Finite Group Databasebeta

← C5×#12220 #11934940

Subgroups — C5×#12220 #11934940

Order 1280.

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

normal subgroups only

This group has 460 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×#2683C211yesmaximal, characteristic
2C5×#2027C211yesmaximal, characteristic
3C5×#1605C212yesmaximal, complemented
4C5×#1354C212yesmaximal, complemented
5C5×#1318C211yesmaximal, characteristic
6C5×#544C2211yescharacteristic
7C2×C5×C22⋊Q8—22no—
8C5×#451—22no—
9C5×#418—21no—
10C5×#396—22no—
11C5×#395C2212yes—
12C5×#390C2211yescharacteristic
13C5×#389—21no—
14C5×#386—21no—
15C5×#343C2211yescharacteristic
16C5×#339—21no—
17C5×#328—22no—
18C5×#324C2212yes—
19#12220C511yesmaximal, characteristic, direct product factor, Sylow
20C22×C5×Q8D412yes—
21C22×C5×D4D411yescharacteristic
22C23×C4×C5—21no—
23C2×C5×Q16—42no—
24C2×C5×M16—21no—
25C5×C4⋊Q8—41no—
26C5×#125—41no—
27C5×C22⋊Q8—22no—
28C5×C22⋊Q8—48no—
29C5×#117—21no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—22no—
32C2×C5×C4⋊C4—22no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4C2311yescharacteristic
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—21no—
38C2×C42×C5D411yescharacteristic
39C5×#103—42no—
40C5×#102—41no—
41C5×#102—41no—
42C5×C22⋊C8—22no—
43C5×#95—22no—
44C5×#95D412yes—
45#2683C2×C511yescharacteristic
46#2027C2×C511yescharacteristic
47#1605C2×C512yescomplemented
48#1354C2×C512yescomplemented
49#1318C2×C511yescharacteristic
50C24×C5—21no—
51C2×C5×Q8—22no—
52C2×C5×Q8—44no—
53C2×C5×Q8—42no—
54C2×C5×D4—21no—
55C2×C5×D4—41no—
56C2×C5×D4—41no—
57C2×C5×D4Q1612yescomplemented
58C22×C4×C5—21no—
59C22×C4×C5—22no—
60C22×C4×C5—22no—
61C22×C4×C5—44no—
62C22×C4×C5—81no—
63C22×C4×C5C2×D411yescharacteristic
64C22×C4×C5C2×D412yes—
65C5×Q16—84no—
66C5×M16—41no—
67C5×M16—41no—
68C2×C5×C8—41no—
69C2×C5×C8—42no—
70C5×C4⋊C4—24no—
71C5×C4⋊C4—21no—
72C5×C4⋊C4—21no—
73C5×C4⋊C4—22no—
74C5×C4⋊C4—41no—
75C5×C4⋊C4—44no—
76C5×C4⋊C4—44no—
77C5×C4⋊C4—42no—
78C5×C4⋊C4—81no—
79C5×C22⋊C4—44no—
80C5×C22⋊C4—44no—
81C5×C22⋊C4—41no—
82C5×C22⋊C4—81no—
83C42×C5—41no—
84C42×C5—41no—
85#544C22×C511yescharacteristic
86C2×C22⋊Q8—22no—
87#451—22no—
88#418—21no—
89#396—22no—
90#395C22×C512yes—
91#390C22×C511yescharacteristic
92#389—21no—
93#386—21no—
94#343C22×C511yescharacteristic
95#339—21no—
96#328—22no—
97#324C22×C512yes—
98C23×C5—22no—
99C23×C5—42no—
100C23×C5—42no—
101C23×C5—41no—
102C23×C5—41no—
103C23×C5#12011yescharacteristic
104C5×Q8—44no—
105C5×Q8—82no—
106C5×Q8—82no—
107C5×D4—42no—
108C5×D4—81no—
109C2×C4×C5—21no—
110C2×C4×C5—21no—
111C2×C4×C5—22no—
112C2×C4×C5—24no—
113C2×C4×C5—22no—
114C2×C4×C5—42no—
115C2×C4×C5—41no—
116C2×C4×C5—41no—
117C2×C4×C5—42no—
118C2×C4×C5—42no—
119C2×C4×C5—42no—
120C2×C4×C5—42no—
121C2×C4×C5—42no—
122C2×C4×C5—41no—
123C2×C4×C5—41no—
124C2×C4×C5—84no—
125C2×C4×C5C2×Q1611yescharacteristic
126C2×C4×C5#13711yescharacteristic
127C5×C8—81no—
128C5×C8—82no—
129C22×Q8C5×D412yes—
130C22×D4C5×D411yescharacteristic
131C23×C4—21no—
132C2×Q16—42no—
133C2×M16—21no—
134C4⋊Q8—41no—
135#125—41no—
136C22⋊Q8—22no—
137C22⋊Q8—48no—
138#117—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4C23×C511yescharacteristic
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C42C5×D411yescharacteristic
148#103—42no—
149#102—41no—
150#102—41no—
151C22⋊C8—22no—
152#95—22no—
153#95C5×D412yes—
154C22×C5—21no—
155C22×C5—21no—
156C22×C5—44no—
157C22×C5—42no—
158C22×C5—82no—
159C22×C5—81no—
160C22×C5—81no—
161C22×C5#45111yescharacteristic
162C22×C5#45511yescharacteristic
163C22×C5B(4,𝔽2)11yescharacteristic
164C4×C5—21no—
165C4×C5—21no—
166C4×C5—42no—
167C4×C5—42no—
168C4×C5—44no—
169C4×C5—82no—
170C4×C5—81no—
171C4×C5—81no—
172C24—21no—
173C2×Q8—22no—
174C2×Q8—44no—
175C2×Q8—42no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4C5×Q1612yescomplemented
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—44no—
184C22×C4—81no—
185C22×C4C2×C5×D411yescharacteristic
186C22×C4C2×C5×D412yes—
187Q16—84no—
188M16—41no—
189M16—41no—
190C2×C8—42no—
191C2×C8—41no—
192C4⋊C4—24no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C22⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—41no—
204C22⋊C4—81no—
205C42—41no—
206C42—41no—
207C2×C5—21no—
208C2×C5—21no—
209C2×C5—82no—
210C2×C5#160211yescharacteristic
211C2×C5#219611yescharacteristic
212C2×C5#220311yescharacteristic
213C23—22no—
214C23—42no—
215C23—42no—
216C23—41no—
217C23—41no—
218C23C5×#12011yescharacteristic
219Q8—44no—
220Q8—82no—
221Q8—82no—
222D4—42no—
223D4—81no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—84no—
240C2×C4C2×C5×Q1611yescharacteristic
241C2×C4C5×#13711yescharacteristic
242C8—81no—
243C8—82no—
244C5#1222011yescharacteristic, direct product factor, Sylow
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22C5×#45111yescharacteristic
253C22C5×#45511yescharacteristic
254C22C5×B(4,𝔽2)11yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—44no—
260C4—82no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2C5×#160211yescharacteristic
267C2C5×#219611yescharacteristic
268C2C5×#220311yescharacteristic
subgroupG/H
center, socleC22×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC22×C5×D4D4
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
3. C5×#1605C2semidirect
4. C5×#1354C2semidirect
19. #12220C5direct (N × H)
47. #1605C2×C5semidirect
48. #1354C2×C5semidirect
57. C2×C5×D4Q16semidirect
179. C2×D4C5×Q16semidirect
244. C5#12220direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#26832
C5×#20272
C5×#16052
C5×#13542
C5×#13182
#122205