Finite Group Databasebeta

← #10615311⋊C3 #14512263

Subgroups — #10615311⋊C3 #14512263

Order 1920.

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

normal subgroups only

This group has 262 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
1C5×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×D20C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×#168C211yesmaximal, characteristic, complemented
4#10615311C311yesmaximal, characteristic, complemented, Hall
5C2×C5×#41⋊C3—21no—
6C2×D5×SL(2,𝔽3)—21no—
7C4×C5×SL(2,𝔽3)C2211yescharacteristic
8C3×#3941—41nomaximal
9#3358⋊C3—51nomaximal, Hall
10C5×#550C2×C311yescharacteristic, complemented
11Q8×D20C2×C311yescharacteristic
12#65361—31no—
13#65359—31no—
14Q8×#168C2×C311yescharacteristic, complemented
15C4×#686—31no—
16#65296—31no—
17C2×C3×D20—41no—
18C5×#41⋊C3—41no—
19C2×C5×SL(2,𝔽3)D411yescharacteristic
20D5×SL(2,𝔽3)—41no—
21C3×C5×C4⋊C4—41no—
22C2×C3×#168—41no—
23C4⋊(#41⋊C3)D511yescharacteristic, complemented
24D4×SL(2,𝔽3)—51no—
25C8×SL(2,𝔽3)—51no—
26C2×C5×#41—21no—
27C2×Q8×D5—21no—
28C5×#127—31no—
29C5×#121—61no—
30C4×C5×Q8C22×C311yescharacteristic
31C4×C5×D4—31no—
32C2×#686—31no—
33#4043—61no—
34C4×D20—31no—
35#4015—31no—
36#3964—32no—
37#3964—31no—
38#3941A411yescharacteristic
39#3939—31no—
40C4⋊#168—31no—
41C4×#168—31no—
42#3358—51noSylow
43C22×C3×D5—81no—
44C2×C3×C4×C5—41no—
45C2×C3×C4×C5—81no—
46C3×D20—42no—
47C3×D20—81no—
48C5×SL(2,𝔽3)D811yescharacteristic, complemented
49C3×#168—81no—
50C2×#41⋊C3—21no—
51C22×SL(2,𝔽3)—101no—
52C4×SL(2,𝔽3)C2×D511yescharacteristic
53C3×#102—201no—
54C5×#41—41no—
55C5×#41—121no—
56C2×C5×Q8C3×D411yescharacteristic
57C2×C5×D4—31no—
58C2×C5×D4—61no—
59C2×C5×D4—61no—
60C22×C4×C5—61no—
61Q8×D5—41no—
62Q8×D5—121no—
63C2×D20C2×A411yescharacteristic
64C2×C4×D5—61no—
65C2×Dic10—61no—
66C5×C4⋊C4—31no—
67C5×C4⋊C4C2×A411yescharacteristic
68C5×C22⋊C4—61no—
69C42×C5—31no—
70#686—62no—
71#685—61no—
72C4⋊Dic5—31no—
73C4⋊Dic5—61no—
74C2×#168—31no—
75C2×#168C2×A411yescharacteristic
76#550C3×D511yescharacteristic, complemented
77D4×Q8—51no—
78#486—151no—
79#474—151no—
80#459—151no—
81C8×Q8—51no—
82C4×D8—151no—
83C22×C3×C5—41no—
84C2×C3×D5—82no—
85C2×C3×D5—161no—
86C2×C3×D5—161no—
87C3×C4×C5—41no—
88C3×C4×C5—41no—
89C3×C4×C5—161no—
90C2×C3×D4—201no—
91#41⋊C3—41no—
92C2×SL(2,𝔽3)—201no—
93C2×SL(2,𝔽3)C22⋊D511yescharacteristic
94C2×C3×C8—201no—
95C3×C4⋊C4—41no—
96C23×C5—61no—
97C22×D5—21no—
98C5×Q8—31no—
99C5×Q8C3×D811yescharacteristic
100C5×D4—61no—
101C5×D4—61no—
102C5×D4—121no—
103C5×D4—121no—
104C5×D4—241no—
105C2×C4×C5—21no—
106C2×C4×C5—31no—
107C2×C4×C5—31no—
108C2×C4×C5—121no—
109C2×C4×C5—121no—
110C2×C4×C5C22×A411yescharacteristic
111C2×Dic5—61no—
112D20—21no—
113D20#41⋊C312yescomplemented
114C4×D5—121no—
115C4×D5—121no—
116Dic10—121no—
117Dic10—121no—
118#168—21no—
119#168—61no—
120C2×#41—21no—
121C22×Q8—101no—
122C2×D8—151no—
123C4⋊Q8—151no—
124#127—31no—
125C22⋊Q8—301no—
126#121—61no—
127C4×Q8C2×C3×D511yescharacteristic
128C4×D4—31no—
129C4×D4—151no—
130#107—151no—
131C4⋊C8—151no—
132#102—51no—
133#102—151no—
134#102—152no—
135C4×C8—151no—
136C2×C3×C5—41no—
137C2×C3×C5—41no—
138C2×C3×C5—41no—
139C3×D5—162no—
140C23×C3—401no—
141C3×D4—202no—
142C3×D4—401no—
143C2×C3×C4—41no—
144C2×C3×C4—81no—
145SL(2,𝔽3)#68611yescharacteristic, complemented
146C3×C8—401no—
147C22×C5—121no—
148C22×C5—121no—
149C22×C5—121no—
150C22×C5D4×A411yescharacteristic
151C2×D5—22no—
152C2×D5—41no—
153C2×D5—41no—
154C4×C5—31no—
155C4×C5—31no—
156C4×C5—41no—
157C4×C5—61no—
158C4×C5C2×#41⋊C311yescharacteristic
159C4×C52−1+4⋊C311yescharacteristic
160Dic5—121no—
161#41—41no—
162#41—121no—
163C2×Q8—201no—
164C2×Q8—301no—
165C2×Q8—601no—
166C2×Q8C3×C22⋊D511yescharacteristic
167C2×D4—31no—
168C2×D4—51no—
169C2×D4—61no—
170C2×D4—61no—
171C22×C4—61no—
172C22×C4—301no—
173D8—302no—
174C2×C8—51no—
175C2×C8—151no—
176C4⋊C4—31no—
177C4⋊C4—151no—
178C4⋊C4—301no—
179C4⋊C4D5×A411yescharacteristic
180C22⋊C4—61no—
181C22⋊C4—301no—
182C42—31no—
183C3×C5—41noHall
184C22×C3—41no—
185C22×C3—402no—
186C22×C3—801no—
187C22×C3—801no—
188C3×C4—41no—
189C3×C4—41no—
190C3×C4—161no—
191C2×C5—241no—
192C2×C5#548⋊C311yescharacteristic
193C2×C5A4×D811yescharacteristic
194C2×C5#577⋊C311yescharacteristic
195D5—42no—
196C23—61no—
197C23—101no—
198Q8—31no—
199Q8—601no—
200Q8—601no—
201Q8C3×#68611yescharacteristic
202D4—52no—
203D4—61no—
204D4—61no—
205D4—101no—
206D4—121no—
207D4—121no—
208D4—241no—
209C2×C4—21no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—121no—
213C2×C4—121no—
214C2×C4—301no—
215C2×C4—601no—
216C2×C4—601no—
217C2×C4C2×D5×A411yescharacteristic
218C8—101no—
219C8—301no—
220C2×C3—41no—
221C2×C3—41no—
222C2×C3—41no—
223C2×C3—802no—
224C5#3358⋊C311yescharacteristic, complemented, Sylow
225C22—102no—
226C22—121no—
227C22—121no—
228C22—121no—
229C22—201no—
230C22—201no—
231C22A4×C22⋊D511yescharacteristic
232C4—31no—
233C4—31no—
234C4—41no—
235C4—61no—
236C4—601no—
237C4D5×#41⋊C311yescharacteristic
238C4#4150⋊C311yescharacteristic
239C3—41noSylow
240C2—202no—
241C2—241no—
242C2A4×#68611yescharacteristic
243C2#66165⋊C311yescharacteristic
244C2#66170⋊C311yescharacteristic
subgroupG/H
centerC22A4×C22⋊D5
commutator, cosocleC4×C5×Q8C22×C3
FrattiniC2×C4C2×D5×A4
FittingC5×#550C2×C3
socleC22×C5D4×A4
hypercenterC4⋊C4D5×A4
hypocenterC5×Q8C3×D8
LCSG ▷ C4×C5×Q8 ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C5×C4⋊(#41⋊C3)C2semidirect
2. SL(2,𝔽3)×D20C2semidirect
3. SL(2,𝔽3)×#168C2semidirect
4. #10615311C3semidirect
10. C5×#550C2×C3semidirect
14. Q8×#168C2×C3semidirect
23. C4⋊(#41⋊C3)D5semidirect
48. C5×SL(2,𝔽3)D8semidirect
76. #550C3×D5semidirect
113. D20#41⋊C3semidirect
145. SL(2,𝔽3)#686semidirect
224. C5#3358⋊C3semidirect

Maximal subgroups

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

subgroupindex
C5×C4⋊(#41⋊C3)2
SL(2,𝔽3)×D202
SL(2,𝔽3)×#1682
#106153113
C3×#39414
#3358⋊C35