Finite Group Databasebeta

← #10617523⋊C3 #14512573

Subgroups — #10617523⋊C3 #14512573

Order 1920.

238 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 48 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#10617523C311yesmaximal, characteristic, complemented, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic
5C3×C22⋊#170—41nomaximal
6Dic5×SL(2,𝔽3)C411yescharacteristic
7Dic5×SL(2,𝔽3)C2211yescharacteristic
8#2984⋊C3—51nomaximal, Hall
9C2×Q8×Dic5C2×C311yescharacteristic
10#65788—32no—
11Q8×#170C2×C312yescomplemented
12#65759—31no—
13#65754—31no—
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—
19C8×SL(2,𝔽3)—52no—
20C22×C5×Q8C3×C411yescharacteristic
21Q8×Dic5—31no—
22Q8×Dic5—31no—
23Q8×Dic5C3×C411yescharacteristic
24Q8×Dic5C22×C311yescharacteristic
25C2×C4⋊Dic5—31no—
26C2×C4×Dic5—31no—
27C22⋊#170—31no—
28C22⋊#170A411yescharacteristic
29#4004—32no—
30#4002—32no—
31C4⋊#170—32no—
32C4×#170—32no—
33#2984—51noSylow
34C23×C3×C5—41no—
35C2×C3×Dic5—41no—
36C2×C3×Dic5—41no—
37C2×C3×Dic5—82no—
38C5×SL(2,𝔽3)M1611yescharacteristic
39C3×#170—82no—
40C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic
41C4×SL(2,𝔽3)—51no—
42C4×SL(2,𝔽3)—51no—
43C3×C22⋊C8—201no—
44C2×C5×Q8—21no—
45C2×C5×Q8—31no—
46C2×C5×Q8—61no—
47C2×C5×Q8C2×C3×C411yescharacteristic
48C22×C4×C5—31no—
49C22×Dic5—31no—
50C22×Dic5C2×A411yescharacteristic
51C2×#170—32no—
52C2×#170C2×A412yes—
53C4⋊Dic5—32no—
54C4⋊Dic5—31no—
55C4⋊Dic5—31no—
56C4×Dic5—31no—
57C4×Dic5—31no—
58C4×Dic5—31no—
59C4×Dic5—31no—
60C2×C4×Q8—51no—
61#446—152no—
62C8×Q8—52no—
63#432—151no—
64#431—151no—
65C22×C3×C5—42no—
66C22×C3×C5—41no—
67C22×C3×C5—81no—
68C22×C3×C5—81no—
69C3×Dic5—42no—
70C3×Dic5—81no—
71C22×C3×C4—201no—
72C2×SL(2,𝔽3)—21no—
73C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
74C2×C3×C8—202no—
75C23×C5C4×A411yescharacteristic
76C5×Q8—31no—
77C5×Q8—61no—
78C5×Q8—61no—
79C5×Q8C3×M1611yescharacteristic
80C2×C4×C5—31no—
81C2×C4×C5—31no—
82C2×C4×C5—61no—
83C2×C4×C5—61no—
84C2×Dic5—22no—
85C2×Dic5—31no—
86C2×Dic5—31no—
87C2×Dic5—62no—
88C2×Dic5C4×A411yescharacteristic
89C2×Dic5C22×A411yescharacteristic
90#170—22no—
91#170—62no—
92C22×Q8C3×AGL(1,𝔽5)11yescharacteristic
93C4×Q8—51no—
94C4×Q8—51no—
95C4×Q8—151no—
96C4×Q8—151no—
97C2×C4⋊C4—151no—
98C2×C42—151no—
99C4⋊C8—152no—
100C4⋊C8—152no—
101C22⋊C8—51no—
102C22⋊C8—151no—
103#97—152no—
104C4×C8—152no—
105C2×C3×C5—41no—
106C2×C3×C5—41no—
107C2×C3×C5—41no—
108C2×C3×C5—82no—
109C23×C3—41no—
110C2×C3×C4—201no—
111C2×C3×C4—201no—
112C2×C3×C4—402no—
113SL(2,𝔽3)#70411yescharacteristic
114C3×C8—402no—
115C22×C5—21no—
116C22×C5—21no—
117C22×C5#131⋊C312yes—
118C22×C5C2×C4×A411yescharacteristic
119C4×C5—31no—
120C4×C5—31no—
121C4×C5—61no—
122Dic5—21no—
123Dic5—61no—
124Dic5—61no—
125Dic52−1+4⋊C312yes—
126C2×Q8—21no—
127C2×Q8—31no—
128C2×Q8—61no—
129C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
130C22×C4—31no—
131C22×C4—51no—
132C22×C4—151no—
133C2×C8—52no—
134C2×C8—152no—
135C4⋊C4—151no—
136C4⋊C4—151no—
137C4⋊C4—152no—
138C42—151no—
139C42—151no—
140C42—151no—
141C42—151no—
142C3×C5—41noHall
143C22×C3—41no—
144C22×C3—42no—
145C22×C3—81no—
146C22×C3—81no—
147C3×C4—202no—
148C3×C4—401no—
149C2×C5—22no—
150C2×C5#519⋊C311yescharacteristic
151C2×C5A4×M1611yescharacteristic
152C2×C5C2×#131⋊C311yescharacteristic
153C23A4×AGL(1,𝔽5)11yescharacteristic
154Q8—31no—
155Q8—61no—
156Q8—61no—
157Q8C3×#70411yescharacteristic
158C2×C4—31no—
159C2×C4—31no—
160C2×C4—51no—
161C2×C4—51no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—102no—
165C2×C4—151no—
166C2×C4—151no—
167C2×C4—302no—
168C8—102no—
169C8—302no—
170C2×C3—41no—
171C2×C3—41no—
172C2×C3—41no—
173C2×C3—82no—
174C5#2984⋊C311yescharacteristic, complemented, Sylow
175C22—21no—
176C22—21no—
177C22#4133⋊C312yes—
178C22C2×A4×AGL(1,𝔽5)11yescharacteristic
179C4—31no—
180C4—31no—
181C4—52no—
182C4—61no—
183C4—101no—
184C4—301no—
185C4—301no—
186C3—41noSylow
187C2—22no—
188C2C2×#4133⋊C311yescharacteristic
189C2#66262⋊C311yescharacteristic
190C2A4×#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. #10617523C3semidirect
11. Q8×#170C2×C3semidirect
174. C5#2984⋊C3semidirect

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
#106175233
C3×C22⋊#1704
#2984⋊C35