Finite Group Databasebeta

← C2×SL(2,𝔽3)⋊#170 #14512485

Subgroups — C2×SL(2,𝔽3)⋊#170 #14512485

Order 1920.

302 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 72 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, direct product factor
3C2×Q8⋊#170—31nomaximal, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic
5C22×C3⋊#170—41nomaximal
6Dic5×SL(2,𝔽3)C411yescharacteristic
7Dic5×SL(2,𝔽3)C2211yescharacteristic
8C2×SL(2,𝔽3)⋊C8—51nomaximal, Hall
9C2×Q8×Dic5D311yescharacteristic
10C2×C4⋊#170—31no—
11C2×C4×#170—31no—
12Q8⋊#170—32no—
13Q8⋊#170—32no—
14C22×C3×Dic5—41no—
15C2×C5×SL(2,𝔽3)C812yescomplemented
16C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
17C2×C3⋊#170—42no—
18C2×C3⋊#170—44no—
19C2×C4×SL(2,𝔽3)—51no—
20SL(2,𝔽3)⋊C8—52no—
21C22×C5×Q8Dic311yescharacteristic
22C22×#170—61no—
23C22×#170—61no—
24Q8×Dic5—31no—
25Q8×Dic5—31no—
26Q8×Dic5Dic311yescharacteristic
27Q8×Dic5C2×D311yescharacteristic
28C2×C4⋊Dic5—31no—
29C2×C4×Dic5—31no—
30C4⋊#170—32no—
31C4⋊#170—61no—
32C4×#170—32no—
33C4×#170—61no—
34C2×Q8⋊C8—151noSylow
35C23×C3×C5—41no—
36C2×C3×Dic5—42no—
37C2×C3×Dic5—42no—
38C2×C3×Dic5—41no—
39C2×C3×Dic5—41no—
40C5×SL(2,𝔽3)C2×C811yescharacteristic, complemented
41C3⋊#170—44no—
42C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic
43C22×C3⋊C8—201no—
44C4×SL(2,𝔽3)—51no—
45C4×SL(2,𝔽3)—51no—
46C2×C5×Q8—32no—
47C2×C5×Q8—31no—
48C2×C5×Q8C3⋊C812yescomplemented
49C2×C5×Q8C2×Dic311yescharacteristic
50C22×C4×C5—31no—
51C22×Dic5—31no—
52C22×Dic5S411yescharacteristic
53C2×#170—62no—
54C2×#170—64no—
55C2×#170—62no—
56C2×#170—122no—
57C4⋊Dic5—31no—
58C4⋊Dic5—31no—
59C4⋊Dic5—61no—
60C4×Dic5—31no—
61C4×Dic5—31no—
62C4×Dic5—31no—
63C4×Dic5—31no—
64C2×C4×Q8—51no—
65C2×C4⋊C8—151no—
66C2×C4×C8—151no—
67Q8⋊C8—152no—
68Q8⋊C8—152no—
69C22×C3×C5—42no—
70C22×C3×C5—42no—
71C22×C3×C5—42no—
72C22×C3×C5—41no—
73C3×Dic5—42no—
74C3×Dic5—41no—
75C3×Dic5—41no—
76C22×C3×C4—201no—
77C2×SL(2,𝔽3)#17012yescomplemented
78C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
79C2×C3⋊C8—202no—
80C2×C3⋊C8—204no—
81C23×C5SL(2,ℤ/4)11yescharacteristic
82C5×Q8—32no—
83C5×Q8—31no—
84C5×Q8—61no—
85C5×Q8C2×C3⋊C811yescharacteristic, complemented
86C2×C4×C5—32no—
87C2×C4×C5—32no—
88C2×C4×C5—31no—
89C2×C4×C5—31no—
90C2×Dic5—31no—
91C2×Dic5—31no—
92C2×Dic5—62no—
93C2×Dic5#22612yes—
94C2×Dic5GL(2,𝔽3)12yes—
95C2×Dic5SL(2,ℤ/4)11yescharacteristic
96C2×Dic5C2×S411yescharacteristic
97#170—64no—
98#170—122no—
99C22×Q8ΓL(1,𝔽16)11yescharacteristic
100C22×C8—301no—
101C22×C8—301no—
102C4×Q8—51no—
103C4×Q8—51no—
104C4×Q8—151no—
105C4×Q8—151no—
106C2×C4⋊C4—151no—
107C2×C42—151no—
108C4⋊C8—152no—
109C4⋊C8—301no—
110C4×C8—152no—
111C4×C8—301no—
112C2×C3×C5—44no—
113C2×C3×C5—41no—
114C2×C3×C5—41no—
115C2×C3×C5—41no—
116C23×C3—41no—
117C2×C3×C4—201no—
118C2×C3×C4—201no—
119C2×C3×C4—202no—
120C2×C3×C4—202no—
121SL(2,𝔽3)C2×#17011yescharacteristic, complemented
122C3⋊C8—204no—
123C22×C5A4⋊C812yes—
124C22×C5#85912yes—
125C22×C5GU(2,𝔽3)12yes—
126C22×C5C2×SL(2,ℤ/4)11yescharacteristic
127C4×C5—31no—
128C4×C5—31no—
129C4×C5—32no—
130Dic5—61no—
131Dic5—61no—
132Dic5#85912yes—
133Dic5C2×#22611yescharacteristic
134Dic5C2×GL(2,𝔽3)11yescharacteristic
135C2×Q8—31no—
136C2×Q8—32no—
137C2×Q8C3⋊#17012yescomplemented
138C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
139C22×C4—31no—
140C22×C4—51no—
141C22×C4—151no—
142C2×C8—302no—
143C2×C8—304no—
144C2×C8—302no—
145C2×C8—602no—
146C4⋊C4—151no—
147C4⋊C4—151no—
148C4⋊C4—301no—
149C42—151no—
150C42—151no—
151C42—151no—
152C42—151no—
153C3×C5—41noHall
154C22×C3—41no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3—42no—
158C3×C4—201no—
159C3×C4—202no—
160C3×C4—201no—
161C2×C5SL(2,𝔽3)⋊C814yescomplemented
162C2×C5C2×A4⋊C811yescharacteristic
163C2×C5C2×#85911yescharacteristic
164C2×C5C2×GU(2,𝔽3)11yescharacteristic
165C23A4⋊AGL(1,𝔽5)11yescharacteristic
166Q8—32no—
167Q8—31no—
168Q8—61no—
169Q8C2×C3⋊#17011yescharacteristic, complemented
170C2×C4—31no—
171C2×C4—31no—
172C2×C4—32no—
173C2×C4—32no—
174C2×C4—51no—
175C2×C4—51no—
176C2×C4—52no—
177C2×C4—52no—
178C2×C4—151no—
179C2×C4—151no—
180C2×C4—302no—
181C8—304no—
182C8—602no—
183C2×C3—44no—
184C2×C3—41no—
185C2×C3—41no—
186C2×C3—41no—
187C5C2×SL(2,𝔽3)⋊C811yescharacteristic, complemented, Sylow
188C22#9188512yes—
189C22#9188612yes—
190C22A4⋊#17012yes—
191C22C2×A4⋊AGL(1,𝔽5)11yescharacteristic
192C4—31no—
193C4—31no—
194C4—32no—
195C4—51no—
196C4—52no—
197C4—51no—
198C4—301no—
199C4—301no—
200C3—41noSylow
201C2SL(2,𝔽3)⋊#17014yesdirect product factor
202C2C2×#9188511yescharacteristic
203C2C2×#9188611yescharacteristic
204C2C2×A4⋊#17011yescharacteristic
subgroupG/H
center, hypercenterC23A4⋊AGL(1,𝔽5)
commutator, hypocenterC5×SL(2,𝔽3)C2×C8
FrattiniC22C2×A4⋊AGL(1,𝔽5)
FittingC22×C5×Q8Dic3
socleC23×C5SL(2,ℤ/4)
prime radicalC22×C5×SL(2,𝔽3)C4
cosocleDic5×SL(2,𝔽3)C22
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)⋊#170C2direct (N × H)
15. C2×C5×SL(2,𝔽3)C8semidirect
40. C5×SL(2,𝔽3)C2×C8semidirect
48. C2×C5×Q8C3⋊C8semidirect
77. C2×SL(2,𝔽3)#170semidirect
85. C5×Q8C2×C3⋊C8semidirect
121. SL(2,𝔽3)C2×#170semidirect
137. C2×Q8C3⋊#170semidirect
161. C2×C5SL(2,𝔽3)⋊C8semidirect
169. Q8C2×C3⋊#170semidirect
187. C5C2×SL(2,𝔽3)⋊C8semidirect
201. C2SL(2,𝔽3)⋊#170direct (N × H)

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
C2×Q8⋊#1703
C22×C3⋊#1704
C2×SL(2,𝔽3)⋊C85