Finite Group Databasebeta

← C5×#41⋊SL(2,𝔽3) #14511391

Subgroups — C5×#41⋊SL(2,𝔽3) #14511391

Order 1920.

446 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C5×Q8⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C5×Q8×#41C311yesmaximal, characteristic, complemented, Hall
3C2×C5×#41⋊C3—41nomaximal
4C4×C5×SL(2,𝔽3)—41nomaximal
5C4×C5×SL(2,𝔽3)—43nomaximal
6#41⋊SL(2,𝔽3)C511yesmaximal, characteristic, direct product factor, Hall
7C5×Q82C2×C311yescharacteristic
8C5×#557—31no—
9C5×#554—33no—
10C5×D4×Q8—31no—
11C5×#533—33no—
12C4×C5×#41—31no—
13C2×C4×C5×Q8—31no—
14C5×#41⋊C3—42no—
15C2×C5×SL(2,𝔽3)—41no—
16C2×C5×SL(2,𝔽3)—43no—
17C2×C5×SL(2,𝔽3)—41no—
18Q8⋊SL(2,𝔽3)C2×C511yescharacteristic, complemented
19C2×C5×#41A411yescharacteristic
20C22×C5×Q8—31no—
21C5×C4⋊Q8—33no—
22C5×C4⋊Q8—33no—
23C5×#125—36no—
24C5×C22⋊Q8—36no—
25C5×C22⋊Q8—33no—
26C5×C22⋊Q8—33no—
27C4×C5×Q8—33no—
28C4×C5×Q8—33no—
29C4×C5×Q8—31no—
30C4×C5×Q8—31no—
31C4×C5×Q8—31no—
32C4×C5×Q8—31no—
33C4×C5×Q8A411yescharacteristic
34C4×C5×Q8A413yes—
35C4×C5×D4—33no—
36C4×C5×D4—33no—
37C5×#117—33no—
38C2×C5×C4⋊C4—33no—
39C2×C42×C5—33no—
40Q8×#41C3×C511yescharacteristic, complemented, Sylow
41C2×C3×C4×C5—161no—
42C5×SL(2,𝔽3)—43no—
43C5×SL(2,𝔽3)—41no—
44C5×SL(2,𝔽3)—41no—
45C2×#41⋊C3—41no—
46C4×SL(2,𝔽3)—41no—
47C4×SL(2,𝔽3)—43no—
48C5×#41—32no—
49C5×#41SL(2,𝔽3)12yescomplemented
50C2×C5×Q8—33no—
51C2×C5×Q8—31no—
52C2×C5×Q8—63no—
53C2×C5×Q8—61no—
54C2×C5×Q8C2×A411yescharacteristic
55C2×C5×Q8C2×A411yescharacteristic
56C2×C5×Q8C2×A413yes—
57C2×C5×D4—31no—
58C22×C4×C5—31no—
59C22×C4×C5—33no—
60C22×C4×C5—33no—
61C5×C4⋊C4—31no—
62C5×C4⋊C4—33no—
63C5×C4⋊C4—36no—
64C5×C4⋊C4—33no—
65C5×C4⋊C4—33no—
66C5×C4⋊C4—33no—
67C5×C4⋊C4—33no—
68C5×C22⋊C4—36no—
69C42×C5—33no—
70C42×C5—33no—
71C42×C5—31no—
72C42×C5—33no—
73Q82C2×C3×C511yescharacteristic
74#557—31no—
75#554—33no—
76D4×Q8—31no—
77#533—33no—
78C4×#41—31no—
79C2×C4×Q8—31no—
80C22×C3×C5—161no—
81C3×C4×C5—162no—
82#41⋊C3—42no—
83C2×SL(2,𝔽3)—41no—
84C2×SL(2,𝔽3)—43no—
85C2×SL(2,𝔽3)—41no—
86C23×C5—31no—
87C5×Q8—31no—
88C5×Q8—31no—
89C5×Q8—43no—
90C5×Q8—63no—
91C5×Q8—63no—
92C5×Q8—121no—
93C5×Q8C2×SL(2,𝔽3)11yescharacteristic
94C5×Q8#41⋊C311yescharacteristic, complemented
95C5×D4—32no—
96C5×D4—32no—
97C2×C4×C5—31no—
98C2×C4×C5—31no—
99C2×C4×C5—32no—
100C2×C4×C5—32no—
101C2×C4×C5—31no—
102C2×C4×C5—33no—
103C2×C4×C5—33no—
104C2×C4×C5—63no—
105C2×C4×C5—63no—
106C2×C4×C5—123no—
107C2×C4×C5C22⋊A411yescharacteristic
108C2×#41C5×A411yescharacteristic
109C22×Q8—31no—
110C4⋊Q8—33no—
111C4⋊Q8—33no—
112#125—36no—
113C22⋊Q8—36no—
114C22⋊Q8—33no—
115C22⋊Q8—33no—
116C4×Q8—33no—
117C4×Q8—33no—
118C4×Q8—31no—
119C4×Q8—31no—
120C4×Q8—31no—
121C4×Q8—31no—
122C4×Q8C5×A411yescharacteristic
123C4×Q8C5×A413yes—
124C4×D4—33no—
125C4×D4—33no—
126#117—33no—
127C2×C4⋊C4—33no—
128C2×C42—33no—
129C2×C3×C5—161no—
130C2×C3×C5—161no—
131C2×C3×C5—161no—
132C2×C3×C4—161no—
133SL(2,𝔽3)—43no—
134SL(2,𝔽3)—41no—
135SL(2,𝔽3)—41no—
136C22×C5—32no—
137C22×C5—61no—
138C22×C5—61no—
139C22×C5C2×C22⋊A411yescharacteristic
140C4×C5—31no—
141C4×C5—31no—
142C4×C5—31no—
143C4×C5—31no—
144C4×C5—63no—
145C4×C5—61no—
146C4×C5—63no—
147C4×C5C22⋊SL(2,𝔽3)12yes—
148#41—32no—
149#41C5×SL(2,𝔽3)12yescomplemented
150C2×Q8—33no—
151C2×Q8—31no—
152C2×Q8—63no—
153C2×Q8—61no—
154C2×Q8C2×C5×A411yescharacteristic
155C2×Q8C2×C5×A411yescharacteristic
156C2×Q8C2×C5×A413yes—
157C2×D4—31no—
158C22×C4—31no—
159C22×C4—33no—
160C22×C4—33no—
161C4⋊C4—31no—
162C4⋊C4—33no—
163C4⋊C4—36no—
164C4⋊C4—33no—
165C4⋊C4—33no—
166C4⋊C4—33no—
167C4⋊C4—33no—
168C22⋊C4—36no—
169C42—33no—
170C42—33no—
171C42—31no—
172C42—33no—
173C3×C5—161noHall
174C22×C3—161no—
175C3×C4—162no—
176C2×C5—62no—
177C2×C5C2×C22⋊SL(2,𝔽3)11yescharacteristic
178C2×C5#594811yescharacteristic
179C2×C5#595011yescharacteristic
180C23—31no—
181Q8—31no—
182Q8—31no—
183Q8—43no—
184Q8—63no—
185Q8—63no—
186Q8—121no—
187Q8C2×C5×SL(2,𝔽3)11yescharacteristic
188Q8C5×#41⋊C311yescharacteristic, complemented
189D4—32no—
190D4—32no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—32no—
194C2×C4—32no—
195C2×C4—31no—
196C2×C4—33no—
197C2×C4—33no—
198C2×C4—63no—
199C2×C4—63no—
200C2×C4—123no—
201C2×C4C5×C22⋊A411yescharacteristic
202C2×C3—161no—
203C2×C3—161no—
204C2×C3—161no—
205C5#41⋊SL(2,𝔽3)11yescharacteristic, direct product factor, Sylow
206C22—32no—
207C22—61no—
208C22—61no—
209C22C2×C5×C22⋊A411yescharacteristic
210C4—31no—
211C4—31no—
212C4—31no—
213C4—31no—
214C4—63no—
215C4—61no—
216C4—63no—
217C4C5×C22⋊SL(2,𝔽3)12yes—
218C3—161noSylow
219C2—62no—
220C2C2×C5×C22⋊SL(2,𝔽3)11yescharacteristic
221C2C5×#594811yescharacteristic
222C2C5×#595011yescharacteristic
subgroupG/H
center, hypercenterC2×C4×C5C22⋊A4
commutator, cosocle, hypocenterQ82C2×C3×C5
FrattiniC22C2×C5×C22⋊A4
FittingC5×Q8×#41C3
socleC22×C5C2×C22⋊A4
LCSG ▷ Q82
UCSC1 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
1. C5×Q8⋊SL(2,𝔽3)C2semidirect
2. C5×Q8×#41C3semidirect
6. #41⋊SL(2,𝔽3)C5direct (N × H)
18. Q8⋊SL(2,𝔽3)C2×C5semidirect
40. Q8×#41C3×C5semidirect
49. C5×#41SL(2,𝔽3)semidirect
94. C5×Q8#41⋊C3semidirect
149. #41C5×SL(2,𝔽3)semidirect
188. Q8C5×#41⋊C3semidirect
205. C5#41⋊SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C5×Q8⋊SL(2,𝔽3)2
C5×Q8×#413
C2×C5×#41⋊C34
C4×C5×SL(2,𝔽3)4
#41⋊SL(2,𝔽3)5