Finite Group Databasebeta

← C2×C4×C5×#226 #14511053

Subgroups — C2×C4×C5×#226 #14511053

Order 1920.

574 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 124 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×#226C211yesmaximal, characteristic
2C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×C5×#859C211yesmaximal, characteristic
4C4×C5×#226C214yesmaximal, direct product factor
5C2×C4×C5×Q16—31nomaximal, Hall
6C22×C5×SL(2,𝔽3)C2211yescharacteristic
7C2×C5×#226C414yesdirect product factor
8C2×C5×#226C2212yes—
9C2×C4×C5×Dic3—41nomaximal
10C4×C5×SL(2,𝔽3)C2212yes—
11C5×#859C2212yes—
12C2×C4×#226C511yesmaximal, characteristic, direct product factor, Hall
13C22×C5×Q16—31no—
14C2×C4×C5×Q8—31no—
15C2×C4×C5×Q8D311yescharacteristic
16C4×C5×Q16—34no—
17C4×C5×Q16—34no—
18C2×C5×#107—31no—
19C2×C5×#103—31no—
20C2×C5×#103—31no—
21C2×C4×C5×C8—31no—
22C22×C3×C4×C5—41no—
23C22×C5×Dic3—41no—
24C22×C5×Dic3—41no—
25C2×C5×SL(2,𝔽3)C2×C412yes—
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C5×#226C2×C414yesdirect product factor
28C4×C5×Dic3—44no—
29C22×#226C2×C511yescharacteristic
30C2×C4×SL(2,𝔽3)C2×C511yescharacteristic
31C2×#859C2×C511yescharacteristic
32C4×#226C2×C514yesdirect product factor
33C22×C5×Q8—31no—
34C22×C5×Q8C2×D311yescharacteristic
35C2×C5×Q16—32no—
36C2×C5×Q16—34no—
37C2×C5×Q16—34no—
38C2×C5×Q16—32no—
39C22×C5×C8—31no—
40C22×C5×C8—31no—
41C4×C5×Q8—32no—
42C4×C5×Q8—34no—
43C4×C5×Q8—62no—
44C4×C5×Q8C2×D312yes—
45C2×C5×C4⋊C4—31no—
46C2×C5×C4⋊C4—31no—
47C2×C5×C4⋊C4—61no—
48C2×C42×C5—31no—
49C2×C42×C5—61no—
50C5×#107—34no—
51C5×#103—34no—
52C5×#103—32no—
53C5×#103—32no—
54C4×C5×C8—34no—
55C2×C4×Q16—31noSylow
56C23×C3×C5—41no—
57C2×C3×C4×C5—44no—
58C2×C3×C4×C5—42no—
59C2×C5×Dic3—44no—
60C2×C5×Dic3—42no—
61C2×C5×Dic3—42no—
62C2×C5×Dic3—44no—
63C5×SL(2,𝔽3)C22×C411yescharacteristic
64C22×SL(2,𝔽3)C22×C511yescharacteristic
65C2×#226C4×C514yesdirect product factor
66C2×#226C22×C512yes—
67C2×C4×Dic3—41no—
68C4×SL(2,𝔽3)C22×C512yes—
69#859C22×C512yes—
70C2×C5×Q8—31no—
71C2×C5×Q8—32no—
72C2×C5×Q8—32no—
73C2×C5×Q8—34no—
74C2×C5×Q8—61no—
75C2×C5×Q8—62no—
76C2×C5×Q8C4×D312yes—
77C2×C5×Q8C22×D311yescharacteristic
78C22×C4×C5—31no—
79C22×C4×C5—31no—
80C22×C4×C5—61no—
81C22×C4×C5—61no—
82C22×C4×C5S411yescharacteristic
83C5×Q16—38no—
84C5×Q16—34no—
85C5×Q16—34no—
86C2×C5×C8—34no—
87C2×C5×C8—32no—
88C2×C5×C8—32no—
89C2×C5×C8—62no—
90C5×C4⋊C4—32no—
91C5×C4⋊C4—32no—
92C5×C4⋊C4—61no—
93C5×C4⋊C4—61no—
94C5×C4⋊C4—64no—
95C42×C5—32no—
96C42×C5—32no—
97C42×C5—64no—
98C22×Q16—31no—
99C2×C4×Q8—31no—
100C2×C4×Q8C5×D311yescharacteristic
101C4×Q16—34no—
102C4×Q16—34no—
103C2×#107—31no—
104C2×#103—31no—
105C2×#103—31no—
106C2×C4×C8—31no—
107C22×C3×C5—42no—
108C22×C3×C5—42no—
109C22×C3×C5—42no—
110C22×C3×C5—41no—
111C3×C4×C5—44no—
112C5×Dic3—44no—
113C5×Dic3—44no—
114C22×C3×C4—41no—
115C22×Dic3—41no—
116C22×Dic3—41no—
117C2×SL(2,𝔽3)C2×C4×C512yes—
118C2×SL(2,𝔽3)C23×C511yescharacteristic
119#226C2×C4×C514yesdirect product factor
120C4×Dic3—44no—
121C23×C5C2×S411yescharacteristic
122C5×Q8—32no—
123C5×Q8—31no—
124C5×Q8—34no—
125C5×Q8—61no—
126C5×Q8—62no—
127C5×Q8—64no—
128C5×Q8C2×C4×D311yescharacteristic
129C2×C4×C5—32no—
130C2×C4×C5—32no—
131C2×C4×C5—31no—
132C2×C4×C5—31no—
133C2×C4×C5—32no—
134C2×C4×C5—64no—
135C2×C4×C5—62no—
136C2×C4×C5—62no—
137C2×C4×C5—62no—
138C2×C4×C5—122no—
139C2×C4×C5#22614yesdirect product factor
140C2×C4×C5C2×S412yes—
141C5×C8—34no—
142C5×C8—62no—
143C22×Q8—31no—
144C22×Q8C2×C5×D311yescharacteristic
145C2×Q16—34no—
146C2×Q16—32no—
147C2×Q16—34no—
148C2×Q16—32no—
149C22×C8—31no—
150C22×C8—31no—
151C4×Q8—32no—
152C4×Q8—34no—
153C4×Q8—62no—
154C4×Q8C2×C5×D312yes—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—61no—
158C2×C42—31no—
159C2×C42—61no—
160#107—34no—
161#103—34no—
162#103—32no—
163#103—32no—
164C4×C8—34no—
165C2×C3×C5—44no—
166C2×C3×C5—41no—
167C2×C3×C5—41no—
168C2×C3×C5—41no—
169C23×C3—41no—
170C2×C3×C4—44no—
171C2×C3×C4—42no—
172C2×Dic3—42no—
173C2×Dic3—42no—
174C2×Dic3—44no—
175C2×Dic3—44no—
176SL(2,𝔽3)C22×C4×C511yescharacteristic
177C22×C5C4×S412yes—
178C22×C5C2×#22612yes—
179C22×C5#98512yes—
180C22×C5C22×S411yescharacteristic
181C4×C5—31no—
182C4×C5—31no—
183C4×C5—32no—
184C4×C5—64no—
185C4×C5—62no—
186C4×C5—122no—
187C4×C5C2×#22614yesdirect product factor
188C2×Q8—31no—
189C2×Q8—32no—
190C2×Q8—32no—
191C2×Q8—34no—
192C2×Q8—61no—
193C2×Q8—62no—
194C2×Q8C4×C5×D312yes—
195C2×Q8C22×C5×D311yescharacteristic
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4C5×S411yescharacteristic
201Q16—38no—
202Q16—34no—
203Q16—34no—
204C2×C8—32no—
205C2×C8—34no—
206C2×C8—32no—
207C2×C8—62no—
208C4⋊C4—32no—
209C4⋊C4—32no—
210C4⋊C4—61no—
211C4⋊C4—61no—
212C4⋊C4—64no—
213C42—32no—
214C42—32no—
215C42—64no—
216C3×C5—41noHall
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—41no—
221C3×C4—44no—
222Dic3—44no—
223Dic3—44no—
224C2×C5C4×#22614yesdirect product factor
225C2×C5C2×C4×S411yescharacteristic
226C2×C5C22×#22611yescharacteristic
227C2×C5C2×#98511yescharacteristic
228C23C2×C5×S411yescharacteristic
229Q8—32no—
230Q8—31no—
231Q8—34no—
232Q8—61no—
233Q8—62no—
234Q8—64no—
235Q8C2×C4×C5×D311yescharacteristic
236C2×C4—32no—
237C2×C4—32no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—32no—
241C2×C4—64no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—122no—
246C2×C4C5×#22614yesdirect product factor
247C2×C4C2×C5×S412yes—
248C8—34no—
249C8—62no—
250C2×C3—41no—
251C2×C3—41no—
252C2×C3—41no—
253C2×C3—44no—
254C5C2×C4×#22611yescharacteristic, direct product factor, Sylow
255C22C4×C5×S412yes—
256C22C2×C5×#22612yes—
257C22C5×#98512yes—
258C22C22×C5×S411yescharacteristic
259C4—31no—
260C4—31no—
261C4—32no—
262C4—64no—
263C4—62no—
264C4—122no—
265C4C2×C5×#22614yesdirect product factor
266C3—41noSylow
267C2C4×C5×#22614yesdirect product factor
268C2C2×C4×C5×S411yescharacteristic
269C2C22×C5×#22611yescharacteristic
270C2C2×C5×#98511yescharacteristic
subgroupG/H
center, hypercenterC22×C4×C5S4
commutator, hypocenterSL(2,𝔽3)C22×C4×C5
FrattiniC22C22×C5×S4
FittingC2×C4×C5×Q8D3
socleC23×C5C2×S4
prime radicalC22×C5×SL(2,𝔽3)C22
cosocleC2×SL(2,𝔽3)C23×C5
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
4. C4×C5×#226C2direct (N × H)
7. C2×C5×#226C4direct (N × H)
12. C2×C4×#226C5direct (N × H)
27. C5×#226C2×C4direct (N × H)
32. C4×#226C2×C5direct (N × H)
65. C2×#226C4×C5direct (N × H)
119. #226C2×C4×C5direct (N × H)
139. C2×C4×C5#226direct (N × H)
187. C4×C5C2×#226direct (N × H)
224. C2×C5C4×#226direct (N × H)
246. C2×C4C5×#226direct (N × H)
254. C5C2×C4×#226direct (N × H)
265. C4C2×C5×#226direct (N × H)
267. C2C4×C5×#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
C2×C5×#8592
C4×C5×#2262
C2×C4×C5×Q163
C2×C4×C5×Dic34
C2×C4×#2265