Finite Group Databasebeta

← #92030⋊C3 #13088595

Subgroups — #92030⋊C3 #13088595

Order 1440.

198 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 34 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)×D15C211yesmaximal, characteristic, complemented
2C5×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C3×D5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#92030C311yesmaximal, characteristic, complemented
5#4150⋊C3—31nomaximal
6C3×C5×SL(2,𝔽3)C2211yescharacteristic
7C3×#1202—41nomaximal
8#1007⋊C3—51nomaximal, Hall
9Q8×D15C2×C311yescharacteristic, complemented
10C5×#239C2×C311yescharacteristic, complemented
11C3×Q8×D5C2×C311yescharacteristic, complemented
12C5×#41⋊C3—31no—
13#6772—31no—
14#6771—31no—
15#6768—31no—
16D3×Dic10—31no—
17D5×SL(2,𝔽3)—21no—
18D5×SL(2,𝔽3)—31no—
19D5×SL(2,𝔽3)D311yescharacteristic, complemented
20C2×C3×D15—41no—
21C2×C32×D5—41no—
22C3×C5×Dic3—41no—
23#4150—31noHall
24C2×C3×SL(2,𝔽3)—51no—
25D3×SL(2,𝔽3)—51no—
26C3⋊(#41⋊C3)D511yescharacteristic, complemented
27C3×C5×Q8C22×C311yescharacteristic
28C4×D15—31no—
29Dic30—31no—
30C5×D12—31no—
31C4×C5×D3—31no—
32C3×D20—121no—
33C3×C4×D5—31no—
34C3×Dic10—31no—
35C5×SL(2,𝔽3)—21no—
36C5×SL(2,𝔽3)C2×D311yescharacteristic
37#1204—31no—
38#1203—31no—
39#1202A411yescharacteristic
40#1201—31no—
41D3×Dic5—32no—
42#1007—51no—
432−1+4⋊C3—151no—
44C2×C32×C5—41no—
45C3×D15—81no—
46C32×D5—81no—
47C5×#41—31no—
48Q8×D5—31no—
49Q8×D5C3×D311yescharacteristic, complemented
50#711—91no—
51#711—91no—
52#709—91no—
53C2×Dic10—91no—
54C3×#67—201no—
55C3×SL(2,𝔽3)C2×D511yescharacteristic
56C2×D15C2×A411yescharacteristic
57C2×C5×D3—31no—
58C2×C3×D5—41no—
59C2×C3×D5—81no—
60C2×C3×D5—121no—
61C2×C3×D5C2×A411yescharacteristic
62C3×C4×C5—31no—
63C3×C4×C5—121no—
64Dic15—31no—
65C3×Dic5—31no—
66C5×Dic3C2×A411yescharacteristic
67C2×C3×Q8—51no—
68#239—151no—
69#239C3×D511yescharacteristic, complemented
70D3×Q8—51no—
71D3×Q8—151no—
72#235—151no—
73#235—151no—
74#41⋊C3—31no—
75C2×SL(2,𝔽3)—51no—
76C2×SL(2,𝔽3)—101no—
77C2×SL(2,𝔽3)—151no—
78C32×C5—41noHall
79C5×Q8C2×C3×D311yescharacteristic
80C5×D4—91no—
81C2×C4×C5—91no—
82C22⋊D5—91no—
83C22⋊D5—91no—
84C2×Dic5—92no—
85D20—31no—
86C4×D5—31no—
87C4×D5—91no—
88Dic10—31no—
89Dic10—91no—
90Dic10—91no—
91C22×C32—201no—
92C2×C3×D3—201no—
93C3×Dic3—41no—
942−1+4—151noSylow
95C2×C3×C5—41no—
96C2×C3×C5—81no—
97C2×C3×C5C22×A411yescharacteristic
98D15—21no—
99C3×D5—21no—
100C3×D5—81no—
101C3×D5—82no—
102C3×D5—241no—
103C5×D3—61no—
104C3×Q8—151no—
105C3×Q8C2×C3×D511yescharacteristic
106C3×D4—601no—
107C2×C3×C4—151no—
108#67—51no—
109#67—151no—
110D12—31no—
111D12—151no—
112C4×D3—31no—
113C4×D3—151no—
114C4×D3—152no—
115Dic6—151no—
116Dic6—151no—
117SL(2,𝔽3)—21no—
118SL(2,𝔽3)D3×D511yescharacteristic
119C22×C5—91no—
120C2×D5—31no—
121C2×D5D3×A411yescharacteristic
122C4×C5—31no—
123C4×C5—31no—
124Dic5—31no—
125Dic5—91no—
126C2×C32—41no—
127C2×C32—401no—
128C3×D3—401no—
129#41—31no—
130#41—451no—
131#41—451no—
132#41—451no—
133C2×Q8—51no—
134C2×Q8—151no—
135C2×Q8—451no—
136C3×C5—41no—
137C3×C5—81no—
138C3×C52−1+4⋊C311yescharacteristic, complemented
139C22×C3—51no—
140C22×C3—201no—
141C22×C3—401no—
142C22×C3—601no—
143C2×D3—31no—
144C2×D3—51no—
145C3×C4—31no—
146C3×C4—121no—
147C3×C4—151no—
148Dic3—151no—
149Dic3D5×A411yescharacteristic
150C2×C5—181no—
151C2×C5C2×D3×A411yescharacteristic
152D5—21no—
153D5—61no—
154C32—41noSylow
155Q8—151no—
156Q8—451no—
157Q8—451no—
158Q8C3×D3×D511yescharacteristic
159D4—91no—
160D4—151no—
161D4—451no—
162D4—451no—
163C2×C4—91no—
164C2×C4—151no—
165C2×C4—451no—
166C2×C4—452no—
167C2×C3—41no—
168C2×C3—81no—
169C2×C3—101no—
170C2×C3—401no—
171C2×C3—402no—
172C2×C3—1201no—
173C2×C3C2×D5×A411yescharacteristic
174D3—61no—
175D3—101no—
176C5#1007⋊C311yescharacteristic, complemented, Sylow
177C22—51no—
178C22—91no—
179C22—151no—
180C4—31no—
181C4—31no—
182C4—151no—
183C4—451no—
184C3—41no—
185C3—81no—
186C3#4150⋊C311yescharacteristic, complemented
187C2—101no—
188C2—181no—
189C2—301no—
190C2D3×D5×A411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D5×A4
commutator, Fitting, cosocle, hypocenterC3×C5×Q8C22×C3
socleC2×C3×C5C22×A4
LCSG ▷ C3×C5×Q8
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. SL(2,𝔽3)×D15C2semidirect
2. C5×C3⋊(#41⋊C3)C2semidirect
3. C3×D5×SL(2,𝔽3)C2semidirect
4. #92030C3semidirect
9. Q8×D15C2×C3semidirect
10. C5×#239C2×C3semidirect
11. C3×Q8×D5C2×C3semidirect
19. D5×SL(2,𝔽3)D3semidirect
26. C3⋊(#41⋊C3)D5semidirect
49. Q8×D5C3×D3semidirect
69. #239C3×D5semidirect
138. C3×C52−1+4⋊C3semidirect
176. C5#1007⋊C3semidirect
186. C3#4150⋊C3semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D152
C5×C3⋊(#41⋊C3)2
C3×D5×SL(2,𝔽3)2
#920303
#4150⋊C33
C3×#12024
#1007⋊C35