Finite Group Databasebeta

← #10617217

Subgroups — #10617217

Order 640.

502 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#704C211yesmaximal, characteristic, complemented
2C2×#4131C211yesmaximal, characteristic, complemented
3#65920C211yesmaximal, characteristic
4C2×C22⋊#170C211yesmaximal, characteristic, complemented
5#65775C211yesmaximal, characteristic, complemented
6#65774C211yesmaximal, characteristic, complemented
7#65773C211yesmaximal, characteristic, complemented
8#65773C211yesmaximal, characteristic, complemented
9D4×#170C211yesmaximal, characteristic, complemented
10#65696C211yesmaximal, characteristic, complemented
11#65695C211yesmaximal, characteristic, complemented
12#65694C211yesmaximal, characteristic, complemented
13#65693C211yesmaximal, characteristic, complemented
14#65692C211yesmaximal, characteristic, complemented
15#65691C211yesmaximal, characteristic, complemented
16C23×Dic5C2211yescharacteristic
17C2×#711C2211yescharacteristic
18C2×#704—24no—
19C2×#704C2211yescharacteristic
20C2×#704C2211yescharacteristic
21C2×#704C2211yescharacteristic
22C2×#704C2211yescharacteristic
23C2×#704C2211yescharacteristic, complemented
24C22×#170C2211yescharacteristic
25C22×#170C2211yescharacteristic
26C22×#170C2211yescharacteristic
27C22×#170C2211yescharacteristic
28C22×#170C2211yescharacteristic, complemented
29#4131—24no—
30C2×#700C2211yescharacteristic, complemented
31C2×D5⋊C8C2211yescharacteristic, complemented
32C5×#120C411yescharacteristic
33#4099C411yescharacteristic
34#4085C411yescharacteristic
35#4085C411yescharacteristic
36#4081C411yescharacteristic
37D4×Dic5C2211yescharacteristic
38D4×Dic5C2211yescharacteristic
39#4032C411yescharacteristic
40#4027C2211yescharacteristic
41#4027C2211yescharacteristic
42#4024C411yescharacteristic
43#4024C411yescharacteristic
44#4023C2211yescharacteristic
45C22⋊#170C2211yescharacteristic
46C22⋊#170C2211yescharacteristic
47C22⋊#170C2211yescharacteristic
48C22⋊#170C2211yescharacteristic
49C22⋊#170C2211yescharacteristic, complemented
50C22⋊#170C2211yescharacteristic, complemented
51#4004C2211yescharacteristic, complemented
52#4004C2211yescharacteristic, complemented
53#4003C2211yescharacteristic, complemented
54#4003C2211yescharacteristic, complemented
55#4002C2211yescharacteristic, complemented
56#4002C2211yescharacteristic, complemented
57C4⋊#170C2211yescharacteristic, complemented
58C4⋊#170C2211yescharacteristic, complemented
59C4×#170C2211yescharacteristic, complemented
60C4×#170C2211yescharacteristic, complemented
61#2933—51nomaximal, Sylow
62C24×C5C2×C411yescharacteristic
63C2×C5×D4C2×C411yescharacteristic
64C2×C5×D4C2×C411yescharacteristic
65C2×C5×D4C2×C411yescharacteristic
66C2×C22⋊D5C2×C411yescharacteristic
67C2×C22⋊D5C2×C411yescharacteristic
68C2×C22⋊D5C2×C411yescharacteristic
69C2×C22⋊D5C2×C411yescharacteristic
70C22×Dic5—22no—
71C22×Dic5—22no—
72C22×Dic5C2×C411yescharacteristic
73C22×Dic5C2×C411yescharacteristic
74C22×Dic5C2×C411yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79#711—22no—
80#711—22no—
81C2×C4×D5C2311yescharacteristic
82C2×Dic10C2×C411yescharacteristic
83#704—22no—
84#704—22no—
85#704—22no—
86#704—22no—
87#704D414yescomplemented
88C2×#170—22no—
89C2×#170—22no—
90C2×#170—22no—
91C2×#170—22no—
92C2×#170—22no—
93C2×#170C2311yescharacteristic
94C2×#170C2311yescharacteristic
95C2×#170C2311yescharacteristic
96C2×#170C2311yescharacteristic
97C2×#170C2311yescharacteristic
98C2×#170C2311yescharacteristic
99C2×#170C2311yescharacteristic
100C2×#170C2311yescharacteristic
101#700—22no—
102D5⋊C8—22no—
103C5×C22⋊C4C2×C411yescharacteristic
104C5×C22⋊C4C2×C411yescharacteristic
105C5×C22⋊C4C2×C411yescharacteristic
106C22⋊Dic5C2×C411yescharacteristic
107C22⋊Dic5C2×C411yescharacteristic
108C22⋊Dic5C2×C411yescharacteristic
109C22⋊Dic5C2×C411yescharacteristic
110C22⋊Dic5C2×C411yescharacteristic
111#685C2×C411yescharacteristic
112#685C2×C411yescharacteristic
113C4⋊Dic5C2×C411yescharacteristic
114C4⋊Dic5C2×C411yescharacteristic
115#683C2×C411yescharacteristic
116#683C2×C411yescharacteristic
117#683C2×C411yescharacteristic
118#683C2×C411yescharacteristic
119C4×Dic5C2311yescharacteristic
120C4×Dic5C2311yescharacteristic
121C2×#131—51no—
122C22×M16—51no—
123#525—51no—
124#436—51no—
125#436—51no—
126#435—51no—
127#435—51no—
128#435—51no—
129#435—51no—
130C8×D4—51no—
131C8×D4—51no—
132#424—51no—
133#408—51no—
134C2×C22⋊C8—51no—
135#405—51no—
136C23×C5—22no—
137C23×C5—22no—
138C23×C5—41no—
139C23×C5C22×C411yescharacteristic
140C23×C5C22×C411yescharacteristic
141C23×C5C22×C411yescharacteristic
142C23×C5C22×C411yescharacteristic
143C22×D5C22×C411yescharacteristic
144C5×D4—22no—
145C5×D4—22no—
146C5×D4—41no—
147C2×C4×C5C22×C411yescharacteristic
148C2×C4×C5C22×C411yescharacteristic
149C2×C4×C5C22×C411yescharacteristic
150C22⋊D5—22no—
151C22⋊D5—22no—
152C22⋊D5—22no—
153C22⋊D5—41no—
154C2×Dic5—22no—
155C2×Dic5—22no—
156C2×Dic5—22no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5C22×C411yescharacteristic
164C2×Dic5C22×C411yescharacteristic
165C2×Dic5C22×C411yescharacteristic
166C2×Dic5C22×C411yescharacteristic
167C2×Dic5C22×C411yescharacteristic
168C2×Dic5C22×C411yescharacteristic
169C2×Dic5C2×D412yes—
170C2×Dic5#4112yes—
171C2×Dic5C2411yescharacteristic
172C4×D5—21no—
173C4×D5—21no—
174Dic10—22no—
175#170—21no—
176#170—21no—
177#170—21no—
178#170—21no—
179#170—21no—
180#170—21no—
181#170C2×D412yes—
182#170C2×D412yes—
183C2×#41—51no—
184C23×C4—51no—
185#131—104no—
186C2×M16—51no—
187C2×M16—51no—
188C2×M16—51no—
189C2×M16—51no—
190C2×M16—51no—
191C2×M16—51no—
192C2×M16—104no—
193C22×C8—51no—
194C22×C8—51no—
195C22×C8—51no—
196C22×C8—51no—
197C22×C8—51no—
198C22×C8—51no—
199#123—51no—
200#123—51no—
201C22⋊Q8—51no—
202C22⋊Q8—51no—
203#121—51no—
204#121—51no—
205#120—51no—
206#120AGL(1,𝔽5)11yescharacteristic
207C4×D4—51no—
208C4×D4—51no—
209C4×D4—51no—
210C4×D4—51no—
211#117—51no—
212C4⋊C8—51no—
213C4⋊C8—51no—
214C4⋊C8—51no—
215C4⋊C8—51no—
216C22⋊C8—51no—
217C22⋊C8—51no—
218C22⋊C8—51no—
219C22⋊C8—51no—
220C22⋊C8—51no—
221C22⋊C8—51no—
222C22⋊C8—51no—
223C22⋊C8—51no—
224#97—51no—
225#97—51no—
226C4×C8—51no—
227C4×C8—51no—
228C22×C5—21no—
229C22×C5—21no—
230C22×C5—21no—
231C22×C5—21no—
232C22×C5—21no—
233C22×C5—21no—
234C22×C5—21no—
235C22×C5—21no—
236C22×C5—21no—
237C22×C5—21no—
238C22×C5—44no—
239C22×C5C4×D412yes—
240C22×C5#13112yescomplemented
241C22×C5C23×C411yescharacteristic
242C2×D5—21no—
243C2×D5—21no—
244C2×D5—21no—
245C4×C5—21no—
246C4×C5—21no—
247C4×C5—21no—
248Dic5—21no—
249Dic5—21no—
250Dic5—21no—
251Dic5—21no—
252Dic5—21no—
253Dic5C4×D412yes—
254Dic5C22×D411yescharacteristic
255Dic5C2×#4111yescharacteristic
256C24C2×AGL(1,𝔽5)11yescharacteristic
257#41—102no—
258#41—102no—
259C2×Q8—51no—
260C2×D4—51no—
261C2×D4—51no—
262C2×D4—51no—
263C2×D4—51no—
264C2×D4C2×AGL(1,𝔽5)11yescharacteristic
265C2×D4C2×AGL(1,𝔽5)11yescharacteristic
266C2×D4C2×AGL(1,𝔽5)11yescharacteristic
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—51no—
270C22×C4—51no—
271C22×C4—51no—
272C22×C4—51no—
273C22×C4—51no—
274C22×C4—51no—
275C22×C4—102no—
276C22×C4—102no—
277M16—54no—
278M16—102no—
279M16—102no—
280M16—102no—
281M16—102no—
282M16—102no—
283C2×C8—51no—
284C2×C8—51no—
285C2×C8—51no—
286C2×C8—51no—
287C2×C8—51no—
288C2×C8—51no—
289C2×C8—51no—
290C2×C8—51no—
291C2×C8—102no—
292C2×C8—102no—
293C2×C8—102no—
294C2×C8—102no—
295C2×C8—102no—
296C2×C8—102no—
297C4⋊C4—51no—
298C4⋊C4—51no—
299C4⋊C4—51no—
300C4⋊C4—51no—
301C4⋊C4—51no—
302C4⋊C4—51no—
303C22⋊C4—51no—
304C22⋊C4—51no—
305C22⋊C4—51no—
306C22⋊C4—51no—
307C22⋊C4—51no—
308C22⋊C4—51no—
309C22⋊C4—51no—
310C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
311C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
312C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
313C42—51no—
314C42—51no—
315C2×C5—22no—
316C2×C5—22no—
317C2×C5—41no—
318C2×C5—41no—
319C2×C5C2×C4×D411yescharacteristic
320C2×C5C2×#13111yescharacteristic
321C2×C5#56811yescharacteristic
322D5—41no—
323C23—22no—
324C23—22no—
325C23—41no—
326C23—51no—
327C23C22×AGL(1,𝔽5)11yescharacteristic
328C23C22×AGL(1,𝔽5)11yescharacteristic
329C23C22×AGL(1,𝔽5)11yescharacteristic
330C23C22×AGL(1,𝔽5)11yescharacteristic
331Q8—102no—
332D4—22no—
333D4—22no—
334D4—41no—
335D4—102no—
336D4—102no—
337D4—102no—
338D4—201no—
339C2×C4—52no—
340C2×C4—52no—
341C2×C4—51no—
342C2×C4—51no—
343C2×C4—51no—
344C2×C4—51no—
345C2×C4—51no—
346C2×C4—51no—
347C2×C4—51no—
348C2×C4—101no—
349C2×C4—101no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—102no—
356C2×C4—102no—
357C2×C4—101no—
358C2×C4—102no—
359C2×C4C22×AGL(1,𝔽5)11yescharacteristic
360C2×C4C22×AGL(1,𝔽5)11yescharacteristic
361C2×C4C22×AGL(1,𝔽5)11yescharacteristic
362C8—52no—
363C8—52no—
364C8—101no—
365C8—101no—
366C8—101no—
367C8—101no—
368C8—101no—
369C8—101no—
370C5#293311yescharacteristic, complemented, Sylow
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—44no—
382C22—101no—
383C22—101no—
384C22—101no—
385C22#413112yescomplemented
386C22D4×AGL(1,𝔽5)12yes—
387C22C23×AGL(1,𝔽5)11yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—51no—
392C4—51no—
393C4—52no—
394C4—101no—
395C4—101no—
396C4—101no—
397C4—101no—
398C4—101no—
399C2—22no—
400C2—22no—
401C2—41no—
402C2—41no—
403C2—201no—
404C2C2×#413111yescharacteristic
405C2#6625611yescharacteristic
406C2C2×D4×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽5)
commutator, socleC22×C5C23×C4
FittingC5×#120C4
prime radical#65920C2
cosocleC2×Dic5C24
hypercenter#120AGL(1,𝔽5)
hypocenterC5#2933
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×#704C2semidirect
2. C2×#4131C2semidirect
4. C2×C22⋊#170C2semidirect
5. #65775C2semidirect
6. #65774C2semidirect
7. #65773C2semidirect
8. #65773C2semidirect
9. D4×#170C2semidirect
10. #65696C2semidirect
11. #65695C2semidirect
12. #65694C2semidirect
13. #65693C2semidirect
14. #65692C2semidirect
15. #65691C2semidirect
23. C2×#704C22semidirect
28. C22×#170C22semidirect
30. C2×#700C22semidirect
31. C2×D5⋊C8C22semidirect
49. C22⋊#170C22semidirect
50. C22⋊#170C22semidirect
51. #4004C22semidirect
52. #4004C22semidirect
53. #4003C22semidirect
54. #4003C22semidirect
55. #4002C22semidirect
56. #4002C22semidirect
57. C4⋊#170C22semidirect
58. C4⋊#170C22semidirect
59. C4×#170C22semidirect
60. C4×#170C22semidirect
87. #704D4semidirect
240. C22×C5#131semidirect
370. C5#2933semidirect
385. C22#4131semidirect

Maximal subgroups

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

subgroupindex
C22×#7042
C2×#41312
#659202
C2×C22⋊#1702
#657752
#657742
#657732
D4×#1702
#656962
#656952
#656942
#656932
#656922
#656912
#29335