Finite Group Databasebeta

← #13034629

Subgroups — #13034629

Order 1280.

616 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
1#10610280C211yesmaximal, characteristic
2#10604764C212yesmaximal
3#10604733C211yesmaximal, characteristic
4#10604727C211yesmaximal, characteristic
5#10604377C212yesmaximal
6#10603887C212yesmaximal, complemented
7#10603884C212yesmaximal, complemented
8#10603852C212yesmaximal, complemented
9#10603851C212yesmaximal, complemented
10C2×#4053C2211yescharacteristic
11C2×#4047C2211yescharacteristic
12#65856C2211yescharacteristic
13#65761C2212yes—
14#65745C2212yes—
15D5⋊#108C2212yes—
16#65721C2212yes—
17#65718C2214yes—
18C5×#405C411yescharacteristic, complemented
19M16×Dic5C411yescharacteristic, complemented
20C2×C8×Dic5C411yescharacteristic, complemented
21#65286C411yescharacteristic, complemented
22#64997C411yescharacteristic, complemented
23#64997C2211yescharacteristic
24#64996C411yescharacteristic, complemented
25#64996C2211yescharacteristic
26#64978C411yescharacteristic, complemented
27#64978C2211yescharacteristic
28C4×C8×D5C411yescharacteristic, complemented
29C4×C8×D5C2211yescharacteristic
30#64897C2212yes—
31#64895C2212yes—
32#64894C2214yes—
33#64863C2214yes—
34#64862C2214yes—
35#20787—51nomaximal, Sylow
36C2×#709C2311yescharacteristic
37#4130C2314yes—
38D5⋊M16C2314yes—
39C2×C5×M16C2×C411yescharacteristic
40C22×C5×C8C2×C411yescharacteristic
41C5×#117C2×C411yescharacteristic
42C2×C4×Dic5C2×C411yescharacteristic
43C2×#681C2×C411yescharacteristic
44C22×#168C2×C411yescharacteristic
45#4053—22no—
46#4053—22no—
47#4047—22no—
48#4047—22no—
49C2×#676C2×C411yescharacteristic
50C2×#676C2×C411yescharacteristic
51C2×#676C2311yescharacteristic
52C2×#676C2311yescharacteristic
53C2×C8×D5C2×C411yescharacteristic
54C2×C8×D5C2×C411yescharacteristic
55C2×C8×D5C2311yescharacteristic
56C2×C8×D5C2311yescharacteristic
57#4039—21no—
58#4033—21no—
59#4027—21no—
60#4027—21no—
61#4018C2×C411yescharacteristic
62#4018C2311yescharacteristic
63C42×D5C2×C411yescharacteristic
64C42×D5C2311yescharacteristic
65#4013—22no—
66#4011—22no—
67#4005—22no—
68#3999—22no—
69#3996—22no—
70#3995—22no—
71#3994—22no—
72#3993—22no—
73#3992—24no—
74C8×AGL(1,𝔽5)—24no—
75C5×#97C2×C411yescharacteristic
76C5×#97C2×C411yescharacteristic
77C4×C5×C8C2×C411yescharacteristic
78C4×C5×C8C2×C411yescharacteristic
79#3947C2×C411yescharacteristic
80#3947C2×C411yescharacteristic
81C8×Dic5C2×C411yescharacteristic
82C8×Dic5C2×C411yescharacteristic
83C8×Dic5C2×C411yescharacteristic
84C8×Dic5C2×C411yescharacteristic
85C8×Dic5C2×C411yescharacteristic
86C8×Dic5C2×C411yescharacteristic
87#3935C2×C411yescharacteristic
88#3935C2×C411yescharacteristic
89C4×#168C2×C411yescharacteristic
90C4×#168C2×C411yescharacteristic
91C4×#131—51no—
92#1841—52no—
93#1833—52no—
94#1827—52no—
95#1813—51no—
96#1812—51no—
97#1772—52no—
98#1757—52no—
99#1747—52no—
100C22×C4×C5C22×C411yescharacteristic
101C2×C22⋊D5#4111yescharacteristic
102C2×C22⋊D5#4111yescharacteristic
103C22×Dic5—21no—
104#709—22no—
105#709—22no—
106C2×D20#4111yescharacteristic
107C2×C4×D5—21no—
108C2×C4×D5—21no—
109C2×C4×D5C22×C411yescharacteristic
110C2×C4×D5C2411yescharacteristic
111C2×Dic10#4111yescharacteristic
112C22⋊AGL(1,𝔽5)—24no—
113#704—24no—
114C4⋊AGL(1,𝔽5)—24no—
115C4×AGL(1,𝔽5)—24no—
116#700—24no—
117D5⋊C8—24no—
118C5×M16—22no—
119C2×C5×C8—21no—
120C2×C5×C8—21no—
121C2×C5×C8C22×C411yescharacteristic
122C2×C5×C8C22×C411yescharacteristic
123C2×C5×C8C22×C411yescharacteristic
124C2×C5×C8C22×C411yescharacteristic
125C5×C4⋊C4—21no—
126C5×C22⋊C4—21no—
127C42×C5C22×C411yescharacteristic
128C42×C5C22×C411yescharacteristic
129#685—21no—
130#685—21no—
131#683—21no—
132#683—21no—
133C4×Dic5—21no—
134C4×Dic5—21no—
135C4×Dic5—21no—
136C4×Dic5C22×C411yescharacteristic
137C4×Dic5C22×C411yescharacteristic
138#681—22no—
139C2×#168—21no—
140C2×#168—21no—
141C2×#168C22×C411yescharacteristic
142C2×#168C22×C411yescharacteristic
143C2×#168C22×C411yescharacteristic
144C2×#168C22×C411yescharacteristic
145#676—22no—
146#676—22no—
147#676—22no—
148#676—22no—
149C8×D5—21no—
150C8×D5—21no—
151C8×D5—21no—
152C8×D5—21no—
153C8×D5#4112yes—
154C8×D5#4112yes—
155C8×D5#4112yes—
156C8×D5#4112yes—
157C2×#131—51no—
158C2×#131—51no—
159C4×#41—51no—
160C2×#108—52no—
161#427—52no—
162#413—52no—
163#410—52no—
164#405—51no—
165#405—51no—
166#405—51no—
167#405—51no—
168#405—51no—
169#405—54no—
170#405AGL(1,𝔽5)11yescharacteristic, complemented
171C4×M16—51no—
172C4×M16—51no—
173C4×M16—51no—
174C2×C4×C8—51no—
175C2×C4×C8—51no—
176C2×C4×C8—51no—
177#344—54no—
178#339—54no—
179#338—52no—
180H(3,ℤ/4)—52no—
181H(3,ℤ/4)—54no—
182C23×C5#11711yescharacteristic
183C22×D5#11711yescharacteristic
184C22×D5C2×#4111yescharacteristic
185C2×AGL(1,𝔽5)—44no—
186C2×C4×C5—21no—
187C2×C4×C5—21no—
188C2×C4×C5—21no—
189C2×C4×C5—21no—
190C2×C4×C5#11711yescharacteristic
191C2×C4×C5C23×C411yescharacteristic
192C22⋊D5—22no—
193C22⋊D5—22no—
194C2×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—42no—
197C2×Dic5#11711yescharacteristic
198C2×Dic5C2×#4111yescharacteristic
199D20—22no—
200C4×D5—21no—
201C4×D5—21no—
202C4×D5—42no—
203C4×D5—42no—
204C4×D5C2×#4112yes—
205C4×D5C2×#4112yes—
206Dic10—22no—
207#170—44no—
208C5×C8—21no—
209C5×C8—21no—
210C5×C8#11711yescharacteristic
211C5×C8#11711yescharacteristic
212C5×C8#11712yes—
213#168—21no—
214#168—21no—
215#168#11711yescharacteristic
216#168#11711yescharacteristic
217#168#11712yes—
218C2×#41—51no—
219#131—102no—
220#131—102no—
221#131—102no—
222#131—102no—
223C2×M16—51no—
224C2×M16—51no—
225C2×M16—51no—
226C2×M16—51no—
227C2×M16—51no—
228C2×M16—54no—
229C2×M16C2×AGL(1,𝔽5)11yescharacteristic
230C22×C8—51no—
231C22×C8—51no—
232C22×C8—51no—
233C22×C8—51no—
234C22×C8—51no—
235C22×C8C2×AGL(1,𝔽5)11yescharacteristic
236C4×Q8—101no—
237C4×D4—101no—
238C4×D4—101no—
239C4×D4—101no—
240#117—51no—
241#117—51no—
242#117—54no—
243#117C2×AGL(1,𝔽5)11yescharacteristic
244C2×C42—51no—
245C2×C42—51no—
246C2×C42—51no—
247#108—102no—
248#108—102no—
249#107—102no—
250#106—102no—
251#101—102no—
252#100—102no—
253#99—102no—
254#99—102no—
255#97—51no—
256#97—51no—
257#97—51no—
258#97—51no—
259#97—104no—
260#97C2×AGL(1,𝔽5)11yescharacteristic
261#97C2×AGL(1,𝔽5)11yescharacteristic
262C4×C8—51no—
263C4×C8—51no—
264C4×C8—51no—
265C4×C8—51no—
266C4×C8—51no—
267C4×C8—51no—
268C4×C8—51no—
269C4×C8—51no—
270C4×C8—104no—
271C4×C8C2×AGL(1,𝔽5)11yescharacteristic
272C4×C8C2×AGL(1,𝔽5)11yescharacteristic
273C22×C5—21no—
274C22×C5—41no—
275C22×C5C2×#11711yescharacteristic
276C2×D5—21no—
277C2×D5—41no—
278C2×D5—41no—
279C2×D5#52912yes—
280AGL(1,𝔽5)—84no—
281C4×C5—21no—
282C4×C5—41no—
283C4×C5—41no—
284C4×C5C2×#11711yescharacteristic
285C4×C5C2×#11711yescharacteristic
286Dic5—22no—
287Dic5—21no—
288Dic5—41no—
289Dic5#52912yes—
290#41—102no—
291#41—102no—
292C2×Q8—51no—
293C2×D4—51no—
294C2×D4—51no—
295C2×D4—51no—
296C22×C4—51no—
297C22×C4—51no—
298C22×C4—101no—
299C22×C4—101no—
300C22×C4—101no—
301C22×C4C22×AGL(1,𝔽5)11yescharacteristic
302M16—22no—
303M16—102no—
304M16—102no—
305M16—102no—
306M16—102no—
307M16—102no—
308M16—104no—
309M16—104no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—51no—
313C2×C8—51no—
314C2×C8—52no—
315C2×C8—52no—
316C2×C8—52no—
317C2×C8—52no—
318C2×C8—51no—
319C2×C8—51no—
320C2×C8—101no—
321C2×C8—101no—
322C2×C8—101no—
323C2×C8—101no—
324C2×C8—101no—
325C2×C8—101no—
326C2×C8—104no—
327C2×C8C22×AGL(1,𝔽5)11yescharacteristic
328C2×C8C22×AGL(1,𝔽5)11yescharacteristic
329C2×C8C22×AGL(1,𝔽5)11yescharacteristic
330C2×C8C22×AGL(1,𝔽5)11yescharacteristic
331C4⋊C4—21no—
332C4⋊C4—101no—
333C4⋊C4—101no—
334C4⋊C4—104no—
335C22⋊C4—21no—
336C22⋊C4—101no—
337C22⋊C4—101no—
338C22⋊C4—104no—
339C42—51no—
340C42—51no—
341C42—101no—
342C42—101no—
343C42—104no—
344C42—101no—
345C42C22×AGL(1,𝔽5)11yescharacteristic
346C42C22×AGL(1,𝔽5)11yescharacteristic
347C2×C5—21no—
348C2×C5—41no—
349C2×C5#228311yescharacteristic
350D5—22no—
351D5—41no—
352C23—51no—
353C23—51no—
354C23#413011yescharacteristic
355Q8—102no—
356D4—102no—
357D4—102no—
358D4—102no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—51no—
364C2×C4—52no—
365C2×C4—52no—
366C2×C4—51no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—101no—
371C2×C4—202no—
372C2×C4—202no—
373C2×C4—202no—
374C2×C4—204no—
375C2×C4#413011yescharacteristic
376C2×C4C23×AGL(1,𝔽5)11yescharacteristic
377C8—21no—
378C8—21no—
379C8—52no—
380C8—51no—
381C8—51no—
382C8—101no—
383C8—101no—
384C8—204no—
385C8#413011yescharacteristic
386C8#413011yescharacteristic
387C8#413012yes—
388C5#2078711yescharacteristic, complemented, Sylow
389C22—21no—
390C22—41no—
391C22—52no—
392C22—101no—
393C22—201no—
394C22—201no—
395C22C2×#413011yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—52no—
400C4—102no—
401C4—101no—
402C4—201no—
403C4—404no—
404C4C2×#413011yescharacteristic
405C4C2×#413011yescharacteristic
406C2—21no—
407C2—41no—
408C2—102no—
409C2—201no—
410C2#1061704611yescharacteristic
subgroupG/H
centerC4C2×#4130
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC5×#405C4
socleC2×C5#2283
prime radicalC2×#709C23
cosocleC2×C4×D5C24
hypercenter#405AGL(1,𝔽5)
hypocenterC5#20787
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4 ◁ #97 ◁ #405

Complements

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

NH ≅ G/Ntype
6. #10603887C2semidirect
7. #10603884C2semidirect
8. #10603852C2semidirect
9. #10603851C2semidirect
18. C5×#405C4semidirect
19. M16×Dic5C4semidirect
20. C2×C8×Dic5C4semidirect
21. #65286C4semidirect
22. #64997C4semidirect
24. #64996C4semidirect
26. #64978C4semidirect
28. C4×C8×D5C4semidirect
170. #405AGL(1,𝔽5)semidirect
388. C5#20787semidirect

Maximal subgroups

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

subgroupindex
#106102802
#106047642
#106047332
#106047272
#106043772
#106038872
#106038842
#106038522
#106038512
#207875