Finite Group Databasebeta

← C3×#91953 #13089061

Subgroups — C3×#91953 #13089061

Order 1440.

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

normal subgroups only

This group has 170 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
1C2×C3×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×C5⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C5⋊#226C211yesmaximal, characteristic, complemented
4#91953C311yesmaximal, characteristic, direct product factor
5C3×#4088—31nomaximal
6C3×#1220—41nomaximal
7C3×C5×SL(2,𝔽3)C2211yescharacteristic
8C3×#983—51nomaximal, Hall
9C2×C3×C5×Q8D311yescharacteristic, complemented
10C3×#709—31no—
11C2×C5×SL(2,𝔽3)—21no—
12C2×C5×SL(2,𝔽3)C2×C311yescharacteristic, complemented
13C5⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
14C5⋊#226C2×C311yescharacteristic, complemented
15C3×#689—31no—
16C3×#689—31no—
17C3×#688—31no—
18C3×#688—31no—
19C3×#681—31no—
20C22×C32×C5—41no—
21C2×C3×D15—41no—
22C3×Dic15—41no—
23#4088—31noHall
24C2×C3×SL(2,𝔽3)D511yescharacteristic, complemented
25C3×GL(2,𝔽3)—51no—
26C3×#226—51no—
27C3×C5×Q8—31no—
28C3×C5×Q8C2×D311yescharacteristic
29C2×C3×C4×C5—31no—
30#1220—41no—
31C3×C22⋊D5—61no—
32C3×D20—31no—
33C3×C4×D5—61no—
34C3×Dic10—31no—
35C5×SL(2,𝔽3)—21no—
36C5×SL(2,𝔽3)C22×C311yescharacteristic
37C3×#168—31no—
38C3×#168—31no—
39#983—51no—
40C3×#137—151no—
41C2×C32×C5—41no—
42C2×C32×C5—81no—
43C3×D15—81no—
44C2×C5×Q8C3×D311yescharacteristic, complemented
45#709—31no—
46#689—31no—
47#689—31no—
48#688—31no—
49#688—31no—
50#681—31no—
51C3×#67—201no—
52C3×SL(2,𝔽3)C2×D511yescharacteristic
53C22×C3×C5—41no—
54C22×C3×C5—81no—
55C22×C3×C5S411yescharacteristic
56C2×D15—41no—
57C2×C3×D5—61no—
58C3×C4×C5—31no—
59C3×C4×C5—31no—
60Dic15—41no—
61C3×Dic5—61no—
62C3×#41—151no—
63C2×C3×Q8D1511yescharacteristic, complemented
64C2×SL(2,𝔽3)—21no—
65C2×SL(2,𝔽3)C3×D511yescharacteristic, complemented
66GL(2,𝔽3)—51no—
67#226—51no—
68C3×Q16—151no—
69C3×Q16—151no—
70C3×SD16—151no—
71C3×SD16—151no—
72C3×M16—151no—
73C32×C5—41noHall
74C5×Q8—31no—
75C5×Q8C2×C3×D311yescharacteristic
76C2×C4×C5—31no—
77C22⋊D5—61no—
78D20—31no—
79C4×D5—61no—
80Dic10—31no—
81#168—31no—
82#168—31no—
83C22×C32—41no—
84C2×C3×D3—201no—
85C3×Dic3—201no—
86#137—151noSylow
87C2×C3×C5—21no—
88C2×C3×C5—41no—
89C2×C3×C5—81no—
90C2×C3×C5—82no—
91C2×C3×C5—81no—
92C2×C3×C5C2×S411yescharacteristic
93D15—81no—
94C3×D5—121no—
95C3×Q8—31no—
96C3×Q8—151no—
97C3×Q8C2×D1511yescharacteristic
98C3×D4—151no—
99C3×D4—301no—
100C2×C3×C4—31no—
101C2×C3×C4—301no—
102#67—201no—
103SL(2,𝔽3)—21no—
104SL(2,𝔽3)C2×C3×D511yescharacteristic
105C3×C8—151no—
106C3×C8—151no—
107C22×C5C3×S411yescharacteristic
108C2×D5—61no—
109C4×C5—31no—
110C4×C5—31no—
111Dic5—61no—
112C2×C32—41no—
113C2×C32—81no—
114C3×D3—401no—
115#41—151no—
116C2×Q8C3×D1511yescharacteristic, complemented
117Q16—151no—
118Q16—151no—
119SD16—151no—
120SD16—151no—
121M16—151no—
122C3×C5—41no—
123C3×C5—81no—
124C3×C5#98311yescharacteristic, complemented
125C22×C3—41no—
126C22×C3—81no—
127C22×C3—301no—
128C22×C3C5⋊S411yescharacteristic
129C2×D3—201no—
130C3×C4—31no—
131C3×C4—31no—
132C3×C4—301no—
133Dic3—201no—
134C2×C5—21no—
135C2×C5C2×C3×S411yescharacteristic
136D5—121no—
137C32—41noSylow
138Q8—31no—
139Q8—151no—
140Q8C2×C3×D1511yescharacteristic
141D4—151no—
142D4—301no—
143C2×C4—31no—
144C2×C4—301no—
145C8—151no—
146C8—151no—
147C2×C3—21no—
148C2×C3—41no—
149C2×C3—81no—
150C2×C3—82no—
151C2×C3—81no—
152C2×C3—601no—
153C2×C3C2×C5⋊S411yescharacteristic
154D3—401no—
155C5C3×#98311yescharacteristic, complemented, Sylow
156C22—301no—
157C22C3×C5⋊S411yescharacteristic
158C4—31no—
159C4—31no—
160C4—301no—
161C3—41no—
162C3—81no—
163C3#9195311yescharacteristic, direct product factor
164C2—21no—
165C2—601no—
166C2C2×C3×C5⋊S411yescharacteristic
subgroupG/H
centerC2×C3C2×C5⋊S4
commutator, cosocle, hypocenterC5×SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×C5⋊S4
FittingC2×C3×C5×Q8D3
socleC2×C3×C5C2×S4
hypercenterC22×C3C5⋊S4
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×SL(2,𝔽3)C2semidirect
2. C3×C5⋊GL(2,𝔽3)C2semidirect
3. C3×C5⋊#226C2semidirect
4. #91953C3direct (N × H)
9. C2×C3×C5×Q8D3semidirect
12. C2×C5×SL(2,𝔽3)C2×C3semidirect
13. C5⋊GL(2,𝔽3)C2×C3semidirect
14. C5⋊#226C2×C3semidirect
24. C2×C3×SL(2,𝔽3)D5semidirect
44. C2×C5×Q8C3×D3semidirect
63. C2×C3×Q8D15semidirect
65. C2×SL(2,𝔽3)C3×D5semidirect
116. C2×Q8C3×D15semidirect
124. C3×C5#983semidirect
155. C5C3×#983semidirect
163. C3#91953direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×SL(2,𝔽3)2
C3×C5⋊GL(2,𝔽3)2
C3×C5⋊#2262
#919533
C3×#40883
C3×#12204
C3×#9835