Finite Group Databasebeta

← C5×#13024 #11935744

Subgroups — C5×#13024 #11935744

Order 1280.

466 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3414C211yesmaximal, characteristic
2C2×C5×#444C211yesmaximal, characteristic, complemented
3C5×#2148C211yesmaximal, characteristic, complemented
4C5×#2143C211yesmaximal, characteristic, complemented
5C5×#1975C211yesmaximal, characteristic, complemented
6C5×#1388C211yesmaximal, characteristic, complemented
7C5×#1377C211yesmaximal, characteristic, complemented
8C2×C5×#135C2211yescharacteristic
9C2×C5×#135C2211yescharacteristic
10C2×C5×#131—21no—
11C5×#510—21no—
12C5×#510C412yescomplemented
13C5×#509—21no—
14C5×#509C412yescomplemented
15C5×#508—22no—
16C2×C5×M32C2211yescharacteristic
17C2×C5×M32C2211yescharacteristic
18C5×#444—22no—
19C5×#444—22no—
20C2×C5×#108C2211yescharacteristic
21C2×C5×#108C2211yescharacteristic
22C2×C5×#104—21no—
23C2×C5×#104—21no—
24C5×#408—21no—
25C2×C5×#97C2211yescharacteristic
26C5×#362—21no—
27C5×#361—21no—
28C5×#359—22no—
29#13024C511yesmaximal, characteristic, direct product factor, Sylow
30C2×C5×#41—21no—
31C2×C5×#41—21no—
32C5×#135—21no—
33C5×#135—21no—
34C5×#135—21no—
35C5×#135—21no—
36C5×#135—21no—
37C5×#135—21no—
38C5×#135C2×C412yes—
39C5×#135D412yes—
40C2×C5×Q16C2×C411yescharacteristic
41C2×C5×Q16D411yescharacteristic
42C2×C5×SD16—21no—
43C2×C5×SD16—21no—
44C2×C5×D8C2×C411yescharacteristic
45C2×C5×D8D411yescharacteristic
46C5×#131—24no—
47C5×#131—42no—
48C2×C5×M16—21no—
49C2×C5×M16—21no—
50C2×C5×M16—41no—
51C22×C5×C8—41no—
52C22×C5×C8D411yescharacteristic
53C22×C5×C8C2311yescharacteristic
54C2×C42×C5D411yescharacteristic
55C5×Q32—22no—
56C5×SD32—22no—
57C5×SD32—22no—
58C5×D16—22no—
59C5×M32—21no—
60C5×M32—21no—
61C5×M32—21no—
62C5×M32C2×C412yes—
63C2×C5×C16—21no—
64C2×C5×C16—21no—
65C5×#108—21no—
66C5×#108—21no—
67C5×#108—21no—
68C5×#108D412yes—
69C5×#104—24no—
70C5×#104—42no—
71C5×C22⋊C8—42no—
72C5×#97—21no—
73C5×#97—21no—
74#3414C2×C511yescharacteristic
75C2×#444C2×C511yescharacteristic, complemented
76#2148C2×C511yescharacteristic, complemented
77#2143C2×C511yescharacteristic, complemented
78#1975C2×C511yescharacteristic, complemented
79#1388C2×C511yescharacteristic, complemented
80#1377C2×C511yescharacteristic, complemented
81C5×#41—22no—
82C5×#41—22no—
83C5×#41—41no—
84C5×#41—41no—
85C5×#41—41no—
86C5×#41—41no—
87C5×#41—41no—
88C5×#41—41no—
89C2×C5×Q8—21no—
90C2×C5×Q8—21no—
91C2×C5×D4—21no—
92C2×C5×D4—21no—
93C2×C5×D4—41no—
94C2×C5×D4—41no—
95C22×C4×C5—21no—
96C22×C4×C5—41no—
97C22×C4×C5—41no—
98C22×C4×C5C2×D411yescharacteristic
99C5×Q16—21no—
100C5×Q16—21no—
101C5×Q16—21no—
102C5×Q16C22⋊C412yes—
103C5×SD16—22no—
104C5×SD16—22no—
105C5×SD16—41no—
106C5×SD16—41no—
107C5×D8—21no—
108C5×D8—21no—
109C5×D8—21no—
110C5×D8C22⋊C412yes—
111C5×M16—22no—
112C5×M16—44no—
113C5×M16—41no—
114C5×M16—41no—
115C5×M16—41no—
116C2×C5×C8—21no—
117C2×C5×C8—21no—
118C2×C5×C8—44no—
119C2×C5×C8—41no—
120C2×C5×C8—41no—
121C2×C5×C8D812yes—
122C2×C5×C8C22×C411yescharacteristic
123C2×C5×C8C2×D411yescharacteristic
124C2×C5×C8C2×D411yescharacteristic
125C2×C5×C8C2×D411yescharacteristic
126C2×C5×C8#4111yescharacteristic
127C2×C5×C8#4111yescharacteristic
128C42×C5—21no—
129C42×C5—21no—
130C5×C16—22no—
131C5×C16—41no—
132C2×#135C22×C511yescharacteristic
133C2×#135C22×C511yescharacteristic
134C2×#131—21no—
135#510—21no—
136#510C4×C512yescomplemented
137#509—21no—
138#509C4×C512yescomplemented
139#508—22no—
140C2×M32C22×C511yescharacteristic
141C2×M32C22×C511yescharacteristic
142#444—22no—
143#444—22no—
144C2×#108C22×C511yescharacteristic
145C2×#108C22×C511yescharacteristic
146C2×#104—21no—
147C2×#104—21no—
148#408—21no—
149C2×#97C22×C511yescharacteristic
150#362—21no—
151#361—21no—
152#359—22no—
153C23×C5—41no—
154C23×C5—41no—
155C23×C5#13511yescharacteristic
156C5×Q8—22no—
157C5×Q8—41no—
158C5×Q8—41no—
159C5×Q8—41no—
160C5×D4—22no—
161C5×D4—42no—
162C5×D4—42no—
163C5×D4—41no—
164C5×D4—41no—
165C5×D4—41no—
166C5×D4—81no—
167C5×D4—81no—
168C2×C4×C5—41no—
169C2×C4×C5—42no—
170C2×C4×C5—42no—
171C2×C4×C5—41no—
172C2×C4×C5—41no—
173C2×C4×C5—81no—
174C2×C4×C5—81no—
175C2×C4×C5—81no—
176C2×C4×C5C4×D411yescharacteristic
177C2×C4×C5#12011yescharacteristic
178C2×C4×C5#12111yescharacteristic
179C2×C4×C5C2×D811yescharacteristic
180C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
182C5×C8—22no—
183C5×C8—42no—
184C5×C8—81no—
185C5×C8C2×C22⋊C411yescharacteristic
186C5×C8C4×D411yescharacteristic
187C5×C8#12111yescharacteristic
188C5×C8#12311yescharacteristic
189C2×#41—21no—
190C2×#41—21no—
191#135—21no—
192#135—21no—
193#135—21no—
194#135—21no—
195#135—21no—
196#135—21no—
197#135C2×C4×C512yes—
198#135C5×D412yes—
199C2×Q16C2×C4×C511yescharacteristic
200C2×Q16C5×D411yescharacteristic
201C2×SD16—21no—
202C2×SD16—21no—
203C2×D8C2×C4×C511yescharacteristic
204C2×D8C5×D411yescharacteristic
205#131—24no—
206#131—42no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—41no—
210C22×C8—41no—
211C22×C8C5×D411yescharacteristic
212C22×C8C23×C511yescharacteristic
213C2×C42C5×D411yescharacteristic
214Q32—22no—
215SD32—22no—
216SD32—22no—
217D16—22no—
218M32—21no—
219M32—21no—
220M32—21no—
221M32C2×C4×C512yes—
222C2×C16—21no—
223C2×C16—21no—
224#108—21no—
225#108—21no—
226#108—21no—
227#108C5×D412yes—
228#104—24no—
229#104—42no—
230C22⋊C8—42no—
231#97—21no—
232#97—21no—
233C22×C5—41no—
234C22×C5—42no—
235C22×C5—82no—
236C22×C5—82no—
237C22×C5—81no—
238C22×C5C4×D811yescharacteristic
239C22×C5#44911yescharacteristic
240C22×C5#46311yescharacteristic
241C4×C5—42no—
242C4×C5—81no—
243C4×C5—81no—
244C4×C5#38611yescharacteristic
245C4×C5#44211yescharacteristic
246C4×C5#44711yescharacteristic
247C4×C5#45911yescharacteristic
248#41—22no—
249#41—22no—
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41—41no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C2×C5×D411yescharacteristic
266Q16—21no—
267Q16—21no—
268Q16—21no—
269Q16C5×C22⋊C412yes—
270SD16—22no—
271SD16—22no—
272SD16—41no—
273SD16—41no—
274D8—21no—
275D8—21no—
276D8—21no—
277D8C5×C22⋊C412yes—
278M16—22no—
279M16—44no—
280M16—41no—
281M16—41no—
282M16—41no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—44no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8C5×D812yes—
289C2×C8C22×C4×C511yescharacteristic
290C2×C8C2×C5×D411yescharacteristic
291C2×C8C2×C5×D411yescharacteristic
292C2×C8C2×C5×D411yescharacteristic
293C2×C8C5×#4111yescharacteristic
294C2×C8C5×#4111yescharacteristic
295C42—21no—
296C42—21no—
297C16—22no—
298C16—41no—
299C2×C5—21no—
300C2×C5—21no—
301C2×C5—21no—
302C2×C5—82no—
303C2×C5—161no—
304C2×C5#187911yescharacteristic
305C23—41no—
306C23—41no—
307C23C5×#13511yescharacteristic
308Q8—22no—
309Q8—41no—
310Q8—41no—
311Q8—41no—
312D4—22no—
313D4—42no—
314D4—42no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—81no—
319D4—81no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4C4×C5×D411yescharacteristic
329C2×C4C5×#12011yescharacteristic
330C2×C4C5×#12111yescharacteristic
331C2×C4C2×C5×D811yescharacteristic
332C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
333C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
334C8—22no—
335C8—42no—
336C8—81no—
337C8C2×C5×C22⋊C411yescharacteristic
338C8C4×C5×D411yescharacteristic
339C8C5×#12111yescharacteristic
340C8C5×#12311yescharacteristic
341C5#1302411yescharacteristic, direct product factor, Sylow
342C22—41no—
343C22—42no—
344C22—82no—
345C22—82no—
346C22—81no—
347C22C4×C5×D811yescharacteristic
348C22C5×#44911yescharacteristic
349C22C5×#46311yescharacteristic
350C4—42no—
351C4—81no—
352C4—81no—
353C4C5×#38611yescharacteristic
354C4C5×#44211yescharacteristic
355C4C5×#44711yescharacteristic
356C4C5×#45911yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—82no—
361C2—161no—
362C2C5×#187911yescharacteristic
subgroupG/H
centerC4×C5#447
commutatorC2×C8C22×C4×C5
Frattini, cosocleC22×C8C23×C5
socleC2×C5#1879
prime radicalC5×#3414C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#444C2semidirect
3. C5×#2148C2semidirect
4. C5×#2143C2semidirect
5. C5×#1975C2semidirect
6. C5×#1388C2semidirect
7. C5×#1377C2semidirect
12. C5×#510C4semidirect
14. C5×#509C4semidirect
29. #13024C5direct (N × H)
75. C2×#444C2×C5semidirect
76. #2148C2×C5semidirect
77. #2143C2×C5semidirect
78. #1975C2×C5semidirect
79. #1388C2×C5semidirect
80. #1377C2×C5semidirect
136. #510C4×C5semidirect
138. #509C4×C5semidirect
341. C5#13024direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34142
C2×C5×#4442
C5×#21482
C5×#21432
C5×#19752
C5×#13882
C5×#13772
#130245