Finite Group Databasebeta

← C2×C5×C4⋊#226 #14511058

Subgroups — C2×C5×C4⋊#226 #14511058

Order 1920.

550 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C22×C5×#226C212yesmaximal
2C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic
3C5×C4⋊#226C214yesmaximal, direct product factor
4C2×C5×#462—31nomaximal, Hall
5C22×C5×SL(2,𝔽3)C2211yescharacteristic
6C2×C5×#226—24no—
7C2×C5×#226C2214yes—
8C2×C5×C4⋊Dic3—41nomaximal
9C4×C5×SL(2,𝔽3)C2212yes—
10C2×C4⋊#226C511yesmaximal, characteristic, direct product factor, Hall
11C22×C5×Q16—32no—
12C2×C5×C4⋊Q8—31no—
13C2×C4×C5×Q8D311yescharacteristic
14C5×#462—34no—
15C5×#462—34no—
16C2×C5×C4⋊C8—31no—
17C2×C5×#103—32no—
18C22×C3×C4×C5—41no—
19C22×C5×Dic3—42no—
20C2×C5×SL(2,𝔽3)D412yes—
21C2×C5×SL(2,𝔽3)C2311yescharacteristic
22C5×#226—24no—
23C5×C4⋊Dic3—44no—
24C22×#226C2×C512yes—
25C2×C4×SL(2,𝔽3)C2×C511yescharacteristic
26C4⋊#226C2×C514yesdirect product factor
27C22×C5×Q8—32no—
28C22×C5×Q8C2×D311yescharacteristic
29C2×C5×Q16—34no—
30C2×C5×Q16—34no—
31C2×C5×Q16—64no—
32C2×C5×Q16—64no—
33C22×C5×C8—32no—
34C5×C4⋊Q8—34no—
35C5×C4⋊Q8—62no—
36C4×C5×Q8—32no—
37C4×C5×Q8C2×D312yes—
38C2×C5×C4⋊C4—31no—
39C2×C5×C4⋊C4—61no—
40C2×C5×C4⋊C4—61no—
41C2×C42×C5—31no—
42C5×C4⋊C8—34no—
43C5×#103—38no—
44C2×#462—31noSylow
45C23×C3×C5—41no—
46C2×C3×C4×C5—44no—
47C2×C3×C4×C5—42no—
48C2×C5×Dic3—44no—
49C2×C5×Dic3—84no—
50C5×SL(2,𝔽3)C2×D411yescharacteristic
51C22×SL(2,𝔽3)C22×C511yescharacteristic
52C2×#226—24no—
53C2×#226C22×C514yes—
54C2×C4⋊Dic3—41no—
55C4×SL(2,𝔽3)C22×C512yes—
56C2×C5×Q8—31no—
57C2×C5×Q8—32no—
58C2×C5×Q8—34no—
59C2×C5×Q8—62no—
60C2×C5×Q8—64no—
61C2×C5×Q8—64no—
62C2×C5×Q8D1212yes—
63C2×C5×Q8C22×D311yescharacteristic
64C22×C4×C5—31no—
65C22×C4×C5—31no—
66C22×C4×C5—62no—
67C22×C4×C5S411yescharacteristic
68C5×Q16—68no—
69C5×Q16—64no—
70C5×Q16—64no—
71C2×C5×C8—34no—
72C2×C5×C8—64no—
73C5×C4⋊C4—32no—
74C5×C4⋊C4—61no—
75C5×C4⋊C4—64no—
76C5×C4⋊C4—64no—
77C42×C5—32no—
78C42×C5—32no—
79C22×Q16—32no—
80C2×C4⋊Q8—31no—
81C2×C4×Q8C5×D311yescharacteristic
82#462—34no—
83#462—34no—
84C2×C4⋊C8—31no—
85C2×#103—32no—
86C22×C3×C5—42no—
87C22×C3×C5—42no—
88C22×C3×C5—42no—
89C22×C3×C5—41no—
90C3×C4×C5—44no—
91C5×Dic3—84no—
92C22×C3×C4—41no—
93C22×Dic3—42no—
94C2×SL(2,𝔽3)C5×D412yes—
95C2×SL(2,𝔽3)C23×C511yescharacteristic
96#226—24no—
97C4⋊Dic3—44no—
98C23×C5C2×S411yescharacteristic
99C5×Q8—32no—
100C5×Q8—31no—
101C5×Q8—61no—
102C5×Q8—64no—
103C5×Q8—64no—
104C5×Q8—124no—
105C5×Q8C2×D1211yescharacteristic
106C2×C4×C5—32no—
107C2×C4×C5—32no—
108C2×C4×C5—31no—
109C2×C4×C5—31no—
110C2×C4×C5—32no—
111C2×C4×C5—62no—
112C2×C4×C5—64no—
113C2×C4×C5—124no—
114C2×C4×C5#22614yescomplemented
115C2×C4×C5C2×S412yes—
116C5×C8—64no—
117C22×Q8—32no—
118C22×Q8C2×C5×D311yescharacteristic
119C2×Q16—34no—
120C2×Q16—34no—
121C2×Q16—64no—
122C2×Q16—64no—
123C22×C8—32no—
124C4⋊Q8—34no—
125C4⋊Q8—62no—
126C4×Q8—32no—
127C4×Q8C2×C5×D312yes—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—61no—
131C2×C42—31no—
132C4⋊C8—34no—
133#103—38no—
134C2×C3×C5—44no—
135C2×C3×C5—41no—
136C2×C3×C5—41no—
137C2×C3×C5—41no—
138C23×C3—41no—
139C2×C3×C4—44no—
140C2×C3×C4—42no—
141C2×Dic3—44no—
142C2×Dic3—84no—
143SL(2,𝔽3)C2×C5×D411yescharacteristic
144C22×C5C4⋊S412yes—
145C22×C5C2×#22612yes—
146C22×C5#98412yes—
147C22×C5C22×S411yescharacteristic
148C4×C5—31no—
149C4×C5—31no—
150C4×C5—32no—
151C4×C5—62no—
152C4×C5—124no—
153C4×C5C2×#22614yescomplemented
154C2×Q8—31no—
155C2×Q8—32no—
156C2×Q8—34no—
157C2×Q8—62no—
158C2×Q8—64no—
159C2×Q8—64no—
160C2×Q8C5×D1212yes—
161C2×Q8C22×C5×D311yescharacteristic
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—62no—
165C22×C4C5×S411yescharacteristic
166Q16—68no—
167Q16—64no—
168Q16—64no—
169C2×C8—34no—
170C2×C8—64no—
171C4⋊C4—32no—
172C4⋊C4—61no—
173C4⋊C4—64no—
174C4⋊C4—64no—
175C42—32no—
176C42—32no—
177C3×C5—41noHall
178C22×C3—42no—
179C22×C3—42no—
180C22×C3—42no—
181C22×C3—41no—
182C3×C4—44no—
183Dic3—84no—
184C2×C5C4⋊#22614yesdirect product factor
185C2×C5C2×C4⋊S411yescharacteristic
186C2×C5C22×#22611yescharacteristic
187C2×C5C2×#98411yescharacteristic
188C23C2×C5×S411yescharacteristic
189Q8—32no—
190Q8—31no—
191Q8—61no—
192Q8—64no—
193Q8—64no—
194Q8—124no—
195Q8C2×C5×D1211yescharacteristic
196C2×C4—32no—
197C2×C4—32no—
198C2×C4—31no—
199C2×C4—31no—
200C2×C4—32no—
201C2×C4—62no—
202C2×C4—64no—
203C2×C4—124no—
204C2×C4C5×#22614yescomplemented
205C2×C4C2×C5×S412yes—
206C8—64no—
207C2×C3—41no—
208C2×C3—41no—
209C2×C3—41no—
210C2×C3—44no—
211C5C2×C4⋊#22611yescharacteristic, direct product factor, Sylow
212C22C5×C4⋊S412yes—
213C22C2×C5×#22612yes—
214C22C5×#98412yes—
215C22C22×C5×S411yescharacteristic
216C4—31no—
217C4—31no—
218C4—32no—
219C4—62no—
220C4—124no—
221C4C2×C5×#22614yescomplemented
222C3—41noSylow
223C2C5×C4⋊#22614yesdirect product factor
224C2C2×C5×C4⋊S411yescharacteristic
225C2C22×C5×#22611yescharacteristic
226C2C2×C5×#98411yescharacteristic
subgroupG/H
center, socleC23×C5C2×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC2×C4×C5×Q8D3
prime radicalC22×C5×SL(2,𝔽3)C22
hypercenterC22×C4×C5S4
hypocenterSL(2,𝔽3)C2×C5×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
3. C5×C4⋊#226C2direct (N × H)
10. C2×C4⋊#226C5direct (N × H)
26. C4⋊#226C2×C5direct (N × H)
114. C2×C4×C5#226semidirect
153. C4×C5C2×#226semidirect
184. C2×C5C4⋊#226direct (N × H)
204. C2×C4C5×#226semidirect
211. C5C2×C4⋊#226direct (N × H)
221. C4C2×C5×#226semidirect
223. C2C5×C4⋊#226direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#2262
C2×C4×C5×SL(2,𝔽3)2
C5×C4⋊#2262
C2×C5×#4623
C2×C5×C4⋊Dic34
C2×C4⋊#2265