Finite Group Databasebeta

← #13034645

Subgroups — #13034645

Order 1280.

1114 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65708C214yesmaximal
2C2×#65681C211yesmaximal, characteristic
3C2×D5⋊(C4⋊C8)C211yesmaximal, characteristic
4C4×#4038C211yesmaximal, characteristic
5#10604191C212yesmaximal, complemented
6#10604190C212yesmaximal, complemented
7#10603784C214yesmaximal, complemented
8C22×#700C2212yes—
9C22×D5⋊C8C2212yes—
10C2×#4038C2212yes—
11C2×C42×D5C2211yescharacteristic
12C2×#4004C2212yes—
13C2×#4004C2218yes—
14C2×C4⋊#170C2212yes—
15C2×C4⋊#170C2218yes—
16C2×C4⋊#170C2211yescharacteristic
17C2×C4⋊#170C2212yes—
18C2×C4⋊#170C2211yescharacteristic
19#65708—216no—
20#65681—24no—
21D5⋊(C4⋊C8)—24no—
22C4×C5×C4⋊C4C411yescharacteristic
23C4⋊C4×Dic5C412yes—
24C4×#685C412yes—
25C4×C4⋊Dic5C411yescharacteristic
26C42×Dic5C2212yes—
27#64955C412yes—
28#64937C2212yes—
29#20793—51nomaximal, Sylow
30C22×C4×D5C2×C411yescharacteristic
31C22×C4×D5C2×C411yescharacteristic
32C22×C4×D5C2311yescharacteristic
33C22×#170C2312yes—
34C22×#170C2312yes—
35C22×#170C2314yes—
36C2×#700—24no—
37C2×#700—28no—
38C2×D5⋊C8—24no—
39C2×D5⋊C8—28no—
40C2×C5×C4⋊C4C2×C412yes—
41C2×C42×C5C2×C412yes—
42C2×C42×C5C2×C411yescharacteristic
43C2×#685C2×C414yes—
44C2×C4⋊Dic5C2×C412yes—
45C2×C4×Dic5C2×C412yes—
46C2×C4×Dic5C2×C414yes—
47C2×C4×Dic5C2×C411yescharacteristic
48C2×C4×Dic5C2311yescharacteristic
49C2×C4×Dic5C2311yescharacteristic
50C2×C4×Dic5C2314yes—
51#4038D418yes—
52#4038Q818yes—
53C42×D5D414yes—
54C42×D5Q814yes—
55#4004—24no—
56#4004—216no—
57C4⋊#170—22no—
58C4⋊#170—22no—
59C4⋊#170—24no—
60C4⋊#170—24no—
61C4⋊#170—216no—
62C5×#95C2×C412yes—
63#3963C2×C412yes—
64#3963C2×C414yes—
65C2×#424—54no—
66C2×#423—51no—
67C22×C4⋊C8—51no—
68C4×#117—51no—
69#1843—54no—
70#1842—52no—
71#1842—52no—
72C23×D5C22×C411yescharacteristic
73C22×C4×C5C22×C414yes—
74C22×C4×C5C22×C411yescharacteristic
75C22×C4×C5C22×C411yescharacteristic
76C22×C4×C5C22×C411yescharacteristic
77C22×Dic5C22×C411yescharacteristic
78C22×Dic5C22×C411yescharacteristic
79C22×Dic5C22×C414yes—
80C22×Dic5C2411yescharacteristic
81C2×C4×D5—22no—
82C2×C4×D5—22no—
83C2×C4×D5—42no—
84C2×C4×D5C4⋊C414yes—
85C2×C4×D5C4⋊C414yes—
86C2×C4×D5C2×D412yes—
87C2×C4×D5C2×D414yes—
88C2×C4×D5C2×Q812yes—
89C2×C4×D5C2×Q814yes—
90C2×#170—24no—
91C2×#170—22no—
92C2×#170—22no—
93C2×#170—24no—
94C2×#170—24no—
95C2×#170—28no—
96#700—48no—
97D5⋊C8—48no—
98C5×C4⋊C4C4⋊C418yes—
99C42×C5—24no—
100C42×C5C4⋊C414yes—
101#685C4⋊C4116yes—
102C4⋊Dic5C4⋊C418yes—
103C4×Dic5—24no—
104C4×Dic5—44no—
105C4×Dic5C4⋊C414yes—
106C4×Dic5C2×D412yes—
107C4×Dic5C2×D412yes—
108C4×Dic5C2×D418yes—
109C4×Dic5C2×Q812yes—
110C4×Dic5C2×Q812yes—
111C4×Dic5C2×Q818yes—
112C22×M16—52no—
113C23×C8—52no—
114C2×#117—52no—
115C22×C42—51no—
116#424—1016no—
117#423—104no—
118C2×C4⋊C8—51no—
119C2×C4⋊C8—51no—
120C2×C4⋊C8—52no—
121C2×C4⋊C8—58no—
122C2×C4⋊C8—58no—
123C2×C4⋊C8—52no—
124C2×C4⋊C8—52no—
125C2×C4⋊C8—104no—
126C4×C4⋊C4—52no—
127C4×C4⋊C4—51no—
128C4×C4⋊C4AGL(1,𝔽5)11yescharacteristic
129C4×C22⋊C4—52no—
130C4×C22⋊C4—52no—
131#376—52no—
132C43—52no—
133C23×C5C23×C411yescharacteristic
134C22×D5—21no—
135C22×D5—21no—
136C22×D5—22no—
137C22×D5C2×C4⋊C414yes—
138C22×D5C2×C4⋊C412yes—
139C2×C4×C5—24no—
140C2×C4×C5—24no—
141C2×C4×C5—22no—
142C2×C4×C5C2×C4⋊C418yes—
143C2×C4×C5C2×C4⋊C412yes—
144C2×C4×C5C2×C4⋊C412yes—
145C2×C4×C5C2×C4⋊C414yes—
146C2×C4×C5C2×C4⋊C412yes—
147C2×C4×C5#13112yes—
148C2×C4×C5#13112yes—
149C2×Dic5—24no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—22no—
153C2×Dic5—24no—
154C2×Dic5C2×C4⋊C412yes—
155C2×Dic5C2×C4⋊C412yes—
156C2×Dic5C2×C4⋊C418yes—
157C2×Dic5C22×D411yescharacteristic
158C2×Dic5C22×D412yes—
159C2×Dic5C22×Q811yescharacteristic
160C2×Dic5C22×Q812yes—
161C4×D5—28no—
162C4×D5—44no—
163C4×D5#12714yes—
164C4×D5C4⋊Q814yes—
165C4×D5C4⋊Q818yes—
166#170—42no—
167#170—42no—
168#170—44no—
169C23×C4—51no—
170C23×C4—51no—
171C23×C4—51no—
172C2×M16—104no—
173C2×M16—108no—
174C22×C8—52no—
175C22×C8—52no—
176C22×C8—54no—
177C22×C8—104no—
178C22×C8—108no—
179#117—58no—
180#117—58no—
181C2×C4⋊C4—52no—
182C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
183C2×C22⋊C4—54no—
184C2×C42—54no—
185C2×C42—51no—
186C2×C42—51no—
187C2×C42—51no—
188C2×C42—54no—
189C2×C42—54no—
190C2×C42—52no—
191C2×C42—54no—
192C2×C42C2×AGL(1,𝔽5)12yes—
193C2×C42C2×AGL(1,𝔽5)11yescharacteristic
194C4⋊C8—102no—
195C4⋊C8—102no—
196C4⋊C8—104no—
197C4⋊C8—104no—
198C4⋊C8—1016no—
199C4⋊C8—104no—
200C4⋊C8—1016no—
201#95—54no—
202#95—52no—
203#95C2×AGL(1,𝔽5)12yes—
204C22×C5C22×C4⋊C411yescharacteristic
205C22×C5C22×C4⋊C412yes—
206C22×C5C2×#13111yescharacteristic
207C22×C5C2×#13111yescharacteristic
208C22×C5#56812yes—
209C2×D5—22no—
210C2×D5—22no—
211C2×D5—24no—
212C2×D5—24no—
213C2×D5#38414yes—
214C4×C5—22no—
215C4×C5—28no—
216C4×C5#38414yes—
217C4×C5#42414yescomplemented
218Dic5—22no—
219Dic5—22no—
220Dic5—28no—
221Dic5C2×#12711yescharacteristic
222Dic5C2×C4⋊Q811yescharacteristic
223Dic5C2×C4⋊Q812yes—
224C24—51no—
225C22×C4—51no—
226C22×C4—52no—
227C22×C4—52no—
228C22×C4—54no—
229C22×C4—54no—
230C22×C4—51no—
231C22×C4—54no—
232C22×C4—51no—
233C22×C4—54no—
234C22×C4—54no—
235C22×C4—102no—
236C22×C4—102no—
237C22×C4—202no—
238C22×C4C22×AGL(1,𝔽5)11yescharacteristic
239C22×C4C22×AGL(1,𝔽5)14yes—
240C22×C4C22×AGL(1,𝔽5)11yescharacteristic
241C22×C4C22×AGL(1,𝔽5)11yescharacteristic
242M16—208no—
243C2×C8—102no—
244C2×C8—102no—
245C2×C8—104no—
246C2×C8—104no—
247C2×C8—104no—
248C2×C8—108no—
249C2×C8—208no—
250C4⋊C4—58no—
251C4⋊C4C4⋊AGL(1,𝔽5)18yes—
252C22⋊C4—516no—
253C42—24no—
254C42—52no—
255C42—52no—
256C42—58no—
257C42—54no—
258C42—52no—
259C42—52no—
260C42—58no—
261C42—104no—
262C42—204no—
263C42C4⋊AGL(1,𝔽5)14yes—
264C2×C5C2×#38411yescharacteristic
265C2×C5C2×#42411yescharacteristic
266C2×C5#290511yescharacteristic
267C2×C5#290612yes—
268C2×C5#290612yes—
269D5—24no—
270C23—54no—
271C23—52no—
272C23—101no—
273C23—101no—
274C23—102no—
275C23C23×AGL(1,𝔽5)11yescharacteristic
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—51no—
280C2×C4—52no—
281C2×C4—51no—
282C2×C4—52no—
283C2×C4—54no—
284C2×C4—54no—
285C2×C4—58no—
286C2×C4—52no—
287C2×C4—52no—
288C2×C4—58no—
289C2×C4—101no—
290C2×C4—104no—
291C2×C4—101no—
292C2×C4—102no—
293C2×C4—104no—
294C2×C4—108no—
295C2×C4—204no—
296C2×C4C2×C4⋊AGL(1,𝔽5)12yes—
297C2×C4C2×C4⋊AGL(1,𝔽5)18yes—
298C2×C4C2×C4⋊AGL(1,𝔽5)12yes—
299C2×C4C2×C4⋊AGL(1,𝔽5)14yes—
300C2×C4C2×C4⋊AGL(1,𝔽5)12yes—
301C2×C4#413112yes—
302C2×C4#413312yes—
303C8—202no—
304C8—202no—
305C8—204no—
306C5#2079311yescharacteristic, complemented, Sylow
307C22—54no—
308C22—102no—
309C22—104no—
310C22—104no—
311C22—102no—
312C22C22×C4⋊AGL(1,𝔽5)11yescharacteristic
313C22C22×C4⋊AGL(1,𝔽5)12yes—
314C22C2×#413111yescharacteristic
315C22C2×#413311yescharacteristic
316C22#6626412yes—
317C4—22no—
318C4—28no—
319C4—51no—
320C4—51no—
321C4—52no—
322C4—102no—
323C4—108no—
324C4—102no—
325C4#6568814yes—
326C4#6570814yescomplemented
327C2—104no—
328C2C2×#6568811yescharacteristic
329C2C2×#6570811yescharacteristic
330C2#1061710811yescharacteristic
331C2#1061713712yes—
332C2#1061718512yes—
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutator, socleC23×C5C23×C4
FittingC4×C5×C4⋊C4C4
prime radicalC23×D5C22×C4
cosocleC22×Dic5C24
hypercenterC4×C4⋊C4AGL(1,𝔽5)
hypocenterC5#20793
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #10604191C2semidirect
6. #10604190C2semidirect
7. #10603784C2semidirect
217. C4×C5#424semidirect
306. C5#20793semidirect
326. C4#65708semidirect

Maximal subgroups

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

subgroupindex
C2×#657082
C2×#656812
C2×D5⋊(C4⋊C8)2
C4×#40382
#106041912
#106041902
#106037842
#207935