Finite Group Databasebeta

← C5×#10500 #11933220

Subgroups — C5×#10500 #11933220

Order 1280.

508 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3205C211yesmaximal, characteristic
2C5×#1982C211yesmaximal, characteristic, complemented
3C5×#1980C211yesmaximal, characteristic, complemented
4C5×#1965C211yesmaximal, characteristic, complemented
5C5×#1565C211yesmaximal, characteristic, complemented
6C5×#1547C211yesmaximal, characteristic, complemented
7C5×#1303C211yesmaximal, characteristic, complemented
8C5×#532C2211yescharacteristic
9C2×C5×#127C2211yescharacteristic
10C5×#493—21no—
11C5×#486—21no—
12C5×#459—21no—
13C5×#459—21no—
14C5×#447—22no—
15C5×#431C2211yescharacteristic
16C5×#423C2211yescharacteristic
17C2×C5×C4⋊C8C2211yescharacteristic
18C2×C5×#104—21no—
19C2×C5×#104—21no—
20C2×C5×#102—22no—
21C2×C5×#100—21no—
22C2×C5×#100—21no—
23C4×C5×M16C2211yescharacteristic
24C4×C5×C4⋊C4C2211yescharacteristic
25C5×#335—21no—
26C5×#334—21no—
27C5×#332—21no—
28C5×#331—21no—
29#10500C511yesmaximal, characteristic, direct product factor, Sylow
30C2×C5×#41—21no—
31C22×C5×D4—21no—
32C22×C5×D4—21no—
33C22×C5×D4D412yes—
34C2×C5×D8—22no—
35C2×C5×D8—22no—
36C2×C5×M16—21no—
37C2×C5×M16D411yescharacteristic
38C2×C5×M16D411yescharacteristic
39C22×C5×C8—21no—
40C5×C4⋊Q8D411yescharacteristic
41C5×#127—21no—
42C5×#127—21no—
43C5×#127—21no—
44C5×#127—21no—
45C5×#127D411yescharacteristic
46C5×#124—21no—
47C5×#121—21no—
48C5×#121—21no—
49C4×C5×D4—21no—
50C4×C5×D4—21no—
51C2×C5×C4⋊C4—21no—
52C2×C42×C5—21no—
53C2×C42×C5C2311yescharacteristic
54C5×C4⋊C8—21no—
55C5×C4⋊C8—21no—
56C5×C4⋊C8—21no—
57C5×C4⋊C8—21no—
58C5×C4⋊C8D411yescharacteristic
59C5×C4⋊C8D411yescharacteristic
60C5×#104—42no—
61C5×#104—42no—
62C5×#102—22no—
63C5×#102—22no—
64C5×#102—44no—
65C5×#100—42no—
66C5×#100—42no—
67C5×C22⋊C8—21no—
68C5×#97—21no—
69C4×C5×C8—21no—
70C4×C5×C8—21no—
71C5×#95—21no—
72#3205C2×C511yescharacteristic
73#1982C2×C511yescharacteristic, complemented
74#1980C2×C511yescharacteristic, complemented
75#1965C2×C511yescharacteristic, complemented
76#1565C2×C511yescharacteristic, complemented
77#1547C2×C511yescharacteristic, complemented
78#1303C2×C511yescharacteristic, complemented
79C24×C5—22no—
80C5×#41—41no—
81C5×#41—41no—
82C5×#41—41no—
83C5×#41—41no—
84C2×C5×Q8—21no—
85C2×C5×D4—21no—
86C2×C5×D4—22no—
87C2×C5×D4—21no—
88C2×C5×D4—22no—
89C2×C5×D4—21no—
90C2×C5×D4—42no—
91C2×C5×D4—42no—
92C2×C5×D4—41no—
93C2×C5×D4—41no—
94C2×C5×D4—41no—
95C2×C5×D4—41no—
96C2×C5×D4—41no—
97C2×C5×D4—41no—
98C2×C5×D4—41no—
99C2×C5×D4—41no—
100C2×C5×D4—41no—
101C2×C5×D4—82no—
102C22×C4×C5—41no—
103C22×C4×C5—41no—
104C22×C4×C5C2×D411yescharacteristic
105C22×C4×C5C2×D411yescharacteristic
106C22×C4×C5#4111yescharacteristic
107C5×D8—82no—
108C5×D8—82no—
109C5×M16—21no—
110C5×M16—21no—
111C5×M16—21no—
112C5×M16—41no—
113C5×M16—41no—
114C5×M16D812yescomplemented
115C2×C5×C8—21no—
116C2×C5×C8—21no—
117C2×C5×C8—21no—
118C2×C5×C8—21no—
119C2×C5×C8—42no—
120C2×C5×C8—41no—
121C2×C5×C8—41no—
122C5×C4⋊C4—22no—
123C5×C4⋊C4—21no—
124C5×C4⋊C4—21no—
125C5×C4⋊C4—41no—
126C5×C22⋊C4—41no—
127C5×C22⋊C4—41no—
128C42×C5—41no—
129C42×C5—41no—
130C42×C5C2×D411yescharacteristic
131C42×C5C2×D411yescharacteristic
132C42×C5#4111yescharacteristic
133C42×C5#4111yescharacteristic
134#532C22×C511yescharacteristic
135C2×#127C22×C511yescharacteristic
136#493—21no—
137#486—21no—
138#459—21no—
139#459—21no—
140#447—22no—
141#431C22×C511yescharacteristic
142#423C22×C511yescharacteristic
143C2×C4⋊C8C22×C511yescharacteristic
144C2×#104—21no—
145C2×#104—21no—
146C2×#102—22no—
147C2×#100—21no—
148C2×#100—21no—
149C4×M16C22×C511yescharacteristic
150C4×C4⋊C4C22×C511yescharacteristic
151#335—21no—
152#334—21no—
153#332—21no—
154#331—21no—
155C23×C5—41no—
156C23×C5—42no—
157C23×C5—42no—
158C23×C5—42no—
159C23×C5—82no—
160C23×C5—82no—
161C23×C5#12111yescharacteristic
162C5×Q8—41no—
163C5×Q8—41no—
164C5×D4—44no—
165C5×D4—41no—
166C5×D4—42no—
167C5×D4—41no—
168C5×D4—42no—
169C5×D4—41no—
170C5×D4—41no—
171C5×D4—81no—
172C5×D4—82no—
173C5×D4—82no—
174C5×D4—81no—
175C5×D4—81no—
176C5×D4—81no—
177C5×D4—81no—
178C5×D4—81no—
179C5×D4—81no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—41no—
183C2×C4×C5—42no—
184C2×C4×C5—41no—
185C2×C4×C5—81no—
186C2×C4×C5—81no—
187C2×C4×C5—81no—
188C2×C4×C5—81no—
189C2×C4×C5#12011yescharacteristic
190C2×C4×C5#12111yescharacteristic
191C2×C4×C5#12111yescharacteristic
192C2×C4×C5#12311yescharacteristic
193C2×C4×C5#12311yescharacteristic
194C2×C4×C5#12411yescharacteristic
195C2×C4×C5C2×D811yescharacteristic
196C2×C4×C5#13511yescharacteristic
197C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
199C5×C8—22no—
200C5×C8—41no—
201C5×C8—41no—
202C5×C8—41no—
203C5×C8—81no—
204C5×C8—81no—
205C2×#41—21no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C5×D412yes—
209C2×D8—22no—
210C2×D8—22no—
211C2×M16—21no—
212C2×M16C5×D411yescharacteristic
213C2×M16C5×D411yescharacteristic
214C22×C8—21no—
215C4⋊Q8C5×D411yescharacteristic
216#127—21no—
217#127—21no—
218#127—21no—
219#127—21no—
220#127C5×D411yescharacteristic
221#124—21no—
222#121—21no—
223#121—21no—
224C4×D4—21no—
225C4×D4—21no—
226C2×C4⋊C4—21no—
227C2×C42—21no—
228C2×C42C23×C511yescharacteristic
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8C5×D411yescharacteristic
234C4⋊C8C5×D411yescharacteristic
235#104—42no—
236#104—42no—
237#102—22no—
238#102—22no—
239#102—44no—
240#100—42no—
241#100—42no—
242C22⋊C8—21no—
243#97—21no—
244C4×C8—21no—
245C4×C8—21no—
246#95—21no—
247C22×C5—21no—
248C22×C5—21no—
249C22×C5—82no—
250C22×C5—81no—
251C22×C5—81no—
252C22×C5—82no—
253C22×C5—82no—
254C22×C5—81no—
255C22×C5—162no—
256C22×C5—162no—
257C22×C5#39411yescharacteristic
258C22×C5#45911yescharacteristic
259C22×C5#46311yescharacteristic
260C4×C5—21no—
261C4×C5—21no—
262C4×C5—21no—
263C4×C5—82no—
264C4×C5—81no—
265C4×C5#45311yescharacteristic
266C4×C5#45511yescharacteristic
267C4×C5#45911yescharacteristic
268C4×C5#46311yescharacteristic
269C4×C5#46611yescharacteristic
270C4×C5#46911yescharacteristic
271C24—22no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276C2×Q8—21no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—82no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C2×C5×D411yescharacteristic
297C22×C4C2×C5×D411yescharacteristic
298C22×C4C5×#4111yescharacteristic
299D8—82no—
300D8—82no—
301M16—21no—
302M16—21no—
303M16—21no—
304M16—41no—
305M16—41no—
306M16C5×D812yescomplemented
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42—41no—
321C42—41no—
322C42C2×C5×D411yescharacteristic
323C42C2×C5×D411yescharacteristic
324C42C5×#4111yescharacteristic
325C42C5×#4111yescharacteristic
326C2×C5—21no—
327C2×C5—21no—
328C2×C5—161no—
329C2×C5—162no—
330C2×C5#203711yescharacteristic
331C2×C5#204811yescharacteristic
332C2×C5#205411yescharacteristic
333C23—41no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—82no—
338C23—82no—
339C23C5×#12111yescharacteristic
340Q8—41no—
341Q8—41no—
342D4—44no—
343D4—41no—
344D4—42no—
345D4—41no—
346D4—42no—
347D4—41no—
348D4—41no—
349D4—81no—
350D4—82no—
351D4—82no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4C5×#12011yescharacteristic
368C2×C4C5×#12111yescharacteristic
369C2×C4C5×#12111yescharacteristic
370C2×C4C5×#12311yescharacteristic
371C2×C4C5×#12311yescharacteristic
372C2×C4C5×#12411yescharacteristic
373C2×C4C2×C5×D811yescharacteristic
374C2×C4C5×#13511yescharacteristic
375C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
376C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
377C8—22no—
378C8—41no—
379C8—41no—
380C8—41no—
381C8—81no—
382C8—81no—
383C5#1050011yescharacteristic, direct product factor, Sylow
384C22—21no—
385C22—21no—
386C22—82no—
387C22—81no—
388C22—82no—
389C22—81no—
390C22—81no—
391C22—82no—
392C22—162no—
393C22—162no—
394C22C5×#39411yescharacteristic
395C22C5×#45911yescharacteristic
396C22C5×#46311yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—81no—
402C4C5×#45311yescharacteristic
403C4C5×#45511yescharacteristic
404C4C5×#45911yescharacteristic
405C4C5×#46311yescharacteristic
406C4C5×#46611yescharacteristic
407C4C5×#46911yescharacteristic
408C2—21no—
409C2—21no—
410C2—162no—
411C2—161no—
412C2C5×#203711yescharacteristic
413C2C5×#204811yescharacteristic
414C2C5×#205411yescharacteristic
subgroupG/H
center, socleC22×C5#394
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC5×#3205C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1982C2semidirect
3. C5×#1980C2semidirect
4. C5×#1965C2semidirect
5. C5×#1565C2semidirect
6. C5×#1547C2semidirect
7. C5×#1303C2semidirect
29. #10500C5direct (N × H)
73. #1982C2×C5semidirect
74. #1980C2×C5semidirect
75. #1965C2×C5semidirect
76. #1565C2×C5semidirect
77. #1547C2×C5semidirect
78. #1303C2×C5semidirect
114. C5×M16D8semidirect
306. M16C5×D8semidirect
383. C5#10500direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#32052
C5×#19822
C5×#19802
C5×#19652
C5×#15652
C5×#15472
C5×#13032
#105005