Finite Group Databasebeta

← C5×#60328 #11983048

Subgroups — C5×#60328 #11983048

Order 1280.

1920 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3483C211yesmaximal, characteristic
2C5×#2991C211yesmaximal, characteristic, complemented
3C5×#2990C213yesmaximal, complemented
4C5×#2988C212yesmaximal, complemented
5C5×#2981C216yesmaximal, complemented
6C5×#2979C213yesmaximal, complemented
7C5×#2971C212yesmaximal, complemented
8C5×#2965C213yesmaximal, complemented
9C5×C8×#41C211yesmaximal, characteristic, complemented
10C5×#2935C216yesmaximal, complemented
11C5×#2931C213yesmaximal, complemented
12C2×C5×#131C2216yes—
13C5×Q82C411yescharacteristic
14C5×#555C413yes—
15C5×#555C416yes—
16C5×#550C411yescharacteristic
17C5×#550C412yes—
18C5×#550C413yes—
19C5×#532C2216yes—
20C5×#532C2213yes—
21C4×C5×#41C2212yes—
22C4×C5×#41C2213yes—
23C4×C5×#41C2211yescharacteristic
24C5×#446C2213yes—
25C5×#446C2211yescharacteristic
26C5×C8×Q8C2211yescharacteristic
27C5×C8×Q8C2212yes—
28C5×C8×Q8C2211yescharacteristic
29C5×#436C2216yes—
30C5×#436C2216yes—
31C5×#436C2213yes—
32C5×#436C2216yes—
33C5×#436C2213yes—
34C5×#435C2216yes—
35C5×#435C22112yes—
36C5×#435C2216yes—
37C5×C8×D4C2213yes—
38C5×C8×D4C2216yes—
39C5×C8×D4C2216yes—
40C5×C8×D4C2216yes—
41C5×C8×D4C2213yes—
42C5×#432C2216yes—
43C5×#431C2216yes—
44C5×#431C2213yes—
45C5×#431C2213yes—
46C5×#423C2213yes—
47C2×C5×C4⋊C8C2216yes—
48C5×#408C2216yes—
49C5×#408C22112yes—
50C4×C5×M16C2216yes—
51C2×C4×C5×C8C2213yes—
52#60328C511yesmaximal, characteristic, direct product factor, Sylow
53C2×C5×#41C2×C412yes—
54C2×C5×#41C2×C411yescharacteristic
55C2×C5×#41C2×C413yes—
56C2×C5×#41C2316yes—
57C5×#131—412no—
58C2×C5×M16C2316yes—
59C2×C5×M16C2316yes—
60C2×C5×M16C2316yes—
61C22×C5×C8C2313yes—
62C22×C5×C8C2313yes—
63C22×C5×C8C23112yes—
64C5×C4⋊Q8C2×C416yes—
65C5×C4⋊Q8C2×C413yes—
66C5×#127C2×C413yes—
67C5×#127C2×C416yes—
68C5×#124C2×C4112yes—
69C5×#124C2×C416yes—
70C5×#121C2×C416yes—
71C5×#121C2×C416yes—
72C5×#121C2×C416yes—
73C5×#121C2×C416yes—
74C5×#121C2×C4112yes—
75C4×C5×Q8C2×C413yes—
76C4×C5×Q8C2×C412yes—
77C4×C5×Q8C2×C411yescharacteristic
78C4×C5×Q8C2311yescharacteristic
79C4×C5×Q8C2311yescharacteristic
80C4×C5×D4C2×C413yes—
81C4×C5×D4C2×C413yes—
82C4×C5×D4C2×C416yes—
83C4×C5×D4C2×C416yes—
84C4×C5×D4C2313yes—
85C4×C5×D4C2316yes—
86C4×C5×D4C2313yes—
87C4×C5×D4C2316yes—
88C5×#117C2×C416yes—
89C5×#117C2×C413yes—
90C5×#117C2×C413yes—
91C5×#117C2×C416yes—
92C2×C42×C5C2313yes—
93C2×C42×C5C2316yes—
94C5×C4⋊C8C2313yes—
95C5×C4⋊C8C2316yes—
96C5×C4⋊C8C2316yes—
97C5×C4⋊C8C2313yes—
98C5×C4⋊C8C2312yes—
99C5×C4⋊C8C2311yescharacteristic
100C5×C4⋊C8C2313yes—
101C5×C22⋊C8C2316yes—
102C5×C22⋊C8C2316yes—
103C5×C22⋊C8C2316yes—
104C5×C22⋊C8C2316yes—
105C5×C22⋊C8C23112yes—
106C5×#97C2312yes—
107C5×#97C2316yes—
108C4×C5×C8C2311yescharacteristic
109C4×C5×C8C2313yes—
110C4×C5×C8C2313yes—
111C4×C5×C8C2313yes—
112C4×C5×C8C2316yes—
113#3483C2×C511yescharacteristic
114#2991C2×C511yescharacteristic, complemented
115#2990C2×C513yescomplemented
116#2988C2×C512yescomplemented
117#2981C2×C516yescomplemented
118#2979C2×C513yescomplemented
119#2971C2×C512yescomplemented
120#2965C2×C513yescomplemented
121C8×#41C2×C511yescharacteristic, complemented
122#2935C2×C516yescomplemented
123#2931C2×C513yescomplemented
124C5×#41—28no—
125C5×#41—24no—
126C5×#41—46no—
127C5×#41—46no—
128C5×#41—46no—
129C2×C5×Q8C22×C411yescharacteristic
130C2×C5×Q8C22×C416yes—
131C2×C5×Q8C22×C411yescharacteristic
132C2×C5×D4C22×C413yes—
133C2×C5×D4C22×C416yes—
134C2×C5×D4C22×C413yes—
135C2×C5×D4C22×C416yes—
136C2×C5×D4C22×C4112yes—
137C2×C5×D4C22×C416yes—
138C22×C4×C5C22×C413yes—
139C22×C4×C5C22×C416yes—
140C22×C4×C5C22×C413yes—
141C22×C4×C5C22×C416yes—
142C22×C4×C5C2413yes—
143C22×C4×C5C2416yes—
144C5×M16—212no—
145C5×M16—46no—
146C5×M16—46no—
147C2×C5×C8—26no—
148C2×C5×C8—43no—
149C2×C5×C8—412no—
150C2×C5×C8C2413yes—
151C2×C5×C8C2411yescharacteristic
152C2×C5×C8C2416yes—
153C2×C5×C8C2413yes—
154C2×C5×C8C2412yes—
155C2×C5×C8C2411yescharacteristic
156C5×C4⋊C4C22×C412yes—
157C5×C4⋊C4C22×C413yes—
158C5×C4⋊C4C22×C411yescharacteristic
159C5×C4⋊C4C22×C416yes—
160C5×C4⋊C4C22×C416yes—
161C5×C4⋊C4C22×C413yes—
162C5×C4⋊C4C22×C413yes—
163C5×C22⋊C4C22×C416yes—
164C5×C22⋊C4C22×C416yes—
165C5×C22⋊C4C22×C4112yes—
166C5×C22⋊C4C22×C416yes—
167C5×C22⋊C4C22×C416yes—
168C42×C5C22×C413yes—
169C42×C5C22×C416yes—
170C42×C5C22×C416yes—
171C42×C5C22×C413yes—
172C42×C5C2413yes—
173C42×C5C2411yescharacteristic
174C42×C5C2412yes—
175C2×#131C22×C516yes—
176Q82C4×C511yescharacteristic
177#555C4×C516yes—
178#555C4×C513yes—
179#550C4×C511yescharacteristic
180#550C4×C512yes—
181#550C4×C513yes—
182#532C22×C513yes—
183#532C22×C516yes—
184C4×#41C22×C511yescharacteristic
185C4×#41C22×C513yes—
186C4×#41C22×C512yes—
187#446C22×C511yescharacteristic
188#446C22×C513yes—
189C8×Q8C22×C511yescharacteristic
190C8×Q8C22×C511yescharacteristic
191C8×Q8C22×C512yes—
192#436C22×C513yes—
193#436C22×C516yes—
194#436C22×C516yes—
195#436C22×C513yes—
196#436C22×C516yes—
197#435C22×C516yes—
198#435C22×C5112yes—
199#435C22×C516yes—
200C8×D4C22×C516yes—
201C8×D4C22×C516yes—
202C8×D4C22×C513yes—
203C8×D4C22×C516yes—
204C8×D4C22×C513yes—
205#432C22×C516yes—
206#431C22×C513yes—
207#431C22×C513yes—
208#431C22×C516yes—
209#423C22×C513yes—
210C2×C4⋊C8C22×C516yes—
211#408C22×C516yes—
212#408C22×C5112yes—
213C4×M16C22×C516yes—
214C2×C4×C8C22×C513yes—
215C23×C5C23×C413yes—
216C23×C5C23×C416yes—
217C5×Q8—22no—
218C5×Q8—46no—
219C5×Q8#13114yescomplemented
220C5×D4—26no—
221C5×D4—212no—
222C5×D4—26no—
223C5×D4—46no—
224C5×D4—412no—
225C5×D4—46no—
226C2×C4×C5—26no—
227C2×C4×C5—212no—
228C2×C4×C5—26no—
229C2×C4×C5—23no—
230C2×C4×C5—26no—
231C2×C4×C5—23no—
232C2×C4×C5—26no—
233C2×C4×C5—46no—
234C2×C4×C5C23×C411yescharacteristic
235C2×C4×C5C23×C412yes—
236C2×C4×C5C23×C413yes—
237C2×C4×C5C23×C412yes—
238C2×C4×C5C23×C413yes—
239C2×C4×C5C23×C416yes—
240C2×C4×C5C23×C413yes—
241C2×C4×C5C23×C411yescharacteristic
242C2×C4×C5C2511yescharacteristic
243C5×C8—23no—
244C5×C8—26no—
245C5×C8—21no—
246C5×C8—23no—
247C5×C8—22no—
248C5×C8H(5,𝔽2)12yes—
249C2×#41C2×C4×C511yescharacteristic
250C2×#41C2×C4×C512yes—
251C2×#41C2×C4×C513yes—
252C2×#41C23×C516yes—
253#131—412no—
254C2×M16C23×C516yes—
255C2×M16C23×C516yes—
256C2×M16C23×C516yes—
257C22×C8C23×C513yes—
258C22×C8C23×C5112yes—
259C22×C8C23×C513yes—
260C4⋊Q8C2×C4×C513yes—
261C4⋊Q8C2×C4×C516yes—
262#127C2×C4×C513yes—
263#127C2×C4×C516yes—
264#124C2×C4×C516yes—
265#124C2×C4×C5112yes—
266#121C2×C4×C516yes—
267#121C2×C4×C516yes—
268#121C2×C4×C516yes—
269#121C2×C4×C516yes—
270#121C2×C4×C5112yes—
271C4×Q8C2×C4×C513yes—
272C4×Q8C2×C4×C511yescharacteristic
273C4×Q8C2×C4×C512yes—
274C4×Q8C23×C511yescharacteristic
275C4×Q8C23×C511yescharacteristic
276C4×D4C2×C4×C513yes—
277C4×D4C2×C4×C513yes—
278C4×D4C2×C4×C516yes—
279C4×D4C2×C4×C516yes—
280C4×D4C23×C513yes—
281C4×D4C23×C513yes—
282C4×D4C23×C516yes—
283C4×D4C23×C516yes—
284#117C2×C4×C513yes—
285#117C2×C4×C513yes—
286#117C2×C4×C516yes—
287#117C2×C4×C516yes—
288C2×C42C23×C513yes—
289C2×C42C23×C516yes—
290C4⋊C8C23×C513yes—
291C4⋊C8C23×C516yes—
292C4⋊C8C23×C516yes—
293C4⋊C8C23×C513yes—
294C4⋊C8C23×C512yes—
295C4⋊C8C23×C511yescharacteristic
296C4⋊C8C23×C513yes—
297C22⋊C8C23×C516yes—
298C22⋊C8C23×C516yes—
299C22⋊C8C23×C516yes—
300C22⋊C8C23×C516yes—
301C22⋊C8C23×C5112yes—
302#97C23×C512yes—
303#97C23×C516yes—
304C4×C8C23×C511yescharacteristic
305C4×C8C23×C513yes—
306C4×C8C23×C513yes—
307C4×C8C23×C513yes—
308C4×C8C23×C516yes—
309C22×C5—23no—
310C22×C5—26no—
311C22×C5—23no—
312C22×C5—26no—
313C22×C5—23no—
314C22×C5—26no—
315C22×C5C24×C411yescharacteristic
316C4×C5—23no—
317C4×C5—26no—
318C4×C5—21no—
319C4×C5—22no—
320C4×C5—23no—
321C4×C5—22no—
322C4×C5C2×#13116yes—
323C4×C5#56812yes—
324C4×C5C2×H(5,𝔽2)11yescharacteristic
325C4×C5#58511yescharacteristic
326#41—24no—
327#41—28no—
328#41—46no—
329#41—46no—
330#41—46no—
331C2×Q8C22×C4×C511yescharacteristic
332C2×Q8C22×C4×C516yes—
333C2×Q8C22×C4×C511yescharacteristic
334C2×D4C22×C4×C513yes—
335C2×D4C22×C4×C516yes—
336C2×D4C22×C4×C513yes—
337C2×D4C22×C4×C516yes—
338C2×D4C22×C4×C5112yes—
339C2×D4C22×C4×C516yes—
340C22×C4C22×C4×C513yes—
341C22×C4C22×C4×C516yes—
342C22×C4C22×C4×C513yes—
343C22×C4C22×C4×C516yes—
344C22×C4C24×C513yes—
345C22×C4C24×C516yes—
346M16—212no—
347M16—46no—
348M16—46no—
349C2×C8—26no—
350C2×C8—412no—
351C2×C8—43no—
352C2×C8C24×C511yescharacteristic
353C2×C8C24×C512yes—
354C2×C8C24×C513yes—
355C2×C8C24×C513yes—
356C2×C8C24×C516yes—
357C2×C8C24×C511yescharacteristic
358C4⋊C4C22×C4×C512yes—
359C4⋊C4C22×C4×C513yes—
360C4⋊C4C22×C4×C511yescharacteristic
361C4⋊C4C22×C4×C516yes—
362C4⋊C4C22×C4×C513yes—
363C4⋊C4C22×C4×C516yes—
364C4⋊C4C22×C4×C513yes—
365C22⋊C4C22×C4×C516yes—
366C22⋊C4C22×C4×C516yes—
367C22⋊C4C22×C4×C5112yes—
368C22⋊C4C22×C4×C516yes—
369C22⋊C4C22×C4×C516yes—
370C42C22×C4×C513yes—
371C42C22×C4×C516yes—
372C42C22×C4×C516yes—
373C42C22×C4×C513yes—
374C42C24×C513yes—
375C42C24×C511yescharacteristic
376C42C24×C512yes—
377C2×C5—43no—
378C2×C5—46no—
379C2×C5C4×H(5,𝔽2)11yescharacteristic
380C2×C5C22×#13111yescharacteristic
381C2×C5C2×#56811yescharacteristic
382C23C23×C4×C513yes—
383C23C23×C4×C516yes—
384Q8—22no—
385Q8—46no—
386Q8C5×#13114yescomplemented
387D4—26no—
388D4—212no—
389D4—26no—
390D4—46no—
391D4—412no—
392D4—46no—
393C2×C4—26no—
394C2×C4—212no—
395C2×C4—26no—
396C2×C4—23no—
397C2×C4—26no—
398C2×C4—23no—
399C2×C4—26no—
400C2×C4—46no—
401C2×C4C23×C4×C511yescharacteristic
402C2×C4C23×C4×C512yes—
403C2×C4C23×C4×C513yes—
404C2×C4C23×C4×C512yes—
405C2×C4C23×C4×C513yes—
406C2×C4C23×C4×C516yes—
407C2×C4C23×C4×C513yes—
408C2×C4C23×C4×C511yescharacteristic
409C2×C4C25×C511yescharacteristic
410C8—23no—
411C8—26no—
412C8—21no—
413C8—23no—
414C8—22no—
415C8C5×H(5,𝔽2)12yes—
416C5#6032811yescharacteristic, direct product factor, Sylow
417C22—23no—
418C22—26no—
419C22—23no—
420C22—26no—
421C22—23no—
422C22—26no—
423C22C24×C4×C511yescharacteristic
424C4—23no—
425C4—26no—
426C4—21no—
427C4—22no—
428C4—23no—
429C4—22no—
430C4C2×C5×#13116yes—
431C4C5×#56812yes—
432C4C2×C5×H(5,𝔽2)11yescharacteristic
433C4C5×#58511yescharacteristic
434C2—43no—
435C2—46no—
436C2C4×C5×H(5,𝔽2)11yescharacteristic
437C2C22×C5×#13111yescharacteristic
438C2C2×C5×#56811yescharacteristic
subgroupG/H
centerC2×C4×C5C25
commutatorC22C24×C4×C5
Frattini, cosocleC2×C4C25×C5
socleC22×C5C24×C4
prime radicalC5×#3483C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2991C2semidirect
3. C5×#2990C2semidirect
4. C5×#2988C2semidirect
5. C5×#2981C2semidirect
6. C5×#2979C2semidirect
7. C5×#2971C2semidirect
8. C5×#2965C2semidirect
9. C5×C8×#41C2semidirect
10. C5×#2935C2semidirect
11. C5×#2931C2semidirect
52. #60328C5direct (N × H)
114. #2991C2×C5semidirect
115. #2990C2×C5semidirect
116. #2988C2×C5semidirect
117. #2981C2×C5semidirect
118. #2979C2×C5semidirect
119. #2971C2×C5semidirect
120. #2965C2×C5semidirect
121. C8×#41C2×C5semidirect
122. #2935C2×C5semidirect
123. #2931C2×C5semidirect
219. C5×Q8#131semidirect
386. Q8C5×#131semidirect
416. C5#60328direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34832
C5×#29912
C5×#29902
C5×#29882
C5×#29812
C5×#29792
C5×#29712
C5×#29652
C5×C8×#412
C5×#29352
C5×#29312
#603285