Finite Group Databasebeta

← #10610668

Subgroups — #10610668

Order 640.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66140C211yesmaximal, characteristic, complemented
2#65866C211yesmaximal, characteristic, complemented
3C5×#418C211yesmaximal, characteristic, complemented
4#65506C211yesmaximal, characteristic, complemented
5#65423C211yesmaximal, characteristic, complemented
6#65416C211yesmaximal, characteristic, complemented
7#65285C211yesmaximal, characteristic, complemented
8#65076C211yesmaximal, characteristic, complemented
9#65074C211yesmaximal, characteristic, complemented
10#65073C211yesmaximal, characteristic, complemented
11#65070C211yesmaximal, characteristic, complemented
12#65064C211yesmaximal, characteristic, complemented
13#65061C211yesmaximal, characteristic, complemented
14#65054C211yesmaximal, characteristic, complemented
15#65048C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C2×#711C2211yescharacteristic
18C2×C5×M16C2211yescharacteristic
19C5×#117C2211yescharacteristic
20#4099—21no—
21C2×C22⋊Dic5—21no—
22#4085—21no—
23#4084C2211yescharacteristic
24#4082C2211yescharacteristic
25#4081—21no—
26D4×Dic5C2211yescharacteristic
27C2×#686C2211yescharacteristic
28C2×#686C2211yescharacteristic
29#4076C2211yescharacteristic
30C4×C22⋊D5C2211yescharacteristic
31#4070C2211yescharacteristic
32C2×#677C2211yescharacteristic, complemented
33C2×C8×D5C2211yescharacteristic, complemented
34#4041C2211yescharacteristic, complemented
35#4039C2211yescharacteristic, complemented
36#4038C2211yescharacteristic, complemented
37#4034C2211yescharacteristic, complemented
38#4031—21no—
39#4030—21no—
40#4021—21no—
41#4018—21no—
42C5×#102C2211yescharacteristic
43C5×#102C2211yescharacteristic
44C5×#102C2211yescharacteristic
45C5×#102C2211yescharacteristic
46#3964C2211yescharacteristic
47#3964C2211yescharacteristic
48C22⋊#168C2211yescharacteristic
49#3953C2211yescharacteristic, complemented
50#3952C2211yescharacteristic, complemented
51#3950C2211yescharacteristic, complemented
52#3948C2211yescharacteristic, complemented
53#3947C2211yescharacteristic, complemented
54#3946C2211yescharacteristic, complemented
55#3942C2211yescharacteristic, complemented
56#3942C2211yescharacteristic, complemented
57#3940C2211yescharacteristic, complemented
58#3939C2211yescharacteristic, complemented
59#3319—51nomaximal, Sylow
60C24×C5—21no—
61C2×C5×D4—22no—
62C2×C5×D4—21no—
63C2×C5×D4—41no—
64C2×C5×D4C2311yescharacteristic
65C2×C5×D4C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C2×C22⋊D5—21no—
68C2×C22⋊D5—21no—
69C2×C22⋊D5—21no—
70C2×C22⋊D5D411yescharacteristic
71C22×Dic5—21no—
72C22×Dic5—21no—
73#711—22no—
74#711—41no—
75C2×D20—21no—
76C2×D20C2311yescharacteristic
77C2×C4×D5—21no—
78C2×C4×D5C2311yescharacteristic
79C2×Dic10—21no—
80C2×Dic10C2311yescharacteristic
81C5×M16—41no—
82C2×C5×C8C2311yescharacteristic
83C2×C5×C8C2311yescharacteristic
84C5×C4⋊C4C2311yescharacteristic
85C5×C4⋊C4C2311yescharacteristic
86C5×C22⋊C4—21no—
87C42×C5—21no—
88C22⋊Dic5—21no—
89C22⋊Dic5—21no—
90C22⋊Dic5—21no—
91C22⋊Dic5—21no—
92C22⋊Dic5—21no—
93C22⋊Dic5D411yescharacteristic
94#686—22no—
95#686—41no—
96#685—21no—
97#685—21no—
98#685—21no—
99#685—21no—
100#685D411yescharacteristic
101C4⋊Dic5C2311yescharacteristic
102C4⋊Dic5C2311yescharacteristic
103#683—21no—
104#683—21no—
105#683D411yescharacteristic
106C4×Dic5—21no—
107C4×Dic5C2311yescharacteristic
108C2×#168C2311yescharacteristic
109C2×#168C2311yescharacteristic
110#677—41no—
111C8×D5—41no—
112#555—51no—
113#546—51no—
114#489—51no—
115#485—51no—
116#482—51no—
117#474—51no—
118#468—51no—
119#466—51no—
120#463—51no—
121#449—51no—
122#447—51no—
123#442—51no—
124#435—51no—
125#418D511yescharacteristic, complemented
126#416—51no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—42no—
130C23×C5—42no—
131C23×C5—81no—
132C23×C5C2×D411yescharacteristic
133C22×D5—21no—
134C22×D5C2×D411yescharacteristic
135C5×D4—21no—
136C5×D4—21no—
137C5×D4—21no—
138C5×D4—42no—
139C5×D4#4112yes—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5C2×D411yescharacteristic
145C2×C4×C5C2411yescharacteristic
146C22⋊D5—41no—
147C22⋊D5—42no—
148C22⋊D5—81no—
149C22⋊D5—81no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—42no—
154C2×Dic5—81no—
155C2×Dic5C2×D411yescharacteristic
156C2×Dic5C2×D411yescharacteristic
157C2×Dic5C2×D411yescharacteristic
158D20—21no—
159D20—21no—
160D20—81no—
161C4×D5—21no—
162C4×D5—21no—
163C4×D5—42no—
164Dic10—21no—
165Dic10—21no—
166Dic10—81no—
167C5×C8—21no—
168C5×C8—21no—
169#168—21no—
170#168—21no—
171C2×#41—51no—
172C22×D4C2×D511yescharacteristic
173C2×SD16—51no—
174C2×D8—51no—
175C2×D8—51no—
176C2×M16C2×D511yescharacteristic
177C22×C8—51no—
178C4⋊Q8—51no—
179#124—51no—
180#124—101no—
181#124—101no—
182#123—101no—
183C22⋊Q8—51no—
184#121—51no—
185#121—51no—
186#121—51no—
187#121—101no—
188#121—101no—
189#120—101no—
190C4×D4—51no—
191C4×D4—51no—
192C4×D4—51no—
193#117—51no—
194#117—101no—
195#117C2×D511yescharacteristic
196C2×C22⋊C4—101no—
197#107—51no—
198#107—51no—
199#106—51no—
200C4⋊C8—51no—
201#103—51no—
202#103—51no—
203#103—51no—
204#102—51no—
205#102—51no—
206#102—51no—
207#102C2×D511yescharacteristic
208#102C2×D511yescharacteristic
209#102C2×D511yescharacteristic
210#102C2×D511yescharacteristic
211C22⋊C8—51no—
212C22⋊C8—51no—
213#97—51no—
214C22×C5—22no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—21no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5—41no—
223C22×C5—84no—
224C22×C5C22×D411yescharacteristic
225C2×D5—21no—
226C2×D5—21no—
227C2×D5—21no—
228C2×D5—41no—
229C2×D5—41no—
230C2×D5—41no—
231C4×C5—21no—
232C4×C5—41no—
233C4×C5—41no—
234C4×C5C2×#4111yescharacteristic
235C4×C5H(5,𝔽2)11yescharacteristic
236Dic5—21no—
237Dic5—21no—
238Dic5—21no—
239Dic5—41no—
240Dic5—41no—
241Dic5—41no—
242C24—21no—
243#41—102no—
244#41—201no—
245C2×Q8—51no—
246C2×Q8—101no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—51no—
251C2×D4—51no—
252C2×D4—101no—
253C2×D4—101no—
254C2×D4—101no—
255C2×D4—101no—
256C2×D4C22×D511yescharacteristic
257C2×D4C22×D511yescharacteristic
258C22×C4—51no—
259C22×C4—101no—
260C22×C4—101no—
261C22×C4—101no—
262C22×C4C22×D511yescharacteristic
263SD16—201no—
264D8—102no—
265D8—201no—
266M16—41no—
267C2×C8—51no—
268C2×C8—51no—
269C2×C8—201no—
270C2×C8C22×D511yescharacteristic
271C2×C8C22×D511yescharacteristic
272C4⋊C4—51no—
273C4⋊C4—51no—
274C4⋊C4—51no—
275C4⋊C4—101no—
276C4⋊C4—101no—
277C4⋊C4C22×D511yescharacteristic
278C4⋊C4C22×D511yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—51no—
281C22⋊C4—51no—
282C22⋊C4—101no—
283C22⋊C4—101no—
284C22⋊C4—101no—
285C22⋊C4—101no—
286C22⋊C4—101no—
287C22⋊C4—101no—
288C22⋊C4—101no—
289C22⋊C4—101no—
290C22⋊C4—101no—
291C42—21no—
292C42—51no—
293C42—101no—
294C2×C5—41no—
295C2×C5—42no—
296C2×C5—81no—
297C2×C5#54611yescharacteristic
298C2×C5#57511yescharacteristic
299C2×C5#57611yescharacteristic
300D5—41no—
301D5—81no—
302C23—21no—
303C23—21no—
304C23—42no—
305C23—42no—
306C23—51no—
307C23—81no—
308C23—101no—
309C23D4×D511yescharacteristic
310Q8—101no—
311Q8—101no—
312Q8—401no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—101no—
318D4—101no—
319D4—201no—
320D4—202no—
321D4—401no—
322D4—401no—
323D4—401no—
324D4#70912yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—51no—
330C2×C4—51no—
331C2×C4—51no—
332C2×C4—101no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—202no—
338C2×C4—202no—
339C2×C4—401no—
340C2×C4D4×D511yescharacteristic
341C2×C4C23×D511yescharacteristic
342C8—21no—
343C8—21no—
344C8—101no—
345C8—101no—
346C5#331911yescharacteristic, complemented, Sylow
347C22—21no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—84no—
357C22—101no—
358C22—101no—
359C22—101no—
360C22—201no—
361C22—201no—
362C22—201no—
363C22C2×D4×D511yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—101no—
368C4—101no—
369C4—101no—
370C4—201no—
371C4—201no—
372C4—201no—
373C4C2×#70911yescharacteristic
374C4#414411yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—201no—
379C2—401no—
380C2#6582411yescharacteristic
381C2#6610711yescharacteristic
382C2#6610811yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#418C2
socleC22×C5C22×D4
hypercenter#418D5
hypocenterC5#3319
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #66140C2semidirect
2. #65866C2semidirect
3. C5×#418C2semidirect
4. #65506C2semidirect
5. #65423C2semidirect
6. #65416C2semidirect
7. #65285C2semidirect
8. #65076C2semidirect
9. #65074C2semidirect
10. #65073C2semidirect
11. #65070C2semidirect
12. #65064C2semidirect
13. #65061C2semidirect
14. #65054C2semidirect
15. #65048C2semidirect
32. C2×#677C22semidirect
33. C2×C8×D5C22semidirect
34. #4041C22semidirect
35. #4039C22semidirect
36. #4038C22semidirect
37. #4034C22semidirect
49. #3953C22semidirect
50. #3952C22semidirect
51. #3950C22semidirect
52. #3948C22semidirect
53. #3947C22semidirect
54. #3946C22semidirect
55. #3942C22semidirect
56. #3942C22semidirect
57. #3940C22semidirect
58. #3939C22semidirect
125. #418D5semidirect
346. C5#3319semidirect

Maximal subgroups

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

subgroupindex
#661402
#658662
C5×#4182
#655062
#654232
#654162
#652852
#650762
#650742
#650732
#650702
#650642
#650612
#650542
#650482
#33195