Finite Group Databasebeta

← #12107608

Subgroups — #12107608

Order 1280.

694 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#4036C211yesmaximal, characteristic
2C5×#1770C211yesmaximal, characteristic
3#10601261C211yesmaximal, characteristic
4#10601236C211yesmaximal, characteristic
5#10600604C212yesmaximal
6#10600603C212yesmaximal
7#10600550C211yesmaximal, characteristic
8C4×C8×Dic5C211yesmaximal, characteristic
9#10600277C211yesmaximal, characteristic
10#10599404C212yesmaximal
11#10599403C212yesmaximal
12C2×#4036C411yescharacteristic
13C2×C5×C4⋊C8C2211yescharacteristic
14C2×C5×#97C2212yes—
15C2×C4×C5×C8C2211yescharacteristic
16C4×C5×C4⋊C4C2211yescharacteristic
17C2×#3947C2212yes—
18C2×#3946C2212yes—
19C2×C8×Dic5C2211yescharacteristic
20C2×C8×Dic5C2211yescharacteristic
21C2×C8×Dic5C2211yescharacteristic
22C2×C8×Dic5C2211yescharacteristic
23#65274C411yescharacteristic
24#65266C411yescharacteristic
25#65262C411yescharacteristic
26C4×C4⋊Dic5C2211yescharacteristic
27C4×C4⋊Dic5C2212yes—
28C4×#683C2212yes—
29C42×Dic5C2211yescharacteristic
30C2×C4⋊#168C2211yescharacteristic
31C2×C4⋊#168C2212yes—
32C2×C4×#168C2211yescharacteristic
33#64944C412yes—
34#64943C412yes—
35C5×#336C2212yes—
36#64770C2212yes—
37#64770C2212yes—
38#64770C2212yes—
39#64770C2212yes—
40#64744C2212yes—
41#21673—51nomaximal, Sylow
42C22×C5×C8C2312yes—
43C22×C5×C8C2311yescharacteristic
44C22×C5×C8C2311yescharacteristic
45C2×C5×C4⋊C4C2×C411yescharacteristic
46C2×C5×C4⋊C4C2×C411yescharacteristic
47C2×C42×C5C2312yes—
48C2×C42×C5C2311yescharacteristic
49C2×C4⋊Dic5C2×C412yes—
50C2×C4⋊Dic5C2×C411yescharacteristic
51C2×C4⋊Dic5C2×C412yes—
52C2×C4⋊Dic5C2×C411yescharacteristic
53C2×#683C2×C412yes—
54C2×#683C2×C412yes—
55C2×C4×Dic5C2×C411yescharacteristic
56C2×C4×Dic5C2×C411yescharacteristic
57C2×C4×Dic5C2×C411yescharacteristic
58C2×C4×Dic5C2×C411yescharacteristic
59C2×C4×Dic5C2311yescharacteristic
60C2×C4×Dic5C2311yescharacteristic
61C2×C4×Dic5C2312yes—
62C22×#168C2312yes—
63C22×#168C2311yescharacteristic
64C22×#168C2311yescharacteristic
65#4036—24no—
66C5×C4⋊C8—22no—
67C5×#97—24no—
68C4×C5×C8Q814yes—
69C5×#95C2×C412yes—
70#3963C2×C412yes—
71#3963C2×C412yes—
72#3963C2×C412yes—
73#3963C2×C412yes—
74#3963C2×C412yes—
75#3947—24no—
76#3946—24no—
77C8×Dic5—22no—
78C8×Dic5—22no—
79C8×Dic5—22no—
80C8×Dic5—22no—
81C4⋊#168—22no—
82C4⋊#168—24no—
83C4×#168—22no—
84C4×#125—51no—
85#1939—51no—
86#1939—51no—
87#1916—52no—
88#1916—52no—
89#1831—51no—
90#1771—52no—
91#1771—52no—
92#1770—51no—
93#1770D511yescharacteristic
94C42×C8—51no—
95C22×C4×C5C22×C412yes—
96C22×C4×C5C22×C412yes—
97C22×C4×C5C22×C411yescharacteristic
98C22×C4×C5C22×C411yescharacteristic
99C22×C4×C5C2411yescharacteristic
100C22×Dic5C22×C412yes—
101C22×Dic5C22×C412yes—
102C22×Dic5C22×C411yescharacteristic
103C22×Dic5C22×C411yescharacteristic
104C22×Dic5C22×C411yescharacteristic
105C22×Dic5C22×C411yescharacteristic
106C2×C5×C8—21no—
107C2×C5×C8—21no—
108C2×C5×C8—22no—
109C2×C5×C8—22no—
110C2×C5×C8—22no—
111C2×C5×C8C2×Q812yes—
112C2×C5×C8C2×Q812yes—
113C2×C5×C8#4112yes—
114C2×C5×C8#4112yes—
115C5×C4⋊C4—22no—
116C5×C4⋊C4—22no—
117C42×C5—22no—
118C42×C5—22no—
119C42×C5C2×Q812yes—
120C42×C5#4112yes—
121C4⋊Dic5—22no—
122C4⋊Dic5—22no—
123C4⋊Dic5—24no—
124C4⋊Dic5—24no—
125#683—24no—
126#683—24no—
127C4×Dic5—22no—
128C4×Dic5—21no—
129C4×Dic5—21no—
130C4×Dic5—21no—
131C4×Dic5—21no—
132C4×Dic5—22no—
133C4×Dic5—22no—
134C4×Dic5—22no—
135C4×Dic5—22no—
136C4×Dic5M1614yes—
137C2×#168—21no—
138C2×#168—21no—
139C2×#168—21no—
140C2×#168—21no—
141C2×#168—22no—
142C2×#168—22no—
143C2×#168—22no—
144C2×#168#4112yes—
145C2×#168#4112yes—
146C2×#125—51no—
147C2×C4⋊C8—52no—
148C2×C4⋊C8—51no—
149C2×C4⋊C8—52no—
150C2×C4⋊C8C2×D511yescharacteristic
151C2×#97—52no—
152C2×#97C2×D512yes—
153C2×C4×C8—51no—
154C2×C4×C8—51no—
155C2×C4×C8—51no—
156C2×C4×C8—51no—
157C2×C4×C8—51no—
158C2×C4×C8C2×D511yescharacteristic
159#389—51no—
160#389—51no—
161#387—52no—
162#387—52no—
163#382—51no—
164C4×C4⋊C4—52no—
165C4×C4⋊C4—52no—
166C4×C4⋊C4—51no—
167C4×C4⋊C4C2×D511yescharacteristic
168C43—51no—
169#336—52no—
170#336—52no—
171#336—52no—
172#336—52no—
173#336—52no—
174#336C2×D512yes—
175C23×C5C23×C411yescharacteristic
176C2×C4×C5—21no—
177C2×C4×C5—21no—
178C2×C4×C5—22no—
179C2×C4×C5—22no—
180C2×C4×C5—22no—
181C2×C4×C5—22no—
182C2×C4×C5—22no—
183C2×C4×C5—22no—
184C2×C4×C5C4×Q812yes—
185C2×C4×C5C4×Q812yes—
186C2×C4×C5C2×M1612yes—
187C2×C4×C5#13112yes—
188C2×C4×C5C22×Q811yescharacteristic
189C2×C4×C5C2×#4111yescharacteristic
190C2×C4×C5C2×#4111yescharacteristic
191C2×C4×C5C2×#4111yescharacteristic
192C2×C4×C5C2×#4111yescharacteristic
193C2×C4×C5C2×#4111yescharacteristic
194C2×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—21no—
197C2×Dic5—21no—
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—21no—
201C2×Dic5—21no—
202C2×Dic5—22no—
203C2×Dic5—22no—
204C2×Dic5—22no—
205C2×Dic5—22no—
206C2×Dic5—22no—
207C2×Dic5—22no—
208C2×Dic5C2×M1612yes—
209C2×Dic5C2×M1612yes—
210C2×Dic5#13112yes—
211C2×Dic5#13112yes—
212C5×C8—22no—
213C5×C8—42no—
214C5×C8#12514yescomplemented
215#168—22no—
216#168—22no—
217#168—42no—
218C22×C8—51no—
219C22×C8—51no—
220C22×C8—52no—
221C22×C8C22×D512yes—
222C22×C8C22×D511yescharacteristic
223C22×C8C22×D511yescharacteristic
224#125—104no—
225C2×C4⋊C4—52no—
226C2×C4⋊C4—52no—
227C2×C4⋊C4—51no—
228C2×C4⋊C4—51no—
229C2×C4⋊C4—52no—
230C2×C4⋊C4—52no—
231C2×C4⋊C4C4×D511yescharacteristic
232C2×C4⋊C4C4×D511yescharacteristic
233C2×C42—51no—
234C2×C42—51no—
235C2×C42—51no—
236C2×C42—51no—
237C2×C42—51no—
238C2×C42—51no—
239C2×C42—52no—
240C2×C42C22×D512yes—
241C2×C42C22×D511yescharacteristic
242C4⋊C8—22no—
243C4⋊C8—104no—
244C4⋊C8—104no—
245C4⋊C8—102no—
246#97—24no—
247#97—104no—
248C4×C8—102no—
249C4×C8—102no—
250C4×C8—102no—
251C4×C8—102no—
252C4×C8—102no—
253C4×C8Dic1014yes—
254#95—52no—
255#95—52no—
256#95—52no—
257#95—52no—
258#95—52no—
259#95C4×D512yes—
260C22×C5C2×C4×Q811yescharacteristic
261C22×C5C4×#4111yescharacteristic
262C22×C5C4×#4111yescharacteristic
263C22×C5C22×M1611yescharacteristic
264C22×C5C2×#13111yescharacteristic
265C22×C5C2×#13111yescharacteristic
266C22×C5C2×#13111yescharacteristic
267C4×C5—22no—
268C4×C5—42no—
269C4×C5—42no—
270C4×C5#44614yes—
271C4×C5C2×#12511yescharacteristic
272C4×C5#52911yescharacteristic
273C4×C5#52911yescharacteristic
274C4×C5#53311yescharacteristic
275Dic5—22no—
276Dic5—22no—
277Dic5—22no—
278Dic5—22no—
279Dic5—42no—
280Dic5—42no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—51no—
285C22×C4—52no—
286C22×C4—52no—
287C22×C4C2×C4×D512yes—
288C22×C4C2×C4×D512yes—
289C22×C4C2×C4×D511yescharacteristic
290C22×C4C2×C4×D511yescharacteristic
291C22×C4C23×D511yescharacteristic
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—52no—
298C2×C8—52no—
299C2×C8—101no—
300C2×C8—101no—
301C2×C8—101no—
302C2×C8—101no—
303C2×C8—102no—
304C2×C8—102no—
305C2×C8—102no—
306C2×C8C2×Dic1012yes—
307C2×C8C2×Dic1012yes—
308C2×C8#71112yes—
309C2×C8#71312yes—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—104no—
313C4⋊C4—104no—
314C4⋊C4—104no—
315C4⋊C4—104no—
316C4⋊C4—102no—
317C4⋊C4—102no—
318C42—22no—
319C42—22no—
320C42—54no—
321C42—102no—
322C42—102no—
323C42—101no—
324C42—101no—
325C42—102no—
326C42—101no—
327C42—101no—
328C42—102no—
329C42—102no—
330C42C2×Dic1012yes—
331C42#70912yes—
332C2×C5C4×#12511yescharacteristic
333C2×C5C2×#44611yescharacteristic
334C2×C5#296011yescharacteristic
335C2×C5#296511yescharacteristic
336C2×C5#296511yescharacteristic
337C2×C5#296611yescharacteristic
338C2×C5#296611yescharacteristic
339C23C22×C4×D511yescharacteristic
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—52no—
349C2×C4—52no—
350C2×C4—52no—
351C2×C4—52no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—101no—
356C2×C4—101no—
357C2×C4—101no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—102no—
361C2×C4—102no—
362C2×C4—102no—
363C2×C4—102no—
364C2×C4—102no—
365C2×C4—102no—
366C2×C4C4×Dic1012yes—
367C2×C4C4×Dic1012yes—
368C2×C4D5×M1612yes—
369C2×C4#405312yes—
370C2×C4C22×Dic1011yescharacteristic
371C2×C4C2×#70911yescharacteristic
372C2×C4C2×#71111yescharacteristic
373C2×C4C2×#71311yescharacteristic
374C2×C4D5×#4111yescharacteristic
375C2×C4D5×#4111yescharacteristic
376C8—22no—
377C8—42no—
378C8—102no—
379C8—102no—
380C8—202no—
381C8#403614yescomplemented
382C5#2167311yescharacteristic, complemented, Sylow
383C22C2×C4×Dic1011yescharacteristic
384C22C4×#71111yescharacteristic
385C22C4×#71311yescharacteristic
386C22C2×D5×M1611yescharacteristic
387C22C2×#405311yescharacteristic
388C22D5×#13111yescharacteristic
389C22D5×#13111yescharacteristic
390C4—22no—
391C4—42no—
392C4—42no—
393C4—102no—
394C4—102no—
395C4—102no—
396C4—102no—
397C4—202no—
398C4—202no—
399C4#6499014yes—
400C4C2×#403611yescharacteristic
401C4#6585111yescharacteristic
402C4#6589111yescharacteristic
403C4#6591411yescharacteristic
404C2C4×#403611yescharacteristic
405C2C2×#6499011yescharacteristic
406C2#1061027311yescharacteristic
407C2#1061124811yescharacteristic
408C2#1061124911yescharacteristic
409C2#1061166911yescharacteristic
410C2#1061167011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D5
commutator, socle, prime radicalC23×C5C23×C4
FittingC5×#1770C2
cosocleC22×C4×C5C24
hypercenter#1770D5
hypocenterC5#21673
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1770

Complements

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

NH ≅ G/Ntype
214. C5×C8#125semidirect
381. C8#4036semidirect
382. C5#21673semidirect

Maximal subgroups

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

subgroupindex
C4×#40362
C5×#17702
#106012612
#106012362
#106006042
#106006032
#106005502
C4×C8×Dic52
#106002772
#105994042
#105994032
#216735