Finite Group Databasebeta

← C2×C3×C5×SL(2,𝔽5) #72219082

Subgroups — C2×C3×C5×SL(2,𝔽5) #72219082

Order 3600.

162 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 28 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
1C3×C5×SL(2,𝔽5)C211yesmaximal, characteristic, direct product factor
2C2×C5×SL(2,𝔽5)C311yesmaximal, characteristic, direct product factor
3C2×C3×C5×SL(2,𝔽3)—51nomaximal
4C2×C3×SL(2,𝔽5)C511yesmaximal, characteristic, direct product factor
5C2×C3×C5×Dic5—61nomaximal
6C5×SL(2,𝔽5)C2×C311yescharacteristic, direct product factor
7C2×C3×C5×Dic3—101nomaximal
8C3×C5×SL(2,𝔽3)—51no—
9C3×SL(2,𝔽5)C2×C511yescharacteristic, direct product factor
10C22×C3×C52—61no—
11C3×C5×Dic5—61no—
12C3×C5×Dic5—61no—
13C2×C3×C5×Q8—51no—
14C2×C5×SL(2,𝔽3)—51no—
15C2×C5×SL(2,𝔽3)—52no—
16C2×SL(2,𝔽5)C3×C511yescharacteristic, direct product factor
17C2×C5×Dic5—61no—
18C22×C32×C5—101no—
19C3×C5×Dic3—101no—
20C3×C5×Dic3—101no—
21C2×C3×C52—62no—
22C2×C3×C52—61no—
23C2×C3×SL(2,𝔽3)—51noHall
24C3×C5×Q8—51no—
25C3×C5×Q8—151no—
26C2×C3×C4×C5—151no—
27C2×C5×Dic3—101no—
28C2×C3×Dic5—61no—
29C5×SL(2,𝔽3)—51no—
30C5×SL(2,𝔽3)—52no—
31SL(2,𝔽5)C2×C3×C511yescharacteristic, direct product factor
32C22×C52—61no—
33C5×Dic5—61no—
34C5×Dic5—61no—
35C2×C32×C5—102no—
36C2×C32×C5—101no—
37C2×C5×Q8—51no—
38C3×C52—61no—
39C2×C3×Dic3—101no—
40C3×SL(2,𝔽3)—51no—
41C22×C3×C5—61no—
42C22×C3×C5—101no—
43C22×C3×C5—122no—
44C22×C3×C5—201no—
45C22×C3×C5A511yesmaximal normal, characteristic
46C3×C4×C5—151no—
47C3×C4×C5—151no—
48C3×Dic5—61no—
49C3×Dic5—61no—
50C5×Dic3—101no—
51C5×Dic3—101no—
52C2×C52—62no—
53C2×C52—61no—
54C2×C3×Q8—51no—
55C2×SL(2,𝔽3)—51no—
56C2×SL(2,𝔽3)—52no—
57C32×C5—101no—
58C5×Q8—51no—
59C5×Q8—151no—
60C2×C4×C5—151no—
61C2×Dic5—61no—
62C22×C32—101no—
63C3×Dic3—101no—
64C3×Dic3—101no—
65C2×C3×C5—61no—
66C2×C3×C5—62no—
67C2×C3×C5—102no—
68C2×C3×C5—101no—
69C2×C3×C5—122no—
70C2×C3×C5—124no—
71C2×C3×C5—202no—
72C2×C3×C5—201no—
73C2×C3×C5SL(2,𝔽5)12yesdirect product factor
74C2×C3×C5C2×A511yescharacteristic
75C52—61noSylow
76C3×Q8—51no—
77C3×Q8—151no—
78C2×C3×C4—151no—
79C2×Dic3—101no—
80SL(2,𝔽3)—51no—
81SL(2,𝔽3)—52no—
82C22×C5—61no—
83C22×C5—122no—
84C22×C5C3×A511yescharacteristic
85C4×C5—151no—
86C4×C5—151no—
87Dic5—61no—
88Dic5—61no—
89C2×C32—101no—
90C2×C32—102no—
91C2×Q8—51noSylow
92C3×C5—61no—
93C3×C5—101no—
94C3×C5—122no—
95C3×C5—201no—
96C3×C5C2×SL(2,𝔽5)11yescharacteristic, direct product factor
97C22×C3—101no—
98C22×C3—201no—
99C22×C3C5×A511yescharacteristic
100C3×C4—151no—
101C3×C4—151no—
102Dic3—101no—
103Dic3—101no—
104C2×C5—61no—
105C2×C5—62no—
106C2×C5—122no—
107C2×C5—124no—
108C2×C5C3×SL(2,𝔽5)12yesdirect product factor
109C2×C5C2×C3×A511yescharacteristic
110C32—101noSylow
111Q8—51no—
112Q8—151no—
113C2×C4—151no—
114C2×C3—101no—
115C2×C3—102no—
116C2×C3—201no—
117C2×C3—202no—
118C2×C3C5×SL(2,𝔽5)12yesdirect product factor
119C2×C3C2×C5×A511yescharacteristic
120C5—61no—
121C5—122no—
122C5C2×C3×SL(2,𝔽5)11yescharacteristic, direct product factor
123C22C3×C5×A511yescharacteristic
124C4—151no—
125C4—151no—
126C3—101no—
127C3—201no—
128C3C2×C5×SL(2,𝔽5)11yescharacteristic, direct product factor
129C2C3×C5×SL(2,𝔽5)12yesdirect product factor
130C2C2×C3×C5×A511yescharacteristic
subgroupG/H
center, Fitting, socle, hypercenterC22×C3×C5A5
commutator, hypocenterSL(2,𝔽5)C2×C3×C5
Frattini, cosocleC2C2×C3×C5×A5
LCSG ▷ SL(2,𝔽5)
UCSC1 ◁ C22×C3×C5

Complements

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

NH ≅ G/Ntype
1. C3×C5×SL(2,𝔽5)C2direct (N × H)
2. C2×C5×SL(2,𝔽5)C3direct (N × H)
4. C2×C3×SL(2,𝔽5)C5direct (N × H)
6. C5×SL(2,𝔽5)C2×C3direct (N × H)
9. C3×SL(2,𝔽5)C2×C5direct (N × H)
16. C2×SL(2,𝔽5)C3×C5direct (N × H)
31. SL(2,𝔽5)C2×C3×C5direct (N × H)
73. C2×C3×C5SL(2,𝔽5)direct (N × H)
96. C3×C5C2×SL(2,𝔽5)direct (N × H)
108. C2×C5C3×SL(2,𝔽5)direct (N × H)
118. C2×C3C5×SL(2,𝔽5)direct (N × H)
122. C5C2×C3×SL(2,𝔽5)direct (N × H)
128. C3C2×C5×SL(2,𝔽5)direct (N × H)
129. C2C3×C5×SL(2,𝔽5)direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×SL(2,𝔽5)2
C2×C5×SL(2,𝔽5)3
C2×C3×C5×SL(2,𝔽3)5
C2×C3×SL(2,𝔽5)5
C2×C3×C5×Dic56
C2×C3×C5×Dic310