Finite Group Databasebeta

← #14512489

Subgroups — #14512489

Order 1920.

246 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 42 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
3#10604969—31nomaximal, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic
5#91241—41nomaximal
6Dic5×SL(2,𝔽3)C411yescharacteristic
7Dic5×SL(2,𝔽3)C2211yescharacteristic
8#73057—51nomaximal, Hall
9C2×Q8×Dic5D311yescharacteristic
10#65759—31no—
11C4×#704—31no—
12Q8⋊#170—32no—
13Q8⋊#170—32no—
14C22×C3×Dic5—41no—
15C2×C5×SL(2,𝔽3)—21no—
16C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
17C2×C3⋊#170—42no—
18C2×C4×SL(2,𝔽3)—51no—
19SL(2,𝔽3)⋊C8—52no—
20C22×C5×Q8Dic311yescharacteristic
21C2×#704—61no—
22Q8×Dic5—31no—
23Q8×Dic5—31no—
24Q8×Dic5Dic311yescharacteristic
25Q8×Dic5C2×D311yescharacteristic
26C2×C4⋊Dic5—31no—
27C2×C4×Dic5—31no—
28C22⋊#170—61no—
29#4002—61no—
30#4002—61no—
31C4⋊#170—32no—
32C4×#170—32no—
33#1491—151noSylow
34C23×C3×C5—41no—
35C2×C3×Dic5—41no—
36C2×C3×Dic5—41no—
37C2×C3×Dic5—82no—
38C5×SL(2,𝔽3)M1611yescharacteristic, complemented
39C3⋊#170—82no—
40C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic
41C4×SL(2,𝔽3)—51no—
42C4×SL(2,𝔽3)—51no—
43#830—201no—
44C2×C5×Q8—21no—
45C2×C5×Q8—31no—
46C2×C5×Q8—61no—
47C2×C5×Q8C2×Dic311yescharacteristic
48C22×C4×C5—31no—
49C22×Dic5—31no—
50C22×Dic5S411yescharacteristic
51#704—64no—
52C2×#170—62no—
53C2×#170—62no—
54C4⋊Dic5—31no—
55C4⋊Dic5—31no—
56C4⋊Dic5—61no—
57C4×Dic5—31no—
58C4×Dic5—31no—
59C4×Dic5—31no—
60C4×Dic5—31no—
61C2×C4×Q8—51no—
62#432—151no—
63C4×M16—151no—
64Q8⋊C8—152no—
65Q8⋊C8—152no—
66C22×C3×C5—42no—
67C22×C3×C5—41no—
68C22×C3×C5—81no—
69C22×C3×C5—81no—
70C3×Dic5—42no—
71C3×Dic5—81no—
72C22×C3×C4—201no—
73C2×SL(2,𝔽3)—21no—
74C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
75C2×C3⋊C8—202no—
76C23×C5SL(2,ℤ/4)11yescharacteristic
77C5×Q8—31no—
78C5×Q8—32no—
79C5×Q8—61no—
80C5×Q8#20811yescharacteristic
81C2×C4×C5—31no—
82C2×C4×C5—31no—
83C2×C4×C5—61no—
84C2×C4×C5—61no—
85C2×Dic5—22no—
86C2×Dic5—31no—
87C2×Dic5—31no—
88C2×Dic5—62no—
89C2×Dic5SL(2,ℤ/4)11yescharacteristic
90C2×Dic5C2×S411yescharacteristic
91#170—64no—
92#170—122no—
93C22×Q8ΓL(1,𝔽16)11yescharacteristic
94C2×M16—301no—
95C4×Q8—51no—
96C4×Q8—51no—
97C4×Q8—151no—
98C4×Q8—151no—
99C2×C4⋊C4—151no—
100C2×C42—151no—
101C4⋊C8—152no—
102C22⋊C8—301no—
103#97—301no—
104#97—301no—
105C4×C8—152no—
106C2×C3×C5—41no—
107C2×C3×C5—41no—
108C2×C3×C5—41no—
109C2×C3×C5—82no—
110C23×C3—41no—
111C2×C3×C4—201no—
112C2×C3×C4—201no—
113C2×C3×C4—402no—
114SL(2,𝔽3)#70411yescharacteristic, complemented
115C3⋊C8—402no—
116C22×C5—21no—
117C22×C5—21no—
118C22×C5GU(2,𝔽3)12yes—
119C22×C5C2×SL(2,ℤ/4)11yescharacteristic
120C4×C5—31no—
121C4×C5—31no—
122C4×C5—61no—
123Dic5—21no—
124Dic5—61no—
125Dic5—61no—
126Dic5#98312yes—
127C2×Q8—21no—
128C2×Q8—31no—
129C2×Q8—61no—
130C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
131C22×C4—31no—
132C22×C4—51no—
133C22×C4—151no—
134M16—304no—
135C2×C8—302no—
136C2×C8—302no—
137C4⋊C4—151no—
138C4⋊C4—151no—
139C4⋊C4—301no—
140C42—151no—
141C42—151no—
142C42—151no—
143C42—151no—
144C3×C5—41noHall
145C22×C3—41no—
146C22×C3—42no—
147C22×C3—81no—
148C22×C3—81no—
149C3×C4—202no—
150C3×C4—401no—
151C2×C5—22no—
152C2×C5#540911yescharacteristic
153C2×C5#542011yescharacteristic
154C2×C5C2×GU(2,𝔽3)11yescharacteristic
155C23A4⋊AGL(1,𝔽5)11yescharacteristic
156Q8—31no—
157Q8—32no—
158Q8—61no—
159Q8#676211yescharacteristic
160C2×C4—31no—
161C2×C4—31no—
162C2×C4—51no—
163C2×C4—51no—
164C2×C4—61no—
165C2×C4—61no—
166C2×C4—102no—
167C2×C4—151no—
168C2×C4—151no—
169C2×C4—302no—
170C8—304no—
171C8—602no—
172C2×C3—41no—
173C2×C3—41no—
174C2×C3—41no—
175C2×C3—82no—
176C5#7305711yescharacteristic, complemented, Sylow
177C22—21no—
178C22—21no—
179C22#9188512yes—
180C22C2×A4⋊AGL(1,𝔽5)11yescharacteristic
181C4—31no—
182C4—31no—
183C4—52no—
184C4—61no—
185C4—101no—
186C4—301no—
187C4—301no—
188C3—41noSylow
189C2—22no—
190C2C2×#9188511yescharacteristic
191C2#1175672211yescharacteristic
192C2#1175672811yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4⋊AGL(1,𝔽5)
commutatorC2×C5×SL(2,𝔽3)C2×C4
FittingC22×C5×Q8Dic3
socleC22×C5C2×SL(2,ℤ/4)
prime radicalC22×C5×SL(2,𝔽3)C4
cosocleDic5×SL(2,𝔽3)C22
hypercenterC23A4⋊AGL(1,𝔽5)
hypocenterC5×SL(2,𝔽3)M16
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)⋊#170C2semidirect
38. C5×SL(2,𝔽3)M16semidirect
114. SL(2,𝔽3)#704semidirect
176. C5#73057semidirect

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
#106049693
#912414
#730575