Finite Group Databasebeta

← C2×C5×#557⋊C3 #14511336

Subgroups — C2×C5×#557⋊C3 #14511336

Order 1920.

566 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 152 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×#557⋊C3C214yesmaximal, direct product factor
2C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×C4×C5×SL(2,𝔽3)C212yesmaximal
4C2×C5×#557C311yesmaximal, characteristic, complemented, Hall
5C22×C5×SL(2,𝔽3)C2211yescharacteristic
6C2×C3×C5×C4⋊C4—41nomaximal
7C4×C5×SL(2,𝔽3)C2212yes—
8C4×C5×SL(2,𝔽3)C2214yes—
9C2×#557⋊C3C511yesmaximal, characteristic, direct product factor, Hall
10C5×#557—34no—
11C5×#557C2×C314yescomplemented
12C2×C5×C4⋊Q8—31no—
13C2×C5×#125—32no—
14C2×C4×C5×Q8—31no—
15C2×C4×C5×Q8C2×C312yes—
16C2×C4×C5×Q8C2×C311yescharacteristic
17C22×C3×C4×C5—41no—
18C22×C3×C4×C5—42no—
19C2×C5×SL(2,𝔽3)Q812yescomplemented
20C2×C5×SL(2,𝔽3)C2311yescharacteristic
21C3×C5×C4⋊C4—44no—
22#557⋊C3C2×C514yesdirect product factor
23C2×C4×SL(2,𝔽3)C2×C511yescharacteristic
24C2×C4×SL(2,𝔽3)C2×C512yes—
25C22×C5×Q8—31no—
26C22×C5×Q8C22×C311yescharacteristic
27C5×C4⋊Q8—34no—
28C5×C4⋊Q8—34no—
29C5×#125—38no—
30C5×#125—38no—
31C4×C5×Q8—34no—
32C4×C5×Q8—34no—
33C4×C5×Q8—34no—
34C4×C5×Q8—32no—
35C4×C5×Q8C22×C314yes—
36C4×C5×Q8C22×C312yes—
37C2×C5×C4⋊C4—31no—
38C2×C5×C4⋊C4—32no—
39C2×C5×C4⋊C4—32no—
40C2×C5×C4⋊C4—32no—
41C2×C5×C4⋊C4A411yescharacteristic
42C2×C42×C5—32no—
43C2×C42×C5—31no—
44C2×#557C3×C511yescharacteristic, complemented, Sylow
45C23×C3×C5—41no—
46C2×C3×C4×C5—44no—
47C2×C3×C4×C5—42no—
48C2×C3×C4×C5—44no—
49C2×C3×C4×C5—84no—
50C5×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
51C22×SL(2,𝔽3)C22×C511yescharacteristic
52C2×C3×C4⋊C4—41no—
53C4×SL(2,𝔽3)C22×C512yes—
54C4×SL(2,𝔽3)C22×C514yes—
55C2×C5×Q8—34no—
56C2×C5×Q8—31no—
57C2×C5×Q8—32no—
58C2×C5×Q8—64no—
59C2×C5×Q8C3×Q812yes—
60C2×C5×Q8C23×C311yescharacteristic
61C22×C4×C5—31no—
62C22×C4×C5—31no—
63C22×C4×C5—32no—
64C22×C4×C5C2×A412yes—
65C22×C4×C5C2×A411yescharacteristic
66C5×C4⋊C4—32no—
67C5×C4⋊C4—34no—
68C5×C4⋊C4—32no—
69C5×C4⋊C4—34no—
70C5×C4⋊C4—38no—
71C5×C4⋊C4—38no—
72C5×C4⋊C4C2×A414yes—
73C42×C5—34no—
74C42×C5—32no—
75C42×C5—34no—
76C42×C5—32no—
77#557—34no—
78#557C2×C3×C514yescomplemented
79C2×C4⋊Q8—31no—
80C2×#125—32no—
81C2×C4×Q8—31no—
82C2×C4×Q8C2×C3×C512yes—
83C2×C4×Q8C2×C3×C511yescharacteristic
84C22×C3×C5—42no—
85C22×C3×C5—42no—
86C22×C3×C5—42no—
87C22×C3×C5—41no—
88C3×C4×C5—44no—
89C3×C4×C5—84no—
90C22×C3×C4—41no—
91C22×C3×C4—42no—
92C2×SL(2,𝔽3)C5×Q812yescomplemented
93C2×SL(2,𝔽3)C23×C511yescharacteristic
94C3×C4⋊C4—44no—
95C23×C5C22×A411yescharacteristic
96C5×Q8—32no—
97C5×Q8—31no—
98C5×Q8—32no—
99C5×Q8—68no—
100C5×Q8C2×C3×Q811yescharacteristic
101C2×C4×C5—24no—
102C2×C4×C5—32no—
103C2×C4×C5—32no—
104C2×C4×C5—31no—
105C2×C4×C5—31no—
106C2×C4×C5—32no—
107C2×C4×C5—34no—
108C2×C4×C5—64no—
109C2×C4×C5—62no—
110C2×C4×C5#41⋊C314yes—
111C2×C4×C5C22×A414yes—
112C2×C4×C5C22×A412yes—
113C22×Q8—31no—
114C22×Q8C22×C3×C511yescharacteristic
115C4⋊Q8—34no—
116C4⋊Q8—34no—
117#125—38no—
118#125—38no—
119C4×Q8—34no—
120C4×Q8—34no—
121C4×Q8—34no—
122C4×Q8—32no—
123C4×Q8C22×C3×C514yes—
124C4×Q8C22×C3×C512yes—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—32no—
127C2×C4⋊C4—32no—
128C2×C4⋊C4—32no—
129C2×C4⋊C4C5×A411yescharacteristic
130C2×C42—32no—
131C2×C42—31no—
132C2×C3×C5—41no—
133C2×C3×C5—41no—
134C2×C3×C5—41no—
135C2×C3×C5—44no—
136C23×C3—41no—
137C2×C3×C4—44no—
138C2×C3×C4—42no—
139C2×C3×C4—44no—
140C2×C3×C4—84no—
141SL(2,𝔽3)C2×C5×Q811yescharacteristic, complemented
142C22×C5Q8×A412yes—
143C22×C5C2×#41⋊C312yes—
144C22×C52−1+4⋊C312yes—
145C22×C5C23×A411yescharacteristic
146C4×C5—24no—
147C4×C5—31no—
148C4×C5—32no—
149C4×C5—31no—
150C4×C5—64no—
151C4×C5—62no—
152C4×C5C2×#41⋊C314yes—
153C2×Q8—34no—
154C2×Q8—31no—
155C2×Q8—32no—
156C2×Q8—64no—
157C2×Q8C3×C5×Q812yes—
158C2×Q8C23×C3×C511yescharacteristic
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—32no—
162C22×C4C2×C5×A412yes—
163C22×C4C2×C5×A411yescharacteristic
164C4⋊C4—32no—
165C4⋊C4—34no—
166C4⋊C4—32no—
167C4⋊C4—34no—
168C4⋊C4—38no—
169C4⋊C4—38no—
170C4⋊C4C2×C5×A414yes—
171C42—34no—
172C42—32no—
173C42—34no—
174C42—32no—
175C3×C5—41noHall
176C22×C3—42no—
177C22×C3—42no—
178C22×C3—42no—
179C22×C3—41no—
180C3×C4—44no—
181C3×C4—84no—
182C2×C5#557⋊C314yesdirect product factor
183C2×C5C2×Q8×A411yescharacteristic
184C2×C5C22×#41⋊C311yescharacteristic
185C2×C5C2×2−1+4⋊C311yescharacteristic
186C23C22×C5×A411yescharacteristic
187Q8—32no—
188Q8—31no—
189Q8—32no—
190Q8—68no—
191Q8C2×C3×C5×Q811yescharacteristic
192C2×C4—24no—
193C2×C4—32no—
194C2×C4—32no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—32no—
198C2×C4—34no—
199C2×C4—64no—
200C2×C4—62no—
201C2×C4C5×#41⋊C314yes—
202C2×C4C22×C5×A414yes—
203C2×C4C22×C5×A412yes—
204C2×C3—41no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3—44no—
208C5C2×#557⋊C311yescharacteristic, direct product factor, Sylow
209C22C5×Q8×A412yes—
210C22C2×C5×#41⋊C312yes—
211C22C5×2−1+4⋊C312yes—
212C22C23×C5×A411yescharacteristic
213C4—24no—
214C4—31no—
215C4—32no—
216C4—31no—
217C4—64no—
218C4—62no—
219C4C2×C5×#41⋊C314yes—
220C3—41noSylow
221C2C5×#557⋊C314yesdirect product factor
222C2C2×C5×Q8×A411yescharacteristic
223C2C22×C5×#41⋊C311yescharacteristic
224C2C2×C5×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC23×C5C22×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC2×C5×#557C3
prime radicalC22×C5×SL(2,𝔽3)C22
hypercenterC2×C5×C4⋊C4A4
hypocenterQ8C2×C3×C5×Q8
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23×C5 ◁ C2×C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C5×#557⋊C3C2direct (N × H)
4. C2×C5×#557C3semidirect
9. C2×#557⋊C3C5direct (N × H)
11. C5×#557C2×C3semidirect
19. C2×C5×SL(2,𝔽3)Q8semidirect
22. #557⋊C3C2×C5direct (N × H)
44. C2×#557C3×C5semidirect
50. C5×SL(2,𝔽3)C2×Q8semidirect
78. #557C2×C3×C5semidirect
92. C2×SL(2,𝔽3)C5×Q8semidirect
141. SL(2,𝔽3)C2×C5×Q8semidirect
182. C2×C5#557⋊C3direct (N × H)
208. C5C2×#557⋊C3direct (N × H)
221. C2C5×#557⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#557⋊C32
C2×C4×C5×SL(2,𝔽3)2
C2×C5×#5573
C2×C3×C5×C4⋊C44
C2×#557⋊C35