Finite Group Databasebeta

← C5×#10216 #11932936

Subgroups — C5×#10216 #11932936

Order 1280.

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

normal subgroups only

This group has 468 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×#3255C211yesmaximal, characteristic, complemented
2C5×#3235C212yesmaximal
3C5×#1963C211yesmaximal, characteristic
4C5×#1506C212yesmaximal, complemented
5C5×#1307C211yesmaximal, characteristic, complemented
6C5×#533C2212yes—
7C5×#532C2211yescharacteristic
8C5×#501—22no—
9C5×#490—22no—
10C5×#482—24no—
11C5×#476—22no—
12C5×#475—22no—
13C5×#474—22no—
14C5×#470—22no—
15C5×#462—22no—
16C5×#432C2212yes—
17C5×#423C2211yescharacteristic
18C2×C5×#104—22no—
19C5×#376C2211yescharacteristic
20C5×#328—24no—
21#10216C511yesmaximal, characteristic, direct product factor, Sylow
22C2×C5×#41—21no—
23C2×C5×Q16—22no—
24C2×C5×M16D412yes—
25C5×C4⋊Q8D412yes—
26C5×C4⋊Q8D412yes—
27C5×C4⋊Q8D411yescharacteristic
28C5×#127D411yescharacteristic
29C5×#125—22no—
30C5×#124—21no—
31C5×C22⋊Q8—22no—
32C5×C22⋊Q8—22no—
33C5×#121—22no—
34C4×C5×Q8—22no—
35C4×C5×Q8—22no—
36C4×C5×D4—22no—
37C5×#117—22no—
38C2×C42×C5—21no—
39C2×C42×C5C2311yescharacteristic
40C5×#107—24no—
41C5×#106—22no—
42C5×#106—24no—
43C5×C4⋊C8—22no—
44C5×C4⋊C8—22no—
45C5×#104—44no—
46C5×#103—24no—
47C5×#102—24no—
48C5×#102—24no—
49C5×C22⋊C8D414yes—
50C5×#97—22no—
51C5×#95—22no—
52#3255C2×C511yescharacteristic, complemented
53#3235C2×C512yes—
54#1963C2×C511yescharacteristic
55#1506C2×C512yescomplemented
56#1307C2×C511yescharacteristic, complemented
57C5×#41—42no—
58C5×#41—42no—
59C2×C5×Q8—22no—
60C2×C5×Q8—21no—
61C2×C5×Q8—22no—
62C2×C5×D4—22no—
63C2×C5×D4—21no—
64C2×C5×D4—41no—
65C22×C4×C5—41no—
66C22×C4×C5—41no—
67C22×C4×C5C2×D411yescharacteristic
68C22×C4×C5C2×D412yes—
69C5×Q16—82no—
70C5×M16—42no—
71C2×C5×C8—22no—
72C2×C5×C8—24no—
73C5×C4⋊C4—22no—
74C5×C4⋊C4—22no—
75C5×C4⋊C4—22no—
76C5×C4⋊C4—22no—
77C5×C4⋊C4—22no—
78C5×C4⋊C4—42no—
79C5×C4⋊C4—42no—
80C5×C22⋊C4—42no—
81C5×C22⋊C4—42no—
82C42×C5—22no—
83C42×C5—41no—
84C42×C5—42no—
85C42×C5C2×D412yes—
86C42×C5C2×D411yescharacteristic
87C42×C5#4111yescharacteristic
88#533C22×C512yes—
89#532C22×C511yescharacteristic
90#501—22no—
91#490—22no—
92#482—24no—
93#476—22no—
94#475—22no—
95#474—22no—
96#470—22no—
97#462—22no—
98#432C22×C512yes—
99#423C22×C511yescharacteristic
100C2×#104—22no—
101#376C22×C511yescharacteristic
102#328—24no—
103C23×C5—41no—
104C23×C5#12711yescharacteristic
105C5×Q8—42no—
106C5×Q8—42no—
107C5×Q8—42no—
108C5×Q8—82no—
109C5×D4—44no—
110C5×D4—42no—
111C5×D4—82no—
112C2×C4×C5—21no—
113C2×C4×C5—21no—
114C2×C4×C5—42no—
115C2×C4×C5—42no—
116C2×C4×C5—42no—
117C2×C4×C5—42no—
118C2×C4×C5—41no—
119C2×C4×C5—82no—
120C2×C4×C5—81no—
121C2×C4×C5—81no—
122C2×C4×C5#12012yes—
123C2×C4×C5#12011yescharacteristic
124C2×C4×C5#12112yes—
125C2×C4×C5#12111yescharacteristic
126C5×C8—44no—
127C5×C8—42no—
128C2×#41—21no—
129C2×Q16—22no—
130C2×M16C5×D412yes—
131C4⋊Q8C5×D411yescharacteristic
132C4⋊Q8C5×D412yes—
133C4⋊Q8C5×D412yes—
134#127C5×D411yescharacteristic
135#125—22no—
136#124—21no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139#121—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×D4—22no—
143#117—22no—
144C2×C42—21no—
145C2×C42C23×C511yescharacteristic
146#107—24no—
147#106—22no—
148#106—24no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151#104—44no—
152#103—24no—
153#102—24no—
154#102—24no—
155C22⋊C8C5×D414yes—
156#97—22no—
157#95—22no—
158C22×C5—21no—
159C22×C5—22no—
160C22×C5—81no—
161C22×C5—82no—
162C22×C5#39211yescharacteristic
163C4×C5—21no—
164C4×C5—22no—
165C4×C5—22no—
166C4×C5—82no—
167C4×C5—81no—
168C4×C5—82no—
169C4×C5—82no—
170C4×C5#45311yescharacteristic
171C4×C5#45411yescharacteristic
172#41—42no—
173#41—42no—
174C2×Q8—22no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×D4—22no—
178C2×D4—21no—
179C2×D4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4C2×C5×D411yescharacteristic
183C22×C4C2×C5×D412yes—
184Q16—82no—
185M16—42no—
186C2×C8—24no—
187C2×C8—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—22no—
198C42—41no—
199C42—42no—
200C42C2×C5×D411yescharacteristic
201C42C2×C5×D412yes—
202C42C5×#4111yescharacteristic
203C2×C5—41no—
204C2×C5—161no—
205C2×C5#200611yescharacteristic
206C2×C5#200912yes—
207C23—41no—
208C23C5×#12711yescharacteristic
209Q8—42no—
210Q8—42no—
211Q8—42no—
212Q8—82no—
213D4—42no—
214D4—44no—
215D4—82no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—82no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4C5×#12012yes—
227C2×C4C5×#12011yescharacteristic
228C2×C4C5×#12112yes—
229C2×C4C5×#12111yescharacteristic
230C8—44no—
231C8—42no—
232C5#1021611yescharacteristic, direct product factor, Sylow
233C22—21no—
234C22—22no—
235C22—81no—
236C22—82no—
237C22C5×#39211yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—81no—
243C4—82no—
244C4—82no—
245C4C5×#45311yescharacteristic
246C4C5×#45411yescharacteristic
247C2—41no—
248C2—161no—
249C2C5×#200611yescharacteristic
250C2C5×#200912yes—
subgroupG/H
center, socleC22×C5#392
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC5×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#3255C2semidirect
4. C5×#1506C2semidirect
5. C5×#1307C2semidirect
21. #10216C5direct (N × H)
52. #3255C2×C5semidirect
55. #1506C2×C5semidirect
56. #1307C2×C5semidirect
232. C5#10216direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#32552
C5×#32352
C5×#19632
C5×#15062
C5×#13072
#102165