Finite Group Databasebeta

← #12109024

Subgroups — #12109024

Order 1280.

1394 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65811C211yesmaximal, characteristic
2C2×C5×#382C211yesmaximal, characteristic, complemented
3C2×#65215C211yesmaximal, characteristic, complemented
4#10602353C214yesmaximal, complemented
5#10600349C214yesmaximal, complemented
6#10600348C212yesmaximal, complemented
7#10600346C212yesmaximal, complemented
8C22×#709—21no—
9C22×C5×C4⋊C4C2212yes—
10C22×C42×C5C2211yescharacteristic
11C2×#4076—21no—
12C2×#4076C2212yes—
13#65811—24no—
14#65811C418yescomplemented
15C2×#4021—21no—
16C2×#4020C2211yescharacteristic
17C2×C4×D20—21no—
18C2×C4×D20C2212yes—
19C2×#4015C2211yescharacteristic
20C5×#382—21no—
21C5×#382—21no—
22C5×#382C2212yes—
23C5×#382C2212yes—
24C2×C5×#95—21no—
25C2×C5×#95—21no—
26C2×C22⋊#168C2212yes—
27C2×#3941C2218yes—
28#65215—24no—
29C2×C4⋊#168C2212yes—
30C2×#3935C2212yes—
31#64749C2218yes—
32#21865—51nomaximal, Sylow
33C23×C4×C5—22no—
34C23×C4×C5D411yescharacteristic
35C23×C4×C5D411yescharacteristic
36C23×C4×C5C2311yescharacteristic
37C22×C22⋊D5—22no—
38C2×#709—28no—
39C2×#709—48no—
40C22×D20—21no—
41C22×D20—21no—
42C22×D20C2312yes—
43C22×C4×D5—22no—
44C22×Dic10—21no—
45C2×C5×C4⋊C4—22no—
46C2×C5×C4⋊C4—44no—
47C2×C5×C4⋊C4C2314yes—
48C2×C42×C5C2×C414yes—
49C2×C42×C5D414yes—
50C2×C42×C5D411yescharacteristic
51C2×C42×C5D411yescharacteristic
52C2×C42×C5C2311yescharacteristic
53C2×C42×C5C2311yescharacteristic
54#4076—28no—
55#4076—24no—
56#4076C2×C418yes—
57C2×#685—22no—
58C2×#685—22no—
59C2×C4⋊Dic5—21no—
60C2×C4⋊Dic5C2312yes—
61C22×#168C2314yes—
62#4021—28no—
63#4020—22no—
64#4020C2×C414yescomplemented
65C4×D20—28no—
66C4×D20—24no—
67C4×D20C2×C418yescomplemented
68#4015—22no—
69#4015C2×C414yescomplemented
70C5×#95—22no—
71C5×#95—22no—
72C5×#95—22no—
73C5×#95—22no—
74C22⋊#168—24no—
75#3941—48no—
76C4⋊#168—24no—
77#3935—24no—
78C2×#532—51no—
79C2×#432—51no—
80C2×#382D511yescharacteristic, complemented
81#1993—52no—
82#1961—52no—
83#1874—54no—
84#1804—54no—
85C24×C5C2×D411yescharacteristic
86C23×D5—22no—
87C22×C4×C5—24no—
88C22×C4×C5—22no—
89C22×C4×C5—22no—
90C22×C4×C5—24no—
91C22×C4×C5—42no—
92C22×C4×C5—44no—
93C22×C4×C5—44no—
94C22×C4×C5—44no—
95C22×C4×C5C22⋊C414yes—
96C22×C4×C5C22⋊C414yes—
97C22×C4×C5C22×C412yes—
98C22×C4×C5C2×D412yes—
99C22×C4×C5C2×D411yescharacteristic
100C22×C4×C5C2×D411yescharacteristic
101C22×C4×C5C2×D411yescharacteristic
102C22×C4×C5C2×D411yescharacteristic
103C22×C4×C5C2×D411yescharacteristic
104C22×C4×C5C2411yescharacteristic
105C2×C22⋊D5—28no—
106C2×C22⋊D5—48no—
107C22×Dic5—22no—
108#709—432no—
109C2×D20—24no—
110C2×D20—24no—
111C2×D20—24no—
112C2×D20—24no—
113C2×D20—22no—
114C2×D20—44no—
115C2×D20—44no—
116C2×D20C22×C414yes—
117C2×C4×D5—28no—
118C2×C4×D5—48no—
119C2×Dic10—24no—
120C2×Dic10—44no—
121C5×C4⋊C4—28no—
122C5×C4⋊C4—44no—
123C42×C5C22⋊C414yes—
124C42×C5C22⋊C414yes—
125C42×C5C22×C412yes—
126C42×C5C22×C412yes—
127C42×C5C2×D412yes—
128C42×C5C2×D412yes—
129#685—28no—
130#685—28no—
131C4⋊Dic5—24no—
132C4⋊Dic5—22no—
133C4⋊Dic5C22×C414yes—
134C2×#168—24no—
135C2×#168—24no—
136C2×#168—24no—
137C22×#41—101no—
138#532—58no—
139#532—104no—
140C2×C4⋊Q8—51no—
141C2×#127—51no—
142C2×#124—101no—
143C2×#121—52no—
144C2×#121—101no—
145C2×C4×D4—52no—
146C2×C4×D4—101no—
147C22×C4⋊C4C2×D512yes—
148C22×C42C2×D511yescharacteristic
149#432—104no—
150C2×C4⋊C8—52no—
151C2×#102—58no—
152C2×C22⋊C8—52no—
153C2×#97—52no—
154#382—21no—
155#382—21no—
156#382C2×D512yes—
157#382C2×D512yes—
158C2×#95—21no—
159C2×#95—21no—
160#340—58no—
161C23×C5—21no—
162C23×C5—21no—
163C23×C5—22no—
164C23×C5C2×C22⋊C412yes—
165C23×C5AGL(1,ℤ/8)14yes—
166C23×C5C22×D411yescharacteristic
167C22×D5—24no—
168C22×D5—42no—
169C22×D5—42no—
170C22×D5—44no—
171C22×D5—44no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—22no—
175C2×C4×C5—22no—
176C2×C4×C5—22no—
177C2×C4×C5—22no—
178C2×C4×C5—28no—
179C2×C4×C5—24no—
180C2×C4×C5—24no—
181C2×C4×C5—44no—
182C2×C4×C5—44no—
183C2×C4×C5—44no—
184C2×C4×C5—44no—
185C2×C4×C5—88no—
186C2×C4×C5C2×C22⋊C412yes—
187C2×C4×C5C2×C22⋊C412yes—
188C2×C4×C5C2×C22⋊C412yes—
189C2×C4×C5C2×C22⋊C412yes—
190C2×C4×C5C2×C22⋊C412yes—
191C2×C4×C5C23×C411yescharacteristic
192C2×C4×C5C22×D411yescharacteristic
193C2×C4×C5H(5,𝔽2)12yes—
194C2×C4×C52−1+412yes—
195C22⋊D5—416no—
196C2×Dic5—24no—
197C2×Dic5—44no—
198D20—28no—
199D20—48no—
200D20—48no—
201D20—44no—
202C4×D5—416no—
203Dic10—48no—
204#168—44no—
205C2×#41—108no—
206C2×#41—208no—
207C22×Q8—101no—
208C22×D4—52no—
209C22×D4—102no—
210C22×D4—101no—
211C22×D4—101no—
212C23×C4—22no—
213C23×C4—102no—
214C23×C4C22⋊D511yescharacteristic
215C23×C4C22⋊D511yescharacteristic
216C23×C4C22×D511yescharacteristic
217C22×C8—54no—
218C4⋊Q8—54no—
219C4⋊Q8—102no—
220#127—54no—
221#127—102no—
222#124—108no—
223#121—58no—
224#121—108no—
225#121—104no—
226C4×D4—58no—
227C4×D4—108no—
228C4×D4—104no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—52no—
232C2×C4⋊C4—101no—
233C2×C4⋊C4C22×D514yes—
234C2×C22⋊C4—102no—
235C2×C22⋊C4—102no—
236C2×C42C4×D514yes—
237C2×C42D2014yes—
238C2×C42C22⋊D511yescharacteristic
239C2×C42C22⋊D511yescharacteristic
240C2×C42C22×D511yescharacteristic
241C2×C42C22×D511yescharacteristic
242C4⋊C8—104no—
243#102—208no—
244C22⋊C8—104no—
245#97—104no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250C22×C5—22no—
251C22×C5—22no—
252C22×C5—24no—
253C22×C5—24no—
254C22×C5#41814yes—
255C22×C5C22×C22⋊C411yescharacteristic
256C22×C5#53511yescharacteristic
257C22×C5#53611yescharacteristic
258C22×C5C2×AGL(1,ℤ/8)12yes—
259C22×C5#57512yes—
260C2×D5—44no—
261C2×D5—44no—
262C2×D5—44no—
263C2×D5—84no—
264C2×D5—84no—
265C4×C5—22no—
266C4×C5—22no—
267C4×C5—22no—
268C4×C5—22no—
269C4×C5—22no—
270C4×C5—84no—
271C4×C5#52012yes—
272C4×C5#53712yes—
273Dic5—44no—
274C24—102no—
275C24C2×C22⋊D511yescharacteristic
276#41—2032no—
277C2×Q8—104no—
278C2×Q8—204no—
279C2×D4—54no—
280C2×D4—108no—
281C2×D4—102no—
282C2×D4—104no—
283C2×D4—104no—
284C2×D4—104no—
285C2×D4—104no—
286C2×D4—204no—
287C2×D4—204no—
288C2×D4—208no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—102no—
298C22×C4—108no—
299C22×C4—208no—
300C22×C4#68514yes—
301C22×C4#68514yes—
302C22×C4C2×C4×D512yes—
303C22×C4C2×D2012yes—
304C22×C4C2×C22⋊D511yescharacteristic
305C22×C4C2×C22⋊D511yescharacteristic
306C22×C4C2×C22⋊D511yescharacteristic
307C22×C4C2×C22⋊D511yescharacteristic
308C22×C4C2×C22⋊D511yescharacteristic
309C22×C4C23×D511yescharacteristic
310C2×C8—104no—
311C2×C8—104no—
312C2×C8—104no—
313C4⋊C4—28no—
314C4⋊C4—44no—
315C4⋊C4—54no—
316C4⋊C4—102no—
317C4⋊C4—104no—
318C22⋊C4—108no—
319C22⋊C4—108no—
320C42#68514yes—
321C42#68514yes—
322C42C2×C4×D512yes—
323C42C2×C4×D512yes—
324C42C2×D2012yes—
325C42C2×D2012yes—
326C2×C5—24no—
327C2×C5#231111yescharacteristic
328C2×C5C2×#41811yescharacteristic
329C2×C5#289611yescharacteristic
330C2×C5#321312yes—
331C2×C5#323412yes—
332D5—84no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—104no—
337C23—202no—
338C23—202no—
339C23—204no—
340C23—204no—
341C23C2×#68512yes—
342C23#409514yes—
343C23C22×C22⋊D511yescharacteristic
344Q8—208no—
345D4—108no—
346D4—208no—
347D4—208no—
348D4—2016no—
349D4—204no—
350C2×C4—28no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—88no—
364C2×C4—104no—
365C2×C4—2016no—
366C2×C4—204no—
367C2×C4C2×#68512yes—
368C2×C4C2×#68512yes—
369C2×C4C2×#68512yes—
370C2×C4C2×#68512yes—
371C2×C4C2×#68512yes—
372C2×C4C22×C4×D511yescharacteristic
373C2×C4C22×D2011yescharacteristic
374C2×C4#414412yes—
375C2×C4#414712yes—
376C8—204no—
377C5#2186511yescharacteristic, complemented, Sylow
378C22—24no—
379C22—22no—
380C22—22no—
381C22—24no—
382C22—204no—
383C22—204no—
384C22—204no—
385C22—404no—
386C22—404no—
387C22#6529014yes—
388C22C22×#68511yescharacteristic
389C22#6613911yescharacteristic
390C22#6615011yescharacteristic
391C22C2×#409512yes—
392C22#6615612yes—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—84no—
399C4—204no—
400C4#6583812yes—
401C4#6584412yes—
402C2—24no—
403C2—404no—
404C2#1061391711yescharacteristic
405C2C2×#6529011yescharacteristic
406C2#1061394411yescharacteristic
407C2#1061467812yes—
408C2#1061475012yes—
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#382C2
socleC23×C5C22×D4
prime radicalC2×#65811C2
cosocleC22×C4×C5C24
hypercenterC2×#382D5
hypocenterC5#21865
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×C5×#382C2semidirect
3. C2×#65215C2semidirect
4. #10602353C2semidirect
5. #10600349C2semidirect
6. #10600348C2semidirect
7. #10600346C2semidirect
14. #65811C4semidirect
64. #4020C2×C4semidirect
67. C4×D20C2×C4semidirect
69. #4015C2×C4semidirect
80. C2×#382D5semidirect
377. C5#21865semidirect

Maximal subgroups

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

subgroupindex
C2×#658112
C2×C5×#3822
C2×#652152
#106023532
#106003492
#106003482
#106003462
#218655