Finite Group Databasebeta

← #12109902

Subgroups — #12109902

Order 1280.

762 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64979C211yesmaximal, characteristic
2#10605956C211yesmaximal, characteristic
3#10605955C211yesmaximal, characteristic
4C5×#1835C211yesmaximal, characteristic, complemented
5#10600558C211yesmaximal, characteristic, complemented
6#10600343C211yesmaximal, characteristic, complemented
7#10600341C211yesmaximal, characteristic, complemented
8#10599670C212yesmaximal
9#10599667C212yesmaximal
10#10599591C212yesmaximal, complemented
11#10599588C212yesmaximal, complemented
12C2×#4038C2211yescharacteristic
13C2×D5×C4⋊C4C2211yescharacteristic
14C2×#4018C2211yescharacteristic
15C2×C4×C5×C8C2211yescharacteristic
16C5×#382C2211yescharacteristic
17C5×#382C2211yescharacteristic
18C2×#3952C2212yes—
19C2×#3950C2212yes—
20C2×#3950C2212yes—
21C2×#3946C2212yes—
22#65276C2212yes—
23#65266C2211yescharacteristic
24#65265C2211yescharacteristic
25C4⋊C4×Dic5C2212yes—
26C2×#3940C2212yes—
27C2×#3939C2212yes—
28#65224C2211yescharacteristic
29#65224C2211yescharacteristic
30C2×#3935C2211yescharacteristic
31#64979C418yescomplemented
32#64959—22no—
33#64957—22no—
34#64944—22no—
35#64936—22no—
36C5×#340C2212yes—
37C5×#340C2212yes—
38#64749C2212yes—
39#64749C2212yes—
40#21991—51nomaximal, Sylow
41C22×C4×D5—21no—
42C22×C4×D5C2311yescharacteristic
43C22×C5×C8C2312yes—
44C2×C5×C4⋊C4C2311yescharacteristic
45C2×C5×C4⋊C4C2311yescharacteristic
46C2×C5×C4⋊C4C2312yes—
47C2×C42×C5C2311yescharacteristic
48C2×#685—21no—
49C2×#685—22no—
50C2×#685D412yes—
51C2×C4⋊Dic5—22no—
52C2×C4⋊Dic5—22no—
53C2×C4⋊Dic5C2311yescharacteristic
54C2×C4⋊Dic5C2311yescharacteristic
55C2×C4⋊Dic5C2312yes—
56C2×#683—21no—
57C2×#683—22no—
58C2×#683D412yes—
59C2×C4×Dic5—21no—
60C2×C4×Dic5—22no—
61C2×C4×Dic5C2311yescharacteristic
62C22×#168C2312yes—
63#4038—42no—
64D5×C4⋊C4—42no—
65#4018C2×C414yes—
66#4018Q814yes—
67C4×C5×C8C2×C414yescomplemented
68C5×#95—22no—
69C5×#95—22no—
70#3963—22no—
71#3963—22no—
72#3963—22no—
73#3952C2×C418yes—
74#3950—42no—
75#3950—42no—
76#3946C2×C418yescomplemented
77#3940—42no—
78#3939—42no—
79#3935C2×C414yescomplemented
80C2×#433—51no—
81#2317—51no—
82#2314—51no—
83#1949—52no—
84#1947—52no—
85#1854—52no—
86#1852—52no—
87#1847—51no—
88#1845—51no—
89#1836—51no—
90#1835D511yescharacteristic, complemented
91C23×D5C2×D411yescharacteristic
92C22×C4×C5—21no—
93C22×C4×C5—21no—
94C22×C4×C5—22no—
95C22×C4×C5C2×D412yes—
96C22×C4×C5C2411yescharacteristic
97C22×Dic5—21no—
98C22×Dic5—22no—
99C22×Dic5—21no—
100C22×Dic5C2×D411yescharacteristic
101C22×Dic5C2×D412yes—
102C2×C4×D5—22no—
103C2×C4×D5—22no—
104C2×C4×D5—42no—
105C2×C4×D5—82no—
106C2×C4×D5C22×C412yes—
107C2×C4×D5C2×Q812yes—
108C2×C5×C8—22no—
109C2×C5×C8—22no—
110C2×C5×C8C22×C414yes—
111C5×C4⋊C4—22no—
112C5×C4⋊C4—22no—
113C5×C4⋊C4—24no—
114C42×C5C22×C412yes—
115C42×C5C2×Q812yes—
116#685—42no—
117#685—82no—
118#685C4⋊C418yes—
119C4⋊Dic5—22no—
120C4⋊Dic5—22no—
121C4⋊Dic5—22no—
122C4⋊Dic5—22no—
123C4⋊Dic5—82no—
124C4⋊Dic5—82no—
125#683—42no—
126#683—82no—
127#683C4⋊C418yes—
128C4×Dic5—42no—
129C4×Dic5—82no—
130C4×Dic5C22×C412yes—
131C4×Dic5C2×Q812yes—
132C2×#168—22no—
133C2×#168—22no—
134C2×#168C22×C414yes—
135C2×#117—51no—
136C2×#117—51no—
137C22×C4⋊C4—51no—
138#433—58no—
139C2×#107—52no—
140C2×#107—52no—
141C2×#107—52no—
142C2×#106—52no—
143C2×C4⋊C8—52no—
144C2×C22⋊C8—52no—
145C2×#97—51no—
146C2×C4×C8C2×D511yescharacteristic
147#389—102no—
148#388—102no—
149#387—102no—
150#385—102no—
151#384—51no—
152#384—51no—
153#384—51no—
154#382—51no—
155#382C2×D511yescharacteristic
156#382C2×D511yescharacteristic
157#380—52no—
158C4×C4⋊C4—52no—
159#340—52no—
160#340—52no—
161#340C2×D512yes—
162#340C2×D512yes—
163C23×C5C22×D411yescharacteristic
164C22×D5—21no—
165C22×D5—21no—
166C22×D5—22no—
167C22×D5—22no—
168C22×D5C2×C4⋊C412yes—
169C2×C4×C5—22no—
170C2×C4×C5—22no—
171C2×C4×C5—41no—
172C2×C4×C5—41no—
173C2×C4×C5—42no—
174C2×C4×C5—42no—
175C2×C4×C5—42no—
176C2×C4×C5—42no—
177C2×C4×C5—42no—
178C2×C4×C5C2×C4⋊C414yes—
179C2×C4×C5C23×C411yescharacteristic
180C2×C4×C5C22×Q811yescharacteristic
181C2×C4×C5H(5,𝔽2)12yes—
182C2×C4×C52−1+412yes—
183C2×Dic5—21no—
184C2×Dic5—21no—
185C2×Dic5—22no—
186C2×Dic5—22no—
187C2×Dic5—41no—
188C2×Dic5—41no—
189C2×Dic5—42no—
190C2×Dic5—42no—
191C2×Dic5—42no—
192C2×Dic5—42no—
193C2×Dic5—42no—
194C2×Dic5C2×C4⋊C414yes—
195C2×Dic5C2×C4⋊C412yes—
196C4×D5—24no—
197C4×D5—24no—
198C4×D5—84no—
199C5×C8—24no—
200#168—24no—
201C23×C4—51no—
202C23×C4—101no—
203C22×C8—52no—
204C22×C8C22×D512yes—
205#117—54no—
206#117—54no—
207#117—202no—
208C2×C4⋊C4—52no—
209C2×C4⋊C4—51no—
210C2×C4⋊C4—51no—
211C2×C4⋊C4—52no—
212C2×C4⋊C4—101no—
213C2×C4⋊C4—102no—
214C2×C4⋊C4—102no—
215C2×C4⋊C4—102no—
216C2×C4⋊C4—202no—
217C2×C4⋊C4C22×D511yescharacteristic
218C2×C4⋊C4C22×D511yescharacteristic
219C2×C4⋊C4C22×D512yes—
220C2×C22⋊C4—52no—
221C2×C22⋊C4—101no—
222C2×C22⋊C4—102no—
223C2×C42—51no—
224C2×C42—101no—
225C2×C42—102no—
226C2×C42C22×D511yescharacteristic
227#107—202no—
228#107—202no—
229#107—202no—
230#106—202no—
231C4⋊C8—58no—
232C22⋊C8—58no—
233#97—54no—
234C4×C8C4×D514yescomplemented
235#95—22no—
236#95—22no—
237#95—102no—
238#95—102no—
239#95—102no—
240C22×C5C22×C4⋊C411yescharacteristic
241C22×C5#53511yescharacteristic
242C22×C5#53611yescharacteristic
243C22×C5#57612yes—
244C22×C5#57812yes—
245C2×D5—22no—
246C2×D5—24no—
247C2×D5—41no—
248C2×D5—41no—
249C2×D5—42no—
250C4×C5—24no—
251C4×C5—81no—
252C4×C5—81no—
253C4×C5—82no—
254C4×C5#52012yes—
255C4×C5#54412yes—
256Dic5—22no—
257Dic5—24no—
258Dic5—81no—
259Dic5—81no—
260Dic5—82no—
261C24—51no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—52no—
266C22×C4—51no—
267C22×C4—52no—
268C22×C4—52no—
269C22×C4—102no—
270C22×C4—102no—
271C22×C4—102no—
272C22×C4—101no—
273C22×C4—101no—
274C22×C4—202no—
275C22×C4—402no—
276C22×C4D4×D512yes—
277C22×C4C23×D511yescharacteristic
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—54no—
281C2×C8—102no—
282C2×C8—102no—
283C2×C8C2×C4×D514yes—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—58no—
288C4⋊C4—102no—
289C4⋊C4—102no—
290C4⋊C4—102no—
291C4⋊C4—102no—
292C4⋊C4—202no—
293C4⋊C4—402no—
294C4⋊C4—402no—
295C4⋊C4—402no—
296C22⋊C4—58no—
297C22⋊C4—202no—
298C22⋊C4—402no—
299C42—52no—
300C42—52no—
301C42—202no—
302C42—402no—
303C42C2×C4×D512yes—
304C42Q8×D512yes—
305C2×C5#231711yescharacteristic
306C2×C5#291312yes—
307C2×C5#322011yescharacteristic
308C2×C5#322311yescharacteristic
309C2×C5#324312yes—
310D5—42no—
311C23—52no—
312C23—102no—
313C23—102no—
314C23—101no—
315C23—101no—
316C23C2×D4×D511yescharacteristic
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—54no—
327C2×C4—52no—
328C2×C4—102no—
329C2×C4—102no—
330C2×C4—101no—
331C2×C4—101no—
332C2×C4—104no—
333C2×C4—104no—
334C2×C4—202no—
335C2×C4—202no—
336C2×C4—202no—
337C2×C4—202no—
338C2×C4—201no—
339C2×C4—201no—
340C2×C4—202no—
341C2×C4—404no—
342C2×C4D5×C4⋊C414yes—
343C2×C4C22×C4×D511yescharacteristic
344C2×C4C2×Q8×D511yescharacteristic
345C2×C4#414912yes—
346C2×C4#415012yes—
347C8—24no—
348C8—104no—
349C5#2199111yescharacteristic, complemented, Sylow
350C22—102no—
351C22—104no—
352C22—201no—
353C22—201no—
354C22—202no—
355C22C2×D5×C4⋊C411yescharacteristic
356C22#6600811yescharacteristic
357C22#6600911yescharacteristic
358C22#6610312yes—
359C22#6610512yes—
360C4—24no—
361C4—81no—
362C4—81no—
363C4—82no—
364C4—102no—
365C4—104no—
366C4—401no—
367C4—402no—
368C4—401no—
369C4#6585712yes—
370C4#6602312yes—
371C2—202no—
372C2#1060595611yescharacteristic
373C2#1061091012yes—
374C2#1061337511yescharacteristic
375C2#1061337811yescharacteristic
376C2#1061338112yes—
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1835C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1835D5
hypocenterC5#21991
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1835

Complements

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

NH ≅ G/Ntype
4. C5×#1835C2semidirect
5. #10600558C2semidirect
6. #10600343C2semidirect
7. #10600341C2semidirect
10. #10599591C2semidirect
11. #10599588C2semidirect
31. #64979C4semidirect
67. C4×C5×C8C2×C4semidirect
76. #3946C2×C4semidirect
79. #3935C2×C4semidirect
90. #1835D5semidirect
234. C4×C8C4×D5semidirect
349. C5#21991semidirect

Maximal subgroups

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

subgroupindex
C2×#649792
#106059562
#106059552
C5×#18352
#106005582
#106003432
#106003412
#105996702
#105996672
#105995912
#105995882
#219915