Finite Group Databasebeta

← C5×#12240 #11934960

Subgroups — C5×#12240 #11934960

Order 1280.

350 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2792C211yesmaximal, characteristic
2C5×#2060C211yesmaximal, characteristic
3C5×#1611C211yesmaximal, characteristic
4C5×#1605C211yesmaximal, characteristic
5C5×#1354C211yesmaximal, characteristic
6C5×#1346C211yesmaximal, characteristic
7C5×#1325C211yesmaximal, characteristic
8C5×#544C2211yescharacteristic
9C5×#483—21no—
10C5×#451—21no—
11C5×#428—21no—
12C5×#398—21no—
13C5×#398C2211yescharacteristic
14C5×#395—21no—
15C5×#395C2211yescharacteristic
16C5×#389—21no—
17C5×#389—21no—
18C5×#387—21no—
19C5×#387—21no—
20C5×#384C2211yescharacteristic
21C5×#343C2211yescharacteristic
22C5×#339—21no—
23C5×#328—21no—
24C5×#328—21no—
25C5×#324C2211yescharacteristic
26C5×#324C2211yescharacteristic
27#12240C511yesmaximal, characteristic, direct product factor, Sylow
28C22×C5×Q8D411yescharacteristic
29C2×C5×Q16—41no—
30C2×C5×M16—21no—
31C5×C4⋊Q8—41no—
32C5×#125—41no—
33C5×C22⋊Q8—21no—
34C5×C22⋊Q8—21no—
35C5×#117—21no—
36C2×C5×C4⋊C4—21no—
37C2×C5×C4⋊C4—21no—
38C2×C5×C4⋊C4—21no—
39C2×C5×C4⋊C4—21no—
40C2×C5×C4⋊C4—21no—
41C2×C5×C4⋊C4—21no—
42C2×C5×C4⋊C4—21no—
43C2×C5×C4⋊C4D411yescharacteristic
44C2×C5×C4⋊C4D411yescharacteristic
45C2×C5×C4⋊C4C2311yescharacteristic
46C2×C42×C5—21no—
47C2×C42×C5D411yescharacteristic
48C5×#107—41no—
49C5×#106—41no—
50C5×#106—41no—
51C5×#103—41no—
52C5×#103—41no—
53C5×C22⋊C8—21no—
54C5×C22⋊C8—21no—
55C5×#95—21no—
56C5×#95—21no—
57C5×#95—21no—
58C5×#95—21no—
59C5×#95—21no—
60C5×#95—21no—
61C5×#95D411yescharacteristic
62C5×#95D411yescharacteristic
63#2792C2×C511yescharacteristic
64#2060C2×C511yescharacteristic
65#1611C2×C511yescharacteristic
66#1605C2×C511yescharacteristic
67#1354C2×C511yescharacteristic
68#1346C2×C511yescharacteristic
69#1325C2×C511yescharacteristic
70C2×C5×Q8—21no—
71C2×C5×Q8—21no—
72C2×C5×Q8—41no—
73C2×C5×Q8—81no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5—21no—
79C22×C4×C5—21no—
80C22×C4×C5C2×D411yescharacteristic
81C22×C4×C5C2×D411yescharacteristic
82C22×C4×C5C2×D411yescharacteristic
83C5×Q16—82no—
84C5×M16—22no—
85C5×M16—41no—
86C2×C5×C8—41no—
87C2×C5×C8—41no—
88C2×C5×C8—41no—
89C5×C4⋊C4—24no—
90C5×C4⋊C4—21no—
91C5×C4⋊C4—21no—
92C5×C4⋊C4—21no—
93C5×C4⋊C4—21no—
94C5×C4⋊C4—41no—
95C5×C4⋊C4—41no—
96C5×C4⋊C4—41no—
97C5×C4⋊C4—41no—
98C5×C4⋊C4—41no—
99C5×C4⋊C4—41no—
100C5×C4⋊C4—41no—
101C5×C4⋊C4—81no—
102C5×C4⋊C4—81no—
103C5×C4⋊C4—81no—
104C5×C4⋊C4—81no—
105C5×C4⋊C4—81no—
106C5×C22⋊C4—41no—
107C42×C5—21no—
108C42×C5—21no—
109C42×C5—41no—
110C42×C5—81no—
111#544C22×C511yescharacteristic
112#483—21no—
113#451—21no—
114#428—21no—
115#398—21no—
116#398C22×C511yescharacteristic
117#395—21no—
118#395C22×C511yescharacteristic
119#389—21no—
120#389—21no—
121#387—21no—
122#387—21no—
123#384C22×C511yescharacteristic
124#343C22×C511yescharacteristic
125#339—21no—
126#328—21no—
127#328—21no—
128#324C22×C511yescharacteristic
129#324C22×C511yescharacteristic
130C23×C5#12011yescharacteristic
131C5×Q8—42no—
132C5×Q8—81no—
133C5×Q8—81no—
134C5×Q8—81no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—22no—
138C2×C4×C5—22no—
139C2×C4×C5—22no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5—41no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—41no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5—41no—
158C2×C4×C5—41no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5#13511yescharacteristic
162C2×C4×C5#13711yescharacteristic
163C5×C8—42no—
164C5×C8—81no—
165C5×C8—81no—
166C22×Q8C5×D411yescharacteristic
167C2×Q16—41no—
168C2×M16—21no—
169C4⋊Q8—41no—
170#125—41no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173#117—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C5×D411yescharacteristic
182C2×C4⋊C4C5×D411yescharacteristic
183C2×C4⋊C4C23×C511yescharacteristic
184C2×C42—21no—
185C2×C42C5×D411yescharacteristic
186#107—41no—
187#106—41no—
188#106—41no—
189#103—41no—
190#103—41no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95C5×D411yescharacteristic
200#95C5×D411yescharacteristic
201C22×C5—21no—
202C22×C5—21no—
203C22×C5#45211yescharacteristic
204C22×C5#45611yescharacteristic
205C22×C5B(4,𝔽2)11yescharacteristic
206C4×C5—21no—
207C4×C5—21no—
208C4×C5—42no—
209C4×C5—42no—
210C4×C5—42no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5—81no—
214C4×C5—81no—
215C4×C5—81no—
216C4×C5—81no—
217C4×C5—81no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—81no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4C2×C5×D411yescharacteristic
229C22×C4C2×C5×D411yescharacteristic
230C22×C4C2×C5×D411yescharacteristic
231Q16—82no—
232M16—22no—
233M16—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C4⋊C4—24no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C22⋊C4—41no—
255C42—21no—
256C42—21no—
257C42—41no—
258C42—81no—
259C2×C5—21no—
260C2×C5—21no—
261C2×C5#160311yescharacteristic
262C2×C5#220011yescharacteristic
263C2×C5#220511yescharacteristic
264C23C5×#12011yescharacteristic
265Q8—42no—
266Q8—81no—
267Q8—81no—
268Q8—81no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4C5×#13511yescharacteristic
296C2×C4C5×#13711yescharacteristic
297C8—42no—
298C8—81no—
299C8—81no—
300C5#1224011yescharacteristic, direct product factor, Sylow
301C22—21no—
302C22—21no—
303C22C5×#45211yescharacteristic
304C22C5×#45611yescharacteristic
305C22C5×B(4,𝔽2)11yescharacteristic
306C4—21no—
307C4—21no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C2—21no—
319C2—21no—
320C2C5×#160311yescharacteristic
321C2C5×#220011yescharacteristic
322C2C5×#220511yescharacteristic
subgroupG/H
center, socleC22×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC23×C5#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C5 ◁ C2×C5×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
27. #12240C5direct (N × H)
300. C5#12240direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#27922
C5×#20602
C5×#16112
C5×#16052
C5×#13542
C5×#13462
C5×#13252
#122405