Finite Group Databasebeta

← #14511701

Subgroups — #14511701

Order 1920.

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

normal subgroups only

This group has 260 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
1C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)⋊#168C211yesmaximal, characteristic, complemented
3C5×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#10599029—31nomaximal, Hall
5C2×D5×SL(2,𝔽3)C411yescharacteristic
6C4×C5×SL(2,𝔽3)C2211yescharacteristic
7C2×#6649—41nomaximal
8Dic5×SL(2,𝔽3)C411yescharacteristic
9#73055—51nomaximal, Hall
10C4×Q8×D5D311yescharacteristic
11#65126—31no—
12#64974—31no—
13C5×Q8⋊C8—31no—
14Q8⋊#168—31no—
15#64704—31no—
16#64676—31no—
17C2×C3×C4×D5—41no—
18C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
19D5×SL(2,𝔽3)—21no—
20C2×C3⋊#168—41no—
21C2×C5×C3⋊C8—41no—
22#6649—44no—
23C2×C4×SL(2,𝔽3)—51no—
24SL(2,𝔽3)⋊C8—51no—
25SL(2,𝔽3)⋊C8D511yescharacteristic, complemented
26C2×Q8×D5Dic311yescharacteristic
27C4×C5×Q8C2×D311yescharacteristic
28Q8×Dic5Dic311yescharacteristic
29C2×#676—61no—
30D5×C4⋊C4—31no—
31#4033—31no—
32C42×D5—31no—
33C4×Dic10—31no—
34C5×C4⋊C8—31no—
35C4×C5×C8—31no—
36#3952—61no—
37#3946—61no—
38#3946—61no—
39C4⋊#168—31no—
40C4×#168—31no—
41#1487—151noSylow
42C22×C3×D5—41no—
43C2×C3×C4×C5—41no—
44C2×C3×Dic5—41no—
45C3×C4×D5—42no—
46C3×C4×D5—42no—
47C5×SL(2,𝔽3)M1611yescharacteristic, complemented
48C3⋊#168—42no—
49C5×C3⋊C8—42no—
50C22×SL(2,𝔽3)—51no—
51C2×#208—201no—
52C4×SL(2,𝔽3)—51no—
53C4×SL(2,𝔽3)C2×D511yescharacteristic
54C2×C5×Q8C2×Dic311yescharacteristic
55Q8×D5—21no—
56Q8×D5—61no—
57C2×C4×D5—31no—
58C2×C4×D5—31no—
59C2×C4×D5S411yescharacteristic
60C2×Dic10—31no—
61C2×C5×C8—61no—
62C2×C5×C8—61no—
63C5×C4⋊C4—31no—
64C42×C5—31no—
65C4⋊Dic5—31no—
66#683—61no—
67C4×Dic5—31no—
68C4×Dic5—31no—
69C4×Dic5—31no—
70C2×#168—61no—
71C2×#168—61no—
72#676—122no—
73C2×C4×Q8—51no—
74#431—151no—
75#423—151no—
76Q8⋊C8—31no—
77Q8⋊C8—151no—
78Q8⋊C8—151no—
79Q8⋊C8—151no—
80C22×C3×C5—41no—
81C2×C3×D5—42no—
82C2×C3×D5—81no—
83C2×C3×D5—81no—
84C3×C4×C5—41no—
85C3×C4×C5—41no—
86C3×Dic5—42no—
87C22×C3×C4—201no—
88C2×SL(2,𝔽3)—101no—
89C2×SL(2,𝔽3)C4×D511yescharacteristic
90#208—204no—
91C2×C3⋊C8—41no—
92C2×C3⋊C8—201no—
93C22×D5SL(2,ℤ/4)11yescharacteristic
94C5×Q8—31no—
95C5×Q8#20811yescharacteristic, complemented
96C2×C4×C5—31no—
97C2×C4×C5—31no—
98C2×C4×C5C2×S411yescharacteristic
99C2×Dic5—31no—
100C2×Dic5—31no—
101C2×Dic5SL(2,ℤ/4)11yescharacteristic
102C4×D5—61no—
103C4×D5—61no—
104C4×D5—121no—
105C4×D5#22612yes—
106C4×D5GL(2,𝔽3)12yes—
107Dic10—32no—
108Dic10—61no—
109C5×C8—121no—
110C5×C8—121no—
111#168—121no—
112#168—121no—
113C22×Q8—51no—
114C2×M16—301no—
115C4×Q8—51no—
116C4×Q8—151no—
117C4×Q8—151no—
118C4×Q8D3×D511yescharacteristic
119C2×C4⋊C4—151no—
120C2×C42—151no—
121C4⋊C8—31no—
122C4⋊C8—151no—
123C4⋊C8—301no—
124C4⋊C8—301no—
125C22⋊C8—301no—
126C4×C8—31no—
127C4×C8—151no—
128C2×C3×C5—41no—
129C2×C3×C5—41no—
130C2×C3×C5—41no—
131C3×D5—82no—
132C23×C3—201no—
133C2×C3×C4—41no—
134C2×C3×C4—202no—
135C2×C3×C4—202no—
136C2×C3×C4—201no—
137SL(2,𝔽3)#67611yescharacteristic, complemented
138C3⋊C8—42no—
139C3⋊C8—202no—
140C22×C5C2×SL(2,ℤ/4)11yescharacteristic
141C2×D5—21no—
142C2×D5—21no—
143C2×D5#85912yes—
144C4×C5—31no—
145C4×C5—31no—
146C4×C5—61no—
147C4×C5C2×#22611yescharacteristic
148C4×C5C2×GL(2,𝔽3)11yescharacteristic
149Dic5—61no—
150Dic5—61no—
151Dic5#85912yes—
152C2×Q8—101no—
153C2×Q8—151no—
154C2×Q8—301no—
155C2×Q8D5×Dic311yescharacteristic
156C22×C4—51no—
157C22×C4—151no—
158C22×C4—151no—
159M16—602no—
160C2×C8—61no—
161C2×C8—61no—
162C2×C8—301no—
163C2×C8—301no—
164C4⋊C4—31no—
165C4⋊C4—151no—
166C4⋊C4—301no—
167C42—31no—
168C42—151no—
169C42—151no—
170C42—151no—
171C3×C5—41noHall
172C22×C3—41no—
173C22×C3—202no—
174C22×C3—401no—
175C22×C3—401no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—202no—
179C2×C5#540911yescharacteristic
180C2×C5C2×#85911yescharacteristic
181C2×C5#542311yescharacteristic
182D5—22no—
183C23—51no—
184Q8—31no—
185Q8—152no—
186Q8—301no—
187Q8#664911yescharacteristic, complemented
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—51no—
191C2×C4—52no—
192C2×C4—52no—
193C2×C4—151no—
194C2×C4—151no—
195C2×C4—301no—
196C2×C4—301no—
197C2×C4—601no—
198C2×C4D5×S411yescharacteristic
199C8—121no—
200C8—121no—
201C8—601no—
202C8—601no—
203C2×C3—41no—
204C2×C3—41no—
205C2×C3—41no—
206C2×C3—402no—
207C5#7305511yescharacteristic, complemented, Sylow
208C22—52no—
209C22—101no—
210C22—101no—
211C22D5×SL(2,ℤ/4)11yescharacteristic
212C4—31no—
213C4—31no—
214C4—52no—
215C4—61no—
216C4—301no—
217C4—301no—
218C4D5×#22611yescharacteristic
219C4D5×GL(2,𝔽3)11yescharacteristic
220C3—41noSylow
221C2—102no—
222C2#1175157811yescharacteristic
223C2D5×#85911yescharacteristic
224C2#1175162411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D5×S4
commutatorC2×C5×SL(2,𝔽3)C2×C4
FittingC4×C5×Q8C2×D3
socleC22×C5C2×SL(2,ℤ/4)
prime radicalC2×D5×SL(2,𝔽3)C4
cosocleC4×C5×SL(2,𝔽3)C22
hypocenterC5×SL(2,𝔽3)M16
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)⋊#168C2semidirect
3. C5×SL(2,𝔽3)⋊C8C2semidirect
25. SL(2,𝔽3)⋊C8D5semidirect
47. C5×SL(2,𝔽3)M16semidirect
95. C5×Q8#208semidirect
137. SL(2,𝔽3)#676semidirect
187. Q8#6649semidirect
207. C5#73055semidirect

Maximal subgroups

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

subgroupindex
C4×D5×SL(2,𝔽3)2
SL(2,𝔽3)⋊#1682
C5×SL(2,𝔽3)⋊C82
#105990293
C2×#66494
#730555