Finite Group Databasebeta

← #12108619

Subgroups — #12108619

Order 1280.

1418 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65811C211yesmaximal, characteristic
2C2×C5×#431C211yesmaximal, characteristic, complemented
3#10613793C211yesmaximal, characteristic, complemented
4#10601251C214yesmaximal, complemented
5#10600753C214yesmaximal, complemented
6#10600565C212yesmaximal, complemented
7#10600564C212yesmaximal, complemented
8C22×#709C2212yes—
9C22×C42×C5C2211yescharacteristic
10C2×#4076—21no—
11C2×#4076—21no—
12C2×#4072—21no—
13#65811—24no—
14#65811C418yescomplemented
15C2×#4021C2212yes—
16C2×#4020C2211yescharacteristic
17C2×C4×D20—21no—
18C2×C4×D20—21no—
19C2×#4015C2211yescharacteristic
20C5×#431—24no—
21C2×C5×C4⋊C8C2212yes—
22C2×C5×C22⋊C8C2212yes—
23C2×C4×C5×C8C2212yes—
24#65501—21no—
25#65501C2212yes—
26C2×#3953C2218yes—
27C2×#3948C2218yes—
28#65225—21no—
29#65224C2212yes—
30C4×C4⋊Dic5—21no—
31C4×C4⋊Dic5C2212yes—
32#21805—51nomaximal, Sylow
33C23×C4×C5D411yescharacteristic
34C23×C4×C5D411yescharacteristic
35C23×C4×C5C2311yescharacteristic
36C22×C22⋊D5—22no—
37C2×#709—24no—
38C2×#709—48no—
39C2×#709C2×C418yes—
40C22×D20—21no—
41C22×D20—21no—
42C22×D20C2312yes—
43C22×C4×D5—22no—
44C22×Dic10C2312yes—
45C22×C5×C8C2314yes—
46C2×C42×C5C2×C414yes—
47C2×C42×C5D411yescharacteristic
48C2×C42×C5D411yescharacteristic
49C2×C42×C5D414yes—
50C2×C42×C5C2311yescharacteristic
51C2×C42×C5C2311yescharacteristic
52C2×C22⋊Dic5—22no—
53#4076—28no—
54#4076—28no—
55C2×#685—22no—
56C2×#685—22no—
57#4072—44no—
58C2×C4⋊Dic5—21no—
59C2×C4⋊Dic5—21no—
60C2×C4⋊Dic5—22no—
61C2×C4⋊Dic5—22no—
62C2×C4⋊Dic5C2314yes—
63C2×C4×Dic5—22no—
64#4021—24no—
65#4021C2×C418yes—
66#4020—22no—
67#4020C2×C414yes—
68C4×D20—28no—
69C4×D20—28no—
70#4015—22no—
71#4015C2×C414yes—
72C5×C4⋊C8—24no—
73C5×C22⋊C8—24no—
74C4×C5×C8—24no—
75#3963—22no—
76#3963—22no—
77#3953—48no—
78#3948—48no—
79C2×#532—51no—
80C2×#431D511yescharacteristic, complemented
81#2285—51no—
82#1983—54no—
83#1957—52no—
84#1956—52no—
85#1788—54no—
86C24×C5C2×D411yescharacteristic
87C23×D5—22no—
88C22×C4×C5—24no—
89C22×C4×C5—22no—
90C22×C4×C5—22no—
91C22×C4×C5—24no—
92C22×C4×C5C22⋊C414yes—
93C22×C4×C5C22⋊C414yes—
94C22×C4×C5C22×C412yes—
95C22×C4×C5C2×D411yescharacteristic
96C22×C4×C5C2×D411yescharacteristic
97C22×C4×C5C2×D411yescharacteristic
98C22×C4×C5C2×D411yescharacteristic
99C22×C4×C5C2×D411yescharacteristic
100C22×C4×C5C2×D412yes—
101C22×C4×C5C2411yescharacteristic
102C2×C22⋊D5—28no—
103C2×C22⋊D5—48no—
104C22×Dic5—22no—
105C22×Dic5—24no—
106#709—416no—
107#709—416no—
108C2×D20—28no—
109C2×D20—24no—
110C2×D20—24no—
111C2×D20—22no—
112C2×D20—44no—
113C2×D20—42no—
114C2×D20—42no—
115C2×D20C22×C414yes—
116C2×C4×D5—28no—
117C2×C4×D5—48no—
118C2×Dic10—28no—
119C2×Dic10—22no—
120C2×Dic10C22×C414yes—
121C2×C5×C8—24no—
122C2×C5×C8—24no—
123C2×C5×C8—24no—
124C42×C5C22⋊C414yes—
125C42×C5C22⋊C414yes—
126C42×C5C22×C412yes—
127C42×C5C22×C412yes—
128C42×C5C2×D412yes—
129C42×C5C2×D412yes—
130C22⋊Dic5—44no—
131#685—28no—
132#685—28no—
133C4⋊Dic5—24no—
134C4⋊Dic5—24no—
135C4⋊Dic5—28no—
136C4⋊Dic5—44no—
137C4⋊Dic5—44no—
138C4×Dic5—44no—
139C22×#41—52no—
140#532—58no—
141#532—104no—
142C2×C4⋊Q8—51no—
143C2×#127—51no—
144C2×#124—52no—
145C2×#121—101no—
146C2×#121—101no—
147C2×C4×D4—101no—
148C2×C4×D4—101no—
149C2×#117—101no—
150C22×C42C2×D511yescharacteristic
151#431—24no—
152C2×C4⋊C8C2×D512yes—
153C2×#103—58no—
154C2×#102—58no—
155C2×C22⋊C8C2×D512yes—
156C2×C4×C8C2×D512yes—
157#384—52no—
158#382—101no—
159#380—52no—
160#380—101no—
161C4×C4⋊C4—52no—
162C4×C4⋊C4—101no—
163C23×C5—22no—
164C23×C5—22no—
165C23×C5—21no—
166C23×C5—21no—
167C23×C5C2×C22⋊C412yes—
168C23×C5C22×D411yescharacteristic
169C22×D5—24no—
170C22×D5—44no—
171C22×D5—44no—
172C22×D5—42no—
173C22×D5—42no—
174C2×C4×C5—22no—
175C2×C4×C5—22no—
176C2×C4×C5—21no—
177C2×C4×C5—21no—
178C2×C4×C5—24no—
179C2×C4×C5—24no—
180C2×C4×C5—28no—
181C2×C4×C5—28no—
182C2×C4×C5C2×C22⋊C412yes—
183C2×C4×C5C2×C22⋊C412yes—
184C2×C4×C5C2×C22⋊C412yes—
185C2×C4×C5C2×C22⋊C412yes—
186C2×C4×C5C2×C22⋊C412yes—
187C2×C4×C5#13514yes—
188C2×C4×C5AGL(1,ℤ/8)12yes—
189C2×C4×C5#13712yes—
190C2×C4×C5C23×C411yescharacteristic
191C2×C4×C5C22×D411yescharacteristic
192C2×C4×C5H(5,𝔽2)14yes—
193C22⋊D5—416no—
194C2×Dic5—24no—
195C2×Dic5—44no—
196C2×Dic5—44no—
197C2×Dic5—44no—
198C2×Dic5—44no—
199D20—28no—
200D20—44no—
201D20—48no—
202D20—48no—
203C4×D5—416no—
204Dic10—28no—
205Dic10—44no—
206C5×C8—44no—
207C2×#41—58no—
208C2×#41—104no—
209C2×#41—208no—
210C22×Q8—52no—
211C22×D4—52no—
212C22×D4—102no—
213C22×D4—101no—
214C22×D4—101no—
215C23×C4—102no—
216C23×C4D2011yescharacteristic
217C23×C4D2011yescharacteristic
218C23×C4C22×D511yescharacteristic
219C22×C8C22×D514yes—
220C4⋊Q8—54no—
221C4⋊Q8—102no—
222#127—54no—
223#127—102no—
224#124—58no—
225#124—104no—
226#121—108no—
227#121—108no—
228C4×D4—108no—
229C4×D4—108no—
230#117—204no—
231C2×C4⋊C4—54no—
232C2×C4⋊C4—101no—
233C2×C4⋊C4—101no—
234C2×C4⋊C4—102no—
235C2×C4⋊C4—102no—
236C2×C22⋊C4—102no—
237C2×C22⋊C4—102no—
238C2×C22⋊C4—102no—
239C2×C42—102no—
240C2×C42C4×D514yes—
241C2×C42D2011yescharacteristic
242C2×C42D2011yescharacteristic
243C2×C42C22⋊D514yes—
244C2×C42C22×D511yescharacteristic
245C2×C42C22×D511yescharacteristic
246C4⋊C8—24no—
247#103—208no—
248#102—208no—
249C22⋊C8—24no—
250C4×C8—24no—
251#95—102no—
252#95—102no—
253C22×C5—22no—
254C22×C5—22no—
255C22×C5—22no—
256C22×C5—42no—
257C22×C5—42no—
258C22×C5C22×C22⋊C411yescharacteristic
259C22×C5#53512yes—
260C22×C5C2×#13512yes—
261C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C5C2×#13711yescharacteristic
263C2×D5—44no—
264C2×D5—44no—
265C2×D5—44no—
266C2×D5—84no—
267C2×D5—84no—
268C4×C5—22no—
269C4×C5—22no—
270C4×C5—22no—
271C4×C5#41614yes—
272C4×C5#41714yes—
273C4×C5#51812yes—
274C4×C5#53512yes—
275Dic5—44no—
276Dic5—84no—
277C24—102no—
278C24C2×D2011yescharacteristic
279#41—2016no—
280#41—2016no—
281C2×Q8—54no—
282C2×Q8—102no—
283C2×Q8—108no—
284C2×D4—54no—
285C2×D4—108no—
286C2×D4—108no—
287C2×D4—102no—
288C2×D4—104no—
289C2×D4—104no—
290C2×D4—208no—
291C2×D4—204no—
292C2×D4—202no—
293C2×D4—202no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—102no—
299C22×C4—108no—
300C22×C4—104no—
301C22×C4—208no—
302C22×C4#68514yes—
303C22×C4#68514yes—
304C22×C4C2×C4×D512yes—
305C22×C4C2×D2011yescharacteristic
306C22×C4C2×D2011yescharacteristic
307C22×C4C2×D2011yescharacteristic
308C22×C4C2×D2011yescharacteristic
309C22×C4C2×D2011yescharacteristic
310C22×C4C2×C22⋊D512yes—
311C22×C4C23×D511yescharacteristic
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C4⋊C4—104no—
316C4⋊C4—104no—
317C4⋊C4—108no—
318C4⋊C4—204no—
319C4⋊C4—204no—
320C22⋊C4—108no—
321C22⋊C4—108no—
322C22⋊C4—204no—
323C42—204no—
324C42#68514yes—
325C42#68514yes—
326C42C2×C4×D512yes—
327C42C2×C4×D512yes—
328C42C2×C22⋊D512yes—
329C42C2×C22⋊D512yes—
330C2×C5—42no—
331C2×C5#230911yescharacteristic
332C2×C5C2×#41611yescharacteristic
333C2×C5C2×#41711yescharacteristic
334C2×C5#320712yes—
335C2×C5#321012yes—
336D5—84no—
337C23—22no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—104no—
342C23—204no—
343C23—202no—
344C23—202no—
345C23—204no—
346C23C2×#68512yes—
347C23C22×D2011yescharacteristic
348Q8—108no—
349Q8—204no—
350D4—108no—
351D4—2016no—
352D4—204no—
353D4—208no—
354D4—208no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—104no—
364C2×C4—2016no—
365C2×C4—204no—
366C2×C4—204no—
367C2×C4—204no—
368C2×C4—204no—
369C2×C4#405014yes—
370C2×C4#405412yes—
371C2×C4#405512yes—
372C2×C4C2×#68512yes—
373C2×C4C2×#68512yes—
374C2×C4C2×#68512yes—
375C2×C4C2×#68512yes—
376C2×C4C2×#68512yes—
377C2×C4C22×C4×D511yescharacteristic
378C2×C4#414914yes—
379C2×C4C22×C22⋊D511yescharacteristic
380C8—44no—
381C5#2180511yescharacteristic, complemented, Sylow
382C22—22no—
383C22—22no—
384C22—22no—
385C22—42no—
386C22—42no—
387C22—204no—
388C22—204no—
389C22—204no—
390C22—404no—
391C22—404no—
392C22#6585912yes—
393C22C2×#405012yes—
394C22C2×#405411yescharacteristic
395C22C2×#405511yescharacteristic
396C22C22×#68511yescharacteristic
397C4—22no—
398C4—22no—
399C4—22no—
400C4—204no—
401C4—404no—
402C4#6540214yes—
403C4#6542214yes—
404C4#6585512yes—
405C4#6616412yes—
406C2—42no—
407C2—404no—
408C2#1061103512yes—
409C2#1061103712yes—
410C2#1061400011yescharacteristic
411C2C2×#6540211yescharacteristic
412C2C2×#6542211yescharacteristic
subgroupG/H
centerC23C22×D20
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#431C2
socleC23×C5C22×D4
prime radicalC2×#65811C2
cosocleC22×C4×C5C24
hypercenterC2×#431D5
hypocenterC5#21805
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
2. C2×C5×#431C2semidirect
3. #10613793C2semidirect
4. #10601251C2semidirect
5. #10600753C2semidirect
6. #10600565C2semidirect
7. #10600564C2semidirect
14. #65811C4semidirect
80. C2×#431D5semidirect
381. C5#21805semidirect

Maximal subgroups

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

subgroupindex
C2×#658112
C2×C5×#4312
#106137932
#106012512
#106007532
#106005652
#106005642
#218055