Finite Group Databasebeta

← C5×#10456 #11933176

Subgroups — C5×#10456 #11933176

Order 1280.

460 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
1C5×#3036C211yesmaximal, characteristic
2C5×#1800C211yesmaximal, characteristic, complemented
3C5×#1567C211yesmaximal, characteristic, complemented
4C5×#1547C211yesmaximal, characteristic, complemented
5C5×#1492C211yesmaximal, characteristic, complemented
6C2×C5×D4⋊C8C211yesmaximal, characteristic, complemented
7C5×#1292C211yesmaximal, characteristic, complemented
8C2×C5×AGL(1,ℤ/8)—21no—
9C5×#532C2211yescharacteristic
10C2×C4×C5×D4C2211yescharacteristic
11C5×#463—22no—
12C5×#461—21no—
13C5×#459—21no—
14C5×#423C2211yescharacteristic
15C5×#423C2211yescharacteristic
16C5×#423C2211yescharacteristic
17C2×C5×C4⋊C8C2211yescharacteristic
18C5×#417—21no—
19C2×C5×#100—21no—
20C2×C4×C5×C8C2211yescharacteristic
21C5×#343—21no—
22C5×#335—22no—
23C5×#332—21no—
24C5×#331—21no—
25C5×D4⋊C8—21no—
26C5×D4⋊C8—21no—
27C5×D4⋊C8—21no—
28C5×D4⋊C8—21no—
29#10456C511yesmaximal, characteristic, direct product factor, Sylow
30C2×C5×#41—21no—
31C22×C5×D4D411yescharacteristic
32C23×C4×C5—21no—
33C5×AGL(1,ℤ/8)—42no—
34C5×AGL(1,ℤ/8)—42no—
35C2×C5×SD16—22no—
36C2×C5×D8—22no—
37C2×C5×M16—21no—
38C2×C5×M16—21no—
39C2×C5×M16—21no—
40C22×C5×C8—21no—
41C22×C5×C8—21no—
42C5×C4⋊Q8D411yescharacteristic
43C5×#127D411yescharacteristic
44C5×#124—21no—
45C5×#121—21no—
46C5×#121—21no—
47C4×C5×D4—21no—
48C4×C5×D4—21no—
49C4×C5×D4—21no—
50C4×C5×D4—21no—
51C4×C5×D4—21no—
52C4×C5×D4—21no—
53C2×C5×C4⋊C4D411yescharacteristic
54C2×C5×C22⋊C4—21no—
55C2×C42×C5C2311yescharacteristic
56C5×C4⋊C8—21no—
57C5×C4⋊C8—21no—
58C5×C4⋊C8—21no—
59C5×C4⋊C8—21no—
60C5×C4⋊C8—21no—
61C5×C4⋊C8—21no—
62C5×C4⋊C8—21no—
63C5×C4⋊C8D412yes—
64C5×#103—22no—
65C5×#102—22no—
66C5×#100—42no—
67C4×C5×C8—21no—
68C4×C5×C8D411yescharacteristic
69C4×C5×C8D411yescharacteristic
70#3036C2×C511yescharacteristic
71#1800C2×C511yescharacteristic, complemented
72#1567C2×C511yescharacteristic, complemented
73#1547C2×C511yescharacteristic, complemented
74#1492C2×C511yescharacteristic, complemented
75C2×D4⋊C8C2×C511yescharacteristic, complemented
76#1292C2×C511yescharacteristic, complemented
77C24×C5—21no—
78C5×#41—41no—
79C5×#41—41no—
80C5×#41—41no—
81C5×#41—41no—
82C2×C5×Q8—21no—
83C2×C5×D4—21no—
84C2×C5×D4—21no—
85C2×C5×D4—21no—
86C2×C5×D4—21no—
87C2×C5×D4—21no—
88C2×C5×D4—21no—
89C2×C5×D4—21no—
90C2×C5×D4—21no—
91C2×C5×D4—41no—
92C2×C5×D4D812yes—
93C22×C4×C5—21no—
94C22×C4×C5—41no—
95C22×C4×C5—41no—
96C22×C4×C5—41no—
97C22×C4×C5—81no—
98C22×C4×C5—81no—
99C22×C4×C5C2×D411yescharacteristic
100C22×C4×C5#4111yescharacteristic
101C22×C4×C5#4111yescharacteristic
102C5×SD16—44no—
103C5×D8—44no—
104C5×M16—41no—
105C5×M16—41no—
106C5×M16—41no—
107C5×M16—41no—
108C5×M16—41no—
109C5×M16—41no—
110C2×C5×C8—21no—
111C2×C5×C8—22no—
112C2×C5×C8—21no—
113C2×C5×C8—42no—
114C2×C5×C8—41no—
115C2×C5×C8—41no—
116C2×C5×C8—41no—
117C2×C5×C8—81no—
118C5×C4⋊C4—21no—
119C5×C4⋊C4—21no—
120C5×C4⋊C4—21no—
121C5×C4⋊C4—21no—
122C5×C22⋊C4—41no—
123C5×C22⋊C4—41no—
124C5×C22⋊C4—41no—
125C5×C22⋊C4—41no—
126C42×C5C2×D411yescharacteristic
127C42×C5C2×D411yescharacteristic
128C42×C5C2×D411yescharacteristic
129C42×C5#4111yescharacteristic
130C2×AGL(1,ℤ/8)—21no—
131#532C22×C511yescharacteristic
132C2×C4×D4C22×C511yescharacteristic
133#463—22no—
134#461—21no—
135#459—21no—
136#423C22×C511yescharacteristic
137#423C22×C511yescharacteristic
138#423C22×C511yescharacteristic
139C2×C4⋊C8C22×C511yescharacteristic
140#417—21no—
141C2×#100—21no—
142C2×C4×C8C22×C511yescharacteristic
143#343—21no—
144#335—22no—
145#332—21no—
146#331—21no—
147D4⋊C8—21no—
148D4⋊C8—21no—
149D4⋊C8—21no—
150D4⋊C8—21no—
151C23×C5—22no—
152C23×C5—41no—
153C23×C5—41no—
154C23×C5—41no—
155C23×C5—81no—
156C23×C5—81no—
157C23×C5#12311yescharacteristic
158C5×Q8—41no—
159C5×Q8—41no—
160C5×D4—22no—
161C5×D4—22no—
162C5×D4—41no—
163C5×D4—41no—
164C5×D4—41no—
165C5×D4—41no—
166C5×D4—41no—
167C5×D4—41no—
168C5×D4—81no—
169C5×D4—81no—
170C5×D4—81no—
171C2×C4×C5—21no—
172C2×C4×C5—21no—
173C2×C4×C5—41no—
174C2×C4×C5—41no—
175C2×C4×C5—41no—
176C2×C4×C5—82no—
177C2×C4×C5—81no—
178C2×C4×C5—81no—
179C2×C4×C5—81no—
180C2×C4×C5—81no—
181C2×C4×C5—81no—
182C2×C4×C5#12011yescharacteristic
183C2×C4×C5#12111yescharacteristic
184C2×C4×C5#12111yescharacteristic
185C2×C4×C5#12111yescharacteristic
186C2×C4×C5#12311yescharacteristic
187C2×C4×C5#12411yescharacteristic
188C2×C4×C5C2×D811yescharacteristic
189C2×C4×C5#13511yescharacteristic
190C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C5#13711yescharacteristic
192C5×C8—22no—
193C5×C8—41no—
194C5×C8—42no—
195C5×C8—81no—
196C5×C8—81no—
197C5×C8—81no—
198C2×#41—21no—
199C22×D4C5×D411yescharacteristic
200C23×C4—21no—
201AGL(1,ℤ/8)—42no—
202AGL(1,ℤ/8)—42no—
203C2×SD16—22no—
204C2×D8—22no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C22×C8—21no—
209C22×C8—21no—
210C4⋊Q8C5×D411yescharacteristic
211#127C5×D411yescharacteristic
212#124—21no—
213#121—21no—
214#121—21no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C2×C4⋊C4C5×D411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C42C23×C511yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8C5×D412yes—
232#103—22no—
233#102—22no—
234#100—42no—
235C4×C8—21no—
236C4×C8C5×D411yescharacteristic
237C4×C8C5×D411yescharacteristic
238C22×C5—21no—
239C22×C5—21no—
240C22×C5—42no—
241C22×C5—82no—
242C22×C5—82no—
243C22×C5—81no—
244C22×C5—81no—
245C22×C5—81no—
246C22×C5—81no—
247C22×C5—81no—
248C22×C5#39411yescharacteristic
249C22×C5#48011yescharacteristic
250C22×C5#48211yescharacteristic
251C4×C5—21no—
252C4×C5—21no—
253C4×C5—21no—
254C4×C5—81no—
255C4×C5—81no—
256C4×C5#44711yescharacteristic
257C4×C5#45211yescharacteristic
258C4×C5#46611yescharacteristic
259C4×C5#46811yescharacteristic
260C4×C5#47111yescharacteristic
261C4×C5#47211yescharacteristic
262C24—21no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4C5×D812yes—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4C2×C5×D411yescharacteristic
285C22×C4C5×#4111yescharacteristic
286C22×C4C5×#4111yescharacteristic
287SD16—44no—
288D8—44no—
289M16—41no—
290M16—41no—
291M16—41no—
292M16—41no—
293M16—41no—
294M16—41no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—81no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C42C2×C5×D411yescharacteristic
312C42C2×C5×D411yescharacteristic
313C42C2×C5×D411yescharacteristic
314C42C5×#4111yescharacteristic
315C2×C5—21no—
316C2×C5—21no—
317C2×C5—82no—
318C2×C5—161no—
319C2×C5#203311yescharacteristic
320C2×C5#205011yescharacteristic
321C2×C5#205111yescharacteristic
322C23—22no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—81no—
327C23—81no—
328C23C5×#12311yescharacteristic
329Q8—41no—
330Q8—41no—
331D4—22no—
332D4—22no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—81no—
340D4—81no—
341D4—81no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4C5×#12011yescharacteristic
354C2×C4C5×#12111yescharacteristic
355C2×C4C5×#12111yescharacteristic
356C2×C4C5×#12111yescharacteristic
357C2×C4C5×#12311yescharacteristic
358C2×C4C5×#12411yescharacteristic
359C2×C4C2×C5×D811yescharacteristic
360C2×C4C5×#13511yescharacteristic
361C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
362C2×C4C5×#13711yescharacteristic
363C8—22no—
364C8—41no—
365C8—42no—
366C8—81no—
367C8—81no—
368C8—81no—
369C5#1045611yescharacteristic, direct product factor, Sylow
370C22—21no—
371C22—21no—
372C22—42no—
373C22—82no—
374C22—82no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22C5×#39411yescharacteristic
381C22C5×#48011yescharacteristic
382C22C5×#48211yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—81no—
387C4—81no—
388C4C5×#44711yescharacteristic
389C4C5×#45211yescharacteristic
390C4C5×#46611yescharacteristic
391C4C5×#46811yescharacteristic
392C4C5×#47111yescharacteristic
393C4C5×#47211yescharacteristic
394C2—21no—
395C2—21no—
396C2—82no—
397C2—161no—
398C2C5×#203311yescharacteristic
399C2C5×#205011yescharacteristic
400C2C5×#205111yescharacteristic
subgroupG/H
center, socleC22×C5#394
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC5×#3036C2
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×#1800C2semidirect
3. C5×#1567C2semidirect
4. C5×#1547C2semidirect
5. C5×#1492C2semidirect
6. C2×C5×D4⋊C8C2semidirect
7. C5×#1292C2semidirect
29. #10456C5direct (N × H)
71. #1800C2×C5semidirect
72. #1567C2×C5semidirect
73. #1547C2×C5semidirect
74. #1492C2×C5semidirect
75. C2×D4⋊C8C2×C5semidirect
76. #1292C2×C5semidirect
369. C5#10456direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#30362
C5×#18002
C5×#15672
C5×#15472
C5×#14922
C2×C5×D4⋊C82
C5×#12922
#104565