Finite Group Databasebeta

← C5×#7215 #11929935

Subgroups — C5×#7215 #11929935

Order 1280.

422 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 82 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×#1821C211yesmaximal, characteristic
2C5×#1463C211yesmaximal, characteristic
3C5×#1436C211yesmaximal, characteristic
4C22×C5×C4⋊C4C2211yescharacteristic
5C22×C42×C5C411yescharacteristic
6C5×#407—21no—
7C5×#407—21no—
8C2×C5×#95—21no—
9C2×C5×#95—21no—
10C2×C5×#95C411yescharacteristic
11C5×#343—21no—
12C5×#343C412yescomplemented
13C5×#324—21no—
14C5×#324C412yescomplemented
15#7215C511yesmaximal, characteristic, direct product factor, Sylow
16C23×C4×C5—21no—
17C23×C4×C5—21no—
18C23×C4×C5C2×C411yescharacteristic
19C23×C4×C5D411yescharacteristic
20C23×C4×C5D411yescharacteristic
21C2×C5×M16—22no—
22C2×C5×M16—41no—
23C2×C5×C4⋊C4—41no—
24C2×C5×C4⋊C4—41no—
25C2×C5×C4⋊C4C2×C411yescharacteristic
26C2×C5×C4⋊C4C2×C411yescharacteristic
27C2×C5×C4⋊C4D411yescharacteristic
28C2×C5×C4⋊C4Q811yescharacteristic
29C2×C42×C5—41no—
30C2×C42×C5—42no—
31C5×C22⋊C8—22no—
32C5×C22⋊C8—41no—
33C5×C22⋊C8—41no—
34C5×C22⋊C8—41no—
35C5×#95—24no—
36C5×#95—41no—
37C5×#95—42no—
38#1821C2×C511yescharacteristic
39#1463C2×C511yescharacteristic
40#1436C2×C511yescharacteristic
41C24×C5C22⋊C411yescharacteristic
42C22×C4×C5—22no—
43C22×C4×C5—22no—
44C22×C4×C5—22no—
45C22×C4×C5—21no—
46C22×C4×C5—21no—
47C22×C4×C5—21no—
48C22×C4×C5—21no—
49C22×C4×C5—21no—
50C22×C4×C5—21no—
51C22×C4×C5—22no—
52C22×C4×C5—22no—
53C22×C4×C5—41no—
54C22×C4×C5—41no—
55C22×C4×C5—41no—
56C22×C4×C5—41no—
57C22×C4×C5—41no—
58C22×C4×C5—41no—
59C22×C4×C5—41no—
60C22×C4×C5—41no—
61C22×C4×C5—41no—
62C22×C4×C5—41no—
63C22×C4×C5—41no—
64C22×C4×C5—41no—
65C22×C4×C5—41no—
66C22×C4×C5—41no—
67C22×C4×C5—41no—
68C22×C4×C5C4211yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C4⋊C411yescharacteristic
72C22×C4×C5C4⋊C411yescharacteristic
73C22×C4×C5C4⋊C411yescharacteristic
74C5×M16—42no—
75C5×M16—42no—
76C5×M16—81no—
77C5×M16—81no—
78C2×C5×C8—42no—
79C2×C5×C8—42no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—41no—
84C42×C5—42no—
85C42×C5—42no—
86C22×C4⋊C4C22×C511yescharacteristic
87C22×C42C4×C511yescharacteristic
88#407—21no—
89#407—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95C4×C511yescharacteristic
93#343—21no—
94#343C4×C512yescomplemented
95#324—21no—
96#324C4×C512yescomplemented
97C23×C5—21no—
98C23×C5—21no—
99C23×C5—21no—
100C23×C5—21no—
101C23×C5—21no—
102C23×C5—21no—
103C23×C5#9511yescharacteristic
104C23×C5#9911yescharacteristic
105C23×C5#9911yescharacteristic
106C2×C4×C5—21no—
107C2×C4×C5—21no—
108C2×C4×C5—24no—
109C2×C4×C5—21no—
110C2×C4×C5—21no—
111C2×C4×C5—21no—
112C2×C4×C5—21no—
113C2×C4×C5—21no—
114C2×C4×C5—21no—
115C2×C4×C5—21no—
116C2×C4×C5—42no—
117C2×C4×C5—42no—
118C2×C4×C5—42no—
119C2×C4×C5—42no—
120C2×C4×C5—42no—
121C2×C4×C5—42no—
122C2×C4×C5—42no—
123C2×C4×C5—42no—
124C2×C4×C5—41no—
125C2×C4×C5—42no—
126C2×C4×C5—41no—
127C2×C4×C5—41no—
128C2×C4×C5—41no—
129C2×C4×C5—41no—
130C2×C4×C5—41no—
131C2×C4×C5—41no—
132C2×C4×C5—41no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—81no—
137C2×C4×C5—81no—
138C2×C4×C5—81no—
139C2×C4×C5—81no—
140C2×C4×C5—81no—
141C2×C4×C5#10412yes—
142C5×C8—82no—
143C5×C8—82no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2×C4×C511yescharacteristic
147C23×C4C5×D411yescharacteristic
148C23×C4C5×D411yescharacteristic
149C2×M16—22no—
150C2×M16—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4C2×C4×C511yescharacteristic
154C2×C4⋊C4C2×C4×C511yescharacteristic
155C2×C4⋊C4C5×D411yescharacteristic
156C2×C4⋊C4C5×Q811yescharacteristic
157C2×C42—41no—
158C2×C42—42no—
159C22⋊C8—22no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163#95—24no—
164#95—41no—
165#95—42no—
166C22×C5—21no—
167C22×C5—21no—
168C22×C5—21no—
169C22×C5—21no—
170C22×C5—21no—
171C22×C5—21no—
172C22×C5—21no—
173C22×C5—21no—
174C22×C5—21no—
175C22×C5—21no—
176C22×C5—21no—
177C22×C5—21no—
178C22×C5—21no—
179C22×C5—21no—
180C22×C5H(3,ℤ/4)11yescharacteristic
181C22×C5#33911yescharacteristic
182C22×C5#34211yescharacteristic
183C22×C5AΣL(1,𝔽16)11yescharacteristic
184C22×C5#35211yescharacteristic
185C22×C5#35511yescharacteristic
186C22×C5#35611yescharacteristic
187C4×C5—22no—
188C4×C5—44no—
189C4×C5—41no—
190C4×C5—41no—
191C4×C5—41no—
192C4×C5—41no—
193C4×C5—41no—
194C4×C5—82no—
195C24C5×C22⋊C411yescharacteristic
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4C42×C511yescharacteristic
223C22×C4C5×C22⋊C411yescharacteristic
224C22×C4C5×C22⋊C411yescharacteristic
225C22×C4C5×C4⋊C411yescharacteristic
226C22×C4C5×C4⋊C411yescharacteristic
227C22×C4C5×C4⋊C411yescharacteristic
228M16—42no—
229M16—42no—
230M16—81no—
231M16—81no—
232C2×C8—42no—
233C2×C8—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C42—42no—
239C42—42no—
240C2×C5—21no—
241C2×C5—21no—
242C2×C5—21no—
243C2×C5—21no—
244C2×C5—21no—
245C2×C5—21no—
246C2×C5#128411yescharacteristic
247C2×C5#139011yescharacteristic
248C2×C5#139411yescharacteristic
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23C5×#9511yescharacteristic
256C23C5×#9911yescharacteristic
257C23C5×#9911yescharacteristic
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—24no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4C5×#10412yes—
294C8—82no—
295C8—82no—
296C5#721511yescharacteristic, direct product factor, Sylow
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22C5×H(3,ℤ/4)11yescharacteristic
312C22C5×#33911yescharacteristic
313C22C5×#34211yescharacteristic
314C22C5×AΣL(1,𝔽16)11yescharacteristic
315C22C5×#35211yescharacteristic
316C22C5×#35511yescharacteristic
317C22C5×#35611yescharacteristic
318C4—22no—
319C4—44no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2C5×#128411yescharacteristic
333C2C5×#139011yescharacteristic
334C2C5×#139411yescharacteristic
subgroupG/H
center, socleC22×C5#342
commutatorC22×C4C42×C5
Frattini, cosocleC22×C4⋊C4C22×C5
prime radicalC24×C5C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
12. C5×#343C4semidirect
14. C5×#324C4semidirect
15. #7215C5direct (N × H)
94. #343C4×C5semidirect
96. #324C4×C5semidirect
296. C5#7215direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#18212
C5×#14632
C5×#14362
#72155