Finite Group Databasebeta

← C5×#226⋊A4 #72277275

Subgroups — C5×#226⋊A4 #72277275

Order 2880.

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

normal subgroups only

This group has 194 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×#5925C311yesmaximal, characteristic, complemented
3C5×#576⋊C3—31nomaximal
4C5×C3⋊(#41⋊C3)—41nomaximal
5C3×C5×#226—41nomaximal
6#226⋊A4C511yesmaximal, characteristic, direct product factor, Hall
7C5×#997—21no—
8C5×#994—31no—
9C5×#994D311yescharacteristic, complemented
10C5×#994C2×C311yescharacteristic
11C5×#986—31no—
12C5×#985—31no—
13C5×#131⋊C3—31no—
14C3×C5×SL(2,𝔽3)—41no—
15C3×C5×SL(2,𝔽3)—41no—
16C5×#576—31noHall
17Ω+(4,𝔽3)C2×C511yescharacteristic, complemented
18C5×#239—41no—
19C5×#41⋊C3—31no—
20C5×#41⋊C3—31no—
21C5×#41⋊C3—61no—
22C5×GL(2,𝔽3)—31no—
23C5×#226A411yescharacteristic, complemented
24C3×C5×Q16—121no—
25#5925C3×C511yescharacteristic, complemented
26#576⋊C3—31no—
27C3×C5×Dic3—161no—
28C5×H(5,𝔽2)—31no—
29C5×H(5,𝔽2)C3×D311yescharacteristic
30C5×AGL(1,ℤ/8)—91no—
31C5×AGL(1,ℤ/8)—91no—
32C5×#135—91no—
33C2×C5×D8—91no—
34C5×#131—31no—
35C3⋊(#41⋊C3)—41no—
36C3×#226—41no—
37C2×C5×A4—241no—
38C3×C5×Q8—41no—
39C3×C5×Q8—41no—
40C3×C5×Q8—121no—
41C5×D12—121no—
42C4×C5×D3—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—
49#994—31no—
50#994C5×D311yescharacteristic, complemented
51#994C2×C3×C511yescharacteristic
52#986—31no—
53#985—31no—
54#131⋊C3—31no—
55C2×C32×C5—161no—
56C5×#41—31no—
57C5×#41—31no—
58C5×#41—61no—
59C5×#41—91no—
60C2×C5×D4—91no—
61C2×C5×D4—91no—
62C2×C5×D4—181no—
63C5×Q16—31no—
64C5×SD16—91no—
65C5×SD16—91no—
66C5×D8—91no—
67C5×D8—92no—
68C5×M16—91no—
69C2×C5×C8—91no—
70C3×SL(2,𝔽3)—41no—
71C3×SL(2,𝔽3)—41no—
72#576—31noSylow
73C2×C5×D3—121no—
74C5×A4—241no—
75C3×C4×C5—121no—
76C3×C4×C5—121no—
77C3×C4×C5—241no—
78C5×Dic3—41no—
79#239—41no—
80#41⋊C3—31no—
81#41⋊C3—31no—
82#41⋊C3—61no—
83GL(2,𝔽3)—31no—
84#226C5×A411yescharacteristic, complemented
85C3×Q16—121no—
86C32×C5—161noHall
87C23×C5—61no—
88C23×C5—181no—
89C5×Q8—31no—
90C5×Q8C3×S411yescharacteristic
91C5×Q8D3×A411yescharacteristic
92C5×D4—91no—
93C5×D4—91no—
94C5×D4—91no—
95C5×D4—181no—
96C5×D4—181no—
97C2×C4×C5—91no—
98C2×C4×C5—181no—
99C5×C8—31no—
100C5×C8—91no—
101C3×Dic3—161no—
102H(5,𝔽2)—31no—
103H(5,𝔽2)C3×C5×D311yescharacteristic
104AGL(1,ℤ/8)—91no—
105AGL(1,ℤ/8)—91no—
106#135—91no—
107C2×D8—91no—
108#131—31no—
109C2×C3×C5—41no—
110C2×C3×C5—41no—
111C2×C3×C5—321no—
112C5×D3—241no—
113C2×A4—241no—
114C3×Q8—41no—
115C3×Q8—41no—
116C3×Q8—121no—
117D12—121no—
118C4×D3—121no—
119SL(2,𝔽3)—81no—
120SL(2,𝔽3)—81no—
121SL(2,𝔽3)C5×S411yescharacteristic
122SL(2,𝔽3)C2×C5×A411yescharacteristic
123C3×C8—121no—
124C22×C5—91no—
125C22×C5—181no—
126C22×C5—241no—
127C22×C5—721no—
128C4×C5—31no—
129C4×C5—31no—
130C4×C5—61no—
131C2×C32—161no—
132#41—31no—
133#41—31no—
134#41—61no—
135#41—91no—
136C2×D4—91no—
137C2×D4—91no—
138C2×D4—181no—
139Q16—31no—
140SD16—91no—
141SD16—91no—
142D8—91no—
143D8—92no—
144M16—91no—
145C2×C8—91no—
146C3×C5—41no—
147C3×C5—41no—
148C3×C5—321no—
149C2×D3—121no—
150A4—241no—
151C3×C4—121no—
152C3×C4—121no—
153C3×C4—241no—
154Dic3—41no—
155C2×C5—181no—
156C2×C5—361no—
157C2×C5A4×S411yescharacteristic
158C32—161noSylow
159C23—61no—
160C23—181no—
161Q8—31no—
162Q8C3×C5×S411yescharacteristic
163Q8C5×D3×A411yescharacteristic
164D4—91no—
165D4—91no—
166D4—91no—
167D4—181no—
168D4—181no—
169C2×C4—91no—
170C2×C4—181no—
171C8—31no—
172C8—91no—
173C2×C3—41no—
174C2×C3—41no—
175C2×C3—321no—
176D3—241no—
177C5#226⋊A411yescharacteristic, direct product factor, Sylow
178C22—91no—
179C22—181no—
180C22—241no—
181C22—721no—
182C4—31no—
183C4—31no—
184C4—61no—
185C3—41no—
186C3—41no—
187C3—321no—
188C2—181no—
189C2—361no—
190C2C5×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×#5925C3semidirect
6. #226⋊A4C5direct (N × H)
9. C5×#994D3semidirect
17. Ω+(4,𝔽3)C2×C5semidirect
23. C5×#226A4semidirect
25. #5925C3×C5semidirect
50. #994C5×D3semidirect
84. #226C5×A4semidirect
177. C5#226⋊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×#59253
C5×#576⋊C33
C5×C3⋊(#41⋊C3)4
C3×C5×#2264
#226⋊A45