Finite Group Databasebeta

← C5×#34978 #11957698

Subgroups — C5×#34978 #11957698

Order 1280.

1834 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2612C214yesmaximal, complemented
2C5×#2571C211yesmaximal, characteristic, complemented
3C5×#2552C212yesmaximal, complemented
4C5×#2551C211yesmaximal, characteristic, complemented
5C5×#2441C214yesmaximal
6C5×#2439C214yesmaximal
7C5×#2396C214yesmaximal
8C4×C5×C4⋊Q8C214yesmaximal, complemented
9C4×C5×#125C214yesmaximal, complemented
10C5×#2285C211yesmaximal, characteristic
11C4×C5×#117C212yesmaximal
12C2×C5×C4⋊Q8C2214yes—
13C2×C5×#125C2214yes—
14C2×C4×C5×Q8C2214yes—
15C2×C5×#117C2212yes—
16C2×C5×#117C2212yes—
17C2×C5×#117C2212yes—
18C22×C42×C5C2211yescharacteristic
19C5×#400C2214yes—
20C5×#399C2214yes—
21C5×#398C2218yes—
22C5×#396C2218yes—
23C5×#395C2214yes—
24C5×#393C2214yes—
25C5×#391C2218yes—
26C5×#389C2214yes—
27C5×#389C2214yes—
28C5×#389C2214yes—
29C5×#389C2214yes—
30C5×#389C2218yes—
31C5×#387C2218yes—
32C5×#387C2218yes—
33C5×#384C2211yescharacteristic
34C5×#384C2211yescharacteristic
35C5×#384C2214yes—
36C5×#382C2212yes—
37C5×#382C2214yes—
38C5×#380C2214yes—
39C4×C5×C4⋊C4C2212yes—
40C4×C5×C4⋊C4C2212yes—
41C4×C5×C4⋊C4C2212yes—
42C4×C5×C4⋊C4C2212yes—
43C4×C5×C4⋊C4C2214yes—
44C4×C5×C4⋊C4C2214yes—
45C4×C5×C4⋊C4C2218yes—
46C4×C5×C22⋊C4C2214yes—
47C4×C5×C22⋊C4C2214yes—
48C5×#376C2214yes—
49C43×C5C2214yes—
50#34978C511yesmaximal, characteristic, direct product factor, Sylow
51C22×C5×Q8C2314yes—
52C23×C4×C5C2312yes—
53C23×C4×C5C2311yescharacteristic
54C5×C4⋊Q8—216no—
55C5×#125—216no—
56C4×C5×Q8—48no—
57C5×#117—28no—
58C5×#117—28no—
59C5×#117—44no—
60C2×C5×C4⋊C4C2312yes—
61C2×C5×C4⋊C4C2312yes—
62C2×C5×C4⋊C4C2314yes—
63C2×C5×C4⋊C4C2314yes—
64C2×C5×C4⋊C4C2318yes—
65C2×C5×C4⋊C4C2318yes—
66C2×C5×C4⋊C4C2318yes—
67C2×C5×C4⋊C4C2312yes—
68C2×C5×C4⋊C4C2312yes—
69C2×C5×C4⋊C4C2318yes—
70C2×C5×C4⋊C4C2314yes—
71C2×C5×C4⋊C4C2312yes—
72C2×C5×C4⋊C4C2312yes—
73C2×C5×C22⋊C4C2314yes—
74C2×C5×C22⋊C4C2314yes—
75C2×C5×C22⋊C4C2314yes—
76C2×C42×C5Q818yescomplemented
77C2×C42×C5C2314yes—
78C2×C42×C5C2314yes—
79C2×C42×C5C2314yes—
80C2×C42×C5C2314yes—
81C2×C42×C5C2314yes—
82C2×C42×C5C2314yes—
83C2×C42×C5C2314yes—
84C2×C42×C5C2311yescharacteristic
85C2×C42×C5C2311yescharacteristic
86C2×C42×C5C2312yes—
87C5×#95C2314yes—
88C5×#95C2314yes—
89C5×#95C2314yes—
90C5×#95C2314yes—
91C5×#95C2318yes—
92C5×#95C2314yes—
93C5×#95C2318yes—
94C5×#95C2314yes—
95C5×#95C2318yes—
96#2612C2×C514yescomplemented
97#2571C2×C511yescharacteristic, complemented
98#2552C2×C512yescomplemented
99#2551C2×C511yescharacteristic, complemented
100#2441C2×C514yes—
101#2439C2×C514yes—
102#2396C2×C514yes—
103C4×C4⋊Q8C2×C514yescomplemented
104C4×#125C2×C514yescomplemented
105#2285C2×C511yescharacteristic
106C4×#117C2×C512yes—
107C24×C5C2411yescharacteristic
108C2×C5×Q8—216no—
109C2×C5×Q8—44no—
110C22×C4×C5—24no—
111C22×C4×C5—24no—
112C22×C4×C5—22no—
113C22×C4×C5—22no—
114C22×C4×C5C2×Q818yes—
115C22×C4×C5C2×Q814yes—
116C22×C4×C5C2414yes—
117C22×C4×C5C2414yes—
118C22×C4×C5C2414yes—
119C22×C4×C5C2414yes—
120C22×C4×C5C2412yes—
121C22×C4×C5C2411yescharacteristic
122C22×C4×C5C2414yes—
123C22×C4×C5C2414yes—
124C22×C4×C5C2412yes—
125C22×C4×C5C2411yescharacteristic
126C5×C4⋊C4—24no—
127C5×C4⋊C4—24no—
128C5×C4⋊C4—216no—
129C5×C4⋊C4—216no—
130C5×C4⋊C4—28no—
131C5×C4⋊C4—216no—
132C5×C4⋊C4—28no—
133C5×C4⋊C4—42no—
134C5×C4⋊C4—42no—
135C5×C4⋊C4—48no—
136C5×C4⋊C4—44no—
137C5×C4⋊C4#4118yes—
138C5×C4⋊C4#4118yes—
139C5×C22⋊C4—28no—
140C5×C22⋊C4—28no—
141C5×C22⋊C4—44no—
142C42×C5—28no—
143C42×C5—28no—
144C42×C5—28no—
145C42×C5—28no—
146C42×C5—28no—
147C42×C5—28no—
148C42×C5—28no—
149C42×C5C2×Q814yes—
150C42×C5C2×Q814yes—
151C42×C5C2×Q818yes—
152C2×C4⋊Q8C22×C514yes—
153C2×#125C22×C514yes—
154C2×C4×Q8C22×C514yes—
155C2×#117C22×C512yes—
156C2×#117C22×C512yes—
157C2×#117C22×C512yes—
158C22×C42C22×C511yescharacteristic
159#400C22×C514yes—
160#399C22×C514yes—
161#398C22×C518yes—
162#396C22×C518yes—
163#395C22×C514yes—
164#393C22×C514yes—
165#391C22×C518yes—
166#389C22×C514yes—
167#389C22×C514yes—
168#389C22×C514yes—
169#389C22×C518yes—
170#389C22×C514yes—
171#387C22×C518yes—
172#387C22×C518yes—
173#384C22×C514yes—
174#384C22×C511yescharacteristic
175#384C22×C511yescharacteristic
176#382C22×C514yes—
177#382C22×C512yes—
178#380C22×C514yes—
179C4×C4⋊C4C22×C514yes—
180C4×C4⋊C4C22×C512yes—
181C4×C4⋊C4C22×C518yes—
182C4×C4⋊C4C22×C514yes—
183C4×C4⋊C4C22×C512yes—
184C4×C4⋊C4C22×C512yes—
185C4×C4⋊C4C22×C512yes—
186C4×C22⋊C4C22×C514yes—
187C4×C22⋊C4C22×C514yes—
188#376C22×C514yes—
189C43C22×C514yes—
190C23×C5—22no—
191C23×C5—22no—
192C23×C5—21no—
193C23×C5—21no—
194C23×C5C22×Q812yes—
195C23×C5C2511yescharacteristic
196C5×Q8—416no—
197C2×C4×C5—22no—
198C2×C4×C5—22no—
199C2×C4×C5—24no—
200C2×C4×C5—24no—
201C2×C4×C5—24no—
202C2×C4×C5—21no—
203C2×C4×C5—24no—
204C2×C4×C5—21no—
205C2×C4×C5—216no—
206C2×C4×C5—28no—
207C2×C4×C5—24no—
208C2×C4×C5—24no—
209C2×C4×C5—24no—
210C2×C4×C5—24no—
211C2×C4×C5—28no—
212C2×C4×C5—28no—
213C2×C4×C5—24no—
214C2×C4×C5—24no—
215C2×C4×C5—24no—
216C2×C4×C5—24no—
217C2×C4×C5C22×Q814yes—
218C2×C4×C5C22×Q812yes—
219C2×C4×C5C22×Q814yes—
220C2×C4×C5C22×Q812yes—
221C2×C4×C5C2×#4118yes—
222C2×C4×C5C2×#4118yes—
223C2×C4×C5C2×#4114yes—
224C2×C4×C5C2×#4114yes—
225C2×C4×C5H(5,𝔽2)12yes—
226C2×C4×C52−1+412yes—
227C22×Q8C23×C514yes—
228C23×C4C23×C511yescharacteristic
229C23×C4C23×C512yes—
230C4⋊Q8—216no—
231#125—216no—
232C4×Q8—48no—
233#117—28no—
234#117—28no—
235#117—44no—
236C2×C4⋊C4C23×C512yes—
237C2×C4⋊C4C23×C512yes—
238C2×C4⋊C4C23×C514yes—
239C2×C4⋊C4C23×C518yes—
240C2×C4⋊C4C23×C518yes—
241C2×C4⋊C4C23×C514yes—
242C2×C4⋊C4C23×C518yes—
243C2×C4⋊C4C23×C512yes—
244C2×C4⋊C4C23×C512yes—
245C2×C4⋊C4C23×C518yes—
246C2×C4⋊C4C23×C514yes—
247C2×C4⋊C4C23×C512yes—
248C2×C4⋊C4C23×C512yes—
249C2×C22⋊C4C23×C514yes—
250C2×C22⋊C4C23×C514yes—
251C2×C22⋊C4C23×C514yes—
252C2×C42C5×Q818yescomplemented
253C2×C42C23×C514yes—
254C2×C42C23×C514yes—
255C2×C42C23×C514yes—
256C2×C42C23×C514yes—
257C2×C42C23×C514yes—
258C2×C42C23×C514yes—
259C2×C42C23×C514yes—
260C2×C42C23×C512yes—
261C2×C42C23×C511yescharacteristic
262C2×C42C23×C511yescharacteristic
263#95C23×C514yes—
264#95C23×C514yes—
265#95C23×C514yes—
266#95C23×C514yes—
267#95C23×C518yes—
268#95C23×C514yes—
269#95C23×C518yes—
270#95C23×C514yes—
271#95C23×C518yes—
272C22×C5—24no—
273C22×C5—22no—
274C22×C5—41no—
275C22×C5—42no—
276C22×C5—41no—
277C22×C5C23×Q811yescharacteristic
278C22×C5C22×#4112yes—
279C22×C5C22×#4112yes—
280C22×C5C2×H(5,𝔽2)11yescharacteristic
281C22×C5C2×2−1+411yescharacteristic
282C4×C5—24no—
283C4×C5—22no—
284C4×C5—28no—
285C4×C5—28no—
286C4×C5—44no—
287C4×C5—44no—
288C4×C5—44no—
289C4×C5—44no—
290C4×C5#53318yes—
291C4×C5#54414yes—
292C24C24×C511yescharacteristic
293C2×Q8—216no—
294C2×Q8—44no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4C2×C5×Q818yes—
300C22×C4C2×C5×Q814yes—
301C22×C4C24×C514yes—
302C22×C4C24×C514yes—
303C22×C4C24×C514yes—
304C22×C4C24×C514yes—
305C22×C4C24×C512yes—
306C22×C4C24×C511yescharacteristic
307C22×C4C24×C514yes—
308C22×C4C24×C514yes—
309C22×C4C24×C512yes—
310C22×C4C24×C511yescharacteristic
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—216no—
314C4⋊C4—216no—
315C4⋊C4—28no—
316C4⋊C4—216no—
317C4⋊C4—28no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—48no—
321C4⋊C4—44no—
322C4⋊C4C5×#4118yes—
323C4⋊C4C5×#4118yes—
324C22⋊C4—28no—
325C22⋊C4—28no—
326C22⋊C4—44no—
327C42—28no—
328C42—28no—
329C42—28no—
330C42—28no—
331C42—28no—
332C42—28no—
333C42—28no—
334C42C2×C5×Q814yes—
335C42C2×C5×Q814yes—
336C42C2×C5×Q818yes—
337C2×C5—42no—
338C2×C5C2×#53312yes—
339C2×C5C2×#54411yescharacteristic
340C2×C5#348111yescharacteristic
341C2×C5#348212yes—
342C2×C5#348311yescharacteristic
343C23—22no—
344C23—22no—
345C23—21no—
346C23—21no—
347C23C22×C5×Q812yes—
348C23C25×C511yescharacteristic
349Q8—416no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—216no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4C22×C5×Q814yes—
371C2×C4C22×C5×Q812yes—
372C2×C4C22×C5×Q814yes—
373C2×C4C22×C5×Q812yes—
374C2×C4C2×C5×#4118yes—
375C2×C4C2×C5×#4118yes—
376C2×C4C2×C5×#4114yes—
377C2×C4C2×C5×#4114yes—
378C2×C4C5×H(5,𝔽2)12yes—
379C2×C4C5×2−1+412yes—
380C5#3497811yescharacteristic, direct product factor, Sylow
381C22—24no—
382C22—22no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22C23×C5×Q811yescharacteristic
387C22C22×C5×#4112yes—
388C22C22×C5×#4112yes—
389C22C2×C5×H(5,𝔽2)11yescharacteristic
390C22C2×C5×2−1+411yescharacteristic
391C4—24no—
392C4—22no—
393C4—28no—
394C4—28no—
395C4—44no—
396C4—44no—
397C4—44no—
398C4—44no—
399C4C5×#53318yes—
400C4C5×#54414yes—
401C2—42no—
402C2C2×C5×#53312yes—
403C2C2×C5×#54411yescharacteristic
404C2C5×#348111yescharacteristic
405C2C5×#348212yes—
406C2C5×#348311yescharacteristic
subgroupG/H
centerC22×C4×C5C24
commutator, Frattini, cosocleC23C25×C5
socleC23×C5C25
prime radicalC24×C5C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#2612C2semidirect
2. C5×#2571C2semidirect
3. C5×#2552C2semidirect
4. C5×#2551C2semidirect
8. C4×C5×C4⋊Q8C2semidirect
9. C4×C5×#125C2semidirect
50. #34978C5direct (N × H)
76. C2×C42×C5Q8semidirect
96. #2612C2×C5semidirect
97. #2571C2×C5semidirect
98. #2552C2×C5semidirect
99. #2551C2×C5semidirect
103. C4×C4⋊Q8C2×C5semidirect
104. C4×#125C2×C5semidirect
252. C2×C42C5×Q8semidirect
380. C5#34978direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#26122
C5×#25712
C5×#25522
C5×#25512
C5×#24412
C5×#24392
C5×#23962
C4×C5×C4⋊Q82
C4×C5×#1252
C5×#22852
C4×C5×#1172
#349785