Finite Group Databasebeta

← #10611656⋊C3 #14512178

Subgroups — #10611656⋊C3 #14512178

Order 1920.

240 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 45 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
1SL(2,𝔽3)×D20C211yesmaximal, characteristic
2#65904⋊C3C211yesmaximal, characteristic, complemented
3C5×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10611656C311yesmaximal, characteristic, complemented, Hall
5C2×D5×SL(2,𝔽3)—21no—
6C4×C5×SL(2,𝔽3)C2211yescharacteristic
7Dic5×SL(2,𝔽3)—21no—
8C3×#3953—41nomaximal
9#3359⋊C3—51nomaximal, Hall
10Q8×D20C2×C311yescharacteristic
11#65904C2×C311yescharacteristic, complemented
12C5×C8×Q8C2×C311yescharacteristic, complemented
13#65140—31no—
14#65130—31no—
15C4×#677—31no—
16#64970—31no—
17C2×C3×D20—41no—
18C2×C5×SL(2,𝔽3)D411yescharacteristic
19D5×SL(2,𝔽3)—41no—
20C2×C3×C5×C8—41no—
21C3×C4⋊Dic5—41no—
22#557⋊C3—51no—
23D4×SL(2,𝔽3)—51no—
24C8×SL(2,𝔽3)D511yescharacteristic, complemented
25C2×Q8×D5—21no—
26C4×C5×Q8C22×C311yescharacteristic
27Q8×Dic5—21no—
28C2×#677—31no—
29#4043—61no—
30#4036—61no—
31C4×D20—31no—
32#4015—31no—
33#4015—31no—
34C4×Dic10—31no—
35C5×C4⋊C8—31no—
36C4×C5×C8—31no—
37#3953A411yescharacteristic
38#3949—31no—
39#3948—32no—
40#3948—31no—
41#3359—51noSylow
42C22×C3×D5—81no—
43C2×C3×C4×C5—41no—
44C2×C3×Dic5—81no—
45C3×D20—42no—
46C3×D20—81no—
47C5×SL(2,𝔽3)SD1611yescharacteristic, complemented
48C3×C5×C8—81no—
49C22×SL(2,𝔽3)—101no—
50C4×SL(2,𝔽3)—101no—
51C4×SL(2,𝔽3)C2×D511yescharacteristic
52C3×#102—201no—
53C2×C5×Q8C3×D411yescharacteristic
54Q8×D5—41no—
55Q8×D5—121no—
56C2×D20C2×A411yescharacteristic
57C2×C4×D5—61no—
58C2×Dic10—31no—
59C2×Dic10—61no—
60C2×C5×C8—31no—
61C2×C5×C8C2×A411yescharacteristic
62C5×C4⋊C4—31no—
63C42×C5—31no—
64#685—61no—
65C4⋊Dic5—31no—
66C4⋊Dic5—61no—
67C4⋊Dic5—61no—
68C4⋊Dic5—61no—
69C4⋊Dic5C2×A411yescharacteristic
70#683—61no—
71C4×Dic5—61no—
72#677—62no—
73#557—51no—
74D4×Q8—51no—
75#487—151no—
76#475—151no—
77#461—151no—
78C8×Q8C3×D511yescharacteristic, complemented
79C4×SD16—151no—
80C22×C3×C5—41no—
81C2×C3×D5—82no—
82C2×C3×D5—161no—
83C2×C3×D5—161no—
84C3×C4×C5—41no—
85C3×C4×C5—41no—
86C3×Dic5—161no—
87C2×C3×D4—201no—
88C2×SL(2,𝔽3)—201no—
89C2×SL(2,𝔽3)D2011yescharacteristic
90C2×C3×C8—41no—
91C3×C4⋊C4—201no—
92C22×D5—21no—
93C5×Q8—31no—
94C5×Q8C3×SD1611yescharacteristic
95C2×C4×C5—31no—
96C2×C4×C5—31no—
97C2×C4×C5C22×A411yescharacteristic
98C2×Dic5—21no—
99C2×Dic5—61no—
100C2×Dic5—61no—
101D20—21no—
102D20#41⋊C312yes—
103C4×D5—121no—
104C4×D5—121no—
105Dic10—61no—
106Dic10—61no—
107Dic10—121no—
108Dic10—121no—
109C5×C8—21no—
110C5×C8—61no—
111C22×Q8—101no—
112C2×SD16—151no—
113C4⋊Q8—151no—
114C4⋊Q8—151no—
115#125—301no—
116C22⋊Q8—301no—
117C4×Q8—101no—
118C4×Q8—151no—
119C4×Q8C2×C3×D511yescharacteristic
120C4×D4—151no—
121#106—151no—
122C4⋊C8—31no—
123#103—152no—
124#103—151no—
125#102—51no—
126C4×C8—31no—
127C2×C3×C5—41no—
128C2×C3×C5—41no—
129C2×C3×C5—41no—
130C3×D5—162no—
131C23×C3—401no—
132C3×D4—202no—
133C3×D4—401no—
134C2×C3×C4—41no—
135C2×C3×C4—401no—
136SL(2,𝔽3)#67711yescharacteristic, complemented
137C3×C8—81no—
138C22×C5D4×A411yescharacteristic
139C2×D5—22no—
140C2×D5—41no—
141C2×D5—41no—
142C4×C5—31no—
143C4×C5—31no—
144C4×C5—61no—
145C4×C5C2×#41⋊C311yescharacteristic
146C4×C52−1+4⋊C311yescharacteristic
147Dic5—41no—
148Dic5—121no—
149Dic5—121no—
150C2×Q8—151no—
151C2×Q8—201no—
152C2×Q8—301no—
153C2×Q8—601no—
154C2×Q8C3×D2011yescharacteristic
155C2×D4—51no—
156C22×C4—301no—
157SD16—302no—
158C2×C8—31no—
159C2×C8D5×A411yescharacteristic
160C4⋊C4—31no—
161C4⋊C4—51no—
162C4⋊C4—151no—
163C4⋊C4—301no—
164C4⋊C4—301no—
165C4⋊C4—301no—
166C4⋊C4—301no—
167C22⋊C4—301no—
168C42—31no—
169C42—301no—
170C3×C5—41noHall
171C22×C3—41no—
172C22×C3—402no—
173C22×C3—801no—
174C22×C3—801no—
175C3×C4—41no—
176C3×C4—41no—
177C3×C4—801no—
178C2×C5#548⋊C311yescharacteristic
179C2×C5A4×SD1611yescharacteristic
180C2×C5#578⋊C311yescharacteristic
181D5—42no—
182C23—101no—
183Q8—31no—
184Q8—301no—
185Q8—301no—
186Q8—601no—
187Q8—601no—
188Q8C3×#67711yescharacteristic
189D4—52no—
190D4—101no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—101no—
194C2×C4—301no—
195C2×C4—301no—
196C2×C4—601no—
197C2×C4—601no—
198C2×C4C2×D5×A411yescharacteristic
199C8—21no—
200C8—61no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—41no—
204C2×C3—802no—
205C5#3359⋊C311yescharacteristic, complemented, Sylow
206C22—102no—
207C22—201no—
208C22—201no—
209C22A4×D2011yescharacteristic
210C4—31no—
211C4—31no—
212C4—61no—
213C4—201no—
214C4—601no—
215C4—601no—
216C4#4147⋊C311yescharacteristic
217C4D5×#41⋊C311yescharacteristic
218C3—41noSylow
219C2—202no—
220C2#65910⋊C311yescharacteristic
221C2A4×#67711yescharacteristic
222C2#66087⋊C311yescharacteristic
subgroupG/H
centerC22A4×D20
commutator, cosocleC4×C5×Q8C22×C3
FrattiniC2×C4C2×D5×A4
FittingC5×C8×Q8C2×C3
socleC22×C5D4×A4
prime radicalSL(2,𝔽3)×D20C2
hypercenterC2×C8D5×A4
hypocenterC5×Q8C3×SD16
LCSG ▷ C4×C5×Q8 ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #65904⋊C3C2semidirect
3. C5×C8×SL(2,𝔽3)C2semidirect
4. #10611656C3semidirect
11. #65904C2×C3semidirect
12. C5×C8×Q8C2×C3semidirect
24. C8×SL(2,𝔽3)D5semidirect
47. C5×SL(2,𝔽3)SD16semidirect
78. C8×Q8C3×D5semidirect
136. SL(2,𝔽3)#677semidirect
205. C5#3359⋊C3semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D202
#65904⋊C32
C5×C8×SL(2,𝔽3)2
#106116563
C3×#39534
#3359⋊C35