Finite Group Databasebeta

← C5×#9310 #11932030

Subgroups — C5×#9310 #11932030

Order 1280.

470 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3037C211yesmaximal, characteristic
2C5×#2983C211yesmaximal, characteristic, complemented
3C5×#1805C211yesmaximal, characteristic, complemented
4C5×#1517C211yesmaximal, characteristic
5C5×#1480C211yesmaximal, characteristic
6C5×#1301C211yesmaximal, characteristic, complemented
7C5×#1286C211yesmaximal, characteristic, complemented
8C2×C5×#137—21no—
9C5×#532C2211yescharacteristic
10C2×C4×C5×Q8C2211yescharacteristic
11C5×#464—22no—
12C5×#462C412yescomplemented
13C5×#460C412yescomplemented
14C5×#446—22no—
15C5×#446—22no—
16C5×#432—22no—
17C5×#431C2211yescharacteristic
18C5×#431C2211yescharacteristic
19C5×#423C2211yescharacteristic
20C2×C5×C4⋊C8C2211yescharacteristic
21C5×#417—21no—
22C2×C5×#103—21no—
23C5×#382C2211yescharacteristic
24C5×#340—21no—
25C5×#327—22no—
26C5×D4⋊C8—22no—
27#9310C511yesmaximal, characteristic, direct product factor, Sylow
28C2×C5×#41—21no—
29C22×C5×Q8D411yescharacteristic
30C5×#137—42no—
31C5×#137—42no—
32C2×C5×Q16—22no—
33C2×C5×SD16—22no—
34C2×C5×M16—21no—
35C22×C5×C8—21no—
36C5×C4⋊Q8C2×C411yescharacteristic
37C5×#127C2×C411yescharacteristic
38C5×#124—21no—
39C5×#121—21no—
40C5×#121—21no—
41C4×C5×Q8—21no—
42C4×C5×Q8—21no—
43C4×C5×Q8C2×C412yes—
44C4×C5×Q8D412yes—
45C4×C5×D4—21no—
46C4×C5×D4—21no—
47C2×C5×C4⋊C4—21no—
48C2×C5×C4⋊C4—21no—
49C2×C5×C4⋊C4D411yescharacteristic
50C2×C42×C5—21no—
51C2×C42×C5C2311yescharacteristic
52C5×C4⋊C8—24no—
53C5×C4⋊C8—21no—
54C5×C4⋊C8—21no—
55C5×C4⋊C8—21no—
56C5×C4⋊C8—21no—
57C5×C4⋊C8—21no—
58C5×C4⋊C8C2×C412yes—
59C5×#103—22no—
60C5×#103—42no—
61C5×#102—22no—
62C5×C22⋊C8—21no—
63C5×C22⋊C8D412yes—
64C5×#97—24no—
65C4×C5×C8—21no—
66C4×C5×C8D412yes—
67C5×#95—21no—
68C5×#95—21no—
69#3037C2×C511yescharacteristic
70#2983C2×C511yescharacteristic, complemented
71#1805C2×C511yescharacteristic, complemented
72#1517C2×C511yescharacteristic
73#1480C2×C511yescharacteristic
74#1301C2×C511yescharacteristic, complemented
75#1286C2×C511yescharacteristic, complemented
76C5×#41—41no—
77C5×#41—41no—
78C5×#41—41no—
79C5×#41—41no—
80C2×C5×Q8—21no—
81C2×C5×Q8—21no—
82C2×C5×Q8—21no—
83C2×C5×Q8—21no—
84C2×C5×Q8—21no—
85C2×C5×Q8—21no—
86C2×C5×Q8C22⋊C412yes—
87C2×C5×D4—21no—
88C2×C5×D4—21no—
89C2×C5×D4—21no—
90C2×C5×D4—41no—
91C22×C4×C5—21no—
92C22×C4×C5—21no—
93C22×C4×C5—41no—
94C22×C4×C5—41no—
95C22×C4×C5C2×D411yescharacteristic
96C22×C4×C5C2×D411yescharacteristic
97C22×C4×C5#4111yescharacteristic
98C5×Q16—44no—
99C5×SD16—44no—
100C5×M16—41no—
101C5×M16—41no—
102C2×C5×C8—22no—
103C2×C5×C8—22no—
104C2×C5×C8—22no—
105C2×C5×C8—41no—
106C2×C5×C8—41no—
107C2×C5×C8—41no—
108C5×C4⋊C4—22no—
109C5×C4⋊C4—21no—
110C5×C4⋊C4—22no—
111C5×C4⋊C4—21no—
112C5×C4⋊C4—21no—
113C5×C4⋊C4—41no—
114C5×C4⋊C4—41no—
115C5×C4⋊C4C22⋊C412yes—
116C5×C22⋊C4—41no—
117C5×C22⋊C4—41no—
118C42×C5—22no—
119C42×C5—22no—
120C42×C5C22×C411yescharacteristic
121C42×C5C2×D411yescharacteristic
122C42×C5C2×D411yescharacteristic
123C42×C5#4111yescharacteristic
124C2×#137—21no—
125#532C22×C511yescharacteristic
126C2×C4×Q8C22×C511yescharacteristic
127#464—22no—
128#462C4×C512yescomplemented
129#460C4×C512yescomplemented
130#446—22no—
131#446—22no—
132#432—22no—
133#431C22×C511yescharacteristic
134#431C22×C511yescharacteristic
135#423C22×C511yescharacteristic
136C2×C4⋊C8C22×C511yescharacteristic
137#417—21no—
138C2×#103—21no—
139#382C22×C511yescharacteristic
140#340—21no—
141#327—22no—
142D4⋊C8—22no—
143C23×C5—41no—
144C23×C5#12111yescharacteristic
145C5×Q8—22no—
146C5×Q8—22no—
147C5×Q8—41no—
148C5×Q8—41no—
149C5×Q8—41no—
150C5×Q8—41no—
151C5×D4—42no—
152C5×D4—41no—
153C5×D4—41no—
154C5×D4—81no—
155C5×D4—81no—
156C5×D4—81no—
157C2×C4×C5—22no—
158C2×C4×C5—22no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5—41no—
164C2×C4×C5—81no—
165C2×C4×C5—81no—
166C2×C4×C5—81no—
167C2×C4×C5—81no—
168C2×C4×C5—81no—
169C2×C4×C5—81no—
170C2×C4×C5C2×C22⋊C411yescharacteristic
171C2×C4×C5C4×D411yescharacteristic
172C2×C4×C5C4×D411yescharacteristic
173C2×C4×C5#12011yescharacteristic
174C2×C4×C5#12111yescharacteristic
175C2×C4×C5#12311yescharacteristic
176C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C5#13711yescharacteristic
180C5×C8—42no—
181C5×C8—42no—
182C5×C8—42no—
183C5×C8—81no—
184C2×#41—21no—
185C22×Q8C5×D411yescharacteristic
186#137—42no—
187#137—42no—
188C2×Q16—22no—
189C2×SD16—22no—
190C2×M16—21no—
191C22×C8—21no—
192C4⋊Q8C2×C4×C511yescharacteristic
193#127C2×C4×C511yescharacteristic
194#124—21no—
195#121—21no—
196#121—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8C2×C4×C512yes—
200C4×Q8C5×D412yes—
201C4×D4—21no—
202C4×D4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C5×D411yescharacteristic
206C2×C42—21no—
207C2×C42C23×C511yescharacteristic
208C4⋊C8—24no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8C2×C4×C512yes—
215#103—22no—
216#103—42no—
217#102—22no—
218C22⋊C8—21no—
219C22⋊C8C5×D412yes—
220#97—24no—
221C4×C8—21no—
222C4×C8C5×D412yes—
223#95—21no—
224#95—21no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—81no—
228C22×C5—81no—
229C22×C5—81no—
230C22×C5#38611yescharacteristic
231C22×C5#46811yescharacteristic
232C22×C5#46911yescharacteristic
233C4×C5—21no—
234C4×C5—21no—
235C4×C5—21no—
236C4×C5—21no—
237C4×C5—42no—
238C4×C5—42no—
239C4×C5—81no—
240C4×C5—161no—
241C4×C5#41711yescharacteristic
242C4×C5#41811yescharacteristic
243C4×C5#46811yescharacteristic
244C4×C5#46911yescharacteristic
245#41—41no—
246#41—41no—
247#41—41no—
248#41—41no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8C5×C22⋊C412yes—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4C2×C5×D411yescharacteristic
265C22×C4C2×C5×D411yescharacteristic
266C22×C4C5×#4111yescharacteristic
267Q16—44no—
268SD16—44no—
269M16—41no—
270M16—41no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4C5×C22⋊C412yes—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—22no—
288C42—22no—
289C42C22×C4×C511yescharacteristic
290C42C2×C5×D411yescharacteristic
291C42C2×C5×D411yescharacteristic
292C42C5×#4111yescharacteristic
293C2×C5—21no—
294C2×C5—21no—
295C2×C5—161no—
296C2×C5#188111yescharacteristic
297C2×C5#189511yescharacteristic
298C2×C5#190511yescharacteristic
299C23—41no—
300C23C5×#12111yescharacteristic
301Q8—22no—
302Q8—22no—
303Q8—41no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307D4—42no—
308D4—41no—
309D4—41no—
310D4—81no—
311D4—81no—
312D4—81no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4C2×C5×C22⋊C411yescharacteristic
327C2×C4C4×C5×D411yescharacteristic
328C2×C4C4×C5×D411yescharacteristic
329C2×C4C5×#12011yescharacteristic
330C2×C4C5×#12111yescharacteristic
331C2×C4C5×#12311yescharacteristic
332C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
333C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
334C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
335C2×C4C5×#13711yescharacteristic
336C8—42no—
337C8—42no—
338C8—42no—
339C8—81no—
340C5#931011yescharacteristic, direct product factor, Sylow
341C22—21no—
342C22—21no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22C5×#38611yescharacteristic
347C22C5×#46811yescharacteristic
348C22C5×#46911yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—42no—
355C4—81no—
356C4—161no—
357C4C5×#41711yescharacteristic
358C4C5×#41811yescharacteristic
359C4C5×#46811yescharacteristic
360C4C5×#46911yescharacteristic
361C2—21no—
362C2—21no—
363C2—161no—
364C2C5×#188111yescharacteristic
365C2C5×#189511yescharacteristic
366C2C5×#190511yescharacteristic
subgroupG/H
center, socleC22×C5#386
commutatorC42C22×C4×C5
Frattini, cosocleC2×C42C23×C5
prime radicalC5×#532C22
LCSG ▷ 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×#2983C2semidirect
3. C5×#1805C2semidirect
6. C5×#1301C2semidirect
7. C5×#1286C2semidirect
12. C5×#462C4semidirect
13. C5×#460C4semidirect
27. #9310C5direct (N × H)
70. #2983C2×C5semidirect
71. #1805C2×C5semidirect
74. #1301C2×C5semidirect
75. #1286C2×C5semidirect
128. #462C4×C5semidirect
129. #460C4×C5semidirect
340. C5#9310direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#30372
C5×#29832
C5×#18052
C5×#15172
C5×#14802
C5×#13012
C5×#12862
#93105