Finite Group Databasebeta

← #10617206⋊C3 #14512527

Subgroups — #10617206⋊C3 #14512527

Order 1920.

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

normal subgroups only

This group has 228 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
1C4×C5⋊(#41⋊C3)C211yesmaximal, characteristic
2SL(2,𝔽3)×#170C212yesmaximal, complemented
3#10617206C311yesmaximal, characteristic, complemented, Hall
4C2×C5⋊(#41⋊C3)C411yescharacteristic
5C4×C5×SL(2,𝔽3)C411yescharacteristic
6C3×C4⋊#170—41nomaximal
7Dic5×SL(2,𝔽3)C2211yescharacteristic
8#2991⋊C3—51nomaximal, Hall
9C4×#713C2×C311yescharacteristic
10Q8×#170C2×C312yes—
11#65705—32no—
12#65681—31no—
13#65680—31no—
14C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
15C2×C3×#170—42no—
16C5⋊(#41⋊C3)—21no—
17C3×C4×Dic5—41no—
18C4×#41⋊C3—51no—
19C8×SL(2,𝔽3)—52no—
20C2×#713C3×C411yescharacteristic
21C2×#700—32no—
22C4×C5×Q8C3×C411yescharacteristic
23Q8×Dic5C22×C311yescharacteristic
24#4039—31no—
25#4038—31no—
26C4×D20—31no—
27C42×D5—31no—
28#4003—32no—
29C4⋊#170—32no—
30C4⋊#170—31no—
31C4⋊#170A411yescharacteristic
32C4×#170—32no—
33#2991—51noSylow
34C2×C3×C4×C5—41no—
35C2×C3×Dic5—41no—
36C2×C3×Dic5—41no—
37C5×SL(2,𝔽3)M1611yescharacteristic, complemented
38C3×#170—82no—
39C2×#41⋊C3—51no—
40C4×SL(2,𝔽3)—51no—
41C4×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic
42C3×C4⋊C8—201no—
43C2×C5×Q8C2×C3×C411yescharacteristic
44#713—21no—
45#713—61no—
46C2×D20—31no—
47C2×C4×D5—31no—
48C2×C4×D5—31no—
49C2×C4×D5—31no—
50C2×#170—32no—
51C2×#170C2×A412yes—
52#700—122no—
53C5×C4⋊C4—31no—
54C42×C5—31no—
55#685—32no—
56C4⋊Dic5—31no—
57C4×Dic5—31no—
58C4×Dic5—31no—
59C4×Dic5C2×A411yescharacteristic
60C4×#41—51no—
61C8×Q8—52no—
62#436—152no—
63#431—151no—
64#423—151no—
65C22×C3×C5—41no—
66C3×C4×C5—42no—
67C3×Dic5—41no—
68C3×Dic5—41no—
69C3×Dic5—81no—
70#41⋊C3—101no—
71C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
72C2×C3×C8—202no—
73C3×C42—201no—
74C22×D5—31no—
75C5×Q8—31no—
76C5×Q8C3×M1611yescharacteristic
77C2×C4×C5—31no—
78C2×C4×C5—31no—
79C2×C4×C5C4×A411yescharacteristic
80C2×Dic5—31no—
81C2×Dic5C4×A411yescharacteristic
82C2×Dic5C22×A411yescharacteristic
83D20—61no—
84D20—61no—
85C4×D5—61no—
86C4×D5—61no—
87C4×D5—62no—
88C4×D5—61no—
89C4×D5—61no—
90#170—22no—
91#170—62no—
92C2×#41—51no—
93C2×M16—152no—
94C4×Q8—51no—
95C4×Q8C3×AGL(1,𝔽5)11yescharacteristic
96C4×D4—151no—
97C4×D4—151no—
98#117—151no—
99C2×C42—151no—
100C4⋊C8—51no—
101C4⋊C8—151no—
102C4⋊C8—152no—
103C22⋊C8—152no—
104C4×C8—152no—
105C2×C3×C5—41no—
106C2×C3×C5—41no—
107C2×C3×C5—41no—
108C2×C3×C4—41no—
109C2×C3×C4—201no—
110C2×C3×C4—201no—
111SL(2,𝔽3)#70011yescharacteristic, complemented
112C3×C8—402no—
113C22×C5C2×C4×A411yescharacteristic
114C2×D5—61no—
115C2×D5—61no—
116C2×D5—61no—
117C4×C5—31no—
118C4×C5—31no—
119C4×C5—61no—
120C4×C5#131⋊C312yes—
121Dic5—21no—
122Dic5—61no—
123Dic5#99411yescharacteristic
124Dic52−1+4⋊C311yescharacteristic
125#41—101no—
126#41—301no—
127C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
128C2×D4—151no—
129C22×C4—151no—
130C22×C4—151no—
131C22×C4—151no—
132M16—602no—
133C2×C8—52no—
134C2×C8—152no—
135C4⋊C4—31no—
136C4⋊C4—151no—
137C22⋊C4—152no—
138C42—31no—
139C42—51no—
140C42—151no—
141C42—151no—
142C3×C5—41noHall
143C22×C3—41no—
144C3×C4—42no—
145C3×C4—201no—
146C3×C4—201no—
147C3×C4—401no—
148C2×C5#520⋊C311yescharacteristic
149C2×C5A4×M1611yescharacteristic
150C2×C5C2×#131⋊C311yescharacteristic
151D5—121no—
152C23—151no—
153Q8—31no—
154Q8C3×#70011yescharacteristic
155D4—301no—
156D4—301no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—51no—
160C2×C4—51no—
161C2×C4—151no—
162C2×C4—301no—
163C2×C4—301no—
164C2×C4—301no—
165C2×C4—301no—
166C2×C4—302no—
167C2×C4A4×AGL(1,𝔽5)11yescharacteristic
168C8—102no—
169C8—302no—
170C2×C3—41no—
171C2×C3—41no—
172C2×C3—41no—
173C5#2991⋊C311yescharacteristic, complemented, Sylow
174C22—301no—
175C22—301no—
176C22—301no—
177C22C2×A4×AGL(1,𝔽5)11yescharacteristic
178C4—31no—
179C4—31no—
180C4—51no—
181C4—51no—
182C4—61no—
183C4—101no—
184C4—301no—
185C4#4133⋊C312yes—
186C3—41noSylow
187C2—601no—
188C2A4×#70011yescharacteristic
189C2C2×#4133⋊C311yescharacteristic
190C2#66266⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×AGL(1,𝔽5)
commutatorC2×C5×Q8C2×C3×C4
FittingC4×C5×Q8C3×C4
socleC22×C5C2×C4×A4
prime radicalC2×C5⋊(#41⋊C3)C4
cosocleQ8×Dic5C22×C3
hypercenterC2×C4A4×AGL(1,𝔽5)
hypocenterC5×Q8C3×M16
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×#170C2semidirect
3. #10617206C3semidirect
37. C5×SL(2,𝔽3)M16semidirect
111. SL(2,𝔽3)#700semidirect
173. C5#2991⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×C5⋊(#41⋊C3)2
SL(2,𝔽3)×#1702
#106172063
C3×C4⋊#1704
#2991⋊C35