Finite Group Databasebeta

← #13016209

Subgroups — #13016209

Order 1280.

686 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65757C211yesmaximal, characteristic
2C22×#4004C211yesmaximal, characteristic
3#10613955C211yesmaximal, characteristic
4#10598725C214yesmaximal, complemented
5C23×#170—21no—
6C2×C5×#117C411yescharacteristic
7C2×#4072C411yescharacteristic
8C22×C4×Dic5C2211yescharacteristic
9C2×C22⋊#170—21no—
10#65757—22no—
11#65757C414yescomplemented
12C2×#4004—22no—
13C2×#4004—21no—
14C2×#4004C414yescomplemented
15C2×#4004C2212yes—
16C2×#4004C2212yes—
17C2×#4002—21no—
18#65266C412yes—
19#65265C2212yes—
20#65231—21no—
21#65231—21no—
22C4×#683—21no—
23C4×#683—21no—
24#8044—51nomaximal, Sylow
25C23×C4×C5C2×C411yescharacteristic
26C23×Dic5D411yescharacteristic
27C23×Dic5D411yescharacteristic
28C22×#170—24no—
29C22×#170—22no—
30C22×#170—22no—
31C22×#170—44no—
32C5×#117—42no—
33C2×C5×C4⋊C4C2×C412yes—
34C2×C5×C22⋊C4—21no—
35C2×C42×C5—21no—
36C2×C22⋊Dic5—21no—
37#4072—42no—
38C2×C4⋊Dic5C2×C412yes—
39C2×#683—22no—
40C2×#683—22no—
41C2×C4×Dic5—21no—
42C2×C4×Dic5C2×C411yescharacteristic
43C2×C4×Dic5C2×C412yes—
44C2×C4×Dic5C2×C412yes—
45C2×C4×Dic5D411yescharacteristic
46C2×C4×Dic5D411yescharacteristic
47C2×C4×Dic5D412yes—
48C2×C4×Dic5Q812yes—
49C2×C4×Dic5C2311yescharacteristic
50C22⋊#170—24no—
51#4004—24no—
52#4004—22no—
53#4004—22no—
54#4004C2×C414yes—
55#4004C2×C414yes—
56#4002—24no—
57#3963—22no—
58#3963—22no—
59C2×#432—51no—
60C22×C4⋊C8—51no—
61#2285—51no—
62#1296—54no—
63C24×C5C22⋊C411yescharacteristic
64C22×C4×C5—22no—
65C22×C4×C5—22no—
66C22×C4×C5—22no—
67C22×C4×C5C4212yes—
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C4⋊C412yes—
70C22×C4×C5C22×C411yescharacteristic
71C22×Dic5—24no—
72C22×Dic5—22no—
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5C22⋊C411yescharacteristic
76C22×Dic5C22⋊C411yescharacteristic
77C22×Dic5C22⋊C412yes—
78C22×Dic5C22⋊C412yes—
79C22×Dic5C4⋊C412yes—
80C22×Dic5C4⋊C412yes—
81C22×Dic5C2×D411yescharacteristic
82C22×Dic5C2×D411yescharacteristic
83C2×#170—24no—
84C2×#170—24no—
85C2×#170—48no—
86C2×#170—44no—
87C2×#170—44no—
88C5×C4⋊C4—42no—
89C5×C22⋊C4—42no—
90C42×C5—42no—
91C22⋊Dic5—42no—
92C4⋊Dic5—42no—
93#683—44no—
94#683—42no—
95#683—42no—
96C4×Dic5—42no—
97C4×Dic5C4212yes—
98C4×Dic5C22⋊C412yes—
99C4×Dic5C22⋊C412yes—
100C4×Dic5C4⋊C412yes—
101C4×Dic5C4⋊C412yes—
102C4×Dic5C4⋊C412yes—
103C4×Dic5C22×C411yescharacteristic
104C4×Dic5C22×C411yescharacteristic
105C4×Dic5C2×D411yescharacteristic
106C4×Dic5C2×Q811yescharacteristic
107C23×C8—101no—
108C2×#117—51no—
109C2×#117AGL(1,𝔽5)11yescharacteristic
110C22×C42—51no—
111#432—54no—
112#432—102no—
113C2×C4⋊C8—52no—
114C2×C4⋊C8—52no—
115C2×C4⋊C8—54no—
116C2×C4⋊C8—102no—
117C2×C4⋊C8—101no—
118C2×C22⋊C8—101no—
119C2×#97—101no—
120#384—52no—
121#382—52no—
122#380—101no—
123#380—101no—
124C4×C4⋊C4—101no—
125C4×C4⋊C4—101no—
126C23×C5—22no—
127C23×C5—22no—
128C23×C5#9512yes—
129C23×C5#10012yes—
130C23×C5#10112yes—
131C23×C5C2×C22⋊C411yescharacteristic
132C2×C4×C5—24no—
133C2×C4×C5—24no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—42no—
141C2×C4×C5#9512yes—
142C2×C4×C5C2×C4211yescharacteristic
143C2×C4×C5C2×C4⋊C411yescharacteristic
144C2×Dic5—22no—
145C2×Dic5—22no—
146C2×Dic5—24no—
147C2×Dic5—22no—
148C2×Dic5—22no—
149C2×Dic5—24no—
150C2×Dic5—24no—
151C2×Dic5—42no—
152C2×Dic5—42no—
153C2×Dic5—42no—
154C2×Dic5#9512yes—
155C2×Dic5#9512yes—
156C2×Dic5C2×C22⋊C411yescharacteristic
157C2×Dic5C2×C22⋊C411yescharacteristic
158C2×Dic5C2×C4⋊C411yescharacteristic
159C2×Dic5C2×C4⋊C411yescharacteristic
160C2×Dic5#13514yes—
161C2×Dic5AGL(1,ℤ/8)12yes—
162C2×Dic5#13712yes—
163#170—44no—
164#170—44no—
165C23×C4—51no—
166C23×C4—51no—
167C23×C4C2×AGL(1,𝔽5)11yescharacteristic
168C22×C8—102no—
169C22×C8—104no—
170C22×C8—102no—
171C22×C8—204no—
172#117—42no—
173#117—202no—
174C2×C4⋊C4—52no—
175C2×C4⋊C4—102no—
176C2×C4⋊C4—102no—
177C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—101no—
180C2×C42—21no—
181C2×C42—52no—
182C2×C42—52no—
183C2×C42—52no—
184C2×C42—52no—
185C2×C42—51no—
186C2×C42—51no—
187C2×C42—51no—
188C2×C42—51no—
189C2×C42—101no—
190C4⋊C8—54no—
191C4⋊C8—54no—
192C4⋊C8—102no—
193C4⋊C8—104no—
194C4⋊C8—102no—
195C22⋊C8—104no—
196#97—104no—
197#95—102no—
198#95—102no—
199C22×C5—24no—
200C22×C5—24no—
201C22×C5—22no—
202C22×C5—22no—
203C22×C5#34314yes—
204C22×C5C2×#9511yescharacteristic
205C22×C5C2×#10011yescharacteristic
206C22×C5C2×#10111yescharacteristic
207C22×C5#41612yes—
208C22×C5#41712yes—
209C4×C5—22no—
210C4×C5—21no—
211C4×C5—21no—
212C4×C5—82no—
213Dic5—22no—
214Dic5—22no—
215Dic5—82no—
216Dic5#41612yes—
217Dic5#41811yescharacteristic
218Dic5#41911yescharacteristic
219Dic5#42712yes—
220Dic5#42812yes—
221C24C22⋊AGL(1,𝔽5)11yescharacteristic
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—51no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—51no—
229C22×C4—52no—
230C22×C4—52no—
231C22×C4—52no—
232C22×C4—52no—
233C22×C4—102no—
234C22×C4—102no—
235C22×C4—104no—
236C22×C4—102no—
237C22×C4C4×AGL(1,𝔽5)12yes—
238C22×C4C4⋊AGL(1,𝔽5)12yes—
239C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
240C22×C4C22×AGL(1,𝔽5)11yescharacteristic
241C2×C8—104no—
242C2×C8—104no—
243C2×C8—204no—
244C2×C8—204no—
245C2×C8—208no—
246C4⋊C4—42no—
247C4⋊C4—202no—
248C4⋊C4—202no—
249C4⋊C4—202no—
250C4⋊C4—204no—
251C22⋊C4—42no—
252C22⋊C4—202no—
253C42—42no—
254C42—51no—
255C42—51no—
256C42—52no—
257C42—51no—
258C42—52no—
259C42—51no—
260C42—52no—
261C42—52no—
262C42—52no—
263C42—52no—
264C42—202no—
265C2×C5—24no—
266C2×C5#155112yes—
267C2×C5#155212yes—
268C2×C5#173511yescharacteristic
269C2×C5#173611yescharacteristic
270C2×C5C2×#34311yescharacteristic
271C23—22no—
272C23—22no—
273C23#400512yes—
274C23(C2×AGL(1,𝔽5))⋊C412yes—
275C23#401312yes—
276C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—52no—
287C2×C4—52no—
288C2×C4—54no—
289C2×C4—51no—
290C2×C4—51no—
291C2×C4—51no—
292C2×C4—51no—
293C2×C4—52no—
294C2×C4—52no—
295C2×C4—102no—
296C2×C4—104no—
297C2×C4—104no—
298C2×C4—102no—
299C2×C4—104no—
300C2×C4—102no—
301C2×C4—102no—
302C2×C4—202no—
303C2×C4—202no—
304C2×C4—202no—
305C2×C4(C2×AGL(1,𝔽5))⋊C412yes—
306C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
307C2×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
308C8—204no—
309C8—204no—
310C5#804411yescharacteristic, complemented, Sylow
311C22—24no—
312C22—22no—
313C22—22no—
314C22—24no—
315C22#6491914yes—
316C22C2×#400511yescharacteristic
317C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
318C22#6579312yes—
319C22#6579412yes—
320C22C2×#401311yescharacteristic
321C4—21no—
322C4—21no—
323C4—22no—
324C4—52no—
325C4—52no—
326C4—51no—
327C4—51no—
328C4—52no—
329C4—82no—
330C4—102no—
331C4—102no—
332C4—402no—
333C2—24no—
334C2#1060436412yes—
335C2#1060437811yescharacteristic
336C2#1060437911yescharacteristic
337C2C2×#6491911yescharacteristic
338C2#1060500512yes—
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC2×C5×#117C4
socleC23×C5C2×C22⋊C4
prime radicalC24×C5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenterC2×#117AGL(1,𝔽5)
hypocenterC5#8044
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
4. #10598725C2semidirect
11. #65757C4semidirect
14. C2×#4004C4semidirect
310. C5#8044semidirect

Maximal subgroups

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

subgroupindex
C2×#657572
C22×#40042
#106139552
#105987252
#80445