Finite Group Databasebeta

← C5×#7810 #11930530

Subgroups — C5×#7810 #11930530

Order 1280.

478 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2733C211yesmaximal, characteristic
2C5×#2096C211yesmaximal, characteristic, complemented
3C5×#2071C212yesmaximal, complemented
4C5×#1272C211yesmaximal, characteristic, complemented
5C5×#1270C212yesmaximal, complemented
6C2×C5×#124—21no—
7C2×C5×#106—21no—
8C2×C5×#103—21no—
9C2×C5×#102—22no—
10C5×#394—21no—
11C5×#392—22no—
12C5×#391C2211yescharacteristic
13C5×#390C2212yes—
14C5×#388—22no—
15C5×#384C2211yescharacteristic
16C5×#340—22no—
17C5×#336C2212yes—
18C5×#336C2211yescharacteristic
19#7810C511yesmaximal, characteristic, direct product factor, Sylow
20C22×C5×Q8D411yescharacteristic
21C22×C5×D4—21no—
22C22×C5×D4—21no—
23C22×C5×D4D412yes—
24C22×C5×C8—21no—
25C22×C5×C8—22no—
26C5×#124—42no—
27C5×#124—42no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4D412yes—
31C2×C5×C4⋊C4D411yescharacteristic
32C2×C5×C22⋊C4—22no—
33C2×C5×C22⋊C4—22no—
34C2×C5×C22⋊C4—22no—
35C2×C5×C22⋊C4—22no—
36C2×C42×C5—21no—
37C2×C42×C5C2311yescharacteristic
38C5×#106—42no—
39C5×#103—42no—
40C5×#102—44no—
41C5×#95—21no—
42C5×#95—22no—
43#2733C2×C511yescharacteristic
44#2096C2×C511yescharacteristic, complemented
45#2071C2×C512yescomplemented
46#1272C2×C511yescharacteristic, complemented
47#1270C2×C512yescomplemented
48C24×C5—22no—
49C2×C5×Q8—21no—
50C2×C5×Q8—41no—
51C2×C5×Q8—41no—
52C2×C5×Q8SD1612yes—
53C2×C5×D4—22no—
54C2×C5×D4—22no—
55C2×C5×D4—42no—
56C2×C5×D4—42no—
57C2×C5×D4—42no—
58C2×C5×D4—81no—
59C2×C5×D4—81no—
60C2×C5×D4—81no—
61C2×C5×D4—82no—
62C22×C4×C5—21no—
63C22×C4×C5—22no—
64C22×C4×C5—21no—
65C22×C4×C5—41no—
66C22×C4×C5C2×D411yescharacteristic
67C22×C4×C5C2×D412yes—
68C2×C5×C8—22no—
69C2×C5×C8—41no—
70C2×C5×C8—41no—
71C2×C5×C8—42no—
72C2×C5×C8—42no—
73C2×C5×C8—42no—
74C5×C4⋊C4—42no—
75C5×C4⋊C4—41no—
76C5×C4⋊C4—41no—
77C5×C4⋊C4—41no—
78C5×C4⋊C4—42no—
79C5×C4⋊C4—42no—
80C5×C22⋊C4—44no—
81C5×C22⋊C4—42no—
82C5×C22⋊C4—42no—
83C5×C22⋊C4—42no—
84C5×C22⋊C4—42no—
85C5×C22⋊C4—82no—
86C42×C5—41no—
87C42×C5—42no—
88C2×#124—21no—
89C2×#106—21no—
90C2×#103—21no—
91C2×#102—22no—
92#394—21no—
93#392—22no—
94#391C22×C511yescharacteristic
95#390C22×C512yes—
96#388—22no—
97#384C22×C511yescharacteristic
98#340—22no—
99#336C22×C512yes—
100#336C22×C511yescharacteristic
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—42no—
110C5×Q8—81no—
111C5×D4—44no—
112C5×D4—82no—
113C5×D4—84no—
114C5×D4—84no—
115C2×C4×C5—21no—
116C2×C4×C5—22no—
117C2×C4×C5—22no—
118C2×C4×C5—21no—
119C2×C4×C5—22no—
120C2×C4×C5—41no—
121C2×C4×C5—41no—
122C2×C4×C5—42no—
123C2×C4×C5—42no—
124C2×C4×C5—41no—
125C2×C4×C5—42no—
126C2×C4×C5—42no—
127C2×C4×C5—82no—
128C2×C4×C5—81no—
129C2×C4×C5C2×SD1611yescharacteristic
130C2×C4×C5#13512yes—
131C2×C4×C5AGL(1,ℤ/8)12yes—
132C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
133C5×C8—42no—
134C5×C8—82no—
135C22×Q8C5×D411yescharacteristic
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C5×D412yes—
139C22×C8—21no—
140C22×C8—22no—
141#124—42no—
142#124—42no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4C5×D412yes—
146C2×C4⋊C4C5×D411yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C42—21no—
152C2×C42C23×C511yescharacteristic
153#106—42no—
154#103—42no—
155#102—44no—
156#95—21no—
157#95—22no—
158C22×C5—82no—
159C22×C5—82no—
160C22×C5—82no—
161C22×C5—82no—
162C22×C5—82no—
163C22×C5—82no—
164C22×C5—82no—
165C22×C5#44912yes—
166C22×C5#45011yescharacteristic
167C22×C5#45311yescharacteristic
168C22×C5#45512yes—
169C22×C5#45811yescharacteristic
170C4×C5—22no—
171C4×C5—22no—
172C4×C5—21no—
173C4×C5—21no—
174C4×C5—42no—
175C4×C5—82no—
176C4×C5—81no—
177C4×C5—161no—
178C24—22no—
179C2×Q8—21no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C5×SD1612yes—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—81no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4—82no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4C2×C5×D411yescharacteristic
197C22×C4C2×C5×D412yes—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C22⋊C4—44no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—82no—
216C42—41no—
217C42—42no—
218C2×C5—162no—
219C2×C5#161212yes—
220C2×C5#161311yescharacteristic
221C2×C5#162411yescharacteristic
222C2×C5#164212yes—
223C2×C5#165911yescharacteristic
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23C5×#12011yescharacteristic
232Q8—42no—
233Q8—81no—
234D4—44no—
235D4—82no—
236D4—84no—
237D4—84no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4C2×C5×SD1611yescharacteristic
253C2×C4C5×#13512yes—
254C2×C4C5×AGL(1,ℤ/8)12yes—
255C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
256C8—42no—
257C8—82no—
258C5#781011yescharacteristic, direct product factor, Sylow
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22C5×#44912yes—
267C22C5×#45011yescharacteristic
268C22C5×#45311yescharacteristic
269C22C5×#45512yes—
270C22C5×#45811yescharacteristic
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—82no—
277C4—81no—
278C4—161no—
279C2—162no—
280C2C5×#161212yes—
281C2C5×#161311yescharacteristic
282C2C5×#162411yescharacteristic
283C2C5×#164212yes—
284C2C5×#165911yescharacteristic
subgroupG/H
center, socleC23×C5#120
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC5×#2733C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2096C2semidirect
3. C5×#2071C2semidirect
4. C5×#1272C2semidirect
5. C5×#1270C2semidirect
19. #7810C5direct (N × H)
44. #2096C2×C5semidirect
45. #2071C2×C5semidirect
46. #1272C2×C5semidirect
47. #1270C2×C5semidirect
258. C5#7810direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#27332
C5×#20962
C5×#20712
C5×#12722
C5×#12702
#78105