Finite Group Databasebeta

← C2×C5×#2371 #11956866

Subgroups — C2×C5×#2371 #11956866

Order 1280.

2422 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#125C211yesmaximal, characteristic
2C5×#2371C2116yesmaximal, direct product factor
3C2×C5×#387C214yesmaximal
4C2×C5×#387C214yesmaximal
5C2×C4×C5×C4⋊C4C214yesmaximal
6C2×C5×#376C212yesmaximal
7C2×C5×#125C4116yescomplemented
8C2×C5×#125C2218yes—
9C22×C5×C4⋊C4C2214yes—
10C22×C5×C4⋊C4C2214yes—
11C22×C5×C4⋊C4C2212yes—
12C22×C42×C5C2214yes—
13C22×C42×C5C2214yes—
14C22×C42×C5C2211yescharacteristic
15C5×#387C22132yes—
16C5×#387C22132yes—
17C4×C5×C4⋊C4C22132yes—
18C5×#376C22116yes—
19C2×C5×#95C2212yes—
20C2×C5×#95C2212yes—
21C2×C5×#95C2214yes—
22C2×C5×#95C2214yes—
23C2×C5×#95C2214yes—
24C2×#2371C511yesmaximal, characteristic, direct product factor, Sylow
25C23×C4×C5C2314yes—
26C23×C4×C5C2311yescharacteristic
27C23×C4×C5C2312yes—
28C23×C4×C5C2314yes—
29C23×C4×C5C2314yes—
30C5×#125C2×C4164yescomplemented
31C2×C5×C4⋊C4—216no—
32C2×C5×C4⋊C4C2×C4132yes—
33C2×C5×C4⋊C4C2×C4116yes—
34C2×C5×C4⋊C4C23116yes—
35C2×C5×C4⋊C4C23116yes—
36C2×C5×C4⋊C4C2318yes—
37C2×C42×C5—216no—
38C2×C42×C5—216no—
39C2×C42×C5C2×C418yes—
40C2×C42×C5C23116yes—
41C2×C42×C5C23116yes—
42C2×C42×C5C2314yes—
43C5×#95C2318yes—
44C5×#95C2318yes—
45C5×#95C23116yes—
46C5×#95C23116yes—
47C5×#95C23116yes—
48C22×#125C2×C511yescharacteristic
49#2371C2×C5116yesdirect product factor
50C2×#387C2×C514yes—
51C2×#387C2×C514yes—
52C2×C4×C4⋊C4C2×C514yes—
53C2×#376C2×C512yes—
54C24×C5C2411yescharacteristic
55C22×C4×C5—28no—
56C22×C4×C5—28no—
57C22×C4×C5—28no—
58C22×C4×C5—28no—
59C22×C4×C5—24no—
60C22×C4×C5—28no—
61C22×C4×C5—28no—
62C22×C4×C5—28no—
63C22×C4×C5—28no—
64C22×C4×C5C22×C418yes—
65C22×C4×C5C22×C414yes—
66C22×C4×C5C22×C4116yes—
67C22×C4×C5#41116yes—
68C22×C4×C5C2418yes—
69C22×C4×C5C2414yes—
70C22×C4×C5C2418yes—
71C22×C4×C5C2412yes—
72C22×C4×C5C2418yes—
73C5×C4⋊C4—232no—
74C5×C4⋊C4C22×C4164yes—
75C5×C4⋊C4C22×C4132yes—
76C42×C5—232no—
77C42×C5—232no—
78C42×C5C22×C4116yes—
79C2×#125C4×C5116yescomplemented
80C2×#125C22×C518yes—
81C22×C4⋊C4C22×C514yes—
82C22×C4⋊C4C22×C514yes—
83C22×C4⋊C4C22×C512yes—
84C22×C42C22×C514yes—
85C22×C42C22×C514yes—
86C22×C42C22×C511yescharacteristic
87#387C22×C5132yes—
88#387C22×C5132yes—
89C4×C4⋊C4C22×C5132yes—
90#376C22×C5116yes—
91C2×#95C22×C512yes—
92C2×#95C22×C512yes—
93C2×#95C22×C514yes—
94C2×#95C22×C514yes—
95C2×#95C22×C514yes—
96C23×C5C23×C412yes—
97C23×C5C2×#4118yes—
98C23×C52−1+414yes—
99C23×C5C2511yescharacteristic
100C2×C4×C5—216no—
101C2×C4×C5—232no—
102C2×C4×C5—28no—
103C2×C4×C5—232no—
104C2×C4×C5—28no—
105C2×C4×C5—28no—
106C2×C4×C5—28no—
107C2×C4×C5—28no—
108C2×C4×C5—24no—
109C2×C4×C5—28no—
110C2×C4×C5—28no—
111C2×C4×C5—28no—
112C2×C4×C5—28no—
113C2×C4×C5—416no—
114C2×C4×C5C23×C418yes—
115C2×C4×C5C23×C414yes—
116C2×C4×C5C23×C4116yes—
117C2×C4×C5C2×#41116yes—
118C23×C4C23×C514yes—
119C23×C4C23×C514yes—
120C23×C4C23×C511yescharacteristic
121C23×C4C23×C512yes—
122C23×C4C23×C514yes—
123#125C2×C4×C5164yescomplemented
124C2×C4⋊C4—216no—
125C2×C4⋊C4C2×C4×C5132yes—
126C2×C4⋊C4C2×C4×C5116yes—
127C2×C4⋊C4C23×C5116yes—
128C2×C4⋊C4C23×C5116yes—
129C2×C4⋊C4C23×C518yes—
130C2×C42—216no—
131C2×C42—216no—
132C2×C42C2×C4×C518yes—
133C2×C42C23×C5116yes—
134C2×C42C23×C5116yes—
135C2×C42C23×C514yes—
136#95C23×C518yes—
137#95C23×C518yes—
138#95C23×C5116yes—
139#95C23×C5116yes—
140#95C23×C5116yes—
141C22×C5C4×#4118yes—
142C22×C5#51914yes—
143C22×C5#552116yes—
144C22×C5C24×C411yescharacteristic
145C22×C5C22×#4114yes—
146C22×C5C2×2−1+412yes—
147C4×C5—28no—
148C4×C5—216no—
149C4×C5—24no—
150C4×C5—216no—
151C4×C5—48no—
152C24C24×C511yescharacteristic
153C22×C4—28no—
154C22×C4—28no—
155C22×C4—28no—
156C22×C4—28no—
157C22×C4—24no—
158C22×C4—28no—
159C22×C4—28no—
160C22×C4—28no—
161C22×C4—28no—
162C22×C4C22×C4×C518yes—
163C22×C4C22×C4×C514yes—
164C22×C4C22×C4×C5116yes—
165C22×C4C5×#41116yes—
166C22×C4C24×C518yes—
167C22×C4C24×C512yes—
168C22×C4C24×C514yes—
169C22×C4C24×C518yes—
170C22×C4C24×C518yes—
171C4⋊C4—232no—
172C4⋊C4C22×C4×C5164yes—
173C4⋊C4C22×C4×C5132yes—
174C42—232no—
175C42—232no—
176C42C22×C4×C5116yes—
177C2×C5#237118yesdirect product factor
178C2×C5C2×C4×#4112yes—
179C2×C5C2×#51911yescharacteristic
180C2×C5C2×#55214yes—
181C23C23×C4×C512yes—
182C23C2×C5×#4118yes—
183C23C5×2−1+414yes—
184C23C25×C511yescharacteristic
185C2×C4—216no—
186C2×C4—232no—
187C2×C4—28no—
188C2×C4—232no—
189C2×C4—28no—
190C2×C4—28no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—24no—
194C2×C4—28no—
195C2×C4—28no—
196C2×C4—28no—
197C2×C4—28no—
198C2×C4—416no—
199C2×C4C23×C4×C518yes—
200C2×C4C23×C4×C514yes—
201C2×C4C23×C4×C5116yes—
202C2×C4C2×C5×#41116yes—
203C5C2×#237111yescharacteristic, direct product factor, Sylow
204C22C4×C5×#4118yes—
205C22C5×#51914yes—
206C22C5×#552116yes—
207C22C24×C4×C511yescharacteristic
208C22C22×C5×#4114yes—
209C22C2×C5×2−1+412yes—
210C4—28no—
211C4—216no—
212C4—24no—
213C4—216no—
214C4—48no—
215C2C5×#237118yesdirect product factor
216C2C2×C4×C5×#4112yes—
217C2C2×C5×#51911yescharacteristic
218C2C2×C5×#55214yes—
subgroupG/H
center, socle, prime radicalC24×C5C24
commutatorC22C24×C4×C5
Frattini, cosocleC23C25×C5
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2371C2direct (N × H)
7. C2×C5×#125C4semidirect
24. C2×#2371C5direct (N × H)
30. C5×#125C2×C4semidirect
49. #2371C2×C5direct (N × H)
79. C2×#125C4×C5semidirect
123. #125C2×C4×C5semidirect
177. C2×C5#2371direct (N × H)
203. C5C2×#2371direct (N × H)
215. C2C5×#2371direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#1252
C5×#23712
C2×C5×#3872
C2×C4×C5×C4⋊C42
C2×C5×#3762
C2×#23715