Finite Group Databasebeta

← #12098056

Subgroups — #12098056

Order 1280.

922 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65122C211yesmaximal, characteristic
2C2×#64974C211yesmaximal, characteristic
3#10605484C211yesmaximal, characteristic
4C5×#1770C211yesmaximal, characteristic, complemented
5#10601259C211yesmaximal, characteristic, complemented
6#10600626C212yesmaximal, complemented
7#10600623C212yesmaximal
8C8×C4⋊Dic5C211yesmaximal, characteristic, complemented
9C4⋊C4×#168C211yesmaximal, characteristic, complemented
10#10599444C212yesmaximal, complemented
11#10599394C212yesmaximal
12C2×#4018C2212yes—
13C2×C42×D5C2211yescharacteristic
14C2×C5×C4⋊C8C2211yescharacteristic
15C2×C5×#97C2212yes—
16C2×C4×C5×C8C2211yescharacteristic
17C4×C5×C4⋊C4C2211yescharacteristic
18C2×#3952C2211yescharacteristic
19C2×#3952C2211yescharacteristic
20C2×#3952C2212yes—
21C2×#3947C2212yes—
22C2×#3946C2211yescharacteristic
23C2×#3946C2211yescharacteristic
24C2×C8×Dic5C2212yes—
25C2×C8×Dic5C2212yes—
26#65276C412yes—
27#65265C412yes—
28C4×C4⋊Dic5C2211yescharacteristic
29C4×#683C2212yes—
30C2×C4⋊#168C2211yescharacteristic
31C2×C4×#168C2212yes—
32C2×C4×#168C2211yescharacteristic
33#65122C418yescomplemented
34#64974C418yescomplemented
35#64956C412yes—
36#64939C412yes—
37C5×#336C2212yes—
38#64770C2212yes—
39#64770C2212yes—
40#64744C2212yes—
41#20538—51nomaximal, Sylow
42C22×C4×D5C2×C411yescharacteristic
43C22×C4×D5C2×C411yescharacteristic
44C22×C4×D5C2311yescharacteristic
45C22×C5×C8C2311yescharacteristic
46C22×C5×C8C2312yes—
47C22×C5×C8C2311yescharacteristic
48C2×C5×C4⋊C4C2×C412yes—
49C2×C42×C5C2312yes—
50C2×C42×C5C2311yescharacteristic
51C2×#685C2×C414yes—
52C2×C4⋊Dic5C2×C412yes—
53C2×#683C2×C414yes—
54C2×#683C2×C414yes—
55C2×C4×Dic5C2×C411yescharacteristic
56C2×C4×Dic5C2×C411yescharacteristic
57C2×C4×Dic5C2312yes—
58C2×C4×Dic5C2311yescharacteristic
59C22×#168C2311yescharacteristic
60C22×#168C2312yes—
61C22×#168C2311yescharacteristic
62#4018—42no—
63#4018—42no—
64C42×D5C2×C414yes—
65C42×D5C2×C414yes—
66C5×C4⋊C8C2×C414yes—
67C5×#97—42no—
68C4×C5×C8C2×C414yes—
69C5×#95C2×C412yes—
70#3963C2×C412yes—
71#3963C2×C414yes—
72#3952C2×C418yes—
73#3952C2×C414yes—
74#3952C2×C414yes—
75#3947—42no—
76#3946C2×C414yes—
77#3946C2×C414yes—
78C8×Dic5—42no—
79C8×Dic5C2×C418yes—
80C4⋊#168C2×C414yes—
81C4×#168—42no—
82C4×#168C2×C414yes—
83C2×#431—51no—
84C2×#431—51no—
85#2285—51no—
86#1770—51no—
87#1770—52no—
88#1770D511yescharacteristic, complemented
89C8×C4⋊C4—51no—
90C8×C4⋊C4—51no—
91C8×C4⋊C4—52no—
92#1750—52no—
93#1750—52no—
94C23×D5C22×C411yescharacteristic
95C22×C4×C5C22×C414yes—
96C22×C4×C5C22×C411yescharacteristic
97C22×C4×C5C22×C411yescharacteristic
98C22×C4×C5C2411yescharacteristic
99C22×Dic5C22×C414yes—
100C22×Dic5C22×C411yescharacteristic
101C22×Dic5C22×C411yescharacteristic
102C22×Dic5C22×C411yescharacteristic
103C2×C4×D5—24no—
104C2×C4×D5—21no—
105C2×C4×D5—21no—
106C2×C4×D5—21no—
107C2×C4×D5—21no—
108C2×C4×D5—24no—
109C2×C4×D5C4214yes—
110C2×C4×D5C4214yes—
111C2×C4×D5C22×C412yes—
112C2×C4×D5C22×C412yes—
113C2×C5×C8—21no—
114C2×C5×C8—22no—
115C2×C5×C8—21no—
116C2×C5×C8—21no—
117C2×C5×C8—22no—
118C2×C5×C8—21no—
119C2×C5×C8C22×C414yes—
120C2×C5×C8C22×C412yes—
121C2×C5×C8C22×C412yes—
122C5×C4⋊C4—24no—
123C42×C5—22no—
124C42×C5—22no—
125C42×C5C22×C412yes—
126C42×C5C22×C412yes—
127#685—44no—
128C4⋊Dic5—24no—
129#683—28no—
130#683—44no—
131C4×Dic5—22no—
132C4×Dic5—22no—
133C4×Dic5C4214yes—
134C4×Dic5C4214yes—
135C4×Dic5C22×C412yes—
136C4×Dic5C22×C412yes—
137C2×#168—21no—
138C2×#168—22no—
139C2×#168—21no—
140C2×#168—22no—
141C2×#168—21no—
142C2×#168—21no—
143C2×#168C22×C412yes—
144C2×#168C22×C412yes—
145C2×#168C22×C414yes—
146C2×#117—52no—
147C22×C42—51no—
148#431—58no—
149#431—58no—
150C2×C4⋊C8—51no—
151C2×C4⋊C8—51no—
152C2×C4⋊C8—51no—
153C2×C4⋊C8C2×D511yescharacteristic
154C2×C22⋊C8—51no—
155C2×C22⋊C8—52no—
156C2×C22⋊C8—51no—
157C2×#97—52no—
158C2×#97C2×D512yes—
159C2×C4×C8—52no—
160C2×C4×C8—51no—
161C2×C4×C8—52no—
162C2×C4×C8—52no—
163C2×C4×C8C2×D511yescharacteristic
164#384—52no—
165#382—52no—
166#380—52no—
167#380—52no—
168C4×C4⋊C4—52no—
169C4×C4⋊C4—51no—
170C4×C4⋊C4C2×D511yescharacteristic
171#336—52no—
172#336—52no—
173#336—52no—
174#336C2×D512yes—
175C23×C5C23×C411yescharacteristic
176C22×D5—21no—
177C22×D5—21no—
178C22×D5—22no—
179C22×D5—22no—
180C22×D5C2×C4212yes—
181C2×C4×C5—22no—
182C2×C4×C5—24no—
183C2×C4×C5—24no—
184C2×C4×C5—24no—
185C2×C4×C5C2×C4212yes—
186C2×C4×C5C2×C4212yes—
187C2×C4×C5#13114yes—
188C2×C4×C5C23×C411yescharacteristic
189C2×C4×C5C23×C411yescharacteristic
190C2×C4×C5H(5,𝔽2)11yescharacteristic
191C2×C4×C5H(5,𝔽2)11yescharacteristic
192C2×C4×C52−1+411yescharacteristic
193C2×C4×C52−1+411yescharacteristic
194C2×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—22no—
197C2×Dic5—24no—
198C2×Dic5—24no—
199C2×Dic5—24no—
200C2×Dic5C2×C4212yes—
201C2×Dic5C2×C4212yes—
202C2×Dic5C2×C4212yes—
203C2×Dic5#13114yes—
204C4×D5—22no—
205C4×D5—22no—
206C4×D5—22no—
207C4×D5—22no—
208C4×D5—28no—
209C4×D5—28no—
210C5×C8—22no—
211C5×C8—22no—
212C5×C8—24no—
213#168—22no—
214#168—22no—
215#168—24no—
216C23×C4—51no—
217C23×C4—51no—
218C23×C4—51no—
219C22×C8—52no—
220C22×C8—51no—
221C22×C8—51no—
222C22×C8C22×D511yescharacteristic
223C22×C8C22×D512yes—
224C22×C8C22×D511yescharacteristic
225#117—202no—
226#117—202no—
227C2×C4⋊C4—54no—
228C2×C4⋊C4—52no—
229C2×C4⋊C4—54no—
230C2×C4⋊C4C4×D512yes—
231C2×C22⋊C4—54no—
232C2×C42—51no—
233C2×C42—51no—
234C2×C42—51no—
235C2×C42—54no—
236C2×C42—54no—
237C2×C42—52no—
238C2×C42C22×D512yes—
239C2×C42C22×D511yescharacteristic
240C4⋊C8—54no—
241C4⋊C8—54no—
242C4⋊C8—54no—
243C4⋊C8C4×D514yes—
244C22⋊C8—58no—
245C22⋊C8—54no—
246C22⋊C8—54no—
247#97—42no—
248#97—202no—
249C4×C8—58no—
250C4×C8—54no—
251C4×C8—202no—
252C4×C8—202no—
253C4×C8C4×D514yes—
254#95—54no—
255#95—52no—
256#95C4×D512yes—
257C22×C5C22×C4211yescharacteristic
258C22×C5#51811yescharacteristic
259C22×C5#51911yescharacteristic
260C22×C5C2×#13112yes—
261C22×C5C2×#13112yes—
262C2×D5—22no—
263C2×D5—22no—
264C2×D5—22no—
265C2×D5—42no—
266C2×D5—42no—
267C4×C5—22no—
268C4×C5—44no—
269C4×C5#40514yes—
270C4×C5#52011yescharacteristic
271C4×C5#52011yescharacteristic
272C4×C5#52011yescharacteristic
273C4×C5#52011yescharacteristic
274Dic5—22no—
275Dic5—22no—
276Dic5—44no—
277Dic5#40514yes—
278C24—51no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—52no—
283C22×C4—52no—
284C22×C4—54no—
285C22×C4—54no—
286C22×C4—54no—
287C22×C4—104no—
288C22×C4—101no—
289C22×C4—101no—
290C22×C4—101no—
291C22×C4—101no—
292C22×C4—104no—
293C22×C4C2×C4×D514yes—
294C22×C4C2×C4×D511yescharacteristic
295C22×C4C2×C4×D511yescharacteristic
296C22×C4C23×D511yescharacteristic
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—52no—
304C2×C8—52no—
305C2×C8—54no—
306C2×C8—101no—
307C2×C8—101no—
308C2×C8—102no—
309C2×C8—101no—
310C2×C8—102no—
311C2×C8—101no—
312C2×C8C2×C4×D514yes—
313C2×C8C2×C4×D512yes—
314C2×C8C2×C4×D512yes—
315C4⋊C4—24no—
316C4⋊C4—108no—
317C4⋊C4—104no—
318C4⋊C4—204no—
319C22⋊C4—204no—
320C42—22no—
321C42—22no—
322C42—52no—
323C42—52no—
324C42—54no—
325C42—54no—
326C42—102no—
327C42—102no—
328C42C2×C4×D512yes—
329C42C2×C4×D512yes—
330C2×C5#227411yescharacteristic
331C2×C5C2×#40511yescharacteristic
332C2×C5C2×#40511yescharacteristic
333C2×C5#297912yes—
334C2×C5#298612yes—
335D5—42no—
336C23—52no—
337C23—102no—
338C23—102no—
339C23—101no—
340C23—101no—
341C23C22×C4×D511yescharacteristic
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—52no—
347C2×C4—54no—
348C2×C4—52no—
349C2×C4—52no—
350C2×C4—108no—
351C2×C4—102no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—102no—
355C2×C4—102no—
356C2×C4—102no—
357C2×C4—102no—
358C2×C4—108no—
359C2×C4—104no—
360C2×C4—104no—
361C2×C4—104no—
362C2×C4C42×D512yes—
363C2×C4C42×D512yes—
364C2×C4#405314yes—
365C2×C4C22×C4×D511yescharacteristic
366C2×C4C22×C4×D511yescharacteristic
367C2×C4#414411yescharacteristic
368C2×C4#414711yescharacteristic
369C2×C4#414911yescharacteristic
370C2×C4#415011yescharacteristic
371C8—22no—
372C8—22no—
373C8—24no—
374C8—102no—
375C8—102no—
376C8—104no—
377C5#2053811yescharacteristic, complemented, Sylow
378C22—102no—
379C22—102no—
380C22—102no—
381C22—202no—
382C22—202no—
383C22C2×C42×D511yescharacteristic
384C22#6587911yescharacteristic
385C22#6590611yescharacteristic
386C22C2×#405312yes—
387C22D5×#13112yes—
388C4—22no—
389C4—44no—
390C4—54no—
391C4—102no—
392C4—102no—
393C4—204no—
394C4#6499714yes—
395C4#6583811yescharacteristic
396C4#6585711yescharacteristic
397C4#6588011yescharacteristic
398C4#6590811yescharacteristic
399C2—202no—
400C2#1060548111yescharacteristic
401C2C2×#6499711yescharacteristic
402C2D5×#40511yescharacteristic
403C2#1061127712yes—
404C2#1061169812yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutatorC22×C5C22×C42
FittingC5×#1770C2
socleC23×C5C23×C4
prime radicalC23×D5C22×C4
cosocleC22×C4×C5C24
hypercenter#1770D5
hypocenterC5#20538
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1770

Complements

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

NH ≅ G/Ntype
4. C5×#1770C2semidirect
5. #10601259C2semidirect
6. #10600626C2semidirect
8. C8×C4⋊Dic5C2semidirect
9. C4⋊C4×#168C2semidirect
10. #10599444C2semidirect
33. #65122C4semidirect
34. #64974C4semidirect
88. #1770D5semidirect
377. C5#20538semidirect

Maximal subgroups

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

subgroupindex
C2×#651222
C2×#649742
#106054842
C5×#17702
#106012592
#106006262
#106006232
C8×C4⋊Dic52
C4⋊C4×#1682
#105994442
#105993942
#205385