Finite Group Databasebeta

← #12116081

Subgroups — #12116081

Order 1280.

1066 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#559C211yesmaximal, characteristic
2Q8×#681C212yesmaximal, complemented
3#10602559C212yesmaximal
4#10602520C214yesmaximal, complemented
5#10602430C211yesmaximal, characteristic, complemented
6C4×#3968C211yesmaximal, characteristic, complemented
7#10602410C212yesmaximal, complemented
8C4×#3965C212yesmaximal, complemented
9C5×#559—22no—
10C5×#559—24no—
11C5×#559—22no—
12C2×C5×#127C2211yescharacteristic
13C2×C5×#124C412yes—
14C2×C5×#124C2212yes—
15C2×C5×#123C414yes—
16C2×C4×C5×Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×D4C412yes—
19C2×C4×C5×D4C2212yes—
20C2×#3968C2212yes—
21C2×#3968C2212yes—
22C2×#3965C2214yes—
23C2×#3965C2214yes—
24#65313—24no—
25#65313—22no—
26Q8×#168—22no—
27#65213C2214yes—
28#65213C2212yes—
29C4×#681C2214yes—
30C4×#681C2212yes—
31#64750C2211yescharacteristic
32#64750C2212yes—
33#64750C2211yescharacteristic
34#22539—51nomaximal, Sylow
35C22×C5×Q8C2×C411yescharacteristic
36C22×C5×Q8C2311yescharacteristic
37C22×C5×D4C2×C412yes—
38C22×C5×D4C2×C412yes—
39C22×C5×D4C2312yes—
40C23×C4×C5C2×C414yes—
41C5×#127—22no—
42C5×#127—22no—
43C5×#124—24no—
44C5×#124—24no—
45C5×#124—24no—
46C5×#124—24no—
47C5×#123—28no—
48C5×#123—28no—
49C4×C5×Q8—22no—
50C4×C5×Q8—22no—
51C4×C5×Q8D412yes—
52C4×C5×Q8D414yes—
53C4×C5×Q8D412yes—
54C4×C5×D4—24no—
55C4×C5×D4—22no—
56C4×C5×D4—22no—
57C4×C5×D4—24no—
58C4×C5×D4—24no—
59C2×C5×C4⋊C4C2×C412yes—
60C2×C5×C4⋊C4C2×C411yescharacteristic
61C2×C5×C4⋊C4C2×C412yes—
62C2×C5×C4⋊C4C2311yescharacteristic
63C2×C5×C22⋊C4C2×C414yes—
64C2×C5×C22⋊C4C2×C414yes—
65C2×C5×C22⋊C4C2×C414yes—
66C2×C42×C5C2×C412yes—
67C2×C42×C5C2312yes—
68C2×C42×C5C2311yescharacteristic
69C2×#681C2314yes—
70C2×#681C2312yes—
71C2×#681C2312yes—
72#3968—24no—
73#3968—42no—
74#3965—28no—
75#3965—44no—
76C4⋊#168—24no—
77C4⋊#168—22no—
78C4⋊#168—24no—
79C4⋊#168—22no—
80#3935—24no—
81#3935—22no—
82C4×#168—24no—
83C4×#168—22no—
84C2×#559D511yescharacteristic
85Q8×M16—52no—
86#1967—54no—
87#1965—52no—
88#1801—51no—
89#1800—52no—
90C4×#101—51no—
91C4×#100—52no—
92C24×C5C22×C414yes—
93C2×C5×Q8—22no—
94C2×C5×Q8—22no—
95C2×C5×Q8—21no—
96C2×C5×Q8—21no—
97C2×C5×Q8—42no—
98C2×C5×Q8C22⋊C412yes—
99C2×C5×Q8C22⋊C412yes—
100C2×C5×D4—22no—
101C2×C5×D4—22no—
102C2×C5×D4—24no—
103C2×C5×D4—24no—
104C2×C5×D4—24no—
105C2×C5×D4—22no—
106C2×C5×D4—22no—
107C2×C5×D4—44no—
108C2×C5×D4—82no—
109C22×C4×C5—24no—
110C22×C4×C5—24no—
111C22×C4×C5—84no—
112C22×C4×C5C22×C412yes—
113C22×C4×C5C22×C411yescharacteristic
114C22×C4×C5C22×C412yes—
115C22×C4×C5C22×C412yes—
116C22×C4×C5C22×C412yes—
117C22×C4×C5C22×C411yescharacteristic
118C22×C4×C5C2411yescharacteristic
119C5×C4⋊C4—24no—
120C5×C4⋊C4—22no—
121C5×C4⋊C4C22⋊C412yes—
122C5×C4⋊C4C22⋊C411yescharacteristic
123C5×C4⋊C4C22⋊C411yescharacteristic
124C5×C4⋊C4C22⋊C414yes—
125C5×C4⋊C4C22⋊C414yes—
126C5×C22⋊C4—24no—
127C5×C22⋊C4—24no—
128C5×C22⋊C4—28no—
129C5×C22⋊C4—28no—
130C42×C5—22no—
131C42×C5—22no—
132C42×C5C2×D414yes—
133C42×C5C2×D414yes—
134C42×C5C2×D412yes—
135C42×C5C2×D412yes—
136#681—22no—
137#681—24no—
138#681—22no—
139#681—22no—
140#681—24no—
141#681#4114yes—
142C2×#168—22no—
143C2×#168—24no—
144C2×#168—22no—
145#559—22no—
146#559—24no—
147#559—22no—
148C2×#127C2×D511yescharacteristic
149C2×#124Dic512yes—
150C2×#124C2×D512yes—
151C2×#123Dic514yes—
152C2×C4×Q8C2×D511yescharacteristic
153C2×C4×Q8C2×D511yescharacteristic
154C2×C4×D4Dic512yes—
155C2×C4×D4C2×D512yes—
156#446—104no—
157#446—102no—
158C8×Q8—102no—
159#423—54no—
160#423—52no—
161C2×#101—52no—
162C2×#101—52no—
163C2×#100—54no—
164C2×#100—54no—
165C4×M16—52no—
166C4×M16—54no—
167#343—51no—
168#343—52no—
169#343—51no—
170C23×C5—24no—
171C23×C5—24no—
172C23×C5—24no—
173C23×C5—44no—
174C23×C5—44no—
175C23×C5C23×C411yescharacteristic
176C5×Q8—24no—
177C5×Q8—42no—
178C5×Q8—42no—
179C5×Q8—42no—
180C5×D4—44no—
181C5×D4—44no—
182C5×D4—44no—
183C5×D4—44no—
184C5×D4—84no—
185C2×C4×C5—22no—
186C2×C4×C5—22no—
187C2×C4×C5—22no—
188C2×C4×C5—22no—
189C2×C4×C5—22no—
190C2×C4×C5—42no—
191C2×C4×C5—41no—
192C2×C4×C5—42no—
193C2×C4×C5—42no—
194C2×C4×C5—84no—
195C2×C4×C5—84no—
196C2×C4×C5C2×C22⋊C411yescharacteristic
197C2×C4×C5C2×C22⋊C414yes—
198C2×C4×C5C2×C22⋊C412yes—
199C2×C4×C5C2×C22⋊C412yes—
200C2×C4×C5C2×C22⋊C412yes—
201C2×C4×C5C2×C22⋊C411yescharacteristic
202C2×C4×C5C22×D412yes—
203C2×C4×C5C2×#4112yes—
204C2×C4×C5H(5,𝔽2)11yescharacteristic
205C2×C4×C52−1+411yescharacteristic
206#168—24no—
207#168—44no—
208#168—42no—
209C22×Q8C2×Dic511yescharacteristic
210C22×Q8C22×D511yescharacteristic
211C22×D4C2×Dic512yes—
212C22×D4C2×Dic512yes—
213C22×D4C22×D512yes—
214C23×C4C2×Dic514yes—
215C2×M16—52no—
216C2×M16—54no—
217C2×M16—52no—
218#127—22no—
219#127—22no—
220#124—24no—
221#124—24no—
222#124—24no—
223#124—24no—
224#123—28no—
225#123—28no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8C22⋊D512yes—
229C4×Q8C22⋊D514yes—
230C4×Q8C22⋊D512yes—
231C4×D4—24no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—24no—
235C4×D4—24no—
236C2×C4⋊C4C2×Dic512yes—
237C2×C4⋊C4C2×Dic511yescharacteristic
238C2×C4⋊C4C2×Dic512yes—
239C2×C4⋊C4C22×D511yescharacteristic
240C2×C22⋊C4C2×Dic514yes—
241C2×C22⋊C4C2×Dic514yes—
242C2×C22⋊C4C2×Dic514yes—
243C2×C42C2×Dic512yes—
244C2×C42C22×D512yes—
245C2×C42C22×D511yescharacteristic
246C4⋊C8—104no—
247C4⋊C8—102no—
248C4⋊C8—104no—
249C4⋊C8—102no—
250#101—104no—
251#101—202no—
252#100—108no—
253#100—204no—
254#97—104no—
255#97—102no—
256C4×C8—104no—
257C4×C8—102no—
258C22×C5—21no—
259C22×C5—21no—
260C22×C5—44no—
261C22×C5—44no—
262C22×C5—44no—
263C22×C5—84no—
264C22×C5—84no—
265C22×C5C22×C22⋊C411yescharacteristic
266C22×C5C4×#4111yescharacteristic
267C22×C5#52011yescharacteristic
268C4×C5—21no—
269C4×C5—24no—
270C4×C5—22no—
271C4×C5—22no—
272C4×C5—22no—
273C4×C5—42no—
274C4×C5—42no—
275C4×C5#41312yes—
276C4×C5#54812yes—
277C4×C5#55012yes—
278C24C22×Dic514yes—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—22no—
283C2×Q8—42no—
284C2×Q8C22⋊Dic512yes—
285C2×Q8C22⋊Dic512yes—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—44no—
294C2×D4—82no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—84no—
298C22×C4C22×Dic512yes—
299C22×C4C22×Dic511yescharacteristic
300C22×C4C22×Dic512yes—
301C22×C4C22×Dic512yes—
302C22×C4C22×Dic512yes—
303C22×C4C22×Dic511yescharacteristic
304C22×C4C23×D511yescharacteristic
305M16—54no—
306M16—102no—
307M16—104no—
308M16—102no—
309M16—102no—
310M16—104no—
311C2×C8—102no—
312C2×C8—104no—
313C2×C8—102no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4C22⋊Dic512yes—
317C4⋊C4C22⋊Dic511yescharacteristic
318C4⋊C4C22⋊Dic511yescharacteristic
319C4⋊C4C22⋊Dic514yes—
320C4⋊C4C22⋊Dic514yes—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—28no—
324C22⋊C4—28no—
325C42—22no—
326C42—22no—
327C42C2×C22⋊D514yes—
328C42C2×C22⋊D512yes—
329C42C2×C22⋊D514yes—
330C42C2×C22⋊D512yes—
331C2×C5—21no—
332C2×C5—21no—
333C2×C5—84no—
334C2×C5#234111yescharacteristic
335C2×C5C2×#41311yescharacteristic
336C2×C5#288811yescharacteristic
337C23—24no—
338C23—24no—
339C23—24no—
340C23—44no—
341C23—44no—
342C23C23×Dic511yescharacteristic
343Q8—24no—
344Q8—42no—
345Q8—42no—
346Q8—42no—
347D4—44no—
348D4—44no—
349D4—44no—
350D4—44no—
351D4—84no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4C2×C22⋊Dic511yescharacteristic
364C2×C4C2×C22⋊Dic514yes—
365C2×C4C2×C22⋊Dic512yes—
366C2×C4C2×C22⋊Dic512yes—
367C2×C4C2×C22⋊Dic512yes—
368C2×C4C2×C22⋊Dic511yescharacteristic
369C2×C4D5×#4112yes—
370C2×C4#414911yescharacteristic
371C2×C4#415011yescharacteristic
372C2×C4C22×C22⋊D512yes—
373C8—104no—
374C8—204no—
375C8—202no—
376C5#2253911yescharacteristic, complemented, Sylow
377C22—21no—
378C22—21no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—84no—
383C22—84no—
384C22#41×Dic511yescharacteristic
385C22#6616111yescharacteristic
386C22C22×C22⋊Dic511yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—21no—
391C4—24no—
392C4—42no—
393C4—42no—
394C4#6552612yes—
395C4#6616512yes—
396C4#6616812yes—
397C2—21no—
398C2—21no—
399C2—84no—
400C2C2×#6552611yescharacteristic
401C2#1061582011yescharacteristic
402C2#1061583111yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic5
commutatorC23×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#559C2
socleC22×C5C22×C22⋊C4
prime radicalC2×C5×#127C22
cosocleC22×C4×C5C24
hypercenterC2×#559D5
hypocenterC5#22539
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. Q8×#681C2semidirect
4. #10602520C2semidirect
5. #10602430C2semidirect
6. C4×#3968C2semidirect
7. #10602410C2semidirect
8. C4×#3965C2semidirect
376. C5#22539semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5592
Q8×#6812
#106025592
#106025202
#106024302
C4×#39682
#106024102
C4×#39652
#225395