Finite Group Databasebeta

← #12005552

Subgroups — #12005552

Order 1280.

1290 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2297C211yesmaximal, characteristic
2C22×#3969C212yesmaximal
3C2×C42×Dic5C211yesmaximal, characteristic, complemented
4C2×#64752C212yesmaximal, complemented
5#10602320C211yesmaximal, characteristic, complemented
6C22×C5×#41C2211yescharacteristic
7C2×C5×#117C2211yescharacteristic
8C22×C42×C5C2211yescharacteristic
9C22×C4×Dic5—22no—
10C22×C4×Dic5C2212yes—
11C22×#681C2212yes—
12C5×#391C412yescomplemented
13C5×#390C412yescomplemented
14C5×#386—22no—
15C4×C5×C22⋊C4—22no—
16C2×#3969—24no—
17C2×#3969—24no—
18C2×#3969C418yescomplemented
19C42×Dic5—24no—
20#64752—24no—
21#64744—22no—
22#10923—51nomaximal, Sylow
23C2×C5×#41—22no—
24C2×C5×#41—22no—
25C2×C5×#41—22no—
26C2×C5×#41—22no—
27C2×C5×#41—22no—
28C2×C5×#41C2×C414yes—
29C22×C5×Q8C2×C411yescharacteristic
30C22×C5×D4—21no—
31C22×C5×D4C2×C411yescharacteristic
32C23×C4×C5—21no—
33C23×C4×C5D412yes—
34C23×C4×C5C2311yescharacteristic
35C23×Dic5—22no—
36C5×#117—22no—
37C5×#117—22no—
38C2×C5×C4⋊C4C2×C412yes—
39C2×C5×C22⋊C4—21no—
40C2×C5×C22⋊C4—24no—
41C2×C42×C5—22no—
42C2×C42×C5—22no—
43C2×C42×C5—21no—
44C2×C42×C5C2×C412yes—
45C2×C42×C5D412yes—
46C2×C4×Dic5—216no—
47C2×C4×Dic5—28no—
48C2×C4×Dic5—22no—
49C2×C4×Dic5—22no—
50C2×C4×Dic5—22no—
51C2×C4×Dic5C2×C414yes—
52C2×C4×Dic5D414yes—
53C2×C4×Dic5D414yes—
54C2×#681—22no—
55C2×#681—22no—
56C2×#681—22no—
57C2×#681—22no—
58C2×#681—22no—
59C2×#681C2×C414yes—
60C22×#168—22no—
61C5×#95—24no—
62#3969—216no—
63C22×#104—52no—
64#2297D511yescharacteristic
65C2×C43—51no—
66C2×#339—52no—
67#1728—51no—
68C24×C5—21no—
69C24×C5C2×D411yescharacteristic
70C5×#41—24no—
71C5×#41—24no—
72C5×#41—24no—
73C5×#41—24no—
74C5×#41—42no—
75C5×#41—42no—
76C5×#41—42no—
77C5×#41—42no—
78C2×C5×Q8—21no—
79C2×C5×Q8—22no—
80C2×C5×Q8—22no—
81C2×C5×Q8C4212yescomplemented
82C2×C5×D4—24no—
83C2×C5×D4—21no—
84C2×C5×D4—22no—
85C2×C5×D4—22no—
86C2×C5×D4—41no—
87C2×C5×D4—42no—
88C2×C5×D4—41no—
89C2×C5×D4C4212yescomplemented
90C22×C4×C5—22no—
91C22×C4×C5—22no—
92C22×C4×C5—28no—
93C22×C4×C5—21no—
94C22×C4×C5—24no—
95C22×C4×C5—22no—
96C22×C4×C5—42no—
97C22×C4×C5—41no—
98C22×C4×C5—41no—
99C22×C4×C5C22⋊C414yes—
100C22×C4×C5C22×C412yes—
101C22×C4×C5C22×C411yescharacteristic
102C22×C4×C5C2×D412yes—
103C22×C4×C5C2×D412yes—
104C22×C4×C5C2×D411yescharacteristic
105C22×C4×C5#4112yes—
106C22×C4×C5#4112yes—
107C22×C4×C5#4112yes—
108C22×Dic5—216no—
109C22×Dic5—24no—
110C22×Dic5—24no—
111C22×Dic5—24no—
112C5×C4⋊C4—24no—
113C5×C22⋊C4—42no—
114C5×C22⋊C4—48no—
115C42×C5—24no—
116C42×C5—24no—
117C42×C5—41no—
118C42×C5—41no—
119C4×Dic5—232no—
120C4×Dic5—24no—
121C4×Dic5—24no—
122C4×Dic5—24no—
123C4×Dic5C22⋊C418yescomplemented
124#681—24no—
125#681—24no—
126#681—42no—
127#681—42no—
128C2×#168—24no—
129C2×#168—42no—
130C2×#168—42no—
131C22×#41C2×D511yescharacteristic
132C22×M16—52no—
133C2×#117C2×D511yescharacteristic
134C22×C42—52no—
135C22×C42—102no—
136C22×C42C2×D511yescharacteristic
137C2×#104—58no—
138C2×#104—104no—
139C2×#104—104no—
140#391Dic512yescomplemented
141#390Dic512yescomplemented
142#386—22no—
143C4×C22⋊C4—22no—
144C43—104no—
145#339—104no—
146#336—102no—
147C23×C5—24no—
148C23×C5—22no—
149C23×C5—41no—
150C23×C5—41no—
151C23×C5—44no—
152C23×C5C2×C22⋊C411yescharacteristic
153C23×C5C4×D412yes—
154C23×C5#12011yescharacteristic
155C23×C5#12112yes—
156C23×C5#12311yescharacteristic
157C5×Q8—24no—
158C5×Q8—42no—
159C5×D4—24no—
160C5×D4—44no—
161C5×D4—44no—
162C5×D4—42no—
163C2×C4×C5—28no—
164C2×C4×C5—28no—
165C2×C4×C5—24no—
166C2×C4×C5—24no—
167C2×C4×C5—22no—
168C2×C4×C5—41no—
169C2×C4×C5—41no—
170C2×C4×C5—44no—
171C2×C4×C5—44no—
172C2×C4×C5—42no—
173C2×C4×C5—44no—
174C2×C4×C5#10418yescomplemented
175C2×C4×C5C2×C4211yescharacteristic
176C2×C4×C5C2×C22⋊C412yes—
177C2×C4×C5#11712yes—
178C2×C4×C5#11711yescharacteristic
179C2×C4×C5C4×D412yes—
180C2×C4×C5C4×D412yes—
181C2×C4×C5C4×D412yes—
182C2×C4×C5C4×D412yes—
183C2×C4×C5C4×D412yes—
184C2×C4×C5#12112yes—
185C2×C4×C5#12112yes—
186C2×C4×C5#12312yes—
187C2×C4×C5#12412yes—
188C2×C4×C5#12411yescharacteristic
189C2×C4×C5#12612yes—
190C2×C4×C5#12711yescharacteristic
191C2×Dic5—216no—
192C2×Dic5—216no—
193C2×Dic5—216no—
194C2×Dic5—28no—
195#168—44no—
196C2×#41—22no—
197C2×#41—22no—
198C2×#41—22no—
199C2×#41—22no—
200C2×#41—22no—
201C2×#41C4×D514yes—
202C22×Q8C2×Dic511yescharacteristic
203C22×D4—21no—
204C22×D4C2×Dic511yescharacteristic
205C23×C4—21no—
206C23×C4—102no—
207C23×C4C22⋊D512yes—
208C23×C4C22×D511yescharacteristic
209C2×M16—54no—
210C2×M16—102no—
211C2×M16—102no—
212C2×M16—102no—
213C2×M16—102no—
214C2×M16—102no—
215C22×C8—102no—
216#117—22no—
217#117—22no—
218C2×C4⋊C4C2×Dic512yes—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—54no—
225C2×C42—54no—
226C2×C42—54no—
227C2×C42—108no—
228C2×C42—1016no—
229C2×C42—102no—
230C2×C42—102no—
231C2×C42—102no—
232C2×C42C2×Dic512yes—
233C2×C42C22⋊D512yes—
234#104—1016no—
235#95—24no—
236C22×C5—22no—
237C22×C5—22no—
238C22×C5—24no—
239C22×C5—24no—
240C22×C5—42no—
241C22×C5—44no—
242C22×C5—82no—
243C22×C5—82no—
244C22×C5C4×C22⋊C411yescharacteristic
245C22×C5#38612yes—
246C22×C5#38611yescharacteristic
247C22×C5#38812yes—
248C22×C5#39011yescharacteristic
249C22×C5C2×#10414yes—
250C4×C5—28no—
251C4×C5—42no—
252C4×C5—82no—
253C4×C5C4×C22⋊C412yes—
254C4×C5#38812yes—
255C4×C5#38812yes—
256C4×C5#39012yes—
257Dic5—216no—
258C24—21no—
259C24C2×C22⋊D511yescharacteristic
260#41—24no—
261#41—24no—
262#41—24no—
263#41—24no—
264#41—42no—
265#41—42no—
266#41—42no—
267#41—42no—
268C2×Q8—21no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8C4×Dic512yescomplemented
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4C4×Dic512yescomplemented
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—1016no—
290C22×C4—104no—
291C22×C4—104no—
292C22×C4—104no—
293C22×C4C22⋊Dic514yes—
294C22×C4C2×C4×D512yes—
295C22×C4#70912yes—
296C22×C4D4×D512yes—
297C22×C4D4×D512yes—
298C22×C4#71112yes—
299C22×C4#71112yes—
300C22×C4C22×Dic511yescharacteristic
301C22×C4C2×C22⋊D511yescharacteristic
302M16—104no—
303M16—104no—
304M16—202no—
305M16—202no—
306C2×C8—104no—
307C2×C8—202no—
308C2×C8—202no—
309C4⋊C4—24no—
310C22⋊C4—42no—
311C22⋊C4—48no—
312C42—24no—
313C42—24no—
314C42—41no—
315C42—41no—
316C42—58no—
317C42—1032no—
318C42—104no—
319C42—104no—
320C42—104no—
321C2×C5—22no—
322C2×C5—22no—
323C2×C5—82no—
324C2×C5#143711yescharacteristic
325C2×C5C4×#10412yes—
326C2×C5#189612yes—
327C2×C5#196212yes—
328C23—24no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—44no—
333C23C4×C22⋊D512yes—
334C23#408111yescharacteristic
335C23#408311yescharacteristic
336C23#408412yes—
337C23C2×C22⋊Dic511yescharacteristic
338Q8—24no—
339Q8—42no—
340D4—24no—
341D4—44no—
342D4—44no—
343D4—42no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—1016no—
356C2×C4—1016no—
357C2×C4—1016no—
358C2×C4—108no—
359C2×C4#396918yescomplemented
360C2×C4#402312yes—
361C2×C4#402512yes—
362C2×C4D5×C22⋊C412yes—
363C2×C4#402712yes—
364C2×C4#402712yes—
365C2×C4#402912yes—
366C2×C4#403012yes—
367C2×C4#403112yes—
368C2×C4C2×C4×Dic511yescharacteristic
369C2×C4#407211yescharacteristic
370C2×C4C4×C22⋊D512yes—
371C2×C4D4×Dic512yes—
372C2×C4D4×Dic512yes—
373C2×C4#408211yescharacteristic
374C2×C4#408512yes—
375C2×C4#408611yescharacteristic
376C8—204no—
377C5#1092311yescharacteristic, complemented, Sylow
378C22—22no—
379C22—22no—
380C22—24no—
381C22—24no—
382C22—42no—
383C22—44no—
384C22—82no—
385C22—82no—
386C22#6523312yes—
387C22#6524512yes—
388C22C4×C22⋊Dic511yescharacteristic
389C22#6550911yescharacteristic
390C22#6551011yescharacteristic
391C22C2×#396914yes—
392C4—28no—
393C4—42no—
394C4—82no—
395C4—1016no—
396C4C22⋊C4×Dic512yes—
397C4#6523412yes—
398C4#6524012yes—
399C4#6524612yes—
400C2—22no—
401C2—22no—
402C2—82no—
403C2#1060233811yescharacteristic
404C2C4×#396912yes—
405C2#1060250412yes—
406C2#1060260812yes—
subgroupG/H
centerC22×C4C2×C22⋊D5
commutatorC2×C4×C5C2×C42
FrattiniC23×C4C22×D5
FittingC5×#2297C2
socleC23×C5C2×C22⋊C4
prime radicalC22×C5×#41C22
cosocleC23×C4×C5C23
hypercenter#2297D5
hypocenterC5#10923
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
3. C2×C42×Dic5C2semidirect
4. C2×#64752C2semidirect
5. #10602320C2semidirect
12. C5×#391C4semidirect
13. C5×#390C4semidirect
18. C2×#3969C4semidirect
81. C2×C5×Q8C42semidirect
89. C2×C5×D4C42semidirect
123. C4×Dic5C22⋊C4semidirect
140. #391Dic5semidirect
141. #390Dic5semidirect
174. C2×C4×C5#104semidirect
271. C2×Q8C4×Dic5semidirect
279. C2×D4C4×Dic5semidirect
359. C2×C4#3969semidirect
377. C5#10923semidirect

Maximal subgroups

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

subgroupindex
C5×#22972
C22×#39692
C2×C42×Dic52
C2×#647522
#106023202
#109235