Finite Group Databasebeta

← C5×#13028 #11935748

Subgroups — C5×#13028 #11935748

Order 1280.

476 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#510C211yesmaximal, characteristic
2C2×C5×#443C211yesmaximal, characteristic
3C2×C5×#362C211yesmaximal, characteristic, complemented
4C5×#2139C211yesmaximal, characteristic
5C5×#1969C211yesmaximal, characteristic, complemented
6C5×#1387C211yesmaximal, characteristic, complemented
7C5×#1376C211yesmaximal, characteristic, complemented
8C2×C5×#135C2211yescharacteristic
9C22×C5×Q16C2211yescharacteristic
10C2×C5×#131—21no—
11C5×#510—22no—
12C5×#510C414yescomplemented
13C2×C5×Q32—21no—
14C2×C5×SD32—21no—
15C2×C5×M32C2211yescharacteristic
16C2×C5×M32C2211yescharacteristic
17C5×#443—22no—
18C5×#443—22no—
19C2×C5×#108C2211yescharacteristic
20C2×C5×#107C2211yescharacteristic
21C2×C5×#104—21no—
22C2×C5×#103—21no—
23C2×C4×C5×C8C2211yescharacteristic
24C5×#391—21no—
25C5×#362—22no—
26C5×#358—21no—
27C5×#357—21no—
28#13028C511yesmaximal, characteristic, direct product factor, Sylow
29C2×C5×#41—21no—
30C22×C5×Q8—21no—
31C5×#135—21no—
32C5×#135—21no—
33C5×#135C2×C412yes—
34C5×#135D412yes—
35C2×C5×Q16—21no—
36C2×C5×Q16—21no—
37C2×C5×Q16—21no—
38C2×C5×Q16—21no—
39C2×C5×Q16—21no—
40C2×C5×Q16C2×C412yes—
41C2×C5×Q16D411yescharacteristic
42C2×C5×SD16—21no—
43C2×C5×D8D411yescharacteristic
44C5×#131—24no—
45C5×#131—42no—
46C2×C5×M16—21no—
47C2×C5×M16—41no—
48C22×C5×C8—41no—
49C22×C5×C8D411yescharacteristic
50C22×C5×C8C2311yescharacteristic
51C2×C5×C4⋊C4—21no—
52C2×C42×C5D411yescharacteristic
53C5×Q32—24no—
54C5×SD32—24no—
55C5×M32—21no—
56C5×M32—21no—
57C5×M32—21no—
58C5×M32C2×C412yes—
59C2×C5×C16—21no—
60C2×C5×C16—21no—
61C5×#108—21no—
62C5×#108D412yes—
63C5×#107—21no—
64C5×#107—21no—
65C5×#104—24no—
66C5×#103—41no—
67C5×#103—41no—
68C4×C5×C8—21no—
69C4×C5×C8—21no—
70C5×#95—42no—
71C2×#510C2×C511yescharacteristic
72C2×#443C2×C511yescharacteristic
73C2×#362C2×C511yescharacteristic, complemented
74#2139C2×C511yescharacteristic
75#1969C2×C511yescharacteristic, complemented
76#1387C2×C511yescharacteristic, complemented
77#1376C2×C511yescharacteristic, complemented
78C5×#41—22no—
79C5×#41—22no—
80C5×#41—41no—
81C5×#41—41no—
82C2×C5×Q8—22no—
83C2×C5×Q8—21no—
84C2×C5×Q8—41no—
85C2×C5×Q8—41no—
86C2×C5×Q8—41no—
87C2×C5×Q8—41no—
88C2×C5×Q8—41no—
89C2×C5×D4—21no—
90C2×C5×D4—41no—
91C22×C4×C5—21no—
92C22×C4×C5—41no—
93C22×C4×C5—41no—
94C22×C4×C5—41no—
95C22×C4×C5C2×D411yescharacteristic
96C5×Q16—22no—
97C5×Q16—22no—
98C5×Q16—21no—
99C5×Q16—41no—
100C5×Q16—41no—
101C5×Q16C22⋊C412yes—
102C5×SD16—22no—
103C5×SD16—22no—
104C5×D8—21no—
105C5×D8C22⋊C412yes—
106C5×M16—22no—
107C5×M16—44no—
108C5×M16—41no—
109C2×C5×C8—21no—
110C2×C5×C8—21no—
111C2×C5×C8—44no—
112C2×C5×C8—41no—
113C2×C5×C8D811yescharacteristic
114C2×C5×C8D811yescharacteristic
115C2×C5×C8C22×C411yescharacteristic
116C2×C5×C8C2×D411yescharacteristic
117C2×C5×C8C2×D411yescharacteristic
118C2×C5×C8C2×D411yescharacteristic
119C2×C5×C8#4111yescharacteristic
120C2×C5×C8#4111yescharacteristic
121C5×C4⋊C4—41no—
122C5×C4⋊C4—41no—
123C42×C5—21no—
124C42×C5—21no—
125C5×C16—22no—
126C5×C16—41no—
127C2×#135C22×C511yescharacteristic
128C22×Q16C22×C511yescharacteristic
129C2×#131—21no—
130#510—22no—
131#510C4×C514yescomplemented
132C2×Q32—21no—
133C2×SD32—21no—
134C2×M32C22×C511yescharacteristic
135C2×M32C22×C511yescharacteristic
136#443—22no—
137#443—22no—
138C2×#108C22×C511yescharacteristic
139C2×#107C22×C511yescharacteristic
140C2×#104—21no—
141C2×#103—21no—
142C2×C4×C8C22×C511yescharacteristic
143#391—21no—
144#362—22no—
145#358—21no—
146#357—21no—
147C23×C5—41no—
148C23×C5AGL(1,ℤ/8)11yescharacteristic
149C5×Q8—22no—
150C5×Q8—41no—
151C5×Q8—42no—
152C5×Q8—42no—
153C5×Q8—81no—
154C5×Q8—81no—
155C5×D4—22no—
156C5×D4—42no—
157C5×D4—42no—
158C5×D4—41no—
159C2×C4×C5—22no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5—42no—
164C2×C4×C5—42no—
165C2×C4×C5—42no—
166C2×C4×C5—81no—
167C2×C4×C5—81no—
168C2×C4×C5—82no—
169C2×C4×C5—81no—
170C2×C4×C5C4×D411yescharacteristic
171C2×C4×C5#12011yescharacteristic
172C2×C4×C5#12111yescharacteristic
173C2×C4×C5C2×D811yescharacteristic
174C2×C4×C5#13511yescharacteristic
175C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
176C5×C8—21no—
177C5×C8—21no—
178C5×C8—42no—
179C5×C8C2×C22⋊C411yescharacteristic
180C5×C8C4×D411yescharacteristic
181C5×C8#12111yescharacteristic
182C5×C8#12311yescharacteristic
183C2×#41—21no—
184C22×Q8—21no—
185#135—21no—
186#135—21no—
187#135C2×C4×C512yes—
188#135C5×D412yes—
189C2×Q16—21no—
190C2×Q16—21no—
191C2×Q16—21no—
192C2×Q16—21no—
193C2×Q16—21no—
194C2×Q16C2×C4×C512yes—
195C2×Q16C5×D411yescharacteristic
196C2×SD16—21no—
197C2×D8C5×D411yescharacteristic
198#131—24no—
199#131—42no—
200C2×M16—21no—
201C2×M16—41no—
202C22×C8—41no—
203C22×C8C5×D411yescharacteristic
204C22×C8C23×C511yescharacteristic
205C2×C4⋊C4—21no—
206C2×C42C5×D411yescharacteristic
207Q32—24no—
208SD32—24no—
209M32—21no—
210M32—21no—
211M32—21no—
212M32C2×C4×C512yes—
213C2×C16—21no—
214C2×C16—21no—
215#108—21no—
216#108C5×D412yes—
217#107—21no—
218#107—21no—
219#104—24no—
220#103—41no—
221#103—41no—
222C4×C8—21no—
223C4×C8—21no—
224#95—42no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—42no—
228C22×C5—81no—
229C22×C5—81no—
230C22×C5#44211yescharacteristic
231C22×C5#44911yescharacteristic
232C22×C5#45911yescharacteristic
233C4×C5—42no—
234C4×C5—42no—
235C4×C5—82no—
236C4×C5—161no—
237C4×C5#38611yescharacteristic
238C4×C5C4×D811yescharacteristic
239C4×C5#44711yescharacteristic
240C4×C5#46311yescharacteristic
241#41—22no—
242#41—22no—
243#41—41no—
244#41—41no—
245C2×Q8—22no—
246C2×Q8—21no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—21no—
253C2×D4—41no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C2×C5×D411yescharacteristic
259Q16—22no—
260Q16—22no—
261Q16—21no—
262Q16—41no—
263Q16—41no—
264Q16C5×C22⋊C412yes—
265SD16—22no—
266SD16—22no—
267D8—21no—
268D8C5×C22⋊C412yes—
269M16—22no—
270M16—44no—
271M16—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—44no—
275C2×C8—41no—
276C2×C8C5×D811yescharacteristic
277C2×C8C5×D811yescharacteristic
278C2×C8C22×C4×C511yescharacteristic
279C2×C8C2×C5×D411yescharacteristic
280C2×C8C2×C5×D411yescharacteristic
281C2×C8C2×C5×D411yescharacteristic
282C2×C8C5×#4111yescharacteristic
283C2×C8C5×#4111yescharacteristic
284C4⋊C4—41no—
285C4⋊C4—41no—
286C42—21no—
287C42—21no—
288C16—22no—
289C16—41no—
290C2×C5—21no—
291C2×C5—21no—
292C2×C5—82no—
293C2×C5#187911yescharacteristic
294C2×C5#221211yescharacteristic
295C2×C5#221411yescharacteristic
296C23—41no—
297C23C5×AGL(1,ℤ/8)11yescharacteristic
298Q8—22no—
299Q8—41no—
300Q8—42no—
301Q8—42no—
302Q8—81no—
303Q8—81no—
304D4—22no—
305D4—42no—
306D4—42no—
307D4—41no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4C4×C5×D411yescharacteristic
320C2×C4C5×#12011yescharacteristic
321C2×C4C5×#12111yescharacteristic
322C2×C4C2×C5×D811yescharacteristic
323C2×C4C5×#13511yescharacteristic
324C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
325C8—21no—
326C8—21no—
327C8—42no—
328C8C2×C5×C22⋊C411yescharacteristic
329C8C4×C5×D411yescharacteristic
330C8C5×#12111yescharacteristic
331C8C5×#12311yescharacteristic
332C5#1302811yescharacteristic, direct product factor, Sylow
333C22—21no—
334C22—21no—
335C22—42no—
336C22—81no—
337C22—81no—
338C22C5×#44211yescharacteristic
339C22C5×#44911yescharacteristic
340C22C5×#45911yescharacteristic
341C4—42no—
342C4—42no—
343C4—82no—
344C4—161no—
345C4C5×#38611yescharacteristic
346C4C4×C5×D811yescharacteristic
347C4C5×#44711yescharacteristic
348C4C5×#46311yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2C5×#187911yescharacteristic
353C2C5×#221211yescharacteristic
354C2C5×#221411yescharacteristic
subgroupG/H
center, socleC22×C5#459
commutatorC2×C8C22×C4×C5
Frattini, cosocleC22×C8C23×C5
prime radicalC2×C5×#135C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C5×#362C2semidirect
5. C5×#1969C2semidirect
6. C5×#1387C2semidirect
7. C5×#1376C2semidirect
12. C5×#510C4semidirect
28. #13028C5direct (N × H)
73. C2×#362C2×C5semidirect
75. #1969C2×C5semidirect
76. #1387C2×C5semidirect
77. #1376C2×C5semidirect
131. #510C4×C5semidirect
332. C5#13028direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5102
C2×C5×#4432
C2×C5×#3622
C5×#21392
C5×#19692
C5×#13872
C5×#13762
#130285