C4⋊#170C418yescomplemented
17C2×C4⋊#170C2212yes—
18C2×C4⋊#170C2211yescharacteristic
19C2×C4⋊#170C2211yescharacteristic
20C2×C4×#170C2214yes—
21C2×C4×#170C2212yes—
22C5×#380C411yescharacteristic
23#65501C411yescharacteristic
24#65266C412yes—
25C4⋊C4×Dic5C2212yes—
26#65232C412yes—
27C22⋊C4×Dic5C2212yes—
28#64940C412yes—
29#64937C2212yes—
30#64918C2214yes—
31#64918C2214yes—
32#20545—51nomaximal, Sylow
33C23×C4×C5C2×C411yescharacteristic
34C23×Dic5C2×C411yescharacteristic
35C23×Dic5C2311yescharacteristic
36C22×#170—44no—
37C22×#170C2×C418yes—
38C22×#170C2312yes—
39C22×#170C2312yes—
40C22×#170C2312yes—
41C22×#170C2312yes—
42C2×C5×C4⋊C4C2×C412yes—
43C2×C5×C22⋊C4C2×C412yes—
44C2×C22⋊Dic5C2×C412yes—
45C2×C4⋊Dic5C2×C412yes—
46C2×C4×Dic5C2×C411yescharacteristic
47C2×C4×Dic5C2×C411yescharacteristic
48C2×C4×Dic5C2×C414yes—
49C2×C4×Dic5C2×C414yes—
50C2×C4×Dic5C2314yes—
51C2×C4×Dic5C2311yescharacteristic
52C2×C4×Dic5C2311yescharacteristic
53C22⋊#170C2×C414yes—
54C22⋊#170C2×C414yes—
55#4004C2×C418yes—
56#4002—44no—
57C4⋊#170C2×C418yes—
58C4⋊#170C2×C414yes—
59C4⋊#170C2×C414yes—
60C4×#170—44no—
61C4×#170C2×C418yes—
62C5×#95C2×C412yes—
63#3963C2×C412yes—
64#3963C2×C414yes—
65#3963C2×C414yes—
66#3963C2×C414yes—
67C2×#431—51no—
68C22×C4⋊C8—51no—
69#2283—51no—
70#1770—52no—
71C8×C4⋊C4—52no—
72#1768—54no—
73#1752—52no—
74C8×C22⋊C4—52no—
75C24×C5C22×C411yescharacteristic
76C22×C4×C5—24no—
77C22×C4×C5C4214yes—
78C22×C4×C5C22×C414yes—
79C22×C4×C5C22×C411yescharacteristic
80C22×C4×C5C22×C411yescharacteristic
81C22×Dic5—22no—
82C22×Dic5—22no—
83C22×Dic5—22no—
84C22×Dic5—22no—
85C22×Dic5C4214yes—
86C22×Dic5C22×C414yes—
87C22×Dic5C22×C411yescharacteristic
88C22×Dic5C22×C411yescharacteristic
89C22×Dic5C22×C411yescharacteristic
90C22×Dic5C22×C412yes—
91C22×Dic5C22×C412yes—
92C22×Dic5C2411yescharacteristic
93C2×#170—24no—
94C2×#170—24no—
95C2×#170—22no—
96C2×#170—22no—
97C2×#170—22no—
98C2×#170—22no—
99C2×#170—48no—
100C2×#170C22×C414yes—
101C2×#170C22×C414yes—
102C2×#170C22×C414yes—
103C2×#170C22×C414yes—
104C5×C4⋊C4—24no—
105C5×C22⋊C4—24no—
106C22⋊Dic5—24no—
107C4⋊Dic5—24no—
108C4×Dic5—24no—
109C4×Dic5—24no—
110C4×Dic5C4214yes—
111C4×Dic5C4214yes—
112C4×Dic5C22×C412yes—
113C4×Dic5C22×C412yes—
114C4×Dic5C22×C412yes—
115C4×Dic5C22×C412yes—
116C23×C8—52no—
117C22×C42—51no—
118#431—58no—
119C2×C4⋊C8—58no—
120C2×C4⋊C8—52no—
121C2×C4⋊C8—51no—
122C2×C4⋊C8—51no—
123C2×C4⋊C8—52no—
124C2×C22⋊C8—51no—
125C2×C22⋊C8—51no—
126C2×#97—54no—
127C2×C4×C8—52no—
128C2×C4×C8—54no—
129#383—52no—
130#382—52no—
131#381—52no—
132#380—51no—
133#380AGL(1,𝔽5)11yescharacteristic
134C4×C4⋊C4—52no—
135C4×C22⋊C4—52no—
136#376—52no—
137#336—54no—
138#336—54no—
139C23×C5—22no—
140C23×C5—22no—
141C23×C5C2×C4212yes—
142C23×C5#13112yes—
143C23×C5#13112yes—
144C23×C5C23×C411yescharacteristic
145C2×C4×C5—22no—
146C2×C4×C5—28no—
147C2×C4×C5—24no—
148C2×C4×C5—24no—
149C2×C4×C5—24no—
150C2×C4×C5C2×C4212yes—
151C2×C4×C5C2×C4212yes—
152C2×C4×C5#13114yes—
153C2×Dic5—22no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—28no—
159C2×Dic5—22no—
160C2×Dic5—22no—
161C2×Dic5—24no—
162C2×Dic5—24no—
163C2×Dic5—24no—
164C2×Dic5—24no—
165C2×Dic5C2×C4212yes—
166C2×Dic5C2×C4212yes—
167C2×Dic5C2×C4212yes—
168C2×Dic5C23×C411yescharacteristic
169C2×Dic5C23×C411yescharacteristic
170C2×Dic5H(5,𝔽2)12yes—
171C2×Dic52−1+412yes—
172#170—24no—
173#170—24no—
174#170—24no—
175#170—24no—
176C23×C4—51no—
177C23×C4—51no—
178C23×C4C2×AGL(1,𝔽5)11yescharacteristic
179C22×C8—52no—
180C22×C8—52no—
181C22×C8—52no—
182C22×C8—52no—
183C22×C8—58no—
184C22×C8—204no—
185C2×C4⋊C4—52no—
186C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
187C2×C22⋊C4—52no—
188C2×C22⋊C4C2×AGL(1,𝔽5)12yes—
189C2×C42—54no—
190C2×C42—54no—
191C2×C42—51no—
192C2×C42—51no—
193C2×C42—51no—
194C2×C42—51no—
195C2×C42—54no—
196C4⋊C8—58no—
197C4⋊C8—54no—
198C4⋊C8—54no—
199C4⋊C8—58no—
200C22⋊C8—54no—
201C22⋊C8—54no—
202#97—204no—
203C4×C8—58no—
204C4×C8—204no—
205#95—52no—
206#95—54no—
207#95—54no—
208#95—54no—
209#95C2×AGL(1,𝔽5)12yes—
210C22×C5—22no—
211C22×C5—24no—
212C22×C5—24no—
213C22×C5—22no—
214C22×C5#40514yes—
215C22×C5C22×C4211yescharacteristic
216C22×C5#52012yes—
217C22×C5C2×#13111yescharacteristic
218C22×C5C2×#13112yes—
219C22×C5C2×#13111yescharacteristic
220C4×C5—22no—
221C4×C5—44no—
222C4×C5#40514yes—
223Dic5—22no—
224Dic5—22no—
225Dic5—22no—
226Dic5—44no—
227Dic5#51811yescharacteristic
228Dic5#51911yescharacteristic
229Dic5#52012yes—
230C24C22×AGL(1,𝔽5)11yescharacteristic
231C22×C4—24no—
232C22×C4—51no—
233C22×C4—52no—
234C22×C4—52no—
235C22×C4—54no—
236C22×C4—51no—
237C22×C4—51no—
238C22×C4—51no—
239C22×C4—54no—
240C22×C4—102no—
241C22×C4—102no—
242C22×C4—102no—
243C22×C4—102no—
244C22×C4C4×AGL(1,𝔽5)14yes—
245C22×C4C22×AGL(1,𝔽5)14yes—
246C22×C4C22×AGL(1,𝔽5)11yescharacteristic
247C22×C4C22×AGL(1,𝔽5)11yescharacteristic
248C2×C8—54no—
249C2×C8—54no—
250C2×C8—54no—
251C2×C8—54no—
252C2×C8—102no—
253C2×C8—104no—
254C2×C8—102no—
255C2×C8—104no—
256C2×C8—102no—
257C2×C8—102no—
258C2×C8—208no—
259C4⋊C4—24no—
260C4⋊C4—104no—
261C22⋊C4—24no—
262C22⋊C4—104no—
263C42—52no—
264C42—52no—
265C42—52no—
266C42—52no—
267C42—54no—
268C42—54no—
269C42—104no—
270C42—104no—
271C2×C5—24no—
272C2×C5#227411yescharacteristic
273C2×C5C2×#40511yescharacteristic
274C2×C5C2×#40511yescharacteristic
275C2×C5#299412yes—
276C2×C5#299412yes—
277C23—22no—
278C23—22no—
279C23C2×C4×AGL(1,𝔽5)12yes—
280C23#413112yes—
281C23#413312yes—
282C23C23×AGL(1,𝔽5)11yescharacteristic
283C2×C4—22no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—51no—
289C2×C4—51no—
290C2×C4—52no—
291C2×C4—52no—
292C2×C4—52no—
293C2×C4—52no—
294C2×C4—52no—
295C2×C4—102no—
296C2×C4—104no—
297C2×C4—102no—
298C2×C4—108no—
299C2×C4—101no—
300C2×C4—101no—
301C2×C4—101no—
302C2×C4—101no—
303C2×C4—102no—
304C2×C4—104no—
305C2×C4—104no—
306C2×C4—104no—
307C2×C4C2×C4×AGL(1,𝔽5)12yes—
308C2×C4C2×C4×AGL(1,𝔽5)12yes—
309C2×C4#413114yes—
310C8—104no—
311C8—104no—
312C8—104no—
313C8—104no—
314C5#2054511yescharacteristic, complemented, Sylow
315C22—22no—
316C22—24no—
317C22—24no—
318C22—22no—
319C22#6570114yes—
320C22C22×C4×AGL(1,𝔽5)11yescharacteristic
321C22C2×#413111yescharacteristic
322C22C2×#413112yes—
323C22C2×#413311yescharacteristic
324C22#6626612yes—
325C4—22no—
326C4—44no—
327C4—51no—
328C4—52no—
329C4—51no—
330C4—102no—
331C4—102no—
332C4—102no—
333C4—204no—
334C4#6569114yes—
335C2—24no—
336C2C2×#6569111yescharacteristic
337C2C2×#6570111yescharacteristic
338C2#1061712711yescharacteristic
339C2#1061724712yes—
340C2#1061727912yes—
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutatorC22×C5C22×C42
FittingC5×#380C4
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC22×Dic5C24
hypercenter#380AGL(1,𝔽5)
hypocenterC5#20545
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
4. #10604315C2semidirect
5. C4⋊C4×#170C2semidirect
8. #10603778C2semidirect
13. #65754C4semidirect
16. C2×C4⋊#170C4semidirect
314. C5#20545semidirect

Maximal subgroups

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

subgroupindex
C2×#657542
C22×C4⋊#1702
#106055942
#106043152
C4⋊C4×#1702
#106042262
C22⋊C4×#1702
#106037782
#205455