Finite Group Databasebeta

← SL(2,𝔽3)×#704 #14512570

Subgroups — SL(2,𝔽3)×#704 #14512570

Order 1920.

276 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×Dic5×SL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)×#170C212yesmaximal, complemented
3Q8×#704C311yesmaximal, characteristic, complemented, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic
5C2×C3×#704—41nomaximal
6Dic5×SL(2,𝔽3)C411yescharacteristic
7Dic5×SL(2,𝔽3)C2211yescharacteristic
8M16×SL(2,𝔽3)—51nomaximal, Hall
9C2×Q8×Dic5C2×C311yescharacteristic
10#65788—32no—
11Q8×#170C2×C312yescomplemented
12#65758—31no—
13C4×#704—31no—
14C22×C3×Dic5—41no—
15C2×C5×SL(2,𝔽3)—21no—
16C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
17C3×#704—44no—
18C2×C3×#170—42no—
19C2×C4×SL(2,𝔽3)—51no—
20C8×SL(2,𝔽3)—52no—
21C22×C5×Q8C3×C411yescharacteristic
22C2×#704—31no—
23C2×#704A411yescharacteristic
24Q8×Dic5—31no—
25Q8×Dic5—31no—
26Q8×Dic5C3×C411yescharacteristic
27Q8×Dic5C22×C311yescharacteristic
28C2×C4⋊Dic5—31no—
29C2×C4×Dic5—31no—
30#4004—32no—
31#4002—32no—
32C4⋊#170—32no—
33C4×#170—32no—
34Q8×M16—51noSylow
35C23×C3×C5—41no—
36C2×C3×Dic5—41no—
37C2×C3×Dic5—42no—
38C2×C3×Dic5—42no—
39C2×C3×Dic5—41no—
40C5×SL(2,𝔽3)M1611yescharacteristic, complemented
41C3×#170—44no—
42C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic
43C2×C3×M16—201no—
44C4×SL(2,𝔽3)—51no—
45C4×SL(2,𝔽3)—51no—
46C2×C5×Q8—21no—
47C2×C5×Q8—31no—
48C2×C5×Q8—61no—
49C2×C5×Q8C2×C3×C411yescharacteristic
50C22×C4×C5—31no—
51C22×Dic5—31no—
52C22×Dic5C2×A411yescharacteristic
53#704—62no—
54#704SL(2,𝔽3)14yesdirect product factor
55C2×#170—32no—
56C2×#170C2×A412yes—
57C4⋊Dic5—32no—
58C4⋊Dic5—31no—
59C4⋊Dic5—31no—
60C4×Dic5—31no—
61C4×Dic5—31no—
62C4×Dic5—31no—
63C4×Dic5—31no—
64C2×C4×Q8—51no—
65#446—152no—
66C8×Q8—52no—
67#423—151no—
68C4×M16—151no—
69C22×C3×C5—42no—
70C22×C3×C5—41no—
71C22×C3×C5—81no—
72C22×C3×C5—81no—
73C3×Dic5—42no—
74C3×Dic5—41no—
75C3×Dic5—41no—
76C22×C3×C4—201no—
77C2×SL(2,𝔽3)—21no—
78C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
79C3×M16—204no—
80C2×C3×C8—202no—
81C23×C5C4×A411yescharacteristic
82C5×Q8—31no—
83C5×Q8—61no—
84C5×Q8—61no—
85C5×Q8C3×M1611yescharacteristic, complemented
86C2×C4×C5—31no—
87C2×C4×C5—31no—
88C2×C4×C5—61no—
89C2×C4×C5—61no—
90C2×Dic5—31no—
91C2×Dic5—31no—
92C2×Dic5—121no—
93C2×Dic5C4×A411yescharacteristic
94C2×Dic5C2×SL(2,𝔽3)12yes—
95C2×Dic5#41⋊C312yes—
96C2×Dic5C22×A411yescharacteristic
97#170—62no—
98#170C2×SL(2,𝔽3)14yescomplemented
99C22×Q8C3×AGL(1,𝔽5)11yescharacteristic
100C2×M16—51no—
101C2×M16—151no—
102C4×Q8—51no—
103C4×Q8—51no—
104C4×Q8—151no—
105C4×Q8—151no—
106C2×C4⋊C4—151no—
107C2×C42—151no—
108C4⋊C8—152no—
109C4⋊C8—152no—
110#97—152no—
111C4×C8—152no—
112C2×C3×C5—41no—
113C2×C3×C5—41no—
114C2×C3×C5—41no—
115C2×C3×C5—82no—
116C23×C3—41no—
117C2×C3×C4—201no—
118C2×C3×C4—201no—
119C2×C3×C4—202no—
120C2×C3×C4—202no—
121SL(2,𝔽3)#70411yescharacteristic, direct product factor
122C3×C8—204no—
123C22×C5—21no—
124C22×C5—21no—
125C22×C5C4×SL(2,𝔽3)12yes—
126C22×C5C2×C4×A411yescharacteristic
127C4×C5—31no—
128C4×C5—31no—
129C4×C5—61no—
130Dic5—61no—
131Dic5—61no—
132Dic5C4×SL(2,𝔽3)12yes—
133Dic5C22×SL(2,𝔽3)11yescharacteristic
134Dic5C2×#41⋊C311yescharacteristic
135C2×Q8—21no—
136C2×Q8—31no—
137C2×Q8—61no—
138C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
139C22×C4—31no—
140C22×C4—51no—
141C22×C4—151no—
142M16—54no—
143M16—302no—
144C2×C8—52no—
145C2×C8—152no—
146C4⋊C4—151no—
147C4⋊C4—151no—
148C4⋊C4—152no—
149C42—151no—
150C42—151no—
151C42—151no—
152C42—151no—
153C3×C5—41noHall
154C22×C3—41no—
155C22×C3—42no—
156C22×C3—81no—
157C22×C3—81no—
158C3×C4—201no—
159C3×C4—201no—
160C3×C4—202no—
161C2×C5—22no—
162C2×C5C2×C4×SL(2,𝔽3)11yescharacteristic
163C2×C5A4×M1611yescharacteristic
164C2×C5#568⋊C311yescharacteristic
165C23A4×AGL(1,𝔽5)11yescharacteristic
166Q8—31no—
167Q8—61no—
168Q8—61no—
169Q8C3×#70411yescharacteristic, complemented
170C2×C4—31no—
171C2×C4—31no—
172C2×C4—51no—
173C2×C4—51no—
174C2×C4—52no—
175C2×C4—52no—
176C2×C4—61no—
177C2×C4—61no—
178C2×C4—151no—
179C2×C4—151no—
180C2×C4—601no—
181C8—54no—
182C8—302no—
183C2×C3—41no—
184C2×C3—41no—
185C2×C3—41no—
186C2×C3—82no—
187C5M16×SL(2,𝔽3)11yescharacteristic, complemented, Sylow
188C22—21no—
189C22—21no—
190C22AGL(1,𝔽5)×SL(2,𝔽3)12yes—
191C22C2×A4×AGL(1,𝔽5)11yescharacteristic
192C4—31no—
193C4—31no—
194C4—51no—
195C4—51no—
196C4—52no—
197C4—61no—
198C4—301no—
199C4—301no—
200C3—41noSylow
201C2—22no—
202C2#66260⋊C311yescharacteristic
203C2C2×AGL(1,𝔽5)×SL(2,𝔽3)11yescharacteristic
204C2A4×#70411yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×AGL(1,𝔽5)
commutatorC2×C5×Q8C2×C3×C4
FittingC22×C5×Q8C3×C4
socleC22×C5C2×C4×A4
prime radicalC22×C5×SL(2,𝔽3)C4
cosocleQ8×Dic5C22×C3
hypercenterC23A4×AGL(1,𝔽5)
hypocenterC5×Q8C3×M16
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×#170C2semidirect
3. Q8×#704C3semidirect
11. Q8×#170C2×C3semidirect
40. C5×SL(2,𝔽3)M16semidirect
54. #704SL(2,𝔽3)direct (N × H)
85. C5×Q8C3×M16semidirect
98. #170C2×SL(2,𝔽3)semidirect
121. SL(2,𝔽3)#704direct (N × H)
169. Q8C3×#704semidirect
187. C5M16×SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C2×Dic5×SL(2,𝔽3)2
SL(2,𝔽3)×#1702
Q8×#7043
C2×C3×#7044
M16×SL(2,𝔽3)5