Finite Group Databasebeta

← #13043461

Subgroups — #13043461

Order 1280.

600 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65411C211yesmaximal, characteristic
2#10604872C211yesmaximal, characteristic
3C2×C22⋊#674C211yesmaximal, characteristic
4#10604747C214yesmaximal, complemented
5#10604746C212yesmaximal, complemented
6#10604741C212yesmaximal, complemented
7#10604710C212yesmaximal, complemented
8#10604709C212yesmaximal, complemented
9C22×C5×M16C411yescharacteristic
10C2×#4072C411yescharacteristic
11C23×#168C2211yescharacteristic
12C2×#3998C2212yes—
13C22×#674C2212yes—
14#65411C411yescharacteristic
15#65411C412yes—
16#65411C411yescharacteristic
17#65411C2211yescharacteristic
18#65411C2211yescharacteristic
19#65411C2212yes—
20C2×#3947C412yes—
21C2×C8×Dic5C2212yes—
22C22⋊#674C2211yescharacteristic
23C22⋊#674C2211yescharacteristic
24C22⋊#674C2212yes—
25C22⋊#674C2212yes—
26C22⋊#674C2212yes—
27#64888C2214yescomplemented
28#64888C2214yescomplemented
29#64886C2214yescomplemented
30#64885C2214yescomplemented
31#33422—51nomaximal, Sylow
32C23×C4×C5C2×C411yescharacteristic
33C2×C5×M16—24no—
34C2×C5×M16C2×C411yescharacteristic
35C2×C5×M16C2×C412yes—
36C2×C5×M16C2×C411yescharacteristic
37C22×C5×C8C2×C412yes—
38C2×C22⋊Dic5C812yes—
39#4072C812yes—
40#4072C811yescharacteristic
41#4072C811yescharacteristic
42#4072C2×C412yes—
43#4072C2×C411yescharacteristic
44#4072C2×C411yescharacteristic
45C2×C4⋊Dic5C811yescharacteristic
46C2×C4⋊Dic5C811yescharacteristic
47C2×C4×Dic5C2×C412yes—
48C22×#168—24no—
49C22×#168C2×C411yescharacteristic
50C22×#168C2×C411yescharacteristic
51C22×#168C2×C411yescharacteristic
52C22×#168C2311yescharacteristic
53C22×#168C2311yescharacteristic
54C22×#168C2311yescharacteristic
55#3998—24no—
56C2×#674—24no—
57C2×#674C2312yes—
58C2×#674C2314yes—
59C2×#674C2312yes—
60#3947C2×C412yes—
61#3947C2×C412yes—
62#3947C2×C412yes—
63#3947C2×C412yes—
64C8×Dic5C2×C412yes—
65C8×Dic5C2×C412yes—
66C8×Dic5C2312yes—
67C8×Dic5C2312yes—
68C2×#405—51no—
69#2164—54no—
70#2163—52no—
71#2162—52no—
72#2151—52no—
73#2112—51no—
74C2×C22⋊C16—51no—
75#2108—52no—
76C24×C5C2×C811yescharacteristic
77C22×C4×C5—22no—
78C22×C4×C5—22no—
79C22×C4×C5C2×C811yescharacteristic
80C22×C4×C5C2×C811yescharacteristic
81C22×C4×C5C2×C811yescharacteristic
82C22×C4×C5C22×C411yescharacteristic
83C22×C4×C5C22×C411yescharacteristic
84C22×C4×C5C22×C411yescharacteristic
85C22×Dic5C2×C812yes—
86C22×Dic5C2×C812yes—
87C5×M16—22no—
88C5×M16—24no—
89C5×M16M1614yes—
90C2×C5×C8—24no—
91C2×C5×C8C22×C412yes—
92C2×C5×C8C22×C412yes—
93C22⋊Dic5C2×C812yes—
94C22⋊Dic5C2×C812yes—
95C22⋊Dic5C2×C812yes—
96C22⋊Dic5C2×C812yes—
97C4⋊Dic5C2×C812yes—
98C4⋊Dic5C2×C812yes—
99C4⋊Dic5C2×C811yescharacteristic
100C4⋊Dic5C2×C811yescharacteristic
101C4⋊Dic5C2×C811yescharacteristic
102C4⋊Dic5C2×C811yescharacteristic
103C4×Dic5C2×C812yes—
104C4×Dic5C2×C812yes—
105C4×Dic5C22×C412yes—
106C4×Dic5C22×C412yes—
107C2×#168—22no—
108C2×#168—22no—
109C2×#168—22no—
110C2×#168—22no—
111C2×#168—22no—
112C2×#168C22×C411yescharacteristic
113C2×#168C22×C411yescharacteristic
114C2×#168C22×C411yescharacteristic
115C2×#168#4114yes—
116C2×#168C2411yescharacteristic
117#674—22no—
118#674—22no—
119#674—24no—
120C22×M16AGL(1,𝔽5)11yescharacteristic
121C23×C8—51no—
122C2×#117—51no—
123C2×M32—52no—
124C22×C16—52no—
125#405—51no—
126#405—51no—
127#405—52no—
128#405—51no—
129#405—52no—
130#405—51no—
131C2×#97—52no—
132C2×C4×C8—52no—
133C4⋊C16—54no—
134C4⋊C16—54no—
135C22⋊C16—51no—
136C22⋊C16—51no—
137C22⋊C16—52no—
138C22⋊C16—52no—
139C22⋊C16—52no—
140#346—54no—
141C4×C16—54no—
142C23×C5—22no—
143C23×C5—22no—
144C23×C5—41no—
145C23×C5C22×C811yescharacteristic
146C23×C5C22×C811yescharacteristic
147C23×C5C22×C811yescharacteristic
148C2×C4×C5—22no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—22no—
154C2×C4×C5—22no—
155C2×C4×C5#11712yes—
156C2×C4×C5C22×C811yescharacteristic
157C2×C4×C5C22×C811yescharacteristic
158C2×C4×C5C22×C811yescharacteristic
159C2×C4×C5C2×M1612yes—
160C2×C4×C5C23×C411yescharacteristic
161C2×Dic5—24no—
162C2×Dic5—42no—
163C2×Dic5C22×C812yes—
164C2×Dic5C22×C812yes—
165C2×Dic5C22×C812yes—
166C2×Dic5C22×C812yes—
167C5×C8—22no—
168C5×C8C2×M1614yes—
169#168—21no—
170#168—21no—
171#168#11712yes—
172#168C2×#4112yes—
173C23×C4C2×AGL(1,𝔽5)11yescharacteristic
174C2×M16—24no—
175C2×M16C2×AGL(1,𝔽5)11yescharacteristic
176C2×M16C2×AGL(1,𝔽5)12yes—
177C2×M16C2×AGL(1,𝔽5)11yescharacteristic
178C22×C8—51no—
179C22×C8—51no—
180C22×C8—51no—
181C22×C8—51no—
182C22×C8—51no—
183C22×C8—51no—
184C22×C8—104no—
185C22×C8C2×AGL(1,𝔽5)12yes—
186#117—51no—
187#117—51no—
188#117—52no—
189#117—52no—
190#117—51no—
191#117—51no—
192C2×C4⋊C4—51no—
193C2×C4⋊C4—51no—
194C2×C22⋊C4—52no—
195C2×C42—52no—
196M32—104no—
197C2×C16—52no—
198C2×C16—54no—
199C2×C16—52no—
200C2×C16—104no—
201#97—52no—
202#97—52no—
203#97—52no—
204#97—52no—
205C4×C8—52no—
206C4×C8—52no—
207C4×C8—52no—
208C4×C8—52no—
209C22×C5—21no—
210C22×C5—21no—
211C22×C5—21no—
212C22×C5—21no—
213C22×C5—21no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—44no—
217C22×C5#43112yes—
218C22×C5#50412yes—
219C22×C5C23×C811yescharacteristic
220C4×C5—21no—
221C4×C5—21no—
222C4×C5#43112yes—
223C4×C5C2×#11711yescharacteristic
224C4×C5C22×M1611yescharacteristic
225Dic5—22no—
226Dic5—22no—
227Dic5—22no—
228Dic5—22no—
229C24D5⋊C811yescharacteristic
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—52no—
233C22×C4—52no—
234C22×C4D5⋊C811yescharacteristic
235C22×C4D5⋊C811yescharacteristic
236C22×C4D5⋊C811yescharacteristic
237C22×C4C22×AGL(1,𝔽5)11yescharacteristic
238C22×C4C22×AGL(1,𝔽5)11yescharacteristic
239C22×C4C22×AGL(1,𝔽5)11yescharacteristic
240M16—22no—
241M16—24no—
242M16#70414yes—
243C2×C8—24no—
244C2×C8—54no—
245C2×C8—51no—
246C2×C8—51no—
247C2×C8—51no—
248C2×C8—51no—
249C2×C8—102no—
250C2×C8—102no—
251C2×C8—102no—
252C2×C8—102no—
253C2×C8—102no—
254C2×C8C22×AGL(1,𝔽5)12yes—
255C2×C8C22×AGL(1,𝔽5)12yes—
256C4⋊C4—52no—
257C4⋊C4—52no—
258C4⋊C4—51no—
259C4⋊C4—51no—
260C4⋊C4—51no—
261C4⋊C4—51no—
262C22⋊C4—52no—
263C22⋊C4—52no—
264C22⋊C4—52no—
265C22⋊C4—52no—
266C42—52no—
267C42—52no—
268C42—52no—
269C42—52no—
270C16—102no—
271C16—102no—
272C16—104no—
273C2×C5—22no—
274C2×C5—22no—
275C2×C5—41no—
276C2×C5C2×#43111yescharacteristic
277C2×C5C2×#50411yescharacteristic
278C2×C5#340711yescharacteristic
279C23—22no—
280C23—22no—
281C23—41no—
282C23C2×D5⋊C811yescharacteristic
283C23C2×D5⋊C811yescharacteristic
284C23C2×D5⋊C811yescharacteristic
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—52no—
293C2×C4—52no—
294C2×C4—52no—
295C2×C4—52no—
296C2×C4—104no—
297C2×C4—202no—
298C2×C4C2×D5⋊C811yescharacteristic
299C2×C4C2×D5⋊C811yescharacteristic
300C2×C4C2×D5⋊C811yescharacteristic
301C2×C4#413012yes—
302C2×C4C2×#70412yes—
303C2×C4C23×AGL(1,𝔽5)11yescharacteristic
304C8—22no—
305C8—52no—
306C8—52no—
307C8—101no—
308C8—101no—
309C8C2×#70414yes—
310C5#3342211yescharacteristic, complemented, Sylow
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22#6572512yes—
320C22#6575412yes—
321C22C22×D5⋊C811yescharacteristic
322C4—21no—
323C4—21no—
324C4—102no—
325C4—102no—
326C4—102no—
327C4—102no—
328C4#6575412yes—
329C4C2×#413011yescharacteristic
330C4C22×#70411yescharacteristic
331C2—22no—
332C2—22no—
333C2—41no—
334C2C2×#6572511yescharacteristic
335C2#1061740411yescharacteristic
336C2C2×#6575411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×AGL(1,𝔽5)
commutator, socleC22×C5C23×C8
FittingC22×C5×M16C4
prime radicalC24×C5C2×C8
cosocleC2×#168C24
hypercenterC22×M16AGL(1,𝔽5)
hypocenterC5#33422
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
4. #10604747C2semidirect
5. #10604746C2semidirect
6. #10604741C2semidirect
7. #10604710C2semidirect
8. #10604709C2semidirect
27. #64888C22semidirect
28. #64888C22semidirect
29. #64886C22semidirect
30. #64885C22semidirect
310. C5#33422semidirect

Maximal subgroups

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

subgroupindex
C2×#654112
#106048722
C2×C22⋊#6742
#106047472
#106047462
#106047412
#106047102
#106047092
#334225