Finite Group Databasebeta

← C5×#11762 #11934482

Subgroups — C5×#11762 #11934482

Order 1280.

622 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2406C211yesmaximal, characteristic
2C2×C5×#343C211yesmaximal, characteristic, complemented
3C5×#1463C211yesmaximal, characteristic, complemented
4C5×#1354C212yesmaximal, complemented
5C5×#1345C212yesmaximal, complemented
6C22×C5×#41C411yescharacteristic
7C22×C5×M16—21no—
8C2×C5×#123—21no—
9C22×C5×C4⋊C4C2211yescharacteristic
10C5×#407—21no—
11C5×#395C411yescharacteristic
12C5×#395C2211yescharacteristic
13C5×#394C411yescharacteristic
14C5×#394C2211yescharacteristic
15C5×#386—21no—
16C5×#385—21no—
17C5×#385—21no—
18C5×#381C411yescharacteristic
19C5×#343—21no—
20C5×#343C2212yes—
21C5×#324—21no—
22C5×#324C2212yes—
23#11762C511yesmaximal, characteristic, direct product factor, Sylow
24C2×C5×#41—41no—
25C2×C5×#41—41no—
26C2×C5×#41—44no—
27C22×C5×Q8C2×C411yescharacteristic
28C22×C5×D4—21no—
29C22×C5×D4C2×C411yescharacteristic
30C23×C4×C5—21no—
31C23×C4×C5C2×C411yescharacteristic
32C23×C4×C5D411yescharacteristic
33C23×C4×C5D411yescharacteristic
34C2×C5×M16—24no—
35C2×C5×M16—22no—
36C2×C5×M16—41no—
37C2×C5×M16—42no—
38C2×C5×M16—41no—
39C22×C5×C8—41no—
40C5×#123—44no—
41C2×C5×C4⋊C4—21no—
42C2×C5×C4⋊C4—21no—
43C2×C5×C4⋊C4—41no—
44C2×C5×C4⋊C4—41no—
45C2×C5×C4⋊C4C2×C411yescharacteristic
46C2×C5×C4⋊C4D411yescharacteristic
47C2×C5×C4⋊C4D411yescharacteristic
48C2×C5×C4⋊C4C2311yescharacteristic
49C2×C5×C22⋊C4—21no—
50C2×C5×C22⋊C4—21no—
51C2×C5×C22⋊C4—21no—
52C2×C5×C22⋊C4—21no—
53C2×C5×C22⋊C4—21no—
54C5×C22⋊C8—22no—
55C5×C22⋊C8—41no—
56C5×#95—21no—
57C5×#95—21no—
58C5×#95—21no—
59C5×#95C2×C411yescharacteristic
60C5×#95C2×C411yescharacteristic
61#2406C2×C511yescharacteristic
62C2×#343C2×C511yescharacteristic, complemented
63#1463C2×C511yescharacteristic, complemented
64#1354C2×C512yescomplemented
65#1345C2×C512yescomplemented
66C24×C5—21no—
67C24×C5C22⋊C411yescharacteristic
68C5×#41—44no—
69C5×#41—44no—
70C5×#41—44no—
71C5×#41—44no—
72C2×C5×Q8—42no—
73C2×C5×Q8—41no—
74C2×C5×D4—24no—
75C2×C5×D4—41no—
76C2×C5×D4—42no—
77C2×C5×D4—41no—
78C2×C5×D4—41no—
79C2×C5×D4—42no—
80C22×C4×C5—21no—
81C22×C4×C5—21no—
82C22×C4×C5—22no—
83C22×C4×C5—21no—
84C22×C4×C5—21no—
85C22×C4×C5—21no—
86C22×C4×C5—21no—
87C22×C4×C5—21no—
88C22×C4×C5—21no—
89C22×C4×C5—21no—
90C22×C4×C5—21no—
91C22×C4×C5—21no—
92C22×C4×C5—42no—
93C22×C4×C5—42no—
94C22×C4×C5—42no—
95C22×C4×C5—42no—
96C22×C4×C5—42no—
97C22×C4×C5C22⋊C411yescharacteristic
98C22×C4×C5C22⋊C411yescharacteristic
99C22×C4×C5C22⋊C411yescharacteristic
100C22×C4×C5C22×C411yescharacteristic
101C22×C4×C5C2×D411yescharacteristic
102C22×C4×C5C2×D411yescharacteristic
103C5×M16—24no—
104C5×M16—44no—
105C5×M16—42no—
106C5×M16—82no—
107C2×C5×C8—44no—
108C2×C5×C8—42no—
109C2×C5×C8—42no—
110C5×C4⋊C4—41no—
111C5×C4⋊C4—41no—
112C5×C4⋊C4—41no—
113C5×C4⋊C4—41no—
114C5×C4⋊C4—42no—
115C5×C4⋊C4—42no—
116C5×C22⋊C4—42no—
117C5×C22⋊C4—42no—
118C5×C22⋊C4—42no—
119C5×C22⋊C4—42no—
120C5×C22⋊C4—41no—
121C5×C22⋊C4—41no—
122C22×#41C4×C511yescharacteristic
123C22×M16—21no—
124C2×#123—21no—
125C22×C4⋊C4C22×C511yescharacteristic
126#407—21no—
127#395C4×C511yescharacteristic
128#395C22×C511yescharacteristic
129#394C4×C511yescharacteristic
130#394C22×C511yescharacteristic
131#386—21no—
132#385—21no—
133#385—21no—
134#381C4×C511yescharacteristic
135#343—21no—
136#343C22×C512yes—
137#324—21no—
138#324C22×C512yes—
139C23×C5—22no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—22no—
143C23×C5—41no—
144C23×C5—41no—
145C23×C5—41no—
146C23×C5—41no—
147C23×C5—41no—
148C23×C5—41no—
149C23×C5—41no—
150C23×C5#9911yescharacteristic
151C23×C5#9911yescharacteristic
152C23×C5C2×C22⋊C411yescharacteristic
153C5×Q8—42no—
154C5×Q8—42no—
155C5×D4—44no—
156C5×D4—42no—
157C5×D4—42no—
158C5×D4—44no—
159C2×C4×C5—22no—
160C2×C4×C5—22no—
161C2×C4×C5—22no—
162C2×C4×C5—22no—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—21no—
166C2×C4×C5—21no—
167C2×C4×C5—22no—
168C2×C4×C5—21no—
169C2×C4×C5—21no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5—41no—
173C2×C4×C5—42no—
174C2×C4×C5—48no—
175C2×C4×C5—42no—
176C2×C4×C5—42no—
177C2×C4×C5—41no—
178C2×C4×C5—41no—
179C2×C4×C5—41no—
180C2×C4×C5—41no—
181C2×C4×C5—41no—
182C2×C4×C5—41no—
183C2×C4×C5—41no—
184C2×C4×C5—41no—
185C2×C4×C5—41no—
186C2×C4×C5—41no—
187C2×C4×C5—41no—
188C2×C4×C5—81no—
189C2×C4×C5#10412yes—
190C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C5#13711yescharacteristic
192C5×C8—44no—
193C5×C8—82no—
194C2×#41—41no—
195C2×#41—41no—
196C2×#41—44no—
197C22×Q8C2×C4×C511yescharacteristic
198C22×D4—21no—
199C22×D4C2×C4×C511yescharacteristic
200C23×C4—21no—
201C23×C4C2×C4×C511yescharacteristic
202C23×C4C5×D411yescharacteristic
203C23×C4C5×D411yescharacteristic
204C2×M16—24no—
205C2×M16—22no—
206C2×M16—41no—
207C2×M16—42no—
208C2×M16—41no—
209C22×C8—41no—
210#123—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C2×C4×C511yescharacteristic
216C2×C4⋊C4C5×D411yescharacteristic
217C2×C4⋊C4C5×D411yescharacteristic
218C2×C4⋊C4C23×C511yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C22⋊C8—22no—
225C22⋊C8—41no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95C2×C4×C511yescharacteristic
230#95C2×C4×C511yescharacteristic
231C22×C5—21no—
232C22×C5—21no—
233C22×C5—21no—
234C22×C5—21no—
235C22×C5—21no—
236C22×C5—21no—
237C22×C5—21no—
238C22×C5—42no—
239C22×C5—42no—
240C22×C5—42no—
241C22×C5—42no—
242C22×C5—42no—
243C22×C5—81no—
244C22×C5—81no—
245C22×C5—81no—
246C22×C5—81no—
247C22×C5AΣL(1,𝔽16)12yes—
248C22×C5C2×#9911yescharacteristic
249C22×C5#41711yescharacteristic
250C22×C5C2×#10411yescharacteristic
251C4×C5—21no—
252C4×C5—22no—
253C4×C5—21no—
254C4×C5—41no—
255C4×C5—41no—
256C4×C5—42no—
257C4×C5—41no—
258C4×C5—41no—
259C4×C5—81no—
260C4×C5—81no—
261C24—21no—
262C24C5×C22⋊C411yescharacteristic
263#41—44no—
264#41—44no—
265#41—44no—
266#41—44no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×D4—24no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4C5×C22⋊C411yescharacteristic
293C22×C4C5×C22⋊C411yescharacteristic
294C22×C4C5×C22⋊C411yescharacteristic
295C22×C4C22×C4×C511yescharacteristic
296C22×C4C2×C5×D411yescharacteristic
297C22×C4C2×C5×D411yescharacteristic
298M16—24no—
299M16—44no—
300M16—42no—
301M16—82no—
302C2×C8—44no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—42no—
317C2×C5—21no—
318C2×C5—21no—
319C2×C5—22no—
320C2×C5—41no—
321C2×C5—82no—
322C2×C5#150411yescharacteristic
323C2×C5C2×AΣL(1,𝔽16)11yescharacteristic
324C2×C5#213311yescharacteristic
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23C5×#9911yescharacteristic
337C23C5×#9911yescharacteristic
338C23C2×C5×C22⋊C411yescharacteristic
339Q8—42no—
340Q8—42no—
341D4—44no—
342D4—42no—
343D4—42no—
344D4—44no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—48no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4C5×#10412yes—
376C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
377C2×C4C5×#13711yescharacteristic
378C8—44no—
379C8—82no—
380C5#1176211yescharacteristic, direct product factor, Sylow
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C5×AΣL(1,𝔽16)12yes—
398C22C2×C5×#9911yescharacteristic
399C22C5×#41711yescharacteristic
400C22C2×C5×#10411yescharacteristic
401C4—21no—
402C4—21no—
403C4—22no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C2—21no—
412C2—21no—
413C2—22no—
414C2—41no—
415C2—82no—
416C2C5×#150411yescharacteristic
417C2C2×C5×AΣL(1,𝔽16)11yescharacteristic
418C2C5×#213311yescharacteristic
subgroupG/H
center, socleC22×C5C2×#99
commutatorC22×C4C22×C4×C5
Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC22×C5×#41C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C24×C5 ◁ C22×C5×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#343C2semidirect
3. C5×#1463C2semidirect
4. C5×#1354C2semidirect
5. C5×#1345C2semidirect
23. #11762C5direct (N × H)
62. C2×#343C2×C5semidirect
63. #1463C2×C5semidirect
64. #1354C2×C5semidirect
65. #1345C2×C5semidirect
380. C5#11762direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#24062
C2×C5×#3432
C5×#14632
C5×#13542
C5×#13452
#117625