Finite Group Databasebeta

← C5×#12250 #11934970

Subgroups — C5×#12250 #11934970

Order 1280.

454 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2819C211yesmaximal, characteristic
2C5×#2061C211yesmaximal, characteristic, complemented
3C5×#1599C212yesmaximal, complemented
4C5×#1345C212yesmaximal, complemented
5C5×#1324C211yesmaximal, characteristic, complemented
6C5×#544C2211yescharacteristic
7C2×C5×#124—21no—
8C5×#450—22no—
9C5×#428—21no—
10C5×#394—21no—
11C5×#394C2212yes—
12C5×#393—22no—
13C5×#392—22no—
14C5×#382C2211yescharacteristic
15C5×#343C2211yescharacteristic
16C5×#339—21no—
17C5×#328—22no—
18C5×#324C2212yes—
19#12250C511yesmaximal, characteristic, direct product factor, Sylow
20C22×C5×Q8—21no—
21C22×C5×D4—21no—
22C22×C5×D4—21no—
23C22×C5×D4D412yes—
24C2×C5×SD16—42no—
25C2×C5×M16—21no—
26C5×C4⋊Q8—41no—
27C5×#125—41no—
28C5×#124—42no—
29C5×#124—42no—
30C5×C22⋊Q8—22no—
31C5×#117—21no—
32C2×C5×C4⋊C4D411yescharacteristic
33C2×C5×C4⋊C4C2311yescharacteristic
34C2×C5×C22⋊C4—22no—
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—22no—
38C2×C42×C5D411yescharacteristic
39C5×#107—41no—
40C5×#106—41no—
41C5×#102—42no—
42C5×C22⋊C8—22no—
43C5×#95—21no—
44C5×#95—22no—
45C5×#95D412yes—
46#2819C2×C511yescharacteristic
47#2061C2×C511yescharacteristic, complemented
48#1599C2×C512yescomplemented
49#1345C2×C512yescomplemented
50#1324C2×C511yescharacteristic, complemented
51C24×C5—22no—
52C2×C5×Q8—22no—
53C2×C5×Q8—42no—
54C2×C5×Q8—81no—
55C2×C5×Q8—81no—
56C2×C5×D4—42no—
57C2×C5×D4—42no—
58C2×C5×D4—82no—
59C2×C5×D4—81no—
60C2×C5×D4—81no—
61C2×C5×D4—82no—
62C2×C5×D4—81no—
63C22×C4×C5—21no—
64C22×C4×C5—22no—
65C22×C4×C5—21no—
66C22×C4×C5C2×D411yescharacteristic
67C22×C4×C5C2×D412yes—
68C5×SD16—84no—
69C5×M16—22no—
70C5×M16—41no—
71C2×C5×C8—41no—
72C2×C5×C8—42no—
73C5×C4⋊C4—21no—
74C5×C4⋊C4—21no—
75C5×C4⋊C4—22no—
76C5×C4⋊C4—41no—
77C5×C4⋊C4—41no—
78C5×C4⋊C4—42no—
79C5×C22⋊C4—44no—
80C5×C22⋊C4—41no—
81C5×C22⋊C4—42no—
82C5×C22⋊C4—42no—
83C5×C22⋊C4—82no—
84C5×C22⋊C4—82no—
85C42×C5—21no—
86C42×C5—41no—
87C42×C5SD1612yescomplemented
88#544C22×C511yescharacteristic
89C2×#124—21no—
90#450—22no—
91#428—21no—
92#394—21no—
93#394C22×C512yes—
94#393—22no—
95#392—22no—
96#382C22×C511yescharacteristic
97#343C22×C511yescharacteristic
98#339—21no—
99#328—22no—
100#324C22×C512yes—
101C23×C5—42no—
102C23×C5—42no—
103C23×C5—42no—
104C23×C5—42no—
105C23×C5—42no—
106C23×C5—42no—
107C23×C5—42no—
108C23×C5#12011yescharacteristic
109C5×Q8—82no—
110C5×Q8—84no—
111C5×D4—44no—
112C5×D4—82no—
113C5×D4—84no—
114C5×D4—84no—
115C2×C4×C5—21no—
116C2×C4×C5—21no—
117C2×C4×C5—22no—
118C2×C4×C5—22no—
119C2×C4×C5—41no—
120C2×C4×C5—41no—
121C2×C4×C5—42no—
122C2×C4×C5—42no—
123C2×C4×C5—42no—
124C2×C4×C5—42no—
125C2×C4×C5—42no—
126C2×C4×C5—41no—
127C2×C4×C5—41no—
128C2×C4×C5—41no—
129C2×C4×C5C2×SD1611yescharacteristic
130C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
131C5×C8—42no—
132C5×C8—82no—
133C22×Q8—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C5×D412yes—
137C2×SD16—42no—
138C2×M16—21no—
139C4⋊Q8—41no—
140#125—41no—
141#124—42no—
142#124—42no—
143C22⋊Q8—22no—
144#117—21no—
145C2×C4⋊C4C5×D411yescharacteristic
146C2×C4⋊C4C23×C511yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C42C5×D411yescharacteristic
152#107—41no—
153#106—41no—
154#102—42no—
155C22⋊C8—22no—
156#95—21no—
157#95—22no—
158#95C5×D412yes—
159C22×C5—21no—
160C22×C5—21no—
161C22×C5—82no—
162C22×C5—82no—
163C22×C5—82no—
164C22×C5—82no—
165C22×C5—82no—
166C22×C5—82no—
167C22×C5—82no—
168C22×C5#45011yescharacteristic
169C22×C5#45611yescharacteristic
170C22×C5B(4,𝔽2)11yescharacteristic
171C4×C5—21no—
172C4×C5—21no—
173C4×C5—42no—
174C4×C5—42no—
175C4×C5—82no—
176C4×C5—81no—
177C4×C5—81no—
178C4×C5—81no—
179C24—22no—
180C2×Q8—22no—
181C2×Q8—42no—
182C2×Q8—81no—
183C2×Q8—81no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—82no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4—82no—
190C2×D4—81no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4C2×C5×D411yescharacteristic
195C22×C4C2×C5×D412yes—
196SD16—84no—
197M16—22no—
198M16—41no—
199C2×C8—41no—
200C2×C8—42no—
201C4⋊C4—22no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C22⋊C4—44no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C42—21no—
214C42—41no—
215C42C5×SD1612yescomplemented
216C2×C5—21no—
217C2×C5—21no—
218C2×C5—162no—
219C2×C5#160411yescharacteristic
220C2×C5#219911yescharacteristic
221C2×C5#220211yescharacteristic
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23C5×#12011yescharacteristic
230Q8—82no—
231Q8—84no—
232D4—44no—
233D4—82no—
234D4—84no—
235D4—84no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4C2×C5×SD1611yescharacteristic
251C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
252C8—42no—
253C8—82no—
254C5#1225011yescharacteristic, direct product factor, Sylow
255C22—21no—
256C22—21no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22C5×#45011yescharacteristic
265C22C5×#45611yescharacteristic
266C22C5×B(4,𝔽2)11yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—81no—
274C4—81no—
275C2—21no—
276C2—21no—
277C2—162no—
278C2C5×#160411yescharacteristic
279C2C5×#219911yescharacteristic
280C2C5×#220211yescharacteristic
subgroupG/H
center, socleC22×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC5×#2819C2
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
2. C5×#2061C2semidirect
3. C5×#1599C2semidirect
4. C5×#1345C2semidirect
5. C5×#1324C2semidirect
19. #12250C5direct (N × H)
47. #2061C2×C5semidirect
48. #1599C2×C5semidirect
49. #1345C2×C5semidirect
50. #1324C2×C5semidirect
87. C42×C5SD16semidirect
215. C42C5×SD16semidirect
254. C5#12250direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#28192
C5×#20612
C5×#15992
C5×#13452
C5×#13242
#122505