Finite Group Databasebeta

← C5×GL(2,𝔽3)⋊A4 #72277265

Subgroups — C5×GL(2,𝔽3)⋊A4 #72277265

Order 2880.

200 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 26 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×Ω+(4,𝔽3)C211yesmaximal, characteristic, complemented
2C5×#5924C311yesmaximal, characteristic, complemented
3C5×#577⋊C3—31nomaximal
4C5×D3×SL(2,𝔽3)—41nomaximal
5C3×C5×GL(2,𝔽3)—41nomaximal
6GL(2,𝔽3)⋊A4C511yesmaximal, characteristic, direct product factor, Hall
7C5×#997—21no—
8C5×2−1+4⋊C3—31no—
9C5×#994D311yescharacteristic, complemented
10C5×#994C2×C311yescharacteristic, complemented
11C5×#985—31no—
12C5×#984—31no—
13C5×#131⋊C3—31no—
14C3×C5×SL(2,𝔽3)—41no—
15C3×C5×SL(2,𝔽3)—41no—
16C5×#577—31noHall
17Ω+(4,𝔽3)C2×C511yescharacteristic, complemented
18C5×D3×Q8—41no—
19C5×#41⋊C3—31no—
20C5×#41⋊C3—31no—
21C2×C5×SL(2,𝔽3)—61no—
22C5×GL(2,𝔽3)A411yescharacteristic, complemented
23C5×#226—31no—
24C3×C5×SD16—121no—
25#5924C3×C511yescharacteristic, complemented
26#577⋊C3—31no—
27C2×C3×C5×D3—161no—
28C5×2−1+4—31no—
29C5×H(5,𝔽2)C3×D311yescharacteristic, complemented
30C5×#137—91no—
31C5×AGL(1,ℤ/8)—91no—
32C5×#135—91no—
33C2×C5×SD16—91no—
34C5×#131—31no—
35D3×SL(2,𝔽3)—41no—
36C3×GL(2,𝔽3)—41no—
37C2×C5×A4—241no—
38C3×C5×Q8—41no—
39C3×C5×Q8—41no—
40C3×C5×D4—121no—
41C4×C5×D3—121no—
42C5×Dic6—121no—
43C5×SL(2,𝔽3)—81no—
44C5×SL(2,𝔽3)—81no—
45C5×SL(2,𝔽3)S411yescharacteristic
46C5×SL(2,𝔽3)C2×A411yescharacteristic
47C3×C5×C8—121no—
48#997—21no—
492−1+4⋊C3—31no—
50#994C5×D311yescharacteristic, complemented
51#994C2×C3×C511yescharacteristic, complemented
52#985—31no—
53#984—31no—
54#131⋊C3—31no—
55C2×C32×C5—161no—
56C3×C5×D3—162no—
57C5×#41—31no—
58C5×#41—31no—
59C5×#41—91no—
60C5×#41—181no—
61C2×C5×Q8—61no—
62C2×C5×Q8—91no—
63C2×C5×D4—91no—
64C5×Q16—91no—
65C5×SD16—31no—
66C5×SD16—91no—
67C5×SD16—92no—
68C5×D8—91no—
69C5×M16—91no—
70C2×C5×C8—91no—
71C3×SL(2,𝔽3)—41no—
72C3×SL(2,𝔽3)—41no—
73#577—31noSylow
74C22×C3×C5—241no—
75C2×C5×D3—41no—
76C5×A4—241no—
77C3×C4×C5—121no—
78C3×C4×C5—121no—
79C5×Dic3—121no—
80D3×Q8—41no—
81#41⋊C3—31no—
82#41⋊C3—31no—
83C2×SL(2,𝔽3)—61no—
84GL(2,𝔽3)C5×A411yescharacteristic, complemented
85#226—31no—
86C3×SD16—121no—
87C32×C5—161noHall
88C23×C5—61no—
89C5×Q8—91no—
90C5×Q8—181no—
91C5×Q8C3×S411yescharacteristic
92C5×Q8D3×A411yescharacteristic
93C5×D4—31no—
94C5×D4—91no—
95C5×D4—91no—
96C5×D4—181no—
97C2×C4×C5—91no—
98C2×C4×C5—181no—
99C2×C4×C5—181no—
100C5×C8—31no—
101C5×C8—91no—
102C2×C3×D3—161no—
1032−1+4—31no—
104H(5,𝔽2)C3×C5×D311yescharacteristic, complemented
105#137—91no—
106AGL(1,ℤ/8)—91no—
107#135—91no—
108C2×SD16—91no—
109#131—31no—
110C2×C3×C5—41no—
111C2×C3×C5—41no—
112C2×C3×C5—321no—
113C2×C3×C5—481no—
114C5×D3—42no—
115C2×A4—241no—
116C3×Q8—41no—
117C3×Q8—41no—
118C3×D4—121no—
119C4×D3—121no—
120Dic6—121no—
121SL(2,𝔽3)—81no—
122SL(2,𝔽3)—81no—
123SL(2,𝔽3)C5×S411yescharacteristic
124SL(2,𝔽3)C2×C5×A411yescharacteristic
125C3×C8—121no—
126C22×C5—61no—
127C22×C5—91no—
128C22×C5—241no—
129C4×C5—31no—
130C4×C5—31no—
131C4×C5—181no—
132C2×C32—161no—
133C3×D3—162no—
134#41—31no—
135#41—31no—
136#41—91no—
137#41—181no—
138C2×Q8—61no—
139C2×Q8—91no—
140C2×D4—91no—
141Q16—91no—
142SD16—31no—
143SD16—91no—
144SD16—92no—
145D8—91no—
146M16—91no—
147C2×C8—91no—
148C3×C5—41no—
149C3×C5—41no—
150C3×C5—321no—
151C22×C3—241no—
152C2×D3—41no—
153A4—241no—
154C3×C4—121no—
155C3×C4—121no—
156Dic3—121no—
157C2×C5—121no—
158C2×C5—181no—
159C2×C5A4×S411yescharacteristic
160C32—161noSylow
161C23—61no—
162Q8—91no—
163Q8—181no—
164Q8C3×C5×S411yescharacteristic
165Q8C5×D3×A411yescharacteristic
166D4—31no—
167D4—91no—
168D4—91no—
169D4—181no—
170C2×C4—91no—
171C2×C4—181no—
172C2×C4—181no—
173C8—31no—
174C8—91no—
175C2×C3—41no—
176C2×C3—41no—
177C2×C3—321no—
178C2×C3—481no—
179D3—42no—
180C5GL(2,𝔽3)⋊A411yescharacteristic, direct product factor, Sylow
181C22—61no—
182C22—91no—
183C22—241no—
184C4—31no—
185C4—31no—
186C4—181no—
187C3—41no—
188C3—41no—
189C3—321no—
190C2—121no—
191C2—181no—
192C2C5×A4×S411yescharacteristic
subgroupG/H
center, socle, hypercenterC2×C5A4×S4
commutator, cosocle, hypocenter#994C2×C3×C5
FrattiniC2C5×A4×S4
FittingC5×H(5,𝔽2)C3×D3
LCSG ▷ #994
UCSC1 ◁ C2×C5

Complements

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

NH ≅ G/Ntype
1. C5×Ω+(4,𝔽3)C2semidirect
2. C5×#5924C3semidirect
6. GL(2,𝔽3)⋊A4C5direct (N × H)
9. C5×#994D3semidirect
10. C5×#994C2×C3semidirect
17. Ω+(4,𝔽3)C2×C5semidirect
22. C5×GL(2,𝔽3)A4semidirect
25. #5924C3×C5semidirect
29. C5×H(5,𝔽2)C3×D3semidirect
50. #994C5×D3semidirect
51. #994C2×C3×C5semidirect
84. GL(2,𝔽3)C5×A4semidirect
104. H(5,𝔽2)C3×C5×D3semidirect
180. C5GL(2,𝔽3)⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C5×Ω+(4,𝔽3)2
C5×#59243
C5×#577⋊C33
C5×D3×SL(2,𝔽3)4
C3×C5×GL(2,𝔽3)4
GL(2,𝔽3)⋊A45