Finite Group Databasebeta

← #14511702

Subgroups — #14511702

Order 1920.

278 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 37 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
1SL(2,𝔽3)×D20C211yesmaximal, characteristic, complemented
2C5×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)⋊#168C211yesmaximal, characteristic, complemented
4#10601828—31nomaximal, Hall
5C2×D5×SL(2,𝔽3)—21no—
6C2×C5×GL(2,𝔽3)—21no—
7C4×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×#6652—41nomaximal
9#73088—51nomaximal, Hall
10Q8×D20D311yescharacteristic, complemented
11C5×#460—31no—
12#65361—31no—
13#65359—31no—
14Q8⋊#168—31no—
15#64703—31no—
16#64702—31no—
17C2×C5×D12—41no—
18C2×C3×D20—41no—
19C2×C5×SL(2,𝔽3)D411yescharacteristic
20D5×SL(2,𝔽3)—41no—
21C5×GL(2,𝔽3)—41no—
22C2×C3⋊#168—41no—
23#6652—44no—
24D4×SL(2,𝔽3)—51no—
25C4⋊GL(2,𝔽3)D511yescharacteristic, complemented
26SL(2,𝔽3)⋊C8—51no—
27C2×Q8×D5—21no—
28C2×C5×SD16—61no—
29C5×#127—31no—
30C4×C5×Q8C2×D311yescharacteristic
31C2×#686—61no—
32#4043—61no—
33C4×D20—31no—
34#4015—31no—
35C5×C4⋊C8—31no—
36C5×#102—61no—
37#3964—62no—
38#3964—61no—
39C4⋊#168—31no—
40C4×#168—31no—
41#1627—151noSylow
42C22×C5×D3—81no—
43C22×C3×D5—81no—
44C2×C3×C4×C5—41no—
45C5×D12—42no—
46C5×D12—81no—
47C3×D20—42no—
48C3×D20—81no—
49C5×SL(2,𝔽3)D811yescharacteristic, complemented
50C3⋊#168—42no—
51C22×SL(2,𝔽3)—101no—
52C2×GL(2,𝔽3)—21no—
53C2×#213—201no—
54C4×SL(2,𝔽3)C2×D511yescharacteristic
55C2×C5×Q8#6711yescharacteristic
56C2×C5×D4—61no—
57C2×C5×D4—61no—
58C2×C5×D4—61no—
59Q8×D5—41no—
60Q8×D5—121no—
61C2×D20S411yescharacteristic
62C2×C4×D5—61no—
63C2×Dic10—61no—
64C5×SD16—121no—
65C5×SD16—121no—
66C2×C5×C8—61no—
67C5×C4⋊C4—31no—
68C42×C5—31no—
69#686—122no—
70#685—61no—
71C4⋊Dic5—32no—
72C4⋊Dic5—31no—
73C2×#168—61no—
74C2×#168—61no—
75D4×Q8—51no—
76#486—151no—
77#460—31no—
78#459—151no—
79#332—151no—
80Q8⋊C8—151no—
81D4⋊C8—151no—
82C22×C3×C5—41no—
83C2×C5×D3—82no—
84C2×C5×D3—161no—
85C2×C5×D3—161no—
86C2×C3×D5—82no—
87C2×C3×D5—161no—
88C2×C3×D5—161no—
89C3×C4×C5—41no—
90C3×C4×C5—41no—
91C2×C3×D4—201no—
92C2×D12—41no—
93C2×SL(2,𝔽3)—201no—
94C2×SL(2,𝔽3)C22⋊D511yescharacteristic
95GL(2,𝔽3)—41no—
96#213—204no—
97C2×C3⋊C8—201no—
98C23×C5—121no—
99C22×D5—21no—
100C5×Q8—31no—
101C5×Q8#21311yescharacteristic, complemented
102C5×D4—121no—
103C5×D4—121no—
104C5×D4—122no—
105C5×D4—121no—
106C5×D4—121no—
107C2×C4×C5—31no—
108C2×C4×C5—31no—
109C2×C4×C5C2×S411yescharacteristic
110C2×Dic5—61no—
111D20—21no—
112D20GL(2,𝔽3)12yescomplemented
113C4×D5—121no—
114C4×D5—121no—
115Dic10—121no—
116Dic10—121no—
117C5×C8—121no—
118#168—121no—
119#168—121no—
120C22×Q8—101no—
121C2×SD16—61no—
122C2×D8—301no—
123C4⋊Q8—151no—
124#127—31no—
125C22⋊Q8—301no—
126C4×Q8D3×D511yescharacteristic
127C4×D4—151no—
128C4⋊C8—31no—
129C4⋊C8—151no—
130#102—61no—
131#102—301no—
132#102—302no—
133C4×C8—151no—
134C2×C3×C5—41no—
135C2×C3×C5—41no—
136C2×C3×C5—41no—
137C3×D5—162no—
138C5×D3—162no—
139C23×C3—401no—
140C22×D3—81no—
141C3×D4—202no—
142C3×D4—401no—
143C2×C3×C4—41no—
144D12—42no—
145D12—81no—
146SL(2,𝔽3)#68611yescharacteristic, complemented
147C3⋊C8—202no—
148C22×C5—241no—
149C22×C5—241no—
150C22×C5—241no—
151C22×C5GL(2,ℤ/4)11yescharacteristic
152C2×D5—22no—
153C2×D5—41no—
154C2×D5—41no—
155C4×C5—31no—
156C4×C5—31no—
157C4×C5—61no—
158C4×C5C2×GL(2,𝔽3)11yescharacteristic
159C4×C5#98311yescharacteristic
160Dic5—121no—
161C2×Q8—201no—
162C2×Q8—301no—
163C2×Q8—601no—
164C2×Q8#120111yescharacteristic
165C2×D4—51no—
166C2×D4—61no—
167C2×D4—61no—
168C2×D4—61no—
169C22×C4—301no—
170SD16—121no—
171SD16—121no—
172D8—602no—
173C2×C8—61no—
174C2×C8—301no—
175C2×C8—301no—
176C4⋊C4—31no—
177C4⋊C4—152no—
178C4⋊C4—151no—
179C22⋊C4—301no—
180C42—31no—
181C3×C5—41noHall
182C22×C3—41no—
183C22×C3—402no—
184C22×C3—801no—
185C22×C3—801no—
186C2×D3—82no—
187C2×D3—161no—
188C2×D3—161no—
189C3×C4—41no—
190C3×C4—41no—
191C2×C5—481no—
192C2×C5#541511yescharacteristic
193C2×C5#542111yescharacteristic
194C2×C5#543111yescharacteristic
195D5—42no—
196C23—101no—
197C23—121no—
198Q8—31no—
199Q8—601no—
200Q8—601no—
201Q8#665211yescharacteristic, complemented
202D4—52no—
203D4—101no—
204D4—121no—
205D4—121no—
206D4—122no—
207D4—121no—
208D4—121no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—301no—
212C2×C4—601no—
213C2×C4—601no—
214C2×C4D5×S411yescharacteristic
215C8—121no—
216C8—601no—
217C8—601no—
218C2×C3—41no—
219C2×C3—41no—
220C2×C3—41no—
221C2×C3—802no—
222D3—162no—
223C5#7308811yescharacteristic, complemented, Sylow
224C22—102no—
225C22—201no—
226C22—201no—
227C22—241no—
228C22—241no—
229C22—241no—
230C22#9190411yescharacteristic
231C4—31no—
232C4—31no—
233C4—61no—
234C4—601no—
235C4#9189611yescharacteristic
236C4D5×GL(2,𝔽3)11yescharacteristic
237C3—41noSylow
238C2—202no—
239C2—481no—
240C2#1175157911yescharacteristic
241C2#1175161311yescharacteristic
242C2#1175162511yescharacteristic
subgroupG/H
centerC22#91904
commutator, cosocleC4×C5×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D5×S4
FittingC4×C5×Q8C2×D3
socleC22×C5GL(2,ℤ/4)
hypocenterC5×SL(2,𝔽3)D8
LCSG ▷ C4×C5×SL(2,𝔽3) ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. SL(2,𝔽3)×D20C2semidirect
2. C5×C4⋊GL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)⋊#168C2semidirect
10. Q8×D20D3semidirect
25. C4⋊GL(2,𝔽3)D5semidirect
49. C5×SL(2,𝔽3)D8semidirect
101. C5×Q8#213semidirect
112. D20GL(2,𝔽3)semidirect
146. SL(2,𝔽3)#686semidirect
201. Q8#6652semidirect
223. C5#73088semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D202
C5×C4⋊GL(2,𝔽3)2
SL(2,𝔽3)⋊#1682
#106018283
C2×#66524
#730885