Finite Group Databasebeta

← C22⋊C4×SL(2,𝔽5) #14514949

Subgroups — C22⋊C4×SL(2,𝔽5) #14514949

Order 1920.

484 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 49 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
1C23×SL(2,𝔽5)C211yesmaximal, characteristic
2C2×C4×SL(2,𝔽5)C212yesmaximal, complemented
3C22×SL(2,𝔽5)—22no—
4C22×SL(2,𝔽5)C412yescomplemented
5C22×SL(2,𝔽5)C2211yescharacteristic
6C4×SL(2,𝔽5)—22no—
7C22⋊C4×SL(2,𝔽3)—51nomaximal, Hall
8C22⋊C4×Dic5—61nomaximal
9C2×SL(2,𝔽5)—22no—
10C2×SL(2,𝔽5)C2×C411yescharacteristic
11C2×SL(2,𝔽5)D412yes—
12C23×SL(2,𝔽3)—51no—
13C2×C4×SL(2,𝔽3)—52no—
14Dic3×C22⋊C4—101nomaximal
15C23×Dic5—61no—
16C2×C5×C22⋊C4—61no—
17C2×C22⋊Dic5—61no—
18C2×C4×Dic5—62no—
19#3963—62no—
20Q8×C22⋊C4—51noSylow
21SL(2,𝔽5)C22⋊C411yescharacteristic, direct product factor
22C23×Dic3—101no—
23C22×SL(2,𝔽3)—51no—
24C22×SL(2,𝔽3)—52no—
25C22×SL(2,𝔽3)—102no—
26C2×C3×C22⋊C4—101no—
27C2×#217—101no—
28C2×C4×Dic3—102no—
29C4×SL(2,𝔽3)—102no—
30C3⋊#95—102no—
31C24×C5—61no—
32C22×C4×C5—62no—
33C22×Dic5—61no—
34C22×Dic5—62no—
35C22×Dic5—62no—
36C22×Dic5—61no—
37C22×Dic5—62no—
38C22×Dic5—122no—
39C22×Dic5—122no—
40C5×C22⋊C4—64no—
41C22⋊Dic5—64no—
42C4×Dic5—122no—
43C4×Dic5—122no—
44C23×Q8—51no—
45C2×C4×Q8—52no—
46#391—152no—
47#380—151no—
48C4×C22⋊C4—151no—
49C24×C3—101no—
50C22×C3×C4—102no—
51C22×Dic3—101no—
52C22×Dic3—101no—
53C22×Dic3—102no—
54C22×Dic3—102no—
55C22×Dic3—102no—
56C22×Dic3—202no—
57C22×Dic3—202no—
58C2×SL(2,𝔽3)—51no—
59C2×SL(2,𝔽3)—52no—
60C2×SL(2,𝔽3)—102no—
61C3×C22⋊C4—104no—
62#217—104no—
63C4×Dic3—202no—
64C4×Dic3—202no—
65C23×C5—62no—
66C23×C5—62no—
67C23×C5—62no—
68C23×C5—61no—
69C23×C5—122no—
70C23×C5—122no—
71C2×C4×C5—64no—
72C2×C4×C5—122no—
73C2×C4×C5—122no—
74C2×Dic5—61no—
75C2×Dic5—62no—
76C2×Dic5—62no—
77C2×Dic5—61no—
78C2×Dic5—62no—
79C2×Dic5—64no—
80C2×Dic5—122no—
81C2×Dic5—124no—
82C2×Dic5—122no—
83C2×Dic5—122no—
84C2×Dic5—122no—
85C2×Dic5—122no—
86C22×Q8—51no—
87C22×Q8—52no—
88C22×Q8—102no—
89C22×Q8—152no—
90C22×Q8—151no—
91C22×Q8—302no—
92C23×C4—151no—
93C4×Q8—102no—
94C4×Q8—302no—
95C2×C4⋊C4—152no—
96C2×C22⋊C4—151no—
97C2×C22⋊C4A511yesmaximal normal, characteristic
98C2×C42—152no—
99#95—152no—
100#95—152no—
101C23×C3—102no—
102C23×C3—102no—
103C23×C3—102no—
104C23×C3—101no—
105C23×C3—202no—
106C23×C3—202no—
107C2×C3×C4—104no—
108C2×C3×C4—202no—
109C2×C3×C4—202no—
110C2×Dic3—101no—
111C2×Dic3—102no—
112C2×Dic3—102no—
113C2×Dic3—101no—
114C2×Dic3—102no—
115C2×Dic3—104no—
116C2×Dic3—202no—
117C2×Dic3—202no—
118C2×Dic3—204no—
119C2×Dic3—202no—
120C2×Dic3—202no—
121C2×Dic3—202no—
122SL(2,𝔽3)—51no—
123C22×C5—64no—
124C22×C5—61no—
125C22×C5—61no—
126C22×C5—62no—
127C22×C5—62no—
128C22×C5—61no—
129C22×C5—122no—
130C22×C5—122no—
131C22×C5—124no—
132C22×C5—124no—
133C4×C5—124no—
134Dic5—61no—
135Dic5—61no—
136Dic5—62no—
137Dic5—122no—
138Dic5—124no—
139C24C2×A511yescharacteristic
140C2×Q8—52no—
141C2×Q8—51no—
142C2×Q8—102no—
143C2×Q8—152no—
144C2×Q8—152no—
145C2×Q8—151no—
146C2×Q8—152no—
147C2×Q8—302no—
148C2×Q8—302no—
149C2×Q8—304no—
150C2×Q8—302no—
151C2×Q8—304no—
152C22×C4—151no—
153C22×C4—152no—
154C22×C4—152no—
155C22×C4—151no—
156C22×C4—152no—
157C22×C4—302no—
158C22×C4—302no—
159C22×C4C2×A512yes—
160C4⋊C4—302no—
161C4⋊C4—302no—
162C22⋊C4—302no—
163C22⋊C4SL(2,𝔽5)14yesdirect product factor
164C42—302no—
165C42—302no—
166C22×C3—104no—
167C22×C3—101no—
168C22×C3—101no—
169C22×C3—102no—
170C22×C3—102no—
171C22×C3—101no—
172C22×C3—202no—
173C22×C3—202no—
174C22×C3—204no—
175C22×C3—204no—
176C3×C4—204no—
177Dic3—101no—
178Dic3—102no—
179Dic3—101no—
180Dic3—202no—
181Dic3—204no—
182C2×C5—61no—
183C2×C5—61no—
184C2×C5—62no—
185C2×C5—61no—
186C2×C5—62no—
187C2×C5—124no—
188C23—22no—
189C23—22no—
190C23C4×A512yes—
191C23#673212yes—
192C23C2×SL(2,𝔽5)12yes—
193C23C22×A511yescharacteristic
194Q8—51no—
195Q8—152no—
196Q8—151no—
197Q8—152no—
198Q8—302no—
199Q8—302no—
200Q8—304no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—151no—
204C2×C4—151no—
205C2×C4—152no—
206C2×C4—152no—
207C2×C4—152no—
208C2×C4—302no—
209C2×C4—302no—
210C2×C4—304no—
211C2×C4—302no—
212C2×C4—302no—
213C2×C4—302no—
214C2×C4—302no—
215C2×C4C2×SL(2,𝔽5)14yescomplemented
216C2×C3—101no—
217C2×C3—101no—
218C2×C3—102no—
219C2×C3—101no—
220C2×C3—102no—
221C2×C3—204no—
222C5—61noSylow
223C22—22no—
224C22—22no—
225C22—24no—
226C22—24no—
227C22C4×SL(2,𝔽5)14yescomplemented
228C22C2×C4×A511yescharacteristic
229C22C2×#673211yescharacteristic
230C22D4×A512yes—
231C22#9188112yes—
232C22C22×SL(2,𝔽5)11yescharacteristic
233C4—24no—
234C4—151no—
235C4—151no—
236C4—152no—
237C4—302no—
238C4—602no—
239C3—101noSylow
240C2—24no—
241C2C22⋊C4×A511yescharacteristic
242C2#1175153411yescharacteristic
243C2C22⋊#673212yes—
244C2C2×C4×SL(2,𝔽5)11yescharacteristic
245C2D4×SL(2,𝔽5)12yes—
subgroupG/H
center, Frattini, socle, cosocleC23C22×A5
commutatorC2×SL(2,𝔽5)C2×C4
Fitting, hypercenterC2×C22⋊C4A5
prime radicalC23×SL(2,𝔽5)C2
hypocenterSL(2,𝔽5)C22⋊C4
LCSG ▷ C2×SL(2,𝔽5) ▷ SL(2,𝔽5)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×SL(2,𝔽5)C2semidirect
4. C22×SL(2,𝔽5)C4semidirect
21. SL(2,𝔽5)C22⋊C4direct (N × H)
163. C22⋊C4SL(2,𝔽5)direct (N × H)
215. C2×C4C2×SL(2,𝔽5)semidirect
227. C22C4×SL(2,𝔽5)semidirect

Maximal subgroups

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

subgroupindex
C23×SL(2,𝔽5)2
C2×C4×SL(2,𝔽5)2
C22⋊C4×SL(2,𝔽3)5
C22⋊C4×Dic56
Dic3×C22⋊C410