Finite Group Databasebeta

← #13034632

Subgroups — #13034632

Order 1280.

608 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10610281C211yesmaximal, characteristic
2#10604773C211yesmaximal, characteristic
3#10604765C211yesmaximal, characteristic
4#10604726C212yesmaximal
5#10604377C212yesmaximal
6#10603885C212yesmaximal, complemented
7#10603884C212yesmaximal, complemented
8#10603853C212yesmaximal, complemented
9#10603852C212yesmaximal, complemented
10C2×#4053C2211yescharacteristic
11C2×#4047C2211yescharacteristic
12#65856C2211yescharacteristic
13#65761C2212yes—
14#65745C2212yes—
15D5⋊#108C2212yes—
16#65721C2212yes—
17#65718C2214yes—
18C5×#405C411yescharacteristic, complemented
19M16×Dic5C411yescharacteristic, complemented
20C2×#3947C411yescharacteristic, complemented
21#65286C411yescharacteristic, complemented
22#64997C411yescharacteristic, complemented
23#64997C2211yescharacteristic
24D5×#97C411yescharacteristic, complemented
25D5×#97C2211yescharacteristic
26#64978C411yescharacteristic, complemented
27#64978C2211yescharacteristic
28C4×#676C411yescharacteristic, complemented
29C4×#676C2211yescharacteristic
30#64897C2212yes—
31#64895C2212yes—
32#64894C2214yes—
33#64863C2214yes—
34#64862C2214yes—
35#20788—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—24no—
48C2×#676C2×C411yescharacteristic
49C2×#676C2×C411yescharacteristic
50C2×#676C2311yescharacteristic
51C2×#676C2311yescharacteristic
52C2×C8×D5C2×C411yescharacteristic
53C2×C8×D5C2×C411yescharacteristic
54C2×C8×D5C2311yescharacteristic
55C2×C8×D5C2311yescharacteristic
56#4039—21no—
57#4033—21no—
58#4027—21no—
59#4027—21no—
60#4018C2×C411yescharacteristic
61#4018C2311yescharacteristic
62C42×D5C2×C411yescharacteristic
63C42×D5C2311yescharacteristic
64#4013—22no—
65#4011—22no—
66#4005—22no—
67#3999—22no—
68#3996—22no—
69#3995—22no—
70#3994—22no—
71#3993—22no—
72#3992—24no—
73C8×AGL(1,𝔽5)—24no—
74C5×#97C2×C411yescharacteristic
75C5×#97C2×C411yescharacteristic
76C4×C5×C8C2×C411yescharacteristic
77C4×C5×C8C2×C411yescharacteristic
78#3947C2×C411yescharacteristic
79#3947C2×C411yescharacteristic
80#3947C2×C411yescharacteristic
81#3947C2×C411yescharacteristic
82#3947C2×C411yescharacteristic
83#3947C2×C411yescharacteristic
84C8×Dic5C2×C411yescharacteristic
85C8×Dic5C2×C411yescharacteristic
86#3935C2×C411yescharacteristic
87#3935C2×C411yescharacteristic
88C4×#168C2×C411yescharacteristic
89C4×#168C2×C411yescharacteristic
90#2875—51no—
91#1841—52no—
92#1833—52no—
93#1827—52no—
94#1813—51no—
95#1812—51no—
96#1773—52no—
97#1757—52no—
98#1747—52no—
99C22×C4×C5C22×C411yescharacteristic
100C2×C22⋊D5#4111yescharacteristic
101C2×C22⋊D5#4111yescharacteristic
102C22×Dic5—21no—
103#709—22no—
104#709—22no—
105C2×D20#4111yescharacteristic
106C2×C4×D5—21no—
107C2×C4×D5—21no—
108C2×C4×D5C22×C411yescharacteristic
109C2×C4×D5C2411yescharacteristic
110C2×Dic10#4111yescharacteristic
111C22⋊AGL(1,𝔽5)—24no—
112#704—24no—
113C4⋊AGL(1,𝔽5)—24no—
114C4×AGL(1,𝔽5)—24no—
115#700—24no—
116D5⋊C8—24no—
117C5×M16—22no—
118C2×C5×C8—22no—
119C2×C5×C8C22×C411yescharacteristic
120C2×C5×C8C22×C411yescharacteristic
121C2×C5×C8C22×C411yescharacteristic
122C2×C5×C8C22×C411yescharacteristic
123C5×C4⋊C4—21no—
124C5×C22⋊C4—21no—
125C42×C5C22×C411yescharacteristic
126C42×C5C22×C411yescharacteristic
127#685—21no—
128#685—21no—
129#683—21no—
130#683—21no—
131C4×Dic5—21no—
132C4×Dic5—21no—
133C4×Dic5—21no—
134C4×Dic5C22×C411yescharacteristic
135C4×Dic5C22×C411yescharacteristic
136#681—22no—
137C2×#168—21no—
138C2×#168—21no—
139C2×#168C22×C411yescharacteristic
140C2×#168C22×C411yescharacteristic
141C2×#168C22×C411yescharacteristic
142C2×#168C22×C411yescharacteristic
143#676—22no—
144#676—22no—
145#676—22no—
146#676—22no—
147C8×D5—22no—
148C8×D5—21no—
149C8×D5—21no—
150C8×D5#4114yes—
151C8×D5#4112yes—
152C8×D5#4112yes—
153C2×#131—51no—
154C2×#131—51no—
155C4×#41—51no—
156C2×#108—52no—
157#427—52no—
158#413—52no—
159#410—52no—
160#405—51no—
161#405—51no—
162#405—51no—
163#405—51no—
164#405—51no—
165#405—54no—
166#405AGL(1,𝔽5)11yescharacteristic, complemented
167C4×M16—51no—
168C4×M16—51no—
169C4×M16—51no—
170C2×#97—51no—
171C2×#97—51no—
172C2×#97—51no—
173#344—54no—
174#339—54no—
175#338—52no—
176H(3,ℤ/4)—52no—
177H(3,ℤ/4)—54no—
178C23×C5#11711yescharacteristic
179C22×D5#11711yescharacteristic
180C22×D5C2×#4111yescharacteristic
181C2×AGL(1,𝔽5)—44no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—21no—
185C2×C4×C5—21no—
186C2×C4×C5#11711yescharacteristic
187C2×C4×C5C23×C411yescharacteristic
188C22⋊D5—22no—
189C22⋊D5—22no—
190C2×Dic5—21no—
191C2×Dic5—21no—
192C2×Dic5—81no—
193C2×Dic5#11711yescharacteristic
194C2×Dic5C2×#4111yescharacteristic
195D20—22no—
196C4×D5—21no—
197C4×D5—21no—
198C4×D5—81no—
199C4×D5—81no—
200C4×D5C2×#4112yes—
201C4×D5C2×#4112yes—
202Dic10—22no—
203#170—44no—
204C5×C8—21no—
205C5×C8—21no—
206C5×C8#11712yes—
207C5×C8#11712yes—
208#168—21no—
209#168—21no—
210#168#11711yescharacteristic
211#168#11711yescharacteristic
212#168#11712yes—
213C2×#41—51no—
214#131—104no—
215#131—102no—
216#131—102no—
217C2×M16—51no—
218C2×M16—51no—
219C2×M16—51no—
220C2×M16—51no—
221C2×M16—51no—
222C2×M16—54no—
223C2×M16C2×AGL(1,𝔽5)11yescharacteristic
224C22×C8—51no—
225C22×C8—51no—
226C22×C8—51no—
227C22×C8—51no—
228C22×C8—51no—
229C22×C8C2×AGL(1,𝔽5)11yescharacteristic
230C4×Q8—101no—
231C4×D4—101no—
232C4×D4—101no—
233C4×D4—101no—
234#117—51no—
235#117—51no—
236#117—54no—
237#117C2×AGL(1,𝔽5)11yescharacteristic
238C2×C42—51no—
239C2×C42—51no—
240C2×C42—51no—
241#108—102no—
242#108—102no—
243#107—102no—
244#106—102no—
245#101—102no—
246#100—102no—
247#99—102no—
248#99—102no—
249#97—51no—
250#97—51no—
251#97—51no—
252#97—51no—
253#97—51no—
254#97—51no—
255#97—51no—
256#97—51no—
257#97—104no—
258#97C2×AGL(1,𝔽5)11yescharacteristic
259#97C2×AGL(1,𝔽5)11yescharacteristic
260C4×C8—51no—
261C4×C8—51no—
262C4×C8—51no—
263C4×C8—51no—
264C4×C8—104no—
265C4×C8C2×AGL(1,𝔽5)11yescharacteristic
266C4×C8C2×AGL(1,𝔽5)11yescharacteristic
267C22×C5—21no—
268C22×C5—41no—
269C22×C5C2×#11711yescharacteristic
270C2×D5—21no—
271C2×D5—41no—
272C2×D5—41no—
273C2×D5#52912yes—
274AGL(1,𝔽5)—84no—
275C4×C5—21no—
276C4×C5—41no—
277C4×C5—41no—
278C4×C5C2×#11711yescharacteristic
279C4×C5C2×#11711yescharacteristic
280Dic5—21no—
281Dic5—41no—
282Dic5—41no—
283Dic5#52912yes—
284#41—102no—
285#41—102no—
286C2×Q8—51no—
287C2×D4—51no—
288C2×D4—51no—
289C2×D4—51no—
290C22×C4—51no—
291C22×C4—51no—
292C22×C4—101no—
293C22×C4—101no—
294C22×C4—101no—
295C22×C4C22×AGL(1,𝔽5)11yescharacteristic
296M16—22no—
297M16—102no—
298M16—102no—
299M16—102no—
300M16—102no—
301M16—102no—
302M16—104no—
303M16—104no—
304C2×C8—22no—
305C2×C8—51no—
306C2×C8—51no—
307C2×C8—54no—
308C2×C8—52no—
309C2×C8—52no—
310C2×C8—51no—
311C2×C8—51no—
312C2×C8—101no—
313C2×C8—101no—
314C2×C8—101no—
315C2×C8—101no—
316C2×C8—102no—
317C2×C8—104no—
318C2×C8C22×AGL(1,𝔽5)11yescharacteristic
319C2×C8C22×AGL(1,𝔽5)11yescharacteristic
320C2×C8C22×AGL(1,𝔽5)11yescharacteristic
321C2×C8C22×AGL(1,𝔽5)11yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—101no—
324C4⋊C4—101no—
325C4⋊C4—104no—
326C22⋊C4—21no—
327C22⋊C4—101no—
328C22⋊C4—101no—
329C22⋊C4—104no—
330C42—51no—
331C42—51no—
332C42—101no—
333C42—101no—
334C42—104no—
335C42—101no—
336C42C22×AGL(1,𝔽5)11yescharacteristic
337C42C22×AGL(1,𝔽5)11yescharacteristic
338C2×C5—21no—
339C2×C5—41no—
340C2×C5#228311yescharacteristic
341D5—22no—
342D5—41no—
343C23—51no—
344C23—51no—
345C23#413011yescharacteristic
346Q8—102no—
347D4—102no—
348D4—102no—
349D4—102no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—51no—
355C2×C4—52no—
356C2×C4—52no—
357C2×C4—51no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—204no—
363C2×C4—401no—
364C2×C4—401no—
365C2×C4—401no—
366C2×C4#413011yescharacteristic
367C2×C4C23×AGL(1,𝔽5)11yescharacteristic
368C8—21no—
369C8—21no—
370C8—51no—
371C8—51no—
372C8—52no—
373C8—101no—
374C8—101no—
375C8—204no—
376C8#413012yes—
377C8#413012yes—
378C5#2078811yescharacteristic, complemented, Sylow
379C22—21no—
380C22—41no—
381C22—52no—
382C22—101no—
383C22—201no—
384C22—201no—
385C22C2×#413011yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—52no—
390C4—101no—
391C4—201no—
392C4—201no—
393C4—404no—
394C4C2×#413011yescharacteristic
395C4C2×#413011yescharacteristic
396C2—21no—
397C2—41no—
398C2—102no—
399C2—201no—
400C2#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#20788
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. #10603885C2semidirect
7. #10603884C2semidirect
8. #10603853C2semidirect
9. #10603852C2semidirect
18. C5×#405C4semidirect
19. M16×Dic5C4semidirect
20. C2×#3947C4semidirect
21. #65286C4semidirect
22. #64997C4semidirect
24. D5×#97C4semidirect
26. #64978C4semidirect
28. C4×#676C4semidirect
166. #405AGL(1,𝔽5)semidirect
378. C5#20788semidirect

Maximal subgroups

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

subgroupindex
#106102812
#106047732
#106047652
#106047262
#106043772
#106038852
#106038842
#106038532
#106038522
#207885